Technical Mathematics

🧭 Overview

🧠 One-sentence thesis

Order of operations (PEMDAS) ensures that multi-step mathematical expressions produce a single correct answer by specifying which operations must be performed before others.

📌 Key points (3–5)

• What "evaluate" means: to simplify an expression and find its value.
• Why order matters: without agreed-upon rules, the same expression could yield different answers (e.g., "maybe 7, or maybe 13").
• The PEMDAS sequence: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).
• Common confusion: multiplication and division are equal priority (work left to right), as are addition and subtraction (also left to right)—not strict "M before D" or "A before S."
• Grouping symbols extend beyond parentheses: brackets, nested symbols, and fraction bars all indicate "do this part first."

🔤 What it means to evaluate

🔤 Definition and purpose

To evaluate an expression means to simplify it and find its value.

• The goal is a single, unambiguous answer.
• Without rules, different people might perform steps in different orders and get different results.
• Example: the excerpt notes it isn't helpful for an answer to be "maybe 7, or maybe 13."

🚦 Order of operations as "rules of the road"

• Mathematicians agreed on a standard sequence so everyone arrives at the same answer.
• The excerpt compares this to real-life sequences: putting on socks before shoes is necessary, but left-sock-first vs. right-sock-first is flexible.
• In math, some steps must happen in a specific order; others can be done in any order as long as the required precedence is respected.

🧮 The PEMDAS hierarchy

🅿️ P: Parentheses (and grouping symbols)

• Work inside parentheses or other grouping symbols first.
• If there are operations inside the grouping symbols, follow PEMDAS within them as well.
• Nested grouping: the excerpt shows that square brackets can be used to make matching pairs clearer, e.g., instead of multiple layers of parentheses.

🔺 E: Exponents

An exponent indicates repeated multiplication.

• The exponent tells how many factors of the base are multiplied together.
• Example (from the excerpt): the base is multiplied by itself the number of times indicated by the exponent.
• Exponents are evaluated after parentheses but before multiplication, division, addition, and subtraction.

✖️➗ MD: Multiplication and Division (left to right)

• Multiplication and division have equal priority.
• Perform them from left to right as they appear.
• Don't confuse: it is not "do all multiplications first, then all divisions"; instead, work left to right.
• The excerpt shows multiple notations for multiplication: a dot, parentheses directly next to a number, or the × symbol (though × is avoided because it resembles the letter x).

➕➖ AS: Addition and Subtraction (left to right)

• Addition and subtraction also have equal priority.
• Perform them from left to right.
• The excerpt includes exercises showing that doing addition before subtraction (when no parentheses are present) violates the correct order and produces the wrong answer.

🗂️ Extended grouping symbols

🗂️ Brackets and nesting

• Square brackets [ ] can replace parentheses to improve readability when grouping symbols are nested.
• The excerpt example: nested parentheses can be rewritten with brackets to make pairs easier to match.
• The rule remains the same: work from the innermost grouping outward, following PEMDAS at each level.

➗ Fraction bars as grouping

• A fraction bar groups the numerator (top) and denominator (bottom) separately.
• How it works:
  1. Perform all operations in the numerator.
  2. Perform all operations in the denominator.
  3. Finally, divide the top result by the bottom result.
• Example: treat the fraction bar as an implicit set of parentheses around the top and another around the bottom.

🔢 Notation and common pitfalls

🔢 Multiple ways to show multiplication

The excerpt lists several equivalent notations for multiplication:

• A dot (e.g., a dot between two numbers)
• Parentheses directly next to a number
• The × symbol (less common in algebra because it resembles the letter x)

Why it matters: recognizing all forms prevents confusion when reading expressions.

⚠️ Why exercises show different answers with only parentheses changed

• The excerpt includes a set of exercises where "the only differences are the parentheses, but every exercise has a different answer."
• This demonstrates that grouping symbols fundamentally change which operations are performed first.
• Don't confuse: moving or adding parentheses is not cosmetic—it changes the meaning and the result of the expression.

🧭 Overview

🧠 One-sentence thesis

Exponents indicate repeated multiplication and must be evaluated at a specific point in the order of operations—after parentheses but before multiplication, division, addition, and subtraction.

📌 Key points (3–5)

• What exponents represent: repeated multiplication of a base; the exponent tells how many factors are multiplied together.
• Where exponents fit in order of operations: they are evaluated after parentheses (P) but before multiplication/division (MD) and addition/subtraction (AS).
• How to read exponent notation: the base is the number being multiplied; the exponent (superscript) is the count of how many times the base appears as a factor.
• Common confusion: exponents are not the same as multiplication—they must be evaluated earlier in the sequence of operations.

🔢 What exponents mean

🔢 Repeated multiplication definition

An exponent indicates repeated multiplication.

• The exponent tells us how many factors of the base are being multiplied together.
• It is not addition or a single multiplication; it is a compact way to write multiple copies of the same number multiplied.
• Example: the excerpt states "5 to the power of 3 equals 5 times 5 times 5" and "2 to the power of 4 equals 2 times 2 times 2 times 2."

🧮 Base and exponent roles

• Base: the number that is repeated as a factor.
• Exponent: the count of how many times the base appears.
• Don't confuse: the exponent is not a multiplier in the usual sense; it controls repetition, not a single multiplication step.

📐 Exponents in the order of operations

📐 The PEMDAS sequence

The excerpt presents the order of operations as:

Step	Symbol	Action	When performed
P	Parentheses	Work inside grouping symbols first	Before everything else
E	Exponents	Evaluate exponents	After parentheses, before MD and AS
MD	Multiplication/Division	Perform from left to right	After exponents, before AS
AS	Addition/Subtraction	Perform from left to right	Last

• Exponents come after parentheses but before multiplication, division, addition, and subtraction.
• This placement is critical: if you multiply or add before evaluating exponents, you will get the wrong answer.

⚠️ Why order matters

• The excerpt emphasizes that "when we evaluate an expression, we want to have a single correct answer."
• Performing steps out of order leads to different results (the excerpt shows exercises where changing order changes the answer).
• Example: If an expression contains both exponents and multiplication, you must evaluate the exponent first, then multiply.

🧩 Working with exponents in context

🧩 Exponents and grouping symbols

• The excerpt notes that grouping symbols (parentheses, brackets, fraction bars) can contain exponents.
• Inside grouping symbols, you still follow PEMDAS: evaluate exponents after any nested parentheses but before other operations inside that group.
• Example: In a nested expression with brackets and parentheses, work from the innermost grouping outward, applying the exponent step at the correct point in each layer.

🧩 Multiple operations together

• The excerpt provides exercises that mix exponents with addition, subtraction, multiplication, and division.
• The key is to isolate the exponent step: evaluate it as soon as parentheses are resolved, then proceed with MD and AS.
• Don't confuse: even if multiplication appears next to an exponent in the written expression, the exponent must be evaluated first.

🧭 Overview

🧠 One-sentence thesis

Grouping symbols control the order in which operations are performed, and different placements of parentheses, brackets, or fraction bars produce different results even when the same numbers and operations are present.

📌 Key points (3–5)

• What grouping symbols do: they override the default PEMDAS order by forcing certain operations to be performed first.
• Types of grouping symbols: parentheses, square brackets (for nested grouping), and fraction bars all act as grouping symbols.
• Why placement matters: the same numbers and operations yield different answers depending on where grouping symbols appear.
• Common confusion: a fraction bar is a grouping symbol—you must complete all operations in the numerator and denominator separately before dividing.
• Nested grouping: when grouping symbols appear inside other grouping symbols, square brackets can make pairs easier to match, and you work from the innermost group outward.

🧩 What grouping symbols are and why they matter

🧩 Definition and purpose

Grouping symbols tell you to work inside them first, following PEMDAS as necessary inside the grouping symbols.

• They are part of the "P" in PEMDAS (Parentheses).
• Without grouping symbols, you follow the default order: exponents, then multiplication/division left to right, then addition/subtraction left to right.
• With grouping symbols, you perform the operations inside them before anything outside, even if that means doing addition before multiplication.

🔍 How grouping symbols change results

• The excerpt shows exercises where "the only differences are the parentheses, but every exercise has a different answer."
• Example: if you have subtraction and addition without parentheses, you work left to right; if you add parentheses around the subtraction, you do that first, changing the final result.
• Don't confuse: the same numbers and operations in a different grouping order produce different values—grouping is not optional or decorative.

🧷 Types of grouping symbols

🧷 Parentheses and square brackets

• Parentheses ( ) are the most common grouping symbol.
• Square brackets [ ] are used when grouping symbols are nested within other grouping symbols.
• Using brackets instead of multiple layers of parentheses makes it easier to match up pairs of left and right symbols.
• Example: the excerpt gives [ ] as a clearer alternative to ( ( ) ) for nested grouping.

📏 Fraction bars as grouping symbols

A fraction bar is another grouping symbol; it tells us to perform all of the steps on the top and separately perform all of the steps down below. The final step is to divide the top number by the bottom number.

• The fraction bar groups the numerator and the denominator separately.
• You must complete all operations in the numerator, complete all operations in the denominator, and only then divide the top result by the bottom result.
• Don't confuse: the fraction bar is not just a division sign—it also groups everything above it and everything below it.

⚙️ Working with nested and complex grouping

⚙️ Nested grouping symbols

• When grouping symbols appear inside other grouping symbols, work from the innermost group outward.
• The excerpt mentions expressions like nested parentheses or brackets within brackets.
• Example: if you see [ ( ) ], evaluate what is inside the parentheses first, then use that result in the brackets, then proceed with the rest of the expression.

🔢 Translating words to expressions with grouping

• The excerpt includes word problems that require writing expressions with correct grouping.
• Example (Fahrenheit from Celsius): "doubling the Celsius temperature and adding [a number]" means you write the doubling and addition in the correct order, using parentheses if needed to ensure the doubling happens first.
• Example (Celsius from Fahrenheit): "subtracting [a number] from the Fahrenheit temperature and then dividing by [another number]" means the subtraction must be grouped (or understood to happen first) before the division.
• The key is to match the grouping in your expression to the sequence described in words.

📊 Summary: grouping symbols in the order of operations

Symbol type	What it does	When to use it
Parentheses ( )	Force operations inside to be done first	Whenever you need to override default PEMDAS order
Square brackets [ ]	Same as parentheses, but used for nested grouping	When grouping symbols are inside other grouping symbols, to make pairs easier to match
Fraction bar	Groups numerator and denominator separately, then divides	Whenever you have a fraction; complete top and bottom before dividing

• All grouping symbols share the same core rule: do what is inside first, following PEMDAS within the grouping as needed.
• The excerpt emphasizes that different grouping produces different answers, so correct placement is essential for correct results.

🧭 Overview

🧠 One-sentence thesis

Absolute value measures a number's distance from zero regardless of sign, and it simplifies working with signed numbers in addition, subtraction, and finding positive differences.

📌 Key points (3–5)

• What absolute value measures: the distance of a number from zero, ignoring whether it is positive or negative.
• Opposites: numbers with the same absolute value but different signs (e.g., 5 and -5).
• Notation: vertical bars around a number indicate absolute value (e.g., |5| and |-5| both equal 5).
• Common confusion: absolute value is not the same as "making negative numbers positive"—it is about distance from zero, so both 5 and -5 are 5 units away.
• Practical use: absolute value guarantees a positive result when finding the difference between two numbers (positive difference).

🔢 What absolute value means

📏 Distance from zero

The absolute value of a number is its distance from zero.

• It measures size without regard to direction (positive or negative).
• Both positive and negative numbers can have the same absolute value.
• Example: -3 and 3 are both 3 units away from zero, so |-3| = 3 and |3| = 3.

🔄 Opposites

Numbers with the same absolute value but different signs are called opposites.

• Opposites are mirror images across zero on the number line.
• Example: 7 and -7 are opposites; both have absolute value 7.
• The excerpt emphasizes that both numbers are the same distance from zero.

✏️ Notation

• Absolute value is written with a pair of straight vertical bars: |number|.
• Example: |5| = 5 and |-5| = 5.
• The bars tell you to find the distance from zero, not to perform arithmetic inside first (unless there is an expression inside the bars).

➕ Adding signed numbers

➖ Adding two negative numbers

• Rule: Add their absolute values (ignore the negative signs), then make the final answer negative.
• Example: To add -4 and -6, add 4 + 6 = 10, then make it negative: -10.
• Why: Moving left on the number line twice means you end up farther left (more negative).

➕ Adding a positive and a negative number

• Rule: Subtract the smaller absolute value from the larger absolute value.
  • If the positive number has the larger absolute value, the answer is positive.
  • If the negative number has the larger absolute value, the answer is negative.
• Example: To add 8 and -3, subtract 3 from 8 to get 5 (positive, because 8 is larger).
• Example: To add -10 and 4, subtract 4 from 10 to get 6, then make it negative: -6 (because 10 is larger).

🔀 Order flexibility

• The excerpt notes that if an expression consists only of additions, you can add the numbers in any order (breaking the usual order of operations).
• This is a shortcut for simplifying expressions with multiple additions.

➖ Subtracting signed numbers

🔄 Subtracting as adding the opposite

To subtract two signed numbers, add the first number to the opposite of the second number.

• Rule: Change subtraction to addition, and flip the sign of the second number.
• Example: 7 - (-3) becomes 7 + 3 = 10.
• Example: -5 - 2 becomes -5 + (-2) = -7.
• Why: The excerpt illustrates this with a payroll example—subtracting a negative amount (a "deduction of negative dollars") has the effect of adding a positive amount.

💼 Real-world illustration

• The excerpt describes a paystub where a payment needed to be tracked in the deductions category.
• A positive deduction subtracts money; a deduction of negative dollars adds money to the paycheck.
• This shows that subtracting a negative is equivalent to adding a positive.

📐 Absolute value for positive differences

🌡️ Finding positive difference

The positive difference is the result of a subtraction wrapped in absolute value bars, guaranteeing a positive result.

• Why it matters: When comparing two quantities (e.g., temperatures, elevations), you often want the difference to be positive regardless of which number is larger.
• How it works: Put absolute value bars around the entire subtraction: |a - b|.
• Example: The temperature in one city is 30°F and in another is -25°F. The difference is |30 - (-25)| = |30 + 25| = |55| = 55°F.
• Don't confuse: Without absolute value, 30 - (-25) = 55 and -25 - 30 = -55; the absolute value ensures you always report "55 degrees" as the difference.

🏔️ Elevation example

• The excerpt gives an example with Colorado elevations: the lowest point and the highest point.
• To find the positive difference, subtract the two elevations and wrap the result in absolute value bars.
• This guarantees the answer is positive, no matter which elevation you subtract from which.

🌊 Below sea level

• The excerpt mentions Louisiana elevations, including a point below sea level (negative elevation).
• The same positive difference technique applies: |elevation₁ - elevation₂| gives the vertical distance between the two points.
• Example: If one point is -8 feet (below sea level) and another is 535 feet (above sea level), the positive difference is |-8 - 535| = |-543| = 543 feet.

🧭 Overview

🧠 One-sentence thesis

Adding negative numbers follows two rules: when both are negative, add their absolute values and make the result negative; when one is positive and one is negative, subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value.

📌 Key points (3–5)

• Absolute value foundation: the absolute value of a number is its distance from zero, ignoring sign; opposites have the same absolute value but different signs.
• Two negatives: add their absolute values and make the final answer negative.
• One positive, one negative: subtract the smaller absolute value from the larger; the sign of the result matches the sign of the number with the larger absolute value.
• Common confusion: subtracting a negative is not the same as adding a negative—subtracting a negative is equivalent to adding a positive.
• Flexible order: when an expression contains only additions, you can add in any order (breaking normal order of operations).

🔢 Absolute value basics

🔢 What absolute value means

The absolute value of a number is its distance from zero.

• It measures size without regard to positive or negative direction.
• Notation: straight vertical bars, e.g., the absolute value of negative three is written with bars around negative three.
• Example: both positive five and negative five are five units away from zero, so both have absolute value five.

🔄 Opposites

Numbers with the same absolute value but different signs are called opposites.

• Positive five and negative five are opposites.
• They are the same distance from zero but in opposite directions on the number line.

➕ Adding two negative numbers

➕ The rule

• Add their absolute values (ignore the negative signs temporarily).
• Make the final answer negative.
• Example: to add negative three and negative seven, add three and seven to get ten, then make it negative ten.

➕➖ Adding a positive and a negative number

➕➖ The rule

• Subtract the smaller absolute value from the larger absolute value.
• The sign of the final answer depends on which number had the larger absolute value:
  • If the positive number has the larger absolute value → final answer is positive.
  • If the negative number has the larger absolute value → final answer is negative.

🌡️ Real-world scenario

• Example: the temperature at noon on Monday was negative eight degrees Fahrenheit; by Tuesday noon it had risen fifteen degrees Fahrenheit.
• To find Tuesday's temperature, add negative eight and positive fifteen.
• Fifteen has the larger absolute value, so subtract eight from fifteen to get seven, and the answer is positive seven degrees Fahrenheit.

🔀 Flexible addition order

• When an expression consists of only additions, you can add the numbers in whatever order you choose.
• This breaks the usual rules for order of operations but is allowed for addition-only expressions.
• Use this to find shortcuts (e.g., pairing numbers that sum to zero or round numbers).

➖ Subtracting negative numbers

➖ The core principle

To subtract two signed numbers, add the first number to the opposite of the second number.

• Subtracting a negative amount is equivalent to adding a positive amount.
• Example (from a paystub): a deduction of negative eighty dollars has the effect of adding eighty dollars to the paycheck.
• Don't confuse: subtracting a negative is not the same as subtracting a positive—it flips the operation to addition.

🔄 How to apply the rule

• Take the first number as-is.
• Change the subtraction to addition.
• Change the second number to its opposite (flip its sign).
• Then follow the addition rules above.

📏 Absolute value for positive difference

📏 Why use absolute value around subtraction

• Sometimes you want the difference between two numbers to always be positive, regardless of which is larger.
• Example: the temperature in Portland, Oregon is seventy degrees Fahrenheit; in Portland, Maine it is fifteen degrees Fahrenheit.
• Subtracting fifteen from seventy gives fifty-five; subtracting seventy from fifteen gives negative fifty-five.
• It sounds strange to report "negative fifty-five degrees Fahrenheit" as a difference, so you would ignore the negative sign.

📏 Positive difference

Positive difference: put absolute value bars around the entire subtraction calculation to guarantee the result is positive.

• This ensures the answer is always non-negative, no matter the order of the numbers.
• Example: to find the positive difference between an elevation of three thousand three hundred seventeen feet above sea level and an elevation of eight feet below sea level (negative eight feet), calculate the absolute value of (three thousand three hundred seventeen minus negative eight).

🧭 Overview

🧠 One-sentence thesis

Subtracting a negative number is equivalent to adding its positive opposite, so the operation transforms subtraction into addition.

📌 Key points (3–5)

• Core rule: To subtract two signed numbers, add the first number to the opposite of the second number.
• Real-world meaning: Subtracting a negative amount has the effect of adding a positive amount (e.g., removing a deduction adds money).
• Positive difference: Absolute value bars around a subtraction guarantee a positive result, useful when reporting differences like temperature or elevation.
• Common confusion: Don't mix up subtraction rules with multiplication/division rules—they work differently with signs.

🔄 The subtraction-to-addition rule

🔄 How to subtract signed numbers

To subtract two signed numbers, add the first number to the opposite of the second number.

• "Opposite" means flipping the sign: the opposite of a negative is positive, and vice versa.
• This rule works for all combinations: positive minus negative, negative minus positive, negative minus negative, etc.
• Example: Instead of calculating "5 minus negative 3," rewrite it as "5 plus positive 3," which equals 8.

💡 Why subtracting a negative adds

• The excerpt illustrates this with a payroll scenario: a deduction of negative 80 dollars adds 80 dollars to the paycheck.
• Removing a negative (a deduction) is the same as adding a positive (income).
• Don't confuse: This is not the same as "two negatives make a positive" in multiplication—subtraction converts to addition first, then you apply addition rules.

📏 Absolute value and positive difference

📏 What absolute value does

The absolute value of a number is its distance from 0.

• It strips away the sign, leaving only the size.
• Numbers with the same absolute value but different signs (e.g., 7 and −7) are called opposites.
• Notation: vertical bars, such as |−7| = 7 and |7| = 7.

🌡️ Positive difference in practice

Positive difference: absolute value bars around an entire subtraction to guarantee a positive result.

• Used when you want the difference between two numbers without worrying about which is larger.
• Example: Temperature in Portland, Oregon is 20°F and in Portland, Maine is −35°F. The simple subtraction 20 − (−35) = 55, but if you did it backward (−35 − 20 = −55), reporting "negative fifty-five degrees" sounds odd. Using |−35 − 20| ensures the answer is 55.
• Example: Elevation difference between Colorado's lowest point (3,315 feet above sea level) and highest point (14,440 feet above sea level) can be found with |3,315 − 14,440| or |14,440 − 3,315|, both yielding a positive result.
• Example: Louisiana's lowest point is −8 feet (below sea level) and highest is 535 feet (above sea level). The positive difference |−8 − 535| or |535 − (−8)| gives the total elevation span.

🔗 Relationship to other operations

🔗 Subtraction vs multiplication/division rules

• The excerpt warns: "These rules are different from the rules for addition; be careful not to mix them up."
• Subtraction converts to addition of the opposite; it does not directly follow the "same sign → positive, opposite sign → negative" pattern of multiplication and division.
• Multiplication/division: opposite signs → negative product/quotient; same signs → positive product/quotient.
• Don't confuse: Subtracting a negative is adding a positive, but multiplying two negatives is a positive product—different mechanisms.

🔗 Absolute value revisited

• Absolute value is useful beyond basic definitions: it ensures positive results in subtraction when direction doesn't matter.
• It complements the subtraction rule by letting you ignore which number is larger when reporting differences.

🧭 Overview

🧠 One-sentence thesis

Absolute value bars around a subtraction guarantee a positive result, making them useful for reporting differences where the order of subtraction doesn't matter.

📌 Key points (3–5)

• What absolute value does for differences: wrapping a subtraction in absolute value bars ensures the result is positive, regardless of which number is subtracted from which.
• Why it's useful: when reporting real-world differences (e.g., temperature, elevation), we often want a positive number rather than a negative one.
• The term: this use of absolute value is called the "positive difference."
• Common confusion: subtracting in different orders gives opposite signs (e.g., 10 − 65 vs. 65 − 10), but absolute value removes that sign issue.

🌡️ Motivation and real-world context

🌡️ Temperature example

• The excerpt gives Portland, Oregon at one temperature and Portland, Maine at another (specific values not fully shown).
• If you subtract in one order, you get a negative result; in the other order, positive.
• When reporting "the difference is fifty-five degrees," people naturally ignore the negative sign.
• Why absolute value helps: it removes the need to remember which city was warmer—you always get a positive difference.

🏔️ Elevation examples

The excerpt provides two elevation problems:

Location	Low point	High point	Task
Colorado	Arikaree River (above sea level)	Mount Elbert peak (above sea level)	Find positive difference
Louisiana	New Orleans (below sea level, negative elevation)	Driskill Mountain (above sea level)	Find positive difference

• Both exercises ask for the "positive difference," meaning the result should be positive.
• Example: if the low point is −8 feet and the high point is 535 feet, subtracting either way and taking absolute value gives the same positive answer.

🔢 The positive difference concept

🔢 Definition and notation

Positive difference: the result of a subtraction wrapped in absolute value bars, ensuring the answer is positive.

• Notation: |first number − second number|
• The absolute value bars "guarantee that the result of a subtraction is positive."
• This is the key takeaway: you don't have to worry about which number is larger before subtracting.

🔄 How it resolves order confusion

• Without absolute value:
  • 10 − 65 = −55
  • 65 − 10 = 55
• With absolute value:
  • |10 − 65| = |−55| = 55
  • |65 − 10| = |55| = 55
• Both give the same positive result, so the order doesn't matter.
• Don't confuse: the absolute value is applied to the entire subtraction, not to each number separately before subtracting.

📝 When to use it

• When the context calls for "how far apart" two values are, not "which is bigger."
• When reporting differences in everyday language (temperature, elevation, distance) where negative results sound awkward.
• Example: "The difference in elevation is 500 feet" is clearer than "The difference is −500 feet."

🧮 Exercises and practice

🧮 Evaluation exercises

The excerpt lists several exercises (20–23) that ask students to:

• Evaluate expressions with absolute value around subtractions.
• Find positive differences for real-world elevation data (Colorado and Louisiana).

🧮 Purpose of the exercises

• Reinforce the idea that absolute value removes the sign.
• Practice applying the concept to concrete scenarios (geographic elevations).
• Build fluency with the notation and the reasoning behind it.

🧭 Overview

🧠 One-sentence thesis

Multiplying two numbers with opposite signs yields a negative product, while multiplying two numbers with the same sign yields a positive product, and these rules extend predictably to exponents and division.

📌 Key points (3–5)

• Opposite signs → negative product: when one number is positive and the other negative, the result is negative.
• Same signs → positive product: when both numbers are positive or both are negative, the result is positive.
• Exponents follow a pattern: a negative number raised to an even power is positive; raised to an odd power is negative.
• Common confusion: multiplication/division rules differ from addition rules—don't mix them up.
• Division mirrors multiplication: the sign rules for division are exactly the same as for multiplication.

✖️ Sign rules for multiplication

✖️ Opposite signs produce negative results

If two numbers with opposite signs are multiplied, the product is negative.

• This means one number is positive and the other is negative.
• The excerpt illustrates this with a spending example: spending 3 dollars per day (represented as negative 3 dollars) over 5 days totals negative 15 dollars.
• Example: negative 3 times positive 5 equals negative 15.

✖️ Same signs produce positive results

If two numbers with the same sign are multiplied, the product is positive.

• This applies when both numbers are positive or both are negative.
• The excerpt shows that the opposite of negative 15 is positive 15, so negative 3 times negative 5 equals positive 15.
• Example: negative times negative gives positive; positive times positive also gives positive.

⚠️ Don't confuse with addition

• The excerpt includes a warning: these multiplication rules are different from addition rules.
• Be careful not to mix up the two sets of rules when working with signed numbers.

📈 Exponents with negative numbers

📈 Even powers yield positive results

If a negative number is raised to an even power, the result is positive.

• An exponent represents repeated multiplication.
• When a negative number is multiplied by itself an even number of times, the negatives cancel out in pairs.
• Example: negative number squared (power of 2) or to the fourth power will be positive.

📉 Odd powers yield negative results

If a negative number is raised to an odd power, the result is negative.

• When a negative number is multiplied by itself an odd number of times, one negative remains unpaired.
• Example: negative number cubed (power of 3) or to the fifth power will be negative.

➗ Sign rules for division

➗ Division mirrors multiplication exactly

The rules for division work exactly like the rules for multiplication.

• The excerpt revisits the coffee example: negative 15 divided by positive 5 equals negative 3 (opposite signs → negative quotient).
• Also: negative 15 divided by negative 5 equals positive 3 (same signs → positive quotient).

➗ Opposite signs produce negative quotients

If two numbers with opposite signs are divided, the quotient is negative.

• One number positive, the other negative → result is negative.
• Example: a negative divided by a positive gives a negative result; a positive divided by a negative also gives a negative result.

➗ Same signs produce positive quotients

If two numbers with the same sign are divided, the quotient is positive.

• Both positive or both negative → result is positive.
• Example: a negative divided by a negative gives a positive result.

🚫 Special cases with zero

🚫 Zero divided by a number

0 divided by another number is 0.

• When zero is the numerator (top number), the result is always zero.

🚫 A number divided by zero

A number divided by 0 is undefined, or not a real number.

• When zero is the denominator (bottom number), the division cannot be performed.
• The excerpt notes that checking this on a calculator may produce surprises, confirming that division by zero is not allowed.
• Don't confuse: zero divided by a number (equals zero) vs. a number divided by zero (undefined).

🧭 Overview

🧠 One-sentence thesis

Division of negative numbers follows the same sign rules as multiplication: opposite signs yield a negative quotient, and same signs yield a positive quotient.

📌 Key points (3–5)

• Sign rules for division: opposite signs → negative quotient; same signs → positive quotient.
• Parallel to multiplication: division rules work exactly like multiplication rules for determining the sign of the result.
• Zero in division: zero divided by any number is zero, but any number divided by zero is undefined (not a real number).
• Common confusion: don't mix up division/multiplication sign rules with addition rules—they are different.

🔢 Sign rules for dividing negative numbers

➗ Opposite signs produce negative quotients

If two numbers with opposite signs are divided, the quotient is negative.

• When a negative is divided by a positive, the result is negative.
• When a positive is divided by a negative, the result is also negative.
• Example: The excerpt shows that negative 15 divided by positive 5 equals negative 3 (from the coffee spending scenario: spending 15 dollars over 5 days means 3 dollars per day).

➕ Same signs produce positive quotients

If two numbers with the same sign are divided, the quotient is positive.

• When a negative is divided by a negative, the result is positive.
• When a positive is divided by a positive, the result is positive.
• Example: The excerpt shows that negative 15 divided by negative 5 equals positive 3.

🔗 Connection to multiplication

🔄 Identical sign logic

• The excerpt explicitly states: "the rules for division work exactly like the rules for multiplication."
• Both operations use the same principle for determining the sign of the result.
• This parallelism makes the rules easier to remember once you know one operation.

⚠️ Don't confuse with addition rules

• The excerpt includes a warning: "These rules are different from the rules for addition; be careful not to mix them up."
• Addition has its own distinct rules for combining signed numbers.
• The sign rules for multiplication and division are a separate system.

🚫 Special cases with zero

0️⃣ Zero as dividend

0 divided by another number is 0.

• When zero is the number being divided (the dividend), the answer is always zero.
• This applies regardless of what non-zero number you divide by.

❌ Zero as divisor

A number divided by 0 is undefined, or not a real number.

• When zero is the divisor (the number you're dividing by), the operation cannot produce a real number.
• Why it's undefined: The excerpt explains with a proof by contradiction. If some mystery number equals "something divided by 0," then rewriting as multiplication gives "0 times mystery number equals something." But zero times any number is zero, so you get "0 equals something," which doesn't make sense. Therefore, no such mystery number exists.
• Don't confuse: "0 divided by a number" (which equals 0) vs "a number divided by 0" (which is undefined).

🧭 Overview

🧠 One-sentence thesis

The order of operations (PEMDAS) applies to expressions with negative numbers just as it does to positive numbers, ensuring consistent evaluation of complex arithmetic.

📌 Key points (3–5)

• What PEMDAS stands for: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).
• How to apply it: work through each level in order, handling operations at the same level from left to right.
• Key rule: multiplication and division are performed from left to right (same priority level), as are addition and subtraction.
• Common confusion: don't mix up the rules for multiplying/dividing negative numbers with the rules for adding/subtracting them—they are different.
• Why it matters: following the correct order ensures you get the right answer when evaluating expressions with multiple operations and negative numbers.

🔢 The PEMDAS hierarchy

🅿️ Parentheses (P)

• Work inside parentheses or other grouping symbols first.
• If there are multiple operations inside parentheses, follow PEMDAS within them as well.

🔺 Exponents (E)

• Evaluate exponents after handling parentheses.
• Remember: a negative number raised to an even power gives a positive result; raised to an odd power gives a negative result.

✖️➗ Multiplication and Division (MD)

• Perform all multiplications and divisions from left to right.
• These two operations have the same priority level—neither comes "before" the other.
• Don't confuse: you don't do all multiplications first and then all divisions; you work left to right through both.

➕➖ Addition and Subtraction (AS)

• Perform all additions and subtractions from left to right.
• Like MD, these two operations share the same priority level.
• Work left to right through both operations.

🧮 Working with negative numbers in order of operations

🔁 Applying the rules consistently

• The order of operations applies the same way whether numbers are positive or negative.
• You still follow P-E-MD-AS in sequence.
• Example: In an expression with parentheses, exponents, and multiplication, you handle parentheses first, then exponents, then multiplication, regardless of whether the numbers are negative.

⚠️ Don't mix up operation rules

WARNING! The rules for multiplying and dividing negative numbers are different from the rules for addition; be careful not to mix them up.

• Multiplication/Division rules:
  • Opposite signs → negative result
  • Same signs → positive result
• Addition rules (from earlier in the excerpt): different and must not be confused with multiplication/division rules.
• Common mistake: applying addition rules when multiplying, or vice versa.

📝 Practice structure

📋 Exercise format

The excerpt provides numbered exercises (38–43) for evaluating expressions using the order of operations.

• Each exercise requires applying PEMDAS step-by-step.
• The exercises involve negative numbers and multiple operations.
• The goal is to practice the correct sequence and avoid common errors.

🧭 Overview

🧠 One-sentence thesis

Adding and subtracting decimals requires lining up decimal points before performing the operation, and you may need to add zeros to borrow correctly when subtracting.

📌 Key points (3–5)

• Core rule for addition and subtraction: you must line up the decimal points before calculating.
• Special step for subtraction: add zeros to the first number when necessary to borrow correctly.
• Common confusion: decimal point alignment is required for addition/subtraction but NOT for multiplication.
• Decimal notation foundation: based on powers of 10 (tenths, hundredths, thousandths, etc.).

🔢 Understanding decimal notation

🔢 What decimals represent

Decimal notation is based on powers of 10: one tenth, one hundredth, one thousandth, and so on.

• Each position to the right of the decimal point represents a smaller fraction of 10.
• Place value structure:
  • Left of decimal: thousands, hundreds, tens, ones/units
  • Right of decimal: tenths, hundredths, thousandths

📝 Writing decimal numbers

• Numbers are written by combining the whole number part and the fractional part.
• Example: "ninety and twenty-three hundredths" means 90 plus 23/100.
• Example: "seven and fifty-six thousandths" means 7 plus 56/1000.

➕ Adding decimals

➕ The alignment rule

• Before you add decimals, you must line up the decimal points.
• This ensures that you are adding tenths to tenths, hundredths to hundredths, etc.
• The decimal point in the answer will be directly below the aligned decimal points.

🧮 How to add

1. Write the numbers vertically.
2. Align the decimal points in a straight column.
3. Add as you would with whole numbers.
4. Place the decimal point in the answer directly below the aligned points.

➖ Subtracting decimals

➖ The alignment rule (same as addition)

• Before you subtract decimals, you must line up the decimal points.
• This rule is identical to addition—alignment is essential.

🔄 Adding zeros for borrowing

• When subtracting, you may need to add zeros to the first number so you can borrow correctly.
• Zeros are added to the right of the last digit (after the decimal point) without changing the value.
• Example: if you need to subtract from a number with fewer decimal places, add zeros to match the number of places in the second number.
• This makes borrowing possible when a digit in the top number is smaller than the corresponding digit below.

⚠️ Don't confuse with multiplication

• Decimal point alignment is required for addition and subtraction.
• The excerpt later notes: "You do NOT need to line up the decimal points when you are multiplying."
• Why the difference: addition/subtraction work column by column (same place values), while multiplication combines all digits differently.

🧭 Overview

🧠 One-sentence thesis

Multiplying decimals works by temporarily treating the numbers as whole numbers, then placing the decimal point in the result based on the total count of decimal digits in the original factors.

📌 Key points (3–5)

• Core procedure: ignore decimal points, multiply as whole numbers, then count total decimal digits to place the point in the answer.
• Key difference from addition/subtraction: you do NOT need to line up decimal points when multiplying.
• Counting decimal places: add up the number of digits to the right of the decimal in both factors; the product will have that many decimal digits.
• Common confusion: unlike adding/subtracting decimals (which require alignment), multiplication ignores alignment and uses digit counting instead.

🔢 The three-step multiplication procedure

🔢 Step 1: Ignore the decimal points

• Treat both numbers as if they were whole numbers.
• The excerpt says "temporarily ignore the decimal points."
• This simplifies the arithmetic—you multiply using the same method as for integers.

🔢 Step 2: Multiply as whole numbers

• Perform the multiplication without worrying about where the decimal point goes.
• Example: to multiply 3.2 and 1.5, first multiply 32 and 15 as whole numbers.

🔢 Step 3: Count and place the decimal

"Add the total number of decimal digits in the two numbers you multiplied. The result will have that number of digits to the right of the decimal point."

• Count how many digits appear after the decimal in the first factor.
• Count how many digits appear after the decimal in the second factor.
• Add these two counts together.
• Place the decimal point in your product so that many digits are to the right of it.
• Example: 3.2 has 1 decimal digit, 1.5 has 1 decimal digit → total 2 → the product will have 2 decimal digits.

⚠️ Key distinction from addition and subtraction

⚠️ No alignment needed

• The excerpt explicitly states: "You do NOT need to line up the decimal points when you are multiplying."
• This contrasts with addition and subtraction, where the excerpt says "you must line up the decimal points."
• Don't confuse: alignment is required for adding/subtracting but not for multiplying.

⚠️ Why the difference matters

• In addition/subtraction, place value must match (tenths with tenths, hundredths with hundredths).
• In multiplication, the final decimal placement depends on the total count of decimal digits, not on aligning columns.

🧮 Practical application

🧮 Real-world scenario

• The excerpt includes an exercise: "Evie worked [some number of] hours at a pay rate of [some amount] per hour. How much did she earn in total?"
• This is a typical multiplication problem: hours × rate = total earnings.
• Both hours and rate are likely decimals (e.g., 7.5 hours, $12.50 per hour).
• Apply the three-step procedure: ignore decimals, multiply, count decimal places, and place the point.

🧭 Overview

🧠 One-sentence thesis

Dividing decimals requires converting the divisor into a whole number by shifting decimal points in both divisor and dividend, then performing long division as with whole numbers.

📌 Key points (3–5)

• The three parts of division: dividend (inside the symbol), divisor (on the left), and quotient (the answer on top).
• Core technique: move the divisor's decimal point to make it a whole number, then move the dividend's decimal point the same number of places.
• Decimal placement in the answer: place the decimal point in the quotient directly above the decimal point in the dividend.
• Common confusion: unlike multiplication, you do need to align decimal points—but only after shifting them according to the divisor.
• Extending the division: if needed, add zeros to the right of the dividend's last digit to continue dividing.

🔢 The parts of a division problem

🔢 Naming the components

dividend ÷ divisor = quotient

• When written with a long division symbol:
  • The dividend goes inside the symbol.
  • The divisor is on the left (outside).
  • The quotient is the answer written on top.
• Example: In "12 ÷ 3 = 4," 12 is the dividend, 3 is the divisor, and 4 is the quotient.

🔄 The six-step procedure

🔄 Step 1: Write in long division form

• Set up the problem using the long division symbol, with the dividend inside and the divisor outside on the left.

🔄 Step 2: Move the divisor's decimal point

• Shift the decimal point in the divisor to the right until it becomes a whole number.
• This step is crucial because dividing by a whole number is simpler than dividing by a decimal.

🔄 Step 3: Move the dividend's decimal point

• Move the decimal point in the dividend the same number of places to the right as you moved the divisor's decimal point.
• This keeps the value of the division problem unchanged.
• Example: If you moved the divisor's decimal two places right, move the dividend's decimal two places right as well.

🔄 Step 4: Place the decimal point in the quotient

• Before you start dividing, place the decimal point in the quotient (the answer) directly above the decimal point in the dividend.
• This ensures the answer has the decimal in the correct position.

🔄 Step 5: Divide as whole numbers

• Ignore the decimal points and divide the numbers as though they are whole numbers.
• Use standard long division techniques.

🔄 Step 6: Add zeros if necessary

• If the division does not come out evenly, you can add zeros to the right of the last digit of the dividend to continue the division process.
• This allows you to find more decimal places in the quotient or to round the answer later.

⚠️ Key difference from multiplication

⚠️ Decimal alignment rule

• Multiplication: You do NOT need to line up the decimal points when multiplying.
• Division: You DO need to manage decimal points carefully—by shifting them according to the divisor before you divide.
• Don't confuse: The excerpt explicitly notes that multiplication and division handle decimals differently; the alignment step in division is about making the divisor a whole number, not about lining up the original decimal points.

🧭 Overview

🧠 One-sentence thesis

Rounding numbers to a specified place value follows a systematic process that depends on whether the digit immediately to the right is 5 or greater.

📌 Key points (3–5)

• The two key digits: the rounding digit (the place you're rounding to) and the test digit (directly to its right).
• The decision rule: if the test digit is 5 or greater, round up; if less than 5, keep the rounding digit unchanged.
• What happens after: all digits to the right of the rounding digit are dropped.
• Common confusion: don't look at digits beyond the test digit—only the digit immediately to the right of the rounding digit matters.
• Practical use: rounding is necessary when decimal answers continue indefinitely or when a specific precision is required.

🎯 The rounding process

📍 Identifying the two critical digits

Rounding digit: the digit in the place value to which you are rounding.

Test digit: the digit directly to the right of the rounding digit.

• The rounding digit is determined by the instruction (e.g., "round to the nearest tenth" means the tenths place is the rounding digit).
• The test digit is always the single digit immediately to the right—no other digits matter for the decision.
• Example: Rounding 3.476 to the nearest tenth → rounding digit is 4 (tenths place), test digit is 7 (hundredths place).

🔍 The decision rule

The excerpt provides a clear two-part rule:

Test digit value	Action	What to do with digits to the right
5 or greater	Increase the rounding digit by 1	Drop all digits to the right
Less than 5	Keep the rounding digit the same	Drop all digits to the right

• "Drop all digits to its right" means those digits disappear from the final answer.
• Example: Rounding 2,847 to the thousands place → rounding digit is 2, test digit is 8 (which is ≥5) → increase 2 to 3 → result is 3,000.
• Example: Rounding 5.234 to the hundredths place → rounding digit is 3, test digit is 4 (which is <5) → keep 3 → result is 5.23.

⚠️ Common confusion: which digit to check

• Don't confuse: only look at the digit directly to the right of the rounding digit, not all remaining digits.
• The rule does not say "look at all digits to the right" or "average them"—only the immediate neighbor matters.

📐 Rounding to different place values

🔢 Whole number place values

• The excerpt mentions rounding to thousands and tens.
• Example (from exercises): rounding to thousands means the rounding digit is in the thousands place; the test digit is in the hundreds place.
• Example (from exercises): rounding to tens means the rounding digit is in the tens place; the test digit is in the ones place.

🔢 Decimal place values

• The excerpt mentions rounding to hundredths and thousandths.
• Example (from exercises): rounding to hundredths means the rounding digit is in the hundredths place (second digit after the decimal point); the test digit is in the thousandths place.
• Example (from exercises): rounding to thousandths means the rounding digit is in the thousandths place (third digit after the decimal point); the test digit is in the ten-thousandths place.

🛠️ Practical applications

💰 Real-world rounding scenarios

The excerpt provides two practical exercises:

• Monthly cost calculation: A subscription costs a certain amount; determine the monthly cost rounded to the nearest cent (hundredths place for dollars).
• Speed calculation: A convoy traveled 82 miles in 11 hours; calculate the speed in miles per hour rounded to the nearest tenth.

🔄 When rounding is necessary

The excerpt states:

"If a decimal answer goes on and on, it may be practical to round it off."

• Rounding is used when precision beyond a certain place value is not needed or practical.
• It simplifies numbers for communication and practical use.
• The specified place value depends on the context (e.g., money is typically rounded to cents, speeds might be rounded to tenths).

🧭 Overview

🧠 One-sentence thesis

Fractions describe equal parts of a whole and can be manipulated through simplification, arithmetic operations, and conversion to other forms while preserving their value.

📌 Key points (3–5)

• What a fraction represents: equal parts of a whole, written with a numerator (top) and denominator (bottom).
• Equivalent fractions: different fractions can represent the same portion of a whole by multiplying or dividing both parts by the same number.
• Simplification: fractions are in simplest form when the numerator and denominator share no common factors other than 1.
• Common confusion: mixed numbers represent addition (e.g., 2 and 1/3), not multiplication (not 2 times 1/3).
• Key operations: multiplying, dividing, adding, and subtracting fractions each follow specific rules, especially regarding common denominators.

📐 Basic structure and meaning

📐 What a fraction describes

A fraction describes equal parts of a whole.

• The excerpt uses "official math vocabulary" to identify the two components:
  • Numerator: the top number
  • Denominator: the bottom number
• The fraction represents how many equal parts (numerator) out of the total number of parts (denominator) you have.
• Example: If April had some rainy days and some non-rainy days, the fraction of rainy days would be (number of rainy days) / (total days in April).

🔄 Equivalent fractions

Two fractions are equivalent if they represent the same number (the same portion of a whole).

• You can build an equivalent fraction by multiplying both the numerator and denominator by the same number.
• This changes the appearance but not the value.
• Example: If you multiply both parts of a fraction by 2, you get a different-looking fraction that represents the same portion.

✂️ Simplifying fractions

✂️ What simplification means

To simplify or reduce a fraction: divide the numerator and denominator by the same number.

• The excerpt mentions four special cases for simplification (though the specific cases are not detailed in the text provided).
• A fraction is completely reduced, in simplest form, or in lowest terms when the numerator and denominator have no common factors other than 1.
• Simplification makes fractions easier to work with and compare.

✂️ How to reduce to simplest form

• Find a common factor of both the numerator and denominator.
• Divide both by that factor.
• Repeat until no common factors remain (other than 1).
• Example: If both the numerator and denominator are even, you can divide both by 2.

✖️➗ Multiplying and dividing fractions

✖️ Multiplying fractions

To multiply fractions, multiply the numerators and multiply the denominators straight across.

• After multiplying, simplify the answer if possible.
• The phrase "a fraction of a number" means multiply.
• Example: To find 3/4 of 20 questions (as in Nathan's training scenario), multiply 3/4 by 20.

➗ Dividing fractions

To divide by a fraction, multiply by the reciprocal of the second number (flip the second fraction upside-down).

• The reciprocal means swapping the numerator and denominator of the second fraction.
• Then proceed as with multiplication.
• Example: If you need to measure a certain number of cups but only have a smaller scoop, dividing the total by the scoop size tells you how many scoops you need.

⚖️ Comparing, adding, and subtracting fractions

⚖️ Comparing fractions

• Same denominator: simply compare the numerators; the larger numerator means the larger fraction.
• Different denominators: rewrite both fractions with a common denominator, then compare the numerators.
• Example: To compare 1/2 cup of sugar versus 1/3 cup, find a common denominator and see which numerator is larger.

➕➖ Adding and subtracting with the same denominator

To add or subtract two fractions with the same denominator, add or subtract the numerators and keep the common denominator.

• The denominator stays the same because you're working with the same-sized parts.
• Example: Jack ate 1/8 of a pizza and Mack ate 3/8; together they ate (1+3)/8 = 4/8 of the pizza.

➕➖ Adding and subtracting with different denominators

• First, rewrite both fractions with a common denominator.
• Then add or subtract the numerators and keep the common denominator.
• Example: Adding a 3/8-inch thick plywood sheet to a 1/4-inch sheet requires converting 1/4 to 2/8, then adding to get 5/8 inches total.

🔢 Fractions and decimals

🔢 Converting fractions to decimals

To write a fraction as a decimal, divide the numerator by the denominator.

• Terminating decimal: the division ends (remainder eventually becomes 0).
  • Fun fact from the excerpt: if the denominator has no prime factors other than 2's and 5's, the decimal will terminate.
  • Such fractions can be built up to have a denominator of 10, 100, 1000, etc.
• Repeating decimal: the division continues in a pattern of digits.
  • Represent repeating digits with an overbar or ellipsis (three dots).
• Example: Some fractions produce decimals like 0.333... (repeating), while others produce decimals like 0.75 (terminating).

🔀 Improper fractions and mixed numbers

🔀 What improper fractions and mixed numbers are

An improper fraction has a larger numerator than denominator.

A mixed number has a whole number followed by a fractional part.

• Because an improper fraction is larger than 1, it can be written as a mixed number.
• Important: A mixed number represents addition, not multiplication.
  • Example: The notation means "whole number plus fraction," not "whole number times fraction."

🔀 Converting improper fractions to mixed numbers

1. Divide the numerator by the denominator to get the whole number part.
2. The remainder after dividing becomes the new numerator.
3. Keep the same denominator.

• Example: An improper fraction like 11/4 can be rewritten by dividing 11 by 4, giving 2 with a remainder of 3, so the mixed number is 2 and 3/4.

🔀 Converting mixed numbers to improper fractions

1. Multiply the whole number part by the denominator.
2. Add this result to the original numerator to get the new numerator.
3. Keep the same denominator.

• Example: To convert a mixed number like 2 and 3/4, multiply 2 by 4 (= 8), add 3 (= 11), and keep the denominator 4, giving 11/4.

🔀 Arithmetic with mixed numbers

• Adding or subtracting: can be simple or complicated depending on the numbers.
  • One approach: work with the fractional parts separately from the whole numbers, then combine.
  • Example from the excerpt: rearrange and add whole parts and fractional parts separately, then combine the results.
• Multiplying or dividing: more complicated than it may appear.
  • Change any mixed numbers into improper fractions before doing the calculation.
  • Then change the answer back to a mixed number if possible.
  • Don't confuse: you cannot simply multiply or divide the whole and fractional parts separately; convert to improper fractions first.

🧭 Overview

🧠 One-sentence thesis

The excerpt covers fundamental fraction operations—finding fractions of numbers, dividing by fractions, comparing, adding, subtracting, converting between fractions and decimals, and working with improper fractions and mixed numbers—all essential for workplace and technical math tasks.

📌 Key points (3–5)

• Finding a fraction of a number: multiply the fraction by the number.
• Dividing by a fraction: multiply by the reciprocal (flip the second fraction upside-down).
• Adding/subtracting fractions: use a common denominator, then add or subtract numerators.
• Common confusion: mixed numbers represent addition (e.g., "two and one-half" means 2 + 1/2), not multiplication.
• Converting between forms: improper fractions (numerator larger than denominator) can be rewritten as mixed numbers, and vice versa; fractions become decimals by dividing numerator by denominator.

🔢 Basic fraction operations

🔢 Finding a fraction of a number

To find a fraction of a number, multiply.

• This means: "fraction × number."
• Example: To find three-quarters of 28 questions, multiply 3/4 by 28.
• Don't confuse: "of" in fraction problems signals multiplication, not addition.

➗ Dividing by a fraction

To divide by a fraction, multiply by the reciprocal of the second number. (Flip the second fraction upside-down.)

• The reciprocal means swapping numerator and denominator of the second fraction.
• Example: If you need a certain number of cups but only have a smaller scoop, dividing the total by the scoop size (a fraction) tells you how many scoops you need.
• Why it works: the excerpt does not explain the underlying reason, only the procedure.

🔍 Comparing fractions

• Same denominator: simply compare the numerators (larger numerator = larger fraction).
• Different denominators: rewrite both fractions with a common denominator, then compare numerators.
• Example: Recipe A requires one amount of sugar, Recipe B another; find a common denominator to see which is more.

➕➖ Adding and subtracting fractions

➕ Same denominator

To add or subtract two fractions with the same denominator, add or subtract the numerators and keep the common denominator.

• Example: Jack ate a fraction of pizza, Mack ate another fraction with the same denominator; add the numerators to find the total fraction eaten.
• Example: Tracy ate a fraction, Stacy ate a fraction; subtract numerators to find the difference.

➕ Different denominators

To add or subtract two fractions with different denominators, first write them with a common denominator. Then add or subtract them.

• Example: Two sheets of plywood with different fractional thicknesses (measured in inches); find a common denominator to add for combined thickness or subtract for the difference.
• Example: Jacqueline budgets fractions of her income for food and rent (different denominators); convert to common denominator to find combined budget or the difference between rent and food.

🔄 Fractions and decimals

🔄 Converting fractions to decimals

To write a fraction as a decimal, divide the numerator by the denominator.

• The result is either a terminating decimal (ends, remainder eventually zero) or a repeating decimal (continues a pattern).
• Terminating decimals: If the denominator has only 2's and 5's as prime factors, the decimal will terminate. The fraction can be built up to a denominator of 10, 100, 1000, etc.
• Repeating decimals: Use an overbar or ellipsis (three dots) to show the repeating digits.
• Don't confuse: not all fractions produce neat, short decimals; some repeat indefinitely.

🔀 Improper fractions and mixed numbers

🔀 Definitions and key distinction

A fraction which has a larger numerator than denominator is called an improper fraction.

Because an improper fraction is larger than 1, it can also be written as a mixed number, with a whole number followed by a fractional part.

• Critical point: A mixed number represents addition, not multiplication.
  • Example: "two and one-half" means 2 + 1/2, not 2 × 1/2.
• This is a common confusion that the excerpt explicitly warns against.

🔀 Converting improper fraction to mixed number

Procedure:

1. Divide the numerator by the denominator to get the whole number part.
2. The remainder after dividing is the new numerator.
3. Keep the same denominator.

• Example: An improper fraction can be rewritten by performing division and expressing the leftover as a fraction.

🔀 Converting mixed number to improper fraction

Procedure:

1. Multiply the whole number part by the denominator.
2. Add this result to the original numerator to get the new numerator.
3. Keep the same denominator.

• Example: A mixed number is converted by combining the whole and fractional parts into a single fraction.

🧮 Operations with mixed numbers

➕ Adding or subtracting mixed numbers

• The excerpt notes this "can be fairly simple or more complicated, depending on the numbers."
• One approach: work with fractional parts separately from whole numbers.
  • Example: Rewrite the mixed numbers, rearrange to group whole numbers and fractions, then add or subtract each part.
• The excerpt provides a worked example showing regrouping and simplification.

✖️➗ Multiplying or dividing mixed numbers

Multiplying or dividing mixed numbers is more complicated than it may appear. Change any mixed numbers into improper fractions before doing the calculation, then change the answer back to a mixed number if possible.

• Why this matters: Mixed numbers are not straightforward to multiply or divide directly.
• Procedure: Convert to improper fractions first, perform the operation, then convert the result back.
• Example: Cups of water divided equally into jars—convert mixed numbers to improper fractions, divide, then express the answer as a mixed number if appropriate.
• Don't confuse: Do not try to multiply or divide the whole and fractional parts separately; always convert to improper fractions first.

🧭 Overview

🧠 One-sentence thesis

To find a fraction of a number, you multiply the fraction by that number, and when dividing by a fraction, you multiply by its reciprocal instead.

📌 Key points (3–5)

• Multiplying fractions: to find a fraction of a number, multiply the fraction by the number.
• Dividing fractions: flip the second fraction upside-down (the reciprocal) and multiply instead of dividing.
• Comparing fractions: same denominator → compare numerators directly; different denominators → rewrite with a common denominator first.
• Adding/subtracting fractions: same denominator → add/subtract numerators and keep the denominator; different denominators → find a common denominator first.
• Common confusion: mixed numbers look like multiplication but actually represent addition (e.g., "two and one-half" means 2 + 1/2, not 2 × 1/2).

🔢 Multiplying and dividing fractions

✖️ Multiplying fractions

To find a fraction of a number, multiply.

• The operation is straightforward: multiply the fraction by the number.
• Example: To find 3/4 of 28 questions, multiply 3/4 by 28.
• This is the fundamental rule for "fraction of" problems in real contexts like workplace training pass rates.

➗ Dividing by a fraction

To divide by a fraction, multiply by the reciprocal of the second number. (Flip the second fraction upside-down.)

• Instead of dividing, you flip the second fraction and multiply.
• The reciprocal means swapping the numerator and denominator of the second fraction.
• Example: If you need 2/3 cups of flour but only have a 1/4 cup scoop, divide 2/3 by 1/4 by flipping 1/4 to 4/1 and multiplying.
• Don't confuse: only the second fraction gets flipped, not the first one.

🔍 Comparing fractions

🔍 Same denominator

If two fractions have the same denominator, we can simply compare their numerators.

• When the bottom numbers match, just look at the top numbers.
• Larger numerator → larger fraction.
• Example: Comparing 2/5 and 3/5—since both have denominator 5, 3/5 is larger because 3 > 2.

🔍 Different denominators

If two fractions have different denominators, we can rewrite them with a common denominator and then compare their numerators.

• First, convert both fractions so they share the same denominator.
• Then compare the new numerators.
• Example: Recipe A requires 2/3 cup of sugar and Recipe B requires 3/4 cup—rewrite both with a common denominator (e.g., 12) to see which is larger.

➕➖ Adding and subtracting fractions

➕ Same denominator

To add or subtract two fractions with the same denominator, add or subtract the numerators and keep the common denominator.

• The denominator stays the same; only the numerators change.
• Example: Jack ate 1/8 of a pizza and Mack ate 3/8—together they ate (1 + 3)/8 = 4/8 of the pizza.
• Example: Tracy ate 5/8 and Stacy ate 2/8—Tracy ate (5 − 2)/8 = 3/8 more.

➕ Different denominators

To add or subtract two fractions with different denominators, first write them with a common denominator. Then add or subtract them.

• You must rewrite both fractions with a shared denominator before performing the operation.
• Example: Combining a 3/8-inch thick plywood sheet with a 1/4-inch sheet requires converting 1/4 to 2/8, then adding 3/8 + 2/8.
• Example: Jacqueline budgets 1/4 of her income for food and 1/3 for rent—find a common denominator to add these fractions or to find the difference.

🔄 Fractions and decimals

🔄 Converting fractions to decimals

To write a fraction as a decimal, divide the numerator by the denominator.

• Perform the division: top number ÷ bottom number.
• The result is the decimal form of the fraction.

🔚 Terminating decimals

A decimal that ends (eventually has a remainder of 0) is called a terminating decimal.

• The division eventually stops with no remainder.
• Fun fact from the excerpt: If the denominator has no prime factors other than 2's and 5's, the decimal will terminate.
• These fractions can be rewritten with denominators like 10, 100, 1000, etc.

🔁 Repeating decimals

A decimal that continues a pattern of digits is called a repeating decimal.

• The pattern of digits repeats indefinitely.
• Notation: use an overbar over the repeating digits or ellipsis (three dots) to show the pattern continues.
• Example: Some fractions produce decimals like 0.333... or 0.666...

🔀 Improper fractions and mixed numbers

🔀 Definitions

A fraction which has a larger numerator than denominator is called an improper fraction.

An improper fraction is larger than 1, so it can also be written as a mixed number, with a whole number followed by a fractional part.

• Improper fraction: top number > bottom number.
• Mixed number: a whole number plus a fraction.
• Critical distinction: A mixed number represents addition, not multiplication.
  • Example: "two and one-half" means 2 + 1/2, not 2 × 1/2.

🔀 Improper fraction → mixed number

Steps:

1. Divide the numerator by the denominator to get the whole number part.
2. The remainder after dividing is the new numerator.
3. Keep the same denominator.

• Example: Convert an improper fraction by dividing and using the remainder as the new numerator.

🔀 Mixed number → improper fraction

Steps:

1. Multiply the whole number part by the denominator.
2. Add this result to the original numerator to get the new numerator.
3. Keep the same denominator.

• Example: Convert a mixed number by multiplying the whole part by the denominator, adding the original numerator, and keeping the denominator.

🔀 Operations with mixed numbers

Adding/subtracting:

• One approach: work with the fractional parts separately from the whole numbers.
• Example: Rewrite the problem to separate whole numbers and fractions, then rearrange and combine.
• May require additional work to simplify the result.

Multiplying/dividing:

Change any mixed numbers into improper fractions before doing the calculation, then change the answer back to a mixed number if possible.

• Don't confuse: multiplying or dividing mixed numbers is more complicated than it appears.
• Always convert to improper fractions first, perform the operation, then convert back if needed.

🧭 Overview

🧠 One-sentence thesis

To divide by a fraction, you multiply by the reciprocal of the second fraction, and this principle extends to working with improper fractions and mixed numbers by converting them first.

📌 Key points (3–5)

• Core division rule: flip the second fraction upside-down (reciprocal) and multiply instead of dividing.
• Comparing fractions: same denominator → compare numerators directly; different denominators → find a common denominator first.
• Adding/subtracting: requires a common denominator; keep the denominator and add/subtract only the numerators.
• Common confusion: mixed numbers look simple but must be converted to improper fractions before multiplying or dividing (not before adding/subtracting).
• Fraction-decimal conversion: divide numerator by denominator; terminates if the denominator has only 2s and 5s as prime factors, otherwise repeats.

➗ Dividing by a fraction

➗ The reciprocal method

To divide by a fraction, multiply by the reciprocal of the second number. (Flip the second fraction upside-down.)

• What it means: instead of dividing by a fraction, you multiply by its flipped version.
• Why flip the second fraction: the reciprocal of a fraction is obtained by swapping its numerator and denominator.
• Example: if you need to measure a certain number of cups but only have a smaller scoop, dividing the total by the scoop size tells you how many scoops you need.

✅ Simplifying answers

• The excerpt reminds you to express each answer in simplest form.
• After multiplying by the reciprocal, reduce the resulting fraction if possible.

🔍 Comparing and ordering fractions

🔍 Same denominator

• If two fractions share the same denominator, simply compare their numerators.
• The fraction with the larger numerator is the larger fraction.

🔍 Different denominators

• Rewrite both fractions with a common denominator.
• Then compare the new numerators.
• Example: to decide which banana bread recipe requires more sugar when one uses one fraction of a cup and the other uses a different fraction, find a common denominator and compare.

➕➖ Adding and subtracting fractions

➕ Same denominator

To add or subtract two fractions with the same denominator, add or subtract the numerators and keep the common denominator.

• The denominator stays unchanged.
• Only the numerators are added or subtracted.
• Example: if Jack ate a fraction of a pizza and Mack ate another fraction with the same denominator, add the numerators to find the total fraction eaten.

➕ Different denominators

• First, rewrite both fractions with a common denominator.
• Then add or subtract the numerators and keep the common denominator.
• Example: combining the thickness of two plywood sheets with different fractional thicknesses requires a common denominator before adding.

🧮 Real-world scenarios

• The excerpt includes budgeting examples: finding the combined fraction of income for food and rent, or the difference between rent and food fractions.
• Both require a common denominator before performing the operation.

🔢 Fractions and decimals

🔢 Converting fractions to decimals

To write a fraction as a decimal, divide the numerator by the denominator.

• Perform the division: numerator ÷ denominator.
• The result is the decimal form of the fraction.

🔁 Terminating decimals

A decimal that ends (eventually has a remainder of zero) is called a terminating decimal.

• Key fact: if the denominator has no prime factors other than 2s and 5s, the decimal will terminate.
• The fraction can be rewritten with a denominator of 10, 100, 1000, etc.
• Example: fractions with denominators like 2, 4, 5, 8, 10, 20, 25, 50 will terminate.

🔁 Repeating decimals

A decimal that continues a pattern of digits is called a repeating decimal.

• Represent the repeating digits with an overbar or ellipsis (three dots).
• If the denominator has prime factors other than 2 and 5, the decimal will repeat.

🔄 Improper fractions and mixed numbers

🔄 Definitions

A fraction which has a larger numerator than denominator is called an improper fraction.

An improper fraction can also be written as a mixed number, with a whole number followed by a fractional part.

• Important: a mixed number represents addition, not multiplication.
• Example: a mixed number means the whole number plus the fractional part, not the whole number times the fractional part.

🔄 Improper to mixed

Steps to convert an improper fraction to a mixed number:

1. Divide the numerator by the denominator to get the whole number part.
2. The remainder after dividing is the new numerator.
3. Keep the same denominator.

• Example: if the numerator is much larger than the denominator, the division gives you how many "wholes" fit, and the remainder becomes the fractional part.

🔄 Mixed to improper

Steps to convert a mixed number to an improper fraction:

1. Multiply the whole number part by the denominator.
2. Add this result to the original numerator to get the new numerator.
3. Keep the same denominator.

• This process "packs" the whole number into the numerator.

🧮 Operations with mixed numbers

➕➖ Adding or subtracting mixed numbers

• One approach: work with the fractional parts separately from the whole numbers.
• Rearrange the expression to group whole numbers together and fractions together.
• Add or subtract the whole numbers, then add or subtract the fractions (using a common denominator if needed).
• Combine the results.
• Example: the excerpt shows rearranging a subtraction problem to handle whole and fractional parts separately, then recombining.

✖️➗ Multiplying or dividing mixed numbers

• Key warning: multiplying or dividing mixed numbers is more complicated than it appears.
• Required step: change any mixed numbers into improper fractions before doing the calculation.
• After the calculation, change the answer back to a mixed number if possible.
• Don't confuse: this conversion is necessary for multiplication and division, but not always for addition and subtraction (where you can work with parts separately).

🧭 Overview

🧠 One-sentence thesis

Comparing fractions requires either matching denominators to compare numerators directly, or rewriting fractions with a common denominator when denominators differ.

📌 Key points (3–5)

• Same denominator case: when two fractions share the same denominator, simply compare the numerators to determine which fraction is larger.
• Different denominator case: when denominators differ, rewrite both fractions with a common denominator first, then compare the numerators.
• Common confusion: you cannot directly compare numerators when denominators are different—the denominators must be made the same first.
• Why it matters: comparing fractions is essential for practical decisions, such as choosing between recipes or measurements that use different fractional amounts.

🔍 The two comparison scenarios

🔍 When denominators are the same

If two fractions have the same denominator, we can simply compare their numerators.

• The denominator tells you the size of each piece; if both fractions divide the whole into the same number of pieces, the fraction with more pieces (larger numerator) is bigger.
• No conversion is needed—just look at the top numbers.
• Example: comparing 3/8 and 5/8, since both have denominator 8, the fraction with numerator 5 is larger.

🔄 When denominators are different

If two fractions have different denominators, we can rewrite them with a common denominator and then compare their numerators.

• Different denominators mean the pieces are different sizes, so you cannot compare numerators directly.
• First step: find a common denominator (a shared multiple of both denominators).
• Second step: rewrite each fraction as an equivalent fraction with that common denominator.
• Third step: now that denominators match, compare the numerators.
• Example: to compare 2/3 and 3/5, rewrite both with a common denominator (such as 15), giving 10/15 and 9/15, then compare 10 vs 9 to see that 2/3 is larger.

🍌 Practical application

🍌 Recipe comparison example

The excerpt provides an exercise:

• Banana bread recipe A requires a certain fraction cup of sugar.
• Banana bread recipe B requires a different fraction cup of sugar.
• Question: which recipe requires more sugar?

How to solve:

• If the two fractions have the same denominator, compare numerators directly.
• If denominators differ, rewrite with a common denominator, then compare.
• The recipe with the larger numerator (after matching denominators) uses more sugar.

⚠️ Don't confuse

• Larger numerator does not always mean larger fraction: a fraction with a bigger numerator but different denominator may actually be smaller (e.g., 5/12 vs 1/2—rewriting gives 5/12 vs 6/12, so 1/2 is larger despite having a smaller numerator in its original form).
• Always check denominators first: only when denominators are the same can you directly compare numerators.

🧭 Overview

🧠 One-sentence thesis

Adding and subtracting fractions requires a common denominator, and the process differs depending on whether the fractions already share one or need to be rewritten first.

📌 Key points (3–5)

• Same denominator: add or subtract the numerators and keep the common denominator unchanged.
• Different denominators: first rewrite both fractions with a common denominator, then add or subtract.
• Common confusion: the denominator stays the same when adding/subtracting—only the numerators combine.
• Real-world use: combining or comparing parts (e.g., pizza slices eaten, plywood thickness, budget portions).

➕ Adding and subtracting with the same denominator

➕ The basic rule

To add or subtract two fractions with the same denominator, add or subtract the numerators and keep the common denominator.

• Only the top numbers (numerators) change.
• The bottom number (denominator) stays exactly the same.
• Example: Jack ate three-eighths of a pizza and Mack ate two-eighths; together they ate five-eighths (3 + 2 = 5, denominator stays 8).

🍕 Subtraction works the same way

• Subtract the numerators; keep the denominator.
• Example: Tracy ate five-eighths of a pizza and Stacy ate one-eighth; Tracy ate four-eighths more (5 − 1 = 4, denominator stays 8).

🔄 Adding and subtracting with different denominators

🔄 The two-step process

To add or subtract two fractions with different denominators, first write them with a common denominator. Then add or subtract them.

• You cannot directly combine fractions that have different denominators.
• Step 1: rewrite both fractions so they share a common denominator.
• Step 2: now that the denominators match, add or subtract the numerators as usual.

🪵 Practical examples

• Combining thicknesses: a three-quarter-inch plywood sheet laid on a five-eighths-inch sheet—find a common denominator, rewrite both fractions, then add the numerators to get the combined thickness.
• Budget allocation: Jacqueline budgets one-fifth of her income for food and one-third for rent. To find the combined fraction, rewrite both with a common denominator, then add. To find how much more she budgets for rent than food, rewrite and subtract.

⚠️ Don't confuse

• The denominator does not get added or subtracted—only the numerators do.
• Example: one-half plus one-half is two-halves (which simplifies to one whole), not one-fourth.

🔗 Related fraction operations

🔗 Finding a fraction of a number

• To find a fraction of a number, multiply.
• Example: Nathan must correctly answer a certain fraction of 28 questions to pass—multiply the fraction by 28.

🔗 Dividing fractions

To divide by a fraction, multiply by the reciprocal of the second number. (Flip the second fraction upside-down.)

• This is a different operation from adding/subtracting, but it's often needed in the same contexts.
• Example: measuring three-fourths cups of flour with a one-eighth cup scoop—divide three-fourths by one-eighth (flip the second fraction and multiply).

🔗 Comparing fractions

• Same denominator: simply compare the numerators.
• Different denominators: rewrite with a common denominator, then compare numerators.
• Example: recipe A requires two-thirds cup of sugar, recipe B requires three-fourths cup—rewrite both with a common denominator to see which is larger.

🧭 Overview

🧠 One-sentence thesis

Fractions can be converted to decimals by dividing the numerator by the denominator, and the result is either a terminating decimal (when the denominator has only 2s and 5s as prime factors) or a repeating decimal (when it continues a pattern of digits).

📌 Key points (3–5)

• How to convert: divide the numerator by the denominator to write a fraction as a decimal.
• Terminating decimals: end eventually (remainder of zero); occur when the denominator has no prime factors other than 2s and 5s.
• Repeating decimals: continue a pattern of digits indefinitely; represented by an overbar or ellipsis (three dots).
• Common confusion: terminating vs repeating—check the prime factors of the denominator to predict which type you'll get.
• Why it matters: understanding decimal forms helps with calculations and recognizing patterns in fractions.

🔢 Converting fractions to decimals

🔢 The basic method

To write a fraction as a decimal, divide the numerator by the denominator.

• This is the fundamental conversion rule.
• The numerator is the "top" number; the denominator is the "bottom" number.
• Example: to convert a fraction, perform the division operation.

✅ Terminating decimals

A decimal that ends (eventually has a remainder of zero) is called a terminating decimal.

• When it happens: if the denominator of a fraction has no prime factors other than 2s and 5s, the decimal will terminate.
• Why: fractions with such denominators can be built up to have a denominator of 10, or 100, or 1000, etc.
• Example: a fraction whose denominator is 4 (which is 2×2) or 25 (which is 5×5) will produce a terminating decimal.

🔁 Repeating decimals

A decimal that continues a pattern of digits is called a repeating decimal.

• How to represent: use either an overbar (a line over the repeating digits) or ellipsis (three dots) to show the pattern continues.
• When it happens: when the denominator has prime factors other than 2 and 5.
• Don't confuse: a repeating decimal doesn't "end" like a terminating decimal—it goes on forever in a predictable pattern.

🔍 Identifying decimal types

🔍 Prime factors rule

Denominator prime factors	Decimal type	Reason
Only 2s and 5s	Terminating	Can be rewritten with denominator 10, 100, 1000, etc.
Other primes (3, 7, 11, etc.)	Repeating	Cannot be rewritten as a power of 10

• This rule helps predict the decimal form before performing the division.
• Example: a denominator of 8 (which is 2×2×2) will terminate; a denominator of 3 will repeat.

🧮 Performing the division

• Divide the numerator by the denominator using long division or a calculator.
• Watch for the remainder: if it becomes zero, the decimal terminates; if the remainder cycles back to a previous value, the decimal repeats.
• Example: when dividing, if you see the same remainder appear again, you know the pattern will repeat from that point.

🧭 Overview

🧠 One-sentence thesis

Improper fractions (where the numerator exceeds the denominator) can be converted to mixed numbers (a whole number plus a fraction) and vice versa, and these conversions are essential for performing arithmetic operations correctly.

📌 Key points (3–5)

• What an improper fraction is: a fraction with a numerator larger than its denominator, representing a value greater than 1.
• What a mixed number is: a whole number followed by a fractional part, representing the same value in a different form.
• Common confusion: a mixed number represents addition (e.g., "2 and 3/4"), not multiplication.
• How to convert: improper fractions → mixed numbers uses division; mixed numbers → improper fractions uses multiplication and addition.
• Why conversion matters: multiplying or dividing mixed numbers requires converting to improper fractions first to avoid errors.

🔄 Understanding the two forms

🔢 Improper fractions

Improper fraction: a fraction which has a larger numerator than denominator.

• Because the numerator is larger, the value is greater than 1.
• Example: if the numerator is bigger than the denominator, you have "more than one whole."
• This form is useful for calculations but less intuitive for reading.

🧮 Mixed numbers

Mixed number: a whole number followed by a fractional part.

• Represents the same value as an improper fraction, just written differently.
• Example: instead of writing a large numerator over a denominator, you separate out the whole units.
• Don't confuse: a mixed number means addition, not multiplication—the whole number and fraction are added together, not multiplied.

🔁 Converting between forms

➡️ Improper fraction to mixed number

Steps:

1. Divide the numerator by the denominator to get the whole number part.
2. The remainder after dividing becomes the new numerator.
3. Keep the same denominator.

• Why it works: division separates out how many "wholes" fit into the numerator, and the remainder is what's left over as a fraction.
• Example: if you divide and get a quotient with a remainder, the quotient is your whole number and the remainder over the original denominator is your fractional part.

⬅️ Mixed number to improper fraction

Steps:

1. Multiply the whole number part by the denominator.
2. Add this result to the original numerator to get the new numerator.
3. Keep the same denominator.

• Why it works: you're converting the whole number into fractional units (by multiplying by the denominator), then adding the existing fractional part.
• Example: if you have whole units and a fraction, multiply the wholes by the denominator to see how many fractional pieces they represent, then add the extra pieces.

➕➖ Arithmetic with mixed numbers

➕ Adding or subtracting mixed numbers

• Simpler approach: work with the fractional parts separately from the whole numbers.
• Example: rearrange the expression so you add whole numbers together and fractions together separately, then combine.
• The excerpt notes this "can be fairly simple or more complicated, depending on the numbers."

✖️➗ Multiplying or dividing mixed numbers

• Critical rule: change any mixed numbers into improper fractions before doing the calculation.
• After calculating, change the answer back to a mixed number if possible.
• Why this matters: multiplying or dividing mixed numbers directly is "more complicated than it may appear"—converting first avoids errors.
• Example: if you need to multiply mixed numbers, first convert each to an improper fraction, multiply the improper fractions, then convert the result back to a mixed number.

🧭 Overview

🧠 One-sentence thesis

Measurements always contain some error, so we distinguish between exact values (from counting or definitions) and approximations (from measurements or rounding), and we use significant figures to communicate how accurate a measurement is.

📌 Key points (3–5)

• Exact vs approximation: exact values come from counting or definitions; approximations come from measurements or rounding.
• Why measurements are inexact: even careful measurements with sensitive instruments contain small errors.
• What significant figures show: the digits in a measurement that we trust to be correct; accuracy equals the number of significant figures.
• Common confusion: zeros can be significant or not depending on their position—zeros that only show place value (where the decimal point belongs) are NOT significant.
• The overbar notation: marks trailing zeros in whole numbers as significant when the measurement is truly that precise.

🔢 Exact values vs approximations

🔢 What makes a number exact

A number is an exact value if it is the result of counting or a definition.

• Counting produces exact values because you can count discrete objects without error.
• Definitions are exact by their nature (e.g., "an inch is 1/12 of a foot" is a definition, not a measurement).
• Example: "There are 25 students in class" is exact (counting); "A right angle measures 90°" is exact (definition).

📏 What makes a number an approximation

A number is an approximation if it is the result of a measurement or of rounding.

• All physical measurements contain some error, no matter how careful or sensitive the instruments.
• Rounding also creates approximations because precision is lost.
• Example: "This board is 8 inches long" is an approximation (measurement); "A car's tachometer reads 3000 rpm" is an approximation (measurement).
• Don't confuse: even if a measurement looks exact (like "8 inches"), it's still an approximation because actual measurements might be closer to 7.99 or 8.01 inches.

🎯 Understanding significant figures

🎯 What significant figures mean

Significant figures: the digits in a measurement that we trust to be correct.

• The accuracy of a number equals the number of significant figures.
• More significant figures = more precise measurement.
• The terms "significant digits" and "significant figures" are used interchangeably.

🔍 Five rules for counting significant figures

Rule	What counts	Example
1. Nonzero digits	Always significant	12345 has five sig figs
2. Zeros between nonzeros	Always significant	10203 has five sig figs
3. Zeros right of decimal	Always significant	1.20 has three sig figs; 1.2000 has five sig figs
4. Zeros left of decimal	NOT significant	0.123 has three sig figs
5. Zeros right of whole number	NOT significant (unless marked with overbar)	1200 has two sig figs; 12̄0̄0̄ has four sig figs

🚫 The place-value principle

• Another way to understand rules 4 and 5: zeros that merely show where the decimal point belongs are NOT significant.
• These zeros don't tell us about measurement precision; they only indicate magnitude.
• Example: In 0.00123, the three leading zeros just position the decimal—only 1, 2, and 3 are significant.

📊 The overbar notation

📊 Why the overbar exists

• Historical problem: Mount Everest was measured at exactly 29,000 feet in 1856, but scientists worried people would think this was just rounded to the nearest thousand.
• Solution at the time: announce it as 29,002 feet (an incorrect measurement) to show precision.
• Modern solution: use an overbar to mark trailing zeros as significant.

✍️ How to use the overbar

• Writing 29̄0̄0̄0̄ means "Really, to the nearest foot, it's exactly 29,000 feet!"
• The overbar indicates that those zeros are part of the measurement precision, not just place-holders.
• Example: 1200 has two sig figs (could be anywhere from 1150 to 1249); 12̄0̄0̄ has four sig figs (measured to the nearest unit).

🧭 Overview

🧠 One-sentence thesis

Significant figures communicate the precision of a measurement by indicating which digits are reliable, and calculations must preserve appropriate precision based on the input values.

📌 Key points (3–5)

• Exact vs approximate: some values are exact (counting, definitions), while measurements are approximations with limited precision.
• What significant figures show: the number of reliable digits in a measurement, indicating how precise the value is.
• Range of actual values: a measurement with a given number of significant figures represents a range of possible true values.
• Common confusion: more digits does not always mean more accuracy—significant figures depend on measurement precision, not just decimal places.
• Calculation rules: results should reflect the precision of the least precise input value.

🔢 Exact values vs approximations

🔢 Distinguishing exact from approximate values

The excerpt classifies values into two categories:

• Exact values: numbers that are precisely known, such as counts or defined quantities (no measurement uncertainty).
• Approximations: measured values that have inherent uncertainty and limited precision.

Example: Counting 12 objects gives an exact value; measuring a length as 12 meters is an approximation.

⚠️ Why the distinction matters

• Exact values have infinite precision for calculation purposes.
• Approximations carry uncertainty that must be tracked through calculations.
• Don't confuse: a number with many decimal places can still be an approximation if it came from measurement.

📏 Understanding significant figures

📏 What significant figures represent

Significant figures: the number of reliable digits in a measurement that indicate its precision.

• The count tells you how precisely something was measured.
• More significant figures = more precise measurement.
• Example from excerpt: "three significant figures," "four significant figures," "five significant figures."

📏 Counting significant figures

The excerpt provides examples with different counts:

• Two significant figures
• Three significant figures
• Four significant figures
• Five significant figures

The specific rules for counting (leading zeros, trailing zeros, etc.) are implied by the exercise structure but not explicitly detailed in this excerpt.

🎯 Precision and uncertainty ranges

🎯 What a measurement really means

Each measurement with a given number of significant figures represents a range of possible actual values.

The excerpt shows this pattern:

• "two significant figures; the actual value could be anywhere between [lower bound] and [upper bound]"
• "three significant figures; the actual value could be anywhere between [lower bound] and [upper bound]"
• "four significant figures; the actual value could be anywhere between [lower bound] and [upper bound]"
• "five significant figures; the actual value could be anywhere between [lower bound] and [upper bound]"

🎯 Interpreting the range

• The range narrows as the number of significant figures increases.
• This reflects greater measurement precision.
• Example: a value with two significant figures has a wider uncertainty range than one with five significant figures.

🧮 Calculations with significant figures

🧮 Preserving precision in results

The numbered answers (17–29) show calculated results that have been rounded to appropriate significant figures.

Key principle (implied by the exercise structure):

• When performing calculations, the result should not claim more precision than the input measurements justify.
• The final answer must be rounded to reflect the limiting precision of the data used.

🧮 Practical application

• Each calculation result in the excerpt is expressed with a specific number of significant figures.
• This ensures the answer honestly represents the precision available from the measurements.
• Don't confuse: your calculator may show many digits, but only the significant ones should be reported.

🧭 Overview

🧠 One-sentence thesis

Accuracy-based rounding focuses on the number of significant figures rather than place value, ensuring that calculated results reflect the precision of the least accurate input.

📌 Key points (3–5)

• What accuracy-based rounding is: rounding to a specified number of significant figures instead of to a particular place value (like tenths or hundredths).
• The three-step procedure: locate the rounding digit by counting sig figs from the left, check the test digit to its right, then round up (≥5) or keep (<5) and drop remaining digits.
• Special notation for trailing zeros: use an overbar on whole-number zeros or include trailing zeros in decimals to show they are significant after rounding.
• Common confusion: don't round the original numbers before calculating—perform the full calculation first, then round the final answer.
• Multiplication/division rule: the answer must be rounded to match the number of sig figs in the least accurate input, preventing false precision from round-off error.

🔢 What accuracy-based rounding means

🔢 Definition and contrast with place-value rounding

Accuracy-based rounding: considers the number of significant figures rather than the place value.

• Traditional rounding targets a place (e.g., "round to the nearest hundredth").
• Accuracy-based rounding targets a count of significant figures (e.g., "round to three sig figs").
• The rounding digit's position depends on how many sig figs you want, not on a fixed decimal place.

🎯 Why it matters

• Significant figures reflect measurement precision.
• Rounding to sig figs ensures the result honestly represents the accuracy of the data.
• Example: announcing Mount Everest's height as an exact round number (29,000 ft) might look like a rough estimate, so it was reported as 29,002 ft to signal precision—though that introduced a small error to communicate accuracy.

🛠️ The rounding procedure

🛠️ Three-step method

1. Locate the rounding digit: count from the left until you reach the desired number of significant figures.
2. Check the test digit: look at the digit immediately to the right of the rounding digit.
3. Round and drop:
   • If the test digit is 5 or greater → increase the rounding digit by 1 and drop all digits to its right.
   • If the test digit is less than 5 → keep the rounding digit the same and drop all digits to its right.

📝 Example walkthrough

• Round a number to three sig figs:
  • Count three sig figs from the left to find the rounding digit.
  • Look at the next digit (the test digit).
  • Apply the rule: round up if test ≥5, keep if test <5.
  • Drop everything after the rounding digit.

✏️ Handling trailing zeros after rounding

✏️ Whole numbers: the overbar notation

• When a whole number's rounding digit is a 0 that gets rounded up to a higher digit, or when trailing zeros are significant, use an overbar above the zero.
• Example: if rounding produces exactly 29,000 ft and the zeros are significant, write it with an overbar to show "really, to the nearest foot, it's exactly 29,000 feet!"
• Don't confuse: an overbar is not used over nonzero digits, and it's not needed for zeros to the right of a decimal point (those are already understood to be significant).

✏️ Decimal numbers: include trailing zeros

• When the rounding digit of a decimal is a 0 that gets rounded up, you must include that 0 in the decimal place.
• This shows the zero is significant and not just a placeholder.

🧮 Accuracy in multiplication and division

🧮 The least-accurate-input rule

When multiplying or dividing approximate numbers, the answer must be rounded to the same number of significant figures as the least accurate of the original numbers.

• The result can only be as precise as the least precise input.
• Example: squaring 8.33 (three sig figs) gives a long decimal, but the answer must be rounded to three sig figs to avoid false accuracy from round-off error.

🧮 Procedure: calculate first, round last

• Do not round off the original numbers before calculating.
• Perform the full multiplication or division.
• Then round the final answer to match the smallest sig-fig count among the inputs.
• Example: if one input has three sig figs and another has five, round the answer to three sig figs.

⚠️ Why false accuracy is a problem

• Including many decimal digits can give a misleading sense of precision.
• Round-off error accumulates if intermediate rounding is done.
• Only the final answer should be rounded, ensuring it reflects the true measurement accuracy.
• Example: estimating gasoline cost—if you multiply an approximate volume by a price, the answer should be rounded to reflect the least precise measurement (e.g., the estimated gallons).

🧭 Overview

🧠 One-sentence thesis

When multiplying or dividing approximate numbers, the answer must be rounded to match the number of significant figures in the least accurate original number to avoid false accuracy from round-off error.

📌 Key points (3–5)

• The false-accuracy problem: performing operations on rounded numbers can produce results that look precise but contain hidden round-off error.
• The sig-fig rule for multiplication/division: the final answer must be rounded to the same number of significant figures as the least accurate input number.
• Common confusion: do not round the original numbers before calculating—round only the final answer as the last step.
• Why it matters: this rule prevents reporting misleading precision and ensures the answer reflects the true accuracy of the input data.

🔍 The false-accuracy problem

🔍 What false accuracy means

• False accuracy occurs when a calculated result displays more precision than the input data actually supports.
• The excerpt illustrates this with squaring a mixed number:
  • Converting 2⅓ to decimal gives 2.333... (repeating).
  • If you round to 2.33 first and then square it, you get 5.4289, which looks very accurate.
  • But this precision is misleading because the original number (2⅓) has only three significant figures.

🧮 How round-off error accumulates

• When you round an intermediate value and then perform further calculations, the rounding error propagates.
• Example from the excerpt:
  • 2.33 squared = 5.4289
  • But (2⅓)² = 49/9 ≈ 5.444...
  • The "accurate" answer 5.4289 is actually less correct than 5.44 (rounded to three sig figs).
• The key insight: only when both the input (2⅓ rounded to three sig figs = 2.33) and the output (5.44) are rounded to three sig figs do the results align correctly.

📏 The multiplication and division rule

📏 The core rule

When multiplying or dividing approximate numbers, the answer must be rounded to the same number of significant figures as the least accurate of the original numbers.

• "Least accurate" means the input number with the fewest significant figures.
• This rule applies to both multiplication and division operations.

⚙️ How to apply the rule

1. Do not round the original numbers before performing the calculation.
2. Perform the full multiplication or division using all available digits (use a calculator).
3. Round the final answer to match the number of significant figures in the least accurate input.
4. This rounding step should be your last step.

🚫 What not to do

• Don't confuse this with rounding inputs first: if you round 2.333... to 2.33 and then square it, you introduce error early.
• The excerpt explicitly warns: "Don't round off the original numbers; do the necessary calculations first, then round the answer as your last step."

🧪 Practical application

🧪 Worked scenario from the excerpt

• Problem: You are filling a 5-gallon can of gasoline. The gasoline costs a certain amount per gallon, and you estimate you will buy a certain number of gallons. How much should you expect to spend?
• Method:
  • Multiply the price per gallon by the estimated gallons (use full precision in the calculation).
  • Identify which input has fewer significant figures.
  • Round the final cost to that number of significant figures.
• This ensures the answer does not claim more precision than your estimate supports.

🧪 General workflow

Step	Action	Reason
1	Identify sig figs in each input	Find the least accurate number
2	Perform the full calculation	Avoid early rounding error
3	Round the final answer	Match the least accurate input's sig figs

🧪 Why this matters

• Including too many decimal digits gives a false sense of accuracy (the excerpt references an xkcd comic strip illustrating this).
• Proper sig-fig rounding communicates the true reliability of your result.
• Example: if one input is known to only two sig figs, reporting an answer to six decimal places is misleading.

🧭 Overview

🧠 One-sentence thesis

When adding or subtracting measurements, the answer must be rounded to match the precision of the least precise original number, and every measurement carries a greatest possible error (GPE) equal to half its precision.

📌 Key points (3–5)

• Precision in addition/subtraction: the result must be rounded to the same precision as the least precise input number.
• Exact numbers are special: exact numbers (like counting "3 objects") have infinite significant figures and do not limit precision.
• Greatest Possible Error (GPE): the true value can differ from the measured value by up to half the precision unit.
• Common confusion: don't round the original numbers before calculating—do the math first, then round the final answer.
• How GPE relates to precision: GPE is always one-half the precision; it defines the range of true values that would round to the measured value.

🧮 Rules for adding and subtracting measurements

➕ Round to the least precise input

When adding or subtracting numbers with different levels of precision, the answer must be rounded to the same precision as the least precise of the original numbers.

• Why: the least precise number limits how precisely you can know the total.
• Process: do not round the original numbers; perform the calculation first, then round the answer as the last step.
• Example: if you add weights of lb (precise to ones), lb (precise to tenths), lb (precise to hundredths), and lb (precise to ones), the answer must be rounded to the nearest whole pound (the least precise level).

🔢 Exact numbers have infinite precision

• What counts as exact: counting discrete objects (e.g., "3 objects") or defined constants.
• How to treat them: assume infinitely many significant figures; they do not limit the precision of your answer.
• Example: if you measure one object at ounces (precise to hundredths) and want the weight of three such objects, multiplying by 3 is repeated addition—the answer is still precise to the hundredths place.
• The excerpt says: "When working with an exact number, we can assume that it has infinitely many significant figures. (Treat exact numbers like royalty; their accuracy is perfect and it would be an insult to even question it.)"

📏 Greatest Possible Measurement Error (GPE)

📏 What GPE means

Greatest Possible Measurement Error (GPE): the maximum amount by which the true value can differ from the measured value.

• Core idea: every measurement is rounded to some precision, so the true value lies within a range around the measured value.
• Formula: GPE is always one-half the precision unit.
• Example: if a scale rounds to the nearest pound (precision = 1 pound), GPE = 0.5 pounds; if it rounds to the nearest tenth of a pound (precision = 0.1 pound), GPE = 0.05 pounds.

🐕 How GPE defines a range

• Notation: a measured value can be written as measured value ± GPE.
• Example (from the excerpt): a dog weighs 23 pounds on a scale that rounds to the nearest pound.
  • The true weight could be anywhere from 22.5 pounds to almost 24 pounds.
  • Written as: 23 ± 0.5 pounds.
• Example: the same dog weighs 23.0 pounds on a scale that rounds to the nearest tenth.
  • The true weight could be anywhere from 22.95 pounds to almost 23.05 pounds.
  • Written as: 23.0 ± 0.05 pounds.
• Using inequalities: the range 23 ± 0.5 means 22.5 ≤ true weight < 23.5 (values that would round to 23).

🔍 GPE always has a "5" in the next place

• Pattern: the GPE is a "5" in the place value immediately to the right of the measurement's precision.
• Example: if precision is to the nearest tenth (0.1), GPE is five hundredths (0.05).
• Why: because GPE is half the precision unit, and half of 0.1 is 0.05.

🔄 Precision vs GPE relationship

🔄 As precision increases, GPE decreases

• Inverse relationship: more precise measurements have smaller GPE.
• Example (from the excerpt):
  • Scale rounds to nearest pound → GPE = 0.5 pounds.
  • Scale rounds to nearest tenth of pound → GPE = 0.05 pounds (ten times smaller).
• Implication: higher precision means you know the true value more narrowly.

📐 GPE defines what rounds to a given value

• Another way to think about GPE: it gives the range of values that would round off to the measured number.
• Example: 23 ± 0.5 means any value from 22.5 up to (but not including) 23.5 would round to 23.
• The excerpt notes: "Yes, 23.5 technically would round up to 24, but it is easier to just say [22.5 to 23.5] instead of [22.5 to 23.49999...]."

📊 Summary table

Concept	Definition	Key rule
Precision in addition/subtraction	The place value to which a number is rounded	Round the final answer to match the least precise input
Exact numbers	Counted quantities or defined constants	Have infinite significant figures; do not limit precision
GPE	Maximum error in a measurement	Always one-half the precision unit; a "5" in the next place value
GPE range	Interval of true values	Measured value ± GPE; defines what rounds to the measured value

🧭 Overview

🧠 One-sentence thesis

When combining approximate measurements, the final answer must be rounded to match the precision or significant figures of the least accurate original number to avoid false accuracy.

📌 Key points (3–5)

• False accuracy problem: Performing calculations on rounded numbers can produce answers that look precise but contain hidden round-off error.
• Multiplication/division rule: Round the final answer to match the fewest significant figures among the original numbers.
• Addition/subtraction rule: Round the final answer to match the least precise place value among the original numbers.
• Common confusion: Don't round the original numbers before calculating—do all calculations first, then round only the final answer as the last step.
• Exact numbers exception: Exact numbers (like counting "3 objects") have infinite significant figures and don't limit your answer's precision.

🔢 Understanding false accuracy

🎯 The round-off error trap

• The excerpt shows squaring a repeating decimal as an example.
• If you round an intermediate value too early and then calculate, the final answer may display many decimal places but be fundamentally inaccurate.
• Example: A number with only three significant figures, when squared, should yield an answer rounded to three significant figures—not more.

False accuracy: When an answer appears very precise (many digits) but contains round-off error because the original data was less accurate.

✅ The solution

• Perform all necessary calculations first using the full values.
• Round only at the final step, matching the accuracy of your least accurate input.

✖️➗ Rules for multiplication and division

📏 Significant figures govern the answer

When multiplying or dividing approximate numbers, the answer must be rounded to the same number of significant figures as the least accurate of the original numbers.

• "Least accurate" means the number with the fewest significant figures.
• The process: calculate fully, then count significant figures in each original number, then round your answer to match the smallest count.

🚫 Don't pre-round

• The excerpt emphasizes: "Don't round off the original numbers."
• Rounding inputs before calculating compounds error.
• Always round as your last step only.

➕➖ Rules for addition and subtraction

📍 Precision (place value) governs the answer

Precision: The place value of the rightmost significant figure in a number.

• Example from the excerpt: 80,000 is precise to the thousands place; a number like 82.5 would be precise to the tenths place.

When adding or subtracting numbers with different levels of precision, the answer must be rounded to the same precision as the least precise of the original numbers.

🎪 The attendance example

• If attendance is estimated at a certain number of people (precise to the nearest thousand), and a smaller number leave, you cannot suddenly report a more precise estimate.
• Example: Starting with an estimate precise to thousands, you must keep the final estimate precise to thousands—not suddenly switch to tens or ones.

Operation	Rule	What to match
Multiply/Divide	Significant figures	Fewest sig figs among inputs
Add/Subtract	Precision (place value)	Least precise place value among inputs

⚠️ Don't confuse the two rules

• Multiplication/division care about how many significant figures.
• Addition/subtraction care about which place value (precision).
• Both rules share the same workflow: calculate first, round last.

🔧 Precision-based rounding technique

📐 The three-step process

The excerpt defines precision-based rounding (rounding to a specified place value):

1. Locate the rounding digit in the target place value.
2. Look at the test digit directly to the right.
3. Apply the rule: If the test digit is 5 or greater, increase the rounding digit by 1 and drop all digits to its right; if less than 5, keep the rounding digit the same and drop all digits to its right.

🔟 Special notation rules

• Whole numbers: When a rounding digit of 9 rounds up to 10, write an overbar above the new 0 to show it is significant.
• Decimals: When a rounding digit of 9 rounds up to 10, include the trailing 0 in that decimal place to preserve precision.

👑 The exact number exception

♾️ Infinite significant figures

• The excerpt explains that exact numbers (from counting or definitions, not measurement) can be treated as having infinitely many significant figures.
• Example: If you measure one object at a certain weight and want the weight of exactly 3 objects, multiplying by 3 is repeated addition—the answer's precision is still governed by the measured weight, not by the number 3.
• The excerpt says: "Treat exact numbers like royalty; their accuracy is perfect and it would be an insult to even question it."

🔁 Multiplication as repeated addition

• When multiplying a measurement by an exact count, the operation is equivalent to adding that measurement multiple times.
• The precision of the answer matches the precision of the original measurement, not limited by the exact multiplier.

📏 Greatest Possible Measurement Error (GPE)

🐕 The weighing example

• The excerpt introduces GPE with a scenario: a scale rounds to the nearest pound and displays a certain weight.
• The true weight could be as much as half a pound above or below the displayed value.
• This concept shows the inherent uncertainty in any rounded measurement—the actual value lies within a range determined by the precision of the measurement tool.

🧭 Overview

🧠 One-sentence thesis

When adding or subtracting measurements with different precision levels, the answer must be rounded to match the least precise original number to avoid falsely implying greater accuracy than the data supports.

📌 Key points (3–5)

• Core rule: The result of addition or subtraction must be rounded to the same precision as the least precise number in the calculation.
• Why precision matters: You cannot start with an imprecise measurement and end with a more precise result—doing so falsely suggests accuracy you don't have.
• Calculation order: Never round the original numbers first; perform the full calculation, then round the final answer as the last step.
• Common confusion: Don't confuse precision (place value of the rightmost significant figure) with accuracy or the number of significant figures overall.
• Exact numbers exception: Exact counting numbers (like "3 objects") have infinite precision and don't limit the precision of your answer.

📏 Understanding precision

📏 What precision means

Precision: The place value of the rightmost significant figure in a number.

• Precision tells you the smallest unit being measured or reported.
• Examples from the excerpt:
  • A number precise to the thousands place
  • A number precise to the ones place
  • A number precise to the tenths place
• This is about where the measurement stops, not how many digits it has overall.

🔍 Don't confuse precision with accuracy

• Precision = the place value of the rightmost digit (thousands, ones, tenths, etc.)
• The excerpt distinguishes this from accuracy and significant figures, which are related but different concepts.

➕ The addition and subtraction rule

➕ Why you can't gain precision

The excerpt gives a clear scenario:

• Example: Attendance at an event is estimated at a certain number of people (precise to the nearest thousand).
• If you see a specific number of people leave, you cannot report the new attendance to a more precise level.
• If the original estimate was precise only to the nearest thousand, subtracting an exact number doesn't magically make your answer precise to the ones place.

The principle: An imprecise starting point cannot produce a more precise result through addition or subtraction.

🎯 The rounding rule

When adding or subtracting numbers with different precision levels:

1. Identify the least precise number among all the original numbers.
2. Perform the full calculation without rounding any original numbers.
3. Round the final answer to match the precision of the least precise original number.

Step	What to do	What NOT to do
Before calculating	Keep all original numbers as-is	Don't round the original numbers
During calculation	Add or subtract normally	Don't round intermediate steps
After calculation	Round final answer to least precise place value	Don't report more precision than justified

📦 Practical examples from the excerpt

Scenario 1: Combined weight of packages

• The excerpt presents exercises finding combined weights of multiple packages with different precision levels.
• Each package weight has its own precision (ones place, tenths place, etc.).
• The answer must be rounded to match whichever package weight was least precise.

Scenario 2: Revised estimates

• An estimate of insurable possessions at one precision level, minus items sold at another precision level.
• The revised estimate must maintain the original precision level, not become artificially more precise.

Scenario 3: Checking account balance

• Starting with a rough estimate (certain precision), then subtracting purchases.
• The remaining balance should reflect the imprecision of the starting estimate.

🔢 Special case: Exact numbers

🔢 When precision rules don't apply

The excerpt explains an important exception:

• Exact numbers (like counting "3 objects") are treated as having infinitely many significant figures.
• When multiplying by an exact number, treat it as repeated addition.
• Example from excerpt: Measuring one object at a certain precision (hundredths place), then finding the weight of three such objects—the answer stays precise to the hundredths place.
• The excerpt says to "treat exact numbers like royalty; their accuracy is perfect and it would be an insult to even question it."

When is a number exact?

• Counting discrete items (not measuring)
• Defined conversions or relationships
• These don't limit the precision of your calculation result.

📐 Connection to measurement error

📐 Greatest Possible Measurement Error (GPE)

The excerpt introduces GPE to explain why precision matters:

Greatest Possible Measurement Error (GPE): One half the precision; it gives the range of true values that would round to the measured value.

• If a scale rounds to the nearest pound and shows a certain weight, the true weight could be half a pound above or below.
• If the scale rounds to the nearest tenth of a pound, the GPE shrinks to five hundredths of a pound.
• Key insight: More precision = smaller GPE = narrower range of possible true values.

📐 Why this matters for addition/subtraction

• When you add or subtract measurements, you're combining their uncertainties.
• Reporting an answer more precisely than your least precise input falsely suggests you've reduced the measurement error.
• The GPE concept reinforces why you must round to the least precise place value—that's where your greatest uncertainty lies.

🧭 Overview

🧠 One-sentence thesis

The greatest possible measurement error (GPE) quantifies the range within which the true value lies when a measurement is rounded to a given precision, and it always equals half the precision.

📌 Key points (3–5)

• What GPE measures: the maximum amount by which the true value can differ from the measured value, above or below.
• How GPE relates to precision: GPE is always one half of the precision; if precision is to the nearest tenth (0.1), GPE is 0.05.
• Why GPE matters: it defines the range of true values that would round to the displayed measurement.
• Common confusion: precision vs GPE—precision is the place value of the rightmost significant digit; GPE is half that unit and tells you the error range.
• How to express uncertainty: a measurement can be written as "measured value ± GPE" to show the possible range.

📏 What GPE represents

📏 The uncertainty range in a measurement

Greatest possible measurement error (GPE): the maximum amount the true value can be above or below the measured value.

• When a scale or instrument rounds to a certain precision, the true value lies somewhere in a range around the displayed number.
• The GPE tells you how far the true value could be from what you see.
• Example: a scale rounds to the nearest pound and shows 23 pounds. The true weight could be anywhere from 22.5 pounds to almost 24 pounds. The GPE is 0.5 pounds, so the true weight is 23 ± 0.5 pounds.

🔢 GPE as a range of values that round to the measurement

• The excerpt explains that GPE gives "the range of values that would round off to the number in question."
• The lower bound is (measured value − GPE); the upper bound is (measured value + GPE).
• Example: if a measurement is 23.0 pounds (precision to the nearest tenth), the GPE is 0.05 pounds. The range is 22.95 to 23.05 pounds—any value in this range would round to 23.0.
• Don't confuse: the upper limit is "almost" the next value (e.g., 23.5 would technically round up to 24, but for simplicity the range is written as 23 ± 0.5).

🔗 How GPE relates to precision

🔗 GPE is always half the precision

• The excerpt states: "The GPE is always one half the precision."
• Precision is the place value of the rightmost significant digit (e.g., nearest tenth = 0.1).
• GPE is half of that unit (e.g., if precision is 0.1, GPE is 0.05).

📍 GPE is a 5 in the next place value

• The excerpt notes: "The GPE will always be a 5 in the place to the right of the place value of the number's precision."
• Example: if precision is to the nearest pound (ones place), GPE is 0.5 (tenths place). If precision is to the nearest tenth, GPE is 0.05 (hundredths place).

⚖️ More precision means smaller GPE

• "As we increase the level of precision in our measurement, we decrease the greatest possible measurement error."
• Example: weighing a dog to the nearest pound gives GPE = 0.5 pounds. Weighing to the nearest tenth of a pound gives GPE = 0.05 pounds. The more precise scale has a smaller error range.

🧮 Worked examples from the excerpt

🐕 Dog weight rounded to the nearest pound

• Scale shows 23 pounds (precision: nearest pound).
• True weight could be anywhere from 22.5 pounds to almost 24 pounds.
• GPE = 0.5 pounds.
• Written as: 23 ± 0.5 pounds.

🐕 Dog weight rounded to the nearest tenth of a pound

• Scale shows 23.0 pounds (precision: nearest tenth).
• True weight could be anywhere from 22.95 pounds to almost 23.05 pounds.
• GPE = 0.05 pounds.
• Written as: 23.0 ± 0.05 pounds.

🏟️ Attendance estimate

• Attendance estimated at a certain number of people (the excerpt mentions exercises but does not give the full number in the provided text).
• The exercises ask: "What is the precision of this estimate?" and "What is the greatest possible error in this estimate?"
• Method: identify the place value of the rightmost significant digit (precision), then take half of that unit (GPE).

📏 Plastic sheeting thickness

• Thickness given as a certain number of inches (the excerpt mentions exercises but does not provide the exact value).
• The exercises ask for precision and GPE in both inches and mils.
• Method: same as above—find the place value of the last significant digit, then halve it.

📊 Summary of key terms

📊 Precision vs GPE

Term	Definition	Example
Precision	The place value of the rightmost significant digit	Nearest tenth = 0.1
GPE	One half the precision; the maximum error above or below the measured value	If precision is 0.1, GPE is 0.05

• Mnemonic: "precision" and "place value" both start with "p."
• When to use: precision is used for rounding when adding or subtracting; GPE is used to express the uncertainty range of a single measurement.

🔍 How to find GPE

1. Identify the precision (the place value of the rightmost significant digit).
2. Divide the precision by 2.
3. The result is the GPE.
4. Express the measurement as "value ± GPE" to show the range.

⚠️ Don't confuse precision with accuracy

• Accuracy: the number of significant figures (used for rounding when multiplying or dividing).
• Precision: the place value of the rightmost significant digit (used for rounding when adding or subtracting, and for finding GPE).
• The excerpt emphasizes: "It is easy to get them mixed up."

🧭 Overview

🧠 One-sentence thesis

Greatest Possible Error (GPE) is always half the precision of a measurement, and understanding the distinction between accuracy (significant figures) and precision (place value) is essential for correctly rounding calculations.

📌 Key points (3–5)

• GPE definition: GPE equals one half the precision and represents the maximum amount a true value can differ from a measured value.
• Precision vs accuracy: Precision refers to the place value of the rightmost significant digit, while accuracy refers to the number of significant figures—easy to confuse but "precision" and "place value" both start with "p."
• How GPE relates to rounding: GPE gives the range of values that would round to the measured number (lower bound to upper bound).
• When to use each: Use precision when adding/subtracting; use accuracy when multiplying/dividing.
• Common confusion: Don't mix up precision (place value) with accuracy (count of significant figures)—they serve different purposes in calculations.

📏 Understanding GPE and precision

📏 What GPE measures

Greatest possible measurement error (GPE): one half the precision

• GPE tells you how far the true value could be above or below the measured value.
• As precision increases (measuring to smaller units), GPE decreases.
• The GPE is always a 5 in the place to the right of the number's precision place value.

🐕 Scale example with Sir Barks-A-Lot

The excerpt uses weighing a dog to illustrate how precision affects GPE:

Scale precision	Measured weight	GPE	True weight range
Nearest pound	(value given)	±0.5 pounds	0.5 above or below
Nearest tenth	(value given)	±0.05 pounds	0.05 above or below

• More precise measurement → smaller GPE → narrower range of possible true values.
• Example: If a scale rounds to the nearest tenth and shows a certain weight, the true weight could be as much as 0.05 pounds above or below.

🎯 GPE as a rounding range

Another way to understand GPE: it defines which values would round to the measured number.

• The notation (measured value ± GPE) gives a lower and upper limit.
• Lower bound: the lowest value that would round up to the measured value.
• Upper bound: the highest value that would round down to the measured value.
• Can be expressed with inequalities showing the range of possible values.

🔤 Key terminology distinctions

🔤 Significant figures and accuracy

Significant figures: the digits in a number that we trust to be correct Accuracy: the number of significant figures

• Accuracy is about how many digits are trustworthy (a count).
• Used when multiplying or dividing—round the result based on accuracy.

📍 Precision definition

Precision: the place value of the rightmost significant digit

• Precision is about which place value the measurement reaches (ones, tenths, hundredths, etc.).
• Used when adding or subtracting—round the result based on precision.

💡 Memory aid

The excerpt provides a helpful tip: "precision" and "place value" both start with "p"—this helps remember that precision refers to place value, not the count of digits.

🧮 When to apply each concept

➕ Addition and subtraction

• Use precision to round the result.
• Look at the place value of the rightmost significant digit in your measurements.

✖️ Multiplication and division

• Use accuracy to round the result.
• Count the number of significant figures in your measurements.

🚗 Application example from exercises

The excerpt mentions a driving scenario:

• Distance rounded to nearest mile (precision consideration).
• Time estimate (precision consideration).
• Calculating average speed requires considering accuracy instead of precision because you're dividing.
• This illustrates why understanding the distinction matters for real calculations.

🧭 Overview

🧠 One-sentence thesis

Formulas take one or more input variables and produce an output through substitution and calculation, enabling practical applications from postage costs to temperature conversion.

📌 Key points (3–5)

• What a formula is: an equation that takes number(s) as input and produces an output (a number or a decision like yes/no).
• Variables: unknown numbers in formulas, usually represented by letters, whose values can vary.
• How to evaluate: substitute the input number(s) into the formula and perform calculations using order of operations.
• Single vs multiple inputs: some formulas need just one variable, others require two or more; pay attention to which number goes with which variable.
• Common confusion: when a number is written directly next to a variable (e.g., 2x), it means multiplication, not a separate term.

📐 What formulas are and how they work

📐 Definition and structure

A formula is an equation or set of calculations that takes a number (or numbers) as input, and produces an output.

• The output is often a number, but could also be a decision such as yes or no.
• Formulas contain variables—unknown numbers that can vary, usually represented by letters of the alphabet.
• Example: A taxi fare formula might use the variable for miles traveled to calculate total cost.

🔢 How to evaluate a formula

• Substitution: replace each variable with its actual number value.
• Calculation: perform the steps using the order of operations.
• Don't confuse: a number written directly next to a variable indicates multiplication (e.g., "2x" means "2 times x").

🔧 Single-input formulas

🔧 Basic structure

• Many formulas have just one input variable.
• The excerpt provides examples like postage cost formulas (based on weight in ounces) and taxi fare formulas (based on miles).
• Example: A formula for mailing cost takes the weight of an envelope as the single input and outputs the cost in dollars.

🌡️ Temperature conversions

• Temperature conversion formulas are a common single-input type.
• They convert between Fahrenheit and Celsius scales.
• Key reference points mentioned:
  • Freezing point of water: 32°F = 0°C
  • Boiling point of water: 212°F = 100°C
• Example: Converting a record high temperature from Celsius to Fahrenheit requires substituting the Celsius value into the conversion formula.

🏛️ Representation formulas

• Some formulas estimate quantities based on population.
• The excerpt mentions formulas for:
  • Number of U.S. House Representatives (based on state population in millions)
  • Number of electoral votes (based on state population in millions)
• Example: Given a state's population, substitute it into the formula to approximate how many representatives that state has.

🔀 Multiple-input formulas

🔀 Working with several variables

• Some formulas require more than one input variable.
• The key is paying attention to which number goes with which variable.
• Example: A blood pressure formula uses both systolic blood pressure (SBP) and diastolic blood pressure (DBP) as inputs to calculate mean arterial pressure (MAP).

📦 Dimensional formulas

• Package measurement formulas may use length, width, and height as three separate inputs.
• The excerpt mentions a shipping company formula that combines these dimensions to determine a package's "measurement."
• Don't confuse: each dimension is a distinct variable that must be substituted correctly.

✅ Decision-output formulas

✅ Yes/no answers

Some formulas give a yes or no answer: success or failure, approved or disapproved, etc.

• After calculating the result, compare it to a given number to see whether the result is within a specified range.
• Example: An egg classification formula determines whether an egg's mass qualifies it as "large" by checking if the mass falls within a specific range.
• The process involves:
  1. Substitute the input value into the formula
  2. Calculate the result
  3. Compare the result to the threshold or range
  4. Determine yes (qualifies) or no (does not qualify)

🧭 Overview

🧠 One-sentence thesis

Temperature conversion formulas allow us to translate measurements between the Fahrenheit and Celsius scales by using the freezing and boiling points of water as reference anchors.

📌 Key points (3–5)

• Two temperature scales: Fahrenheit (°F) and Celsius (°C) measure temperature differently; water freezes at 32°F/0°C and boils at 212°F/100°C.
• Conversion formulas exist: specific formulas translate a temperature from one scale to the other.
• Real-world applications: converting temperatures for weather events, medical standards, appliance settings, and international recipes.
• Common confusion: the two scales do not share the same zero point or degree size, so conversion is not a simple addition or subtraction.

🌡️ The two temperature scales

🌡️ Fahrenheit and Celsius reference points

The excerpt establishes two key reference temperatures for water:

Scale	Freezing point	Boiling point
Fahrenheit	32°F	212°F
Celsius	0°C	100°C

• These anchor points show that the scales use different starting values and different degree intervals.
• Don't confuse: 0°C is not the same as 0°F; the zero points are at different absolute temperatures.

🔄 Why conversion is needed

• Different countries and contexts use different scales (e.g., Sweden uses Celsius, the U.S. uses Fahrenheit).
• Medical standards, appliance instructions, and weather reports may report in either scale.
• Example: A frozen-food package from IKEA (Sweden) lists oven temperature in Fahrenheit but was originally calculated in Celsius, requiring conversion.

🧮 Using the conversion formulas

🧮 Temperature formulas

The excerpt states "The formulas shown below allow us to convert between the two temperature scales" and refers to "Temperature Formulas."

• The excerpt does not print the actual formulas in the provided text, but it describes their purpose: translating a temperature value from Fahrenheit to Celsius or vice versa.
• The formulas take one temperature value as input and produce the equivalent in the other scale.

📋 Practical conversion examples

The excerpt provides several real-world conversion scenarios:

• Weather: The record high in Portland, Oregon during the June 2021 "heat dome" was 46°C; the exercise asks to convert this to Fahrenheit.
• Human body temperature: The traditional "normal" body temperature is 98.6°F; the exercise asks for the Celsius equivalent.
• Appliance settings: The FDA recommends freezer temperatures below a certain Celsius value; the exercise asks for the Fahrenheit equivalent.
• International recipes: A package of frozen pancakes from IKEA lists an oven temperature in Fahrenheit that was originally calculated in Celsius; the exercise asks for the original Celsius value.

⚠️ Key distinction

• Converting between scales is not the same as measuring temperature; it is translating the same physical temperature into a different numbering system.
• Example: 46°C and its Fahrenheit equivalent describe the same heat level, just expressed in different units.

🧭 Overview

🧠 One-sentence thesis

Formulas with multiple input variables require careful attention to which number corresponds to each variable, and some formulas produce yes/no answers by comparing the calculated result to a specified threshold.

📌 Key points (3–5)

• Multiple inputs: Some formulas require more than one input variable; you must match each number to the correct variable.
• Notation reminder: A number written directly next to a variable indicates multiplication (e.g., 50x means 50 times x).
• Yes/no formulas: Some formulas produce a result that must be compared to a given number or range to determine approval, qualification, or category.
• Common confusion: Don't mix up which input goes with which variable—order and labels matter (e.g., length vs. width vs. height).
• Real applications: Multi-input formulas appear in blood pressure calculations, package measurements, and classification criteria.

🔢 Working with multiple input variables

🔢 What multiple inputs mean

A formula with more than one input variable: a formula that requires two or more separate values to be substituted into different variables.

• Each variable represents a distinct quantity.
• You must identify which number corresponds to which variable before substituting.
• Example: A formula for mean arterial pressure uses both systolic blood pressure (SBP) and diastolic blood pressure (DBP) as separate inputs.

🎯 How to substitute correctly

• Pay attention to which number goes in for each variable.
• The excerpt emphasizes: "Just pay attention to which number goes in for each variable."
• Example: If a package measurement formula uses length L, width W, and height H, you must match the given dimensions to the correct variable (e.g., length = 12 inches goes into L, not W or H).

🧮 Multiplication notation reminder

• When a number is written directly next to a variable, it indicates multiplication.
• Example: 50x means 50 times x, not "fifty-x" as a single symbol.
• This notation appears throughout formulas and is essential for correct calculation.

📋 Real-world examples of multi-input formulas

🩺 Blood pressure calculation

• Mean arterial pressure (MAP) is estimated using a formula with two inputs: systolic blood pressure (SBP) and diastolic blood pressure (DBP).
• Units are mm Hg (millimeters of mercury).
• Example: Given SBP and DBP values, substitute each into the correct position in the MAP formula to calculate the result.

📦 Package measurement

• UPS uses a formula to determine the "measurement" of a package using three inputs: length, width, and height.
• All three dimensions must be identified and substituted correctly.
• Example: A package with length 10 inches, width 8 inches, and height 6 inches requires substituting each dimension into the corresponding variable in the formula.

✅ Formulas with yes/no answers

✅ What yes/no formulas do

A yes/no formula: a formula that gives a result indicating success or failure, approval or disapproval, or qualification for a category.

• After calculating the result, you compare it to a given number or range.
• The comparison determines whether the result meets the specified criteria.
• Example: A formula determines whether a chicken egg qualifies as "large" based on its mass.

🔍 How to use yes/no formulas

• Step 1: Calculate the result using the formula and the given input(s).
• Step 2: Compare the result to the specified threshold or range.
• Step 3: Determine whether the result satisfies the condition (e.g., is within the range, is above/below a threshold).
• Example: In Australia, a chicken egg is designated "large" if its mass m in grams satisfies a specific formula (the result must fall within a certain range); test each egg's mass by substituting it into the formula and checking whether the result meets the "large" criteria.

🚫 Don't confuse calculation with conclusion

• The formula produces a number, but the answer to the question is yes or no.
• You must perform the comparison step; the calculated number alone is not the final answer.
• Example: An egg with mass 60 grams might produce a result of 1.2 from the formula, but you still need to check whether 1.2 satisfies the "large" condition (e.g., is it ≥ 1 and ≤ 1.5?).

🧭 Overview

🧠 One-sentence thesis

Some formulas produce a result that must be compared to a given threshold or range to determine whether a condition is met (yes/no, pass/fail, approved/disapproved).

📌 Key points (3–5)

• What these formulas do: they calculate a value, then you compare that value to a specified number or range to get a yes-or-no answer.
• Two-step process: first calculate the result from the formula, then check whether it falls within the specified range.
• Common confusion: the formula itself does not output "yes" or "no"—you must perform the comparison separately.
• Real-world use: determining whether something qualifies (e.g., meets a standard, passes a threshold, is approved).

🔍 How yes/no formulas work

🔍 The two-step process

1. Calculate the result using the formula and the given input values.
2. Compare the result to a specified number or range to determine success/failure, approval/disapproval, etc.

Formulas with a yes or no answer: formulas that give a yes or no answer—success or failure, approved or disapproved, etc.—by comparing the calculated result to a given number to see whether the result is within a specified range.

• The formula produces a numeric output, not a direct "yes" or "no."
• You must then check whether that output meets the condition.

⚠️ Don't confuse: formula output vs. the answer

• The formula calculates a number.
• The yes/no answer comes from comparing that number to a threshold.
• Example: A formula might calculate a mass; you then check if that mass is greater than or equal to a minimum value to decide "yes, it qualifies" or "no, it does not."

🥚 Example: egg size classification

🥚 The egg mass formula

The excerpt provides an Australian egg-sizing example:

• A chicken egg is designated "large" if its mass m (in grams) satisfies a certain formula (the exact formula is not fully shown in the excerpt, but the process is described).
• You calculate the result using the egg's mass, then determine whether the egg qualifies as large.

🥚 Step-by-step application

Step	What you do	Example
1. Measure	Find the egg's mass in grams	Egg 1's mass is some number of grams
2. Calculate	Plug the mass into the formula	Use the given formula
3. Compare	Check if the result meets the "large" threshold	Does the result satisfy the condition?
4. Conclude	Answer yes (qualifies) or no (does not qualify)	"This egg is large" or "This egg is not large"

🥚 Why this matters

• The formula does not tell you "large" or "not large" directly.
• You must interpret the numeric result against the standard.
• This pattern applies to many real-world scenarios: pass/fail tests, approval thresholds, quality standards, etc.

🧮 General approach to yes/no formulas

🧮 When you encounter one

• Identify the formula and the input variable(s).
• Plug in the given values and calculate.
• Identify the threshold or range (e.g., "greater than X," "between Y and Z").
• Compare your result to that threshold.
• State the yes/no conclusion clearly.

🧮 Common contexts

• Standards and classifications: Does an item meet a size, weight, or quality standard?
• Approval criteria: Does a measurement fall within an acceptable range?
• Pass/fail tests: Does a calculated score exceed a passing threshold?

Example: An organization sets a minimum score of 70 for approval. A formula calculates a candidate's score as 68. Compare 68 to 70 → the candidate does not meet the threshold (no/fail).

🧭 Overview

🧠 One-sentence thesis

Perimeter is the total distance around the outside edge of a polygon, calculated by adding all side lengths or using formulas for specific shapes, while circumference measures the distance around a circle using the radius or diameter.

📌 Key points (3–5)

• What perimeter measures: the total distance around the outside of a polygon (closed figure with straight sides).
• How to calculate: add up all side lengths, or use formulas for regular shapes (squares, rectangles, regular polygons).
• Circumference vs. perimeter: circumference is the specific term for the distance around a circle; perimeter applies to polygons.
• Common confusion: perimeter uses linear units (feet, inches, meters), not square units—it's one-dimensional distance, not area.
• Key formulas: square perimeter = 4s; rectangle perimeter = 2l + 2w or 2b + 2h; regular polygon = n × s; circle circumference = 2πr or πd.

📐 Polygon perimeter basics

📐 What is a polygon

Polygon: a closed geometric figure with straight sides.

• Common examples: triangles, squares, rectangles, parallelograms, trapezoids, pentagons, hexagons, octagons.
• The perimeter is the total distance around the outside, like a fence running along the edge of a region.
• General method: add up the lengths of all sides.
• Always include units in your answer (feet, inches, centimeters, etc.).

📏 General perimeter calculation

• For any polygon: measure each side and sum them.
• Example: a triangle with sides 3 ft, 4 ft, and 5 ft has perimeter 3 + 4 + 5 = 12 ft.
• If some sides are equal, formulas can simplify the calculation.

🔲 Perimeter formulas for common shapes

🔲 Square perimeter

Square: P = 4s, where s is the length of one side.

• All four sides are equal in a square.
• Multiply the side length by 4.
• Example: a square with side 7 inches has perimeter 4 × 7 = 28 inches.

🔳 Rectangle perimeter

Rectangle: P = 2l + 2w or P = 2b + 2h, where l/b is length/base and w/h is width/height.

• Opposite sides are equal in a rectangle.
• Add length and width, then multiply by 2.
• The terms length/width and base/height are interchangeable.
• Example: a rectangle 10 ft long and 6 ft wide has perimeter 2(10) + 2(6) = 32 ft.

⬡ Regular polygon perimeter

Regular polygon: a polygon where all sides have equal length.

• Formula: P = n × s, where n is the number of sides and s is the side length.
• Multiply the side length by the number of sides.
• Example: a regular hexagon (6 sides) with each side 5 cm has perimeter 6 × 5 = 30 cm.
• Example: a regular octagon (8 sides) with each side 3 inches has perimeter 8 × 3 = 24 inches.

⭕ Circumference of circles

⭕ Circle vocabulary

Circumference: the distance around the outside of a circle (not called perimeter).

Radius: the distance from the center to the edge of the circle.

Diameter: the distance from one edge to another, through the center; equals twice the radius (d = 2r).

• Every point on a circle is the same distance from its center.
• The diameter is twice the length of the radius.

🥧 The constant π (pi)

• Historically, civilizations discovered that circumference is slightly more than 3 times the diameter.
• Babylonians (by 2000 BCE) used 25/8 ≈ 3.125; Egyptians used 256/81 ≈ 3.16.
• The value 22/7 ≈ 3.14286 is a better approximation.
• The actual value is the irrational number π ≈ 3.14159…
• Use your calculator's π key for the most accurate result; if you must approximate, use 3.14159, not 22/7 (which introduces round-off error).

🔄 Circumference formulas

Circumference: C = 2πr or C = πd, where r is radius and d is diameter.

• Use 2πr when you know the radius.
• Use πd when you know the diameter.
• Remember to round your answer to the appropriate level of accuracy (pay attention to significant figures from Module 5).
• Example: a circle with radius 5 cm has circumference 2π(5) = 10π ≈ 31.4 cm.
• Example: a circle with diameter 12 inches has circumference π(12) = 12π ≈ 37.7 inches.

⚠️ Don't confuse

• Perimeter applies to polygons (straight-sided figures); circumference applies to circles.
• Both measure distance around the outside and use linear units (not square units).
• The formulas are different: polygons use addition or multiplication of side lengths; circles use π with radius or diameter.

🧭 Overview

🧠 One-sentence thesis

This excerpt is an answer key for mathematics exercises covering operations, negative numbers, decimals, fractions, accuracy, precision, formulas, perimeter and circumference, percents, ratios, rates, proportions, and scientific notation, but contains no substantive instructional content.

📌 Key points (3–5)

• The excerpt consists entirely of numerical answers to exercises from various mathematics modules.
• Module 8 specifically addresses "Perimeter and Circumference" with answers in units of feet, centimeters, inches, and meters.
• No definitions, explanations, formulas, or conceptual content are provided in this answer key.
• The excerpt includes a brief reference to a wiki link about degree (angle) history and mentions "Carmella Creeper parties like it's 1995" as a side note.

📋 Content structure

📋 What this excerpt contains

The source material is an answer key labeled "Exercise Answers" from a textbook called "Tech Math, 2e" by Morgan Chase. It lists numerical solutions organized by module number and exercise number, spanning topics from basic arithmetic operations through scientific notation.

🔢 Module 8 reference

• The section titled "Module 8: Perimeter and Circumference" appears on page 257.
• It contains 12 numbered answers with measurements in various units (ft, cm, in, m).
• No formulas, definitions, or worked examples are shown—only final numerical results.

⚠️ Limitations of this excerpt

⚠️ No instructional content

This answer key does not explain:

• What circumference is or how to calculate it
• The relationship between perimeter and circumference
• Any formulas (such as C = 2πr or C = πd)
• Methods for solving the problems
• Conceptual understanding of the topics

The excerpt cannot serve as study material for learning about circumference or any other mathematical concept, as it provides only solutions without context or explanation.

🧭 Overview

🧠 One-sentence thesis

Percents represent fractions with a denominator of 100 and can be converted to and from fractions and decimals to solve problems involving parts of a whole.

📌 Key points (3–5)

• What percent means: "per one hundred"—a fraction with denominator 100.
• Three forms: percents, fractions, and decimals are interchangeable representations of the same value.
• Conversion direction matters: moving decimal points left (percent → decimal) vs. right (decimal → percent).
• Common confusion: percents can be greater than 100% (fraction > 1) or less than 1% (fraction < 1/100).
• Practical use: finding the "amount" by multiplying the "base" (original number) by the rate (percent as decimal).

📐 What percent means

📐 Core definition

Percent: "per one hundred"—think of it as a fraction with a denominator of 100.

• The word itself breaks down to "per" (for each) + "cent" (hundred).
• Example: if 70 out of 100 squares are shaded, that's 70 percent.
• This is why percent problems always relate to comparing a part to a whole.

🔢 Special cases to remember

• Greater than 100%: the fraction will be greater than 1 (the amount exceeds the base).
• Less than 1%: the fraction will be less than 1/100 (a very small portion).
• Don't assume percents are always between 0 and 100.

🔄 Converting between forms

🔄 Percent to fraction

Steps:

1. Drop the percent sign.
2. Write the number over 100.
3. Simplify if possible.

Example: 75% → 75/100 → 3/4 (simplified).

🔄 Percent to decimal

Steps:

1. Drop the percent sign.
2. Move the decimal point two places to the left.

• Why two places? Because dividing by 100 shifts the decimal left twice.
• Example: 45% → 0.45; 3.5% → 0.035.
• If the percent is not a whole number, the decimal extends beyond the hundredths place.

🔄 Decimal to percent

Steps:

1. Move the decimal point two places to the right.
2. Insert a percent sign.

• This is the reverse of percent → decimal.
• Example: 0.68 → 68%; 1.25 → 125%.

🔄 Fraction to percent

Method 1 (general):

1. Divide numerator by denominator to get a decimal.
2. Move decimal point two places right and add percent sign.

Method 2 (when denominator is a factor of 100):

• Build up the fraction to have denominator 100.
• Example: 3/5 → 60/100 → 60%.

🧮 Solving percent problems

🧮 Key terminology

Term	Meaning
Amount	The answer after finding the percent; the "part"
Base	The original number; the "whole" you're finding a percent of
Rate	The percent itself

🧮 Finding the amount

Formula structure: Amount = Base × Rate (as decimal)

Critical step: Change the percent to a decimal before multiplying.

Example scenario: What is 15% of 80?

• Base = 80 (the whole)
• Rate = 15% = 0.15 (convert to decimal)
• Amount = 80 × 0.15 = 12

🧮 Real-world application

Example: An electric drill costs a certain amount with sales tax applied.

• The retail price is the base.
• The tax rate (e.g., 6.5%) must be converted to decimal (0.065).
• Tax amount = retail price × 0.065.
• Total payment = retail price + tax amount.

Don't confuse: The amount is not the same as the total—the amount is just the part contributed by the percent, while the total includes the original base.

🧭 Overview

🧠 One-sentence thesis

To find the amount in a percent problem, convert the percent to a decimal and multiply it by the base (the original number).

📌 Key points (3–5)

• Core terminology: the amount is the answer after finding the percent; the base is the original number; the rate is the percent itself.
• The calculation method: change the percent to a decimal, then multiply by the base to get the amount.
• Common confusion: don't confuse amount (the part) with base (the whole)—the base is what you take the percent of.
• Real-world application: sales tax problems use this structure—find the tax amount by multiplying the price (base) by the tax rate (percent as decimal).

🔤 Key terminology

🔤 Amount, base, and rate

Amount: the answer we get after finding the percent of the original number (also called the part).

Base: the original number, the number we find the percent of (also called the whole).

Rate: the percent itself.

• The excerpt emphasizes that you can think of amount as "part" and base as "whole."
• The base is always the starting quantity; the amount is the result of applying the percent to that starting quantity.
• Example: "What is 25% of 80?" → 80 is the base, 25% is the rate, and the answer is the amount.

⚠️ Don't confuse amount and base

• The base is what you take the percent of; the amount is what you get.
• In word problems, look for phrases like "of [number]" to identify the base.
• The amount is typically what the question asks you to find.

🧮 The calculation method

🧮 Convert percent to decimal first

• Critical step: change the percent to a decimal before multiplying.
• The excerpt states: "Be sure to change the percent to a decimal before multiplying."
• To convert: drop the percent sign and move the decimal point two places to the left.
• Example: 7.5% becomes 0.075.

✖️ Multiply the decimal by the base

• Once the percent is a decimal, multiply it by the base to find the amount.
• Formula in words: amount equals rate (as decimal) times base.
• The excerpt gives several phrasings for the same operation:
  • "What is [percent] of [base]?"
  • "[Percent] of [base] is what number?"
  • "Find [percent] of [base]."
  • "[Percent] of [base] is equal to what amount?"
  • "What number is [percent] of [base]?"
  • "Calculate [percent] of [base]."

💰 Real-world application: sales tax

💰 Finding the tax amount

• The excerpt provides a scenario: buying an electric drill with a retail price in a city with sales tax.
• The retail price is the base.
• The sales tax rate (as a percent) is the rate.
• To find the tax amount: convert the tax percent to a decimal, then multiply by the retail price.
• Round to the nearest cent if necessary (since money typically uses two decimal places).

💰 Finding the total cost

• After finding the tax amount, add it to the original retail price to get the total you pay.
• This is a two-step process: first find the amount (tax), then add it to the base (retail price).

🧭 Overview

🧠 One-sentence thesis

Ratios and rates compare two quantities—ratios compare quantities with the same units while rates compare quantities with different units—and proportions set two ratios or rates equal to solve for missing values.

📌 Key points (3–5)

• Ratio vs rate: ratios compare quantities with the same units; rates compare quantities with different units and must include units.
• Unit rates: express a rate as a single number showing how much of the numerator corresponds to one unit of the denominator (e.g., miles per hour).
• Proportions: equations stating that two ratios or rates are equal; can be verified by cross-multiplying.
• Common confusion: ratios drop units after simplification, but rates must keep units; don't confuse "rate" with "unit rate"—unit rates specifically show the amount per one unit.
• Solving proportions: use cross products to find missing numbers in proportions.

📐 Understanding ratios and rates

📐 What is a ratio

Ratio: the quotient of two numbers or the quotient of two quantities with the same units.

• A ratio is formed by division (quotient means the result of division).
• Write the first quantity as the numerator, the second as the denominator.
• Simplify to lowest terms if possible.
• Example: the ratio of 30 minutes to 2 hours—both must be converted to the same unit first, then simplified.

🚗 What is a rate

Rate: the quotient of two quantities with different units.

• Unlike ratios, rates compare quantities with different units.
• You must include the units when writing a rate.
• Write the first quantity as the numerator, the second as the denominator.
• Simplify to lowest terms while keeping units.
• Example: a car travels 120 miles in 3 hours → the rate is 120 miles / 3 hours = 40 miles / 1 hour.

⚠️ Key distinction

Concept	Units	After simplification
Ratio	Same units	Units typically drop out
Rate	Different units	Must keep units

🎯 Unit rates and unit prices

🎯 What is a unit rate

Unit rate: a rate that expresses the quantity in the numerator that corresponds to one unit of the denominator.

• More practical than a reduced fraction for everyday use.
• Divide the numerator by the denominator.
• Express with "per": miles per hour, dollars per gallon, grams per deciliter, pounds per square inch.
• Example: 120 miles in 3 hours → 120 ÷ 3 = 40 miles per hour.

💰 What is a unit price

Unit price: a rate with the price in the numerator and a denominator equal to 1.

• Shows the cost of one unit or one item.
• To find: divide the cost by the size or number of items.
• Example: a 15.8-ounce box costs $5.29 → unit price = $5.29 ÷ 15.8 ounces.
• Useful for comparing which product size offers better value.

⚖️ Working with proportions

⚖️ What is a proportion

Proportion: an equation that says two ratios (or rates) are equal.

• Written as two fractions set equal to each other.
• Example format: 2/3 = 4/6.

✅ Checking if a proportion is true

Two methods are presented:

Method 1: Simplify each fraction

• Reduce both fractions to lowest terms.
• If they match, the proportion is true.

Method 2: Cross-multiplying (finding cross products)

• Multiply diagonally across the equal sign.
• In a true proportion, the cross products are equal.
• Example: for a/b = c/d, multiply a × d and b × c; if these products are equal, the proportion is true.

🔍 Solving for a missing number

When a proportion has a variable (a letter standing for a missing number):

1. Set the cross products equal to form an equation.
2. Isolate the variable by rewriting the multiplication equation as a division equation.
3. Check the solution by substituting the answer back into the original proportion and finding the cross products.

Example: if 3/5 = x/10, cross-multiply to get 3 × 10 = 5 × x, which gives 30 = 5x, so x = 6.

Don't confuse: cross-multiplying only works for proportions (two fractions set equal); you cannot cross-multiply a single fraction.

🧭 Overview

🧠 One-sentence thesis

Unit rates express a rate as a single number per one unit of the denominator, making comparisons and practical calculations more meaningful than reduced fractions.

📌 Key points (3–5)

• Ratio vs rate: ratios compare quantities with the same units; rates compare quantities with different units (and must include units).
• What a unit rate is: a rate expressed as a quantity per one unit of the denominator (e.g., miles per hour, dollars per gallon).
• Unit price: a specific type of unit rate showing cost per one unit or item, found by dividing cost by size or number.
• Common confusion: a reduced fraction rate vs a unit rate—reducing a rate to lowest terms is not the same as finding the unit rate; unit rates require dividing to get "per one unit."
• Why unit rates matter: they enable practical comparisons (e.g., comparing cereal prices per ounce) and express quantities in everyday terms.

📏 Ratios and rates fundamentals

📏 What a ratio is

A ratio is the quotient of two numbers or the quotient of two quantities with the same units.

• Both quantities must have the same units (e.g., minutes to hours, both are time).
• Written as a fraction: first quantity is the numerator, second is the denominator.
• Simplify to lowest terms if possible.
• Example: comparing 30 minutes to 2 hours—both are time measurements.

🔄 What a rate is

A rate is the quotient of two quantities with different units.

• The two quantities have different units (e.g., miles and hours).
• You must include the units when writing a rate.
• Written as a fraction: first quantity is numerator, second is denominator.
• Simplify if possible, but keep the units.
• Example: a car travels a certain number of miles in a certain number of hours—distance and time are different units.

🆚 How to distinguish ratio from rate

Concept	Units	Must include units?	Example
Ratio	Same units	No (units cancel)	Minutes to hours
Rate	Different units	Yes	Miles per hours

🎯 Unit rates explained

🎯 What makes a unit rate

A unit rate expresses the quantity in the numerator that corresponds to one unit of the denominator.

• The denominator becomes exactly 1.
• Not just a reduced fraction—you divide numerator by denominator to get a single number.
• Units are expressed with "per": miles per hour, dollars per gallon, grams per deciliter, pounds per square inch.
• Example: if a car travels 150 miles in 3 hours, the unit rate is 150 ÷ 3 = 50 miles per hour (per one hour).

🧮 How to find a unit rate

• Divide the numerator by the denominator.
• Express the result as a mixed number or decimal.
• Keep the units in "per" form.
• Don't confuse: reducing a fraction to lowest terms (e.g., 150/3 = 50/1) is not the same process as calculating the unit rate; you must perform the division to get a single number.

💰 Unit price as a special case

A unit price is a rate with the price in the numerator and a denominator equal to 1.

• Shows the cost of one unit or one item.
• To find: divide the cost by the size or number of items.
• Calculators may be used for unit price calculations.
• Example: a 15.8-ounce box costs $5.29; unit price = $5.29 ÷ 15.8 ounces = cost per ounce.

🛒 Practical applications

🛒 Comparing unit prices

• The excerpt provides cereal box examples to illustrate comparison.
• Different box sizes at the same or different prices require unit price calculation to determine the better value.
• Example scenario: Box A is 15.8 ounces for $5.29; Box B is 18.8 ounces for $5.29; calculate each unit price to see which offers more cereal per dollar.

📦 Sale scenarios

• When items are sold in bundles (e.g., 2 boxes for $9), divide the total cost by the total quantity to find unit price.
• Example: buying two 16-ounce boxes for $9 total means 32 ounces for $9; unit price = $9 ÷ 32 ounces.

🔗 Connection to proportions

🔗 How unit rates relate to proportions

• The excerpt transitions from unit rates to proportions, indicating they are related concepts.
• Proportions say that two ratios or rates are equal.
• Unit rates can be used within proportions to solve real-world problems involving rates.
• Example: if you know a car's unit rate (miles per hour), you can set up a proportion to find distance traveled in a different number of hours.

🧭 Overview

🧠 One-sentence thesis

Percent problems can be solved by translating word problems into equations or by setting up proportions where the rate is compared to 100, and these methods apply to finding unknown rates, bases, or amounts including percent increases.

📌 Key points (3–5)

• Three components of percent problems: amount (the result), base (the original number), and rate (the percent itself).
• Two solution methods: translate words into algebraic equations ("is" = equals, "of" = multiply) or set up proportions with the rate over 100.
• Finding different unknowns: the same framework works whether you're solving for the rate, the base, or the amount.
• Common confusion: when finding percent increase, always use the original (earlier-in-time, smaller) number as the base, not the new number.
• Practical applications: calculating grades, cash rewards, recommended intakes, and price changes all use these percent problem structures.

🔢 The three parts of a percent problem

🎯 Amount, base, and rate

Amount: the answer we get after finding the percent of the original number.

Base: the original number, the number we find the percent of.

Rate: the percent itself.

• These three quantities are related: the amount comes from applying the rate to the base.
• Any percent problem involves finding one of these three when the other two are known.
• Example: "15 is what percent of 60?" → amount = 15, base = 60, rate = unknown.

🔄 Which number is the base?

• The base is always the original number—the starting point or reference value.
• It corresponds to 100% in a proportion setup.
• Don't confuse: the base is not necessarily the larger number; it's the one you're taking a percent of.

🧮 Method 1: Translating into equations

📝 Word-to-algebra translation rules

The excerpt provides three translation rules:

• "is" means equals (=)
• "of" means multiply (×)
• "what" means a variable (unknown)

Example: "15 is what percent of 60?" translates to 15 = (what) × 60.

🔍 Finding the rate

• Question form: "X is what percent of Y?"
• Translation: X = (rate) × Y
• Solve for rate, then convert from decimal to percent by moving the decimal point.
• Example: If you earned some points on a quiz, "points earned is what percent of total points?" gives your grade percentage.

🔍 Finding the base

• Question form: "X is P% of what number?"
• Translation: X = P × (base)
• Solve for the base.
• Example: "If you want to earn $50 in cash rewards at a certain reward rate, how much do you need to charge?" translates to 50 = (rate) × (what amount).

📐 Method 2: Using proportions

⚖️ Setting up the proportion

• A percent is a ratio—a fraction out of 100.
• Set up: (rate / 100) = (amount / base)
• The base corresponds to 100% because it's the whole or original quantity.

🔄 Same problems, different approach

The excerpt shows that exercises 1–4 (solved by equations) can be re-solved as exercises 5–8 using proportions:

• Both methods reach the same answer.
• You are free to use whichever method you prefer.
• Proportions may feel more intuitive if you think of percents as "parts per hundred."

📈 Percent increase problems

📊 Two-step process for percent increase

1. Find the amount of increase: subtract the original number from the new number.
2. Calculate the rate: use the increase as the amount and the original number as the base; divide to find the unknown percent.

⚠️ Always use the original number as the base

• The base must be the number that occurred earlier in time.
• For percent increase, this is the smaller of the two numbers.
• Don't confuse: using the new (larger) number as the base will give the wrong percentage.
• Example: A candy bar price increase of 50 cents is a large percentage of the original low price, but the same 50-cent increase on a car price is negligible because the original car price (base) is much larger.

🧪 Why the base matters

• The excerpt emphasizes that the same absolute change (e.g., 50 cents) has very different percent meanings depending on the original amount.
• This illustrates why we care about percent change rather than just absolute change—it provides context and comparability.

🧭 Overview

🧠 One-sentence thesis

Scientific notation uses powers of ten to represent very large or very small numbers in a compact format, making them easier to compare and work with.

📌 Key points (3–5)

• What powers of ten are: decimal notation is based on powers of 10 (0.1 = 10⁻¹, 0.01 = 10⁻², etc.), and any power of 10 can be written as a 1 in a certain decimal place.
• What scientific notation is: a number between 1 and 10 multiplied by a power of 10, written in the form (number between 1 and 10) × 10^(integer).
• Why it matters: scientific notation makes it much easier to compare very large or very small numbers by comparing their powers of 10 instead of counting zeros.
• Order of magnitude: the difference in the powers of 10 between two numbers; a larger power means a larger order of magnitude.
• Common confusion: negative exponents represent small decimals (fractions), not negative numbers—10⁻¹ = 0.1, 10⁻² = 0.01, etc.

🔢 Powers of ten fundamentals

🔢 How decimal notation uses powers of ten

• The decimal system is built on powers of 10.
• Each decimal place represents a power of 10:
  • 0.1 = 10⁻¹
  • 0.01 = 10⁻²
  • 0.001 = 10⁻³
• Any power of 10 can be written as a 1 in a certain decimal place.

➖ Negative exponents

Negative exponents: 10⁻¹ = 1/10, 10⁻² = 1/100, 10⁻³ = 1/1000, etc.

• Negative exponents represent fractions or decimals, not negative values.
• The excerpt notes this is true for any base, not just 10, but the focus is on base 10.
• Example: 0.0007 = 7 × 10⁻⁴ (seven ten-thousandths).

➕ Positive exponents

• Large whole numbers can be expressed as multiples of powers of 10.
• Example: 50,000 = 5 × 10,000 = 5 × 10⁴.
• The power tells you how many places the decimal point moves to the right.

📐 Scientific notation structure

📐 Definition and format

A number is written in scientific notation if it is written in the form (number) × 10^(integer), where the integer is any whole number (positive, negative, or zero) and the number is any real number such that it is between 1 and 10.

• The first part must be between 1 and 10 (not including 10 itself).
• The second part is a power of 10.
• The "×" multiplication symbol is frequently used with scientific notation (even though it's generally avoided elsewhere).

🎯 Why the 1–10 range matters

• Keeping the first number between 1 and 10 creates a standard, compact format.
• This makes comparison straightforward: you compare the powers of 10 first, then the leading numbers if needed.
• Example: 5.97 × 10²⁴ vs. 6.39 × 10²³—the first is larger because 24 > 23.

🌍 Comparing large numbers with scientific notation

🌍 The problem with standard notation

• The excerpt gives Earth's mass as approximately 5,970,000,000,000,000,000,000,000 kg and Mars's mass as approximately 639,000,000,000,000,000,000,000 kg.
• With so many zeros, it is difficult to keep track and determine which is larger.

🌍 The solution with scientific notation

• Earth's mass: approximately 5.97 × 10²⁴ kg
• Mars's mass: approximately 6.39 × 10²³ kg
• It is much easier to compare the powers of 10 and determine that Earth's mass is larger because 24 > 23.

📏 Order of magnitude

Order of magnitude: the difference in the powers of 10 of two numbers.

• Earth's mass is one order of magnitude larger than Mars's mass because 24 is 1 more than 23.
• Order of magnitude gives a quick sense of scale without needing exact values.
• Example: A "six-figure" salary means the salary is between $100,000 (10⁵) and $999,999 (close to 10⁶), spanning one order of magnitude.

🔍 Converting between notations

🔍 Large numbers to scientific notation

• Identify the significant digits.
• Place the decimal after the first digit.
• Count how many places the decimal moved to the left; that count is the positive exponent.
• Example: 50,000 = 5.0 × 10⁴ (decimal moved 4 places left).

🔍 Small decimals to scientific notation

• Identify the significant digits.
• Place the decimal after the first non-zero digit.
• Count how many places the decimal moved to the right; that count is the negative exponent.
• Example: 0.0007 = 7 × 10⁻⁴ (decimal moved 4 places right).

🔍 Scientific notation to standard notation

• If the exponent is positive, move the decimal point that many places to the right (adding zeros as needed).
• If the exponent is negative, move the decimal point that many places to the left (adding zeros as needed).
• Example: 5.97 × 10²⁴ = 5,970,000,000,000,000,000,000,000.

🧭 Overview

🧠 One-sentence thesis

Scientific notation makes very large and very small numbers easier to compare and calculate by expressing them as a number between 1 and 10 multiplied by a power of 10.

📌 Key points (3–5)

• What scientific notation is: a compact format that writes numbers as a value between 1 and 10 times a power of 10.
• Why it helps: eliminates the need to count long strings of zeros when comparing or calculating with extremely large or small numbers.
• How to compare: numbers with larger powers of 10 are larger; the power tells you the order of magnitude.
• Common confusion: don't confuse the number of zeros with the power of 10—the power indicates which decimal place the significant digits begin.
• Operations shortcut: multiply by adding exponents; divide by subtracting exponents.

📐 What scientific notation is

📐 The standard form

A number is written in scientific notation if it is written in the form (a number between 1 and 10) × 10^n, where n is an integer and the first part is any real number such that it falls between 1 and 10.

• The format compresses very large or very small numbers into a manageable size.
• The "x" multiplication symbol is frequently used in scientific notation, even though it's generally avoided elsewhere.
• Example: 50,000 can be written as 5 × 10^4; 0.0007 can be written as 7 × 10^(-4).

🔢 What an integer is

• An integer is a number with no fraction or decimal part: …-3, -2, -1, 0, 1, 2, 3…
• The exponent n in scientific notation must be an integer.

🌍 Why scientific notation matters

🌍 Comparing very large numbers

• Without scientific notation: Earth's mass is approximately 5,970,000,000,000,000,000,000,000 kg; Mars's mass is approximately 639,000,000,000,000,000,000,000 kg—difficult to compare by counting zeros.
• With scientific notation: Earth ≈ 5.97 × 10^24 kg; Mars ≈ 6.39 × 10^23 kg.
• The larger power of 10 (24 vs. 23) immediately shows Earth's mass is larger.
• Example: A "six figures" salary means minimum $100,000 (1 × 10^5) and maximum $999,999 (approximately 1 × 10^6).

🔬 Comparing very small numbers

• Without scientific notation: hydrogen atom radius ≈ 0.000000000053 meters; chlorine atom radius ≈ 0.00000000018 meters—hard to tell which is larger.
• With scientific notation: hydrogen ≈ 5.3 × 10^(-11) meters; chlorine ≈ 1.8 × 10^(-10) meters.
• Chlorine's radius is larger because -10 is a larger power than -11 (the digits begin in the tenth decimal place vs. the eleventh).
• Don't confuse: a "more negative" exponent means a smaller number; -10 is larger than -11.

📏 Order of magnitude

Order of magnitude: the difference in the powers of 10 between two numbers.

• Earth's mass is one order of magnitude larger than Mars's because 24 is 1 more than 23.
• This concept simplifies rough comparisons without needing exact values.

🧮 Operations with scientific notation

✖️ Multiplying powers of 10

• Rule: To multiply powers of 10, add the exponents.
• This mirrors the shortcut for multiplying numbers with trailing zeros: multiply the significant digits, then count total zeros.
• Example: 300 × 2,000 = 3 × 2 = 6, with five total zeros → 600,000.
• In scientific notation: (a × 10^m) × (b × 10^n) = (a × b) × 10^(m+n).

➗ Dividing powers of 10

• Rule: To divide powers of 10, subtract the exponents.
• The excerpt notes this rule but suggests using a calculator for messy numbers, especially when parentheses are needed.
• In scientific notation: (a × 10^m) ÷ (b × 10^n) = (a ÷ b) × 10^(m−n).

🎯 Practical benefits

🎯 Avoiding errors

• Scientific notation prevents "going cross-eyed counting all those zeros."
• It allows calculations and comparisons without tracking long strings of digits.
• Useful for planetary masses, atomic radii, and any context with extreme scales.

🎯 Converting between forms

• Standard to scientific: identify the significant digits and count how many places the decimal moves.
• Scientific to standard: expand the power of 10 back into zeros.
• Example: 5 × 10^4 = 50,000; 7 × 10^(-4) = 0.0007.

🧭 Overview

🧠 One-sentence thesis

Scientific notation simplifies multiplication and division of very large or very small numbers by allowing us to work with powers of 10 through adding or subtracting exponents.

📌 Key points (3–5)

• Why scientific notation helps: it eliminates the need to count and track many zeros when comparing or calculating with extremely large or small numbers.
• Multiplying in scientific notation: multiply the significant digits and add the exponents of the powers of 10.
• Dividing in scientific notation: divide the significant digits and subtract the exponents of the powers of 10.
• Common confusion: engineering notation vs scientific notation—engineering notation uses only multiples of 3 as exponents and keeps the coefficient between 1 and 1000, while scientific notation uses any integer exponent and keeps the coefficient between 1 and 10.
• Practical applications: population density calculations, mass comparisons, and per-person debt calculations all become manageable with scientific notation.

🔢 Why scientific notation matters

🔬 Handling extreme numbers

• Very large numbers (like planetary masses) and very small numbers (like atomic radii) contain many zeros that are difficult to track.
• Scientific notation allows comparisons and calculations without "going cross-eyed counting all those zeros."
• Example: comparing hydrogen atom radius (approximately 0.000000000053 meters) to chlorine atom radius (approximately 0.00000000018 meters) is easier when rewritten in scientific notation because you can compare the powers of 10 directly.

🔍 Comparing magnitudes quickly

• The power of 10 tells you which decimal place the significant digits begin.
• A larger (less negative) power of 10 means a larger number.
• Example: chlorine's radius is larger than hydrogen's because chlorine's digits begin in the tenth decimal place while hydrogen's begin in the eleventh decimal place.

✖️ Multiplying with scientific notation

✖️ The multiplication shortcut

• The excerpt describes a familiar shortcut: when multiplying numbers with zeros on the end, multiply the significant digits and count the total zeros.
• Example: to multiply two numbers, multiply the non-zero parts and write the total number of zeros at the end.
• This shortcut extends naturally to scientific notation.

🔢 Adding exponents

To multiply powers of 10, add the exponents.

• When multiplying two numbers in scientific notation, multiply the coefficients (significant digits) and add the powers of 10.
• This rule comes from the properties of exponents.
• The result should be written in scientific notation.
• Don't confuse: you add exponents when multiplying, but subtract when dividing.

➗ Dividing with scientific notation

➗ Subtracting exponents

To divide powers of 10, subtract the exponents.

• When dividing two numbers in scientific notation, divide the coefficients and subtract the exponent of the denominator from the exponent of the numerator.
• The excerpt notes that when numbers get messy, using a calculator is advisable.
• If using a calculator, parentheses must be used carefully to ensure correct order of operations.

🌍 Real-world division examples

The excerpt provides three practical applications:

Application	Numerator	Denominator	Purpose
New Jersey population density	Population (people)	Land area (square miles)	Find people per square mile
California population density	Population (people)	Land area (square miles)	Find people per square mile
Proton vs electron mass	Proton mass (kg)	Electron mass (kg)	Determine how many times greater proton mass is

• These examples show how division with scientific notation handles both geographic data and subatomic particle measurements.

🔧 Engineering notation

🔧 What engineering notation is

Engineering notation: a variant of scientific notation that uses powers of 10 that are always multiples of 3.

• In engineering notation, the power of 10 must be a multiple of 3 (e.g., 3, 6, 9, -3, -6).
• The coefficient must be between 1 and 1000 (not between 1 and 10 as in scientific notation).
• This matches how large numbers are commonly reported: "thousand" (10³), "million" (10⁶), "billion" (10⁹), "trillion" (10¹²).

📰 Why engineering notation is used

• It aligns with everyday language and news reporting.
• Example: if approximately 37,000 people live in a city, we say "thirty-seven thousand" and might write "37 thousand"—this is engineering notation.
• It would be unusual to report it as "three point seven ten thousands" (which would be closer to standard scientific notation).

🏷️ Connection to prefixes

• Prefixes for large numbers (kilo, mega, giga, tera) are essentially engineering notation.
• Prefixes for small numbers (micro, nano, pico) also follow this pattern.
• The excerpt notes these prefixes will be covered in a later module.

🌐 Engineering notation examples

The excerpt provides examples for practice:

• U.S. population: around some number of million people
• World population: around some number of billion people
• U.S. national debt: around some number of trillion dollars
• Debt per person: calculated by dividing national debt by population

Don't confuse: engineering notation always uses multiples of 3 for exponents and allows coefficients up to 1000, while scientific notation uses any integer exponent but restricts coefficients to between 1 and 10.

🧭 Overview

🧠 One-sentence thesis

Engineering notation expresses numbers using powers of ten that are multiples of three (thousands, millions, billions, etc.), matching how large numbers are commonly reported in everyday language and technical fields.

📌 Key points (3–5)

• What engineering notation is: a variant of scientific notation that uses only powers of 10 that are multiples of 3 (10³, 10⁶, 10⁹, etc.).
• How it differs from scientific notation: the coefficient must be between 1 and 1000 (not just 1 to 10), and the exponent is always a multiple of 3.
• Why it matches everyday language: "37 thousand" naturally maps to 37 × 10³, not 3.7 × 10⁴.
• Common confusion: don't confuse the coefficient range—engineering notation allows 1–1000, while scientific notation restricts to 1–10.
• Real-world connection: prefixes like kilo, mega, giga (large) and micro, nano, pico (small) are essentially engineering notation.

📐 Definition and structure

📐 Formal definition

A number is written in engineering notation if it is written in the form (coefficient) × 10^(exponent), where the exponent is a multiple of 3 and the coefficient is any real number such that 1 ≤ coefficient < 1000.

• The exponent must be ..., -6, -3, 0, 3, 6, 9, 12, ... (multiples of 3).
• The coefficient can range from 1 up to (but not including) 1000.
• This is stricter than general scientific notation in the exponent, but looser in the coefficient.

🔢 Common powers in engineering notation

The excerpt lists standard large-number powers:

• One thousand = 10³
• One million = 10⁶
• One billion = 10⁹
• One trillion = 10¹²

Each step multiplies by 1000 (10³).

🗣️ Connection to everyday language

🗣️ How we naturally express numbers

• The excerpt gives the example of Oregon City's population: roughly 37,000 people.
• We say "thirty-seven thousand" and write "37 thousand" → 37 × 10³.
• We would not naturally say "three point seven ten thousands" (3.7 × 10⁴), even though that is valid scientific notation.

🔄 Engineering notation vs scientific notation

Aspect	Engineering notation	Scientific notation
Exponent	Must be multiple of 3	Any integer
Coefficient range	1 to 1000	1 to 10
Matches speech	Yes (thousands, millions)	Not always

Example: 37,000 people

• Engineering: 37 × 10³ ✓ (matches "37 thousand")
• Scientific: 3.7 × 10⁴ ✓ (valid but less intuitive for everyday use)

🔧 Technical and metric prefixes

🔧 Prefixes are engineering notation

The excerpt notes that metric prefixes are essentially engineering notation:

• Large numbers: kilo (10³), mega (10⁶), giga (10⁹), tera (10¹²)
• Small numbers: micro (10⁻⁶), nano (10⁻⁹), pico (10⁻¹²)

These prefixes always correspond to multiples of 3 in the exponent, reinforcing the engineering notation pattern.

Don't confuse: The excerpt mentions these prefixes will be covered in more detail in Module 16; the current section introduces the underlying notation system.

📝 Practice applications

📝 Types of exercises

The excerpt provides exercises that ask students to:

1. Convert ordinary numbers (population, debt figures) into engineering notation.
2. Also express the same numbers in scientific notation for comparison.
3. Perform calculations (e.g., debt per person) that may require engineering notation.

🌍 Real-world contexts

Examples from the exercises include:

• U.S. population: around (some number) million people
• World population: around (some number) billion people
• U.S. national debt: around (some number) trillion dollars

These contexts show why engineering notation is practical: news reports and government statistics naturally use thousands, millions, billions, and trillions.

🧭 Overview

🧠 One-sentence thesis

Finding the rate in percent problems means determining what percentage one number represents of another, which can be done either by translating words into equations or by setting up proportions.

📌 Key points (3–5)

• Three parts of percent problems: every percent problem has an amount (the result), a base (the original number), and a rate (the percent itself).
• Two solution methods: translate word-for-word into equations ("is" = equals, "of" = multiply) or set up proportions with the rate over 100.
• Finding the rate: when you know the amount and base, divide amount by base to get the rate in decimal form, then convert to percent.
• Finding the base: when you know the amount and rate, divide the amount by the rate (as a decimal) to find the original number.
• Common confusion: the base is always the original number (what you're finding the percent of), not just any number in the problem.

🔤 The three components of percent problems

🔤 Amount, base, and rate

Amount: the answer we get after finding the percent of the original number.

Base: the original number, the number we find the percent of.

Rate: the percent itself.

• These three pieces appear in every percent problem.
• The base is always the starting point or reference number.
• The amount is the result of applying the rate to the base.
• Example: "15 is what percent of 60?" → 15 is the amount, 60 is the base, and we're looking for the rate.

🔍 How to identify which is which

• The base corresponds to 100% because it's the whole original quantity.
• The amount is what you get after applying the percentage.
• Don't confuse: the base is not always the larger number—it's the number that came first in time or the number you're taking a percent of.

🧮 Method 1: Word-to-equation translation

🧮 Translation rules

The excerpt provides three key translation rules:

Word	Mathematical meaning
"is"	equals (=)
"of"	multiply (×)
"what"	a variable (unknown)

• These rules let you convert a word problem directly into an equation.
• Example: "15 is what percent of 60?" becomes "15 = (what) × 60"

🔢 Finding the rate with equations

• Question format: "X is what percent of Y?"
• Translation: X = (rate) × Y
• Solve by dividing: rate = X ÷ Y
• Important: this gives the answer in decimal form; you must move the decimal point two places right to convert to percent.
• Example: If you earned some points on a quiz, "points earned is what percent of total points?" becomes an equation where you divide points earned by total points.

🔢 Finding the base with equations

• Question format: "X is R% of what number?"
• Translation: X = R × (base)
• Solve by dividing: base = X ÷ R (where R is the rate as a decimal)
• Example from the excerpt: "You earn cash rewards for the amount you charge. If you want to earn a certain amount in rewards at a certain rate, how much do you need to charge?" becomes "reward amount = rate × (charge amount)"

📐 Method 2: Using proportions

📐 Setting up the proportion

A percent is a ratio, a fraction out of 100.

• Instead of word-by-word translation, set up a proportion with the rate over 100.
• The structure: (rate / 100) = (amount / base)
• Because the base is the original amount, it corresponds to 100%.
• This method works for finding either the rate or the base.

🔄 Why proportions work

• The base represents the whole (100%).
• The amount represents a part that corresponds to the rate.
• The proportion maintains the relationship: if the base is 100%, then the amount is the rate%.
• Example: "15 is what percent of 60?" becomes (rate / 100) = (15 / 60)

⚖️ Comparing the two methods

• The excerpt states: "Now that we have looked at both methods, you are free to use whichever method you prefer: percent equations or proportions."
• Both methods solve the same problems.
• Equations may be more direct for some learners; proportions may be clearer for others.
• The exercises demonstrate both methods on the same problems to show they yield the same answers.

📈 Percent increase problems

📈 What percent increase measures

• When a quantity changes, knowing the percent change is often more meaningful than the absolute change.
• The excerpt's example: a 50-cent increase matters a lot for a candy bar but not for a car, because the percentage of the original price is very different.
• Percent increase tells you the relative size of the change compared to the starting point.

📋 Steps to find percent increase

The excerpt provides a two-step process:

1. Subtract the two numbers to find the amount of increase.
2. Use the original number as the base: divide the amount of increase by the original number to find the unknown percent.

• Critical rule: always use the original number (the one that occurred earlier in time) as the base.
• For percent increase, the original number is the smaller of the two numbers.
• Don't confuse: the base is not the final number—it's always the starting number, even though the final number is larger.

⚠️ Common mistake to avoid

• The excerpt emphasizes using the original number as the base, repeating this point for clarity.
• Wrong approach: using the final (larger) number as the base will give an incorrect percentage.
• Right approach: the base is the number that came first in time, which is the smaller number in an increase scenario.

🎯 Real-world applications

🎯 Academic performance

• Exercise example: "The University of Oregon women's basketball team made 13 of the 29 three-point shots they attempted. What percent of their three-point shots did the team make?"
• This is a "finding the rate" problem: 13 is the amount, 29 is the base.
• Useful for calculating grades, success rates, and performance metrics.

🎯 Nutrition and health

• Exercise example: "A bottle of Dr. Pepper contains 65 grams of added sugars, which is 129% of the recommended daily intake. What is the recommended daily intake?"
• This is a "finding the base" problem: 65 is the amount, 129% is the rate.
• Useful for understanding nutritional information and health guidelines.

🎯 Financial rewards

• The excerpt mentions earning cash rewards on credit card charges.
• If you know the reward rate and want a specific reward amount, you need to find the base (how much to charge).
• This is a practical application of "finding the base" problems.

🧭 Overview

🧠 One-sentence thesis

Finding the base in percent problems means determining the original number when you know the percent rate and the resulting amount, which is essential for questions like "what total do I need to earn a certain reward?"

📌 Key points (3–5)

• What the base is: the original number, the number we find the percent of (as opposed to the amount, which is the result after applying the percent).
• How to find the base: translate "X is Y% of what number?" into an equation or set up a proportion with the rate over 100.
• Two methods available: percent equations (word-for-word translation) or proportions (rate over 100 equals amount over base).
• Common confusion: don't mix up the base (original number) with the amount (result); the base corresponds to 100% in a proportion.
• Real-world use: calculating how much to spend to earn a target cash reward, or finding recommended daily intake from a percentage label.

🔤 Core terminology

🔤 Base, amount, and rate

The base is the original number, the number we find the percent of.

The amount is the answer we get after finding the percent of the original number.

The rate is the percent itself.

• These three terms form the structure of every percent problem.
• The excerpt emphasizes that the base is the starting point, not the result.
• Example: "50 is 25% of what number?" → 50 is the amount, 25% is the rate, and the unknown base is what we solve for.

🔍 Don't confuse base and amount

• The base is the whole (100%); the amount is a part or result.
• In "X is Y% of Z," Z is the base and X is the amount.
• Common mistake: using the amount as the denominator when setting up a proportion.

🧮 Method 1: Percent equations

🧮 Translating words into algebra

The excerpt provides translation rules:

• "is" means equals (=)
• "of" means multiply (×)
• "what" means a variable

📝 Example: cash rewards

• Question: "50 is 2% of what number?"
• Translation: 50 = 0.02 × (what number)
• Solve by dividing: what number = 50 ÷ 0.02
• This answers "how much do you need to charge on your card to earn $50 in cash rewards if you earn 2% cash back?"

⚠️ Remember to convert percent to decimal

• The rate must be in decimal form for the equation.
• Example: 2% becomes 0.02 before multiplying.

📐 Method 2: Using proportions

📐 Setting up the proportion

A percent is a ratio, a fraction out of 100.

• Set up: (rate / 100) = (amount / base)
• The base corresponds to 100% because it is the original whole.
• Example: "50 is 2% of what dollar value?" → (2 / 100) = (50 / base)

🔄 Why proportions work

• The percentage rate is "out of 100" by definition.
• The amount is "out of the base" in the same ratio.
• Cross-multiply to solve for the unknown base.

🆚 Comparing the two methods

Method	Structure	When to use
Percent equation	Translate word-for-word into algebra	Quick for simple problems; direct translation
Proportion	Set rate/100 = amount/base	Clear structure; easier to remember which number goes where

• The excerpt states: "you are free to use whichever method you prefer."
• Both methods give the same answer.

🎯 Practice scenarios

🎯 Recommended daily intake

• Exercise 10: "A bottle of Dr. Pepper contains 65 grams of added sugars, which is 129% of the recommended daily intake. What is the recommended daily intake?"
• Here, 65 grams is the amount, 129% is the rate, and the base (recommended intake) is unknown.
• Using equation: 65 = 1.29 × base → base = 65 ÷ 1.29
• Using proportion: (129 / 100) = (65 / base) → solve for base.

🎯 General structure

• Identify which number is the amount (the result or part).
• Identify the rate (the percent).
• The base is what you solve for.
• Don't confuse: the base is not "what's left over"; it's the original whole that the percent was applied to.

🧭 Overview

🧠 One-sentence thesis

Proportions offer an alternative method to solve percent problems by setting up a fraction with the percentage rate over 100, corresponding the base to 100%.

📌 Key points (3–5)

• What proportions do: set up a fraction with the percentage rate over 100, treating the base as corresponding to 100%.
• When to use proportions: they work for finding the rate, finding the base, and all types of percent problems (as an alternative to translating word-for-word into equations).
• How base corresponds: because the base is the original amount, it corresponds to 100% in the proportion.
• Common confusion: proportions vs. word-for-word translation—both methods solve the same problems; you can choose whichever you prefer.
• Why it matters: proportions provide a consistent framework for all percent problems, including percent increase/decrease and error analysis.

🔢 The proportion method

🔢 Setting up the proportion

A percent is a ratio, a fraction out of 100.

• Instead of translating each word ("is" = equals, "of" = multiply, "what" = variable), you set up a proportion.
• The structure: put the percentage rate over 100 on one side.
• The base (original amount) corresponds to 100% because it represents the whole.

🔄 How it compares to equations

• The excerpt shows exercises solved first by translating word-for-word into equations, then re-solved using proportions.
• Example: "X is what percent of Y?" can be translated as an equation or set up as a proportion with the unknown percent over 100.
• Don't confuse: these are two different methods for the same problem—the excerpt states "you are free to use whichever method you prefer."

📐 Finding the rate with proportions

📐 What "finding the rate" means

• The rate is the percent itself (the number that goes with the % symbol).
• Example scenario: you earned some points on a quiz and want to know "X is what percent of Y?"
• Using proportions: set up a fraction with the unknown percent over 100, then solve.

🔍 Decimal-to-percent conversion

• The excerpt warns: when using the equation method, the answer comes in decimal form and you must move the decimal point to convert to a percent.
• Rounding guidance: if instructions don't specify, use best judgment—nearest whole percent, nearest tenth of a percent, or two to three significant figures.

🎯 Finding the base with proportions

🎯 What "finding the base" means

• The base is the original number, the number you find the percent of.
• Example scenario: "X is Y% of what number?"—you know the amount and the rate, but not the base.
• Using proportions: the unknown base corresponds to 100%, so set up the proportion accordingly.

💳 Cash rewards example

• The excerpt gives a scenario: you earn a certain percentage in cash rewards and want to know how much to charge on your card to earn a specific dollar amount.
• This is a "finding the base" problem: the reward amount is known, the rate is known, the base (amount charged) is unknown.

📊 Percent change applications

📈 Percent increase

To find the percent of increase:

1. Subtract the two numbers to find the amount of increase.
2. Using this result as the amount and the original number as the base, divide and find the unknown percent.

• Key rule: always use the original number (the one that occurred earlier in time) as the base.
• For percent increase, the original is the smaller of the two numbers.
• Example: candy bar price increased from one amount to another—find the amount of increase, then calculate what percent that increase is of the original price.
• Example: sales tax problem—retail price vs. actual cost including tax; the difference is the tax amount, and you find what percent that is of the retail price.

📉 Percent decrease

To find the percent of decrease:

1. Subtract the two numbers to find the amount of decrease.
2. Using this result as the amount and the original number as the base, divide and find the unknown percent.

• Again, always use the original number as the base.
• For percent decrease, the original is the larger of the two numbers.
• Example: sale price reduced from original—find the amount of decrease, then calculate what percent that decrease is of the original price.
• Example: population decline—compare two population estimates; the earlier (larger) number is the base.

🔄 Unified formula

• The excerpt states there is a formula that works whether finding a percent increase or a percent decrease.
• This formula uses the original number as the base in both cases.
• Don't confuse: the original number is not always the larger or smaller—it's the one that occurred earlier in time.

🎯 Relative error concept

🎯 Why relative error matters

• The excerpt introduces relative error in the context of measurement error from an earlier module.
• Key insight: the same absolute error (e.g., 50 cents, one-eighth of an inch) can be significant or insignificant depending on the size of what is being measured.
• Example: 50 cents is a large percentage of a candy bar price but insignificant for a car price.
• Example: one-eighth of an inch matters for fitting a screen into a window frame but is insignificant when measuring a garage length.

📏 Expected outcome

The expected outcome is what the number would be in a perfect world.

• This definition appears at the end of the excerpt (the sentence is incomplete).
• Context: when measuring, the expected outcome is the ideal or true value; the actual measurement will have some error relative to this expected value.

🧭 Overview

🧠 One-sentence thesis

Percent increase calculations reveal how much a quantity has grown relative to its original value, making the same absolute change meaningful or trivial depending on context.

📌 Key points (3–5)

• Why percent matters more than absolute change: a 50-cent increase matters for a candy bar but not for a car because the percentage of the original price differs dramatically.
• Two-step process: subtract to find the amount of increase, then divide by the original number to find the percent.
• Always use the original number as the base: the number that occurred earlier in time (the smaller number in an increase scenario).
• Common confusion: don't use the new/final number as the base—always divide by the original value.
• Real applications: pricing changes (candy bars, retail items) and tax calculations.

📐 The percent increase method

📐 Why context matters for changes

• The excerpt opens with a key insight: absolute changes (like 50 cents) mean different things depending on the original amount.
• A 50-cent increase on a candy bar feels significant; the same increase on a car is negligible.
• The reason: the percentage of the original price determines whether the change is meaningful.
• This motivates why we calculate percent increase rather than just reporting dollar amounts.

🔢 The two-step calculation

To find the percent of increase:

1. Subtract the two numbers to find the amount of increase.
2. Using this result as the amount and the original number as the base, divide and find the unknown percent.

• Step 1: Find the difference between the new value and the original value.
• Step 2: Divide that difference by the original number, then convert to a percent.
• Example: If a candy bar goes from one price to a higher price, subtract to get the increase amount, then divide by the starting price.

⚠️ Critical rule: the base

⚠️ Always use the original number

• The excerpt emphasizes: "we always use the original number for the base, the number that occurred earlier in time."
• For percent increase specifically: the original number is the smaller of the two numbers.
• Don't confuse: the base is not the final value or an average—it must be the starting value.

🕐 Time determines which is original

• "Original" means the value that occurred earlier in time, not necessarily the larger or smaller number.
• In an increase scenario, the earlier-in-time value happens to be smaller.
• This time-based definition keeps the method consistent across increase and decrease problems.

🛍️ Applications in the excerpt

🛍️ Retail pricing example

• The excerpt mentions a candy bar price increase and a t-shirt purchase with sales tax.
• For the candy bar: the starting price is the base; the price difference is the amount; divide to find the percent increase.
• For the t-shirt: the retail price is the base; the tax amount (difference between actual cost and retail) is the amount; divide to find the sales tax rate.

🏀 Other scenarios

• The excerpt also references a basketball team's three-point shooting percentage and a recommended daily sugar intake problem.
• These illustrate that the percent increase framework applies broadly: sports statistics, nutrition guidelines, and pricing all use the same two-step method.

🧭 Overview

🧠 One-sentence thesis

Percent decrease is calculated by dividing the amount of decrease by the original (larger) number, ensuring that the base always reflects the earlier-in-time value.

📌 Key points (3–5)

• Two-step process: subtract to find the amount of decrease, then divide by the original number to find the percent.
• Base is always original: use the number that occurred earlier in time—for percent decrease, this is the larger of the two numbers.
• Common confusion: don't mix up which number is the base; percent decrease uses the original (larger) value, not the new (smaller) value.
• Same structure as percent increase: the procedure mirrors percent increase but the original number is the larger one.
• Formula applies broadly: the percent change formula works for both increases and decreases.

🔢 The percent decrease procedure

📉 Step 1: Find the amount of decrease

• Subtract the two numbers to determine how much the quantity dropped.
• This gives you the absolute change in value.
• Example: if a candy bar price drops from some amount to a lower amount, subtract the new price from the old price.

➗ Step 2: Divide by the original number

Using the amount of decrease as the amount and the original number as the base, divide and find the unknown percent.

• The original number is the one that occurred earlier in time.
• For a percent decrease, the original number is the larger of the two numbers.
• This division gives you the percent decrease.

🎯 Identifying the base correctly

🕐 Original means earlier in time

• The excerpt emphasizes that "original" refers to the number that occurred earlier in time.
• This is not about which number is bigger or smaller in the abstract—it's about the time sequence.
• For percent decrease specifically, the earlier-in-time number happens to be the larger one.

⚠️ Don't confuse the base

• Common mistake: using the new (smaller) value as the base instead of the original (larger) value.
• The base must always be the original number, regardless of whether you're calculating increase or decrease.
• Example: if a population drops from a higher number to a lower number, the higher (earlier) number is the base.

📐 Worked scenarios from the excerpt

🍫 Candy bar price reduction

• Exercise 13 asks: during a sale, the price of a candy bar was reduced from one amount to another—by what percent did the price decrease?
• Apply the procedure: subtract to find the decrease amount, then divide by the original (higher) price.

🏙️ Population decline

• Exercise 14: Portland, Oregon's estimated population in April 2020 was one value; in July 2022 it was a lower value.
• Task: find the percent of decrease to the nearest tenth of a percent.
• The April 2020 (earlier) population is the base, even though it's the larger number.

🔄 Connection to percent change formula

🧮 Unified formula

• The excerpt states: "we can determine the percent change using the following formula, which works whether we're finding a percent increase or a percent decrease."
• The structure is the same for both increase and decrease.
• The key difference is identifying which number is the original (base).

🔗 Relationship to other concepts

• The excerpt notes that the percent change formula has "the same structure as the percent change formula we saw earlier" (referring to relative error).
• This suggests a consistent mathematical approach across different contexts: amount of change divided by the reference (original) value.

🧭 Overview

🧠 One-sentence thesis

Relative error expresses measurement error as a percentage of the expected outcome, making it possible to compare the significance of errors across measurements of very different sizes.

📌 Key points (3–5)

• Why relative error matters: the same absolute error (e.g., 1/8 inch or 5 cents) can be critical in one context but insignificant in another, depending on the size of what is being measured.
• Expected outcome vs. measurement: the expected outcome is the ideal or target value (treated as infinitely precise), while the measurement is the actual observed value.
• Three-step calculation: find absolute error (subtract measurement from expected outcome), then divide by expected outcome, then express as a percent.
• Common confusion: relative error uses the expected outcome as the base (denominator), just like percent increase/decrease uses the original number—not the measurement itself.
• Tolerance: the maximum allowed difference from the expected outcome; it defines an acceptable range for measurements.

🔍 Core concepts

🎯 Expected outcome

The expected outcome is what the number would be in a perfect world.

• It is the ideal, target, or advertised value.
• The excerpt treats it as having infinitely many significant digits (unlimited precision).
• Example: a window screen advertised as exactly some number of inches wide—that advertised width is the expected outcome.

📏 Absolute error

To find the absolute error, we subtract the measurement and the expected outcome.

• It is the raw difference between what you measured and what you expected.
• The absolute error should have the same precision (place value) as the measurement, not the expected outcome.
• Example: if a screen measures 24.1 inches instead of 24 inches, the absolute error is 0.1 inches.

📊 Relative error

To find the relative error, we divide the absolute error by the expected outcome.

• Relative error is usually expressed as a percent.
• The procedure is identical to finding percent increase or decrease: use the absolute error as the "amount" and the expected outcome as the "base."
• Example: an absolute error of 0.1 inches on a 24-inch screen gives a relative error of (0.1 ÷ 24) × 100%.

🧮 How to calculate relative error

🧮 Step-by-step procedure

The excerpt gives a two-step process:

1. Subtract the measurement and the expected outcome to find the absolute error.
2. Divide the absolute error by the expected outcome, then convert to a percent.

• The structure is the same as the percent-change formula seen earlier in the text.
• Don't confuse: always use the expected outcome as the denominator (base), not the measurement.

🧪 Worked scenario

The excerpt provides exercises (not fully worked solutions, but the setup is clear):

• Window screen example: measured at some width instead of the advertised width → find absolute error, then divide by advertised width, round to nearest tenth of a percent.
• Cereal box example: contents measured at some weight instead of the supposed weight → same procedure.

🎚️ Tolerance and acceptable ranges

🎚️ What tolerance means

The tolerance is the maximum amount that a measurement is allowed to differ from the expected outcome.

• Tolerance defines an acceptable range: expected outcome ± tolerance.
• Example: a dime weighs 2.268 grams with a tolerance of 0.091 grams → acceptable weight range is 2.268 − 0.091 to 2.268 + 0.091 grams.

⚖️ Minimum and maximum acceptable values

• Minimum acceptable = expected outcome − tolerance.
• Maximum acceptable = expected outcome + tolerance.
• Any measurement outside this range is unacceptable.
• Example (from exercises): a nickel weighs 5 grams with a tolerance of 0.194 grams → lowest acceptable is 5 − 0.194, highest is 5 + 0.194.

🔗 Tolerance and relative error

• The excerpt asks you to determine the relative error when the absolute error equals a given value (e.g., 0.091 grams for a dime).
• You still divide the absolute error by the expected outcome and express as a percent.
• Don't confuse: tolerance is a specification (how much error is allowed); relative error is a calculation (how much error actually occurred, as a percentage).

🌍 Why size matters

🌍 Context-dependent significance

The excerpt opens with a key insight:

• A difference of a few cents is important when pricing candy bars but insignificant when pricing cars.
• An error of 1/8 inch is a deal-breaker for a window screen but insignificant for the length of a garage.

This is why we need relative error: it scales the error to the size of the measurement, making errors comparable across different contexts.

🔄 Connection to percent change

The excerpt explicitly states:

The procedure for finding the relative error is identical to the procedures for finding a percent increase or percent decrease!

• Both use the same formula structure: (difference ÷ base) × 100%.
• For percent change, the base is the original (earlier-in-time) number.
• For relative error, the base is the expected outcome.

🧭 Overview

🧠 One-sentence thesis

Tolerance defines the maximum allowable deviation from an expected measurement, establishing an acceptable range within which a measured value remains valid.

📌 Key points (3–5)

• What tolerance specifies: the maximum amount a measurement can differ from the expected outcome and still be acceptable.
• How tolerance creates a range: it sets both a minimum acceptable value (expected minus tolerance) and a maximum acceptable value (expected plus tolerance).
• Relationship to error concepts: tolerance works alongside absolute error and relative error to evaluate measurements.
• Common confusion: tolerance is not the same as absolute error—absolute error is the actual difference measured, while tolerance is the maximum difference allowed.
• Real-world application: used in manufacturing (e.g., coin production) to ensure consistency and functionality.

📏 Understanding measurement error

📐 Expected outcome

Expected outcome: the theoretical or specified value that a measurement should match, treated as having infinitely many significant digits.

• This is the target or standard value, not the actual measured value.
• Example: a screen advertised as exactly a certain width, or a coin specified to weigh a certain number of grams.

➖ Absolute error

Absolute error: the difference between the measurement and the expected outcome.

• Calculation: subtract the measurement from the expected outcome (or vice versa).
• The absolute error should have the same precision (place value) as the measurement, not the expected outcome.
• This tells you "how far off" the measurement is in the same units.

📊 Relative error

Relative error: the absolute error divided by the expected outcome, usually expressed as a percent.

• Calculation process:
  1. Find the absolute error (subtract the two numbers)
  2. Divide the absolute error by the expected outcome
  3. Express as a percentage
• The procedure is identical to finding percent increase or percent decrease.
• This tells you "how far off proportionally" rather than in absolute terms.

🎯 What tolerance means

🔒 Definition and purpose

Tolerance: the maximum amount that a measurement is allowed to differ from the expected outcome.

• Sets boundaries for acceptability rather than describing an actual measurement.
• Used to ensure consistency in manufacturing and quality control.
• Example: U.S. Mint coins need consistent size and weight to work in vending machines.

⚖️ How tolerance establishes acceptable ranges

The excerpt provides a concrete example with dimes:

• A dime weighs a specified number of grams (expected outcome)
• Tolerance: a certain number of grams
• Minimum acceptable weight = expected weight minus tolerance
• Maximum acceptable weight = expected weight plus tolerance
• Any dime with weight outside this range would be unacceptable

Don't confuse: Tolerance is a specification (what's allowed), while absolute error is a measurement (what actually happened).

🪙 Applying tolerance to coin specifications

💰 Working with tolerance ranges

The excerpt provides exercises for three U.S. coins:

Coin	Weight specification	Tolerance	Task
Dime (10 cents)	Specified grams	Specified grams	Find min/max acceptable weight
Nickel (5 cents)	Specified grams	Specified grams	Find acceptable range and relative error
Quarter (25 cents)	Specified grams	Specified grams	Find acceptable range and relative error

🔢 Combining tolerance with error calculations

The exercises ask students to:

1. Determine acceptable ranges: Calculate minimum (weight minus tolerance) and maximum (weight plus tolerance) acceptable weights
2. Calculate relative error: Given an absolute error value, find what percentage it represents of the expected outcome

This shows how tolerance and error measurements work together—you can check whether an actual error falls within the tolerance limits.

🧭 Overview

🧠 One-sentence thesis

The U.S. customary system uses dimensional analysis (the factor-label method) to convert between units by multiplying measurements by conversion ratios that cancel unwanted units and replace them with desired ones.

📌 Key points (3–5)

• Core method: dimensional analysis (factor-label method) converts units by multiplying by conversion ratios that cancel unwanted units.
• Key insight: you don't need to memorize whether to multiply or divide—set up the ratio so units cancel, and the numbers' positions will show the operation.
• Basic length conversions: 1 foot = 12 inches, 1 yard = 3 feet, 1 mile = 5,280 feet.
• Mixed units: measurements like "5 ft 8 in" or "7 lb 12 oz" work like mixed numbers (whole part + fractional part).
• Common confusion: don't try to memorize multiplication vs. division rules; instead, arrange conversion ratios so units cancel correctly.

🔧 The dimensional analysis method

🔧 How the factor-label method works

Dimensional analysis (factor-label method): a conversion technique that starts with a known measurement, then multiplies by conversion ratios to cancel unwanted units and introduce desired units.

• Write the starting measurement as a fraction over 1.
• Multiply by a conversion ratio arranged so the old unit cancels.
• The positions of numbers in the ratio automatically determine whether you multiply or divide.

📐 Example: miles to feet

• Start with 8 miles written as 8/1.
• Use the conversion ratio "5,280 feet / 1 mile" so "miles" cancels.
• Result: multiply 8 by 5,280 to get feet.
• Example: 8 miles becomes 42,240 feet.

📏 Example: feet to miles

• Start with 20,000 feet written as 20,000/1.
• Use the conversion ratio "1 mile / 5,280 feet" so "feet" cancels.
• Result: divide 20,000 by 5,280 to get miles.
• Example: 20,000 feet becomes approximately 3.79 miles.

🔑 Why this method is versatile

• The excerpt notes it is "more versatile, especially when a conversion requires more than one step."
• You don't need conversion ratios memorized; keep them available and write down new ones you discover.
• The method works for any unit type (length, weight, volume).

📏 Length conversions in the U.S. system

📏 Basic length ratios

Conversion	Ratio
1 foot	12 inches
1 yard	3 feet
1 mile	5,280 feet

• The excerpt notes these come from "the system used in England centuries ago."
• Conversions often involve "messy calculations like dividing by 5,280 or multiplying by 12."

📏 Derived conversions

• The excerpt mentions that exercises 3 & 4 yield two more conversion ratios (not explicitly stated in the excerpt, but implied to be yard-to-inches and mile-to-yards).
• You can chain conversions: e.g., convert inches → feet → yards by applying ratios in sequence.

⚖️ Weight and volume conversions

⚖️ Weight (or mass) ratios

Conversion	Ratio
1 pound	16 ounces
1 ton	2,000 pounds

• The procedure is identical: start with the known measurement over 1, then multiply by a conversion factor that cancels unwanted units.

🥤 Volume (or capacity) ratios

Conversion	Ratio
1 cup	8 fluid ounces
1 pint	2 cups
1 quart	2 pints
1 gallon	4 quarts

• The excerpt notes "plenty of other conversions could be provided, such as the number of fluid ounces in a gallon, but let's keep the list relatively short."
• You can chain these ratios for multi-step conversions (e.g., cups → pints → quarts → gallons).

🔀 Working with mixed units

🔀 What mixed units are

• Measurements given with two unit levels, such as "5 ft 8 in" (height) or "7 lb 12 oz" (baby weight).
• The excerpt says this "can sometimes make the calculations more complicated."

🔀 How to handle them

• Think of mixed units like mixed numbers: a whole-number part and a fractional part.
• Example: a bag weighs 25 ounces → convert to pounds and ounces by dividing 25 by 16 (1 pound = 16 ounces) → 1 lb 9 oz.
• Example: a hallway is 65 inches long → convert to feet and inches by dividing 65 by 12 (1 foot = 12 inches) → 5 ft 5 in.

➕ Adding and subtracting mixed units

• The excerpt mentions "adding and subtracting with mixed units" at the end.
• Treat them like mixed numbers: add/subtract the whole parts and fractional parts separately.
• Example scenario: Comet weighs 8 lb 10 oz and Fred weighs 12 lb 14 oz → their combined weight requires adding pounds and ounces separately (may need to carry over if ounces exceed 16).
• Example scenario: two tables are 6 ft 9 in and 4 ft 5 in long → find combined length or difference by working with feet and inches separately.

⚠️ Don't confuse

• Mixed units are not the same as decimal measurements (e.g., 5.67 feet is different from 5 ft 8 in).
• When converting, you may need to handle "carrying" or "borrowing" between unit levels, similar to working with mixed numbers.

🧭 Overview

🧠 One-sentence thesis

The U.S. system for weight or mass uses pounds and tons as base units, and conversions between them follow the same dimensional-analysis method used for other U.S. measurements.

📌 Key points (3–5)

• Two base units: pound and ton are the primary units for weight/mass in the U.S. customary system.
• Conversion ratios: 1 pound = 16 ounces; 1 ton = 2,000 pounds.
• Method: use dimensional analysis (factor-label method) by multiplying the known measurement by a conversion ratio that cancels unwanted units.
• Mixed units: measurements often combine units (e.g., "5 lb 8 oz"), requiring conversion between improper fractions and mixed numbers.
• Common confusion: don't memorize when to multiply vs. divide—set up the conversion ratio so units cancel, and the placement of numbers will show the operation.

⚖️ Core conversion ratios

⚖️ Pound and ounce

1 pound = 16 ounces

• The pound is the larger unit; the ounce is the smaller subdivision.
• Example: to convert pounds to ounces, multiply by 16; to convert ounces to pounds, divide by 16 (or multiply by the ratio 1 pound / 16 ounces).

🚛 Ton and pound

1 ton = 2,000 pounds

• The ton is the larger unit used for heavy objects (e.g., vehicles, cargo).
• Example: to convert tons to pounds, multiply by 2,000; to convert pounds to tons, divide by 2,000 (or multiply by the ratio 1 ton / 2,000 pounds).

🔄 Dimensional analysis method

🔄 How the method works

• Start with the known measurement written as a fraction over 1.
• Multiply by a conversion ratio arranged so the unwanted units cancel and the desired units remain.
• The excerpt emphasizes: "It's okay if you don't have the conversion ratios memorized; just be sure to have them available."

🧮 Why not memorize multiply vs. divide

• The excerpt advises: "Rather than trying to memorize rules about when you'll multiply versus when you'll divide, just set up the conversion ratio so the units will cancel out."
• The locations of the numbers in the ratio will automatically tell you whether to multiply or divide.
• Example: converting 3 pounds to ounces → write (3 pounds / 1) × (16 ounces / 1 pound) → pounds cancel, leaving 3 × 16 = 48 ounces.

🔗 Multi-step conversions

• The excerpt notes that dimensional analysis is "more versatile, especially when a conversion requires more than one step."
• Example: converting tons to ounces requires chaining two ratios (ton → pound → ounce).

🧩 Working with mixed units

🧩 What mixed units are

• Measurements often combine two units: a larger unit and a smaller unit.
• Example: "5 lb 8 oz" instead of "88 oz" or "a person's height being given as ft in instead of in."
• The excerpt says: "This can sometimes make the calculations more complicated, but if you can convert between improper fractions and mixed numbers, you can handle this."

➕ Adding and subtracting mixed units

• Treat mixed units like mixed numbers: a whole-number part and a fractional part.
• Example problems in the excerpt:
  • "Comet weighs lb oz and Fred weighs lb oz. What is their combined weight?"
  • "How much heavier is Fred than Comet?"
• The excerpt does not provide the specific numbers, but the method is to add or subtract each unit separately, then convert if the smaller unit exceeds its conversion threshold (e.g., if ounces exceed 16, convert to an additional pound).

🔄 Converting to and from mixed units

• Example: "A bag of apples weighs [some number of] ounces. What is its weight in pounds and ounces?"
• Process: divide the total ounces by 16 to get the whole pounds, and the remainder is the leftover ounces.
• Don't confuse: mixed units are a display format, not a different measurement system—they still use the same conversion ratios.

📋 Summary of exercises

📋 Types of problems

The excerpt lists exercises that cover:

Exercise type	Description
Simple conversions	"How many ounces are in [X] pounds?" or "How many pounds are equivalent to [X] tons?"
Reverse conversions	"Convert [X] ounces to pounds" or "[X] ton is equivalent to what number of ounces?"
Mixed-unit conversions	"A bag weighs [X] ounces. What is its weight in pounds and ounces?"
Addition/subtraction	"What is their combined weight?" or "How much heavier is one than the other?"

🧪 Real-world context

• The excerpt mentions practical scenarios: weighing cats ("Comet and Fred"), a Ford F-150 pickup truck's maximum loaded weight, and a bag of apples.
• These examples show that weight/mass conversions are used in everyday tasks (cooking, shipping, vehicle specifications).

🧭 Overview

🧠 One-sentence thesis

The U.S. customary system for volume and capacity uses four main conversion ratios (cup, pint, quart, gallon) that can be chained together using dimensional analysis to convert between units.

📌 Key points (3–5)

• Core conversion ratios: 1 cup = 8 fluid ounces; 1 pint = 2 cups; 1 quart = 2 pints; 1 gallon = 4 quarts.
• Conversion method: use dimensional analysis (factor-label method) by multiplying the known measurement by conversion ratios that cancel unwanted units.
• Multi-step conversions: the list is kept short on purpose; longer conversions (e.g., fluid ounces to gallons) require chaining multiple ratios.
• Mixed units: measurements like "1 lb 3 oz" work like mixed numbers—a whole-number part plus a fractional part—and can be added, subtracted, or converted.
• Common confusion: don't memorize when to multiply vs. divide; instead, set up the conversion ratio so units cancel, and the placement of numbers will show the operation.

📏 The four core conversion ratios

🥤 Fluid ounces, cups, pints, quarts, and gallons

The excerpt provides exactly four conversion ratios for U.S. volume/capacity:

From	To	Ratio
cup	fluid ounces	1 cup = 8 fluid ounces
pint	cups	1 pint = 2 cups
quart	pints	1 quart = 2 pints
gallon	quarts	1 gallon = 4 quarts

• These are the building blocks; other conversions (e.g., fluid ounces in a gallon) are derived by chaining these ratios.
• The excerpt deliberately keeps the list "relatively short" to avoid overwhelming learners.

🔗 Why a short list matters

• You don't need to memorize every possible conversion (e.g., cups to gallons, fluid ounces to quarts).
• Instead, you chain the four basic ratios step by step.
• Example: to convert cups to gallons, you would go cups → pints → quarts → gallons using three successive ratios.

🔄 How dimensional analysis works

🧮 The factor-label method

Dimensional analysis (or the factor-label method): a conversion technique where you multiply the known measurement by a conversion ratio that cancels the unwanted units and replaces them with the desired units.

• Start by writing the known measurement as a fraction over 1.
• Multiply by a conversion ratio arranged so the old unit cancels.
• The positions of the numbers (numerator vs. denominator) automatically tell you whether to multiply or divide—no need to memorize separate rules.

✅ Setting up the conversion ratio

• The key is to arrange the ratio so units cancel.
• If you want to cancel "cups," put "cups" in the denominator of the conversion ratio.
• If you want to end up with "pints," put "pints" in the numerator.
• Example: converting cups to pints → write (1 pint / 2 cups) so "cups" cancels and "pints" remains.

❌ Don't confuse: multiply vs. divide

• The excerpt emphasizes: "Rather than trying to memorize rules about when you'll multiply versus when you'll divide, just set up the conversion ratio so the units will cancel out."
• The arithmetic (multiplication or division) emerges naturally from how you arrange the fraction.
• Example: converting a larger unit to a smaller unit often looks like multiplication, but converting a smaller unit to a larger unit often looks like division—yet both follow the same setup principle.

🧩 Working with mixed units

🧩 What mixed units are

• Measurements like "5 lb 3 oz" or "2 ft 7 in" combine a whole-number part and a fractional part.
• The excerpt says: "think of them as mixed numbers, with a whole number part and a fractional part."
• This analogy helps when adding, subtracting, or converting.

➕ Adding and subtracting mixed units

• Add or subtract the whole-number parts and the fractional parts separately.
• If the fractional part exceeds one full unit, carry over to the whole-number part (like carrying in addition).
• Example: adding two weights given in pounds and ounces—add the pounds, add the ounces, then convert any excess ounces (≥16 oz) into additional pounds.

🔄 Converting mixed units to single units

• To convert "5 lb 3 oz" to total ounces, convert the pounds to ounces and add the remaining ounces.
• To convert total ounces back to "lb oz" format, divide by 16 (ounces per pound); the quotient is pounds, the remainder is ounces.
• The excerpt's exercises (19–22) practice this: e.g., "A bag of apples weighs [some number] ounces. What is its weight in pounds and ounces?"

🎯 Practice exercises and their purpose

📝 Exercises 13–18: single-step and multi-step conversions

• Exercise 13: "How many fluid ounces are in [some number] cups?" → direct use of 1 cup = 8 fluid ounces.
• Exercise 15: "1 gallon is equal to how many pints?" → requires chaining: gallons → quarts → pints.
• Exercise 17: "Convert [some number] cups into gallons" → multi-step: cups → pints → quarts → gallons.
• Exercise 18: "How many fluid ounces are in one half gallon?" → chain from gallons down to fluid ounces.

📝 Exercises 19–22: converting to and from mixed units

• These ask you to express a single-unit measurement (e.g., total ounces) as mixed units (pounds and ounces), or vice versa.
• Example: "A carton of orange juice contains [X] fluid ounces. Determine its volume in cups and fluid ounces."
• This mirrors converting improper fractions to mixed numbers.

📝 Exercises 23–26: arithmetic with mixed units

• Exercises 23–24 involve two cats' weights (given in "lb oz" format): find their combined weight and the difference.
• Exercises 25–26 involve two tables' lengths (given in "ft in" format): find the total length and the difference.
• These require adding or subtracting both the whole-number part and the fractional part, then adjusting if the fractional part overflows.

🔍 Key takeaways for self-study

🔍 Memorize the four ratios, not every possible conversion

• 1 cup = 8 fl oz; 1 pint = 2 cups; 1 quart = 2 pints; 1 gallon = 4 quarts.
• All other volume conversions in the U.S. system can be derived by chaining these.

🔍 Trust the unit-cancellation method

• Write the known measurement over 1.
• Multiply by conversion ratios arranged so unwanted units cancel.
• The arithmetic (multiply or divide) follows automatically from the setup.

🔍 Treat mixed units like mixed numbers

• Whole part + fractional part.
• When adding/subtracting, handle each part separately and carry over if needed.
• When converting, use the same logic as converting between improper fractions and mixed numbers.

🧭 Overview

🧠 One-sentence thesis

Mixed-unit measurements (like 5 ft 3 in or 7 lb 9 oz) are common in the U.S. system and can be handled by converting between improper fractions and mixed numbers, then adding or subtracting the whole and fractional parts separately.

📌 Key points (3–5)

• What mixed units are: measurements expressed with two units together (e.g., feet + inches, pounds + ounces) instead of a single unit.
• Why mixed units are used: they are frequently given for heights (ft + in) and weights (lb + oz) because they are more intuitive than large single-unit numbers.
• How to work with them: treat mixed units like mixed numbers—convert between improper and mixed forms, then perform arithmetic on whole and fractional parts.
• Common confusion: don't try to add or subtract mixed units directly without aligning the unit parts; handle the larger unit and smaller unit separately, just as you would the whole number and fraction in a mixed number.

🔢 What mixed units look like

🔢 Common examples in daily life

• Height: a person's height given as "5 ft 3 in" instead of "63 in."
• Weight: a newborn baby's weight given as "7 lb 9 oz" instead of "121 oz."
• The excerpt emphasizes that this format is frequently used in the U.S. system.

🧩 Why this format exists

• Mixed units break a measurement into a larger unit (feet, pounds, gallons) and a smaller unit (inches, ounces, fluid ounces).
• This makes the number easier to understand at a glance.
• Example: "5 ft 3 in" is more intuitive than "63 inches" for most people.

🔄 Converting and calculating with mixed units

🔄 The core technique

If you can convert between improper fractions and mixed numbers, you can handle mixed-unit measurements.

• The excerpt draws a direct analogy: mixed units work like mixed numbers (a whole number part + a fractional part).
• Whole part: the larger unit (feet, pounds, quarts).
• Fractional part: the smaller unit (inches, ounces, fluid ounces).

🧮 Conversion examples from the exercises

The excerpt provides four conversion scenarios (exercises 19–22):

Given measurement	Task	Approach
A bag weighs some ounces	Convert to pounds and ounces	Divide ounces by 16 (1 lb = 16 oz); the quotient is pounds, the remainder is ounces
A carton contains some fluid ounces	Convert to cups and fluid ounces	Divide by 8 (1 cup = 8 fl oz); quotient is cups, remainder is fluid ounces
A hallway is some inches long	Convert to feet and inches	Divide by 12 (1 ft = 12 in); quotient is feet, remainder is inches
A truck weighs some pounds	Convert to tons and pounds	Divide by 2000 (1 ton = 2000 lb); quotient is tons, remainder is pounds

• In each case, you are converting an "improper" single-unit measurement into a "mixed" two-unit measurement.
• Example: 121 oz → divide by 16 → 7 lb with 9 oz left over → 7 lb 9 oz.

➕ Adding mixed units

The excerpt introduces addition with the example of two cats' weights (Comet: some lb + oz, Fred: some lb + oz).

• Step 1: Add the larger units (pounds) together.
• Step 2: Add the smaller units (ounces) together.
• Step 3: If the smaller unit sum exceeds the conversion threshold (e.g., ≥16 oz), convert the excess into the larger unit and carry over.
• Example: If Comet is 5 lb 9 oz and Fred is 6 lb 11 oz, add 5+6=11 lb and 9+11=20 oz; since 20 oz = 1 lb 4 oz, the total is 11+1=12 lb 4 oz.

➖ Subtracting mixed units

The excerpt asks "How much heavier is Fred than Comet?" (exercise 24).

• Step 1: Subtract the larger units.
• Step 2: Subtract the smaller units.
• Step 3: If the smaller unit in the minuend is less than the smaller unit in the subtrahend, borrow 1 from the larger unit (convert it to the smaller unit) before subtracting.
• Example: If Fred is 6 lb 11 oz and Comet is 5 lb 9 oz, subtract 6−5=1 lb and 11−9=2 oz → difference is 1 lb 2 oz.

📏 Length examples

The excerpt also provides two exercises with tables (exercises 25–26):

• Adding lengths: Two tables (some ft + in each) placed end to end → add feet, add inches, carry over if inches ≥12.
• Subtracting lengths: Difference in length → subtract feet, subtract inches, borrow if needed.
• The procedure is identical to weight; only the units change.

🧠 Key analogy: mixed numbers

🧠 Why the analogy helps

• The excerpt states: "it may help to think of them as mixed numbers, with a whole number part and a fractional part."
• Whole number part = larger unit (feet, pounds, gallons).
• Fractional part = smaller unit (inches, ounces, fluid ounces).
• This mental model lets you apply the same arithmetic rules you already know for mixed numbers.

🚫 Don't confuse

• Don't treat mixed units as simple decimals or try to add/subtract without aligning units.
• You must handle the two unit parts separately and convert/carry/borrow as needed, just as you would with the whole and fractional parts of a mixed number.

🧭 Overview

🧠 One-sentence thesis

The metric system simplifies unit conversion by using prefixes based on powers of 10, allowing conversions through simple decimal point movements rather than complex multiplication.

📌 Key points (3–5)

• Why the metric system exists: designed to be based on the natural world and use powers of 10 instead of irregular numbers like 12, 16, or 5,280.
• How conversion works: all metric conversions are done by moving the decimal point, no calculator needed.
• Common prefixes: kilo- (thousand), centi- (hundredth), milli- (thousandth) relate to the base unit in predictable ways.
• Common confusion: the prefix order matters—moving from larger to smaller units (e.g., meters to centimeters) moves the decimal right; smaller to larger moves it left.
• Base units: meter for length, gram for mass—the same prefix system applies to both.

📏 The meter and its prefixes

📏 What a meter is

The base unit of length is the meter, which is a bit longer than a yard (three feet).

• The meter serves as the foundation for all metric length measurements.
• All other length units are defined by their relationship to the meter using prefixes.

🔢 Key length prefixes

Prefix	Meaning	Relationship to meter	Real-world comparison
kilo- (km)	one thousand	1 kilometer = 1,000 meters	Around six tenths of a mile
centi- (cm)	one hundredth	1 centimeter = 1/100 meter; 1 meter = 100 centimeters	Roughly the thickness of a pen
milli- (mm)	one thousandth	1 millimeter = 1/1000 meter; 1 meter = 1,000 millimeters	Roughly the thickness of a credit card

🎯 Choosing appropriate units

The excerpt provides guidance on selecting reasonable measurements:

• Car length: meters
• Notebook height: centimeters
• Distance to next town: kilometers
• Adult height: meters or centimeters
• Glass pane thickness: millimeters

Don't confuse: The same object can be measured in different units; choose the one that gives a practical number (not too large, not too small).

🔄 Converting between metric units

🔄 The prefix table system

The excerpt presents a systematic table:

kilo- (k) → hecta- (h) → deka- (da) → [base unit] → deci- (d) → centi- (c) → milli- (m)

• Each step in the table represents one decimal place movement.
• Moving right (toward smaller units) = move decimal point right.
• Moving left (toward larger units) = move decimal point left.

Note: deka- is abbreviated "da" (not just "d") to avoid confusion with deci-.

✨ Why it's easy

• The excerpt emphasizes: "converting between metric units incredibly easy because all we need to do is move the decimal point."
• No multiplication or division by irregular numbers (like 12, 16, 5,280) is needed.
• Example: Converting 5.83 meters to centimeters requires moving two places right (meter → deci- → centi-), giving 583 centimeters.

🧮 Conversion examples from the excerpt

The excerpt provides practice scenarios:

• Converting a truck length (meters) to centimeters and millimeters
• Converting kilometers to meters and centimeters (using the approximation that one mile ≈ 1.609 kilometers)
• Converting paper length (millimeters) to meters and centimeters
• Converting building height (meters) to kilometers and dekameters

⚖️ Mass measurements with grams

⚖️ The gram as base unit

The base unit for mass is the gram, which is about the mass of a paper clip.

• A kilogram is 1,000 grams (approximately 2.2 pounds, as mentioned for the next module).
• Milligrams are used for measuring active ingredients in medicines.

🔁 Same conversion system

The excerpt provides an identical prefix table for mass:

kilo- (kg) → hecta- (hg) → deka- (dag) → gram (g) → deci- (dg) → centi- (cg) → milli- (mg)

• The excerpt explicitly states: "This table is identical to the previous table; the only difference is that the base unit 'meter' has been replaced by 'gram'."
• This means converting metric units of mass uses exactly the same decimal-point-movement process as length.

📦 Practical mass examples

The excerpt gives real-world scenarios:

• Apple mass: grams
• Adult man mass: kilograms
• Medicine active ingredient: milligrams
• Vehicle weight: kilograms
• Flour bag conversions (kilograms to grams and milligrams)
• Beverage contents (grams to centigrams, milligrams, and kilograms)
• Sodium content (milligrams to centigrams and grams)

Don't confuse: Weight and mass are used somewhat interchangeably in the excerpt, though technically they differ; the metric system measures mass in grams.

🧭 Overview

🧠 One-sentence thesis

The metric system uses the gram as the base unit for mass, and converting between metric mass units requires only moving the decimal point because all units are related by powers of ten.

📌 Key points (3–5)

• Base unit: the gram is the fundamental unit of mass, roughly equal to the mass of a paper clip.
• Common units: kilogram (1,000 grams) for larger masses, milligram for tiny amounts like medicine ingredients.
• Conversion method: identical to length conversions—simply move the decimal point left or right using the prefix table.
• Common confusion: the conversion table for mass (gram-based) is structurally identical to the length table (meter-based); only the base unit name changes.
• Practical applications: everyday objects range from milligrams (medicine) to kilograms (people, food).

📏 The gram and its relatives

📏 What is a gram

Gram: the base unit for mass in the metric system, approximately the mass of a paper clip.

• The gram serves as the reference point for all other mass measurements.
• It is a relatively small unit, suitable for measuring lightweight objects.

⚖️ Kilogram and milligram

Kilogram (kg)

• Equals 1,000 grams (the prefix kilo- means one thousand).
• Used for measuring heavier objects: people, bags of flour, vehicles.
• Example: an adult man's mass is measured in kilograms, not grams or milligrams.

Milligram (mg)

• Equals one thousandth of a gram (the prefix milli- means one thousandth).
• Commonly used for active ingredients in medicines.
• Example: the amount of active ingredient in a pain relief pill is measured in milligrams.

🔄 Converting metric mass units

🔄 The prefix table for mass

kilo- (kg)	hecta- (hg)	deka- (dag)	gram (g)	deci- (dg)	centi- (cg)	milli- (mg)

• This table is identical to the length conversion table; only the base unit changes from "meter" to "gram."
• Converting metric units of mass uses exactly the same process as converting metric units of length.

🔢 How to convert by moving the decimal

• Count the number of places between the starting unit and the target unit in the table.
• Move the decimal point that many places in the same direction.
• No calculator required—the metric system is designed for easy mental conversion.

Example: Converting 2.3 kilograms to grams requires moving three places to the right (kilo → hecta → deka → gram), so 2.3 kg = 2,300 g.

🎯 Choosing appropriate units

🎯 Matching units to real-world objects

The excerpt provides guidance on selecting reasonable measurements:

Object	Appropriate unit	Why
Apple	grams	Medium-sized everyday object
Adult man	kilograms	Large mass, thousands of grams
Medicine ingredient	milligrams	Very small, precise amount
Pickup truck	kilograms	Very large mass

🚫 Don't confuse unit scales

• Using milligrams for a person's mass or kilograms for medicine would be unreasonable.
• The metric prefix system allows you to choose the most practical unit for the scale of what you're measuring.
• Example: a five-pound bag of flour is about 2.3 kilograms, not 2,300,000 milligrams (even though they're equivalent).

🧭 Overview

🧠 One-sentence thesis

The liter is the base unit of volume in the metric system, and converting between volume units follows the same decimal-shift pattern as length and mass conversions.

📌 Key points (3–5)

• Base unit: the liter is slightly larger than one quart; the milliliter is also commonly used (1,000 milliliters = 1 liter).
• Visual reference: one liter equals the volume of a cube with 10 cm sides.
• Connection across units: volume, length, and mass are linked—1 cubic centimeter = 1 milliliter, and 1 milliliter of water = 1 gram.
• Conversion method: use the same prefix table (kilo-, hecto-, deka-, base, deci-, centi-, milli-) and move the decimal point left or right.
• Common confusion: don't confuse centiliters (cL) with milliliters (mL)—they differ by one decimal place.

📏 The liter and its relationship to other units

📦 What is a liter?

Liter: the base unit of volume in the metric system, slightly larger than one quart.

• The liter is the starting point for all metric volume measurements.
• It is not a tiny unit; it's comparable to everyday liquid containers (e.g., a large bottle of soda).

🧊 Visual model: the 10 cm cube

• One liter is equivalent to a cube with each side measuring 10 centimeters.
• This gives a concrete spatial sense: 10 cm × 10 cm × 10 cm = 1 liter.
• Example: imagine a box that fits in your hand—that's roughly one liter.

🔗 How volume, length, and mass connect

The excerpt explains that metric units are interconnected:

Unit type	Measure	Relationship
Volume	1 cubic centimeter	= 1 milliliter
Volume	1 milliliter	(of water)
Mass	1 gram	= mass of 1 mL of water

• This means you can move between volume and mass for water without separate conversion factors.
• Example: 500 mL of water weighs 500 grams.

🔄 Converting metric volume units

📊 The prefix table for volume

The excerpt provides the same table structure used for length and mass:

kilo- (kL)	hecta- (hL)	deka- (daL)	liter (L)	deci- (dL)	centi- (cL)	milli- (mL)

• Each step left multiplies by 10; each step right divides by 10.
• The table is "identical to the previous tables; just move the decimal point left or right to convert the units."

➡️ How to convert: move the decimal

• To convert to a smaller unit (e.g., liters → milliliters): move the decimal point to the right.
• To convert to a larger unit (e.g., centiliters → liters): move the decimal point to the left.
• No calculator required for these conversions—just count the steps in the table.

Example from the excerpt:

• 50 centiliters to liters: move two places left → 0.50 liters.
• 1.75 liters to milliliters: move three places right → 1,750 mL.

🧪 Practical scenarios

The excerpt includes real-world contexts:

• Car gas tank: measured in liters (not milliliters).
• Liquid cough medicine dosage: measured in milliliters (not liters).
• Sparkling water bottle: labeled 50 cL (centiliters).
• Saline solution bag: labeled 1.05 liters, convertible to deciliters.
• Orange juice carton: 1.75 liters = 1,750 mL.
• One cup (8 fluid ounces): approximately 250 milliliters = 0.25 liters.

Example: Two mini bottles on IcelandAir—one labeled 50 mL, another 5 cL. Since 5 cL = 50 mL, the two bottles are the same size.

🔍 Common confusions and comparisons

⚠️ Centiliters vs milliliters

• Don't confuse cL and mL—they are one step apart on the table.
• 1 cL = 10 mL.
• Example: 50 mL and 5 cL are equivalent (both equal 0.05 liters).

🔢 Comparing volumes across units

The excerpt asks about engine displacement:

• Yamaha Majesty scooter: 125 cc (cubic centimeters).
• Chevrolet Spark: 1.4 L (liters).
• Since 1 cc = 1 mL and 1 L = 1,000 mL, 1.4 L = 1,400 cc.
• Ratio: 1,400 ÷ 125 ≈ 11:1.

🥤 Bottle equivalence

• How many 500 mL bottles equal one 2-liter bottle?
• 2 liters = 2,000 mL.
• 2,000 ÷ 500 = 4 bottles.

🧭 Overview

🧠 One-sentence thesis

Converting between U.S. and metric length units requires using specific conversion ratios, some exact and some approximate, with attention to significant figures to ensure consistent results.

📌 Key points (3–5)

• Exact vs approximate ratios: some conversions (inch to cm, foot to m, yard to m) are exact because the yard is defined as exactly a certain number of meters; mile-to-kilometer conversions are approximate (rounded to four significant figures).
• How to convert: write the measurement as a fraction over 1, then multiply by the conversion ratio so units cancel.
• Rounding matters: as long as you don't try to keep more than four significant figures in your result, different conversion ratios (e.g., ft to m or m to ft) will give the same rounded answer.
• Common confusion: using both directions of a ratio (e.g., ft to m vs m to ft) should yield consistent results when properly rounded—verify conversions by working backward.
• Why it matters: the metric system is increasingly used in a global economy, so converting accurately between systems is essential.

📏 Conversion ratios for length

📐 Exact conversions

The excerpt states that these ratios are exact because a yard is defined to be exactly a certain number of meters:

• 1 inch = (some value) cm (exact)
• 1 foot = (some value) cm (exact)
• 1 foot = (some value) m (exact), or the reciprocal m to ft (exact)
• 1 yard = (some value) m (exact), or the reciprocal m to yd (exact)

The excerpt notes that a foot is exactly a certain number of meters and an inch is exactly a certain number of centimeters because of the definition of the yard.

🌍 Approximate conversions

• 1 mile ≈ (some value) km or km to mi (approximate)
• These are rounded to four significant figures.

Don't confuse: exact ratios come from definitions; approximate ratios are rounded measurements.

🔄 How to perform conversions

🔢 The cancellation method

The excerpt demonstrates the process:

1. Write the measurement as a fraction over 1.
2. Multiply by the conversion ratio so that units cancel.
3. Calculate and round to the appropriate number of significant figures.

Example (from the excerpt): To verify a conversion, start with the measurement as a fraction, use the conversion ratio to cancel units, then round to three significant figures to check consistency.

✅ Verifying conversions

The excerpt shows that you can verify a conversion by using both directions of a ratio:

• Convert using one ratio (e.g., ft to m).
• Convert using the reciprocal ratio (e.g., m to ft).
• Both should give results that round to the same value.

Example (from the excerpt): The text verifies a conversion by first using one ratio, rounding to three significant figures, then using the approximate reciprocal ratio. Both conversions give results that round to the same value, confirming accuracy.

Don't confuse: different ratios are acceptable as long as rounding is consistent—"as long as we don't try to keep more than four sig figs in our result, we can use either conversion ratio and get the same result."

🧮 Significant figures and rounding

🎯 The four-sig-fig rule

The excerpt states: "The numbers in these conversion ratios are usually difficult to work with, so we will use a calculator whenever necessary and pay attention to rounding."

• Conversion ratios that are not exact are rounded to four significant figures.
• When performing conversions, do not try to keep more than four significant figures in your result.
• Using a calculator is recommended because the numbers are difficult to work with.

🔁 Consistency through rounding

• If you round properly, using either direction of a conversion ratio (e.g., ft to m or m to ft) will yield the same final answer.
• The excerpt emphasizes checking that results "round to" the expected value, not that they are exactly equal before rounding.

Example (from the excerpt): After converting and rounding to three significant figures, "everything looks good; both conversions give us a result that rounds to [the same value]."

🌐 Why conversions matter

🌍 Global economy context

The excerpt states:

"Converting between the U.S. system and metric system is important in today's global economy; like it or not, the metric system is infiltrating our lives."

• The metric system is increasingly present in everyday life.
• Accurate conversion is necessary for international communication and commerce.

📝 Practical advice

• "If you discover other conversion ratios that aren't provided here, write them down!"
• The excerpt encourages building a personal reference of useful ratios.

🧭 Overview

🧠 One-sentence thesis

Converting weight or mass between U.S. and metric systems requires approximate conversion ratios rounded to four significant figures, and these conversions are essential for practical applications in today's global economy.

📌 Key points (3–5)

• Conversion ratios are approximate: unlike some length conversions that are exact, weight/mass conversions use rounded values (four significant figures).
• Two main conversion ratios: ounces to grams (oz ≈ g) and kilograms to pounds (kg ≈ lb or lb ≈ kg).
• Practical applications: verifying product labels, converting recipes, and calculating real-world values like the worth of gold coatings.
• Common confusion: the exact conversion (1 pound = 0.45359237 kilograms) exists but is impractical; rounded four-significant-figure values are used instead.
• Calculator use encouraged: the numbers in these conversions are difficult to work with, so calculators are recommended throughout.

📏 Understanding the conversion ratios

📏 Why these conversions are approximate

• The excerpt explicitly states: "These conversions are approximate."
• Technically, one pound equals exactly 0.45359237 kilograms, but this level of precision is impractical.
• The material uses four significant figures as the standard for rounding.
• Don't confuse: "approximate" doesn't mean inaccurate for everyday use—it means rounded to a practical level of precision.

📏 The two key conversion pairs

Conversion	Ratio	Direction
Ounces ↔ Grams	oz ≈ g	Small units
Kilograms ↔ Pounds	kg ≈ lb or lb ≈ kg	Larger units

• These ratios can be used in either direction depending on what you're converting from and to.
• Example: converting 5 kilograms to pounds uses the kg ≈ lb ratio.

🛠️ Practical applications

🛠️ Verifying product labels

• Real-world scenario: A shelving system rated for a total weight in pounds also shows metric equivalents in kilograms.
• The exercises ask students to convert the pound measurements and check if they match the printed kilogram values.
• This teaches both conversion skills and quality-control thinking.

🍳 Recipe conversions

• Converting cooking measurements between systems is a common need.
• Example from exercises: converting 4 ounces of flour to grams for a gravy recipe.
• Example: converting half a pound of ground beef to grams.
• Example: converting grams of protein in a smoothie recipe to ounces.

💰 Calculating real-world values

• The excerpt includes exercises about calculating the dollar value of gold coatings on replica coins.
• Process: gold amount given in milligrams → convert to ounces → multiply by price per ounce.
• Example scenario: A coin coated in a certain number of milligrams of pure gold, with gold priced at dollars per ounce—what is the gold's actual value?
• This demonstrates why unit conversion matters for evaluating commercial claims.

🔄 Working with the conversions

🔄 Using conversion ratios as fractions

• The excerpt shows that conversion ratios can be written as fractions to cancel units.
• You write the measurement as a fraction over 1, then multiply by the conversion ratio arranged so unwanted units cancel.
• This systematic approach ensures correct unit cancellation.

🔄 Rounding and significant figures

• Pay attention to rounding throughout the conversion process.
• The material consistently uses four significant figures for the conversion ratios themselves.
• Final answers should be rounded appropriately based on the precision of the original measurement.
• Don't confuse: keeping more than four significant figures in results can lead to false precision when using approximate ratios.

🌍 Context and importance

🌍 Why these conversions matter

"Converting between the U.S. system and metric system is important in today's global economy; like it or not, the metric system is infiltrating our lives."

• The excerpt frames this as a practical necessity, not just an academic exercise.
• Cross-border commerce (example: buying supplies in Quebec while living in Vermont) requires these conversions.
• International standards (example: TSA airline liquid limits in milliliters) require U.S. travelers to understand metric measurements.

🌍 When to use calculators

• The excerpt explicitly permits and encourages calculator use throughout the module.
• Reason: "The numbers in these conversion ratios are usually difficult to work with."
• This is practical math—accuracy matters more than mental arithmetic speed.

🧭 Overview

🧠 One-sentence thesis

Volume and capacity conversions between metric and customary units rely on approximate ratios rounded to four significant figures, enabling practical calculations for everyday scenarios like purchasing products, travel, and following regulations.

📌 Key points (3–5)

• Core conversion ratios: fluid ounces to milliliters, liters to fluid ounces, liters to quarts, and gallons to liters are the fundamental relationships.
• Approximation nature: all conversion ratios are approximations rounded to four significant figures, not exact values.
• Practical applications: conversions appear in consumer products (toilets, beverages), travel regulations (TSA liquid limits), automotive contexts (motor oil, fuel efficiency), and cross-border shopping.
• Common confusion: different countries use different spellings and units—Canada uses "litres" and "kilometres" rather than "liters" and "kilometers."
• Rate conversions: some problems require converting compound rates (e.g., miles per gallon to kilometres per litre), not just simple quantities.

📏 Fundamental conversion ratios

📏 The four key relationships

The excerpt provides these approximate conversions:

From	To	Conversion
fluid ounces	milliliters	1 fl oz ≈ 29.57 mL
liters	fluid ounces	1 L ≈ 33.81 fl oz
liters	quarts	1 L ≈ 1.057 qt
gallons	liters	1 gal ≈ 3.785 L

• All values are rounded to four significant figures.
• These are not exact conversions but practical approximations for everyday use.

🔢 Why approximations matter

• The excerpt emphasizes that "these conversion ratios are approximations."
• Four significant figures provide sufficient precision for most practical purposes without unnecessary complexity.
• Example: when verifying product labels or following regulations, this level of precision is adequate.

🛒 Consumer and travel applications

🚽 Product specifications

• Real-world products display both metric and customary measurements.
• Example: a toilet uses 4.8 liters of water per flush—consumers may need to convert this to gallons to understand water usage in familiar terms.
• Cross-border shopping requires conversions when product specifications use different unit systems.

✈️ Regulatory compliance

• TSA airline regulations limit carry-on liquids to 100 milliliters.
• Travelers need to convert this limit to fluid ounces to understand how much they can pack.
• The excerpt instructs rounding "to the nearest tenth" for practical use.

🥤 Beverage containers

• Common container sizes are given in one system but may need conversion.
• Example: a 12-ounce can contains a certain number of milliliters—useful for comparing serving sizes across different labeling systems.

🚗 Automotive and rate conversions

🛢️ Motor oil and fuel

• Large quantities like motor oil are sold in both gallons and liters.
• Example: verifying that 6 gallons of motor oil equals 22.7 liters helps confirm product labeling accuracy.
• Gas stations in different countries measure fuel differently (gallons vs. litres).

🧮 Compound rate conversions

• Some conversions involve rates, not just simple quantities.
• Example scenario from the excerpt: a car gets 26 miles per gallon—when traveling in Canada, this needs conversion to kilometres per litre.
• This requires converting both the numerator (miles to kilometres) and denominator (gallons to litres) separately.
• Don't confuse: converting a rate requires two separate conversions, not just one.

🗺️ Travel distance calculations

• Combining tank capacity with fuel efficiency requires multiple conversions.
• Example: a 14.2-gallon tank at 26 miles per gallon determines total travel distance—converting to kilometres requires using both distance and volume conversion ratios.

🌍 International variations

🌍 Spelling and terminology differences

• The excerpt explicitly notes Canadian spelling variations: "kilometres" instead of "kilometers" and "litres" instead of "liters."
• These are the same units with different regional spellings, not different measurement systems.
• Don't confuse: spelling differences don't change the conversion ratios—1 liter equals 1 litre.

🧭 Overview

🧠 One-sentence thesis

Converting rates (measurements with units in both numerator and denominator) requires applying conversion factors to both parts of the rate to transform units like miles per gallon into kilometres per litre or speed measurements across different unit systems.

📌 Key points (3–5)

• What rate conversion means: transforming measurements that have units in both numerator and denominator (e.g., miles per gallon, kilometers per hour).
• How to convert rates: apply conversion factors to both the numerator unit and the denominator unit separately.
• Real-world applications: fuel efficiency (mpg to km/L), speed (mph to km/h), and consumption rates (gallons per minute).
• Common confusion: rate conversions require two separate unit conversions (one for numerator, one for denominator), not just one.
• Why it matters: enables comparison and calculation across different measurement systems (e.g., driving in Canada with U.S. units).

🚗 Converting fuel efficiency rates

⛽ Miles per gallon to kilometres per litre

The excerpt presents a scenario where Maxine drives across Canada with a car that gets 26 miles per gallon, but Canada measures fuel in litres.

The conversion challenge:

• Start with: 26 miles per gallon
• Need to convert: miles → kilometres (numerator) AND gallons → litres (denominator)
• This requires two separate conversion factors applied to one rate

Why this matters:

• Different countries use different units for fuel efficiency
• Cannot directly compare "26 mpg" to Canadian fuel ratings without conversion
• Example: To understand fuel costs in Canada, Maxine must know her car's efficiency in kilometres per litre

🛣️ Distance calculations with converted units

The excerpt asks how many kilometres Maxine can travel on a full tank (14.2-gallon capacity) at 26 miles per gallon.

The approach:

• First calculate total distance in original units (miles)
• Then convert the result to kilometres
• Alternatively: convert the rate first, then multiply by converted tank capacity

✈️ Converting speed and related rates

🚀 Speed conversions across unit systems

The excerpt uses an F-fighter jet example traveling at Mach (times the speed of sound, which is given as miles per hour).

Multiple rate conversions needed:

• Top speed in miles per hour (starting point)
• Distance traveled per minute (requires converting hours to minutes in denominator)
• Flight duration calculations (distance divided by speed)

⏱️ Time-based rate conversions

When converting speed to "distance per minute," the denominator changes from hours to minutes.

Key insight:

• The same speed can be expressed in different time units
• Example: miles per hour → miles per minute requires dividing by 60 (since 60 minutes = 1 hour)
• Don't confuse: the jet's speed doesn't change, only how we express it

⛽ Fuel consumption rates

📊 Multiple ways to express efficiency

Rate type	What it measures	Example from excerpt
Miles per gallon	Distance traveled per unit fuel	Jet's fuel efficiency after flying a certain distance
Gallons per mile	Fuel consumed per unit distance	Reciprocal of miles per gallon
Gallons per minute	Fuel consumed per unit time	Combining fuel efficiency with speed

🔄 Reciprocal rates

The excerpt asks to rewrite fuel efficiency from "miles per gallon" to "gallons per mile."

Understanding reciprocals:

• These express the same relationship from opposite perspectives
• Miles per gallon: how far you go on one gallon
• Gallons per mile: how much fuel you use to go one mile
• One is the mathematical reciprocal of the other (flip the fraction)

⏲️ Combining rates for consumption over time

The excerpt asks "how many gallons of fuel does the jet consume in one minute?"

This requires combining two rates:

• Fuel efficiency (gallons per mile)
• Speed (miles per minute)
• Result: gallons per minute (a consumption rate over time)

Why this matters:

• Helps calculate total fuel needed for a timed flight
• Example: If you know consumption per minute and flight duration, you can find total fuel used

🧭 Overview

🧠 One-sentence thesis

Time conversions use standard ratios for units up to weeks, but larger units like months and years require approximations because calendars are not perfectly uniform.

📌 Key points (3–5)

• Standard time ratios: seconds, minutes, hours, days, and weeks have fixed conversion factors (60 seconds per minute, 24 hours per day, etc.).
• Approximation needed for large units: months and years don't convert exactly because months have different lengths and Earth's orbit isn't exactly 365 days.
• Rate conversions: when converting rates (units in numerator and denominator), both parts must be converted separately.
• Common confusion: expecting exact conversions for "minutes in a month" when calendar irregularities make this impossible—use best judgment instead.
• Measurement prefixes: large-scale prefixes (kilo-, mega-, giga-, tera-) follow powers of 1,000, moving the decimal point three places per step.

⏰ Basic time conversion ratios

⏰ Fixed conversion factors

The excerpt provides standard time conversions:

Unit	Conversion
1 minute	60 seconds
1 hour	60 minutes
1 day	24 hours
1 week	7 days
1 year	365 days

• These ratios are exact and consistent for units up to weeks.
• Example: To find minutes in a 365-day year, multiply 365 days × 24 hours/day × 60 minutes/hour.

🌍 Why larger units are problematic

The excerpt explains three calendar irregularities:

• Not all months have the same number of days.
• A year is not exactly a whole number of weeks.
• Earth's orbit around the Sun is not exactly 365 days.

Implication: Questions like "how many minutes are in one month?" cannot have a single exact answer—you must use best judgment (e.g., assume 30 days, or use a specific month's length).

🔄 Converting rates with multiple units

🔄 How rate conversion works

Rate conversion: when units appear in both numerator and denominator, convert each part separately.

• The excerpt notes: "Previously when we were converting units, we began with units in the numerator only. If we need to convert a rate, however, we'll begin with units in both the numerator and denominator."
• Example: Converting speed from meters per second to kilometers per hour requires converting meters to kilometers (numerator) and seconds to hours (denominator).

✈️ Multi-step rate problems

The excerpt uses an F-fighter jet example (Mach speed, fuel consumption):

• Given: top speed in miles per hour, range in miles, fuel capacity in gallons.
• You can calculate: miles per minute (distance per time), miles per gallon (distance per fuel), gallons per minute (fuel per time).
• Each calculation requires identifying which units to convert and in what order.

Don't confuse: Miles per gallon vs. gallons per mile—these are reciprocals; one measures efficiency (distance per fuel), the other measures consumption (fuel per distance).

📏 Measurement prefixes for large units

📏 The prefix scale

The excerpt provides a chart of large-unit prefixes:

Prefix	Symbol	Multiplier	Power of 10
kilo-	k	thousand	1,000
mega-	M	million	1,000,000
giga-	G	billion	1,000,000,000
tera-	T	trillion	1,000,000,000,000

• Each step is 1,000 times the previous (powers of 10³).
• Moving one place in the chart = moving the decimal point three places.
• Example: 1 megabyte = 1,000 kilobytes = 1,000,000 bytes.

🔤 Capitalization matters

The excerpt warns: "capitalization is important."

• Megagram (Mg) with capital M = one million grams (also called a metric ton).
• Milligram (mg) with lowercase m = one thousandth of a gram.
• Don't confuse: M vs. m represent completely different scales (mega- vs. milli-).

💾 Computer memory note

The excerpt acknowledges an inconsistency:

• Computer memory is counted in powers of 2, not 10.
• Engineers originally defined 1 kilobyte as 1,024 bytes (2¹⁰) because it's close to 1,000.
• For this material, treat prefixes as powers of 1,000, not 1,024.

🔢 Applying conversions to real scenarios

🔢 Storage capacity comparisons

Example from the excerpt: comparing floppy disk storage (kilobytes in the 1980s vs. megabytes in the 1990s) or hard drives (gigabytes vs. terabytes).

• To find "by what factor" capacity increased, divide the larger unit by the smaller after converting to the same unit.
• Example: A terabyte hard drive vs. a gigabyte hard drive—since 1 TB = 1,000 GB, the factor is 1,000 times greater.

⚡ Power and resource rates

The excerpt includes nuclear reactor examples:

• Megawatts of electricity per number of homes → watts per home.
• Megawatts per gallons of water per day → watts per gallon.
• These require converting the prefix (mega- to base unit) and then dividing by the other quantity.

🧭 Overview

🧠 One-sentence thesis

Converting rates requires applying conversion factors to both the numerator and denominator units, enabling us to express speeds, fuel efficiency, and other ratios in different measurement systems.

📌 Key points (3–5)

• What makes rate conversion different: rates have units in both numerator and denominator, unlike simple unit conversions that have units only in the numerator.
• How to convert rates: apply appropriate conversion factors to each part of the rate separately.
• Real-world applications: speed conversions (meters per second to miles per hour), fuel efficiency (miles per gallon to gallons per mile), and power rates (watts per home, watts per gallon).
• Common confusion: don't confuse converting a single measurement with converting a rate—rates require handling two dimensions simultaneously.
• Measurement prefixes matter: large-scale conversions use prefixes (kilo-, mega-, giga-, tera-) where each step represents a factor of 1,000.

🔄 What makes rates different

🔄 Rates vs simple measurements

• The excerpt emphasizes that "Previously when we were converting units, we began with units in the numerator only."
• Rates are different: "If we need to convert a rate, however, we'll begin with units in both the numerator and denominator."
• Example: Converting 100 meters is simpler than converting 100 meters per second, because the rate includes both distance and time.

📐 Why both parts matter

• A rate expresses a relationship between two quantities.
• Both the numerator (what you're measuring) and denominator (what you're measuring it against) may need conversion.
• Don't confuse: changing only one part of a rate gives you a different type of measurement entirely.

🏃 Speed and distance rate conversions

🏃 Converting speed measurements

The excerpt provides a worked example with Usain Bolt's 100-meter dash:

• Starting point: 100 meters in 9.58 seconds
• The exercises ask to convert this to kilometers per hour and miles per hour
• This requires converting both the distance unit (meters) and the time unit (seconds)

✈️ Multi-step rate problems

The F-15 fighter jet example shows how rates connect:

• Top speed given as Mach 1.8 (1.8 times the speed of sound, which is 767 miles per hour)
• From speed, you can calculate distance traveled in a given time
• From distance and fuel consumed, you can calculate fuel efficiency (miles per gallon)
• Fuel efficiency can be inverted (gallons per mile)
• Combining rates gives fuel consumption per time (gallons per minute)

The excerpt notes: "The more information we know, the more things we can figure out."

⛽ Reciprocal rates

⛽ Flipping numerator and denominator

The jet fuel example demonstrates converting between two forms:

• Miles per gallon (distance per fuel)
• Gallons per mile (fuel per distance)

These are reciprocals of each other—they express the same relationship from opposite perspectives.

🔁 When to use each form

• Miles per gallon: useful for understanding range and efficiency
• Gallons per mile: useful for calculating fuel consumption for a specific distance
• Example: If you know gallons per mile and miles traveled, you can directly calculate total fuel used.

📏 Measurement prefixes for large units

📏 The prefix hierarchy

The excerpt provides a systematic table of large-unit prefixes:

Prefix	Symbol	Multiplier	Numeric value
tera-	T	trillion	1,000,000,000,000
giga-	G	billion	1,000,000,000
mega-	M	million	1,000,000
kilo-	k	thousand	1,000
[base]	—	one	1

🔢 Powers of 1,000

• Each prefix is 1,000 times the next smaller prefix
• Moving one place in the chart = moving the decimal point three places
• The excerpt notes these are "multiples of 3, just as with the engineering notation"

💾 Computer memory example

The excerpt uses bytes to illustrate:

• 1 kilobyte = 1,000 bytes
• 1 megabyte = 1,000 kilobytes = 1,000,000 bytes
• 1 gigabyte = 1,000 megabytes = 1,000,000 kilobytes
• 1 terabyte = 1,000 gigabytes = 1,000,000 megabytes

Important note: The excerpt acknowledges that computer memory is actually counted in powers of 2, not 10, but states "we will consider these prefixes to be powers of 10, not 2" for consistency.

⚡ Power and production rates

The nuclear reactor example shows prefix use in rates:

• "One module will produce 77 megawatts of electricity"
• This can be converted to electricity per home (watts per home)
• Another rate: "a 77 megawatt module could produce about 20 million gallons of clean water per day"
• This gives watts per gallon when combined

Don't confuse: Capitalization matters—megagram (Mg) with capital M versus milligram (mg) with lowercase m are vastly different units.

⚠️ Approximations and context

⚠️ When exact conversions aren't possible

The excerpt notes challenges with time conversions:

• "Not all months have the same number of days"
• "A year is not exactly 52 weeks"
• "The time it takes for the Earth to orbit the Sun is not exactly 365 days"

Therefore: "It doesn't make sense to expect an exact answer to a question like 'how many minutes are in one month?' We will have to use our best judgment in situations such as these."

🎯 Rounding and significant figures

• The excerpt states "conversion ratios are approximations rounded to four significant figures"
• This reminds us that conversions carry inherent approximation
• Precision depends on the context and the original measurement's precision

🧭 Overview

🧠 One-sentence thesis

Measurement prefixes for large units (kilo-, mega-, giga-, tera-) follow a consistent pattern where each step represents a factor of 1,000, making it straightforward to convert between units by moving the decimal point three places.

📌 Key points (3–5)

• The prefix ladder: tera- (trillion), giga- (billion), mega- (million), kilo- (thousand), then the base unit—each step is 1,000 times the next smaller prefix.
• Powers of 10 in multiples of 3: these prefixes correspond to engineering notation, with exponents that are multiples of 3.
• Decimal point rule: moving one place in the chart equals moving the decimal point three places.
• Common confusion: capitalization matters—Mg (megagram) vs. mg (milligram) are very different units.
• Computer memory caveat: there can be inconsistencies because computer memory uses powers of 2, not 10, though the excerpt treats prefixes as powers of 10.

📏 The prefix system

📊 The four large prefixes

The excerpt presents four prefixes for large measurements:

Prefix	Symbol	Meaning	Power of 10
tera-	T	trillion	1,000,000,000,000
giga-	G	billion	1,000,000,000
mega-	M	million	1,000,000
kilo-	k	thousand	1,000

• Each prefix is exactly 1,000 times the next smaller prefix.
• The base unit (meter, gram, byte, etc.) comes after kilo-.

🔢 Powers of 10 in multiples of 3

• These prefixes align with engineering notation from an earlier module.
• The exponents increase or decrease by 3 at each step.
• This pattern makes calculations more systematic than arbitrary conversion factors.

🔄 Converting between prefixes

↔️ The three-place rule

Moving one place in the chart is equivalent to moving the decimal point three places.

• Going up the chart (toward larger prefixes) = divide by 1,000 = move decimal left 3 places.
• Going down the chart (toward smaller prefixes) = multiply by 1,000 = move decimal right 3 places.
• Example: 1 megabyte = 1,000 kilobytes = 1,000,000 bytes.

🔠 Capitalization is critical

• Don't confuse: Mg (megagram, capital M) vs. mg (milligram, lowercase m).
• Megagram is also called a metric ton.
• The case of the letter changes the meaning by a factor of one billion.

💾 Computer memory example

💾 Byte hierarchy

The excerpt uses computer memory to illustrate the prefix system:

• 1 kilobyte = 1,000 bytes
• 1 megabyte = 1,000 kilobytes = 1,000,000 bytes
• 1 gigabyte = 1,000 megabytes = 1,000,000 kilobytes, etc.
• 1 terabyte = 1,000 gigabytes = 1,000,000 megabytes, etc.

⚠️ The powers-of-2 caveat

• Computer engineers originally defined 1 kilobyte as 1,024 bytes (which is 2 to the 10th power).
• This happened because 1,024 is very close to 1,000.
• Important: the excerpt states it will treat these prefixes as powers of 10, not powers of 2, for consistency.
• There can be inconsistencies in how different people understand these prefixes for computer memory.
• A reference is provided at physics.nist.gov for further explanation.

🧮 Real-world applications

🧮 Storage capacity comparisons

• Example from exercises: comparing floppy disk storage (kilobytes in the 1980s vs. megabytes in the 1990s) to modern hard drives (terabytes).
• The prefix system allows quick calculation of how many times greater one capacity is than another.

⚡ Energy and power measurements

• Example: nuclear reactor modules measured in megawatts.
• Example: nuclear weapon destructive power measured in kilotons (thousands of tons of TNT) or megatons (millions of tons of TNT).
• The prefix system applies across many scientific and engineering domains, not just computer memory.

🧭 Overview

🧠 One-sentence thesis

Smaller measurement prefixes (milli-, micro-, nano-, pico-) decrease by factors of 1,000 each step, and converting between them requires moving the decimal point three places per prefix level.

📌 Key points (3–5)

• What the prefixes represent: milli- = thousandth, micro- = millionth, nano- = billionth, pico- = trillionth of the base unit.
• How the pattern works: each prefix is 1,000 times smaller than the previous one, just like the larger prefixes (kilo-, mega-, giga-, tera-) but in the opposite direction.
• Common confusion: negative exponents can be tricky—it helps to think in reverse (e.g., 1 meter = 1,000 millimeters rather than 1 millimeter = 0.001 meters).
• Symbol notation: micro- uses the Greek letter μ (mu), but may appear as "u" or "mc" in web or plaintext contexts.
• Why it matters: these prefixes are essential for expressing very small quantities in fields like networking, radiation measurement, and wavelength analysis.

📏 The prefix scale

📏 Four smaller prefixes

The excerpt presents four prefixes for quantities smaller than the base unit:

Prefix	Symbol	Meaning	Power of 10
milli-	m	thousandth	one divided by 1,000
micro-	μ (or mc)	millionth	one divided by 1,000,000
nano-	n	billionth	one divided by 1,000,000,000
pico-	p	trillionth	one divided by 1,000,000,000,000

• Each step down the table divides by 1,000.
• The pattern mirrors the larger prefixes (kilo-, mega-, giga-, tera-) but goes in the opposite direction.

🔤 Symbol notation for micro-

The symbol for micro- is the Greek letter μ (pronounced "myoo").

• Because μ can be difficult to type or display, you may see "u" or "mc" used instead in web-based or plaintext technical articles.
• Example: micrometer might appear as "μm," "um," or "mcm" depending on the context.

🔄 Converting between prefixes

🔄 The 1,000× rule

• Each prefix is 1,000 times (or one-thousandth) the next prefix.
• Moving one place in the chart = moving the decimal point three places.
• This is the same pattern as the larger prefixes, just in the opposite direction.

🔄 Thinking in reverse

The excerpt notes that negative exponents can be complicated, so it helps to think about conversions in reverse:

• 1 meter = 1,000 millimeters = 1,000,000 micrometers = 1,000,000,000 nanometers = 1,000,000,000,000 picometers
• 1 second = 1,000 milliseconds = 1,000,000 microseconds = 1,000,000,000 nanoseconds = 1,000,000,000,000 picoseconds

Don't confuse: Instead of thinking "1 millimeter = 0.001 meters" (which involves a decimal), think "1 meter = 1,000 millimeters" (which uses whole numbers and is easier to work with mentally).

Example: To convert 5 milliseconds to microseconds, recognize that 1 millisecond = 1,000 microseconds, so 5 milliseconds = 5,000 microseconds.

🧪 Real-world applications

🧪 Network latency

The excerpt mentions an article comparing network latency times:

• A certain link adds milliseconds to round-trip time
• Another link adds microseconds
• The exercise asks to rewrite all times in the same unit (either all milliseconds or all microseconds)

This shows why understanding prefix conversions matters: comparing quantities requires a common unit.

🧪 Wavelengths

The excerpt gives two examples:

• Red light wavelength: around nanometers
• Infrared radiation wavelength: approximately micrometers

The exercise asks for the ratio of these wavelengths, which requires converting to the same prefix first.

🧪 Radiation measurement

Nuclear radiation is measured in units called Sieverts, but because this unit is too large to be practical when discussing people's exposure to radiation, milliSieverts and microSieverts are more commonly used.

• Chernobyl cleanup workers (1986): exposed to an estimated dose in milliSieverts
• Typical chest x-ray: exposes a person to microSieverts
• The exercise asks how many chest x-rays' worth of radiation the workers received, requiring conversion between milliSieverts and microSieverts.

Example: If workers received 100 milliSieverts and one x-ray is 100 microSieverts, first convert: 100 milliSieverts = 100,000 microSieverts. Then divide: 100,000 ÷ 100 = 1,000 x-rays' worth.

🧭 Overview

🧠 One-sentence thesis

Angle measurement is essential in many fields, and understanding how to classify angles, apply geometric rules, and convert between measurement systems enables practical problem-solving in construction, surveying, and other technical work.

📌 Key points (3–5)

• What an angle is: either the figure formed when two line segments share a common endpoint, or a measure of rotation.
• Classification by size: angles are categorized as acute, right, obtuse, straight, or reflexive based on their degree measure relative to benchmarks (90°, 180°, 360°).
• Key geometric rules: triangle angles always sum to 180°; parallel lines crossed by a third line create predictable angle patterns.
• Common confusion: angles can be measured in decimal degrees or in the DMS (degrees-minutes-seconds) system, where 1 minute = 1/60 of a degree and 1 second = 1/60 of a minute.
• Why it matters: angle measurement applies to construction, surveying, physical therapy, and navigation (latitude/longitude).

📐 Understanding angles

📐 Two ways to visualize an angle

The excerpt presents two complementary definitions:

• Geometric figure: the shape formed when two line segments share a common endpoint.
• Rotation measure: how much turning has occurred.

Example: A full rotation around a circle is 360°, so a half rotation (U-turn) is 180°, and a quarter turn is 90°.

🏷️ Angle classifications

Angles are named based on their size relative to key benchmarks:

Type	Measure	Description
Acute	Between 0° and 90°	Smaller than a right angle
Right	Exactly 90°	Quarter turn; perpendicular lines
Obtuse	Between 90° and 180°	Larger than right, smaller than straight
Straight	Exactly 180°	Half rotation; forms a line
Reflexive	Between 180° and 360°	More than half rotation

⊥ Perpendicular lines

Lines that form a 90° angle are called perpendicular.

• The excerpt gives a medical example: a needle should be perpendicular to the body surface for an intramuscular injection.
• This means the needle forms a right angle with the skin.

🔺 Angles in triangles

🔺 The triangle angle sum rule

The sum of the angles of every triangle is 180°.

• This is a fundamental property that applies to every triangle, regardless of shape.
• If you know two angles, you can always find the third by subtracting from 180°.

Example: If a triangle has angles of 60° and 70°, the third angle must be 180° - 60° - 70° = 50°.

⚖️ Equal sides create equal angles

If any sides of a triangle have equal lengths, then the angles opposite those sides will have equal measures.

• This rule helps solve for unknown angles in isosceles or equilateral triangles.
• "Opposite" means the angle across from a particular side.

∥ Parallel lines and angles

∥ What parallel lines are

Two lines that point in the exact same direction and will never cross are called parallel lines.

🔀 Angle patterns when a line crosses parallel lines

When a third line crosses two parallel lines, predictable angle relationships emerge:

• All four acute angles formed will be equal in measure.
• All four obtuse angles formed will be equal in measure.
• Any acute angle and any obtuse angle will have a combined measure of 180°.

This pattern allows you to find multiple unknown angles if you know just one angle measure.

🕐 The DMS measurement system

🕐 What DMS means

The degrees-minutes-seconds or DMS system divides a degree using fractions out of 60.

• 1 minute = 1/60 of a degree
• 1 second = 1/60 of a minute, which means 1/3600 of a degree
• These units work exactly like time (think of 1 degree as 1 hour)

🔄 Converting DMS to decimal degrees

The excerpt describes the conversion process:

• Enter the calculation: degrees + (minutes divided by 60) + (seconds divided by 3600)
• Round the result to the fourth decimal place if necessary

Example: The excerpt mentions latitude/longitude in DMS format like N [degrees] [minutes] [seconds] W [degrees] [minutes] [seconds], which can be converted to decimal form for use in modern mapping systems.

Don't confuse: DMS is an older method; decimal degrees are more common now, but both systems measure the same thing—just with different notation.

🧭 Overview

🧠 One-sentence thesis

The sum of all angles in any triangle always equals 180°, and when sides are equal in length, their opposite angles are also equal in measure.

📌 Key points (3–5)

• The fundamental rule: every triangle's three angles add up to exactly 180°.
• Equal sides mean equal angles: if a triangle has sides of equal length, the angles opposite those sides have equal measures.
• How to find unknown angles: use the 180° rule and the equal-sides-equal-angles property to calculate missing angle measurements.
• Common confusion: don't assume all triangles have equal angles—only the sum is always the same (180°); individual angle sizes vary by triangle type.

📐 The core triangle angle rule

📐 Sum of angles equals 180°

The sum of the angles of every triangle is 180°.

• This is true for every triangle, regardless of shape or size.
• It means if you know two angles, you can always find the third by subtracting from 180°.
• Example: if a triangle has angles of 60° and 70°, the third angle must be 180° - 60° - 70° = 50°.

🔍 Why this matters

• The 180° rule is the foundation for solving unknown angles in triangles.
• You only need partial information (one or two angles) to determine all angles.
• This property applies whether the triangle is acute, obtuse, right, equilateral, isosceles, or scalene.

🔗 The equal-sides-equal-angles relationship

🔗 When sides match, angles match

If any sides of a triangle have equal lengths, then the angles opposite those sides will have equal measures.

• "Opposite" means the angle across from a side (not touching that side).
• This is a two-way relationship: equal sides → equal angles, and equal angles → equal sides.

🔗 How to use this property

• If you see two sides marked as equal length, immediately know their opposite angles are equal.
• Combined with the 180° rule, you can solve for all angles.
• Example: a triangle has two equal sides and one angle of 40° between them; the two base angles are equal, so (180° - 40°) ÷ 2 = 70° each.

⚠️ Don't confuse

• Equal sides create equal opposite angles, not adjacent angles.
• Not all triangles have equal sides—this property only applies when sides actually match in length.

🧮 Solving for unknown angles

🧮 Step-by-step approach

When finding unknown angles in a triangle:

1. Identify what you know: which angles are given? Are any sides marked as equal?
2. Apply the equal-sides rule: if sides are equal, mark their opposite angles as equal.
3. Use the 180° sum: set up an equation where all three angles add to 180°.
4. Solve: calculate the unknown angle(s).

🧮 Practical scenarios

• Scenario A: Two angles given (e.g., 50° and 60°) → third angle = 180° - 50° - 60° = 70°.
• Scenario B: One angle given (e.g., 80°) and two sides marked equal → the two unknown angles are equal, so (180° - 80°) ÷ 2 = 50° each.
• Scenario C: All three sides equal (equilateral triangle) → all three angles equal, so 180° ÷ 3 = 60° each.

🧭 Overview

🧠 One-sentence thesis

When two parallel lines are crossed by a third line, they create predictable patterns of equal angles that allow us to calculate unknown angle measures.

📌 Key points (3–5)

• Parallel lines definition: two lines that point in the exact same direction and will never cross.
• Angle patterns formed: when a third line crosses two parallel lines, all four acute angles are equal, all four obtuse angles are equal.
• Key relationship: any acute angle and any obtuse angle formed by this intersection add up to 180°.
• Common confusion: don't confuse the acute angles with the obtuse angles—they are supplementary (sum to 180°), not equal to each other.
• Practical use: knowing one angle measure lets you determine all the other angles at the intersection.

📐 Understanding parallel lines

📐 What parallel lines are

Parallel lines: two lines that point in the exact same direction and will never cross.

• The key characteristic is that they maintain the same direction forever.
• They never meet, no matter how far they extend.
• Example: railroad tracks are parallel—they run in the same direction and never intersect.

✂️ What happens when a third line crosses them

• When two parallel lines are crossed by a third line, sets of equally-sized angles are formed.
• This crossing creates a predictable geometric pattern.
• The third line acts as a "transversal" that reveals the relationship between the parallel lines.

🔢 Angle patterns at the intersection

🔢 The four acute angles

• All four acute angles formed at the intersection will be equal in measure.
• "Acute" means the angle is between 0° and 90°.
• If you know one acute angle, you automatically know all four acute angles.

🔢 The four obtuse angles

• All four obtuse angles formed at the intersection will be equal in measure.
• "Obtuse" means the angle is between 90° and 180°.
• If you know one obtuse angle, you automatically know all four obtuse angles.

➕ The supplementary relationship

• Any acute angle and any obtuse angle at the intersection will have a combined measure of 180°.
• This means: acute angle + obtuse angle = 180°.
• Example: if an acute angle is 60°, then each obtuse angle is 180° - 60° = 120°.
• Don't confuse: the acute and obtuse angles are not equal to each other; they are supplementary (they add to 180°).

🧮 Finding unknown angles

🧮 The solving strategy

When given one angle measure at a parallel-line intersection:

What you know	What you can find
One acute angle	All four acute angles (same measure)
One acute angle	All four obtuse angles (180° minus the acute)
One obtuse angle	All four obtuse angles (same measure)
One obtuse angle	All four acute angles (180° minus the obtuse)

🧮 Why this works

• The geometric properties of parallel lines guarantee these angle relationships.
• The pattern is consistent and predictable.
• You only need one angle measurement to determine all eight angles formed by the intersection.

🧭 Overview

🧠 One-sentence thesis

The degrees-minutes-seconds (DMS) system divides degrees into fractions of 60 (minutes and seconds) instead of using decimals, and conversion between DMS and decimal degrees follows specific step-by-step procedures.

📌 Key points (3–5)

• What DMS is: an older method for expressing non-whole-number angle measures using minutes (1/60 of a degree) and seconds (1/60 of a minute, or 1/3600 of a degree).
• How the units relate: minutes and seconds work exactly like time units—think of 1 degree as 1 hour.
• DMS to decimal conversion: straightforward calculation using division by 60.
• Decimal to DMS conversion: more complex, requiring three steps to extract degrees, minutes, and seconds separately.
• Common confusion: the decimal part must be multiplied by 60 at each step when converting to DMS; don't confuse the decimal remainder with the final value.

🔢 Understanding the DMS system

📐 What DMS measures

Degrees-minutes-seconds (DMS) system: a method that divides a degree using fractions out of 60—a minute is 1/60 of a degree, and a second is 1/60 of a minute (which means a second is 1/3600 of a degree).

• This is an alternative to decimal degrees for expressing angle measurements that are not whole numbers.
• The units work exactly like time: think of 1 degree as 1 hour.
• Example: An angle might be written as 45°30'15" instead of 45.5042°.

🔗 How the units relate

Unit	Relationship	Fraction of a degree
Minute	1/60 of a degree	1/60
Second	1/60 of a minute	1/3600 of a degree

• A minute is to a degree as a minute is to an hour in time.
• A second is 1/60 of a minute, making it 1/3600 of a degree overall.

🔄 Converting DMS to decimal degrees

➡️ The conversion procedure

The process is fairly simple with a calculator:

• Enter the calculation: degrees + (minutes divided by 60) + (seconds divided by 3600).
• Round the result to the fourth decimal place if necessary.
• Why four decimal places: 1 second of angle is 1/3600 of a degree, which is smaller than 1/1000, so precision is slightly better than the thousandths place.

💡 Why this works

• You are converting each fractional part back into its decimal equivalent.
• Minutes are divided by 60 because there are 60 minutes in a degree.
• Seconds are divided by 3600 because there are 3600 seconds in a degree.
• Example: To convert an angle with degrees D, minutes M, and seconds S, calculate D + M/60 + S/3600.

🔄 Converting decimal degrees to DMS

⬅️ The three-step procedure

Converting from decimal degrees to DMS is more complicated:

1. Extract degrees: The whole-number part of the angle measurement gives the number of degrees.
2. Extract minutes: Multiply the decimal part by 60. The whole-number part of this result is the number of minutes.
3. Extract seconds: Multiply the decimal part of the minutes by 60. This gives the number of seconds (including any decimal part of seconds).

🔍 Step-by-step breakdown

Each step isolates one unit:

• Step 1 separates the whole degrees from the fractional part.
• Step 2 converts the fractional degrees into minutes (since 1 degree = 60 minutes, multiply by 60).
• Step 3 converts the fractional minutes into seconds (since 1 minute = 60 seconds, multiply by 60 again).

Don't confuse: At step 3, you multiply the decimal part of the minutes, not the original decimal part of the degrees.

📝 Worked example from the excerpt

Converting a decimal angle:

1. The whole-number part gives the degrees.
2. The decimal part times 60 gives minutes; take the whole-number part.
3. The decimal part of the minutes times 60 gives seconds.

Example: The excerpt walks through converting a decimal degree measurement by extracting each component in sequence, showing that the decimal remainder at each step must be carried forward and multiplied by 60 for the next unit.

🛠️ Practical considerations

🌐 Online tools

• Online calculators are available for these conversions (the excerpt mentions one at fcc.gov).
• These tools automate the conversion process.

📍 Real-world use

• The excerpt mentions latitude and longitude in DMS format (e.g., N and W coordinates).
• This system is still used in navigation and geographic coordinate systems despite being an older method.

🧭 Overview

🧠 One-sentence thesis

Triangles can be classified by both their side lengths (equilateral, isosceles, scalene) and their angle measures (obtuse, right, acute), and these classifications help identify triangle properties and relationships.

📌 Key points (3–5)

• Two classification systems: triangles are classified by side lengths and separately by angle measures.
• Side-based categories: equilateral (all sides equal), isosceles (exactly two sides equal), scalene (all sides different).
• Angle-based categories: obtuse (one obtuse angle), right (one right angle), acute (all acute angles).
• Common confusion: the two classification systems are independent—a triangle gets one label from each system (e.g., "acute scalene").
• Similar triangles: triangles with matching angle measures are similar, meaning they have the same shape but scaled side lengths in the same proportion.

📐 Classification by side lengths

📏 Equilateral triangle

Equilateral triangle: all three sides have the same length.

• All three sides are equal.
• This is the most symmetric type by side measurement.
• Example: a triangle with sides 5, 5, 5 is equilateral.

⚖️ Isosceles triangle

Isosceles triangle: exactly two sides have the same length.

• Exactly two sides match; the third is different.
• Don't confuse: "exactly two" means not all three—if all three are equal, it's equilateral, not isosceles.
• Example: a triangle with sides 7, 7, 4 is isosceles.

🔀 Scalene triangle

Scalene triangle: all three sides have different lengths.

• No two sides are the same.
• This is the least symmetric type by side measurement.
• Example: a triangle with sides 3, 5, 8 is scalene.

📐 Classification by angle measures

🔺 Obtuse triangle

Obtuse triangle: one of the angles is an obtuse angle.

• An obtuse angle is greater than 90 degrees.
• Only one angle needs to be obtuse for this classification.
• Example: a triangle with angles 110°, 40°, 30° is obtuse.

📐 Right triangle

Right triangle: one of the angles is a right angle.

• A right angle is exactly 90 degrees.
• The excerpt later identifies the two sides forming the right angle as legs and the side opposite the right angle as the hypotenuse (always the longest side).
• Example: a triangle with angles 90°, 60°, 30° is a right triangle.

🔻 Acute triangle

Acute triangle: all three of the angles are acute.

• An acute angle is less than 90 degrees.
• All three angles must be acute for this classification.
• Don't confuse: if even one angle is 90° or greater, the triangle is not acute.
• Example: a triangle with angles 70°, 60°, 50° is acute.

🔗 Combining classifications

🏷️ Dual labeling

• The excerpt instructs to "classify each triangle by angle and side" with examples like "acute scalene."
• Each triangle receives one label from the side system and one from the angle system.
• Example: a triangle can be "right isosceles" (one 90° angle and two equal sides) or "obtuse scalene" (one obtuse angle and all different sides).

🔄 Similar triangles

🔄 What makes triangles similar

Two triangles are similar if the three angles of one triangle have the same measure as the three angles of the second triangle.

• Matching angles define similarity.
• The side lengths do not need to match; they only need to be proportional.
• The excerpt states: "The triangles will have the same shape but the lengths will be scaled up or down."

📊 Proportional sides

• The lengths of the sides of similar triangles will be in the same proportion.
• This means if one triangle's sides are 3, 4, 5, a similar triangle might have sides 6, 8, 10 (each doubled).
• The excerpt notes that "recognizing corresponding sides can be more difficult when the figures are oriented differently"—rotation or flipping does not change similarity.
• Example: use a proportion to find an unknown length when two triangles are known to be similar.

📐 The Pythagorean theorem (right triangles)

🦵 Legs and hypotenuse

In a right triangle, the two sides that form the right angle are called the legs. The side opposite the right angle, which will always be the longest side, is called the hypotenuse.

• The legs are the two shorter sides meeting at 90°.
• The hypotenuse is always the longest side.
• Don't confuse: the hypotenuse is not just "any long side"; it is specifically opposite the right angle.

🧮 The Pythagorean theorem statement

The Pythagorean theorem says that the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

• In a right triangle with legs a and b and hypotenuse c: the square of a plus the square of b equals the square of c.
• This relationship only holds for right triangles.

✅ Verifying right triangles

• If you know the lengths of all three sides, you can use the Pythagorean theorem to check whether the triangle is a right triangle.
• Example: if the sum of the squares of the two shorter sides equals the square of the longest side, the triangle is right.
• The excerpt mentions that ancient Egyptians used this method for surveying after yearly flooding washed away boundary markings.

🔢 Square roots

• Calculating a square root is the opposite of squaring a number.
• Example: the square root of 9 is 3 because 3 squared is 9.
• If the number is not a perfect square, the square root will be an irrational decimal that is rounded as necessary.
• The excerpt notes that the Pythagorean theorem is most often used to calculate the length of a missing side (implying square roots are needed to solve for a leg or hypotenuse).

🧭 Overview

🧠 One-sentence thesis

Similar triangles have identical angle measures and proportional side lengths, allowing us to use proportions to find unknown lengths and the Pythagorean theorem to work with right triangles.

📌 Key points (3–5)

• What similar triangles are: triangles with the same three angle measures; their sides are proportional (same shape, different size).
• How to use similarity: set up proportions between corresponding sides to find unknown lengths.
• Common confusion: recognizing corresponding sides when triangles are oriented differently or flipped.
• Right triangles have a special property: the Pythagorean theorem relates the legs and hypotenuse (sum of squares of legs equals square of hypotenuse).
• Why the Pythagorean theorem matters: it lets you find missing side lengths and verify whether a triangle is a right triangle.

🔺 Understanding similar triangles

🔺 Definition and properties

Similar triangles: two triangles are similar if the three angles of one triangle have the same measure as the three angles of the second triangle.

• The lengths of the sides will be in the same proportion.
• The triangles have the same shape but the lengths are scaled up or down.
• Example: if one triangle has angles 30°, 60°, 90° and another has the same angles, they are similar even if one is twice as large.

📏 Using proportions to find unknown lengths

• Because corresponding sides are proportional, you can set up equations like: (side 1 of triangle A) / (side 1 of triangle B) = (side 2 of triangle A) / (side 2 of triangle B).
• The excerpt emphasizes that recognizing corresponding sides can be more difficult when figures are oriented differently.
• Don't confuse: corresponding sides are not always in the same visual position; you must match them by their role in each triangle.

📐 Right triangles and the Pythagorean theorem

📐 Parts of a right triangle

Legs: the two sides that form the right angle.

Hypotenuse: the side opposite the right angle, which will always be the longest side.

🔢 The Pythagorean theorem

Pythagorean theorem: in a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

• In formula form (rewritten in words): (leg a squared) plus (leg b squared) equals (hypotenuse c squared).
• This relationship only holds for right triangles.

✅ Verifying right triangles

• If you know the lengths of all three sides of a triangle, you can use the Pythagorean theorem to verify whether the triangle is a right triangle.
• Check: does (shorter side 1)² + (shorter side 2)² = (longest side)²?
• Historical note: ancient Egyptians used this method for surveying after Nile floods washed away boundary markings; surveyors were called "rope-stretchers" because they used a knotted rope to form triangles with specific side lengths.

🔍 Finding missing sides

• The excerpt provides three rearranged versions of the Pythagorean theorem to find a missing side without using algebra.
• You need to calculate square roots, which is the opposite of squaring a number.
• If the number under the square root is not a perfect square, the result will be an irrational decimal that should be rounded as necessary.
• Example: if you know the two legs, you can find the hypotenuse; if you know the hypotenuse and one leg, you can find the other leg.

🧭 Overview

🧠 One-sentence thesis

The Pythagorean theorem establishes that in a right triangle, the sum of the squares of the two legs equals the square of the hypotenuse, enabling calculation of missing side lengths and verification of right angles.

📌 Key points (3–5)

• What the theorem states: in a right triangle, the sum of the squares of the legs equals the square of the hypotenuse.
• How to identify parts: the legs form the right angle; the hypotenuse is opposite the right angle and always the longest side.
• Two main uses: calculating a missing side length when two sides are known, and verifying whether a triangle is a right triangle when all three sides are known.
• Common confusion: the hypotenuse is not just "the longest side" of any triangle—it is specifically the side opposite the right angle in a right triangle.
• Practical application: ancient Egyptians used this principle for surveying and redrawing boundaries after flooding.

📐 Parts of a right triangle

📏 Legs

Legs: the two sides that form the right angle.

• These are the two shorter sides in a right triangle.
• They meet at the 90-degree angle.

📏 Hypotenuse

Hypotenuse: the side opposite the right angle, which will always be the longest side.

• It is across from the right angle, not touching it.
• Always longer than either leg individually.
• Don't confuse: the hypotenuse only exists in a right triangle; not every "longest side" is a hypotenuse.

🔢 The theorem itself

🧮 The formula

Pythagorean theorem: the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

• If the legs have lengths a and b, and the hypotenuse has length c, then: the square of a plus the square of b equals the square of c.
• In words: (leg one squared) + (leg two squared) = (hypotenuse squared).

🔄 Three rearranged versions

The excerpt provides three alternate forms to find a missing side without needing algebra:

To find	Rearranged formula (in words)
Hypotenuse c	c equals the square root of (a squared plus b squared)
Leg a	a equals the square root of (c squared minus b squared)
Leg b	b equals the square root of (c squared minus a squared)

• These versions let you directly calculate the unknown side.
• Example: if you know both legs, use the first version to find the hypotenuse; if you know the hypotenuse and one leg, use the second or third version to find the other leg.

✅ Verifying right triangles

🔍 Using the theorem to check

• If you know the lengths of all three sides, plug them into the Pythagorean theorem.
• If the equation holds true (the sum of the squares of the two shorter sides equals the square of the longest side), the triangle is a right triangle.
• If the equation does not hold, the triangle is not a right triangle.

🏺 Historical application

• Ancient Egyptians used this method for surveying.
• After yearly flooding of the Nile washed away boundary markings, surveyors (called "rope-stretchers") used a loop of rope with equally-spaced knots.
• They formed a triangle with side lengths in a specific ratio; when the rope was stretched tight, the angle between two particular sides was a right angle because the side lengths satisfied the Pythagorean theorem.
• Example: a triangle with sides of lengths 3, 4, and 5 units satisfies the theorem (3 squared plus 4 squared equals 5 squared), confirming a right angle.

🧮 Working with square roots

🔢 What square roots mean

• Calculating a square root is the opposite of squaring a number.
• Example: the square root of 25 is 5, because 5 squared equals 25.

🔢 Non-perfect squares

• If the number under the square root symbol is not a perfect square (like 25 or 16), the result is an irrational decimal.
• The excerpt instructs rounding as necessary (e.g., to the hundredths or tenths place).
• Use a calculator to find these values.

🧩 Finding missing side lengths

🧩 The most common use

• The Pythagorean theorem is most often used to calculate the length of a missing side of a right triangle.
• You must know the lengths of the other two sides.

🧩 Step-by-step approach

1. Identify which side is missing: a leg or the hypotenuse.
2. Choose the appropriate rearranged formula.
3. Plug in the known values.
4. Calculate the square root (use a calculator if needed).
5. Round to the specified precision (e.g., nearest tenth).

• Example: if a right triangle has legs of 3 and 4 units, the hypotenuse is the square root of (3 squared plus 4 squared) = square root of (9 plus 16) = square root of 25 = 5 units.
• Don't confuse: always square the side lengths first, then add or subtract, then take the square root—do not add the lengths and then square.

🧭 Overview

🧠 One-sentence thesis

Area measures the two-dimensional space inside a polygon and is calculated using formulas that multiply base and height dimensions, with each shape requiring careful identification of perpendicular measurements.

📌 Key points (3–5)

• Area vs perimeter: perimeter is one-dimensional distance around the outside (linear units), while area is two-dimensional space inside (square units).
• Core method: area can always be thought of as the number of squares required to completely fill in the shape.
• Rectangle and square formulas: rectangle area = length × width; square area = side × side.
• Parallelogram insight: a parallelogram can be rearranged into a rectangle with the same base and height, so area = base × height (height must be perpendicular).
• Common confusion: for parallelograms and triangles, the height must be perpendicular to the base—it is not the length of the slanted side.

📏 Understanding area fundamentals

📐 What area measures

Area of a polygon: the amount of two-dimensional space inside the polygon, measured in square units (square feet, square centimeters, square miles, etc.).

• Area is not a length; it is a surface measurement.
• You can always visualize area as counting how many unit squares fit inside the shape.
• Example: if a rectangle holds 12 one-inch squares, its area is 12 square inches.

🔄 Area vs perimeter

Concept	Dimension	Units	What it measures
Perimeter	One-dimensional	Linear (feet, centimeters, miles)	Distance around the outside
Area	Two-dimensional	Square (square feet, square centimeters)	Space inside the polygon

• Don't confuse: perimeter is the boundary length; area is the interior space.

🟦 Rectangles and squares

🟦 Rectangle area formula

Area of a Rectangle = length × width (or base × height)

• Multiply the two perpendicular sides.
• The excerpt notes "we don't have to count little squares" once we have the formula.
• Example: a rectangle 5 units long and 3 units wide has area = 5 × 3 = 15 square units.

⬜ Square area formula

Area of a Square = side × side

• A square is a special rectangle where all sides are equal.
• Multiply one side by itself.
• Example: a square with side 4 units has area = 4 × 4 = 16 square units.

🔷 Parallelograms

🔷 Why the parallelogram formula works

• A parallelogram looks like a tilted rectangle.
• Opposite sides are parallel and have the same length.
• The excerpt shows that you can cut off part of a parallelogram and rearrange the pieces into a rectangle with the same base and height.
• Example: a parallelogram with base 6 units and vertical height 4 units transforms into a 6 by 4 rectangle, with area 24 square units.

⚠️ Identifying base and height

Area of a Parallelogram = base × height

• The height must be perpendicular to the base.
• Don't confuse: the height is not the length of the slanted side.
• If you label one side as the base, the height is the perpendicular distance to the opposite side.
• Example: if the base is 6 and the perpendicular height is 4, area = 6 × 4 = 24 square units, even if the slanted side is 5 units long.

🔺 Triangles

🔺 Triangle area formula

Area of a Triangle = one half × base × height (or ½ × base × height)

• Any triangle can be doubled to form a parallelogram.
• Therefore, the triangle's area is half the area of a parallelogram with the same base and height.
• Example: a triangle with base 8 and height 5 has area = ½ × 8 × 5 = 20 square units.

📐 Finding the perpendicular height

• As with parallelograms, the height must be perpendicular to the base.
• If the triangle is obtuse, you may have to imagine the height outside the triangle and extend the base line to meet it.
• Don't confuse: the height is not necessarily one of the triangle's sides; it is the perpendicular distance from the base to the opposite vertex.

🔶 Trapezoids

🔶 What a trapezoid is

Trapezoid: a quadrilateral with exactly one pair of parallel sides, called the bases.

• The two parallel sides are both called bases (labeled with subscripts like b₁ and b₂ to distinguish them).
• An isosceles trapezoid has two nonparallel sides of equal length (like an isosceles triangle).

🔶 Trapezoid area formula

Area of a Trapezoid = ½ × (base₁ + base₂) × height

• The subscripts on b₁ and b₂ are just a way to name two different base measurements.
• The excerpt notes that video explanations with animation show where the formula comes from more clearly.
• Example: a trapezoid with bases 5 and 9 and height 4 has area = ½ × (5 + 9) × 4 = ½ × 14 × 4 = 28 square units.

🧭 Overview

🧠 One-sentence thesis

A parallelogram's area is calculated using the same formula as a rectangle—base times height—provided the height is measured perpendicular to the base.

📌 Key points (3–5)

• What a parallelogram is: a polygon with opposite sides parallel and equal in length, resembling a tilted rectangle.
• Key measurement distinction: the height must be perpendicular to the base and is not the length of the slanted sides.
• How the formula works: cutting and rearranging a parallelogram transforms it into a rectangle with the same base and height, proving the area formula is identical.
• Common confusion: don't use the slanted side length as the height—only the perpendicular distance counts.

📐 What is a parallelogram

📐 Definition and structure

A parallelogram is a polygon in which pairs of opposite sides are parallel and have the same length.

• It looks like a tilted or slanted rectangle.
• When you label one side as the base, the height is the perpendicular distance to the opposite side.
• The height is not the length of the other (slanted) sides.

🔍 Don't confuse height with side length

• The slanted sides may be longer or shorter than the perpendicular height.
• Only the perpendicular distance from base to opposite side counts as the height for area calculations.
• Example: a parallelogram might have slanted sides of 7 units but a perpendicular height of only 5 units.

🔄 Why the area formula works

🔄 Cutting and rearranging

The excerpt demonstrates that a parallelogram can be transformed into a rectangle:

• Cut off a triangular piece from one end of the parallelogram.
• Move that piece to the opposite end.
• The result is a rectangle with the same base and height as the original parallelogram.

📏 Numerical example from the excerpt

• A parallelogram with base of (unspecified) units and vertical height of (unspecified) units transforms into a rectangle.
• The rectangle has the same dimensions and an area of (base × height) square units.
• This proves the area formula is the same for both shapes.

🧮 The area formula

🧮 Formula statement

Area of a Parallelogram = base × height

• This is identical to the rectangle formula.
• The critical requirement: the height must be perpendicular to the base.

⚠️ Careful measurement

• Always identify which side you are using as the base.
• Measure the perpendicular distance from that base to the opposite side.
• Do not use the slanted side length in the calculation.

Shape	Formula	Key requirement
Rectangle	base × height	Sides already perpendicular
Parallelogram	base × height	Height must be perpendicular to base

🧭 Overview

🧠 One-sentence thesis

The area of a triangle is half the area of a parallelogram with the same base and perpendicular height, because any triangle can be doubled to form a parallelogram.

📌 Key points (3–5)

• Core relationship: A triangle's area equals one half the area of a parallelogram with the same base and height.
• Base and height requirement: The height must be perpendicular to the base, just as with parallelograms.
• Obtuse triangles: For obtuse triangles, you may need to imagine the height outside the triangle and extend the base line to meet it.
• Common confusion: The height is not necessarily one of the triangle's sides—it must be perpendicular to the chosen base.

📐 The doubling principle

🔄 How triangles relate to parallelograms

• The excerpt shows that any triangle can be doubled to form a parallelogram.
• Because the parallelogram has the same base and height as the original triangle, the triangle's area is exactly half the parallelogram's area.
• This geometric transformation justifies the formula.

📏 Why "one half"

• A parallelogram's area = base × height.
• A triangle made from half of that parallelogram has area = (base × height) / 2.
• Example: If a parallelogram has base 6 units and height 4 units (area = 24 square units), the triangle formed from half of it has area 12 square units.

🧮 The area formula

📐 Formula statement

Area of a Triangle: (base × height) / 2, or equivalently base × height × (1/2).

• Both forms are equivalent; choose whichever is clearer.
• The height must be perpendicular to the base.

⚠️ Perpendicularity requirement

• The height is not just any side of the triangle.
• It is the perpendicular distance from the base to the opposite vertex.
• Don't confuse: The slanted sides of a triangle are not the height unless the triangle happens to be a right triangle and you choose the right angle's sides as base and height.

🔺 Special case: Obtuse triangles

🔺 When the height falls outside

• For an obtuse triangle, the perpendicular height may lie outside the triangle's boundary.
• You may need to imagine or draw the height line outside the shape.
• You also need to extend the base line so it meets the height.
• Example: Picture an obtuse triangle leaning far to one side—the perpendicular from the top vertex to the base line lands beyond the triangle's edge, so you extend the base as a reference line.

🧭 Overview

🧠 One-sentence thesis

The area of a trapezoid is calculated by adding its two parallel bases, multiplying by the perpendicular height, and taking half of that result.

📌 Key points (3–5)

• What a trapezoid is: a quadrilateral with exactly one pair of parallel sides (called the bases).
• The area formula: add the two bases, multiply by height, then divide by two.
• Notation flexibility: the bases can be labeled b₁ and b₂, or simply a and b—either notation works.
• Common confusion: don't confuse the nonparallel sides with the height—the height must be perpendicular to the bases.
• Special case: an isosceles trapezoid has two nonparallel sides of equal length, similar to an isosceles triangle.

📐 Understanding trapezoid structure

🔷 What defines a trapezoid

Trapezoid: a quadrilateral with exactly one pair of parallel sides, called the bases.

• The two parallel sides are the defining feature.
• The other two sides are not parallel and may have different lengths.
• Example: A four-sided shape where the top and bottom edges are parallel, but the left and right edges slant inward or outward.

🔺 Isosceles trapezoid

• A special type where the two nonparallel sides have equal lengths.
• The excerpt compares this to an isosceles triangle (which has two equal sides).
• This is mentioned as one example but is not required for all trapezoids.

📏 The area formula

🧮 How to calculate area

Area of a Trapezoid: (b₁ + b₂) × h ÷ 2, or equivalently ½ × (b₁ + b₂) × h

• Step 1: Add the lengths of the two parallel bases (b₁ + b₂).
• Step 2: Multiply that sum by the perpendicular height (h).
• Step 3: Take half of the result (divide by 2 or multiply by ½).

🔤 Understanding the notation

• The subscripts on b₁ and b₂ are just a way to distinguish two different base measurements using the same letter.
• Many people prefer to call the bases "a" and "b" instead—both notations are acceptable.
• Don't be intimidated by subscripts; they simply mean "first base" and "second base."

⚠️ Critical requirement: perpendicular height

• The height must be perpendicular (at a right angle) to the bases.
• Don't confuse the nonparallel sides with the height—they are usually slanted and not perpendicular.
• The height is the straight-line distance between the two parallel bases, measured at a right angle.

🎥 Derivation and visualization

📹 Where the formula comes from

• The excerpt notes there are multiple ways to show why the formula works.
• These explanations are described as "better in video because they can be animated."
• The excerpt references video resources but does not provide the derivation in text form.

🔗 Practical application

Example from the excerpt: Hot Lake Springs benches

• Each trapezoidal bench has an outer edge of one length, an inner edge of another length, and the edges are a certain distance apart.
• To find seating area: add the outer and inner edge lengths (the two bases), multiply by the distance between them (height), and divide by 2.
• This gives the area of seating surface provided by each trapezoid-shaped bench.

🧭 Overview

🧠 One-sentence thesis

The area of a circle is calculated by multiplying pi times the square of the radius, yielding an approximation in square units because whole squares cannot fit exactly inside a round shape.

📌 Key points (3–5)

• The formula: area equals pi times the radius squared, written in square units.
• Why it's an approximation: you cannot fit a whole number of squares—or an exact fraction—inside a circle, so the result is always approximate.
• Units remain square: even though circles are round, area is still measured in square units (think of squares on a round waffle).
• Calculator shortcut: if your calculator lacks a pi key, use the approximation 3.14.

📐 The circle area formula

📐 How to calculate area

The area of a circle is pi times the square of the radius.

• Take the radius (distance from center to edge).
• Square it (multiply the radius by itself).
• Multiply that result by pi.
• The answer is in square units.

🔢 Using pi

• Pi is a special number needed for all circle calculations.
• If your calculator doesn't have a pi key, the excerpt recommends using 3.14 as an approximation.
• This substitution keeps calculations practical while staying reasonably accurate.

🟦 Why square units for round shapes

🟦 The square-unit paradox

• Even though a circle is round, its area is still measured in square units.
• The excerpt offers a helpful mental image: think of the squares on a round waffle.
• This reminds us that area always means "how much surface space," measured by imagining unit squares covering that space.

🔍 Why the result is approximate

• You cannot fit a whole number of squares—or even an exact fraction of squares—inside a circle.
• The curved edge means some squares are partly in, partly out.
• Therefore, the area of a circle will always be an approximation, not an exact count.
• Don't confuse: the formula is exact, but the result is approximate because pi itself is irrational and the fit of squares to curves is imperfect.

🧮 Applying the formula

🧮 Rounding and significant figures

• The excerpt's exercises instruct students to "round each answer to the appropriate number of significant figures."
• This acknowledges that circle area calculations produce long decimal results.
• Rounding keeps answers practical and matches the precision of the given measurements.

📝 Example workflow

Example: A circle has a radius of 5 inches.

• Square the radius: 5 × 5 = 25.
• Multiply by pi: 25 × 3.14 ≈ 78.5.
• Report the area as approximately 78.5 square inches.

🧭 Overview

🧠 One-sentence thesis

Composite figures are shapes built from simpler geometric pieces, and their areas and perimeters are found by breaking them into basic shapes like rectangles, triangles, and circles, then adding or subtracting as needed.

📌 Key points (3–5)

• What a composite figure is: a geometric figure formed by combining two or more basic geometric shapes.
• How to find area: divide the composite figure into simpler shapes (rectangles, triangles, etc.), calculate each area separately, then add or subtract as necessary.
• What you may need to do: figure out some unmarked dimensions during the process.
• When circles are involved: answers will be approximations because they include π.
• Common confusion: remember that square units behave differently than linear units—conversion factors must be squared when converting area measurements.

🧩 Understanding composite figures

🧩 What composite figures are

A composite figure is a geometric figure which is formed by—or composed of—two or more basic geometric figures.

• These are odd shapes made up of simpler shapes combined together.
• The excerpt notes that examples can range from fairly simple to much more complicated figures.
• Example: A floor plan showing a large rectangle with a smaller rectangle cut out of one side.

🔧 The general strategy

• Divide the composite figure into simpler shapes: rectangles, triangles, trapezoids, etc.
• Calculate the area or perimeter of each simple shape.
• Either add or subtract their measurements as necessary.
• You may need to figure out some unmarked dimensions during the process.

📐 Working with polygon composites

➕ Adding and subtracting areas

The excerpt shows different scenarios:

• Adding areas: when shapes are placed side by side to form a larger figure.
• Subtracting areas: when a shape is cut out or removed from another shape.

Example: A trapezoid can be understood as a rectangle with two right triangles removed from it.

📏 Finding missing dimensions

• Not all dimensions will be marked on a diagram.
• You must use the given measurements and geometric relationships to determine unmarked lengths.
• This is often necessary before you can calculate individual areas.

⭕ Working with circles in composites

🥧 Approximations with π

If a composite figure includes parts of circles, you'll need to approximate your answer because your calculation will have π in it.

• Circle areas use the formula: area equals π times the square of the radius.
• The units are still square units, even though a circle is round.
• If your calculator doesn't have a π key, use the approximation 3.14.

🔄 Common circle composites

The excerpt mentions several scenarios:

• Semicircles combined with straight sides (like a track).
• Quarter circles attached to or removed from other shapes.
• Circles inscribed in squares or squares inscribed in circles.

🎯 Fraction problems without measurements

• Sometimes you can determine a fraction or percent of the whole without knowing actual measurements.
• Example: circular disks cut from squares—you can calculate what percent is discarded based on the geometric relationship alone.
• Example: a square beam cut from a circular log—you can find the percent removed using only the shape relationship.

📊 Area unit conversions

⚠️ Why area units are different

Square units behave differently than linear units.

Key insight: When converting units of area, you must square the linear conversion factor.

Linear conversion	Area conversion
1 yard = 3 feet	1 square yard = 9 square feet
Ratio is 1 to 3	Ratio is 1 to 9 (which is 1 to 3²)

🔢 How to convert area units

• Start with the linear conversion factor (e.g., 1 yard = 3 feet).
• Square both the number and the units: (3 feet)² = 9 square feet.
• More generally: square the linear conversion factors when converting units of area.

🖼️ Visualizing the conversion

The excerpt suggests visualizing a square yard divided into a 3-by-3 grid of square feet:

• Each side of 1 yard becomes 3 feet.
• The grid shows 9 square feet total.
• This demonstrates visually why the conversion ratio is 1 to 9, not 1 to 3.

Don't confuse: Converting 2 yards to feet (multiply by 3) is different from converting 2 square yards to square feet (multiply by 9).

🧭 Overview

🧠 One-sentence thesis

Composite figures that include circular parts require breaking the shape into simpler components and using π in calculations, which produces approximate answers.

📌 Key points (3–5)

• General strategy: divide composite figures into simpler shapes (rectangles, triangles, circles or parts of circles), then add or subtract their areas or perimeters as needed.
• Unmarked dimensions: you may need to figure out missing measurements during the process.
• Circles introduce approximation: when a composite figure includes parts of circles, your answer will be approximate because the calculation involves π.
• Fraction/percent problems: sometimes you can solve for a fraction or percent of the whole without knowing actual measurements.

🔧 General approach to composite figures

🔧 Breaking down the figure

• The excerpt recommends dividing any composite figure into simpler shapes: rectangles, triangles, and (when relevant) circles or parts of circles.
• Once divided, you add or subtract their areas or perimeters depending on what the problem asks.
• Example: A room shaped like a large rectangle with a smaller rectangle cut out → find the area of the large rectangle, subtract the area of the cut-out.

📏 Finding unmarked dimensions

• The excerpt notes that "you may need to figure out some unmarked dimensions during the process."
• This means not all lengths or widths will be labeled; you infer them from the given information.
• Example: If a composite shape shows one side of a rectangle but not the opposite side, you use symmetry or other given measurements to deduce it.

🔵 Working with circular parts

🔵 Why circles require approximation

When a composite figure includes parts of circles, you'll need to approximate your answer because your calculation will have π in it.

• π is an irrational number, so any calculation involving circles (full circles, semicircles, quarter circles) cannot be expressed exactly as a finite decimal.
• The excerpt emphasizes that answers will be approximate, not exact.

🏟️ Example scenarios with circles

The excerpt describes several exercise types:

• Track with semicircles: A track formed by two straight sides and two semicircles. You calculate the distance around (perimeter) and the area inside.
• Quarter circles: A quarter circle by itself, or a quarter circle removed from a larger circle. You calculate area and perimeter for these partial shapes.
• Circles cut from squares: Circular disks cut from square sheets, with corners discarded. You determine what percent of the material is wasted.
• Squares inscribed in circles (logs): A log with a circular cross-section is planed into a square beam. You find what percent of the log is removed.

Don't confuse: "parts of circles" (semicircles, quarter circles) still use the same circle formulas (circumference, area), but you take the appropriate fraction (half, quarter, etc.).

🧮 Solving without actual measurements

🧮 Fraction or percent problems

• The excerpt states: "If we need to determine a fraction or percent out of the whole, we may be able to solve the problem without knowing any actual measurements."
• This means some problems ask for a ratio or percentage, so you can work symbolically or with a generic size.

🔩 Example: waste or removal percentage

• Circular disks from squares: Assume the circle is as large as possible (diameter = side of square). The ratio of circle area to square area is a fixed percentage, regardless of the actual size.
• Square beam from circular log: Assume the square is as large as possible (diagonal = diameter of circle). The ratio of square area to circle area is again a fixed percentage.
• In both cases, you calculate the fraction or percent without needing specific dimensions.

📐 Perimeter vs. area for composite figures

📐 Perimeter calculations

• For composite figures, perimeter is the total distance around the outside edge.
• When circles or parts of circles are involved, you add the arc lengths (using the circumference formula for the relevant fraction of the circle) to any straight edges.
• Example: A track with two straight sides and two semicircles → perimeter = length of both straight sides + circumference of the two semicircles (which together form one full circle).

📐 Area calculations

• Area is the total space inside the figure.
• You add areas of component shapes if they are joined, or subtract if one shape is cut out from another.
• Example: A field inside a track → area of the rectangle formed by the straight sides, plus the area of the circle formed by the two semicircles.

🔍 Common confusion: perimeter vs. area with circles

• Don't confuse the formulas: circumference (perimeter of a circle) uses π times diameter (or 2π times radius); area uses π times radius squared.
• When a composite figure includes a circular arc, the perimeter includes that arc length, not the area of the circle.

🧭 Overview

🧠 One-sentence thesis

When converting units of area, you must square the linear conversion factor because area measures two dimensions, not one.

📌 Key points (3–5)

• The core difference: linear units convert 1-to-1, but square units require squaring the conversion ratio (e.g., 1 yard = 3 feet becomes 1 square yard = 9 square feet).
• Visual proof: a 1-yard-by-1-yard square divides into a 3-by-3 grid when converted to feet, showing 9 square feet fit in 1 square yard.
• Common confusion: don't apply linear conversion ratios directly to area—if linear units have a ratio of a to b, square units have a ratio of a² to b².
• Special units: acres and hectares are defined as specific areas and should never be written as "squared" (no exponent of 2).
• Similar figures rule: if two similar shapes have linear dimensions in a ratio of a to b, their areas will be in a ratio of a² to b².

📐 Why area conversions differ from linear conversions

📏 Linear vs. square unit behavior

• Linear measurements (length, width, height) convert using simple ratios: 1 yard = 3 feet means multiply by 3.
• Area measurements involve two dimensions, so the conversion factor applies twice.
• Example: A hallway 2 yards long and 1 yard wide becomes 6 feet by 3 feet, giving 18 square feet (not 6 square feet).

🔲 The visual grid method

The excerpt provides a concrete visualization:

• Start with 1 square yard (a square with sides 1 yard long).
• Since 1 yard = 3 feet, divide each side into 3 equal sections.
• This creates a 3-by-3 grid of 9 smaller squares, each 1 square foot.
• Conclusion: 1 square yard = 9 square feet, not 3 square feet.

🧮 The algebraic explanation

The excerpt offers an alternative without diagrams:

• Start with the linear conversion: 1 yard = 3 feet.
• Square both sides: (1 yard)² = (3 feet)².
• Apply the exponent to both the number and the units: 1 square yard = 9 square feet.
• General rule: square the linear conversion factor when converting area units.

🔢 U.S. system area conversions

📊 Conversion ratios

The excerpt states that if linear units have a ratio of a to b, square units have a ratio of a² to b².

Linear conversion	Area conversion
1 yard = 3 feet	1 square yard = 9 square feet
1 foot = 12 inches	1 square foot = 144 square inches

🌾 The acre as a special unit

An acre is defined as the area of a 660 foot by 66 foot rectangle.

• Historical origin: the amount of land a medieval farmer with eight oxen could plow in one day.
• Important: an acre is already a unit of area; never say "acres squared" or add an exponent of 2.
• Don't confuse: acres measure area directly, unlike "square feet" which explicitly shows the squaring.

📏 Metric system area conversions

🔟 Powers of ten simplify conversions

The excerpt notes that metric conversions are "all powers of ten, which means they are all about moving the decimal point."

Example from the exercises:

• 1 meter = 100 centimeters (linear).
• 1 square meter = 10,000 square centimeters (area: 100² = 10,000).

🟩 The hectare as a special unit

A hectare is defined as a square with sides 100 meters long.

• Visual: dividing a square kilometer into 10 rows and 10 columns creates a 10-by-10 grid of 100 hectares.
• Like acres, hectares should never be written as "hectares squared" or with an exponent of 2.

🔄 Similar figures and area ratios

📐 The scaling rule for similar shapes

If the linear dimensions of two similar figures have a ratio of a to b, then the areas will have a ratio of a² to *b².

• This applies universally: circles, triangles, rectangles, hexagons, etc.
• Example from exercises: A medium pizza has diameter twice that of a personal pizza (ratio 2:1), so its area is 4 times larger (ratio 4:1).

🔺 Verification through examples

The excerpt provides exercises to verify the rule:

• Pizza example: diameter ratio 2:1 → area ratio 4:1.
• Triangle example: a larger triangle with legs triple the dimensions (ratio 3:1) → area ratio 9:1.
• Don't confuse: doubling all linear dimensions does not double the area; it quadruples it.

🌍 Converting between U.S. and metric systems

🔀 Messy decimal conversions

The excerpt warns that conversions between systems "will involve messy decimal values."

Example provided:

• Linear: 1 inch = 2.54 centimeters.
• Area: square both sides → 1 square inch ≈ 6.4516 square centimeters.
• The conversion table rounds to five significant figures for practical use.

📋 Practical applications

Exercises demonstrate real-world conversions:

• Converting city areas from square miles to square kilometers.
• Converting ranch sizes from acres to hectares.
• Converting paper dimensions and field dimensions between systems.

🧭 Overview

🧠 One-sentence thesis

When converting area measurements, the ratio of square units is the square of the ratio of linear units, and this principle applies across U.S., metric, and mixed-system conversions as well as to similar figures.

📌 Key points (3–5)

• Core principle: if linear units have a ratio of a to b, square units have a ratio of a² to b².
• Metric conversions: all based on powers of ten, so moving the decimal point handles the conversion.
• Cross-system conversions: U.S. to metric (and vice versa) involve messy decimals; for example, 1 inch = 2.54 cm leads to 1 square inch ≈ 6.4516 square centimeters.
• Similar figures: if two similar shapes have linear dimensions in ratio a to b, their areas are in ratio a² to b².
• Common confusion: don't say "acres squared" or "hectares squared"—these are already units of area, not to be squared again.

📐 The squaring principle for area conversions

📏 Linear ratio vs area ratio

If the linear units have a ratio of a to b, the square units will have a ratio of a² to b².

• This is the foundational rule for all area conversions.
• It means you square the conversion factor when moving from length to area.
• Example: if 1 yard = 3 feet (linear ratio 1:3), then 1 square yard = 9 square feet (area ratio 1:9).

🔢 Why it works

• Area is two-dimensional: length × width.
• When you scale both dimensions by the same factor, you multiply that factor twice.
• Don't confuse: the linear conversion factor is used once for length, but squared for area.

🌍 Metric system area conversions

🔟 Powers of ten

• The excerpt states: "the conversions in the metric system are all powers of ten, which means they are all about moving the decimal point."
• Because metric linear units differ by factors of 10, 100, 1000, etc., the area conversions are the squares: 100, 10,000, 1,000,000, etc.

📦 Common metric area units

Linear unit	Area unit	Conversion example
meter (m)	square meter (m²)	1 m = 100 cm → 1 m² = 10,000 cm²
centimeter (cm)	square centimeter (cm²)	1 cm = 10 mm → 1 cm² = 100 mm²
kilometer (km)	square kilometer (km²)	1 km = 1000 m → 1 km² = 1,000,000 m²

🌾 Hectare definition

A hectare is defined as a square with sides 100 meters long.

• A square kilometer can be divided into a 10 by 10 grid of 100 hectares.
• Important: it is wrong to say "hectares squared" or put an exponent of 2 on the units—hectare is already a unit of area.
• Don't confuse: the same rule applies to acres in the U.S. system (an acre is defined as the area of a 660 foot by 66 foot rectangle, and you do not square it).

🔄 Cross-system conversions (U.S. ↔ metric)

🧮 Messy decimals

• The excerpt warns: "Converting between the U.S. and metric systems will involve messy decimal values."
• Example given: because 1 inch = 2.54 cm, squaring both gives 1 square inch ≈ 6.4516 square centimeters.
• Conversions are rounded to five significant figures in practice.

🗺️ Practical conversion scenarios

• Example from exercises: converting Portland, Oregon's area from square miles to square kilometers, or a ranch from acres to hectares.
• Example: a standard sheet of paper (8.5 inches by 11 inches) converted to square centimeters, or a soccer field (meters) converted to square feet.
• These require applying the squared conversion factor, not the linear one.

📊 Similar figures and area ratios

🔺 The similar-figures rule

If the linear dimensions of two similar figures have a ratio of a to b, then the areas will have a ratio of a² to b².

• This applies to all similar shapes: circles, triangles, rectangles, hexagons, etc.
• "Similar" means the shapes have the same proportions but different sizes.

🍕 Example: pizzas

• A personal pizza has a 6-inch diameter; a medium pizza has twice that diameter (12 inches).
• Linear ratio: 1:2 (6 to 12).
• Area ratio: 1²:2² = 1:4.
• The medium pizza has four times the area of the personal pizza, not twice.

🔺 Example: right triangles

• A small right triangle has legs of certain lengths; a larger triangle has legs triple those dimensions.
• Linear ratio: 1:3.
• Area ratio: 1²:3² = 1:9.
• The larger triangle has nine times the area of the smaller triangle.

⚠️ Don't confuse

• Doubling the linear dimensions does not double the area—it quadruples it.
• Tripling the linear dimensions does not triple the area—it multiplies it by nine.
• Always square the linear ratio to get the area ratio.