Mathematics for Elementary Teachers

Introduction to Arithmetic Models

Introduction

🧭 Overview

🧠 One-sentence thesis

Mental and physical models help learners and teachers understand what arithmetic operations actually mean.

📌 Key points (3–5)

Why models matter: they provide concrete representations of abstract operations.
Purpose in learning: models support understanding of arithmetic beyond memorization.
Purpose in teaching: models give educators tools to explain operations meaningfully.
Foundation for the book: the approach used throughout builds on these models.

🎯 The role of models in arithmetic

🧠 What models provide

Models: mental and physical representations that show what operations mean.

Models are not just calculation shortcuts; they reveal the underlying meaning of operations.
They bridge the gap between abstract symbols (like +, −, ×, ÷) and concrete understanding.
The excerpt emphasizes that models are useful for both learning and teaching contexts.

📚 Why understanding matters

Arithmetic involves operations that can seem mechanical without models.
Models help answer "what does this operation actually do?" rather than just "how do I compute it?"
Example: A model might show addition as combining groups, helping learners see why the operation works the way it does.

🔧 Application in this book

🔧 The "Dots and Boxes" approach

The book uses a specific modeling approach called "Dots and Boxes" throughout.
This approach comes from James Tanton and is used with permission.
The method appears in this part and continues through the rest of the book.

🎓 For learners and teachers

For learners: models provide ways to visualize and internalize what operations do.
For teachers: models offer concrete tools to explain arithmetic concepts.
The excerpt positions models as helpful for both roles, emphasizing their dual purpose.

Problem or Exercise?

🧭 Overview

🧠 One-sentence thesis

The main activity of mathematics is solving problems—which require exploration and understanding—not just practicing exercises that rehearse known techniques.

📌 Key points (3–5)

What distinguishes problems from exercises: problems require figuring out an approach without knowing in advance what techniques to use; exercises practice a known skill.
Problems involve uncertainty: understanding what is being asked, false starts, mistakes, and lots of scratch paper are part of the process.
Context matters: what counts as a problem versus an exercise depends on the solver's background knowledge.
Common confusion: exercises feel like "doing math," but the ultimate goal is developing skills (through exercises) to solve harder, more interesting problems.
Why both matter: exercises build skills; problems are "playing the game"—the real point of learning mathematics.

🎯 What makes something a problem vs. an exercise

🧩 Problems: exploration and uncertainty

Problem: you probably don't know at first how to approach solving it; you don't know what mathematical ideas might be used in the solution.

Part of solving a problem is understanding what is being asked and knowing what a solution should look like.
Problems often involve:
- False starts
- Making mistakes
- Lots of scratch paper
Example: The broken clock question (arranging clock numbers into consecutive sums) requires figuring out an approach without a template.

🔁 Exercises: practicing a known skill

Exercise: you are often practicing a skill; you may have seen a teacher demonstrate a technique or read a worked example, then practice on very similar assignments.

The goal is mastering that skill through repetition.
You already know the technique; you're applying it to similar cases.
Example: "What is the product of 4,500 and 27?" practices multiplication; the method is already known.

⚖️ Context-dependent distinction

Don't confuse: the same question can be a problem for one person and an exercise for another.
The excerpt gives an example:
- For a young student just learning addition, "Fill in the blank: ___" might be a problem.
- For someone with more background, that same question is an exercise.
The distinction depends on the solver's prior knowledge and experience.

🏀 Why both problems and exercises matter

🏋️ Exercises as skill-building

Exercises develop and refine techniques so that you can tackle harder problems later.
The excerpt compares this to practicing sports skills:
- Hitting hundreds of forehands in tennis to place them accurately
- Breaking down swimming strokes for efficiency
- Controlling the ball during quick turns in soccer
- Shooting free throws in basketball
- Catching high fly balls in baseball
Each drill isolates and strengthens a component skill.

🎮 Problems as "playing the game"

The ultimate goal: develop more and better skills (through exercises) so that you can solve harder and more interesting problems.
The excerpt emphasizes: "In mathematics, solving problems is playing the game!"
Just as you practice sports skills to play the actual game, you practice math techniques to solve real problems.
Don't confuse: doing exercises is not the end goal; it's preparation for problem-solving.

📝 Recognizing problems vs. exercises in practice

🔍 Clues that something is a problem (for you)

You don't immediately know which technique to use.
You need to interpret what the question is really asking.
You expect to try multiple approaches or make mistakes.
Example from the excerpt: "Arrange the digits 1–6 into a 'difference triangle' where each number below is the difference of the two above it" (requires experimentation).

🔍 Clues that something is an exercise (for you)

You recognize the question type and know the procedure.
You've practiced similar problems before.
The main challenge is careful execution, not figuring out the approach.
Example from the excerpt: "A soccer coach began the year with a $500 budget and spent $450 by December. How much was not spent?" (straightforward subtraction).

🧠 Self-assessment tip

The excerpt suggests: after labeling questions as problem or exercise, compare your answers with a partner.

Different people may classify the same question differently based on their background.
This reinforces that the distinction is relative to the solver's knowledge.

Problem Solving Strategies

🧭 Overview

🧠 One-sentence thesis

Solving mathematical problems (as opposed to exercises) requires creative strategies and practice rather than a fixed recipe, and building a toolbox of flexible approaches helps you tackle unfamiliar challenges.

📌 Key points (3–5)

Problem vs exercise: A problem is something you don't immediately know how to solve; an exercise applies a known method.
Pólya's four steps: Understand the problem, make a plan, carry out the plan, and look back—but steps 1 and 2 require creativity and experience.
No single strategy works every time: Problem solving is more art than science; you need multiple tools and must try different approaches.
Common confusion: Don't assume constraints not stated in the problem (e.g., assuming a clock must break radially when the problem doesn't require it).
Patterns need explanation: Finding a pattern is not enough; you must explain why it works and verify it will continue.

🎯 Problem vs Exercise

🎯 What makes something a problem

A problem is a question you don't immediately know how to solve; there is no simple recipe.
An exercise applies a method or formula you already know.
Example: "What is the product of 4,500 and 27?" is an exercise (you know the multiplication procedure). "Arrange digits 1–6 into a difference triangle where each number below is the difference of the two above it" is a problem (no obvious procedure).

🧠 Why the distinction matters

Problems require creativity, experimentation, and strategy.
You get better at solving problems by building background knowledge and practicing.
The excerpt emphasizes: "Unlike exercises, there is never a simple recipe for solving a problem."

🗺️ Pólya's Four-Step Method

📖 The four steps

George Pólya published How to Solve It in 1945, proposing:

Understand the problem
Make a plan
Carry out the plan
Look back on your work (How could it be better?)

❓ The challenge

Steps 1 and 2 are "particularly mysterious"—how do you actually "make a plan"?
You need tools in your toolbox and experience to draw upon.
The excerpt states: "This is where math becomes a creative endeavor (and where it becomes so much fun)."
No list of strategies will ever be complete; the best way to improve is to learn background material well and solve many problems.

🧰 Core Problem-Solving Strategies

🌟 Strategy 1: Wishful Thinking

What it means: Don't be afraid to change the problem.
Ask "what if" questions:
- What if the picture was different?
- What if the numbers were simpler?
- What if I just made up some numbers?
Important: Go back to the original problem at the end.
This strategy helps you get started when you're stuck.

🚀 Strategy 2: Try Something!

"If you are really trying to solve a problem, the whole point is that you do not know what to do right out of the starting gate."

Put pencil to paper and try something—this helps you understand the problem.
Mess around with it to figure out what is going on.
Equally important: If what you tried first doesn't work, try something else!
The excerpt calls this "the most important problem solving strategy of all."

🖼️ Strategy 3: Draw a Picture

Some problems are obviously geometric, but even non-geometric problems can benefit from visual thinking.
Example (Payback problem): Draw a square to represent all of Alex's money, then shade portions to show what he gave away.
Can you represent something in the situation with a picture?

🔢 Strategy 4: Make Up Numbers

When a problem has no specific numbers, the numbers probably aren't important—so invent them.
You can work forwards (assume a starting amount) or backwards (assume an ending amount and reverse-engineer).
Critical warning: Remember what the original problem asked! Don't answer based on your made-up numbers; re-read the problem and answer the actual question.

🪜 Strategy 5: Try a Simpler Problem

Pólya said: "If you can't solve a problem, then there is an easier problem you can solve: find it."
Example (Squares on a Chess Board): An 8×8 board is big—try 1×1, 2×2, 3×3 first.
Working with simpler cases gives insight and helps you devise a plan.

📋 Strategy 6: Work Systematically

Keep track of what you've figured out and what changes as the problem gets more complicated.
Use a table to organize information.
Example: Track how many 1×1, 2×2, 3×3 squares appear on each board size.

size of board	# of 1×1 squares	# of 2×2 squares	# of 3×3 squares
1 by 1	1	0	0
2 by 2	4	1	0
3 by 3	9	4	1

🧩 Strategy 7: Use Manipulatives to Help You Investigate

Sometimes drawing a picture isn't enough; actual materials you can move help a lot.
Example: Cut out 1×1, 2×2, 3×3 squares and move them across the chess board systematically.
This ensures you count everything once and don't count anything twice.

🔍 Strategy 8: Look for and Explain Patterns

When numbers are very large (e.g., 100×100 chess board), counting by hand is impractical.
Find a pattern in smaller cases, then extend it with a calculation.
Critical: Describe the patterns you see, then explain and justify them—how can you be sure they will continue?
Don't confuse: finding a pattern vs. proving it works.

🧮 Strategy 9: Find the Math, Remove the Context

Problems often have details that are unimportant, at least for getting started.
Identify the underlying math problem, solve it, then return to the original question.
Example (Broken Clock): Instead of worrying about how the clock breaks, focus on finding consecutive numbers that sum to the correct total.
After solving the math, check if it translates back to the original problem.

🔓 Strategy 10: Check Your Assumptions

It's easy to add extra constraints that aren't in the problem.
Ask yourself: "Am I constraining my thinking too much?"
Example (Broken Clock): Many people assume the clock must break radially (like pie slices) because the first solution does, but the problem doesn't require that.
The clock might break in other ways; don't limit yourself unnecessarily.

⚠️ Beware of Patterns!

⚠️ Patterns can be misleading

The "Look for Patterns" strategy is appealing but requires caution.
Don't forget the "and Explain" part: Not all patterns are obvious, and not all will continue.
Example (Dots on a Circle): Connecting 2 dots gives 2 pieces; 3 dots give 4 pieces. The pattern might suggest doubling, but you must verify this holds for larger cases.

🛡️ How to use patterns safely

Always describe the pattern clearly.
Explain and justify why the pattern works.
Verify it continues before applying it to larger cases.
The excerpt warns: "Not all patterns are obvious, and not all of them will continue."

🎓 Becoming a Better Problem Solver

🎓 The path to improvement

Problem solving is "more art than science."
The best way to become skilled: learn background material well and solve a lot of problems.
Learn how other people solved problems to build your strategy toolbox.
No strategy list will ever be complete; this is just a start.

🎯 The ultimate goal

At the end, you should be able to declare: "Here's my answer, and here is why I know it is correct!"
Never ask "Is this right?"—instead, prove to yourself (and others) that your answer is correct.
Pólya's step 4 (Look back) helps: How could your work be better? What did you learn?

Beware of Patterns!

🧭 Overview

🧠 One-sentence thesis

Patterns are powerful problem-solving tools, but they must be explained and tied to the problem's structure—not just observed—because not all patterns continue as expected.

📌 Key points (3–5)

Patterns can mislead: A pattern may appear to hold for several steps but then break down (e.g., the "Dots on a Circle" problem).
Observation alone is not enough: Mathematicians require an explanation of why a pattern occurs, not just that it appears to work.
Common confusion: Assuming the first pattern you notice is the only one—many different rules can generate the same initial sequence (e.g., 2, 4, 8, …).
How to verify a pattern: Tie the pattern to the problem's structure using clear reasoning; use variables to describe the pattern precisely.
Check your assumptions: Problems may have hidden constraints you add yourself (e.g., assuming the clock must break radially).

⚠️ Why patterns can fail

⚠️ The Dots on a Circle problem

Problem: Place n dots on a circle and connect every pair of dots. How many pieces does the circle divide into?

The natural approach: try small cases and look for a pattern.
What seems to happen: For 2, 3, 4, 5 dots, the number of pieces appears to double each time (2 → 4 → 8 → 16 → 32).
The breakdown: With 6 dots, you get only 30 or 31 pieces—not 32—no matter how you arrange them.
Lesson: The doubling pattern simply does not continue, even though it worked for the first several steps.

🔢 Multiple patterns from the same start

The sequence 2, 4, 8, … can be continued in many valid ways:

Pattern rule	Next terms	Explanation
Repeating cycle	2, 4, 8, 2, 4, 8, …	Cycle through 2, 4, 8 and repeat
Multiply previous two	2, 4, 8, 32, 256, 8192, …	Each term = product of the previous two
Powers of 2	2, 4, 8, 16, 32, 64, …	Double each time
Add increasing even numbers	2, 4, 8, 14, 22, 32, …	Add 2, then 4, then 6, then 8, etc.

Don't confuse: The first few terms matching does not mean the pattern is unique or will continue in the way you expect.
Without context or explanation, you cannot know which rule is "correct."

🔗 How to verify a pattern

🔗 Tie the pattern to the problem structure

Not enough: "I see the pattern continues this way."
Required: Explain why the pattern must continue based on the problem's constraints and structure.
Example: In the Chess Board problem, you might notice that a 5×5 board has 5² squares of size 1×1, 4² squares of size 2×2, etc.
The question: How do you know this pattern holds for larger boards or doesn't break down like "Dots on a Circle"?

🧮 Use a variable to describe the pattern clearly

Problem Solving Strategy 11 (Use a Variable!): Give quantities like "the number of squares" or "the number of dots" a name; they become much easier to work with.

Instead of saying "the number of k×k squares on an n×n board," use variables k and n explicitly.
Example from Chess Board:
- On a 5×5 board, there are (5 – k + 1)² squares of size k×k.
- On an n×n board, there are (n – k + 1)² squares of size k×k.
Writing the pattern with variables forces you to be precise and makes it easier to explain.

📐 Explain why the pattern works

Chess Board solution (from the excerpt):

Let n = side length of the board, k = side length of the square.
The square must fit: k ≤ n.
Horizontal movement: Place the k×k square in the upper left. It occupies k spaces, leaving (n – k) spaces to the right. You can slide it right (n – k) times, giving (n – k + 1) positions (counting the start).
Vertical movement: Shift the square down one row and repeat. There are (n – k) rows below, so (n – k + 1) total rows (counting the top).
Total squares of size k×k: (n – k + 1) rows × (n – k + 1) positions per row = (n – k + 1)².
Total squares on the board: Sum (n – k + 1)² for all k from 1 to n.

This explanation shows why the pattern must hold—it is not just an observation.

🧩 Check your assumptions

🧩 Hidden constraints you add yourself

Problem Solving Strategy 10 (Check Your Assumptions): When solving problems, it is easy to limit your thinking by adding extra assumptions that are not in the problem. Ask yourself: Am I constraining my thinking too much?

Clock problem example: If the first solution breaks the clock radially (all pieces meet at the center, like slicing a pie), many people assume all solutions must be radial.
Reality: The problem does not require radial breaks; the clock might break into non-radial pieces.
Lesson: Re-read the problem and identify assumptions you have added that are not stated.

🔍 When to stop looking

The excerpt asks: "How do I know when I am done? When should I stop looking?"
Answer: You are done when you have both:
1. Found a pattern or solution.
2. Explained why it works and verified it solves the original problem, not just a simplified version.
Example: Solving "which consecutive numbers sum to a target" is not the same as solving "how to break the clock into those pieces."

🛠️ Problem-solving strategies recap

The excerpt lists 11 strategies so far:

Wishful Thinking
Try Something!
Draw a Picture
Make Up Numbers
Try a Simpler Problem
Work Systematically
Use Manipulatives to Help you Investigate
Look for and Explain Patterns (emphasis on explain)
Find the Math, Remove the Context
Check Your Assumptions
Use a Variable

🧠 Key mindset for patterns

Mathematicians love patterns and get excited by them.
But: Mathematicians are also very skeptical of patterns.
If you cannot explain why a pattern would occur, do not just believe it will continue.
Always tie the pattern back to the problem's structure.

Careful Use of Language in Mathematics

🧭 Overview

🧠 One-sentence thesis

Mathematics is a social endeavor that requires precise, unambiguous language to communicate solutions and convince others that your work is complete and correct.

📌 Key points (3–5)

Problem solving has three components: solving the problem, convincing yourself it's correct, and convincing someone else it's correct—without the last step, you haven't really solved the problem.
Mathematical statements must be either true or false: they cannot be questions, commands, opinions, or paradoxes; they must be complete grammatical sentences.
"Or" and "and" have precise meanings: in mathematics, "or" always means "one or the other or both" (inclusive), while "and" means "both are true"—this differs from everyday English usage.
Common confusion—universal vs existential statements: universal statements ("every," "for all") require only one counterexample to disprove but need exhaustive checking or logical reasoning to prove; existential statements ("there is," "sometimes") need only one example to prove but require exhaustive checking or logical reasoning to disprove.
Conditional statements are false only in one case: "if hypothesis then conclusion" is false only when the hypothesis is true and the conclusion is false; in all other cases (including false hypothesis), the statement is true.

🗣️ Why mathematical communication matters

🗣️ Mathematics as a social endeavor

Mathematics is not just about solving problems on scratch paper—it requires sharing and verification.
The three essential components of problem solving work together:
1. Solving: scratch paper and careful thinking
2. Self-verification: checking your own work and asking yourself questions
3. Communicating: writing up the solution carefully or presenting it to others
If you cannot complete step 3 (convincing someone else), you have not truly solved the problem.
This is why precise language matters: mathematical communication cannot rely on context or implied meanings.

📖 Reading mathematics aloud

A good test for mathematical writing: try to read it out loud.
Even equations should read naturally, like English sentences.
Example: "1/2 = 2/4" reads as "one-half (subject) equals (verb) two-fourths (object)."
This is a complete, grammatically correct sentence, making it a valid mathematical statement.

📝 Mathematical statements

📝 What counts as a mathematical statement

A mathematical statement is a complete sentence that is either true or false, but not both at once.

Must be a complete, grammatically correct sentence (subject, verb, usually an object).
Cannot be a question, command, or matter of opinion.
Must have a definite truth value—either true or false (even if you don't know which).
Part of a mathematician's work is figuring out which sentences are true and which are false.

❌ What fails to be a mathematical statement

Examples from the excerpt that fail:

Questions: "Is your dog friendly?" (not a statement)
Commands: "Go to bed" (not a statement)
Opinions: "Blue is the prettiest color" or "UH Manoa is the best college in the world" (matters of opinion, not definite truth values)
Paradoxes: "This sentence is false" (self-contradictory—if true then false, if false then true; cannot be used as a mathematical statement)

Examples that succeed:

"60 is an even number" (true)
"Honolulu is the capital of Hawaii" (true)
"1/2 = 2/4" (true)
"All roses are red" (false, but still a valid statement)
"There are a total of 204 squares on an 8 × 8 chess board" (has a definite truth value)

🔍 Don't confuse: everyday sentences vs mathematical statements

Everyday sentences can be vague, context-dependent, or opinion-based.
Mathematical statements must have no ambiguity—they are definitively true or false.
You may not know which truth value applies, but the statement itself must have exactly one.

🔤 Precise meanings of "and" and "or"

🔤 Mathematical "or" vs everyday "or"

In mathematics, the word "or" always means "one or the other or both."

Everyday English: "After work, I will go to the beach, or I will do my grocery shopping" usually means exclusive or (one but not both).
Mathematical usage: "or" is always inclusive—it allows for both options to be true simultaneously.
Example: "x is odd or x is even" is true for x = 3 (odd), true for x = 4 (even), and would be true if a number could somehow be both (though that's impossible for integers).
Some people use "and/or" in everyday speech to clarify inclusive or, but in mathematics "or" alone always means this.

🔤 Mathematical "and"

The word "and" always means "both are true."

Both parts of the statement must be true for the whole statement to be true.
Example: "x is odd and x is even" is false for x = 3 (only odd, not even) and false for any integer (no integer is both).
Example: "x · 1 = x and x · 0 = x" is false for x = 3 because while 3 · 1 = 3 is true, 3 · 0 = 3 is false.

🔍 Don't confuse: context-dependent vs precise meanings

In everyday conversation, we rely on context to understand whether "or" is exclusive or inclusive.
In mathematics, we cannot rely on context or assumptions—the meaning must be unambiguous.
Always interpret "or" as inclusive and "and" as requiring both conditions.

🌐 Universal and existential statements

🌐 Universal statements

Universal statements claim something is always true, no matter what. They use phrases like:

"Every time…"
"For all numbers…"
"For every choice…"
"It's always true that…"

To prove a universal statement false: find one counterexample where it fails.

Example: "Every odd number is prime" is false—counterexample: 9 is odd but not prime.

To prove a universal statement true: either check every single case (often impossible with infinitely many cases) or find a logical reason why it must always be true.

Example: "Addition of real numbers is commutative" requires logical reasoning, not checking every pair of numbers.

🌐 Existential statements

Existential statements claim there is at least one example where the statement is true. They use phrases like:

"Sometimes…"
"There is some number…"
"For some choice…"
"At least once…"

To prove an existential statement true: find just one example where it works.

Example: "There is some number x such that x + 1 = 7" is true—example: x = 6.

To prove an existential statement false: show it fails in every single case (often impossible) or find a logical reason why it cannot be true.

🔍 Don't confuse: what it takes to prove vs disprove

Statement type	To prove TRUE	To prove FALSE
Universal ("every," "for all")	Check all cases OR logical reasoning	Find ONE counterexample
Existential ("there is," "sometimes")	Find ONE example	Check all cases OR logical reasoning

The burden of proof is opposite for the two types.
Universal statements are vulnerable to a single counterexample.
Existential statements are proven by a single example.

🔍 Hidden quantifiers

Some statements hide the quantifier in the meaning of the words.
Example: "Addition of real numbers is commutative" means "For all real numbers a and b, a + b = b + a" (universal).
Example: "The points (1, 1), (2, 1), and (3, 0) all lie on the same line" is a universal statement about those specific points (checking if they all satisfy a condition).

🔀 Conditional statements

🔀 Structure of conditional statements

A conditional statement can be written in the form "If some statement then some statement," where the first statement is the hypothesis and the second statement is the conclusion.

Not all conditional statements are written in "if/then" form, but they can be rewritten that way.
Example: "An integer n is even if it is a multiple of 2" can be rewritten as "If n is a multiple of 2, then n is even."
Example: "The team wins when JJ plays" can be rewritten as "If JJ plays, then the team wins."

🔀 When is a conditional statement true or false?

Key principle: Think of a conditional statement like a promise. When is the promise broken?

A conditional statement is false only when the hypothesis is true and the conclusion is false.

In all other cases, the statement is true:

Hypothesis	Conclusion	Statement truth value	Why
True	True	TRUE	Promise kept
True	False	FALSE	Promise broken
False	True	TRUE	Promise not broken (didn't claim anything about this case)
False	False	TRUE	Promise not broken (didn't claim anything about this case)

🔀 Understanding false hypothesis cases

Example from the excerpt: "If I win the lottery, then I'll give each of my students $1,000."

If I do NOT win the lottery (false hypothesis), the statement is true regardless of whether I give money or not.
I only break my promise if I win the lottery (true hypothesis) and then don't give the money (false conclusion).

Another example: "If you live in Honolulu, then you live in Hawaii."

For someone who lives in Honolulu: hypothesis true, conclusion true → statement true.
For someone who lives in Los Angeles: hypothesis false → statement automatically true (doesn't matter where they live).
The statement is really a universal statement: "Everyone who lives in Honolulu lives in Hawaii."

🔀 Proving conditional statements false

To show a universal conditional statement is false, give a specific instance where the hypothesis is true and the conclusion is false.

Example: "If you are a good swimmer, then you are a good surfer."

Find one person who is a good swimmer (true hypothesis) but not a good surfer (false conclusion).
That one counterexample proves the statement false.

🔍 Don't confuse: false hypothesis with false statement

A conditional statement with a false hypothesis is still TRUE (not false).
Example: "If 2 + 2 = 5, then all odd numbers are prime" is TRUE because the hypothesis is false.
Example: "If 2 + 2 = 4, then all odd numbers are prime" is FALSE because the hypothesis is true but the conclusion is false.
The truth of the overall statement depends on the combination of hypothesis and conclusion truth values, not just one or the other.

✍️ Writing and explaining mathematical work

✍️ The goal of mathematical writing

Mathematical writing is different from poetry or English papers.
The goal is clarity, not flowery description.
If your reader does not understand, you have not done a good job.
Writing is an essential part of the problem-solving process—it could be considered "Pólya's step 5."

✍️ Professional context

Professional mathematicians write journal articles, books, and grant proposals.
Teachers explain mathematical ideas to students both in writing and orally.
Explaining your work is essential to completing the problem-solving process.

Explaining Your Work

🧭 Overview

🧠 One-sentence thesis

Mathematical writing is fundamentally about clarity and justification—you must explain your reasoning so clearly that your reader understands and accepts your solution, not just present an answer.

📌 Key points (3–5)

Mathematics is social: solving a problem is incomplete until you explain it to someone else and they understand and agree.
Goal is clarity, not length: mathematical writing prioritizes clear explanation over elaborate description; conciseness and correctness matter more than volume.
Justification is required: you cannot simply state an answer—you must show how you know it is correct and explain your reasoning.
Common confusion: scratch work vs. final write-up—the messy trial-and-error process is not what you share; you must rewrite your solution cleanly after solving.
Reader-centered approach: making the reader's job easier (clear answer, defined variables, logical flow) is a primary goal.

📝 The social nature of mathematics

🤝 Why explanation is essential

The excerpt states that "you have not really solved the problem until you have explained your work to someone else, and they sign off on it."
Mathematics is described as "a social endeavor" even when you work alone.
Professional mathematicians write articles, books, and proposals; teachers explain ideas to students both in writing and orally.
The excerpt suggests explaining your work "probably should have been Pólya's step 5" in the problem-solving process.

🎯 The goal of mathematical writing

The goal of mathematical writing is not florid description, but clarity.

If your reader does not understand, you have not done a good job.
This is different from poetry or English papers—the emphasis is on making the solution understandable, not on elaborate language.

🛠️ Preparing your solution

🗑️ Do not turn in scratch work

When solving problems (not exercises), you will have many false starts, things that don't work, and mistakes.
Once you find an idea that solves the problem, that scribbled paper is not what you turn in.
You must take that idea and "write it up carefully, neatly, and clearly."
Don't confuse: the messy exploration process vs. the polished final explanation—only the latter should be shared.

📋 (Re)state the problem

Do not assume your reader knows what problem you are solving, even if it is the teacher who assigned it.
If the problem has a long description, you can summarize it, but make sure the question is clear.
You don't need to rewrite it word-for-word or give all details, but the problem must be stated.

✅ Clearly give the answer

State the answer prominently; it's not a bad idea to give it right up front, then show the justifying work.
This helps the reader know what you are trying to justify as they read, making their job easier.
The answer should be clearly stated somewhere and easy to find.

🔍 Content requirements

✔️ Be correct

Everyone makes mistakes while working, but the final write-up (after solving) must be correct.
"The best writing in the world cannot save a wrong approach and a wrong answer."
Check your work carefully; ask someone else to read your solution with a critical eye.

🧾 Justify your answer

You cannot simply give an answer and expect the reader to "take your word for it."
You must explain how you know your answer is correct.
This means showing your work, explaining your reasoning, and justifying what you say.

✂️ Be concise

"There is no bonus prize for writing a lot in math class."
Think clearly and write clearly.
If you find yourself going on and on, stop, rethink what you want to say, and start over.

🧮 Using mathematical tools

🔢 Use variables and equations

An equation can be much easier to read than a long paragraph describing a calculation.
Mathematical writing often has fewer words and more equations than other kinds of writing.

🏷️ Define your variables

Always say what a variable stands for before you use it.
If you use an equation, say where it comes from and why it applies to this situation.
Do not make your reader guess.
Example: Instead of writing "Let x be the solution," write "Let x represent the number of items purchased."

🖼️ Use pictures

If pictures helped you solve the problem, include nice versions in your final solution.
Even if you didn't draw a picture to solve it, a picture might still help your reader understand.
Remember: your goal is to help the reader understand.

📐 Presentation and formatting

📖 Use correct spelling and grammar

Proofread your work.
A good test: read your work aloud, including equations and calculations—there should be complete, natural-sounding sentences.
Be especially careful with pronouns; avoid using "it" and "they" for mathematical objects—use the names of the objects (or variables) instead.

🎨 Format clearly

Do not write one long paragraph; separate your thoughts.
Put complicated equations on a single displayed line rather than in the middle of a paragraph.
Do not write too small.
Do not make your reader struggle to read and understand your work.

🤝 Acknowledge collaborators

If you worked with someone else on solving the problem, give them credit.

📊 Evaluating solutions (example analysis)

🔬 The eight-digit problem

The excerpt presents a problem about finding the largest eight-digit number using digits 1, 1, 2, 2, 3, 3, 4, 4 with specific separation rules, then shows nine student solutions for evaluation.

📝 Range of solution quality

The excerpt asks readers to score solutions on a scale of 1 to 5 using the criteria above. The solutions demonstrate varying levels of:

Quality aspect	What varies across solutions
Justification	Some give no reasoning; others explain step-by-step logic
Clarity	Some are cryptic abbreviations; others use complete sentences
Process explanation	Some show trial-and-error thinking; others present only final logic
Formatting	Some use visual aids (spacing, diagrams); others are dense text

Solution 1 gives the answer with minimal justification ("b/c the #s...all separated").
Solution 3 provides detailed step-by-step reasoning for each digit placement.
Solution 8 uses clear formatting with progressive steps showing the number being built.
Don't confuse: showing your thinking process (good) vs. turning in unorganized scratch work (bad)—solutions that explain trial-and-error clearly are still acceptable if well-organized.

The Last Step

🧭 Overview

🧠 One-sentence thesis

After solving a mathematical problem, asking new questions is as valuable and difficult as solving the original problem, and it is a core mathematical habit that professional mathematicians practice.

📌 Key points (3–5)

Core claim: Asking good questions is as valuable (and as difficult) as solving mathematical problems, though this is rarely acknowledged outside professional mathematics.
When to ask: After solving a problem and explaining your solution, ask yourself "What other questions can I ask?"
Types of follow-up questions: Generalize parameters (e.g., different board sizes), change constraints (e.g., rectangles instead of squares), or find all possibilities instead of just one.
Common confusion: Problem-solving discussions (from Polya to Common Core) focus on solving problems, but rarely emphasize that asking good questions is equally important.

🔍 The overlooked skill

🔍 What is rarely acknowledged

Many sources (Polya, Common Core State Standards, and others) discuss problem solving in mathematics.
However, one fact is rarely acknowledged except by professional mathematicians: asking good questions is as valuable and as difficult as solving mathematical problems.
This is not just a nice-to-have skill; it is central to mathematical practice.

🧠 The habit to develop

After solving a mathematical problem and explaining your solution to someone else, it is a very good mathematical habit to ask yourself: What other questions can I ask?

This is positioned as "the last step" in problem-solving—not finishing when you have an answer, but continuing to explore.
The habit applies after you have both solved and explained the problem.

🎯 Types of follow-up questions

🎯 Generalizing parameters

Squares on a Chess Board example: The original problem asks how many squares of any size are on an 8 × 8 chess board (answer: not 64, much bigger).
Obvious follow-up: What about a 10 × 10 board? Or 100 × 100? Or any size?
This type of question changes a specific number to a variable or explores different scales.

🔄 Changing the object or constraint

Rectangles instead of squares: How many rectangles of any size and shape can you find on an 8 × 8 chess board?
- The excerpt notes this is "a lot harder" because rectangles come in many different dimensions (1 × 2, 5 × 3, etc.).
- Example: Instead of counting only squares, count all possible rectangles—this requires a different counting strategy.
Triangles in a picture: How many triangles of any size and shape can you find in a given picture?
- This shifts from squares/rectangles to a completely different shape.

🔢 Finding all possibilities

Broken Clock example: The original problem asks if you can break a clock into pieces so the sums of numbers on each piece are consecutive numbers, and asks for one other way beyond the given example.
Follow-up question: "Find every possible way to break the clock into some number of pieces so that the sums of the numbers on each piece are consecutive numbers. Justify that you have found every possibility."
This shifts from finding one solution to finding all solutions and proving completeness.

🧩 Why this matters

🧩 Professional mathematical practice

The excerpt explicitly states that professional mathematicians recognize asking good questions as valuable and difficult.
This suggests that the skill is not just pedagogical but reflects real mathematical work.

🧩 Active engagement

The excerpt includes a "Think / Pair / Share" prompt: Choose a problem from the Problem Bank (preferably one you have worked on) and ask what follow-up or similar questions you could ask.
This reinforces that the skill is meant to be practiced, not just understood in theory.

🧩 Don't confuse: solving vs. asking

Solving a problem: Finding an answer to a given question.
Asking a question: Identifying new, interesting questions after solving the original one.
The excerpt emphasizes that both are equally valuable and difficult, but asking questions is often overlooked in problem-solving discussions.

Dots and Boxes

🧭 Overview

🧠 One-sentence thesis

The "exploding dots" game reveals that place value systems work by grouping dots according to different rules, and changing the grouping rule (such as 2←1, 1←3, or 1←10) generates different number representations including binary and decimal notation.

📌 Key points (3–5)

Core mechanism: whenever a box accumulates a threshold number of dots, they "explode" and become one dot in the box to the left, creating positional value.
Different rules create different bases: the 1←2 rule produces binary (base two), the 1←10 rule produces decimal, and other rules produce other base systems.
Position determines value: in the 1←2 system, boxes from right to left are worth 1, 2, 4, 8, etc. (powers of two); in 1←3 they are worth 1, 3, 9, 27, etc.
Common confusion: the code "1001" in base two is not "one thousand and one"—it means one 8, zero 4s, zero 2s, and one 1, totaling nine dots.
Real-world application: computers use binary (1←2) because transistors are either on (1) or off (0), and 64 bits can represent numbers from 0 to over 18 quintillion.

🎲 The exploding dots mechanism

🎲 The 2←1 rule (original example)

The 2←1 Rule: Whenever there are two dots in a single box, they "explode," disappear, and become one dot in the box to the left.

Start with dots in the rightmost box.
Repeatedly find any box with two or more dots, explode pairs into the left box, until no box has two or more dots.
Example: Nine dots → after explosions → code 1001 (one dot in the leftmost box, zero, zero, one in the rightmost).
Example: Seven dots → code 111.

🔄 Other grouping rules

The excerpt introduces variations:

Rule	Threshold	What it means
2←1	2 dots	Two dots explode into one dot to the left
1←3	3 dots	Three dots explode into one dot to the left
1←4	4 dots	Four dots explode into one dot to the left
1←5	5 dots	Five dots explode into one dot to the left
1←9	9 dots	Nine dots explode into one dot to the left
1←10	10 dots	Ten dots explode into one dot to the left

Example: Fifteen dots in the 1←3 system → code 120 (one dot in the third box, two in the second, zero in the first).
Example: Twenty dots in the 1←3 system → code 202.

⚠️ Validity of codes

In the 1←3 system, each box can only hold 0, 1, or 2 dots (because three would explode).
Question from the excerpt: "Is it possible for a collection of dots to have 1←3 code 2031?" The answer is no, because the digit "3" cannot appear—three dots would have already exploded.

🔢 Binary numbers (the 1←2 system)

🔢 How position determines value

In the 1←2 rule, two dots in the rightmost box (value 1 each) become one dot in the next box → that box is worth 2.
Two dots worth 2 each explode into one dot worth 4.
Two dots worth 4 each explode into one dot worth 8.
Pattern: boxes from right to left are worth 1, 2, 4, 8, 16, 32, … (each box is double the previous).

🧮 Reading binary codes

The code 1001 in base two means: (1 × 8) + (0 × 4) + (0 × 2) + (1 × 1) = 9 dots.
The code 1010 in base two means: (1 × 8) + (0 × 4) + (1 × 2) + (0 × 1) = 10 dots.
Don't confuse: when reading 1001₂, say "one zero zero one base two," not "one thousand and one" (because "thousand" is a decimal concept).

🔤 Definition and notation

Binary numbers or base two numbers: numbers written in the 1←2 code.

A subscript "two" indicates base two: 1001₂ means nine dots.
Binary numbers only contain the digits 0 and 1, because any box with two or more dots would explode.

💻 Why computers use binary

Computers represent numbers in binary because transistors (the basic units) are either on (1) or off (0).
One transistor stores one bit of information.
Eight bits make a byte.
A typical home computer CPU uses 64-bit registries (8 bytes).
With 64 bits, the 1←2 rule can represent numbers from 0 to 18,446,744,073,709,551,615.

🧩 Other base systems

🧩 The 1←3 system (base three)

Three dots in one box explode into one dot in the next box.
Boxes from right to left are worth 1, 3, 9, 27, … (each box is triple the previous).
Example: code 120 in base three means (1 × 9) + (2 × 3) + (0 × 1) = 15 dots.
Valid digits: only 0, 1, 2 (because three would explode).

🔟 The 1←10 system (decimal)

Ten dots in one box explode into one dot in the next box.
This is the familiar decimal system.
Boxes from right to left are worth 1, 10, 100, 1000, …
Example: thirteen dots → code 13 in base ten.
Example: two hundred thirty-eight dots → code 238 in base ten.
Example: five thousand eight hundred thirty-three dots → code 5833 in base ten.

🎯 General pattern

The rule "1←N" creates base N.
In base N, boxes are worth 1, N, N², N³, … from right to left.
Valid digits in base N: 0 through N−1 (because N dots would explode).

🛠️ Working with the system

🛠️ Converting from code to number of dots

Method: multiply each digit by its box value and add.
Example: 100101₂ = (1 × 32) + (0 × 16) + (0 × 8) + (1 × 4) + (0 × 2) + (1 × 1) = 37 dots.
The excerpt suggests testing by "unexploding" dots (reversing the process).

🛠️ Converting from dots to code

Method: repeatedly explode dots according to the rule until no box exceeds the threshold.
The excerpt challenges readers to find a general method "without actually going through the exploding dot process."
Example tasks: find the 1←2 code for two dots, seventeen dots, sixty-three dots, one hundred dots.

🛠️ Practice problems

The excerpt includes exercises such as:

Find the 1←4 code for thirteen dots.
Find the 1←5 code for thirteen dots and for five dots.
Find the 1←9 code for thirteen dots and for thirty dots.
What number of dots has 1←3 code 101?
What number of dots has 1←3 code 1022?

Other Rules

🧭 Overview

🧠 One-sentence thesis

Changing the explosion rule from "2 dots explode into 1" to other numbers (like 3, 4, 5, 9, or 10) creates different number systems that encode the same quantity of dots in different ways.

📌 Key points (3–5)

The core mechanism: instead of the 2←1 rule, you can use 1←3 (three dots explode into one), 1←4, 1←5, etc., and each rule produces a different code for the same number of dots.
How codes differ: the same quantity (e.g., thirteen dots) will have different codes depending on which explosion rule you use.
Special case—1←10: this rule produces codes that match our everyday decimal numbers.
Common confusion: the number of dots stays the same; only the code representation changes when you switch rules.
Why it matters: understanding different explosion rules reveals that our familiar base-ten system is just one choice among many possible number systems.

🎲 How different explosion rules work

🎲 The 1←3 rule

The 1←3 Rule: Whenever there are three dots in a single box, they "explode," disappear, and become one dot in the box to the left.

Instead of pairs exploding (as in 2←1), now triples explode.
Example: Fifteen dots in the 1←3 system produces code 120.
- You keep exploding groups of three until no box has three or more dots.
The excerpt shows that twenty dots has 1←3 code 202.

🔢 Other explosion rules (1←4, 1←5, 1←9)

1←4 rule: four dots in a box explode into one dot in the next box to the left.
1←5 rule: five dots explode into one.
1←9 rule: nine dots explode into one.
Each rule produces a unique code for the same collection of dots.
Example: Thirteen dots will have different codes under 1←3, 1←4, 1←5, and 1←9 systems.

🔟 The 1←10 rule—our familiar system

1←10 rule: ten dots in a box explode into one dot in the next box.
The excerpt asks:
- What is the 1←10 code for thirteen dots?
- What is the 1←10 code for thirty-seven dots?
- What is the 1←10 code for two hundred thirty-eight dots?
- What is the 1←10 code for five thousand eight hundred and thirty-three dots?
The "Think / Pair / Share" prompt hints that students should recognize something familiar happening in Problem 6.
Key insight: the 1←10 system produces the codes we use every day (our decimal place-value system).

🧮 Understanding what the codes mean

🧮 Each box has a value

Just as in the 2←1 system (where boxes represent 1, 2, 4, 8…), each explosion rule assigns values to boxes.
In a 1←3 system, boxes would represent powers of three (1, 3, 9, 27…).
In a 1←10 system, boxes represent powers of ten (1, 10, 100, 1000…).

✅ Checking validity of codes

The excerpt asks: "Is it possible for a collection of dots to have 1←3 code 2031?"
This requires understanding the rule: in a 1←3 system, no box can have 3 or more dots (they would have exploded).
So valid 1←3 codes can only contain digits 0, 1, or 2.
Don't confuse: a code like 2031 contains a "3," which means three dots are still in one box—but they should have exploded under the 1←3 rule, so this code is impossible.

🔄 Converting between dots and codes

🔄 From dots to code

Start with all dots in the rightmost box.
Apply the explosion rule repeatedly until no box has enough dots to explode.
Read the final configuration from left to right to get the code.

🔄 From code to dots

Each digit in the code tells you how many dots are in that box.
Multiply each digit by its box value and add them up.
Example (from the excerpt): In the 1←3 system, code 120 means 1 dot in a high-value box, 2 dots in the next box, and 0 in the rightmost box, totaling fifteen dots.

🔄 Practice problems structure

The excerpt provides systematic practice:

Problem	Rule	Task
Problem 2	1←3	Find codes for 20, 13, 25 dots; find dot count for code 1022; check if code 2031 is valid
Problem 3	1←4	Describe the rule; find code for 13 dots
Problem 4	1←5	Find codes for 13 and 5 dots
Problem 5	1←9	Find codes for 13 and 30 dots
Problem 6	1←10	Find codes for 13, 37, 238, and 5833 dots

Binary Numbers

🧭 Overview

🧠 One-sentence thesis

The 1←2 "exploding dots" rule generates binary (base two) numbers by assigning each position a power-of-two value, and this same positional logic extends to any base system.

📌 Key points (3–5)

The 1←2 rule: two dots in any box "explode" and become one dot in the box to the left, creating a positional system where each box is worth double the previous box (1, 2, 4, 8, …).
Binary representation: numbers in base two use only the digits 0 and 1 because the rule never allows two or more dots to remain in a single box.
Reading binary correctly: 1001₂ is read "one zero zero one base two," not "one thousand and one," because place values are powers of two, not powers of ten.
Common confusion: the same numeral (e.g., "1001") means different quantities in different bases—1001₂ = nine dots, but 1001₁₀ = one thousand and one dots.
Why it matters: computers use binary because transistors have two states (on/off), and the same positional logic applies to any base b system (base three, base four, base ten, etc.).

🔢 How the 1←2 rule builds place values

💥 The explosion mechanism

The 1←2 Rule: Whenever there are two dots in a single box, they "explode," disappear, and become one dot in the box to the left.

Each original dot is worth "one."
Two dots in the right-most box → one dot in the next box to the left, so that box is worth two.
Two dots in the "2" box → one dot in the next box, so that box is worth two 2's = four.
Two dots in the "4" box → one dot in the next box, so that box is worth two 4's = eight.
This pattern continues: each box to the left doubles in value.

📦 Box values grow by powers of two

Box position (right to left)	Value
1st (rightmost)	1
2nd	2
3rd	4
4th	8
5th	16
nth	2 to the power of (n minus 1)

Example: Nine dots in the 1←2 system has code 1001₂.
- Check: one dot in the "8" box + zero in the "4" box + zero in the "2" box + one in the "1" box = 8 + 0 + 0 + 1 = 9. ✓
Example: Ten dots has code 1010₂.
- Check: 8 + 0 + 2 + 0 = 10. ✓

🧮 Binary numbers and their properties

🔤 Definition and notation

Binary numbers or base two numbers: numbers written in the 1←2 code.

A subscript "two" indicates base two: 1001₂ means "the number of dots that has 1←2 code 1001," which is nine.
Important: Read 1001₂ as "one zero zero one base two," not "one thousand and one," because "thousand" is a base-ten concept.

🎯 Why only 0 and 1?

The 1←2 rule never allows two or more dots to remain in a single box—they always explode into the next box.
Therefore, each box can only contain zero dots (digit 0) or one dot (digit 1).
This is why binary numbers contain only the digits 0 and 1.

🔄 General methods

To find the number of dots from a binary number:

Multiply each digit by its box value (power of two) and add them up.
Example: 100101₂ = (1 × 32) + (0 × 16) + (0 × 8) + (1 × 4) + (0 × 2) + (1 × 1) = 32 + 4 + 1 = 37 dots.

To find the binary code from a number of dots (without exploding):

The excerpt asks students to develop this method themselves.
The goal is to determine which powers of two sum to the target number, then place 1's in those positions and 0's elsewhere.

💻 Why binary matters: computers

🖥️ Transistors and bits

Computers represent numbers in binary because the basic units are transistors, which are either on (1) or off (0).
One transistor stores one bit of information.
Eight bits make a byte.
A typical home computer's central processing unit uses 64-bit registries (8 bytes).

📊 Capacity of 64 bits

Using the 1←2 rule, 64 bits can represent numbers from 0 through 18,446,744,073,709,551,615.
This enormous range comes from the exponential growth of powers of two.

🌐 Other bases: extending the logic

🔢 The 1←b rule generalizes

Numbers written in the 1←b system are called base b numbers. In a base b number system, each place represents a power of b.

The same "exploding dots" logic applies to any base.
In the 1←3 system: three dots in one box → one dot in the box to the left.
In the 1←4 system: four dots in one box → one dot in the box to the left.
In the 1←10 system: ten dots in one box → one dot in the box to the left.

🧩 Base three (1←3 system)

Box values (right to left): 1, 3, 9, 27, 81, …
Each box is worth three times the previous box.
Example: The 1←3 code for fifteen is 120₃.
- Check: (1 × 9) + (2 × 3) + (0 × 1) = 9 + 6 + 0 = 15. ✓

🧩 Base four (1←4 system)

Box values (right to left): 1, 4, 16, 64, 256, …
Each box is worth four times the previous box.

🧩 Base ten (1←10 system)

Box values (right to left): 1, 10, 100, 1,000, 10,000, …
Each box is worth ten times the previous box.
This is the standard decimal system we use every day.
Example: In the number 7,842:
- The "7" represents seven groups of 1,000.
- The "8" represents eight groups of 100.
- The "4" represents four groups of 10.
- The "2" represents two groups of 1.
Why we use base ten: The excerpt implies this is the standard system for writing numbers (likely because humans have ten fingers, though the excerpt does not state this reason explicitly).

🔍 Don't confuse: same digits, different bases

The numeral "120" means different quantities depending on the base:
- 120₃ = (1 × 9) + (2 × 3) + (0 × 1) = 15 dots.
- 120₁₀ = (1 × 100) + (2 × 10) + (0 × 1) = 120 dots.
Always check the subscript to know which base is being used.

Other Bases

🧭 Overview

🧠 One-sentence thesis

Any whole number can be represented in different base systems (base 2, base 3, base 4, etc.), where the base determines how many dots trigger an "explosion" to the next position and each position represents a power of that base.

📌 Key points (3–5)

What a base-b system means: In a 1←b system, b dots in one box are worth one dot in the box one place to the left; each position represents a power of b.
Reading vs saying base numbers: When reading binary like 1001₂, say "one zero zero one base two," not "one thousand and one," because "thousand" is not a binary concept.
Positional value structure: The right-most place is ones (b⁰), the second is b (b¹), the third is b² (b times b), the fourth is b³, and so on.
Common confusion—notation: A subscript (e.g., 102₃) indicates the code for a number of dots in that base, not the actual quantity; without a subscript, assume base ten.
Two conversion methods: Convert base ten to base b either left-to-right (find largest power of b first) or right-to-left (repeatedly divide by b and track remainders).

🔢 Understanding base systems

🔢 What the 1←b rule means

In a 1←b system, b dots in one box are worth one dot in the box one place to the left.

This "explosion" rule defines the base.
Example: In 1←3 (base three), three dots in a box become one dot in the next box to the left.
Example: In 1←4 (base four), four dots in a box become one dot in the next box to the left.

📦 Box values and powers of b

Each box position corresponds to a power of the base b.
The right-most box is always the ones place (b⁰ = 1).
The second box from the right is the b place (b¹ = b).
The third box is the b² place (b times b).
The fourth box is the b³ place (b times b times b), and so on.

Example (base three):

Right-most box: 1
Second box: 3
Third box: 9 (which is 3 × 3)
Fourth box: 27 (which is 3 × 3 × 3)

🔤 Which digits are used

Base two (binary): only digits 0 and 1
Base three: digits 0, 1, 2
Base four: digits 0, 1, 2, 3
Base b in general: digits 0 through b−1

The excerpt explains that binary numbers only contain 0 and 1 because in the 1←2 system, you can have at most one dot in a box before it "explodes" into the next position.

🏷️ Notation and reading

Whenever dealing with numbers in different bases, use a subscript to indicate the base so there is no confusion.

102₃ is read "one-zero-two base three" (the base three code for eleven dots).
222₄ is read "two-two-two base four" (the base four code for forty-two dots).
54321₁₀ is a base ten number (you can say "fifty-four thousand three hundred and twenty-one").
If no subscript is written, assume base ten.
Don't confuse: The subscript tells you the code (representation), not the actual number of dots.

🔄 Converting from base b to base ten

🔄 The multiplication method

To find how many dots a base b number represents, multiply each digit by its positional value (the power of b for that position) and add them up.

Example from the excerpt:

The number 4132₅ in base five represents:
- 4 dots in the b³ box: 4 × 5³ = 4 × 125 = 500
- 1 dot in the b² box: 1 × 5² = 1 × 25 = 25
- 3 dots in the b¹ box: 3 × 5 = 15
- 2 dots in the ones box: 2 × 1 = 2
- Total: 500 + 25 + 15 + 2 = 542 dots

🧮 Why this works

Each position holds a certain number of dots, and each dot in that position is worth a power of b.
You are simply "unexploding" the dots by calculating the total value without drawing the picture.

➡️ Converting from base ten to base b

➡️ Method 1: Left-to-right (largest power first)

Steps:

Start with your base ten number n.
Find the largest power of b that is less than or equal to n (call it b^k).
Determine how many times b^k fits into n without going over (call this digit a); put a in the b^k position.
Subtract a × b^k from n to find how many dots remain.
Repeat with the remaining dots and the next smaller power of b.
Continue until all dots are accounted for; fill any unused positions with 0.

Example from the excerpt (321₁₀ to base five):

Powers of five: 1, 5, 25, 125, 625…
Largest power ≤ 321 is 125 (5³).
321 ÷ 125 = 2 remainder 71 → put 2 in the 125 box.
71 ÷ 25 = 2 remainder 21 → put 2 in the 25 box.
21 ÷ 5 = 4 remainder 1 → put 4 in the 5 box.
1 ÷ 1 = 1 remainder 0 → put 1 in the 1 box.
Result: 2241₅

⬅️ Method 2: Right-to-left (repeated division)

Steps:

Divide the base ten number by b to get a quotient and a remainder.
The remainder is the right-most digit in the base b number.
If the quotient is less than b, it becomes the next digit to the left; otherwise, repeat step 1 with the quotient.
Continue filling in remainders from right to left until the quotient is zero or less than b.

Example from the excerpt (712₁₀ to base seven):

712 ÷ 7 = 101 remainder 5 → right-most digit is 5
101 ÷ 7 = 14 remainder 3 → next digit is 3
14 ÷ 7 = 2 remainder 0 → next digit is 0
2 < 7 → left-most digit is 2
Result: 2035₇

Check: 2×7³ + 0×7² + 3×7¹ + 5×7⁰ = 2×343 + 0 + 21 + 5 = 686 + 21 + 5 = 712 ✓

🔍 Comparing the two methods

Method	Direction	Key idea	When to use
Method 1	Left-to-right	Find largest power of b first, then work down	Good for understanding place value structure
Method 2	Right-to-left	Repeatedly divide by b, track remainders	Often faster for computation

Both methods produce the same answer; choose whichever makes more sense to you.

💻 Why base systems matter

💻 Binary and computers

For computers, numbers are always represented in binary. The basic units are transistors which are either on (1) or off (0).

A transistor stores one bit of information (either 0 or 1).
Eight bits make a byte.
A typical home computer's central processing unit uses 64-bit registries (8 bytes).
Using the 1←2 rule, 64 bits can represent numbers from 0 through 18,446,744,073,709,551,615.

📐 Positional number systems

A positional number system has unique symbols for 1 through b−1 (and 0), where b is the base. The positional value of each symbol depends on its position: first position is face value, second is b times face value, third is b² times face value, and so on. The value of a number is the sum of the positional values of its digits.

Both base ten (our everyday system) and base b numbers are positional systems.
The position of a digit determines its value, not just the digit itself.
Example: In base ten, the "7" in 7,842 represents 7 groups of 1,000; the "8" represents 8 groups of 100; the "4" represents 4 groups of 10; the "2" represents 2 groups of 1.

🆚 Contrast with additive systems

In an additive number system, the value of a written number is the sum of the face values of the symbols. The only symbol necessary is a symbol for 1, though many additive systems contain other symbols.

Roman numerals are an example of an additive system (with a subtraction twist).
In Roman numerals: I=1, V=5, X=10, L=50, C=100, D=500, M=1,000.
Example: MMXIII = 1,000 + 1,000 + 10 + 1 + 1 + 1 = 2,013.
Roman numerals use subtraction when a smaller symbol appears before a larger one (e.g., IX = 9, not VIIII).
Don't confuse: Positional systems use position to determine value; additive systems simply add (or subtract) face values.
Very large numbers become impractical in additive systems (one million would require one thousand M's).

Number Systems

🧭 Overview

🧠 One-sentence thesis

Different number systems—positional (like base ten and other bases), additive (like Roman numerals), and mixed systems—organize and represent quantities in distinct ways, and understanding these systems reveals that properties like "even" depend on quantity itself, not on how we write numbers.

📌 Key points (3–5)

Positional vs additive systems: positional systems use place value where position determines value (base ten, base b), while additive systems sum face values of symbols (Roman numerals).
Base conversion methods: you can convert base ten to another base by repeatedly dividing by the base and recording remainders, or by "exploding" groups.
Historical context: many cultures developed their own systems (Babylonian base 60, Mayan base 20, Hawaiian mixed base 4/10) before the Hindu-Arabic system spread globally.
Common confusion: evenness is a property of the quantity itself, not the representation—the rule for recognizing even numbers changes depending on the base you're using.
Why zero matters: early positional systems used blank spaces instead of zero, creating ambiguity about whether a number was 23, 203, or 2003.

📐 Positional number systems

📐 What makes a system positional

A positional number system is one way of writing numbers. It has unique symbols for 1 through b – 1, where b is the base of the system. Modern positional number systems also include a symbol for 0. The positional value of each symbol depends on its position in the number.

The first position has face value
The second position is b times the face value
The third position is b squared times the face value
The value of a number is the sum of all positional values

🔄 How position determines value

Same symbol means different amounts depending on where it sits
Example: In base ten, the "2" in 203 represents two hundreds, not just two
This is fundamentally different from additive systems where symbols always mean the same amount

🔢 Converting between bases

🔢 The division method

Divide the base ten number by b to get a quotient and remainder
Put the remainder in the right-most space
If the quotient is less than b, it goes one spot to the left; otherwise repeat step 1 with the quotient
Fill in remainders from right to left

💥 The "explode" method

Group dots according to the target base
When you have enough groups to make a full set of the base, "explode" them one box to the left
Leave behind any dots that don't form a complete group
Repeat until fewer than b dots remain in each box
Example: Converting 101 to base seven—group into 7s, explode groups left, leave 3 behind; repeat until done

✅ Checking your work

Convert the answer back to base ten to verify
The excerpt emphasizes this as something "we can (and should!) check"

🏛️ Additive number systems

🏛️ How additive systems work

In an additive number system, the value of a written number is the sum of the face values of the symbols that make up the number. The only symbol necessary for an additive number system is a symbol for 1, however many additive number systems contain other symbols.

🏺 Roman numerals as an example

Symbols: I (1), V (5), X (10), L (50), C (100), D (500), M (1,000)
The number 2013 is MMXIII: two thousands, one ten, three ones
Efficiency trick: if a smaller symbol appears left of a larger one, subtract instead of add (IX = 9, not VIIII)

⚠️ Limitations

Very large numbers become impractical—one million would need one thousand M's
Without subtraction rules, even medium numbers require many symbols

🌍 Historical number systems

🌍 Early positional systems

Babylonians: used base 60
Mayans: used base 20
Both developed before they had a symbol for zero
Used blank spaces instead, which created ambiguity

📚 Spread of Hindu-Arabic system

Muḥammad ibn Mūsā al-Khwārizmī described the Hindu-Arabic system in 825 CE in "On the Calculation with Hindu Numerals"
Not well-known in Europe at the time
Fibonacci's "Liber Abaci" introduced it to European audiences with business applications
Marked the beginning of a reawakening of European mathematics
Now used nearly exclusively throughout the globe

🌺 Hawaiian mixed system

Pre-contact Hawaiians used two different systems depending on what they counted
Sometimes used base 4 or a mixed base-10 and base-4 system
Theory: objects like fish and taro were bundled in groups of 4
Four also had spiritual significance in Hawaiian culture
Humans have 5 fingers but 4 gaps between fingers—natural for carrying items

🔢 How the Hawaiian system worked

Instead of powers of 10, numbers broke down into sums of 4 times powers of 10
Key values: 40 (kanahā), 400 (lau), 4,000 (mano), 40,000 (kini), 400,000 (lehu)
Example: three 40,000's, eight 400's, and one equals 3 × 40,000 + 8 × 400 + 1 = 123,201

🎯 Even and odd numbers

🎯 The fundamental definition

Some number of dots is even if I can divide the dots into pairs, and every dot has a partner. Some number of dots is odd if, when I try to pair up the dots, I always have a single dot left over with no partner.

This definition is about the quantity itself, not how we write it
A number is either even or odd—this property doesn't change when you represent that quantity in different bases

🔍 Why the base-ten trick works

The excerpt explains four key steps:

Every base ten number looks like (some multiple of ten) + (ones digit)
Every multiple of ten is even, since ten equals two times five, and two times a whole number is always even
Your whole number is (even number) + (ones digit)
Even plus even is even, and even plus odd is odd, so the whole number's evenness depends only on the ones digit

🔄 Rules change with different bases

In base seven, the rule for recognizing even numbers is different from base ten
In base four, the rule is different again
Don't confuse: the evenness of the quantity stays the same, but how you recognize it from the written form changes
The rules differ because the place values are different—what counts as "a multiple of the base" changes

📊 Comparison of system types

System type	How value is determined	Examples	Key advantage	Key limitation
Positional	Position determines value; sum of positional values	Base 10, base 4, base 60, base 20	Compact representation; efficient for large numbers	Requires understanding of place value
Additive	Sum of face values of all symbols	Roman numerals	Intuitive; easy to add symbols	Impractical for very large numbers
Mixed	Combination of approaches	Hawaiian (base 4/10 mix)	Can match cultural practices or practical needs	More complex rules

Even Numbers

🧭 Overview

🧠 One-sentence thesis

Even numbers can be recognized by looking at the ones digit in base ten, and this trick works differently in other bases because the structure of each base determines which place values are even.

📌 Key points (3–5)

Definition of even/odd: a number is even if dots can be paired with no leftovers; odd if one dot is always left without a partner.
Evenness is a property of quantity: whether a number is even or odd doesn't change when you represent it in different bases.
Base-ten shortcut: you can tell if a base-ten number is even by checking only the ones digit, because every multiple of ten is even.
Common confusion: the rule for recognizing even numbers changes in different bases (e.g., base seven vs. base four) because the place values themselves have different evenness properties.
Why the shortcut works: any number is (some multiple of the base) + (ones digit); if the base is even, the rule depends only on the ones digit.

🔢 What even and odd mean

🔢 Definition by pairing

Some number of dots is even if I can divide the dots into pairs, and every dot has a partner. Some number of dots is odd if, when I try to pair up the dots, I always have a single dot left over with no partner.

This is the fundamental definition—it's about the quantity itself, not how you write it.
You physically or mentally try to pair up all the dots; if you succeed with no leftovers, the number is even.
Example: 6 dots can be paired as (•,•), (•,•), (•,•) → even. 7 dots leave one unpaired → odd.

🌍 Evenness doesn't depend on the base

The excerpt emphasizes: "It's a property of the quantity and doesn't change when you represent that quantity in different bases."
Whether you write a number in base ten, base seven, or base four, the underlying quantity of dots is still either even or odd.
Don't confuse: the rule for recognizing even numbers changes with the base, but the property of being even does not.

🧩 Why the base-ten shortcut works

🧩 Structure of base-ten numbers

Every base-ten number looks like: (some multiple of ten) + (ones digit).
Example: 273 = 27×10 + 3.
The excerpt breaks this into two parts: an even part (the multiple of ten) and the ones digit.

🔟 Why multiples of ten are even

The excerpt states: "Every multiple of ten is an even number, since 10 = 2×5 and two times a whole number is always even."
Because 10 itself is even, any multiple of 10 is also even.
This means the (some multiple of ten) part is always even, so the evenness of the whole number depends only on the ones digit.

➕ Even plus even, even plus odd

The excerpt explains the last step: "Even plus even is even, and even plus odd is odd."
Your whole number is: (even number) + (ones digit).
If the ones digit is even, you have even + even = even.
If the ones digit is odd, you have even + odd = odd.
The excerpt asks you to use the pairing definition to justify these rules.

Why even + even = even:

If you have two even piles of dots, each pile pairs perfectly. Combining them still pairs perfectly → even.

Why even + odd = odd:

An even pile pairs perfectly; an odd pile has one leftover. Combining them leaves that one leftover → odd.

What about odd + odd?

The excerpt asks: "What about odd + odd? Is that odd or even?"
Two odd piles each have one leftover. Combining them gives two leftovers, which pair together → even.

🔄 Recognizing even numbers in other bases

🔄 Base seven

Problem 14 asks you to write 0–15 in base seven, circle the even numbers, and find a rule.
The excerpt does not give the answer, but it prompts you to discover the pattern yourself.
Key idea: in base seven, the place values are powers of 7 (7, 49, 343…). Since 7 is odd, multiples of 7 are odd, so the rule will be different from base ten.

🔄 Base four

Problem 15 repeats the exercise for base four.
In base four, the place values are powers of 4 (4, 16, 64…). Since 4 is even, multiples of 4 are even.
The excerpt asks: "How can you tell if a number is even when it's written in base four?"
The rule will depend on whether the base itself (4) is even or odd.

🔍 Why rules differ across bases

The excerpt asks: "Why are the rules for recognizing even numbers different in different bases?"
Answer (from the logic above): the rule depends on whether the base is even or odd.
- If the base is even (like 10 or 4), every multiple of the base is even, so you only need to check the ones digit.
- If the base is odd (like 7), multiples of the base are odd, so the rule must account for all digits.
The excerpt prompts: "For either your base four rule or your base seven rule, can you explain why it works that way?"

📊 Summary table: even recognition across bases

Base	Is the base even or odd?	What the excerpt implies
Base ten	Even (10 = 2×5)	Check only the ones digit; multiples of 10 are even
Base four	Even (4 = 2×2)	Similar logic: multiples of 4 are even, so check the ones digit
Base seven	Odd (7 is prime)	Multiples of 7 are odd, so the rule will be more complex

The excerpt does not state the exact rules for base four or base seven; it asks you to discover and justify them through the problems.
The key takeaway: the structure of the base (even or odd) determines how you recognize even numbers in that base.

Introduction to Fractions

Introduction

🧭 Overview

🧠 One-sentence thesis

Fractions are fundamentally the answer to a division problem—specifically, the amount each person receives when items are shared equally—and this "pies per child" interpretation provides a clearer foundation than traditional teaching methods.

📌 Key points (3–5)

Core definition: A fraction represents the result of a division problem (numerator divided by denominator).
The "pies per child" model: A fraction tells you how much one whole person gets when items are shared equally among a group.
Why fractions are hard: You must think about them in many different ways depending on the problem context.
Common confusion: Traditional teaching often starts with "one-half" as splitting one pie between two children, but the general principle (any number of pies shared among any number of children) is more powerful.
Teaching insight: Multiple mental models are essential for teachers to explain fractions effectively.

🥧 The core model: fractions as division

🥧 What a fraction really means

A fraction is the answer to a division problem.

The fraction a/b is equivalent to the division problem a ÷ b.
It represents the amount one whole person receives when items are shared equally.
This is not about "parts of a whole" in the abstract—it's about the concrete result of fair sharing.

📐 The "pies per child" interpretation

The excerpt uses a consistent scenario: pies shared equally among children.

Numerator: total number of pies
Denominator: total number of children
The fraction itself: pies per child (how much one child gets)

Example: 6 pies shared among 3 children

Division: 6 ÷ 3 = 2
Fraction: 6/3 = 2
Meaning: each child receives 2 pies

Don't confuse: The fraction is not "6 out of 3" or "6 parts labeled 3"—it is the result of the sharing process.

🔢 How the model works across different cases

✅ Whole-number results

When division yields a whole number, the fraction equals that number:

Pies	Children	Fraction	Pies per child
10	2	10/2	5
8	2	8/2	4
5	5	5/5	1

Each example shows that the fraction represents a concrete quantity: the amount one child receives.
When the numerator is larger than the denominator, each child gets more than one pie.
When numerator equals denominator, each child gets exactly one pie.

🍰 Fractional results: the case of "one-half"

The excerpt highlights one special case:

1 pie among 2 children yields 1/2, called "one-half."
This is the same principle as the whole-number cases, just with a result less than one.
The excerpt notes this "is actually saying something"—it represents how fractions are usually taught, but now grounded in the division interpretation.

Why this matters: Traditional teaching often starts with "split one thing in half," but the "pies per child" model shows that 1/2 is just one instance of the general rule (1 ÷ 2 = 1/2).

🧠 Why fractions are challenging

🧩 Multiple ways of thinking required

The excerpt opens by acknowledging that fractions are "one of the hardest topics to teach (and learn!) in elementary school."

The core difficulty: You must think about fractions in many different ways depending on the problem.
No single mental model covers all uses of fractions.
The excerpt promises to explore "some different ways to understand the idea of fractions" throughout the chapter.

👥 The need for multiple mental models

Teachers should have lots of mental models—lots of ways to explain the same concept.

One explanation will not reach every student.
The "pies per child" approach is presented as one powerful model, but not the only one.
Understanding why fractions are hard helps teachers prepare better explanations.

Reflection prompt from the excerpt: Think about your own experience or friends' struggles with fractions—what made the topic harder than other elementary school math topics?

Addition: Dots and Boxes

🧭 Overview

🧠 One-sentence thesis

The dots-and-boxes model provides an intuitive, place-value-based way to understand addition by representing numbers as collections of dots in columns and combining them, making the standard algorithm's "carrying" process transparent through regrouping (exploding dots).

📌 Key points (3–5)

Core model: Numbers are represented as dots organized in place-value boxes (ones, tens, hundreds), making quantity and place value visible.
Addition as combining: Add by merging dots from the same place-value columns, then regroup when a column has ten or more dots.
Regrouping (explosions): When a box accumulates ten dots, "explode" them into one dot in the next higher place-value box.
Common confusion: The standard algorithm's "carrying" is actually regrouping—ten ones become one ten, ten tens become one hundred—not a mysterious process.
Two approaches compared: Dots-and-boxes is easy to understand (add in any order, explode at the end); the standard algorithm is efficient (work right-to-left, regroup as you go).

🔢 The dots-and-boxes representation

🔢 What numbers look like

Counting numbers: the numbers used for counting (1, 2, 3, 4, 5…), sometimes called natural numbers.

Whole numbers: the counting numbers together with zero.

In base-10, each number is shown as dots arranged in place-value boxes: ones, tens, hundreds, etc.
Example from the excerpt: 273 is represented with dots in three boxes—2 dots in hundreds, 7 in tens, 3 in ones.
The physical arrangement makes place value concrete and visible.

📦 Why boxes matter

Each box corresponds to a power of ten in the base-10 system.
Dots in the same box have the same place value.
This visual structure supports understanding of regrouping and the standard algorithm.

➕ How addition works with dots and boxes

➕ Combining collections

Addition as combining: Start with two collections of dots (two numbers) and combine them to form one bigger collection.

Place the two numbers' dots in their respective place-value columns.
Combine dots from the same columns.
Example from excerpt: 273 + 512 means combining 2 hundreds with 5 hundreds (gives 7 hundreds), 7 tens with 1 ten (gives 8 tens), and 3 ones with 2 ones (gives 5 ones), yielding 785.

💥 Explosions (regrouping)

When a place-value box accumulates ten or more dots, "explode" ten dots into one dot in the next higher box.
Example from excerpt: 163 + 489 initially gives 5 hundreds, 14 tens, and 12 ones.
The answer "5 | 14 | 12" (pronounced "five hundred and fourteeny-tenty twelvety") needs regrouping to be understood.
Explode 10 of the 12 ones into 1 ten (leaving 2 ones); explode 10 of the 15 tens into 1 hundred (leaving 5 tens).
Final answer: 652 ("six hundred fifty two").

🔄 Flexibility of the method

In dots-and-boxes, you can add columns in any order, then do all explosions at the end.
This contrasts with the standard algorithm's strict right-to-left sequence.
The flexibility makes the underlying logic clearer for learners.

🎯 Connecting to the standard algorithm

🎯 What the standard algorithm does

The standard algorithm works from right to left, regrouping (carrying) as you go.
Students are taught to write a small digit above the next column when carrying.
Example from excerpt: In 163 + 489, students add 3 + 9 = 12, write 2 in the ones place, and put a small "1" above the tens column.

🔍 Why it seems mysterious

The standard algorithm's notation (the small carried digit) can appear arbitrary without understanding the underlying regrouping.
The dots-and-boxes model reveals what's happening: the "1" being carried is actually one exploded dot moving to the next place-value box.
Don't confuse: "Carrying" is not moving a digit arbitrarily—it's regrouping ten units of one place value into one unit of the next higher place value.

⚖️ Trade-offs between methods

Method	Advantage	Why
Standard algorithm	Efficient	Work right-to-left, regroup immediately, minimal writing
Dots and boxes	Easy to understand	Visual, flexible order, shows place value and regrouping clearly

The excerpt explicitly states: "Why do we like the standard algorithm? Because it is efficient. Why do we like the dots and boxes method? Because it is easy to understand."
Teachers benefit from knowing both: use dots-and-boxes to build conceptual understanding, then transition to the efficient standard algorithm.

🧠 Mental models and teaching

🧠 Multiple models for operations

The excerpt emphasizes that teachers should have "lots of mental models—lots of ways to explain the same concept."
Different students may understand operations differently; having multiple explanations helps reach more learners.
The dots-and-boxes approach is one powerful model for addition (and subtraction, mentioned later in the excerpt).

🎨 Working with the model

Students can draw pictures, use physical materials (manipulatives), or just think about the model to reason out answers.
The model helps with unfamiliar problems: visualize the dots and boxes to work through the logic.
Example: The excerpt includes exercises in different bases (not just base-10), encouraging students to work directly with the dots-and-boxes representation without converting.

Subtraction: Dots and Boxes

🧭 Overview

🧠 One-sentence thesis

The dots-and-boxes method for subtraction models taking away quantities by removing dots from place-value boxes and "unexploding" (regrouping) higher-place dots when a box doesn't have enough dots to remove.

📌 Key points (3–5)

What subtraction means here: start with one collection of dots (one number) and take some dots away from it.
How to subtract: remove the required number of dots from each box (ones, tens, hundreds, etc.), working in any order you like.
When you need regrouping: if a box doesn't have enough dots to take away, "unexplode" one dot from the next higher box to get ten dots in the current box.
Common confusion: "borrowing" vs "unexploding"—both terms describe regrouping, but unexploding makes the place-value exchange clearer (one hundred becomes ten tens, etc.).
Standard algorithm vs dots-and-boxes: the standard algorithm works right-to-left and regroups as you go; dots-and-boxes lets you work in any order and unexplode as needed.

🧮 Basic subtraction: take-away model

🧮 What subtraction as take-away means

Subtraction as take-away: start with one collection of dots (one number) and take some dots away.

Unlike addition (which combines two collections into one bigger collection), subtraction starts with a single collection and removes part of it.
The dots-and-boxes representation shows the starting number, then you physically remove dots from each box.

✅ Simple example: 376 – 125

Start with 376 represented in boxes: 3 dots in hundreds, 7 in tens, 6 in ones.
Take away 125 means:
- Remove 1 dot from the hundreds box → 2 hundreds left.
- Remove 2 dots from the tens box → 5 tens left.
- Remove 5 dots from the ones box → 1 one left.
Answer: 251.
You can work through the boxes in any order; no regrouping is needed because each box has enough dots to remove.

🔄 Regrouping: unexploding dots

🔄 When you can't take away enough dots

Sometimes a box doesn't have enough dots to remove the required amount.
Example: 921 – 551
- Start with 921: 9 hundreds, 2 tens, 1 one.
- You need to take away 5 tens, but the tens box only has 2 dots.
Solution: "unexplode" one dot from the hundreds box.
- One hundreds dot becomes ten tens dots.
- Now the tens box has 12 dots (the original 2 plus 10 from unexploding).
- You can now remove 5 tens, leaving 7 tens.
- The hundreds box now has one fewer dot (8 instead of 9).

🔄 Completing the example: 921 – 551

After unexploding, the representation is effectively 8 hundreds, 12 tens, 1 one.
Take away 5 hundreds → 3 hundreds left.
Take away 5 tens (from the 12) → 7 tens left.
Take away 1 one → 0 ones left.
Answer: 370.

🔄 Don't confuse: "borrowing" vs "unexploding"

The standard algorithm calls this process "borrowing," but that term is misleading.
A better term is regrouping or unexploding: you are converting one unit of a higher place value into ten units of the next lower place value.
Example: one hundred = ten tens; one ten = ten ones.

📐 Standard algorithm vs dots-and-boxes

📐 How the standard algorithm works

The standard algorithm requires you to work right to left (ones, then tens, then hundreds, etc.).
You regroup (unexplode) as you go, whenever a column doesn't have enough to subtract.
Example for 921 – 551:
- Start at the ones: 1 – 1 = 0.
- Move to tens: need to subtract 5 from 2, so regroup one hundred into ten tens.
- Rewrite 921 as "8 hundreds, 12 tens, 1 one" (shown by crossing out and rewriting digits).
- Subtract 5 from 12 → 7 tens.
- Subtract 5 from 8 hundreds → 3 hundreds.
- Answer: 370.

📐 How dots-and-boxes differs

Feature	Standard algorithm	Dots-and-boxes
Direction	Must work right to left	Work in any order you like
Regrouping	Regroup as you go, column by column	Unexplode dots as needed, whenever a box runs short
Why use it	Efficient for computation	Easy to understand the place-value logic

📐 Why both methods matter

Standard algorithm: efficient and fast once you know the steps.
Dots-and-boxes: makes the underlying place-value structure visible and helps learners understand why regrouping works.
The excerpt emphasizes that dots-and-boxes is "easy to understand," while the standard algorithm is "efficient."

🧪 Practice and extensions

🧪 Working in other bases

The excerpt includes exercises that ask students to use dots-and-boxes in bases other than 10 (e.g., base 8).
The same unexploding logic applies: in base 8, one dot in a higher box becomes eight dots in the next lower box.
Students are encouraged to draw pictures and work directly in the given base, not convert to base 10.

🧪 Think/Pair/Share prompts

The excerpt includes collaborative problem-solving prompts where students solve subtraction problems by imagining or drawing dots-and-boxes.
These exercises reinforce the concept that regrouping is a place-value exchange, not a mysterious "borrowing" trick.

Multiplication: Dots and Boxes

🧭 Overview

🧠 One-sentence thesis

The dots-and-boxes model extends from addition and subtraction to multiplication by treating multiplication as repeated addition, where each place value accumulates groups of dots that can be "exploded" (regrouped) into higher place values.

📌 Key points (3–5)

Multiplication as repeated addition: multiplying a number by 4 means adding that number to itself four times, which can be visualized by combining dots in each place-value box.
How regrouping works: when a box accumulates enough dots (e.g., 10 in base 10), they "explode" into one dot in the next higher box.
Notation flexibility: the excerpt uses both × and · for multiplication; · is clearer with variables, and 4N is shorthand for 4 · N.
Common confusion: don't confuse the dots-and-boxes method with the standard algorithm—both represent the same regrouping, but dots-and-boxes makes the place-value logic visible.
Works in any base: the method applies to base 8, base 5, or any other base, not just base 10.

🔢 Multiplication as repeated addition

🔢 Core idea

Multiplication as repeated addition: if we have some number and multiply it by 4, we mean adding that number to itself four times.

The excerpt defines multiplication through addition: 3 × 4 = 4 + 4 + 4.
In the dots-and-boxes model, multiplying 243192 by 4 means combining four copies of that number.
Each place value is handled separately:
- 2 + 2 + 2 + 2 = 8 ones
- 9 + 9 + 9 + 9 = 36 tens
- 1 + 1 + 1 + 1 = 4 hundreds
- and so on.

💥 Explosions (regrouping)

When a box accumulates more dots than the base allows, those dots "explode" into the next higher place.
Example: in base 10, 36 tens means 3 hundreds and 6 tens; the 30 tens explode into 3 hundreds.
This is the same as "carrying" in the standard algorithm, but the dots-and-boxes picture shows why it happens.

🧮 Notation and variables

🧮 Multiplication symbols

The excerpt introduces three notations for multiplication:

Notation	When to use	Why
×	General multiplication	Can be confused with the letter x when variables are present
·	Multiplication with variables	Clearer than × in expressions like 4 · N
4N	Shorthand for 4 · N	Only when multiplying variables by a quantity; not for 3 · 4

Don't confuse: 34 is the number thirty-four, not 3 · 4.

🧩 Using an addition table for multiplication

Problem 5 presents a strange addition table with symbols A, B, C.
To solve 5C (which means C + C + C + C + C), you use the addition table repeatedly without assigning numeric values to A, B, or C.
This reinforces that multiplication is repeated addition, even when working with abstract symbols.

🎯 Working in different bases

🎯 Base 8 example

Jenny's method (from Problem 3) works in any base.
The excerpt asks: "Can you adapt Jenny's method to solve these problems? Write your answers in base eight. Try to work directly in base eight rather than converting to base 10 and back again!"
The key is to remember the explosion rule for that base: in base 8, 8 dots in one box explode into 1 dot in the next box.

🎯 Base 5 example

Problem 7 asks students to work with base five numbers (a 1 ← 5 system).
Example: to compute 424 (base five) ÷ 11 (base five), draw dots-and-boxes pictures in base 5 and perform the division without converting to base 10.
Don't confuse: the place values in base 5 are different from base 10 (ones, fives, twenty-fives, etc.).

🧠 Why dots-and-boxes vs. standard algorithm

🧠 Comparison

Method	Strength	Weakness
Dots and boxes	Easy to understand; shows why regrouping happens; fun to draw	Takes more time and space to write
Standard algorithm	Quick, compact, works every time	Can feel mysterious without understanding the underlying place-value logic

The excerpt states: "Why do we like the standard algorithm? Because it is quick, not too much to write down, and it works every time."
"Why do we like the dots and boxes method? Because it is easy to understand. (And drawing dots and boxes is kind of fun!)"

🧠 When to use each

Use dots-and-boxes to build understanding and see the place-value structure.
Use the standard algorithm for efficiency once the concept is clear.
Both methods represent the same mathematical process; the choice depends on the learning goal.

Division: Dots and Boxes

🧭 Overview

🧠 One-sentence thesis

The quotative model of division asks "how many groups of the divisor fit into the dividend," and the dots-and-boxes method visualizes this by finding groups at each place value, unexploding boxes when necessary to continue grouping.

📌 Key points (3–5)

Quotative model definition: division as splitting the dividend into equal-sized groups determined by the divisor.
Core mechanism: count how many groups of the divisor fit into each place-value box, unexploding (trading) higher-place dots into lower-place dots when needed.
Remainders: leftover dots that cannot form another complete group represent the remainder.
Common confusion: the standard algorithm and dots-and-boxes are the same process—both count groups at each place value and handle leftovers by regrouping—but dots-and-boxes makes the place-value logic visible.
Why it matters: the method clarifies why the standard algorithm works and extends naturally to other bases (e.g., base five).

🧮 The quotative model of division

🧮 What the quotative model means

In the quotative model of division, you are given a dividend and you are asked to split it into equal-sized groups, where the size of the group is given by the divisor.

The question "3906 ÷ 3" means "How many groups of 3 fit into 3906?"
You are not splitting into a given number of groups; you are counting how many groups of a given size you can make.
Example: if the dividend is 3906 and the divisor is 3, you draw 3906 in dots-and-boxes and look for groups of three dots.

🔍 Dividend and divisor roles

Term	Role	In 3906 ÷ 3
Dividend	The total quantity to be divided	3906
Divisor	The size of each group	3
Quotient	The number of groups that fit	(to be found)

🎨 Dots-and-boxes division step-by-step

🎨 Simple case: 3906 ÷ 3

Draw 3906 in dots-and-boxes: three dots in the thousands box, nine in the hundreds, zero in the tens, six in the ones.
Draw the divisor 3 as three dots in a single box.
Count groups of three at each place value:
- Thousands: one group of 3
- Hundreds: three groups of 3
- Tens: zero groups of 3
- Ones: two groups of 3
Read the answer from the number of groups at each place: 1 thousand + 3 hundreds + 0 tens + 2 ones = 1302.

🔄 Unexploding when stuck: 402 ÷ 3

Draw 402: four dots in the hundreds, zero in the tens, two in the ones.
Look for groups of three:
- Hundreds: one group of 3, leaving one dot.
- Tens: zero dots, so no groups.
Key step: unexplode the leftover hundreds dot into ten dots in the tens box.
Now the tens box has ten dots; find three groups of 3, leaving one dot.
Unexplode that leftover tens dot into ten dots in the ones box (now 2 + 10 = 12 ones).
Ones: four groups of 3.
Answer: 1 hundred + 3 tens + 4 ones = 134.

Don't confuse: "unexploding" is the same as regrouping or trading one higher-place-value unit for ten lower-place-value units; it is not inventing new dots, just changing their representation.

🧩 Multi-digit divisor: 156 ÷ 12

The divisor 12 is usually written as one dot in the tens box and two dots in the ones box.
Look for this pattern (one ten and two ones) in the dividend 156:
- Tens: one group of 12 (one ten and two ones).
- After removing that group, three groups of 12 remain in the ones place.
Answer: 1 ten + 3 ones = 13.

Note: The excerpt says "with an unexplosion this would be twelve dots in the tens box," meaning you could also think of 12 as twelve dots in a single box, but the standard representation is more efficient.

🔢 The standard algorithm connection

🔢 How the standard algorithm mirrors dots-and-boxes

The standard algorithm uses a table to track groups and leftovers at each place value.
Example for 402 ÷ 3:
- Guess 100 groups of 3 (300); subtract from 402, leaving 102.
- Guess 30 groups of 3 (90); subtract from 102, leaving 12.
- Guess 4 groups of 3 (12); subtract from 12, leaving 0.
- Add the guesses: 100 + 30 + 4 = 134.
This is the same as counting groups at each place value in dots-and-boxes, but written as repeated subtraction.

🆚 Dots-and-boxes vs. standard algorithm

Method	Strength
Dots-and-boxes	Easy to understand; makes place-value logic visible; "kind of fun"
Standard algorithm	Quick; less to write; works every time

Common confusion: The two methods are not different procedures; they are different notations for the same counting-and-regrouping process.

➗ Division with remainders

➗ What a remainder means

A remainder is the number of dots left over after forming as many complete groups as possible.
Example: 403 ÷ 3
- 402 ÷ 3 = 134 with no remainder.
- 403 has one extra dot that cannot form another group of 3.
- Answer: 134 groups with remainder 1, written as 403 ÷ 3 = 134 R1.
This means 403 = 134 × 3 + 1.

🧩 Multi-place-value remainders: 263 ÷ 12

After counting groups of 12 in 263, one dot remains in the tens box and one in the ones box.
The remainder is not "two dots"; it is eleven (one ten and one one).
Answer: 21 groups of 12 with remainder 11.

Don't confuse: The remainder is a number, not a count of leftover dots in separate boxes; you must read the place values of the leftover dots to find the remainder.

🌍 Extension to other bases

🌍 Base five division

The excerpt asks to compute division in base five (a 1 ← 5 system) without converting to base 10.
Example: 424₅ ÷ 11₅
- Draw 424₅ in a 1 ← 5 dots-and-boxes system (place values are powers of 5).
- Draw 11₅ as one dot in the fives box and one in the ones box.
- Count groups at each place value, unexploding when necessary.
- Write the answer in base five.

Why this matters: The dots-and-boxes method works in any base because it is based on place-value structure, not on memorized base-ten facts.

🔟 Dividing by powers of ten

The excerpt asks what happens when dividing by 10 or 100.
10 is one dot in the tens box; 100 is one dot in the hundreds box.
Dividing by 10 means "how many groups of one ten fit?"—this shifts all place values down by one.
Example: 2130 ÷ 10 = 213 (each place value moves one position to the right).

Key insight: Dividing by 10 or 100 is a place-value shift, visible in the dots-and-boxes picture as removing one or two empty boxes on the right.

Number Line Model

🧭 Overview

🧠 One-sentence thesis

The number line model represents arithmetic operations as movements along a line, where addition and subtraction are forward and backward steps, and multiplication and division are repeated movements that land on specific positions.

📌 Key points (3–5)

What the model is: a measurement approach where numbers are abstract quantities on a line, with a basic unit assigned to the number one.
How operations work: addition walks forward twice; subtraction walks forward then backward; multiplication repeats forward steps; division counts how many repeated steps reach a target.
Key distinction: addition vs multiplication—addition is two separate forward walks; multiplication is repeating the same forward walk multiple times.
Common confusion: subtraction direction—you always walk forward first (the minuend), then walk backward (the subtrahend), not backward from zero.
Extension to area: multiplication can also be visualized as a rectangle's area, where factors become length and width.

📏 Measurement foundation

📏 What a measurement model is

In a measurement model, you pick a basic unit—a quantity (length, area, or volume) that you assign to the number one—and then assign numbers to other quantities based on how many of your basic unit fit inside.

The excerpt focuses on length as the quantity.
The number line already has the basic unit marked off (the distance between 0 and 1).
This is different from the dots-and-boxes model (a grouping/counting model); here numbers represent measurable distances.

🚶 Zed the walker

The excerpt introduces Zed, a person who stands on the number line.
Zed's step distance is exactly one unit.
Always starts at 0 and always faces the positive direction (towards 1) for whole-number operations.
Where Zed lands after following rules determines the answer.

➕➖ Addition and subtraction

➕ Addition on the number line

Process: walk forward (right) the first addend's number of steps, then walk forward again the second addend's number of steps.
Where Zed lands is the sum.
Example: 3 + 4 means start at 0, walk forward 3 steps, then walk forward 4 more steps → land at 7.

➖ Subtraction on the number line

Process: walk forward (right) the minuend's number of steps, then walk backward (left) the subtrahend's number of steps.
Where Zed lands is the difference.
Example: 11 – 3 means start at 0, walk forward 11 steps, then walk backward 3 steps → land at 8.
Don't confuse: you do not start by walking backward from 0; you walk forward first, then reverse direction.

🤔 Why forward and backward make sense

The excerpt asks: "Why does it make sense to walk forward for addition and walk backwards for subtraction?"
Connection to other models: forward is like "combining" (adding more); backward is like "take away" (removing some).
The excerpt also raises problems like 6 – 9 or 1 – 7, where walking backward from a small number would go below zero (these require extending the model to negative numbers).

✖️➗ Multiplication and division

✖️ Multiplication as repeated addition

Multiplication is really repeated addition.

Definition used: 3 × 4 = 4 + 4 + 4 (add the second factor to itself, the first factor's number of times).
Process: start at 0, face positive; walk forward the second factor's steps; repeat that process the first factor's number of times.
Where Zed lands is the product.
Example: 3 × 4 means repeat "walk forward 4 steps" three times → land at 12.

➗ Division as quotative grouping

The excerpt uses the quotative model: "How many groups of [divisor] fit into [dividend]?"
On the number line: "Zed takes [divisor] steps at a time. If Zed lands at [dividend], how many times did he take [divisor] steps?"
Process: start at 0, face positive; walk forward the divisor's steps; repeat until you land at the dividend; count how many times you repeated.
The count is the quotient.
Example: 15 ÷ 5 means walk forward 5 steps repeatedly until reaching 15 → takes 3 repetitions → quotient is 3.

🔄 Comparing multiplication and division

Operation	What Zed does	What you count
Multiplication	Repeat a fixed forward walk	Where you land (product)
Division	Repeat a fixed forward walk until target	How many repetitions (quotient)

Both involve repeating the same step size; multiplication tells you the landing spot, division tells you the number of repetitions needed to reach a given spot.

🟦 Area model for multiplication

🟦 From line to rectangle

The excerpt introduces a second way to visualize multiplication: using area instead of length.
Basic unit: one square.
Example: 4 × 3 can be pictured as 4 groups of 3 squares lined up (number line style) or as a rectangle with 4 rows and 3 squares per row.

📐 How the area model works

4 × 3 becomes a rectangle with length 3 and width 4.
The product (12) is the total number of unit squares inside the rectangle.
This is also the area of the rectangle (since each square is one unit).
Don't confuse: the area model is still multiplication, but it uses two-dimensional arrangement instead of one-dimensional repeated steps.

Area Model for Multiplication

🧭 Overview

🧠 One-sentence thesis

The area model interprets multiplication as a rectangle whose dimensions are the factors and whose total number of unit squares (area) is the product, revealing why the standard algorithm and alternative methods work.

📌 Key points (3–5)

What the area model is: multiplication visualized as a rectangle with length and width equal to the factors; the product is the total number of unit squares inside.
Connection to the standard algorithm: the area model and the traditional multiplication algorithm compute the same partial products—just organized differently.
Alternative methods exist: lattice multiplication and the lines-and-intersections method are disguised versions of the same underlying area logic.
Common confusion: stacking squares vs. lining them up—both represent the same multiplication; the area model simply rearranges groups into rows and columns.
Why it matters: understanding the area model explains why multiplication algorithms work and makes mental computation strategies clearer.

📐 What the area model represents

📐 From groups to rectangles

The excerpt starts with the familiar "groups" interpretation: 4 × 3 means "4 groups, with 3 squares in each group."
Instead of lining up the groups in a row, you can stack them into a rectangle.
Example: 4 × 3 becomes a rectangle with 4 rows and 3 squares in each row.

The area model: multiplication as a rectangle with length equal to one factor and width equal to the other; the product is the total number of unit squares (the area).

🔢 The product as area

Each small square represents one unit.
The total count of squares inside the rectangle equals the product.
Example: a 4-by-3 rectangle contains 12 unit squares, so 4 × 3 = 12.

🧮 How the area model connects to algorithms

🧮 Vera's picture method

The excerpt describes Vera drawing a picture to compute 15 × 17.
She breaks the rectangle into smaller pieces (partial products) and adds them up.
This is the same logic as the standard algorithm—just visualized.

🧮 The standard algorithm in disguise

The traditional multiplication layout (e.g., 83 × 27) computes partial products:
- Multiply each digit of one number by each digit of the other.
- Add the results, aligning by place value.
The area model does exactly this: each partial product corresponds to a sub-rectangle.
Example: 83 × 27 splits into (80 + 3) × (20 + 7), which gives four smaller rectangles: 80×20, 80×7, 3×20, and 3×7.

Don't confuse: the standard algorithm and the area model are not different methods—they are the same calculation organized in different formats.

🔀 Alternative multiplication methods

🔀 Lines and intersections

To compute 22 × 13, draw vertical lines (two sets: 2 lines, then 2 lines for the digits in 22) and horizontal lines (two sets: 1 line, then 3 lines for the digits in 13).
Count intersection points in four regions and add diagonally.
The result appears as digits of the product.
Possible glitch: if a diagonal sum exceeds 9, you must carry (e.g., 246 × 32 gives "6 thousands, 16 hundreds, 26 tens, 12 ones" → carry to get 7,872).

🔀 Lattice multiplication (galley method)

Draw a grid (e.g., 2 × 3 for 43 × 218).
Write one number's digits along the top, the other's along the right side.
In each cell, write the product of the column digit and row digit, splitting tens and units across a diagonal.
Add along diagonals, carrying as needed, to read off the final product.

Why these work: both methods are the standard algorithm in disguise—they compute the same partial products and organize them spatially instead of vertically.

🧩 The role of diagonals

In the lattice method, diagonal lines separate place values (tens vs. units within each partial product).
Adding along diagonals groups digits by the same overall place value (ones, tens, hundreds, etc.).
This mimics the carrying and alignment in the standard algorithm.

🧪 Practice and verification

🧪 Using pictures to compute

The excerpt asks readers to draw area-model pictures for problems like 23 × 37, 8 × 43, and 371 × 42.
Break each factor into place-value parts, draw sub-rectangles, and sum the areas.
Example: 23 × 37 = (20 + 3) × (30 + 7) → four rectangles: 20×30, 20×7, 3×30, 3×7.

🧪 Comparing methods

The excerpt encourages computing the same product (e.g., 23 × 14) using both Vera's rectangle method and the standard algorithm.
Identify where each method computes the same partial products.
This reinforces that all methods are equivalent—they just organize the work differently.

Don't confuse: different layouts do not mean different mathematics; the underlying multiplication facts and place-value logic remain the same.

Properties of Operations

🧭 Overview

🧠 One-sentence thesis

The properties of operations (commutative, associative, identity, distributive) are not arbitrary rules but logical consequences that can be explained through models and definitions of addition, subtraction, multiplication, and division.

📌 Key points (3–5)

Operations are connected: subtraction is "missing addend" addition (c – b = a means c = a + b), and division is "missing factor" multiplication (c ÷ b = a means c = a · b).
Properties need explanation, not just examples: checking specific numbers never proves a universal statement; you need general reasoning based on definitions and models.
Common confusion: examples vs. explanations—examples demonstrate a property, but only a general argument (using models or definitions) explains why it always works.
Not all properties transfer: addition is commutative and associative, but subtraction is neither; multiplication is commutative and associative, but division is neither.
Two division models exist: quotative (how many groups of a given size?) vs. partitive (what size for a given number of groups?).

🔗 How operations connect to each other

🔗 Subtraction as missing addend

Subtraction problem c – b = a is the same mathematical fact as the addition problem c = a + b.

Instead of "take away b from c," think "what do I add to b to get c?"
This connection lets you use an addition table to solve subtraction problems.
Example: 27 – 13 = ____ is the same question as 27 = 13 + ____.

🔗 Division as missing factor

Division problem c ÷ b = a is the same mathematical fact as the multiplication problem c = a · b.

Instead of "how many groups of b fit into c?" think "what do I multiply b by to get c?"
This connection lets you use a multiplication table to solve division problems.
Example: 27 ÷ 3 = ____ is the same question as 27 = ____ × 3.

🧮 Properties of addition and subtraction

➕ Addition is commutative

For any two whole numbers a and b, a + b = b + a.

Why it's true (combining model):

Put a red dots and b blue dots in a box: you have a + b total dots.
Put b blue dots and a red dots in a box: you have b + a total dots.
The two boxes have the same total because you can match up all the red dots (a in each) and all the blue dots (b in each) one-to-one.

Why it's true (measurement model):

A segment of length a combined with length b gives a + b.
Rotate the segment upside down: now it's length b combined with length a, giving b + a.
Same segment, just flipped—so the lengths are equal.

➕ Addition is associative

For any three whole numbers a, b, and c, (a + b) + c = a + (b + c).

When adding three numbers in order, grouping doesn't matter.
The parentheses tell you which two to add first, but the final sum is the same either way.

0️⃣ Zero is an identity for addition

For any whole number n, n + 0 = n and 0 + n = n.

Adding zero (in either order) leaves the number unchanged.
This is what "identity" means: the operation returns the original value.

➖ Subtraction is NOT commutative

Universal statements need to hold for every case; one counterexample disproves them.
Counterexample: 4 – 3 = 1, but 3 – 4 ≠ 1 (it equals –1).
Therefore subtraction is not commutative.
Don't confuse: there are special cases where a – b = b – a (e.g., when a = b), but that doesn't make subtraction commutative in general.

➖ Subtraction is NOT associative

Check whether (a – b) – c always equals a – (b – c).
Counterexample: (16 – 4) – 2 = 12 – 2 = 10, but 16 – (4 – 2) = 16 – 2 = 14.
One counterexample is enough to conclude subtraction is not associative.

0️⃣ Zero is NOT an identity for subtraction

For 0 to be an identity, both n – 0 and 0 – n would need to equal n.
n – 0 = n is true (taking away nothing leaves n unchanged).
But 0 – n ≠ n (taking away n from zero does not give n back).
Since the property fails in one direction, 0 is not an identity for subtraction.

🔢 Properties of multiplication and division

✖️ Multiplication is commutative

For any two whole numbers a and b, a · b = b · a.

Why it's true (area model):

a · b represents a rows with b squares each.
b · a represents b rows with a squares each.
These are the same rectangle, just rotated—so the total number of squares is the same.

✖️ Multiplication is associative

For any three whole numbers a, b, and c, (a · b) · c = a · (b · c).

When multiplying three numbers in order, grouping doesn't matter.
The parentheses tell you which two to multiply first, but the final product is the same.

🎁 Distributive property

For any three whole numbers x, y, and z, x · (y + z) = x · y + x · z.

Multiplication distributes over addition.
Example: 8 · (20 + 3) = 8 · 20 + 8 · 3 = 160 + 24 = 184.
This is the foundation of many mental math strategies.

1️⃣ One is an identity for multiplication

For any whole number m, m × 1 = m and 1 × m = m.

Why it's true (repeated addition):

m × 1 means add m to itself one time, which is just m.
1 × m means add m to itself one time, which is also m.

Why it's true (area model):

m × 1 is m rows with one square each = m squares total.
1 × m is one row of m squares = m squares total.

➗ Division is NOT commutative

Check whether a ÷ b always equals b ÷ a.
Counterexample: 8 ÷ 4 = 2, but 4 ÷ 8 ≠ 2 (it equals 0.5).
Division is not commutative.

➗ Division is NOT associative

Check whether (a ÷ b) ÷ c always equals a ÷ (b ÷ c).
Counterexample: (16 ÷ 4) ÷ 2 = 4 ÷ 2 = 2, but 16 ÷ (4 ÷ 2) = 16 ÷ 2 = 8.
Division is not associative.

➗ Division does NOT distribute over addition

Check whether a ÷ (b + c) always equals (a ÷ b) + (a ÷ c).
Test with examples to find counterexamples.
Division does not distribute over addition.

1️⃣ One is NOT an identity for division

For 1 to be an identity, both n ÷ 1 and 1 ÷ n would need to equal n.
n ÷ 1 = n is true.
But 1 ÷ n ≠ n (except when n = 1).
Since the property fails in one direction, 1 is not an identity for division.

0️⃣ Special role of zero

0️⃣ Zero property of multiplication

For every whole number n, n × 0 = 0 and 0 × n = 0.

Why it's true (repeated addition):

n × 0 means add 0 to itself n times: 0 + 0 + ... + 0 = 0.
0 × n means add n to itself zero times: nothing at all = 0.

Don't confuse: the zero property (multiplying by 0 gives 0) is very different from being an identity (which leaves the number unchanged).

🚫 Division by zero is undefined

Turn division into multiplication: 5 ÷ 0 = ____ means 5 = ____ × 0.
But any number times 0 equals 0, never 5—so there's no answer.
For 0 ÷ 0 = ____, we'd need 0 = ____ × 0, which is true for every number—so there's no unique answer.
Because these problems have either no solution or infinitely many solutions, we say division by 0 is undefined.

👨‍👩‍👧‍👦 Four fact families

👨‍👩‍👧‍👦 Addition/subtraction families

One fact generates four related equations through commutativity and the addition-subtraction connection.
Example family: 2 + 3 = 5, 3 + 2 = 5, 5 – 3 = 2, 5 – 2 = 3.
These are really one fact expressed four ways, using commutativity of addition and the definition that subtraction is the inverse of addition.

👨‍👩‍👧‍👦 Multiplication/division families

One fact generates four related equations through commutativity and the multiplication-division connection.
Example family: 2 · 3 = 6, 3 · 2 = 6, 6 ÷ 3 = 2, 6 ÷ 2 = 3.
These are really one fact expressed four ways, using commutativity of multiplication and the definition that division is the inverse of multiplication.

🎯 Two models of division

🎯 Quotative division (how many groups?)

When you know the size of each group, quotative division finds the number of groups.

Question form: "How many groups of [size] are there in [total]?"
Example: 20 ÷ 4 means "How many groups of 4 are in 20?" Answer: 5 groups.
Word problem: "David made 36 cookies and packaged them in boxes of 9. How many boxes did he use?"

🎯 Partitive division (what size?)

When you know the number of groups, partitive division finds the size of each group.

Question form: "[Total] is [number] groups of what size?"
Example: 20 ÷ 4 means "20 is 4 groups of what size?" Answer: size 5.
Word problem: "David made 36 cookies to share with 12 people. How many cookies did each person get?"

Why both matter: Students need exposure to both types of problems to fully understand division; recognizing the two models helps teachers diagnose difficulties and choose appropriate examples.

📦 Division with remainder

Not all division problems come out evenly.
23 ÷ 4 = 5 R3 means 23 = 5 · 4 + 3.
The remainder (3) is what's left over after making as many complete groups as possible.
Context determines the right answer: sometimes round down (10), sometimes round up (11), sometimes use a decimal (10.75).

Division Explorations

🧭 Overview

🧠 One-sentence thesis

Division algorithms work consistently across different base systems because the underlying place-value structure (powers of the base) remains valid regardless of which base you choose.

📌 Key points (3–5)

Base-agnostic division: Anu's division method works in any base system (1←x) because it operates on the place-value structure itself, not specific digit values.
Place values as powers: In a 1←x system, boxes represent powers of x (1, x, x², x³...), just as base-10 uses powers of ten and base-5 uses powers of five.
Same algorithm, different interpretations: The identical symbolic calculation represents different numerical division problems depending on which base x equals.
Common confusion: The same written symbols (like "2556 ÷ 12") mean completely different numerical values in different bases—2556₁₀ ≠ 2556₁₁.
Verification across bases: A division performed symbolically in the abstract 1←x system can be checked by substituting any specific base value for x.

🔢 Base systems and place value

🔢 What a 1←x system means

A 1←x system: a place-value notation where each position represents a power of x.

In base-10 (1←10): positions are 1, 10, 100, 1000, 10000...
In base-5 (1←5): positions are 1, 5, 25, 125, 625...
In Anu's mystery system (1←x): positions are 1, x, x², x³, x⁴...

📦 How to read numbers in 1←x notation

When Anu writes a number using boxes/positions, each digit multiplies the corresponding power of x:

The number with digits in positions means: (coefficient of x²) · x² + (coefficient of x¹) · x¹ + (coefficient of x⁰) · x⁰
Example: If Anu writes "2556" in her system, it means 2·x³ + 5·x² + 5·x¹ + 6·x⁰
Don't confuse: "2556" looks the same in any base notation, but represents completely different quantities depending on x.

➗ Division in the abstract system

➗ Anu's division method

The excerpt shows Anu computing a division problem using her dots-and-boxes method without revealing what x is:

She performs the division algorithm using the place-value structure
The calculation works symbolically: she divides, finds quotient and remainder
The method doesn't require knowing the actual base value

✅ Checking the division

Problem 28 asks to verify Anu's division by computing the product:

If division is correct, then (quotient) × (divisor) + (remainder) should equal the original dividend
This check works in any base because multiplication and division are inverse operations
Example: If the division claims A ÷ B = Q with remainder R, then Q · B + R must equal A

🔄 Multiple interpretations of one calculation

🔄 Same symbols, different values

The excerpt demonstrates that Anu's single symbolic calculation represents different numerical problems depending on x:

Base (x value)	What "2556" means	What "12" means	What the division represents
x = 10 (base-10)	2556₁₀ = 2556	12₁₀ = 12	2556 ÷ 12 = 213
x = 11 (base-11)	2556₁₁ = 3328₁₀	12₁₁ = 13₁₀	3328 ÷ 13 = 256

🧮 Converting between representations

When Anu reveals x = 11, her "2556" becomes:

2·11³ + 5·11² + 5·11¹ + 6·11⁰
= 2·1331 + 5·121 + 5·11 + 6·1
= 2662 + 605 + 55 + 6
= 3328 in base-10

Similarly, "12" in base-11 means 1·11 + 2 = 13 in base-10.

🎯 Universal validity

Problem 31 shows a division that works for multiple base values:

The same symbolic division is correct when x = 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11
Each choice of x yields a different numerical division problem in base-10
All are correct because the place-value algorithm is base-independent
Don't confuse: This doesn't mean the numerical answers are the same—the problems themselves are different.

🧪 Practice applications

🧪 Working in specific bases

The Problem Bank extends these ideas:

Problems 32–37 ask students to perform operations (addition, subtraction, multiplication, division) directly in various bases
Key instruction: "Don't translate to base 10 and then calculate there—try to work in base [specified base]"
This reinforces understanding of how the algorithm operates on place-value structure

🔍 Remainder patterns

Problem 49 explores remainders:

When dividing by a number n, possible remainders are 0, 1, 2, ..., n-1
There are exactly n possible remainders when dividing by n
This is true in any base system

🎲 Edge cases

Problem 31 part 5 asks: "What is this saying for x = 0?"

This probes the boundary of the system—base-0 is undefined or degenerate
Highlights that the place-value system requires a base ≥ 2 to be meaningful

Introduction to Algebraic Thinking

Introduction

🧭 Overview

🧠 One-sentence thesis

Algebra is essential for elementary students because it teaches abstraction, problem-solving, and the ability to understand and explain why mathematical operations work, not just how to perform them.

📌 Key points (3–5)

Why algebra matters: it is the language of science and technology, a tool for solving problems, and a way to think abstractly about operations.
What algebraic thinking means: thinking about operations (addition, subtraction, multiplication, division) separately from calculating with specific numbers; understanding and explaining why operations work, not just how.
When it starts: algebraic thinking begins in kindergarten (Common Core includes "Operations and Algebraic Thinking" from the start), not just in 8th grade.
Common confusion: algebra is not only symbol manipulation in later grades—it is abstraction and generalization that students already do informally before school.
Teacher's role: provide experiences in abstraction and generalization so that formal algebra feels natural later.

🎯 Why elementary teachers need algebra

🌍 Algebra as a language and tool

Language of science and technology: algebra helps students make sense of the world, interact productively with technology, and succeed in other fields.
Problem-solving tool: if you can "algebratize" a problem (translate it into algebraic form), it often leads you to a solution.
Don't confuse: algebra is not just a high-school subject—it is a way of thinking that applies across disciplines.

🧠 Algebra as abstract thinking

Algebra is a tool for thinking about operations like addition, subtraction, multiplication, and division separate from doing calculations on particular numbers.

It helps you:
- Understand and explain why operations work the way they do.
- Describe their properties clearly.
- Manipulate expressions to see the bigger picture.
Example: instead of only computing 3 + 5 = 8, you think about the property that addition is commutative (order doesn't matter) and why that is true.

🧒 Algebraic thinking starts early

🌱 Kindergarten to elementary

The Common Core Standards include "Operations and Algebraic Thinking" beginning in kindergarten.
Everyone who shows up to school has already learned a lot about abstraction and generalization—the fundamental ideas in algebra.
Students are already capable of learning to formalize these ideas.

🎓 The teacher's job

Provide students with more experiences in abstraction and generalization in a mathematical context.
Goal: make these ideas seem natural when students reach a class called "Algebra."
Don't confuse: teaching algebra in elementary school does not mean teaching 8th-grade symbol manipulation; it means building the thinking habits that underlie algebra.

🔗 Connection to fractions (from preceding excerpt)

🔢 Fractions as an example of abstraction

The excerpt on fractions (preceding this section) illustrates the shift from concrete to abstract thinking:

A single fraction is actually a whole infinite class of pairs of numbers that we consider "equivalent."

Mathematicians think of a fraction not as a single symbol but as a class of equivalent pairs.
Example: the fraction one-half is represented by infinitely many pairs (1 and 2, 2 and 4, 3 and 6, etc.).
This is a "hefty shift of thinking": a "number" changes from a specific combination of symbols to a whole class of combinations deemed equivalent.

🧮 Defining operations abstractly

Mathematicians define addition and multiplication of fractions using rules (e.g., the rule for adding fractions with different denominators).
They must then prove that choosing different equivalent representations leads to the same answer.
Example: adding fractions using different equivalent forms (like 1/2 vs. 2/4) should give equivalent results—this is not immediately obvious and must be proven.
This approach is "abstract, dry, and not at all the best first encounter" for students, but it is the most honest mathematical foundation.

🤔 Real-world meaning vs. formal definition

The formal approach avoids the question of what a fraction "really means" in the real world.
For elementary students, teachers must balance:
- Concrete models (like "Pies Per Child") that give intuitive meaning.
- Gradual introduction to the idea that fractions are classes of equivalent representations.
Don't confuse: the formal definition is the endpoint, not the starting point, for learners.

What is a Fraction?

🧭 Overview

🧠 One-sentence thesis

Fractions are difficult to teach and learn because they require multiple mental models, but thinking of a fraction as the answer to a division problem—specifically, "pies per child"—provides a concrete foundation for understanding them.

📌 Key points (3–5)

Core difficulty: Fractions require thinking about them in many different ways depending on the problem context.
Division interpretation: A fraction is the answer to a division problem—the numerator divided by the denominator.
"Pies per child" model: The fraction represents how much one whole child receives when a certain number of pies is shared equally among a certain number of children.
Common confusion: Fractions are usually taught starting with "one-half" (1 pie among 2 children), but the general principle applies to any division scenario.
Why it matters: Teachers need multiple mental models to explain fractions effectively; the division/sharing model is one foundational approach.

🥧 The division interpretation of fractions

🥧 Fraction as division answer

A fraction is equivalent to a division problem: the numerator divided by the denominator.

The fraction represents "the number of pies one whole child receives when [denominator] kids share [numerator] pies equally."
This is not just a symbolic rule; it describes a real-world sharing scenario.
Example: 6 pies shared equally among 3 children yields 2 pies per kid, written as 6 ÷ 3 = 2.

🔢 How the model works

Numerator: the total number of pies to be shared.
Denominator: the number of children sharing.
Result: the amount each child receives (pies per child).
The fraction notation captures this relationship: numerator ÷ denominator = pies per child.

📐 Examples of the "pies per child" model

📐 Whole-number results

When the division yields a whole number, the fraction simplifies to that number:

Pies	Children	Calculation	Pies per child
10	2	10 ÷ 2	5
8	2	8 ÷ 2	4
5	5	5 ÷ 5	1

These examples show that fractions are not always "parts of a whole" in the traditional sense—they can represent whole quantities.
The model works the same way regardless of whether the answer is a whole number or not.

🍰 Fractional results

When the division does not yield a whole number, the fraction remains in fractional form:

1 pie among 2 children yields one-half (1 ÷ 2), written as 1/2.
This is the typical starting point for teaching fractions, but the excerpt emphasizes it is just one instance of the general division principle.
Don't confuse: "one-half" is not a special case requiring a different definition—it follows the same "pies per child" logic as all other fractions.

🧠 Why fractions are hard to teach and learn

🧠 Multiple mental models required

The excerpt states that fractions are "one of the hardest topics to teach (and learn!) in elementary school."
The reason: "you have to think about them in a lot of different ways, depending on the problem at hand."
Teachers need "lots of mental models—lots of ways to explain the same concept."
The division/sharing model is one such model, but students and teachers must be flexible in how they conceptualize fractions.

🔄 Context-dependent thinking

Different problems may require thinking of fractions as:
- Parts of a whole (traditional model).
- Results of division (the model presented here).
- Ratios or rates.
- Other interpretations depending on the situation.
This flexibility is what makes fractions challenging but also powerful once mastered.

The Key Fraction Rule

🧭 Overview

🧠 One-sentence thesis

A fraction represents the answer to a division problem—specifically, the amount each person receives when a quantity is shared equally.

📌 Key points (3–5)

Core definition: A fraction is the result of dividing a numerator (what you have) by a denominator (how many share it).
The "pies per child" model: fractions express how much one whole person gets when items are divided equally.
Why fractions are hard: you must think about them in many different ways depending on the problem context.
Common confusion: fractions are not just "parts of a whole" in isolation—they answer a specific division question (how much per person/unit).
Key insight: even simple fractions like one-half (1/2) mean "1 pie shared among 2 children yields one-half pie per child."

🥧 The pies-per-child model

🥧 What a fraction represents

A fraction is equivalent to a division problem: it represents the number of pies one whole child receives when children share pies equally.

The numerator is what you have (the pies).
The denominator is how many share it (the children).
The fraction is the answer: how much each child gets.

🔢 How the model works

The excerpt gives these examples:

Pies	Children	Fraction	Pies per child
6	3	6/3	2
10	2	10/2	5
8	2	8/2	4
5	5	5/5	1
1	2	1/2	one-half

In every case, the fraction answers: "How much does one child receive?"
Example: 6 pies shared among 3 children → each child gets 2 pies → 6/3 = 2.
Example: 1 pie shared among 2 children → each child gets one-half pie → 1/2 = "one-half."

🧩 Why "one-half" is meaningful

The excerpt emphasizes that 1/2 is not just a symbol; it tells a story.
It means: when 1 pie is divided equally among 2 children, each child receives one-half of a pie.
This is "how fractions are usually taught"—the final example grounds the abstract symbol in a concrete sharing scenario.

🤔 Why fractions are difficult

🤔 Multiple mental models required

The excerpt states that fractions are "one of the hardest topics to teach (and learn!) in elementary school."
The reason: "you have to think about them in a lot of different ways, depending on the problem at hand."
Teachers need "lots of mental models—lots of ways to explain the same concept."

🔄 Context-dependent thinking

A fraction can represent:
- A division result (pies per child).
- A part of a whole.
- A ratio or rate.
The excerpt focuses on the division interpretation as a foundational model.
Don't confuse: the same fraction symbol may require different mental pictures in different problems.

🧮 Fractions as division

🧮 The equivalence rule

The excerpt explicitly states: "The fraction [numerator/denominator] is equivalent to the division problem [numerator ÷ denominator]."
This is the "key fraction rule": every fraction is a division waiting to be performed.
Example: 6/3 is the same as asking "6 divided by 3," which equals 2.

🎯 What the division means

The division is not abstract; it answers a sharing question.
Numerator: the total amount to be shared.
Denominator: the number of recipients.
Result: the amount per recipient.
Example: 5 pies among 5 kids → 5/5 = 1 pie per kid.

🚫 Common confusion

Don't think of the numerator and denominator as separate, unrelated numbers.
They form a single division problem: the numerator is being divided by the denominator.
The fraction is the answer to that division, not just "a number over another number."

Adding and Subtracting Fractions

🧭 Overview

🧠 One-sentence thesis

Adding and subtracting fractions requires understanding that fractions represent amounts per child (not pies or children themselves), and when denominators differ, we must first rewrite fractions with a common denominator before combining the amounts.

📌 Key points (3–5)

What a fraction really is: not a pie or a child, but an amount of pie per child—something more subtle than the objects themselves.
Adding same denominators: when denominators match, add the numerators because you are adding amounts received by children in the same-sized group (e.g., 2/7 + 3/7 = 5/7).
Different denominators problem: fractions with unlike denominators cannot be added directly; you must first find equivalent fractions with a common denominator.
Common confusion: students may think "add numerators and add denominators" (e.g., 2/7 + 3/7 = 5/14) because they confuse adding pies and children with adding the amounts per child.
Why same denominators are essential: the "same 7 kids" can share 2 pies and then 3 more pies, totaling 5 pies among 7 kids; different group sizes require rewriting first.

🥧 What fractions actually represent

🥧 Not pies, not children—amounts per child

The excerpt emphasizes that a fraction is not a pie and not a child.
Instead:

A fraction is an amount of pie per child.
You cannot add pies, you cannot add children; you must add the amounts individual kids receive.
Example: 2/7 means "the amount of pie that one child gets when two pies are shared by seven children."

🚫 Why "add pies and add kids" fails

It is tempting to say 2/7 + 3/7 = 5/14 (adding numerators and denominators separately).
But 5/14 would represent 5 pies among 14 kids—a different scenario entirely.
The correct answer is 5/7, because the same 7 kids share a total of 5 pies.

➕ Adding fractions with the same denominator

➕ The basic rule

When denominators are the same, add the numerators:
- 2/7 + 3/7 = 5/7
- 4 tenths + 3 tenths + 8 tenths = 15 tenths
- 82 sixty-fifths + 91 sixty-fifths = 173 sixty-fifths
The excerpt says this "seems just as easy as adding apples."

🧒 Why it works: the Pies Per Child Model

Imagine 7 kids sharing pies.
First, they share 2 pies → each child gets 2/7.
Then they share 3 more pies → each child gets an additional 3/7.
Total per child: the same as if the 7 kids had shared 5 pies to begin with → 5/7.
General case: c kids first share a pies, then share b more pies. Total per child is a/c + b/c = (a + b)/c.
Key insight: it does not matter that the kids first share a pies and then b pies; the total is the same as sharing all (a + b) pies at once.

➖ Subtracting with the same denominator

Subtraction works the same way: subtract the numerators.
Example: 5/7 − 2/7 = 3/7.
The "same denominator" fact is essential: the explanation relies on the same group of kids sharing pies in sequence.

🔀 Adding fractions with different denominators

🔀 The challenge

Example: What is 2/5 + 1/3?
In words: Poindexter is part of a team of 5 kids that shares 2 pies, then later part of a team of 3 kids that shares 1 pie. How much pie does Poindexter receive in total?
The excerpt calls this "a very difficult problem."

🔄 The solution: find a common denominator

Write each fraction in equivalent forms using the key fraction rule (multiply numerator and denominator by the same number).
For 2/5 and 1/3, list equivalent fractions:
- 2/5 = 4/10 = 6/15 = 8/20 = ...
- 1/3 = 2/6 = 3/9 = 4/12 = 5/15 = ...
Notice that 6/15 and 5/15 share the same denominator.
Now add: 6/15 + 5/15 = 11/15.
You do not need to list all forms; if you can see a common denominator right away, use it.

📐 Another example: 3/8 + 3/10

Valerie is part of a group of 8 kids who share 3 pies, then part of a group of 10 kids who share 3 different pies.
Find a common denominator (e.g., 40):
- 3/8 = 15/40
- 3/10 = 12/40
Add: 15/40 + 12/40 = 27/40.

❌ Common mistake: "add the denominators"

A student (Cassie) suggests: "When denominators are the same, we add the numerators. So when numerators are the same, shouldn't we just add the denominators?"
- Example: 3/8 + 3/10 = 3/18 (?).
Why this is wrong: adding denominators means combining the number of kids, not the amounts per child.
The excerpt asks: "What do you think of Cassie's suggestion? Does it make sense?"
Answer: It does not make sense because fractions are amounts per child, not a count of pies or children.

🔍 Why the same denominator is essential

🔍 Where the "same kids" assumption is used

The explanation for adding same-denominator fractions relies on imagining "the same 7 kids" sharing pies in two rounds.
If the denominators differ, the groups are different sizes, so you cannot directly combine the amounts.
You must first rewrite both fractions so they represent amounts for groups of the same size (common denominator).

🔍 Subtraction with different denominators

The same logic applies: find a common denominator first, then subtract the numerators.
Example: 5/6 − 1/4 requires rewriting both fractions with a common denominator (e.g., 12):
- 5/6 = 10/12
- 1/4 = 3/12
- 10/12 − 3/12 = 7/12.

🧩 Limitations of the Pies Per Child Model

🧩 The model's scope

The "Pies Per Child Model" works well for:
- Understanding what a fraction is (amount per child).
- Equivalent fractions (same amount per child with different numbers of pies and kids).
- Adding and subtracting fractions (combining amounts per child).
It does not work for multiplication: the excerpt asks, "What would 2/3 × 3/7 mean?" and states there is "no way to use this model to make sense of multiplying fractions."

🔄 Switching models is confusing

The excerpt acknowledges that students must switch concepts and models as they learn more about fractions.
"We keep switching concepts and models, and speak of fractions in each case as though all is naturally linked and obvious. None of this is obvious, it is all absolutely confusing."
This is "one of the reasons that fractions can be such a difficult concept to teach and to learn in elementary school."

What is a Fraction? Revisited

🧭 Overview

🧠 One-sentence thesis

Fractions require switching between different models—from "pies per child" to measurement on a number line to area—and this fundamental shift in perspective is one reason fractions are so difficult to teach and learn.

📌 Key points (3–5)

Model limitations: The "Pies Per Child" division model works for addition and subtraction but cannot explain multiplication of fractions.
Units matter: Every fraction has a hidden unit (the "whole"), and problems become difficult when different fractions refer to different wholes.
Common confusion: Students face multiple models (division, measurement, area) presented as if they are naturally linked, but the connections are not obvious.
Ordering strategies: Comparing fractions relies on intuitive methods like benchmarks (0, ½, 1), same denominators/numerators, and equivalent fractions.
Area model for multiplication: Multiplying fractions corresponds to finding the area of a rectangle with fractional side lengths.

🔄 Why we must switch models

🥧 The Pies Per Child model and its limits

The "Pies Per Child Model" treats a fraction as the answer to a division problem—for example, 2/3 is the result of sharing two pies among three children.

This model works well for understanding:
- What fractions mean
- Equivalent fractions
- Adding and subtracting fractions
Pies can be any shape: round, square, triangular, or squiggly.
The breakdown: There is no way to use this model to make sense of multiplying fractions.
Example: What would "3/4 times 2/5 pies per child" mean? The division interpretation fails here.

🌀 The confusion students face

Teachers switch concepts and models but speak as though all is naturally linked and obvious.
Don't confuse: The models are not obviously connected; the shift is "fundamentally perturbing."
This is one reason fractions are difficult to teach and learn in elementary school.
Students must learn to recognize which model applies to which operation.

📏 Units and the measurement model

📦 Every fraction has a hidden unit

Units are always attached to a fraction, even if hidden—the "whole" or unit is the amount that equals 1.

When you see 1/2, ask yourself: "half of what?"
The answer is your unit.
In the Pies Per Child model, the unit was consistently one whole pie.
Key insight: Fractions are always relative to some whole, and that whole can change.

🔀 Different units in the same problem

One thing that makes fraction problems difficult: fractions in the problem may be given in different units (parts of different wholes).
Example: Mr. Li shows a picture, and students give different answers:
- Kendra: 5
- Dylan: 5/6
- Kiana: 5/12
- Nate: 10/12
Everyone is right because each student chose a different unit.
To justify each answer, identify what each student thought was the unit in the picture.

📐 Fractions as portions of a segment

Thinking of fractions as "portions of a segment" allows us to place them on a number line.
On a number line, the unit is clear: it is the distance between 0 and 1.
This measurement model makes it easier to compare the relative size of fractions.
Fractions farther to the right are larger, just like whole numbers.

🔢 Ordering and comparing fractions

🎯 Benchmark strategies

Greater than 1: A fraction is greater than 1 if its numerator is greater than its denominator (you have more pieces than needed to make one whole).
Greater than 1/2: A fraction is greater than 1/2 if the numerator is more than half the denominator; equivalently, if twice the numerator is bigger than the denominator.
Use benchmarks to organize fractions: 0 to 1/2, 1/2 to 1, and greater than 1.

⚖️ Comparing with same denominators or numerators

Situation	Rule	Reason	Example
Same denominators	Compare numerators; fractions in same order as numerators	Pieces are the same size; more pieces = bigger number	3/7 < 5/7
Same numerators	Compare denominators; fractions in reverse order of denominators	Bigger denominator = smaller pieces; same number of smaller pieces = smaller total	3/7 > 3/11
Numerator = denominator − 1	Fractions in same order as denominators	Each is a pie with one piece missing; bigger denominator = smaller missing piece = more remaining	6/7 > 11/12
Numerator = denominator − constant	Fractions in same order as denominators	Same reasoning: missing the same number of pieces, but smaller pieces	3/7 < 8/12 (both missing 4 pieces)

🔄 Using equivalent fractions

Find equivalent fractions that let you compare numerators or denominators.
Then apply one of the rules above.
Example: To compare 3/5 and 5/8, note that 3/5 is just a bit smaller than 5/8 on the number line.

🔲 The area model for multiplication

🟦 Building the area model

The product of two fractions corresponds to an area problem, just as multiplying whole numbers does.

For whole numbers: 23 × 37 is the area (number of 1×1 squares) of a 23-by-37 rectangle.
For fractions: 4/7 × 2/3 should also correspond to an area.

🧮 Example: 4/7 × 2/3

Start with a segment of length 1 unit.
Build a square with 1 unit on each side; its area is 1 square unit.
Divide the top segment into three equal pieces (each piece is 1/3).
Divide the side segment into seven equal pieces (each piece is 1/7).
Use those marks to divide the whole square into small, equal-sized rectangles.
Each small rectangle has one side measuring 1/3 and another side measuring 1/7.
The product 4/7 × 2/3 is the area of a rectangle that is 4/7 units tall and 2/3 units wide.

🎨 Why the area model works

The area model provides a visual way to understand fraction multiplication.
It connects to the whole-number area model students already know.
Don't confuse: This is a different model from "pies per child"; it is not about division but about measuring area with fractional dimensions.

🔢 Arithmetic sequences with fractions

📊 What is an arithmetic sequence?

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is a constant.

Examples with whole numbers:
- Pattern A: 2, 5, 8, 11, 14, ... (common difference: 3)
- Pattern B: 10, 20, 30, 40, ... (common difference: 10)
The same concept applies to fractions.

🧩 Checking if a fraction sequence is arithmetic

Calculate the difference between consecutive terms.
If the difference is constant, the sequence is arithmetic.
Example problem: Find two fractions between 1/4 and 3/4. Are the resulting four fractions in an arithmetic sequence?
Another problem: Find two fractions between 1/4 and 3/4 so the resulting four numbers are in an arithmetic sequence.

🎯 Creating arithmetic sequences with fractions

You can insert fractions between two given fractions to form an arithmetic sequence.
The common difference must be chosen so that the spacing is equal.
Example: Find three fractions between 1/4 and 3/4 so the resulting five numbers are in an arithmetic sequence.

Multiplying Fractions

🧭 Overview

🧠 One-sentence thesis

Multiplying fractions can be understood through area models, the "pies per child" interpretation, and unitizing, all of which explain why the rule "multiply numerators and multiply denominators" works.

📌 Key points (3–5)

Area model foundation: The product of two fractions corresponds to the area of a rectangle whose sides have fractional lengths.
The multiplication rule: To multiply fractions, multiply the numerators together and multiply the denominators together: (a/b) × (c/d) = (a×c)/(b×d).
Why the rule works: The area model shows that dividing one side into b pieces and the other into d pieces creates b×d total rectangles, and shading a pieces on one side and c on the other shades a×c rectangles.
Common confusion: "of" in word problems sometimes means multiplication (e.g., "2/3 of 4/7 of a cake") but not always—context matters, not just keyword translation.
Multiple interpretations: Multiplying a fraction by a whole number can be understood through the fraction rule, the "pies per child" model, or unitizing (counting equal pieces).

🖼️ The area model for fraction multiplication

📐 Building the rectangle

The area model extends the whole-number multiplication approach to fractions:

Start with a square that is 1 unit on each side (area = 1 square unit).
To multiply (4/7) × (2/3), divide the top side into 3 equal pieces and the left side into 7 equal pieces.
This divides the whole square into small, equal-sized rectangles.
Mark off 4/7 on one side and 2/3 on the other side.
The shaded rectangle (bounded by these fractional lengths) represents the product.

Example: For (4/7) × (2/3), the unit square is divided into 21 equal rectangles total, and the shaded region contains 8 of them, so (4/7) × (2/3) = 8/21.

🔢 Why the denominator is b×d

The top segment divided into b pieces creates b columns.
The side segment divided into d pieces creates d rows.
A rectangle with b columns and d rows has b×d total pieces (using the area model for whole-number multiplication).

🎨 Why the numerator is a×c

The excerpt asks readers to explain why a×c rectangles are shaded when you shade a pieces on one side and c pieces on the other—this follows from the grid structure created by the divisions.

🥧 Multiplying fractions by whole numbers

🧮 Using the fraction rule

Elementary students often learn to treat a whole number as a fraction with denominator 1.

Example: To compute 2 × (3/7), write 2 as 2/1, then multiply: (2/1) × (3/7) = (2×3)/(1×7) = 6/7.

🍰 The "pies per child" interpretation

The fraction a/b means the amount each child gets when b children evenly share a pies.

(3/7) means 7 children share 3 pies.
2 × (3/7) means doubling the amount each child gets, which is the same as doubling the number of pies: 7 children share 6 pies, so the answer is 6/7.

🧩 Unitizing (counting equal pieces)

(3/7) means "I have 7 equal pieces (of something) and I take 3 of them."
2 × (3/7) means "do that twice": take 3 pieces, then 3 pieces again, for a total of 6 pieces.
There are still 7 equal pieces in the whole, so the answer is 6/7.

Don't confuse: All three methods—fraction rule, pies per child, and unitizing—give the same answer, but they offer different mental models. The excerpt notes that which method feels more natural may depend on the particular numbers or personal preference.

📏 The general multiplication rule

📝 Statement of the rule

Multiplying Fractions: (a/b) × (c/d) = (a×c)/(b×d). You may then simplify the answer, but it is always equivalent to this form.

🔍 Why the rule works (area model explanation)

The area model justifies the rule step-by-step:

Build a rectangle with one side of length a/b and the other of length c/d.
Start with a 1×1 square.
Divide the top into b equal pieces and shade a of them (length a/b).
Divide the left side into d equal pieces and shade c of them (length c/d).
Divide the whole square into equal-sized rectangles using the tick marks.
Shade the rectangle bounded by the shaded segments.

Result:

Total rectangles = b×d (from the grid structure).
Shaded rectangles = a×c (from the shaded segments on each side).
So the answer is (a×c)/(b×d).

🧠 Roy's reasoning using the key fraction rule

Roy explains the fraction rule (a/b) = a × (1/b) by reasoning with multiplication:

Multiplying by 1 does not change a number: a = a × 1.
In general, a = a × (b/b) because b/b = 1.
So a = (a × b)/b.
Since (a × b)/b = a, dividing both sides by b (conceptually) gives a/b = a × (1/b).
By the same reasoning, (1/b) = 1 × (1/b), so a/b = a × (1/b).

The excerpt asks readers to extend Roy's reasoning to explain the general rule a/b = a × (1/b) for positive whole numbers.

⚠️ Word problems and the "of" keyword

📖 Two different problems

The excerpt contrasts two problems to show that "of" does not always mean multiplication:

Problem	Description	Does "of" mean multiply?
Pam had 2/3 of a cake and ate 1/2 of it. How much total cake did she eat?	"1/2 of 2/3" means multiply: (1/2) × (2/3)	Yes
On Monday, Pam ate 2/3 of a cake. On Tuesday, she ate 1/2 of a cake. How much total cake did she eat?	"2/3 of a cake" and "1/2 of a cake" are separate amounts to add	No

🚨 Don't memorize keyword rules

Students are often taught to treat "of" as multiplication.
The excerpt warns: this makes sense in some cases but not others.
Important: Read carefully and understand what the problem is asking; do not memorize rules about "translating" word problems based on keywords alone.

🔧 Extending the area model

🔢 Fractions greater than 1

The excerpt poses a problem: How can you extend the area model for fractions greater than 1?

Examples given (without solutions in the excerpt):

(3/2) × (4/3)
(5/4) × (7/5)
(9/8) × (11/9)

The excerpt does not provide the answer but encourages readers to try drawing pictures for these cases.

Dividing Fractions: Meaning

🧭 Overview

🧠 One-sentence thesis

Dividing fractions can be understood through meaningful models—finding equal-sized groups, using common denominators, or rewriting as missing-factor multiplication—rather than relying solely on the "invert and multiply" rule.

📌 Key points (3–5)

Quotative model for fractions: division asks "How many groups of this size can I find in that amount?" just as with whole numbers.
Visual rectangle method: draw rectangles divided by both denominators, shade the fractions, and count how many copies of the divisor fit into the dividend.
Common denominator shortcut: when two fractions share a denominator, divide the numerators directly.
Missing factor approach: rewrite division as a multiplication question ("what times the divisor equals the dividend?").
Common confusion: the "invert and multiply" rule works, but understanding why requires seeing division as grouping or missing factors, not just memorizing the algorithm.

🍰 The quotative model for fraction division

🍰 What "groups of equal size" means

Quotative model: division asks "How many groups of [divisor size] can I find in [dividend]?"

For whole numbers: 18 ÷ 3 means "How many groups of 3 can I find in 18?"
For fractions: 6 ÷ two-thirds means "How many groups of two-thirds can I find in 6?"
The model is the same; only the size of the groups changes.

🖼️ Simple example: 6 ÷ two-thirds

Draw 6 pies.
Mark off groups of two-thirds (each group is two-thirds of one pie).
Count the groups: you find nine equal groups.
Conclusion: 6 ÷ two-thirds = 9.

🧩 Harder example: three-fourths ÷ one-third

Question: "How many groups of one-third can I find in three-fourths?"
A rough sketch shows two full groups of one-third inside three-fourths, plus a little bit left over.
The answer is more than 2, but we need a precise method to find the exact remainder.

📐 The rectangle method for exact answers

📐 How to set up the rectangle

Draw two identical rectangles.
Divide each rectangle into rows equal to the denominator of the first fraction and columns equal to the denominator of the second fraction.
Example: for three-fourths ÷ one-third, use 4 rows (from three-fourths) and 3 columns (from one-third).

🎨 Shading and counting

Shade three-fourths of the first rectangle (three rows out of four).
Shade one-third of the second rectangle (one column out of three).
Now ask: "How many copies of the shaded one-third fit into the shaded three-fourths?"

🔢 Finding the quotient

One-third equals four small squares (one column).
In the three-fourths rectangle, count groups of four squares:
- Two complete groups of four squares (two groups of one-third).
- One square left over, which is one-fourth of a group of four.
Conclusion: three-fourths ÷ one-third = 2 and one-fourth.

Don't confuse: the leftover piece is measured as a fraction of the group size you're looking for, not of the original rectangle.

🔗 The common denominator method

🔗 The shortcut rule

Common denominator method: If two fractions have the same denominator, divide the numerators directly.

In symbols: (a over d) ÷ (b over d) = a ÷ b.
Example: six-fifths ÷ two-fifths = 6 ÷ 2 = 3.

🛠️ Why it works

When both fractions count pieces of the same size (same denominator), the question becomes "How many groups of b pieces fit into a pieces?"
The denominator cancels out; only the numerator counts matter.

🔄 What if denominators differ?

The method still works if you first rewrite both fractions with a common denominator.
Example: to compute three-fourths ÷ one-third, rewrite as nine-twelfths ÷ four-twelfths, then divide 9 ÷ 4.
This approach always produces an exact answer.

🧮 The missing factor method

🧮 Rewriting division as multiplication

Any division problem can be rewritten as a missing-factor multiplication question.
Example: 18 ÷ 3 becomes "What times 3 equals 18?"
For fractions: a ÷ b becomes "What times b equals a?"

✅ When it works nicely

Some fraction divisions have obvious whole-number or simple-fraction answers.
Example: two-thirds ÷ one-third rewrites as "What times one-third equals two-thirds?" Answer: 2.
This method builds on what you already know about multiplying fractions.

⚠️ When it's less obvious

Example: three-fourths ÷ two-fifths rewrites as "What times two-fifths equals three-fourths?"
There isn't an immediately obvious ratio of whole numbers to fill in the blank.
The excerpt notes this limitation and promises to resolve it later (the resolution is not included in this excerpt).

Don't confuse: the missing factor method is conceptually powerful (it connects division to multiplication), but it doesn't always yield a quick numerical answer without further techniques.

Dividing Fractions: Invert and Multiply

🧭 Overview

🧠 One-sentence thesis

The "invert and multiply" rule for dividing fractions emerges naturally from simplifying ugly fractions by multiplying numerator and denominator by the denominators of the component fractions.

📌 Key points (3–5)

Core insight: Dividing fractions can be rewritten as simplifying an "ugly fraction" (a fraction whose numerator or denominator is itself a fraction).
The simplification technique: Multiply both the numerator and denominator by the denominators of the inner fractions to clear them out.
How invert-and-multiply emerges: When you simplify a division problem written as a fraction, you end up multiplying the first fraction by the reciprocal (upside-down version) of the second.
Common confusion: Students may correctly apply the rule but not understand why it works—the excerpt emphasizes building understanding through the missing factor method and ugly fraction simplification.
Multiple methods exist: The excerpt presents five methods (rectangle pictures, common denominator, missing factor, ugly fraction simplification, and invert-and-multiply), each with different strengths for understanding vs. computation.

🔄 From missing factor to ugly fractions

🔄 The missing factor approach

The excerpt explains that any division problem can be rewritten as a missing factor multiplication problem.
Example: the division problem "a ÷ b" becomes "b × ? = a".
For fractions, this means you ask what number times the divisor gives you the dividend.
Limitation: This method "doesn't always work out so nicely" when there isn't an obvious whole-number ratio to fill in the blank.

🧩 Ugly fractions as division

A fraction is the answer to a division problem.

The excerpt shows that a division like 3/5 ÷ 4/7 can be written as the "ugly fraction" (3/5) / (4/7).
This fraction has fractions in both the numerator and denominator, making it hard to interpret directly.
The goal is to simplify this ugly fraction into a friendlier form that clearly shows how much each person/unit gets.

🧮 The simplification technique

🧮 Multiply to clear denominators

The excerpt demonstrates a systematic process:

Identify the denominators of the inner fractions (e.g., if you have (3/5) / (4/7), the inner denominators are 5 and 7).
Multiply both the numerator and denominator of the ugly fraction by these numbers.
This clears out the inner fractions, leaving only whole numbers.

Why this works: Multiplying numerator and denominator by the same number is the same as multiplying by 1, so the value doesn't change—you're just rewriting it in a clearer form.

📐 Step-by-step example from the excerpt

For 3/5 ÷ 4/7, written as (3/5) / (4/7):

First multiply numerator and denominator each by 5 (to clear the 5 in 3/5).
Then multiply numerator and denominator each by 7 (to clear the 7 in 4/7).
The result is (3 × 7) / (4 × 5) = 21/20.

The excerpt also shows you can multiply by both denominators "at the same time" for efficiency.

⚡ Streamlined version

For 5/9 ÷ 8/11:

Write it as (5/9) / (8/11).
Multiply numerator and denominator by 9 and by 11 simultaneously.
Numerator becomes: (5/9) × 9 × 11 = 5 × 11 = 55.
Denominator becomes: (8/11) × 9 × 11 = 8 × 9 = 72.
Result: 55/72.

Don't confuse: You're not changing the value of the fraction, only its appearance—this is an application of the "key fraction rule" (multiplying top and bottom by the same thing).

🔁 Discovering invert and multiply

🔁 Janine's observation

The excerpt presents a student named Janine who notices a pattern while simplifying (a/b) / (c/d):

After multiplying numerator and denominator by b and d, she gets (a × d) / (b × c).
She realizes this is the same as a/b × d/c.
Her conclusion: "Dividing one fraction by another is the same as multiplying the first fraction with the second fraction upside down!"

🧪 Testing the pattern

The excerpt asks readers to verify Janine's claim by:

Working out several examples using the simplification method.
Comparing the results to what you get by inverting the second fraction and multiplying.
Considering whether this is always true, not just in specific examples.

Key question from the excerpt: "Is dividing two fractions always the same as multiplying the two fractions with the second one turned upside down?"

🎯 Why it works

When you simplify (a/b) / (c/d):

Multiply numerator and denominator by b: you get (a) / (b × c/d).
Multiply numerator and denominator by d: you get (a × d) / (b × c).
This is exactly a/b × d/c, which is the first fraction times the reciprocal of the second.

The "invert and multiply" rule is not arbitrary—it's a shortcut that emerges from the logic of simplifying ugly fractions.

📚 Summary of methods

📚 Five methods for dividing fractions

The excerpt lists all the methods covered:

Method	What it involves	Strength
Rectangle picture	Draw and shade rectangles to visualize groups	Good for understanding meaning
Common denominator	Rewrite with same denominator, divide numerators	Works when fractions share or can share a denominator
Missing factor	Rewrite division as multiplication with a blank	Reinforces relationship between operations
Ugly fraction simplification	Multiply numerator and denominator to clear inner fractions	Shows why invert-and-multiply works
Invert and multiply	Flip the second fraction and multiply	Fastest for computation

🤔 Pedagogical considerations

The excerpt asks readers to reflect on:

Which method is easiest to understand why it works.
Which method is easiest to use in computations.
Benefits and drawbacks of each method, both as a future teacher and as someone solving problems.

Don't confuse: Ease of computation vs. ease of understanding—the fastest method (invert and multiply) may not be the most transparent for learning.

👥 Student misconceptions

The excerpt includes two student examples (Jessica and Isaac) who apply methods in ways that may or may not be correct. Teachers are asked to:

Determine if the student's solution is correct.
Explain what is going on with the solution.
Decide what to do as the student's teacher.

This emphasizes that understanding the process matters, not just getting the right answer.

Dividing Fractions: Problems

🧭 Overview

🧠 One-sentence thesis

Understanding when and why to divide fractions requires recognizing division situations in real-world contexts and distinguishing between quotative division (how many groups?) and partitive division (what size is each group?).

📌 Key points (3–5)

Why division with fractions is hard to recognize: fractions themselves are already answers to division problems, making "dividing divisions" conceptually complex.
Strategy for creating fraction division problems: start with whole-number division problems, then sensibly replace numbers with fractions.
Quotative vs partitive with fractions: most fraction division word problems ask "how many groups of this size?" (quotative), but some ask "what size is each group?" (partitive).
Common confusion: multiplying by one-half (e.g., halving a recipe) is NOT the same as dividing fractions—it's multiplication, not division.
Zero cases: zero divided by any number equals zero, but dividing by zero is undefined because no multiplication fact can solve it.

🔍 Recognizing division vs multiplication

❌ The recipe confusion

A common mistake when asked to create a division problem for fractions:

Wrong example: "My recipe calls for 3/4 cups of flour, but I only want to make half a recipe. How much flour should I use?"
Why it's wrong: This problem asks you to cut the recipe in half, which means dividing by 2 or multiplying by 1/2—not dividing fractions.
The excerpt emphasizes this is a multiplication problem, not a division problem.

🎯 Why fraction division is conceptually difficult

The excerpt identifies two reasons:

Fractions are already answers to division problems, so dividing fractions means "dividing and then dividing some more."
Fractions make division problems look more complicated than whole-number versions.

🛠️ Creating fraction division problems

📝 The whole-number strategy

Strategy: Write a problem that involves division of whole numbers, and then see if you can change the numbers to fractions in a sensible way.

This approach helps because:

You first establish the division structure with familiar numbers.
Then you adapt the context so fractions fit naturally.
The division meaning (quotative or partitive) stays the same.

🧵 Quotative division examples (how many groups?)

The excerpt provides several parallel examples showing whole numbers converted to fractions:

Whole-number version	Fraction version	Why it's division
10 feet of ribbon, cut into 2-inch pieces	5/6 feet of ribbon, cut into 1/2-foot pieces	Making equal groups of 1/2 foot each, asking how many groups
Clock rings every 15 minutes over 120 minutes	Alarm every half hour during 1 1/2 hour movie	Making equal groups of 1/2 hour each, asking how many groups
6-gallon tank, 3-gallon bucket	3/4 gallon tank, 1/2 gallon bucket	Making equal groups of 1/2 gallon each, asking how many buckets
6 cups flour, 2-cup scoop	3/4 cups flour, 1/3 cup measure	Making equal groups of 1/3 cup, asking how many times to fill

All of these are quotative division: you have a total amount and are dividing it into equal-sized groups, asking "how many groups?"

🏃 Partitive division example (what size per group?)

The excerpt highlights one different type:

Whole-number version: "I ran 12 miles and went around the same route 3 times. How long was the route?"
Fraction version: "I ran 3/4 miles before I twisted my ankle. I only finished half the race. How long was the race course?"

This is partitive division because:

For whole numbers: "20 divided by 4" asks "20 is 4 groups of what size?"
For fractions: "3/4 divided by 1/2" asks "3/4 is half a group of what size?"
You know the number of groups and the total; you're finding the size of each group.

🔢 Special case: fractions involving zero

0️⃣ Zero in the numerator

The fraction 0 divided by any non-zero number equals 0.

Pies-per-child explanation:

If there are zero pies among eleven kids, each child gets zero pies.
No matter how many children there are, zero pies means no one gets any pie.

Missing factor explanation:

Think of 0/11 as asking: "0 equals 11 times what?"
The only answer is 0, because 11 times 0 equals 0.
Therefore 0/11 = 0.

⚠️ Zero in the denominator

Dividing by 0 is undefined.

Why it doesn't work:

The excerpt connects this to the zero property of multiplication: any number times 0 equals 0.
If you try to solve "11 divided by 0," you're asking "11 equals 0 times what?"
No number works, because 0 times anything is always 0, never 11.
Using the connection between fractions and division, and division and multiplication, there is no solution.

Don't confuse: Students often learn "dividing by 0 is undefined" as a rule, but the excerpt emphasizes understanding why it makes sense through the connection to multiplication facts.

Fractions involving zero

🧭 Overview

🧠 One-sentence thesis

Division by zero is undefined because fractions with zero in the denominator either have no meaningful value or too many possible values, while zero in the numerator always equals zero.

📌 Key points (3–5)

Zero in the numerator: Any fraction with 0 on top (like 0/11) equals zero, because zero items shared among any number of people gives zero per person.
Zero in the denominator (non-zero numerator): Fractions like 11/0 are undefined because no number satisfies the corresponding multiplication check.
Zero divided by zero: The fraction 0/0 is also undefined, but for a different reason—every number passes the multiplication check, so there are too many possible values.
Common confusion: Students often learn "dividing by zero is undefined" as a rule without understanding why; the key is connecting fractions to multiplication facts.
Safe rule: Zero as a numerator is fine; zero as a denominator must always be avoided.

🥧 Zero in the numerator

🥧 What it means

A fraction with zero in the numerator (like 0/11) represents zero items shared among some number of recipients.

Using the "pies per child" model: if there are zero pies to share among eleven children, each child gets zero pies.
It doesn't matter how many children there are—no pie is no pie.
Example: 0/5, 0/100, 0/1000 all equal zero.

✅ Why it equals zero

The excerpt connects fractions to multiplication through a "missing factor" problem:

The fraction 0/11 asks: what number times 11 equals 0?
The only answer is 0, because 0 × 11 = 0.
This justifies that 0/11 = 0.

🔢 General pattern

The excerpt concludes that for any non-zero number in the denominator, zero divided by that number equals zero.

🚫 Zero in the denominator (non-zero numerator)

🚫 Why it's undefined

Fractions like 11/0 ask: what number times 0 equals 11?

The zero property of multiplication states that any number times zero equals zero.
There is no number that, when multiplied by zero, gives 11 (or any non-zero result).
Therefore, 11/0 has no meaningful value—it is undefined.

🔗 Connection to multiplication

The excerpt emphasizes using the connection between fractions, division, and multiplication:

A fraction represents a division problem.
Division connects to a missing-factor multiplication problem.
When the denominator is zero, the multiplication problem has no solution.

📋 Examples from the excerpt

The excerpt lists several undefined expressions:

There is no number equal to 11/0
There is no number equal to 1/0
There is no number equal to 237/0

All are undefined because they correspond to impossible multiplication problems.

❓ The special case: zero divided by zero

❓ Why 0/0 is also undefined

The fraction 0/0 presents a different problem than other zero-denominator cases.

The excerpt presents three students' claims:

Cyril says 0/0 = 0 because 0 × 0 = 0 ✓
Ethel says 0/0 = 1 because 1 × 0 = 0 ✓
Wonhi says 0/0 = 17 because 17 × 0 = 0 ✓

All three pass the multiplication check! In fact, any number works.

🔄 Too many vs. no solutions

The excerpt contrasts two types of problems:

Expression	Problem	Reason it's undefined
11/0 (non-zero/zero)	No meaningful value	No number satisfies the multiplication check
0/0 (zero/zero)	Too many possible values	Every number satisfies the multiplication check

⚠️ Don't confuse

A fraction like 11/0 fails because there's no solution.
The fraction 0/0 fails because there are infinitely many solutions.
Both are undefined, but for opposite reasons.

✔️ The safe rule

✔️ What's allowed

The excerpt concludes with a clear guideline:

Zero as a numerator is perfectly fine (it equals zero).
Zero as a denominator must never be allowed.
Dividing by zero is "simply too problematic to be done."

🎯 Why this matters

Understanding why division by zero is undefined (rather than just memorizing the rule) connects to deeper mathematical reasoning:

It reinforces the relationship between fractions, division, and multiplication.
It shows that mathematical rules have logical foundations.
It helps distinguish between different types of undefined expressions.

Egyptian Fractions

🧭 Overview

🧠 One-sentence thesis

Egyptian fractions provide a practical and historical method for expressing any fraction as a sum of distinct unit fractions (fractions with numerator 1), which reduces complexity in sharing and calculation.

📌 Key points (3–5)

What Egyptian fractions are: any fraction rewritten as a sum of unit fractions (numerator = 1) with all different denominators.
Practical motivation: sharing pies among children using fewer, larger pieces instead of many tiny pieces of the same size.
Historical context: the Rhind Papyrus (circa 1650 B.C.) shows Egyptians worked primarily with unit fractions and insisted denominators be different.
Common confusion: Egyptian fractions are not about finding any sum of fractions, but specifically unit fractions with distinct denominators—you cannot repeat the same denominator.
Why it matters: this approach simplifies physical division tasks and appears in advanced algebra when working with complex fractional expressions.

🥧 Practical sharing problem

🥧 The pie-sharing scenario

The excerpt introduces the problem: share 7 pies equally among 12 kids.

The standard answer is each child gets 7/12 of a pie, but this "has little intuitive feel" and requires cutting each pie into many small pieces.
A more practical approach:
- First, give each child 1/2 of a pie (12 halves from 6 pies).
- One pie remains; divide it into twelfths and give each child 1/12.
- Result: each child receives 1/2 + 1/12, which equals 7/12.
Why this is better: fewer cuts, less mess, and each child gets a small number of relatively large pieces.

Example: To share 5 pies among 6 children, you could give each child a large piece (e.g., 1/2 or 2/3) first, then divide the remainder into smaller unit fractions, rather than cutting everything into sixths.

🔍 Key insight

The goal is to replace a single fraction (like 7/12) with a sum of unit fractions that are easier to produce physically and conceptually.

📜 Historical background

📜 The Rhind Papyrus

The Rhind Papyrus is an ancient account of Egyptian mathematics named after Alexander Henry Rhind, who acquired it in 1858 in Luxor, Egypt; it dates back to around 1650 B.C.

The papyrus was copied by a scribe named Ahmes from an even older lost text (reign of king Amenehat III).
Ahmes is described as "the earliest known contributor to the field of mathematics."
The document covers fractions, volume, area, pyramids, and more.
Opening quote: "Accurate reckoning for inquiring into things, and the knowledge of all things, mysteries…all secrets."

🏺 Egyptian fraction rules

The Egyptians had a specific way of working with fractions:

They worked primarily with unit fractions (fractions with numerator 1).
They insisted on writing all fractions as sums of unit fractions.
All denominators must be different—no repeating denominators allowed.

Example: Egyptians would not write 3/10 as 1/10 + 1/10 + 1/10 (repeating denominators). Instead, they wrote it as a sum like 1/5 + 1/10 (distinct denominators).

Don't confuse: Egyptian fractions are not just any sum of fractions; they must be unit fractions with all different denominators.

🧮 Definition and examples

🧮 Formal definition

To write a fraction as an Egyptian fraction, you must rewrite the fraction as:

a sum of unit fractions (numerator is 1), and

the denominators must all be different.

📐 Worked examples from the excerpt

Original fraction	Egyptian fraction form	Notes
3/10	(exact form not given, but must be distinct unit fractions)	Cannot use 1/10 + 1/10 + 1/10
5/7	(exact form not given, but must be distinct unit fractions)	Must find unit fractions that sum to 5/7
7/12	1/2 + 1/12	From the pie-sharing example

The excerpt emphasizes: "You should check that the sums above give the correct resulting fractions!"

🔢 General patterns

The excerpt poses questions about finding general rules:

For fractions of the form 2/n (where n is odd): can you find a pattern for writing these as Egyptian fractions?
For fractions bigger than 1/2: may need more than two unit fractions in the sum.
For any fraction at all: can you find a general algorithm?

(The excerpt does not provide the answers, only the problems.)

🔗 Connection to algebra

🔗 Complex fractional expressions

The excerpt notes that Egyptian fraction techniques relate to simplifying complicated algebra expressions.

Advanced algebra students encounter expressions with fractions in both numerator and denominator.
The "key fraction rule" (same technique from "Dividing Fractions: Invert and Multiply") helps simplify these.
Strategy: multiply numerator and denominator by a common expression to eliminate nested fractions.

Example (from excerpt): Given a complex fraction, multiply both numerator and denominator by an appropriate term to make the expression "much less scary."

🧪 Why fractions matter in science and math

Expressions involving fractions "cannot be rewritten as a decimal" in many cases.
"Expressions like this arise in numerous applications."
Important for math and science students to work with fractions in fraction form, without always converting to decimals.

Don't confuse: The algebraic simplification is not about Egyptian fractions per se, but uses the same principle of rewriting fractions into more manageable forms.

Algebra Connections

🧭 Overview

🧠 One-sentence thesis

Complicated algebraic fraction expressions can be simplified by multiplying both numerator and denominator by carefully chosen terms, allowing students to work with fractions in symbolic form rather than always converting to decimals.

📌 Key points (3–5)

Core technique: multiply numerator and denominator by the same expression to simplify complex fractions.
Why it matters: many applications produce expressions that cannot be rewritten as decimals, so working with fractions in symbolic form is essential.
Connection to earlier material: uses the same "key fraction rule" from the chapter on dividing fractions (invert and multiply).
Common confusion: the goal is not to eliminate fractions but to make them "less scary" and easier to manipulate algebraically.

🔧 The simplification technique

🔧 Multiplying by a strategic choice

The excerpt demonstrates multiplying both numerator and denominator by the same term—this does not change the value of the fraction.
The key is choosing what to multiply by: look at the structure of the expression to find what will cancel or simplify terms.
Example: For an expression with a term like "1 over (a plus b over c)" in the denominator, multiply numerator and denominator each by c to clear the nested fraction.

🔗 Link to the key fraction rule

The excerpt states this is "exactly the same technique we used in the chapter on 'Dividing Fractions: Invert and Multiply.'"
The underlying principle: multiplying numerator and denominator by the same nonzero quantity preserves equivalence.
Don't confuse: this is not about converting to decimals; it is about rewriting the fraction in a more workable symbolic form.

🧮 Why symbolic fractions matter

🧮 When decimals are not an option

Expressions like this arise in numerous applications, so it is important for math and science students to be able to work with fractions in fraction form, without always resorting to converting to decimals.

The excerpt emphasizes that some expressions "cannot be rewritten as a decimal."
Working symbolically is a necessary skill in advanced algebra and science contexts.
Example: An expression involving variables (like a, b, c) has no single decimal value; it must remain in fraction form.

🎯 Making expressions "less scary"

The goal is to transform a complicated-looking fraction into one that is "much less scary" or "friendlier."
This does not mean eliminating the fraction entirely—just reorganizing it so the structure is clearer and easier to work with.
The excerpt shows that after simplification, the resulting expression is still a fraction but is more manageable.

📚 Examples from the excerpt

📚 First example walkthrough

Start with a complex fraction (the excerpt does not give the exact starting form, but describes the process).
Multiply numerator and denominator each by a chosen term (the excerpt suggests multiplying by a specific expression).
The result is a simpler-looking fraction.
The excerpt notes: "Do you see why this is a good choice?" indicating that the choice of multiplier is strategic, not arbitrary.

📚 Second example walkthrough

Given another expression, the excerpt suggests multiplying numerator and denominator "each by [one term] and then each by [another term]."
This shows that sometimes multiple steps are needed: apply the technique more than once to fully simplify.
Each step uses the same core idea: multiply top and bottom by the same thing.

📚 Third example hint

For a different expression, the excerpt asks "Why?" after suggesting a particular multiplier.
This reinforces that understanding why a choice works is part of mastering the technique—not just following a recipe.

🧩 Historical context: Egyptian fractions

🧩 What Egyptian fractions are

To write a fraction as an Egyptian fraction, you must rewrite the fraction as: a sum of unit fractions (that means the numerator is 1), and the denominators must all be different.

The Rhind Papyrus (circa 1650 B.C.) shows that ancient Egyptians expressed fractions as sums of distinct unit fractions.
A unit fraction has numerator 1 (e.g., 1/2, 1/3, 1/5).
Egyptians did not write fractions like 3/10 directly; they decomposed them (e.g., 3/10 might be written as 1/5 + 1/10, though the excerpt does not give this exact decomposition).

🧩 Why this is relevant

The excerpt includes problems asking students to rewrite fractions as sums of distinct unit fractions and to find general rules or algorithms.
This historical approach contrasts with modern symbolic manipulation but shares the theme of rewriting fractions in different forms.
Don't confuse: Egyptian fractions are about addition of unit fractions; the algebra section is about simplifying complex fractions by multiplication.

🤔 What is a fraction, really?

🤔 Multiple models

The excerpt notes that "we have no single model that makes sense of fractions in all contexts."
Sometimes a fraction is an action ("Cut this in half"), sometimes a quantity ("We each get 2/3 of a pie"), and sometimes a number on the number line.
This multiplicity of meanings can be confusing but reflects the richness of the concept.

🤔 The mathematician's view

Mathematicians think of a fraction as a pair of numbers (a, b) written as a/b, where b ≠ 0.
A single fraction represents an infinite class of equivalent pairs: a/b is shorthand for all pairs that look like (k·a)/(k·b) for any nonzero k.
Addition and multiplication are then defined by specific rules, and mathematicians must prove that different representations yield equivalent results.
The excerpt acknowledges this is "abstract, dry, and not at all the best first encounter" for students, but it is "the best one can do if one is to be completely honest."

🤔 The real-world question

The excerpt ends by posing: "So… what is a fraction, really? How do…" (text cuts off).
This signals that the question of what fractions mean in the real world remains open and context-dependent, even after formal definitions are established.

What is a Fraction? Part 3

What is a Fraction? Part

🧭 Overview

🧠 One-sentence thesis

Mathematicians define a fraction as an infinite class of equivalent pairs of numbers, which requires proving that operations on fractions yield consistent results regardless of which equivalent representation is chosen.

📌 Key points (3–5)

Multiple models, no single answer: fractions can be actions, quantities, or numbers on a number line, but no one model fits all contexts.
Formal definition: a fraction is not just one pair of numbers but an entire infinite class of equivalent pairs (e.g., 1/2 = 2/4 = 3/6 = …).
Operations must be proven consistent: when mathematicians define addition and multiplication of fractions, they must prove that different equivalent representations yield equivalent answers.
Common confusion: a fraction looks like a single symbol (e.g., 3/5), but it actually represents infinitely many equivalent pairs; this shift from "specific symbol" to "class of symbols" is a major conceptual leap.
Why this matters: the formal approach is abstract and not ideal for first learners, but it is the only completely honest mathematical foundation.

🔄 The multiple faces of fractions

🎭 Context-dependent meanings

Fractions appear in different roles depending on the situation:

Action: "Cut this in half" — a fraction describes an operation to perform.
Quantity: "We each get 2/3 of a pie" — a fraction describes an amount.
Number: A tick on the number line between whole numbers — a fraction is a position.

Key insight: No single model makes sense of fractions in all contexts.

🚫 Why simple definitions fail

One might try to define a fraction as just a pair of numbers a and b (written as a/b), where b is not zero.

Problem: This is incomplete because infinitely many pairs represent the same fraction.

Example: 1/2, 2/4, 3/6, 4/8, … all represent the same fraction.

Don't confuse: A fraction is not one pair of numbers; it is an entire collection of equivalent pairs.

🧮 The formal mathematical definition

📐 Fraction as equivalence class

A fraction is an infinite class of pairs of numbers that are considered equivalent.

Written as a/b where b ≠ 0.
Really shorthand for all pairs that look like (ka)/(kb) for all k.

The conceptual shift:

Before: a "number" is a specific combination of symbols.
Now: a "number" (fraction) is a whole class of combinations deemed equivalent.

The excerpt calls this "a hefty shift of thinking."

➕ Defining addition

Mathematicians define fraction addition by the rule:

a/b + c/d = (ad + bc) / (bd)

This definition is motivated by models like "Pies Per Child."
Critical requirement: Must prove that choosing different equivalent representations for a/b and c/d leads to equivalent final answers.

Example: The excerpt notes it is not immediately obvious that different representations give equivalent answers (though they do).

✖️ Defining multiplication

The product of fractions is defined as:

a/b × c/d = (ac) / (bd)

Again, must prove that equivalent fractions yield equivalent products.
After defining operations, mathematicians verify all arithmetic axioms hold (commutativity, associativity, distributive law, identity elements, etc.).

🎓 Why this approach matters (and its limits)

🏗️ Mathematical honesty

The formal approach is "the best one can do if one is to be completely honest."

It provides a rigorous foundation.
It ensures consistency across all equivalent representations.
It establishes that fractions obey the same arithmetic rules as whole numbers.

🚸 Not for beginners

The excerpt explicitly states this approach is:

Abstract
Dry
"Not at all the best first encounter to offer students on the topic of fractions"

What it avoids: The formal definition completely sidesteps the question of what a fraction really means in the "real world."

🤔 The open question

The excerpt ends by asking: "So… what is a fraction, really? How do you think about them? And what is the best way to talk about them with elementary school students?"

This question acknowledges the tension between mathematical rigor and pedagogical accessibility.

🔗 Connection to algebra (from excerpt context)

🧩 Fractions in advanced contexts

The excerpt includes examples showing that fraction techniques apply to complicated algebraic expressions.

Multiplying numerator and denominator by the same expression (the "key fraction rule") simplifies complex forms.
Many expressions cannot be rewritten as decimals, so working with fractions in fraction form is essential for math and science students.

Why this matters: Understanding fractions as formal objects (not just decimal approximations) is necessary for higher mathematics.

Introduction to Geometry

Introduction

🧭 Overview

🧠 One-sentence thesis

The excerpt does not contain substantive content about geometry beyond the etymology of the word and a quotation.

📌 Key points (3–5)

The excerpt consists primarily of mathematics problems unrelated to geometry (decimal operations, word problems, and a card game).
A new section titled "Geometry" begins at the end with only a quotation and the start of an etymology note.
No geometric concepts, definitions, or explanations are present in the excerpt.

📄 Content summary

📄 What the excerpt contains

The source excerpt includes:

Problems 33–38: exercises on decimal multiplication, division, estimation, and word problems (pizza cost, unit conversion, scale models, rope length, paycheck error, card game strategy).
Geometry section header: a quotation attributed to Henri Poincaré and the incomplete sentence "The word 'geometry' comes from the..."

⚠️ Missing substantive content

The "Introduction" to geometry has not yet begun in a meaningful way.
The excerpt cuts off mid-sentence before explaining the etymology or any geometric ideas.
No definitions, theorems, visual reasoning principles, or geometric concepts are provided.

💬 The Poincaré quotation

💬 What it says

"Geometry is the art of good reasoning from bad drawings." – Henri Poincaré

This suggests that geometric reasoning relies on logical deduction rather than the precision of diagrams.
The quotation implies that even imperfect or schematic drawings can support rigorous conclusions.
Note: The excerpt does not elaborate on this idea or connect it to any geometric content.

Borders on a Square

🧭 Overview

🧠 One-sentence thesis

Counting the colored border squares of a large square in multiple ways develops algebraic thinking by helping students see that different expressions can represent the same quantity and that the equals sign expresses a relationship, not a sequence of operations.

📌 Key points (3–5)

The core problem: calculate how many unit squares lie along the border of a 10×10 square (or an n×n square) without counting one-by-one, using different reasoning strategies.
Multiple valid methods: students can break down the border in different ways (e.g., four sides minus corners, or four corners plus four edges), leading to different but equivalent expressions.
Algebraic thinking in elementary school: generalizing from a specific case (10×10) to any size square (n×n) is a key algebraic skill accessible to young students.
Common confusion about "=": the equals sign is a relationship (two sides have the same value), not an operation or a left-to-right instruction meaning "and the answer is…"
Why it matters: understanding equality and multiple representations prepares students for formal algebra and helps them see structure in mathematics.

🟥 The border-counting problem

🟥 The 10×10 square setup

Problem: A large square is made of 100 smaller unit squares (10×10). The unit squares along the border are colored red. How many red squares are there?

The challenge is to find the count without counting one-by-one.
Students must describe their reasoning clearly and justify why their answer is correct.
This problem has many correct solution strategies, each leading to a different arithmetic expression.

🧮 Justin's solution: four sides minus double-counted corners

Justin calculated 10 × 4 − 4.

His reasoning:

There are 10 squares along each of the four sides, so 10 × 4 = 40.
But each corner square belongs to two sides, so each corner has been counted twice.
Subtract the 4 corners once to correct the double-counting: 40 − 4 = 36.

Why it works:

The method explicitly accounts for overlap at the corners.
Color-coding in Justin's picture shows which squares are counted in which step.

Don't confuse: "10 × 4" does not mean "the total number of border squares"; it is an intermediate step that over-counts, and the "− 4" is a correction.

🎨 Multiple strategies for the same count

🎨 Five different calculations

The excerpt presents five students' methods for counting the border squares of the 10×10 square. Each student's calculation is different, but all are correct.

Student	Calculation	What it represents
Valerie	(not shown in detail)	A different decomposition of the border
Kayla	(not shown in detail)	Another way to group the border squares
Linda	(not shown in detail)	Yet another grouping strategy
Mark	(not shown in detail)	A different breakdown
Allan	(not shown in detail)	Another valid method

Task for students (Problem 3):

For each calculation, write a justification and draw a picture.
Use color-coding to show which parts of the border correspond to which parts of the expression.

Key insight:

Different ways of "seeing" the border lead to different arithmetic expressions.
All expressions are equal because they count the same set of squares.
This is the heart of algebraic thinking: recognizing that different forms can represent the same quantity.

🔢 Generalizing to an n×n square

Problem 4 and Problem 5 ask students to:

Adapt two techniques to count the border of a 15×15 square (Problem 4).
Adapt two techniques to count the border of an n×n square (Problem 5).

Why generalization matters:

Moving from a specific number (10) to a variable (n) is a fundamental algebraic skill.
Students learn to see the structure of the problem, not just the answer for one case.
Example: Justin's method generalizes to n × 4 − 4 or equivalently 4(n − 1) for an n×n square.

Don't confuse: The variable n is not "any number you feel like"; it represents the side length of the square, and the expression must work for all valid side lengths.

🔄 Working backwards: from border count to square size

🔄 Can you build a square from a given number of border squares?

Problem 6 reverses the question:

If you have 64 red squares, can you use all of them to make the border of a larger square? If yes, what are the dimensions?
Same question for 30 red squares.
Same question for 256 red squares.

What this requires:

Understanding the relationship between the number of border squares and the side length.
Testing whether a given number "fits" the pattern.
Example: If the border count is 4(n − 1), then 64 = 4(n − 1) implies n − 1 = 16, so n = 17. A 17×17 square has 64 border squares.

🧪 General rules

The excerpt asks students to describe:

Rule 1: If you have a large n×n square with the border colored red, how many red squares will there be? (Justify with words and a picture.)
Rule 2: If you have a certain number of red squares, is there a quick test to decide if you can use all of them to make the border of a large square? Can you tell how big the square will be?

Why these rules matter:

They formalize the relationship between side length and border count.
They require students to think about the structure of the problem in general terms, not just compute answers for specific cases.

🟰 Understanding the equals sign

🟰 What "=" really means

The symbol "=" expresses a relationship. It is not an operation in the way that + and × are operations. It should not be read left-to-right, and it definitely does not mean "… and the answer is …"

Key points:

The equals sign says that the expressions on both sides have the same value.
It does not mean "do this calculation next" or "the result is."
Misusing "=" makes mathematical work hard to understand and can lead to incorrect reasoning.

Example of misuse (Kim's solution to Problem 7):

Problem: Akira had some money, his grandmother gave him $1.50, he bought a book for $3.20, and he had $2.30 left. How much did he start with?
Kim wrote: 1.50 + 3.20 = 4.70 + 2.30 = 7 − 1.50 = 5.50 − 3.20 = 2.30, so the answer is 4.
What's wrong: The chain of equalities is false. It is not true that 1.50 + 3.20 = 4.70 + 2.30 (the left side is 4.70, the right side is 7).
Kim is using "=" to mean "and then I do this next step," which is incorrect.
Although she got the correct numerical answer (4), her written work does not make mathematical sense.

Don't confuse: "=" is not a one-way arrow or a command to compute; it is a two-way relationship asserting that both sides are equal.

🧩 Always true, sometimes true, or never true?

Problem 8 asks students to examine equations and decide:

Is the statement always true, sometimes true, or never true?

What this teaches:

An equation can be an identity (always true, like 2 + 3 = 5 or x + 0 = x).
An equation can have solutions (sometimes true, like x + 1 = 5, which is true only when x = 4).
An equation can be a contradiction (never true, like 0 = 1).

Problem 9 extends this:

Fill in the blank to make an equation always true, always false, or sometimes true.

Why it matters:

Students learn that "=" is not just about computing an answer; it is about expressing a relationship that may or may not hold.
This prepares them to understand solving equations (finding values that make the relationship true).

🔍 What does "solve" mean?

Problem 10 asks:

If someone asked you to solve the equations in Problem 8, what would you do in each case and why?

Key insight:

"Solving" an equation that is always true (an identity) means recognizing that it holds for all values.
"Solving" an equation that is sometimes true means finding the specific values that make it true.
"Solving" an equation that is never true means recognizing that no solution exists.

Don't confuse: "Solve" does not always mean "find a number"; it means "determine for which values (if any) the equation is true."

🌱 Why this matters for elementary teachers

🌱 Algebra starts in kindergarten

The Common Core Standards include "Operations and Algebraic Thinking" beginning in kindergarten.
Students arrive at school already capable of abstraction and generalization.
Elementary teachers provide experiences that make algebraic ideas feel natural before students reach a class called "Algebra."

🌱 Algebra as a tool for thinking

Language of science and technology: Algebraic thinking helps students make sense of the world and interact with technology.
Problem-solving tool: "Algebratizing" a problem often leads to a solution.
Abstract thinking: Algebra helps students understand why operations work, describe their properties, and see the bigger picture beyond specific calculations.

Example from the excerpt:

The border-counting problem is not "algebra" in the sense of solving for x, but it is algebraic thinking: recognizing patterns, generalizing from specific cases, and understanding that different expressions can represent the same quantity.

Don't confuse: Algebraic thinking is not the same as manipulating symbols in high school algebra; it is the foundation of reasoning about operations, relationships, and structure.

Careful Use of Language in Mathematics: =

🧭 Overview

🧠 One-sentence thesis

The equals symbol "=" represents a relationship between two quantities that have the same value, not an operation or a left-to-right instruction to calculate an answer, and using it correctly is essential for clear mathematical communication and understanding.

📌 Key points (3–5)

What "=" means: It expresses a relationship (equality), not an operation like + or ×, and should not be read as "and the answer is."
Common misuse: Students often string together calculations with "=" incorrectly, writing statements that are mathematically false even when they arrive at correct numerical answers.
Balance scale analogy: A two-pan balance scale models equality—when balanced, the weights on both sides are equal, just as both sides of an equation represent the same value.
Always/sometimes/never true: Equations can be always true (identities), sometimes true (conditional), or never true (contradictions), depending on what values make both sides equal.
Why precision matters: Correct use of "=" is essential for your own clarity and for teaching future students to understand algebraic thinking.

🔍 What "=" really means

🔍 Definition and core concept

The symbol "=" expresses a relationship between two quantities or expressions that have the same value.

It is not an operation (unlike + or ×).
It should not be read left-to-right as a sequence of steps.
It does not mean "… and the answer is …".
Both sides of the equation represent the same quantity or value.

❌ Common misuse example

The excerpt presents Kim's solution to a word problem:

Kim's work: "1.50 + 3.20 = 4.70 + 2.30 = 7"

Kim got the correct numerical answer (7).
However, her calculation doesn't make mathematical sense.
It is true that 1.50 + 3.20 = 4.70.
It is not true that 4.70 + 2.30 = 7 equals 1.50 + 3.20.
She incorrectly chained together calculations with "=", making her solution hard to understand.

Don't confuse: Getting the right answer with writing mathematically correct statements—you can arrive at a correct number while writing false equations.

⚖️ The balance scale model

⚖️ How the model works

The excerpt uses a two-pan balance scale to illustrate equality:

Place one object in each pan.
If one side is lower, that side holds heavier objects.
If the two sides are balanced, the objects on each side weigh the same.

This physical model helps visualize what "=" means: both sides have equal weight (equal value).

🔺 Working with balanced scales

The problems present scenarios where:

Different shapes (triangles, circles, squares, stars) have consistent weights within each shape type.
The scale is balanced, meaning left side = right side.
You can determine relationships between shapes by comparing what balances.

Example reasoning: If three triangles balance with two circles, and the scale is balanced, then the total weight of three triangles equals the total weight of two circles.

🔗 Connection to equations

Balance scale problems model algebraic equations:

Each side of the scale represents one side of an equation.
"Balanced" means the two sides are equal.
You can manipulate both sides equally (add/remove the same weight from both sides) and maintain balance.
This mirrors solving equations by performing the same operation on both sides.

📐 Types of equations

📐 Always, sometimes, or never true

Problem 8 asks students to classify equations as:

Always true: The equation holds for all possible values (an identity).
Sometimes true: The equation holds only for certain values (a conditional equation).
Never true: The equation cannot be satisfied by any value (a contradiction).

🔢 Filling in the blank (Problem 9)

The excerpt asks students to make an equation with a blank:

Fill in something that makes it always true (any expression equivalent to the other side).
Fill in something that makes it always false (an expression that can never equal the other side).
Fill in something that makes it sometimes true and sometimes false (a variable or expression that equals the other side only for specific values).

This exercise reinforces that "=" is about relationship and equivalence, not just calculation.

🧮 What "solve" means

Problem 10 asks: "If someone asked you to solve the equations in Problem 8, what would you do?"

For an always-true equation, there's nothing to solve—it's an identity.
For a sometimes-true equation, solving means finding which values make it true.
For a never-true equation, the solution is "no solution" or the empty set.

Don't confuse: "Solving" an equation with "evaluating" whether a statement is true—not all equations need solving.

🎯 Why precision matters

🎯 For your own work

Using "=" correctly makes your mathematical work clear and easily understood by others.
Incorrect use creates confusion and can make correct reasoning appear wrong.
Mathematical communication requires precision in symbols as well as words.

👩‍🏫 For teaching future students

The excerpt emphasizes:

"For your future students to understand the meaning of the = symbol and use it correctly, it is essential that you are clear and precise in your use of it."
Students often misunderstand "=" as a command to calculate rather than as a statement of equality.
Teachers must model correct usage to help students develop proper algebraic thinking.

Growing Patterns

🧭 Overview

🧠 One-sentence thesis

Growing patterns can be understood in multiple valid ways by decomposing figures into different structural components, and each way of seeing the pattern leads to a different but equivalent method for calculating the number of tiles or toothpicks in any figure.

📌 Key points (3–5)

Multiple valid perspectives: Different students (Ali, Michael, Kelli, Hy) see the same tile pattern growing in different ways, all correct.
From visual to algebraic: Each way of seeing the pattern corresponds to a method for calculating the number of elements in the nth figure.
Key questions to ask: How many tiles/toothpicks for the 5th, 10th, 100th, or nth figure? Can you build a figure with exactly 25 or 100 tiles?
Common confusion: There is not one "right" way to see a pattern—different decompositions (e.g., counting rows vs. counting added pieces) yield different expressions that are mathematically equivalent.
Matching representations: Patterns can be described visually, algebraically (expressions), numerically (tables), and verbally (words); the goal is to match equivalent representations.

🔍 Multiple ways to see a pattern

👁️ Different student perspectives

The excerpt presents a pattern made from square tiles and shows how different students visualized its growth:

Ali's picture, Michael's picture, Kelli's picture, and Hy's picture each decompose the same pattern differently.
Each student's drawing reflects a unique structural understanding (e.g., seeing rows, columns, added layers, or other groupings).
The excerpt emphasizes: "Describe in words how each student saw the pattern growing."

🧩 Why multiple perspectives matter

Each way of seeing leads to a different calculation method for the number of tiles.
Example: One student might count "two rows of n tiles plus a corner," while another counts "add 3 tiles in a C-shape each time."
Don't confuse: Different methods are not contradictory—they are equivalent algebraic expressions derived from different visual decompositions.

🔢 Calculating tile counts

🔢 Specific figures (5th, 10th, 100th)

The excerpt asks students to calculate:

How many tiles for the 5th figure?
How many tiles for the 10th figure?
How many tiles for the 100th figure?
Each answer must be justified based on how the pattern grows (i.e., tied to the visual decomposition).

🧮 General formula for the nth figure

"How can you compute the number of tiles in any figure in the pattern?"

The goal is to express the count as a function of the figure number n.
Example (from Hy's method): "How would Hy calculate the number of tiles needed to build any figure in the pattern?"
The formula depends on the decomposition chosen (e.g., "two times n minus one" vs. "three times n plus one").

❓ Reverse questions: Can you build with exactly X tiles?

The excerpt poses:

"Could you make one of the figures in the pattern using exactly 25 tiles? If yes, which figure? If no, why not?"
"Could you make one of the figures in the pattern using exactly 100 tiles?"
These require solving for n given a total count, and justifying whether the solution is a whole number (i.e., whether that figure exists in the pattern).

🧩 Worked example: Hy's perspective

🖼️ Hy's decomposition

The excerpt highlights Hy's unique way of seeing the pattern:

"Hy saw the pattern in a different way from everyone else in class."
His picture shows a specific structural breakdown (details not fully visible in the excerpt, but implied to be distinct).

🧮 Hy's calculations

The excerpt asks:

How would Hy calculate the 10th figure?
How would Hy calculate the 100th figure?
How would Hy calculate any figure?

Each answer must be based on Hy's visual decomposition, showing that the method of counting follows directly from how the pattern is seen.

🎯 Matching representations

🔗 Four types of representations

The "Matching Game" section presents patterns in multiple forms:

Representation	What it includes
Algebraic expressions	Formulas labeled (a) through (o) (specific expressions not shown in excerpt)
Visual patterns	Patterns 1–7 made from tiles or toothpicks
Tables of numbers	Tables A–G showing input (figure number) and output (tile/toothpick count)
Descriptions in words	Verbal explanations (1–9) of how to count elements in a pattern

🔍 How to match

The task is to find equivalent representations: which algebraic expression matches which visual pattern, table, and verbal description?
The excerpt notes: "There may be more than one algebraic expression to match a given pattern, or more than one pattern to match a given description."
Example: A verbal description like "Each arm of the L shape has the same number of tiles as the figure number, but we've counted the corner twice, so subtract one" corresponds to a specific algebraic form and a specific visual pattern.

🧠 Key insight from verbal descriptions

The excerpt provides detailed verbal descriptions (1–9) that illustrate different decomposition strategies:

Description 1: "Count horizontal and vertical toothpicks separately. Horizontal: there are two rows of n toothpicks... The vertical toothpicks are just the same."
Description 2: "To get a figure from the previous one, you add three toothpicks in a C shape... The total is three times the figure number, plus one extra."
Description 4: "Each arm of the L shape has the same number of tiles as the figure number. But then we've counted the corner twice, so subtract one."
Description 7: "The stars are in a square, and the sides of the square have the same number of stars as the figure number."
Description 9: "There are the same number of squares as the figure number, and each square uses four toothpicks. But then I've double-counted the toothpicks..." (excerpt cuts off).

Don't confuse: "Three times n plus one" and "two times n minus one" may describe different patterns, or they may describe the same pattern seen in different ways—context and visual structure determine equivalence.

🧪 Systematic problem-solving approach

📋 Standard questions for each pattern

For Problems 16–20 (visual patterns provided but not detailed in the excerpt), students are instructed to:

Describe in words and pictures how the pattern grows.
Calculate tiles for the 10th figure and justify based on growth.
Calculate tiles for the 100th figure.
Describe how to find the number of tiles in any figure, justified by the growth pattern.
Determine if a figure can be made with exactly 25 tiles (which figure, or why not).
Determine if a figure can be made with exactly 100 tiles (which figure, or why not).

🔑 Justification requirement

Every answer must be justified based on how the pattern grows.
This ties the algebraic formula back to the visual or structural decomposition, reinforcing that formulas are not arbitrary but derived from the pattern's structure.

🔗 Connection to the equals sign

🤔 Reflective question

The excerpt includes a "Think / Pair / Share" prompt:

"What do Problems 11–14 above have to do with the '=' symbol?"

Problems 11–14 involve balanced scales with shapes of different weights (orange triangles, green circles, purple squares).
The connection: just as a balanced scale shows two sides are equal in weight, different algebraic expressions for the same pattern are equal in value—they represent the same quantity seen in different ways.
Example: If one student's method gives "2n - 1" and another gives a different expression, both equal the same tile count for figure n, just as both sides of a balanced scale weigh the same.

Don't confuse: The equals sign does not mean "the answer comes next"; it means "these two quantities are the same," whether in a scale problem or in equivalent pattern formulas.

Matching Game

🧭 Overview

🧠 One-sentence thesis

This activity teaches students to recognize the same mathematical pattern across multiple representations—visual diagrams, algebraic expressions, numerical tables, and verbal descriptions—by matching equivalent forms and justifying their reasoning.

📌 Key points (3–5)

What the task requires: match patterns shown visually, algebraically, numerically (in tables), and verbally so that equivalent representations are grouped together.
Multiple valid matches: more than one algebraic expression may describe the same pattern, and more than one pattern may fit a given description.
Justification is essential: students must explain why their matches make sense based on how the pattern grows.
Common confusion: different-looking expressions can represent the same pattern (e.g., counting the same structure in different ways).
Connection to structural algebra: the activity emphasizes understanding what variables represent and how symbols express meaning in a situation, not just solving for unknowns.

🎯 The matching task structure

🧩 Four types of representations

The excerpt provides patterns in four formats:

Algebraic expressions: labeled (a) through (o), using variables to describe quantities
Visual patterns: labeled Pattern 1 through Pattern 7, showing tile or toothpick arrangements
Tables of numbers: labeled Table A through Table G, showing input-output pairs
Descriptions in words: numbered 1 through 11, explaining how patterns grow in everyday language

🔗 The matching goal

Students must identify which representations describe the same underlying pattern.

Key challenge: The excerpt explicitly notes there may be more than one correct match—multiple algebraic expressions might fit one visual pattern, or several patterns might match one verbal description.

Why justification matters: Because multiple answers are possible, students must explain their reasoning based on how they see the pattern growing.

🔢 Sample pattern elements

📊 Table examples

The excerpt shows several input-output tables:

Table	Outputs for inputs 1, 2, 3, 4	Pattern type
Table A	1, 4, 9, 16	Square numbers
Table B	10, 15, 20, 25	Adding 5 each time
Table C	1, 3, 5, 7	Odd numbers
Table D	3, 5, 7, 9	Odd numbers starting at 3
Table E	4, 7, 10, 13	Adding 3 each time
Table F	4, 10, 16, 22	Adding 6 each time
Table G	2, 4, 6, 8	Even numbers

🗣️ Verbal description examples

The word descriptions explain growth mechanisms:

Description 4: "Each arm of the 'L' shape has the same number of tiles as the figure number. But then we've counted the corner of the 'L' twice, so we have to subtract one."
Description 5: "The stars are in two equal rows, and each row has the same number of stars as the figure number."
Description 9: Describes squares made of toothpicks where adjacent squares share edges, requiring subtraction to avoid double-counting.

Each description reveals a different way of visualizing and counting elements in a growing pattern.

🧮 Connection to algebraic thinking

🏗️ Structural vs procedural algebra

The excerpt distinguishes two approaches:

Procedural algebra: using symbols to solve for unknown quantities (e.g., "solving for x")

Structural algebra: using symbols to express meaning in a situation, where variables represent varying quantities rather than single unknowns

🎯 Why this activity emphasizes structural thinking

Variables represent figure numbers or case numbers in growing patterns, not single unknown values
The same variable can take on different values
Expressions give information about how many elements are needed for any particular figure
Students must answer: "What does the variable mean? What does it represent?"

Example from the excerpt: If an expression represents tiles in a pattern, students evaluate it at specific values to understand what those numbers tell them about the pattern's growth, then identify where each part of the expression appears in the visual pattern.

🔄 Multiple representations deepen understanding

By matching the same pattern across formats, students learn that:

Different algebraic expressions can describe the same growth (counting the same structure in different ways)
Visual patterns can be "seen" and counted in multiple valid ways
Tables capture the numerical output of the pattern's rule
Verbal descriptions make the growth mechanism explicit in everyday language

Don't confuse: Having multiple correct matches doesn't mean "anything goes"—each match must be justified by showing how the representations describe the same underlying growth pattern.

Structural and Procedural Algebra

🧭 Overview

🧠 One-sentence thesis

Algebra encompasses both procedural skills (solving for unknowns) and structural thinking (using symbols to express meaning and relationships), with the structural view being essential for understanding patterns and real-world situations.

📌 Key points (3–5)

Two views of algebra: procedural algebra focuses on solving for unknown quantities, while structural algebra uses symbols to express meaning and relationships in situations.
Variables have different meanings: a symbol can represent a single unknown quantity or a varying quantity that takes on different values.
Equations serve multiple purposes: they can represent problems to solve, relationships between quantities, or mathematical identities (always-true statements).
Common confusion: distinguishing when a variable is "unknown and fixed" versus "varying across cases"—in growing patterns, the variable represents different figure numbers, not a single mystery value.
Elementary applications: even young students engage with algebraic thinking when they work with unknown quantities in simple addition/subtraction problems.

🔄 Two approaches to algebra

🔧 Procedural algebra

Procedural algebra: the traditional view focused on solving equations to find unknown quantities.

Most people associate algebra with "solving for x."
The goal is always to determine what the unknown value is.
Elementary students encounter this when solving problems like "blank + 5 = 10" or finding pairs of numbers that add to 10.
Example: Given an equation with one unknown, work through steps to isolate and find that single value.

🏗️ Structural algebra

Structural algebra: using symbols to express meaning in a situation, where variables represent varying quantities rather than single unknowns.

The key question is: "What does the x mean? What does it represent?"
Variables can take on different values, and expressions give information about relationships.
Example: In growing patterns, a letter represents the "figure number" or "case number," and the expression tells you how many tiles or toothpicks are needed for that particular figure.
Don't confuse: structural algebra is not about finding one answer but about describing how quantities relate as they vary.

🎯 Why structural thinking matters

It's described as more important than just procedural skills.
It helps students understand patterns, relationships, and real-world situations.
Most of the chapter's work involves structural algebra—building expressions that describe how patterns grow.

🔢 Understanding variables

🎭 Multiple meanings of symbols

Variables can represent different things depending on context:

Meaning	Description	Example use
Unknown quantity	A single fixed value we need to find	Solving "x + 5 = 10"
Varying quantity	A value that changes across different cases	Figure number in a pattern

The same letter can mean different things in different problems.
Understanding what a variable represents is crucial for structural thinking.

📐 Variables in patterns

When an expression like "3n + 2" represents tiles in a pattern, n is the figure number (varying).
Evaluating at different values (n = 1, n = 2, etc.) tells you about different stages of the pattern.
You should be able to see where each number in the expression appears in the physical pattern.
Example: If the expression is "3n + 2," you should identify which part of the tile pattern corresponds to "3n" and which corresponds to "2."

📊 Understanding equations

🎯 Three types of equations

Equations can represent different things:

Problems to solve: Traditional procedural questions where you find an unknown value.
Relationships between quantities: Expressions like "A = s²" showing how area relates to side length.
Identities: Mathematical truths that are always true for every value, like "(x + 1)² = x² + 2x + 1."

⚖️ Balance scale models

Balance puzzles provide a concrete way to understand equations.
Bags with unknown numbers of blocks represent variables.
The balanced scale represents equality (both sides equal).
Example: A scale with one bag and 3 blocks on the left, and 7 blocks on the right, represents "x + 3 = 7."
Solving means determining how many blocks are in each bag.

🔍 Special equation cases

Some equations lead to results like "0 = 0" when solved—these are identities, true for all values.
Some equations lead to contradictions—these have no solution.
Don't confuse: an identity is not "unsolvable"; it's a statement that's always true, so there's nothing to solve for.

🧮 Connecting representations

🌍 Real-world contexts

Any algebraic expression should connect to a real-world situation or concrete scenario.
Example: "2x + 5" could represent the total cost when items cost 2 dollars each and there's a 5-dollar fee.
Students should practice translating between situations, expressions, and equations.

🎨 Multiple representations for the same problem

The same situation can be described with different variable choices.
Example: In the rock piles problem, you can let x represent pile A, pile B, or pile C—each choice leads to different but equivalent equations.
The choice of what the variable represents affects how you write the equations, but the solution to the real problem remains the same.

🔗 Blending structural and procedural thinking

Many problems involve both types of thinking.
You might use structural thinking to set up an equation that represents a situation, then use procedural thinking to solve it.
Example: Understanding what variables represent in a balance puzzle (structural) and then solving for the unknown (procedural).

Review of Dots & Boxes Model

🧭 Overview

🧠 One-sentence thesis

The Dots & Boxes model provides a visual method for understanding place value in any base system by representing numbers as dots that "explode" into the next position according to a base-specific rule, and extends naturally to decimals by adding boxes to the right of the ones place.

📌 Key points (3–5)

Core mechanism: In a 1←b system, when b dots accumulate in one box, they explode and become one dot in the box to the left.
Base notation: Each position represents a power of the base (rightmost is ones, next is b, then b², etc.), and subscripts indicate which base a number is written in.
Common confusion: The written code (like 1001₂) is not the quantity itself but the representation of that quantity in a particular base system.
Decimal extension: Boxes to the right of the decimal point represent reciprocals of powers of ten (1/10, 1/100, 1/1000, etc.).
Conversion principle: Numbers can be converted between bases by repeatedly applying the explosion rule or by understanding place values as powers.

🎯 The explosion mechanism

💥 How the 1←2 rule works

The 1←2 Rule: Whenever there are two dots in a single box, they "explode," disappear, and become one dot in the box to the left.

This is the fundamental operation for base 2 (binary).
You keep exploding until no box has two or more dots.
Example: Nine dots in base 2 becomes 1001₂ after all explosions complete.
The process: place all dots in the rightmost box, then repeatedly explode pairs moving leftward.

🔢 How the 1←3 rule works

The 1←3 Rule: Whenever there are three dots in a single box, they "explode," disappear, and become one dot in the box to the left.

This is the fundamental operation for base 3.
Example: Fifteen dots becomes 120₃ after all explosions.
The same principle applies: keep exploding until no box has three or more dots.

🔄 General 1←b rule

In a base b system, b dots in one box explode to become one dot in the next box left.
Different bases need different numbers of digits: base 7 needs digits 0-6, base 12 needs digits 0-11.
The explosion threshold determines the base.

📐 Place value and powers

📊 What each position represents

Position	Value	Base 10 example	General base b
Rightmost	Ones (b⁰)	1	1
Second	b place (b¹)	10	b
Third	b² place	100	b²
Fourth	b³ place	1000	b³

Each place represents a power of the base.
Moving left, each position is worth b times the previous position.
The rightmost place is always the ones place (any number to the zero power equals 1).

🏷️ Reading base notation

When you see a subscript, you are seeing the code for some number of dots.

102₃ means "one-zero-two base three" and represents eleven dots total.
222₄ means "two-two-two base four" and represents forty-two dots total.
Without a subscript, assume base ten.
Don't confuse: 102₃ ≠ 102₁₀; they represent different quantities.

✖️ Multiplying by the base

Multiplying a whole number in base b by b simply appends a zero to the right end.
In base 10: any number times ten adds a zero (e.g., 47 × 10 = 470).
This works because you're shifting every dot one box to the left, leaving the rightmost box empty.

🔟 Extending to decimals

➡️ Boxes to the right of the decimal

Boxes can extend infinitely to the right as well as left.
A decimal point separates boxes to the right of the ones place.
If the first box right of the decimal has value x, then ten x's equal 1, so x = 1/10.
The next box has value y where ten y's equal 1/10, so y = 1/100.
Pattern: boxes represent 1/10, 1/100, 1/1000, etc. (reciprocals of powers of ten).

🎨 Visualizing decimal values

0.3 represents three dots in the 1/10 box, which equals 3/10.
0.007 represents seven dots in the 1/1000 box, which equals 7/1000.
Each digit's position determines what fraction it represents.

Example: 0.3 is three groups of one-tenth.

🔄 Two approaches to reading decimals

Approach 1 (add fractions): 0.31 means 3/10 + 1/100, which equals 30/100 + 1/100 = 31/100.

Approach 2 (unexplode): Unexplode the three dots in the 1/10 position to get 30 dots in the 1/100 position, plus the original 1 dot = 31/100.

Both approaches give the same answer.
Unexploding is the reverse of exploding: one dot in a box becomes b dots in the box to the right.

📖 Reading decimals aloud

"Point six" versus "six tenths" for 0.6.
The excerpt suggests "six tenths" is more helpful for understanding because it explicitly states the fraction value.
This connects the decimal notation directly to its fractional meaning.

🔀 Converting between representations

➡️ Fractions to decimals

If a fraction's denominator is a power of ten, conversion is straightforward.
Example: 3/4 = 75/100 (using the key fraction rule to rewrite with denominator 100) = 0.75.
The numerator tells you how many dots to place in the corresponding decimal position.

⬅️ Decimals to fractions

Each decimal position corresponds to a specific power-of-ten denominator.
Read the decimal value and write it over the appropriate power of ten.
Simplify if possible.

Example: 0.31 = 31/100 (cannot simplify further).

🎯 Multiple valid representations

The same fraction can be drawn with dots in different positions.
Example: 3/10 can be shown as three dots in the 1/10 box, or as 30 dots in the 1/100 box (after unexploding).
Both are correct because they represent the same quantity.
Don't confuse: different dot arrangements can represent the same value if related by explosions/unexplosions.

Decimals

🧭 Overview

🧠 One-sentence thesis

Decimals extend the "Dots & Boxes" place-value model to the right of the ones place, representing fractions with denominators that are powers of the base, and can be understood through explosions and unexplosions just like whole numbers.

📌 Key points (3–5)

What boxes to the right mean: In a 1←10 system, boxes to the right of the decimal point represent reciprocals of powers of ten (1/10, 1/100, 1/1000, etc.).
How decimals represent fractions: Each decimal can be converted to a fraction by recognizing the place values or by using unexplosions to combine values into a single denominator.
Common confusion: 0.9 and 1.0 appear different but represent the same number—explosions and unexplosions show all their "Dots & Boxes" pictures are equivalent.
Decimals work in any base: The same logic applies to other bases (base-5, base-4, etc.), where boxes to the right represent reciprocals of powers of that base, though "decimal point" becomes "radix point."
Connection to division: A fraction like 3/4 is the answer to a division problem and can be rewritten as a decimal by converting the denominator to a power of ten.

📦 Extending the Dots & Boxes model

📦 Why extend to the right

Previously, the "Dots & Boxes" model had boxes extending infinitely to the left (for larger place values).
The excerpt asks: why not have boxes extending to the right as well?
Working with a 1←10 rule, boxes to the right represent smaller fractional values.

🔢 Notation: the decimal point

The decimal point separates boxes to the right of the ones place. (In base ten, "dec" means "ten.")

The first box to the right has value x where ten x's equal 1, so x = 1/10.
The next box to the right has value y where ten y's equal x = 1/10, so y = 1/100.
Continuing this pattern, boxes to the right represent 1/10, 1/100, 1/1000, and so on—reciprocals of powers of ten.

🖼️ Picture examples

0.3: Three dots in the 1/10 box → three groups of 1/10 = 3/10.
0.007: Seven dots in the 1/1000 box → seven groups of 1/1000 = 7/1000.
Example: If a fraction simplifies (e.g., 5/10 = 1/2), the decimal can also simplify.

🔄 Converting between decimals and fractions

🔄 Fractions to decimals using powers of ten

If a fraction's denominator can be rewritten as a power of ten, it converts easily to a decimal.
Example from the excerpt: 3/4 can be rewritten using the key fraction rule to get a denominator of 100, so 3/4 = 75/100 = 0.75.
The excerpt shows 12 3/4: rewrite 3/4 as 75/100, so 12 3/4 = 12.75.

🔄 Decimals to fractions: two approaches

The excerpt uses 0.31 as an example:

Approach 1 (add fractions):

0.31 has three dots in the 1/10 box and one dot in the 1/100 box.
3/10 + 1/100 = 30/100 + 1/100 = 31/100.

Approach 2 (unexplode):

Unexplode the three dots in the 1/10 position to produce 30 dots in the 1/100 position.
Now all dots are in the 1/100 box: 31 dots → 31/100.
This approach shows the fraction directly without needing a common denominator.

🧩 Multiple representations

The excerpt shows that 0.9 and 1.0 can have multiple "Dots & Boxes" pictures.
Example: 0.9 can be shown as nine dots in the 1/10 box, or (after unexplosions) as 90 dots in the 1/100 box, etc.
Similarly, 1.0 can be shown as one dot in the ones box, or (after explosions) as ten dots in the 1/10 box, etc.
Key insight: One unexplosion can show the first picture of 0.9 is equivalent to the second picture of 1.0; several unexplosions show all four pictures are equivalent.
Don't confuse: Different-looking decimals (like 0.9 and 1.0) can represent the same number when explosions and unexplosions are applied.

🌐 Decimals in other bases (x-mals)

🌐 The radix point

The prefix "dec" in "decimal point" means ten, so in other bases the point is called a radix point.
Example: In base-5, the point separates boxes representing powers of 5 from boxes representing reciprocals of powers of 5.

🌐 Values in other bases

In a base-b system, boxes to the left of the ones place represent positive powers of b.
Boxes to the right represent reciprocals: 1/b, 1/b², 1/b³, etc.
Example from the excerpt: In base-5, the first box to the right is 1/5, the second is 1/25, and so on.

🌐 Converting x-mals to base-10 fractions

The excerpt includes a dialogue between Tami and Courtney working on a base-5 number:

Courtney's mistake: She thought a two-place base-5 number should convert to a fraction with denominator 10 (like base-10 two-place decimals have denominator 100).
Tami's reasoning: The base-5 number is less than one (no dots in the ones place), but Courtney's fraction is almost two, so they can't be the same.
Lesson: In base-5, two places to the right means the denominator is 5² = 25, not 10.
Example: To convert a base-5 "decimal" to a base-10 fraction, use the place values 1/5, 1/25, etc., then add fractions or unexplode.

Base	First box right of point	Second box right of point
10	1/10	1/100
5	1/5	1/25
4	1/4	1/16
6	1/6	1/36

➗ Division and decimals

➗ Fractions as division problems

A fraction is the answer to a division problem.

Example from the excerpt: 6 pies shared equally among 3 children yields 2 pies per child, written as 6 ÷ 3 = 2.
The fraction 6/3 is equivalent to the division problem 6 ÷ 3; it represents the number of pies one whole child receives.
Similarly: 10/2 = 5 pies per kid, 8/2 = 4 pies per kid, 5/5 = 1 pie per kid.
This connection means any fraction can be thought of as a division problem, which can then be converted to a decimal if the denominator is (or can be rewritten as) a power of the base.

X-Mals

🧭 Overview

🧠 One-sentence thesis

The "Dots & Boxes" model extends naturally to represent fractional place values (radix points) in any number base, enabling division and decimal-like expansions through systematic unexplosion and grouping.

📌 Key points (3–5)

Radix points generalize decimals: Just as base-10 uses a decimal point for tenths/hundredths, any base can use a radix point for fractional place values (e.g., "pentimal" for base-5).
Division creates fractional representations: Performing division in Dots & Boxes by repeatedly unexploding dots and forming groups produces decimal-like expansions.
Terminating vs repeating patterns: Fractions with denominators containing only factors of the base (2s and 5s in base-10) terminate; others repeat infinitely.
Common confusion—period vs remainder: The period (length of repeating block) is determined by when remainders repeat during division, not by the denominator itself.
Cross-base insight: The same fraction behaves differently in different bases—what terminates in one base may repeat in another.

🔢 Radix points in different bases

🔢 What a radix point represents

Radix point: The general term for the point separating whole-number places from fractional places in any base system.

In base-10, we call it a "decimal point" (from Latin "decem" = ten).
In base-5, it could be called a "pentimal point."
Boxes to the left of the radix point represent positive powers of the base.
Boxes to the right represent reciprocals (negative powers) of the base.

🧮 Place values in other bases

In a base-b system:

The first box right of the radix point represents 1/b.
The second box represents 1/b².
The third box represents 1/b³, and so on.

Example: In base-5, the boxes right of the radix point represent 1/5, 1/25, 1/125, etc.

Don't confuse: The place values change with the base—1/5 in base-5 is not the same position as 1/10 in base-10.

🔄 Division and decimal expansions

🔄 Division as grouping in Dots & Boxes

The excerpt shows that division can be performed by:

Starting with the dividend in Dots & Boxes form.
Forming groups equal to the divisor.
When no more groups fit, unexplode a dot into the next box to the right.
Continue forming groups and unexploding until done (or a pattern emerges).

Example: To compute 1 ÷ 8 in base-10:

Start with 1 dot in the ones box.
Unexplode to get 10 dots in the 1/10 box → one group of 8, remainder 2.
Unexplode the 2 dots to get 20 in the 1/100 box → two groups of 8, remainder 4.
Unexplode the 4 dots to get 40 in the 1/1000 box → five groups of 8, remainder 0.
Result: 1/8 = 0.125.

🔁 When division repeats forever

If during division the same remainder appears in the same position again, the process will repeat infinitely.

Example: Computing 1 ÷ 3:

Unexplode 1 dot → 10 dots in 1/10 box → three groups of 3, remainder 1.
Unexplode that 1 dot → 10 dots in 1/100 box → three groups of 3, remainder 1.
The remainder of 1 keeps recurring, so the pattern repeats: 1/3 = 0.333...

Notation: A vinculum (horizontal bar) indicates repeating digits: 0.3̄ means 0.333...

🎯 Terminating vs repeating decimals

🎯 What makes a decimal terminate

A unit fraction 1/n has a terminating decimal expansion in base-10 if and only if the denominator n can be factored into only 2s and 5s.

Why: Because 2 × 5 = 10, any combination of 2s and 5s can be converted to a power of 10.

Example: 1/8 = 1/2³ can be rewritten as 125/1000 by multiplying numerator and denominator by 5³.

Fraction	Denominator factors	Terminates?
1/8	2³	Yes
1/25	5²	Yes
1/6	2 × 3	No (has factor 3)

🔁 What makes a decimal repeat

If the denominator has prime factors besides 2 and 5, the decimal expansion will repeat infinitely.

Period definition:

Period: The smallest number of digits that repeat in a repeating decimal.

Example: 1/6 = 0.1̄6̄ has period 1 (just "6" repeats), while 6/7 = 0.857142̄ has period 6.

⏱️ Why repetition is guaranteed

When dividing 1 by n, the possible remainders are limited to 0, 1, 2, ..., n−1.

If remainder becomes 0, the division terminates.
If remainder never becomes 0, you must eventually see a repeated remainder (since there are only n−1 non-zero options).
Once a remainder repeats, the entire division process repeats from that point.

Key insight: The period of 1/n must be less than n, because you'll see a repeated remainder within n steps.

🌐 Cross-base behavior

🌐 Same fraction, different bases

The excerpt demonstrates that the same fraction (expressed in base-10) behaves differently when converted to other bases.

Example from the excerpt: The fraction 1/2 (in base-10):

In base-10: 1/2 = 0.5 (terminates).
In base-5: The division 1 ÷ 2 in base-5 produces a different pattern.
In base-3: Yet another pattern emerges.

🔍 General rule for termination

A fraction terminates in base-b if and only if the denominator (after simplification) contains only prime factors that are also factors of b.

Example: In base-6 (= 2 × 3), fractions with denominators containing only 2s and 3s will terminate.

Don't confuse: A fraction that repeats in base-10 might terminate in another base, and vice versa.

🧮 Operations on decimals

➕ Adding and subtracting with place value

The excerpt emphasizes lining up the decimal points when adding or subtracting.

Why this works: Each box in Dots & Boxes represents a specific place value—you can only combine dots in the same box (same place value).

Example: Adding 1.6 + 4.89 requires:

Aligning the ones, tenths, hundredths places.
Adding within each place.
Performing explosions if needed.

✖️ Multiplying decimals

The standard algorithm:

Multiply as if both numbers were whole numbers (ignore decimal points).
Count total digits to the right of decimal points in both factors.
Place the decimal point in the product so it has that many digits to the right.

Why this works: Writing decimals as fractions shows that multiplying a/(10^m) by b/(10^n) gives (a×b)/(10^(m+n)).

➗ Dividing decimals

The standard algorithm:

Move the divisor's decimal point to the end (make it a whole number).
Move the dividend's decimal point the same number of places.
Divide the new dividend by the new whole-number divisor.

Why this works: Multiplying both dividend and divisor by the same power of 10 doesn't change the quotient's value.

📏 Orders of magnitude and Fermi problems

📏 Thinking multiplicatively about large numbers

The excerpt emphasizes that people often think additively about large numbers when they should think multiplicatively.

Example: One billion is not "a million plus a little more"—it's 1,000 times one million.

🧪 Fermi problem approach

Fermi problem: A problem requiring estimation and reasoning to find an approximate answer when exact data is unavailable.

Steps illustrated in the excerpt:

Define terms clearly.
Write down what you know (or can reasonably estimate).
Make educated guesses with justification.
Do simple calculations to reach an order-of-magnitude answer.

Example from excerpt: Estimating the number of elementary teachers in Hawaii by:

Estimating population (~1,000,000).
Estimating fraction who are K-5 students.
Estimating average class size.
Calculating number of classes ≈ number of teachers.

Key insight: The goal is not exact precision but getting the right order of magnitude (is it hundreds? thousands? millions?).

Division and Decimals

🧭 Overview

🧠 One-sentence thesis

Fractions can be understood as division problems solved through a "Dots & Boxes" place-value model, which reveals why some fractions produce terminating decimals while others repeat infinitely.

📌 Key points (3–5)

Fractions as division: A fraction like 3/5 represents the answer to "3 divided by 5" — the amount each person gets when sharing 3 items among 5 people.
Dots & Boxes division method: Division is performed by finding groups of the divisor in a place-value picture, "unexploding" (trading) dots from higher place values when needed.
Terminating vs repeating decimals: Some fractions (like 1/8 = 0.125) finish cleanly, while others (like 1/3 = 0.333...) create an infinite repeating pattern.
Common confusion: The vinculum notation (bar over digits) means "repeat forever," not "approximately" — 0.3̄ truly equals 1/3, not just a rounded approximation.
Works in any base: The same division reasoning applies in base 5 or any other base, not just base 10.

🧮 Fractions as division problems

🥧 The sharing interpretation

A fraction represents the answer to a division problem — specifically, the amount one person receives when items are shared equally.

Basic idea: 6 pies ÷ 3 children = 2 pies per child, written as 6/3 = 2.
Unit fractions: When sharing 1 pie among 2 children, each gets 1/2; among 3 children, each gets 1/3; among 5 children, each gets 1/5.
General fractions: 3/5 means sharing 3 pies among 5 children — each child receives "three-fifths."
Example: The picture of three shaded parts out of five total represents 3 ÷ 5 = 3/5.

🔢 Division in Dots & Boxes

The excerpt shows division as "How many groups of the divisor fit into the dividend?"

3906 ÷ 3: Look for groups of 3 dots in the place-value picture.
- 1 group of 3 in thousands → 1000
- 3 groups of 3 in hundreds → 300
- 0 groups of 3 in tens → 0
- 2 groups of 3 in ones → 2
- Answer: 1302
With remainders (1024 ÷ 3): When a box doesn't have enough dots, "unexplode" a dot from the next higher place (trade 1 ten for 10 ones, etc.).
- Result: 341 groups of 3, plus 1 extra dot remaining.

🔄 Converting fractions to decimals

💡 The unexploding technique

To find the decimal form of a fraction, perform division using place values to the right of the decimal point.

Example: 1/8

Start with 1 dot in the ones box; look for groups of 8 (none found).
Unexplode into tenths: 1 becomes 10 tenths → 1 group of 8, leaving 2 tenths.
Unexplode into hundredths: 2 tenths become 20 hundredths → 2 groups of 8, leaving 4 hundredths.
Unexplode into thousandths: 4 hundredths become 40 thousandths → 5 groups of 8, no remainder.
Result: 1 group in tenths, 2 in hundredths, 5 in thousandths → 1/8 = 0.125

✅ Checking your work

The excerpt emphasizes verifying: 0.125 × 8 should equal 1.

🔁 Repeating decimals

♾️ When division never ends

Some fractions create an infinite cycle because the remainder keeps returning to the same value.

Example: 1/3

Start with 1 in ones; unexplode to 10 tenths → 3 groups of 3, leaving 1 tenth.
Unexplode to 10 hundredths → 3 groups of 3, leaving 1 hundredth.
Unexplode to 10 thousandths → 3 groups of 3, leaving 1 thousandth.
Pattern: Always 3 groups with 1 dot remaining → the cycle repeats forever.
Result: 1/3 = 0.333... (infinitely many 3s)

📏 Vinculum notation

The vinculum (horizontal bar) represents infinitely repeating digits.

Notation	Meaning	Full expansion
0.3̄	Repeat 3 forever	0.333333...
0.4̄1̄2̄	Repeat 412 forever	0.412412412...

Important distinction: The ellipsis "..." means "keep going forever" — humans can imagine this but cannot actually write all the digits.

🔄 Complex repeating patterns

Example: 6/7

The division process cycles through different remainders.
Eventually returns to the starting remainder (6 in the ones box).
Once you return to the beginning, the entire sequence repeats.
Result: 6/7 = 0.857142̄ (the sequence 857142 repeats forever)

Don't confuse: Returning to the original remainder signals the start of the repeating cycle, not an error in calculation.

🌐 Division in other bases

🔢 Base-5 example

The same "Dots & Boxes" division method works in any base, not just base 10.

Key difference: In base 5, "unexploding" means trading 1 dot for 5 dots in the next place (not 10).

Example shown: In base 5, a division problem produces groups in the "five-ths" place, "twenty-five-ths" place, etc.

🔄 Repeating in other bases

Example: 1432₅ ÷ 13₅

After finding whole groups, 2 dots remain.
Unexplode 1 dot → 1 dot in original box + 5 in next box.
Form a group of 13₅ (uses 1 + 3 dots).
2 dots remain again → the cycle repeats.
Result: A repeating "decimal" (more accurately, "five-mal") in base 5.

The excerpt emphasizes: The same infinite-repetition logic applies — when remainders cycle back, the pattern repeats forever.

More x-mals

More x -Mals

🧭 Overview

🧠 One-sentence thesis

Division in non-decimal bases produces repeating "decimal" expansions just as in base ten, and the "Dots & Boxes" model reveals why the same division process cycles forever when remainders repeat.

📌 Key points (3–5)

Division in other bases: the "Dots & Boxes" model for division works in any base, not just base ten.
Repeating patterns emerge: when dividing in base five (or any base), leftover dots can be unexploded indefinitely, creating repeating digit sequences.
Base-specific notation: the result of division in base b is written as a "decimal" (or "trimal," "quadimal," etc.) in that base.
Common confusion: the fraction and its decimal expansion look different in different bases, but they represent the same quantity—translate both the fraction and the base to check correctness.
Why it matters: understanding division and repeating expansions in arbitrary bases deepens place-value reasoning and reveals universal patterns in number systems.

🎨 The "Dots & Boxes" model in other bases

🎨 How the model works in base five

The "Dots & Boxes" model: a visual representation of division where dots are grouped and "unexploded" (traded) between place-value boxes according to the base.

In base ten, one dot in a box can be unexploded into ten dots in the next box to the right.
In base five, one dot unexplodes into five dots in the next box.
The excerpt shows dividing 1432₅ by 13₅ using this model.

🔁 The repeating cycle

The excerpt describes the step-by-step process:

Start with two dots left over (remainder 2).
Unexplode one dot → you have one dot in the original box and five dots in the box to the right.
Form a group of 13₅ (which uses one dot from the original box and three from the next box).
Two dots are left over again.
Repeat the same steps forever.

Why it repeats: each cycle leaves the same remainder (two dots), so the process never terminates.

🔍 Where to see the equation in the picture

The excerpt asks readers to identify parts of the equation in the visual model:

Where is the dividend? The original collection of dots (1432₅).
Where is the divisor? The size of each group being formed (13₅).
Where are the quotient and remainder? The number of complete groups and the leftover dots.

Don't confuse: the same division problem looks different in different bases, but the underlying grouping logic is identical.

🔢 Translating between bases

🔢 What the base-five equation says in base ten

The excerpt presents an equation in base five and asks:

What is 1432₅ in base ten?
What is 13₅ in base ten?
Translate the entire equation to base ten and verify it is correct.

How to translate: expand each base-five numeral using powers of five.

Example: 13₅ = 1×5 + 3×1 = 8 in base ten.

🧮 Division in base nine and other bases

Problem 2 asks:

Compute a division in base ten and show the repeating decimal.
Compute the same division in base nine and write the answer as a "nonimal" (a playful name for a base-nine "decimal").

Key idea: the division algorithm and the repeating behavior work the same way, but the digits and cycle length depend on the base.

🔁 Patterns in repeating expansions

🔁 Computing 1 ÷ 3 in multiple bases

Problem 3 asks for the decimal expansion of 1 ÷ 3 in:

Base ten → 0.333...
Base three → a "trimal"
Base four → a "quadimal"
Base six → a "heximal"

The excerpt prompts:

Describe any patterns you notice.
Do you have a conjecture of a general rule?
Can you prove your general rule is true?

What to look for: how the repeating block changes with the base, and whether certain bases produce terminating vs. repeating expansions.

🔁 The fraction 1/7 in different bases

Problem 4 explores the fraction 1/7 (given in base ten):

Find its decimal expansion in base ten.
Rewrite 1/7 as a division problem in base four, then find the base-four expansion.
Repeat for base five.
Repeat for base seven.
Check Barry's claim: in base fifteen, the expansion is a specific repeating pattern.

Why this matters: the same fraction has different expansions in different bases, but the underlying value is the same.

Don't confuse: "1/7" written in base ten is not the same as "1/7" written in another base—you must translate the numerator and denominator separately.

🔁 Expanding fractions in various bases

Problem 5 asks for the "decimal" expansion of several fractions (given in base ten) in different bases.

The excerpt prompts:

Do you notice any patterns?
Any conjectures?

Goal: discover general rules about when expansions terminate or repeat, and how the base affects the cycle.

🧩 Challenge problems

🧩 Reverse problem: from expansion to fraction

Problem 6 (Challenge) asks:

What fraction has a given repeating decimal expansion?
How do you know you are right?

Strategy: use the repeating pattern to set up an equation and solve for the fraction (a technique often used in base ten, now applied to other bases).

🧩 Terminating or repeating?

The excerpt ends with a heading "Terminating or Repeating?" but does not provide the content.

Implied question: given a fraction and a base, can you predict whether the expansion will terminate or repeat?

Hint from the problems: look for patterns related to the prime factors of the denominator and the base.

📋 Summary table: division in different bases

Base	Example division	Key feature	Notation nickname
5	1432₅ ÷ 13₅	Unexplode into 5 dots	"Decimal" in base 5
9	(from Problem 2)	Unexplode into 9 dots	"Nonimal"
3	1 ÷ 3 in base 3	Unexplode into 3 dots	"Trimal"
4	1 ÷ 3 in base 4	Unexplode into 4 dots	"Quadimal"
6	1 ÷ 3 in base 6	Unexplode into 6 dots	"Heximal"

Common thread: the "Dots & Boxes" model and the division algorithm are universal; only the grouping size (the base) changes.

Terminating or Repeating Decimals

Terminating or Repeating?

🧭 Overview

🧠 One-sentence thesis

A fraction's decimal representation either terminates or repeats forever, and whether it terminates depends entirely on whether the denominator (in lowest terms) has only 2s and 5s as prime factors.

📌 Key points (3–5)

Two possibilities only: when you write a fraction as a decimal, it either terminates (stops) or repeats forever in a pattern; it cannot go on forever without repeating.
What determines terminating vs repeating: a unit fraction 1/n terminates if and only if n can be factored into only 2s and 5s; otherwise it repeats forever.
Why powers of 2 and 5 matter: you can always turn denominators with only 2s and 5s into powers of 10 by multiplying by the right combination of 2s or 5s, which gives a finite decimal.
Common confusion: irrational numbers (like 0.101001000100001...) go on forever without repeating, but they can never be written as a fraction of two whole numbers—fractions always either terminate or repeat.
Period of repeating decimals: when a unit fraction 1/n repeats, the period (length of the repeating block) is always less than n.

🔢 The two types of decimal expansions

🔢 Terminating decimals

A decimal terminates when it stops after a finite number of digits.

Example from the excerpt: some fractions produce decimals that end, like certain simple fractions.
These are the fractions that can be rewritten with a power of 10 in the denominator.

🔁 Repeating decimals

A decimal repeats when it goes on forever in a repeating pattern.

Example from the excerpt: some fractions produce decimals that continue infinitely but with a repeating block of digits.
The excerpt emphasizes these are the only two things that can happen when you write a fraction as a decimal.

🚫 Irrational numbers (not fractions)

You can imagine a decimal that goes on forever but doesn't repeat itself (example given: 0.101001000100001...).
These numbers can never be written as a nice fraction a/b where a and b are whole numbers.
They are called irrational numbers because they cannot be expressed as a ratio of two whole numbers.
Don't confuse: repeating decimals are fractions; non-repeating infinite decimals are not fractions.

🔑 What makes a decimal terminate

🔑 The rule for unit fractions

A unit fraction is a fraction that has 1 in the numerator; it looks like 1/n for some whole number n.

The excerpt focuses on unit fractions to explore the terminating vs repeating question.
The key insight: a unit fraction 1/n has a terminating decimal if and only if n has only 2s and 5s as prime factors.

🔟 Why 2s and 5s are special

Powers of 10 are built from 2s and 5s: 10 = 2 × 5, 100 = 2² × 5², etc.
If the denominator has only 2s and 5s, you can always form an equivalent fraction with a power of 10 in the denominator.
Example from the excerpt: if you have a denominator with only 2s, you can "turn them into 10s" by multiplying by enough 5s (and vice versa).
The excerpt gives this pattern for 1/(2^k): you can multiply numerator and denominator by 5^k to get a power of 10 in the denominator.

📐 How many decimal places

Problem 9 in the excerpt asks: the unit fraction 1/(2^k) will have a decimal representation with a certain number of decimal digits.
The representation will be equivalent to a fraction with a power of 10 in the denominator.
The number of decimal places is determined by the power of 10 needed.

🧮 Examples with powers of 2 and 5

Denominator type	What happens	Why
Power of 2 only	Terminates	Multiply by enough 5s to make a power of 10
Power of 5 only	Terminates	Multiply by enough 2s to make a power of 10
Mix of 2s and 5s	Terminates	Already can be turned into a power of 10
Other prime factors	Repeats forever	Cannot be turned into a power of 10

🔄 When decimals repeat forever

🔄 Denominators with other prime factors

If the denominator has prime factors besides 2s and 5s, you cannot turn it into a power of 10.
Powers of 10 have only 2s and 5s as prime factors, so no amount of multiplying will work.
Therefore, the decimal expansion will go on forever.
But the excerpt emphasizes: it will have a repeating pattern, not just go on randomly.

📏 The period of a repeating decimal

The period of a repeating decimal is the smallest number of digits that repeat.

Example from the excerpt: if the repeating part is just the single digit 3, the period is one.
Another example: if the smallest repeating part is six digits, the period is six.
You can think of the period as the length of the string of digits under the vinculum (the horizontal bar that indicates the repeating digits).

🎯 Why it must repeat

The excerpt uses "Dots & Boxes" division to explain why the decimal must repeat.
When you divide to find the decimal expansion, you repeatedly:
- Unexplode dots (multiply the remainder by 10)
- Form groups (divide by the denominator)
- See how many dots are left (find the remainder)
Key insight: when you get the same remainder again, the process will repeat forever from that point.
Example from the excerpt: computing 1/6, you get remainder 4, then remainder 4 again, so the pattern repeats.

🔢 Upper bound on the period

Problem 13 asks for a convincing argument that the period of 1/n will be less than n.
The reason (implicit in the "Dots & Boxes" explanation): there are only n possible remainders (0 through n-1).
If the remainder is 0, the decimal terminates.
Otherwise, you must eventually get a repeated remainder (since there are fewer than n non-zero remainders).
Once a remainder repeats, the entire process repeats from that point.
Therefore, the period cannot be longer than n-1 (and is usually much shorter).

🌐 Decimals in other bases

🌐 The same principles apply

Problem 14 asks about decimal expansions in other bases (not just base 10).
The excerpt prompts a conjecture: if you write 1/n in base b, when will that expansion be finite vs infinite repeating?
The pattern: it depends on whether n can be factored into only the prime factors of b.
Example: in base 10, the special primes are 2 and 5 (since 10 = 2 × 5).
In base 9, the special primes would be 3 (since 9 = 3²).
In base 4, the special prime would be 2 (since 4 = 2²).

🧪 Testing the pattern

The excerpt asks students to find decimal expansions in various bases and look for patterns.
The goal is to generalize: the rule about 2s and 5s in base 10 extends to other bases with their own special prime factors.
Don't confuse: "decimal" in another base is sometimes called "nonimal" (base 9), "trimal" (base 3), "quadimal" (base 4), or "heximal" (base 6) in the excerpt, but the principles are the same.

Operations on Decimals

🧭 Overview

🧠 One-sentence thesis

Decimal operations follow the same logical principles as fraction operations but can be performed more efficiently by understanding place value and using algorithms that align decimal points or adjust them systematically.

📌 Key points (3–5)

Core principle: Decimals can always be converted to fractions for operations, but direct decimal methods are often faster when you understand place value.
Addition/subtraction rule: Line up decimal points because you must add/subtract like place values (ones with ones, tenths with tenths, etc.).
Multiplication insight: The number of decimal places in the product equals the sum of decimal places in both factors, based on how fractions work.
Division strategy: Move decimal points in both divisor and dividend by the same amount to create an equivalent problem with a whole-number divisor.
Common confusion: Multiplying by 10 does NOT simply append a zero for decimals—the decimal point appears to "move" one place to the right instead.

➕ Adding and subtracting decimals

📦 The Dots & Boxes model

The excerpt uses a "Dots & Boxes" place-value model where each box represents a place (ones, tens, hundreds, etc.).
Addition: Draw dots for each number, then "explode" (regroup) when a box has ten or more dots—move ten dots from one box to one dot in the next higher place.
Subtraction: Start with the minuend's dots, then remove (take away) the subtrahend's dots; if a box doesn't have enough dots, "unexplode" (unborrow) a dot from the next higher place into ten dots in the current place.

Example from excerpt: For 921 – 551, you start with 9 hundreds, 2 tens, 1 one; to subtract 5 tens when you only have 2, you unexplode one hundred into ten tens, leaving 3 hundreds and 12 tens, then remove 5 tens to get 7 tens.

🎯 Why "line up the decimal points"

Elementary students are taught to add and subtract decimals by "lining up the decimal points."

This shorthand works because you must add or subtract digits in the same place value.
The Dots & Boxes model shows that tenths must combine with tenths, hundredths with hundredths, etc.
Lining up decimal points automatically aligns all corresponding place values.
Don't confuse: You are NOT lining up the rightmost digits; you are aligning by place value.

🚫 Common student mistake

The excerpt describes Chloe adding two decimals and getting an incorrect answer (specific numbers not fully shown in context).

Likely error: Not aligning decimal points, treating decimals like whole numbers.
Teaching fix: Use the Dots & Boxes model to help students see that each place value is a separate "box" and must be combined with its matching box.

✖️ Multiplying decimals

🔟 Multiplying by powers of 10

The excerpt emphasizes that the whole-number rule "append a zero" does NOT work for decimals.

New rule for both whole numbers and decimals:

Multiplying by 10 shifts every digit one place to the left (makes each digit ten times larger).
Visually, the decimal point appears to move one place to the right.
Multiplying by 100 shifts two places, by 1000 shifts three places, etc.
Dividing by 10 shifts one place to the right (decimal point appears to move left).

Don't confuse: You are not "moving the decimal point" as a physical action—you are changing the place value of every digit; the decimal point's position relative to the digits changes as a result.

🧮 Understanding decimal multiplication through place value

Using number sense instead of rote computation:

Example from excerpt: To compute 321 × 0.4, a student reasons:

I know 321 × 4 = 1284.
Since I want to multiply by 0.4 (which is one-tenth of 4), my answer should be one-tenth of 1284.
So 321 × 0.4 = 128.4.

This uses the associative property: 321 × 0.4 = 321 × (4 × 0.1) = (321 × 4) × 0.1.

📏 The standard multiplication algorithm

Three-step process:

Step	Action	Reason
1	Multiply as if both numbers are whole numbers (ignore decimal points)	Simplifies computation
2	Count total decimal places in both factors (add them together to get n)	Tracks how many times you've divided by 10
3	Place decimal point in product so there are n digits to its right	Adjusts for the actual place values

Why it works (based on fractions):

If a number has 2 decimal places, it's like a fraction with 100 in the denominator (divided by 10 twice).
If another has 3 decimal places, it's like a fraction with 1000 in the denominator.
Multiplying these fractions gives a denominator with 5 zeros (100 × 1000 = 100,000).
So the product needs 5 decimal places.

Example: The excerpt asks students to explain why the number of zeros in the denominator of a decimal-as-fraction equals the number of digits right of the decimal point—this is the place-value foundation.

➗ Dividing decimals

🎯 The core challenge

The Dots & Boxes model struggles with division by decimals because "how do you make groups of 0.3 dots?"

The excerpt guides students to see that division can be rewritten using equivalent fractions.

🔄 Key equivalence principle

The exercises establish:

Two fractions are equivalent if you multiply (or divide) both numerator and denominator by the same number.
Therefore: dividing by a decimal is equivalent to dividing by a whole number if you scale both dividend and divisor appropriately.

Example pattern from exercises:

Why does 12 ÷ 3 give the same result as 120 ÷ 30?
Because both represent the same fraction: 12/3 = 120/30 (multiply top and bottom by 10).

📐 The standard division algorithm

Three-step process:

Move the decimal point of the divisor to the end (make it a whole number).
Move the decimal point of the dividend the same number of positions in the same direction.
Divide the new dividend by the new whole-number divisor using standard methods.

Why it works:

Moving both decimal points the same distance is equivalent to multiplying both dividend and divisor by the same power of 10.
This creates an equivalent fraction (same quotient).
Since any finite decimal can be converted to a whole number by multiplying by an appropriate power of 10, you can always transform the divisor into a whole number.
Dividing by a whole number is straightforward using standard methods.

Don't confuse: "Moving the decimal point" is shorthand for "multiplying by a power of 10"—you're creating an equivalent problem, not changing the original numbers arbitrarily.

🔍 Justification requirement

The excerpt repeatedly asks students to "carefully explain why" the algorithm works, emphasizing that:

Step 1 and 2 together multiply both numbers by the same power of 10.
This preserves the quotient (the ratio stays the same).
Step 3 then solves an equivalent but simpler problem.

Orders of Magnitude

🧭 Overview

🧠 One-sentence thesis

Understanding orders of magnitude—the multiplicative rather than additive differences between large numbers—is essential for making sense of big numbers and solving estimation problems (Fermi problems) through logical reasoning and educated guesses.

📌 Key points (3–5)

Multiplicative vs additive thinking: People naturally think additively (a billion is "a million plus more") but must think multiplicatively (a billion is 1,000 times a million) to understand large numbers.
Orders of magnitude: A number is "on the order of" a power of 10 when it falls between that power and the next (e.g., between 10 trillion and 100 trillion means the ten-trillions place).
Fermi problems: Named after physicist Enrico Fermi, these are estimation problems that seem unanswerable but can be solved by defining terms, using known facts, making reasonable guesses, and doing simple calculations.
Common confusion: Gut instincts about large numbers are often wildly wrong—most people underestimate the difference between a million and a billion.
Third kind of knowledge: Beyond knowing facts or knowing where to find them, you can figure out new knowledge yourself through reasoning and estimation.

🔢 Understanding large numbers

🔢 The million vs billion gap

The excerpt uses age in seconds to illustrate how poorly people grasp large numbers:

One million seconds = 1,000,000 seconds
One billion seconds = 1,000,000,000 seconds
The key insight: one billion is 1,000 times one million, not just "a bit more"
Example: If your million-second birthday answer was calculated, you could multiply by 1,000 to get your billion-second birthday—the difference is multiplicative, not additive.

🧮 Why intuition fails

People have good instincts for small, everyday numbers but very bad instincts about big numbers.
The excerpt asks readers to guess before calculating: "About a day? A week? A month? A year?"
Most people are surprised by the actual answers because they think additively instead of multiplicatively.

📏 Orders of magnitude concept

📏 What "order of" means

"On the order of 10 trillion dollars" means more than 10 trillion but less than 100 trillion.

If you write the number in dots-and-boxes notation, the dots would extend as far left as the ten-trillions place.
This is a way of expressing approximate scale without needing exact figures.

💰 National debt example

The excerpt uses US debt (summer 2013, on the order of 10 trillion dollars) to practice:

Scenario 1: Paying back one penny every second—would this pay off the debt in your lifetime?
Scenario 2: Paying down $35 billion every quarter (four times per year)—how many years to pay off the total?
These problems require understanding the multiplicative relationship between pennies, billions, and trillions.

🎯 Fermi problems methodology

🎯 What Fermi problems are

Fermi problems involve using your knowledge, making educated guesses, and doing reasonable calculations to come up with an answer that might at first seem unanswerable.

Named after physicist Enrico Fermi.
They create a "third kind of knowledge"—not facts you know or can look up, but answers you figure out for yourself.
Even when answers could be googled, the exercise teaches estimation and reasoning skills.

🛠️ The four-step process

The excerpt provides a systematic approach:

Step	What to do	Purpose
1. Define your terms	Be specific about what you're counting	Avoid ambiguity
2. Write down what you know	List relevant facts and data	Establish starting points
3. Make reasonable guesses/estimates	Fill in missing information with educated assumptions	Bridge knowledge gaps
4. Do simple calculations	Use basic arithmetic to reach an answer	Arrive at order-of-magnitude estimate

📚 Detailed example: Hawaii teachers

The excerpt walks through estimating K–5 teachers in Hawaii:

Step 1 - Define terms: Classroom teachers (not administrators, subs, aides) with permanent positions in grades K–5.

Step 2 - Known facts: Hawaii population is about 1,000,000 people (not exact, but order of magnitude).

Step 3 - Reasonable guesses:

Assume population is distributed across age ranges (with adjustments for college-age bump and fewer elderly).
K–5 covers about 6 years of the 0–9 age range.
If 125,000 people are aged 0–9, that's about 12,500 per year of age.
Total K–5 students: approximately 75,000.
Average class size: 25 students (accounting for variation by grade and school type).

Step 4 - Calculate: 75,000 students ÷ 25 students per class = 3,000 classrooms = 3,000 teachers (since elementary has one teacher per class).

🤔 Making simplifying assumptions

The excerpt emphasizes that estimation requires reasonable simplifications:

"We'll assume people don't live past 80. Of course some people do! But we're all about making simplifying assumptions right now."
Adjustments can be made for known factors (e.g., university creating a "bump" in 20–29 age range).
The goal is order of magnitude, not precision.

🧪 Practice problems provided

🧪 Jelly bean problems

The excerpt offers concrete visualization challenges:

Area: A million jelly beans tiling the floor—classroom size? Football field? Bigger?
Height: A million jelly beans stacked—as tall as a person? Tree? Skyscraper?
Scale comparison: How many jelly beans to reach the moon?
Then repeat for a billion jelly beans to reinforce the multiplicative difference.

🧪 Other Fermi problems listed

The excerpt provides multiple practice scenarios:

University parking revenue per year
Annual tourists visiting Waikiki
Gas saved if 10% of people carpooled
Energy in a chocolate bar for mountain climbing
Monthly pizza consumption at a university
Cost of free daycare for all four-year-olds in the US
Number of books in a university library
Creating your own Fermi problem

Each requires the same methodology: define, list knowns, estimate unknowns, calculate.

Introduction

🧭 Overview

🧠 One-sentence thesis

The excerpt does not contain substantive content—it consists of geometry problem prompts, a brief photo credit note, and an incomplete sentence about Polynesian history.

📌 Key points (3–5)

Tangram problems: assign values to tangram pieces based on different wholes (large triangle or small triangle).
Triangle classification exercises: sketch or explain why certain combinations (e.g., right + scalene, obtuse + equilateral) are possible or impossible.
Vertical angles and triangle properties: use reasoning about angle sums and side lengths to justify geometric facts.
Real-world applications: triangles provide structural support (SSS congruence); symmetry can combine reflection, rotation, and translation.
Incomplete narrative: the excerpt ends mid-sentence about historians and Polynesian history in the 1950s–1960s, with no further explanation.

📐 Tangram value assignments

🔺 Large triangle as one whole

Problem 4 asks: if the large tangram triangle is "one whole," what value should each of the seven tangram pieces have?
Students must justify their answers by comparing the areas of the pieces.
The excerpt does not provide the solution or the tangram diagram.

🔻 Small triangle as one whole

Problem 5 asks: if a small tangram triangle is "one whole," assign values to all seven pieces.
Again, justification is required but no diagram or answer is given.

🔺 Triangle classification problems

🔍 Combining angle and side properties

The excerpt lists four problem sets (Problems 19–22) that ask whether certain triangle types can exist:

Angle type	Side type	Problems ask
Right	Scalene / Isosceles / Equilateral	Sketch or explain why not
Acute	Scalene / Isosceles / Equilateral	Sketch or explain why not
Obtuse	Scalene / Isosceles / Equilateral	Sketch or explain why not
Equiangular	Scalene / Isosceles / Equilateral	Sketch or explain why not

Students must determine which combinations are geometrically possible.
Example: a right triangle can be scalene or isosceles, but not equilateral (because a right angle and two 60° angles cannot coexist).
The excerpt does not provide answers or explanations.

⚠️ Common confusion

Don't confuse "equiangular" with "equilateral": equiangular means all angles are equal (which forces the triangle to be equilateral), but the problem asks students to reason through this.

📐 Reasoning about angles and sides

🔄 Vertical angles (Problem 23)

Vertical angles: angles opposite each other when two lines intersect (e.g., angles A and D, or B and C in the diagram).

The problem asks students to explain why vertical angles must have the same measure.
Hint provided: consider the sum of angles A and B, and use that reasoning.
The excerpt does not give the full explanation, but the hint suggests using the fact that adjacent angles on a line sum to 180°.

📏 Triangle side length constraints (Problems 24–25)

Problem 24: given a triangle with some known side lengths, determine whether x = 4 cm or x = 20 cm is possible, and list three possible values.
Problem 25: determine whether x = 3 cm or x = 8 cm makes the triangle isosceles, and list three impossible values.
Students must focus on "what you know for sure" (triangle inequality, isosceles definition) rather than what the picture looks like.
The excerpt does not show the diagrams or provide solutions.

❌ Finding mistakes (Problem 26)

A diagram shows three triangles (△ABC, △ABD, △CBD) with side lengths and angle measurements.
The problem states there are "lots of mistakes" and asks students to identify them using triangle properties.
Students must explain what is wrong and why.
The excerpt does not describe the mistakes or the diagram details.

🏗️ Real-world applications and symmetry

🏛️ Triangular supports (Problem 27)

Why triangles are sturdy: SSS (side-side-side) congruence makes triangles rigid and stable.
Triangles are used in architecture and design for buildings, bridges, and other structures.
Students are asked to photograph and describe triangular supports in their neighborhood or campus.

🔁 Multiple symmetries (Problem 28)

The problem asks students to find or create designs with combinations of:

Reflection symmetry and rotational symmetry
Reflection symmetry and translational symmetry
Rotational symmetry and translational symmetry

Don't confuse: these are distinct types of symmetry, but a single design can exhibit more than one type.

🚢 Hōkūle'a voyage note

🌊 Brief context

The excerpt includes a short quote: "We sail for peace, for the love of our planet and with the desire to leave the children of the world a hopeful and hopeful future." —Hōkūle'a crew
A photo credit note states that images come from the Hōkūle'a press room and are used non-commercially.
The section titled "Introduction" begins: "In the 1950's and 1960's, historians couldn't agree on how the Polynesian..." and then stops mid-sentence.
No substantive content about the voyage or the historical debate is provided in this excerpt.

Tangrams

🧭 Overview

🧠 One-sentence thesis

Tangrams are a seven-piece geometric puzzle that teaches reasoning about shapes and spatial relationships by requiring all pieces to fit together without overlapping to form various designs.

📌 Key points (3–5)

What tangrams are: a seven-piece geometric puzzle dating back to the Song Dynasty in China (about 1100 AD).
The core rule: use all seven pieces to form a shape, fitting together like puzzle pieces lying flat with no overlapping allowed.
Two types of challenges: "real life" objects (cats, people) versus purely mathematical objects (rectangles).
Connection to geometry: tangrams illustrate that geometry is about reasoning and justifying solutions, not trusting appearances—you must figure out what you know for sure and why.
Common confusion: "appears to be" or "looks like" is not sufficient in geometry; you need logical reasoning to determine what is actually true.

🧩 What tangrams are and how they work

🧩 The puzzle structure

Tangrams: a seven-piece geometric puzzle that dates back at least to the Song Dynasty in China (about 1100 AD).

The puzzle consists of exactly seven geometric pieces.
Players must make a careful copy, cut out the pieces, and use them to solve various shape-building challenges.
Each challenge is separate and requires all seven pieces.

📏 The fundamental rules

Use all seven pieces to form the target shape.
Pieces should fit together like puzzle pieces, sitting flat on the table.
No overlapping of pieces is allowed.
Tracing around solutions helps remember what you have done and provides a record of your work.

🎯 Types of tangram challenges

🐱 "Real life" objects

Challenges include building shapes that represent everyday objects like cats and people.
These designs use all seven pieces arranged to suggest recognizable forms.

📐 Mathematical objects

Challenges include building purely geometric shapes like rectangles.
These designs focus on mathematical properties rather than representational imagery.

🤔 Comparing difficulty

The excerpt poses reflection questions:

Which problems were easier: "real life" objects or purely mathematical objects?
What made one kind of problem easier or harder?

The excerpt does not provide answers but encourages learners to think about the different cognitive demands of representational versus abstract geometric reasoning.

🧠 Connection to geometric reasoning

🧠 Geometry as reasoning, not appearance

Geometry is the art of good reasoning from bad drawings. (Henri Poincaré)

This insight should guide study: never trust a drawing.
One line segment might appear longer than another, or an angle might look like 90 degrees.
"Appears to be" and "looks like" are not good enough.
You must reason through the situation and figure out what you know for sure and why you know it.

🔍 What geometry is really about

Geometry is probably the oldest field of mathematics, useful for calculating lengths, areas, and volumes of everyday objects.
The name comes from ancient Greek: "geo" (Earth) + "metron" (measurement).
What's really important in geometry is reasoning, making sense of problems, and justifying your solutions.
Example: In tangrams, you cannot simply eyeball whether pieces fit; you must understand their geometric properties (angles, side lengths, shapes) to solve puzzles correctly.

⚠️ Don't confuse visual intuition with proof

Visual appearance can be misleading.
Tangrams require spatial reasoning and verification, not just guessing based on how things look.
The discipline of using all seven pieces with no overlapping forces precise geometric thinking rather than approximate visual matching.

📐 Basic geometric objects (context)

📐 Foundational concepts

The excerpt lists reflection questions about basic geometric objects (without providing answers):

What is a point?
What is a line? A segment? A ray?
What is a plane?
What is a circle?
What is an angle?
Which of these basic objects can be measured? How are they measured? What kinds of tools are useful?

These questions frame the study of geometry as understanding fundamental building blocks and their properties.

🔺 Triangle classifications (preview)

🔺 Classification by sides

Type	Definition
Scalene	All sides have different lengths
Isosceles	Two sides have the same length
Equilateral	All three sides have the same length

📐 Classification by angles

Type	Definition
Acute	All interior angles measure less than 90°
Obtuse	One interior angle measures more than 90°
Right	One interior angle measures exactly 90°
Equiangular	(Definition cut off in excerpt)

🎓 Purpose of vocabulary

The point of learning geometry is not to learn a lot of vocabulary.
It is useful to use correct terms for objects so we can communicate clearly.
The excerpt provides this "quick dictionary" to support clear communication about geometric properties.

Triangles and Quadrilaterals

🧭 Overview

🧠 One-sentence thesis

Three side lengths uniquely determine a triangle's shape (SSS congruence), but the same is not true for quadrilaterals, which can be "squished" into different shapes with the same four side lengths.

📌 Key points (3–5)

Triangle classification: triangles are classified by sides (scalene, isosceles, equilateral) and by angles (acute, obtuse, right, equiangular).
Angle sum in triangles: the sum of the angles in any triangle is 180° in Euclidean (flat) geometry; this is Euclid's fifth axiom.
Triangle Inequality: the sum of the lengths of any two sides in a triangle must be greater than the length of the third side.
SSS Congruence for triangles: if two triangles have the same three side lengths, they are congruent (exactly the same shape and size).
Common confusion: quadrilaterals with the same four side lengths are not necessarily congruent—they can be squished or rearranged into different shapes, unlike triangles.

📐 Classifying triangles

📏 Classification by sides

Triangles are grouped by how their side lengths compare:

Type	Definition
Scalene	All sides have different lengths
Isosceles	Two sides have the same length
Equilateral	All three sides have the same length

Use tick marks to indicate which sides are equal, because drawings may not be precise.
Remember: "geometry is the art of good reasoning from bad drawings"—you cannot always trust your eyes.

📐 Classification by angles

Triangles are also grouped by their interior angles:

Type	Definition
Acute	All interior angles measure less than 90°
Obtuse	One interior angle measures more than 90°
Right	One interior angle measures exactly 90°
Equiangular	All interior angles have the same measure

A little square symbol is used to indicate a right angle in drawings.
Tick marks or written measurements show when angles are supposed to be equal, even if the drawing looks imperfect.

🔍 Notation: tick marks and symbols

Tick marks: Mathematicians use tick marks to indicate when sides and angles are supposed to be equal.

If two sides have the same number of tick marks, they must be treated as equal, even if the drawing looks different.
Example: a right triangle drawing uses a small square at the 90° angle to mark it clearly.

📊 Angle sum in triangles

🧩 The 180° rule

The sum of the angles in any triangle is 180°.

The excerpt describes an activity: cut out a triangle, tear off the three corners, and place the vertices together—they form what looks like a straight line (180°).
But drawings are not exact, so how can we be sure it's exactly 180° and not 178° or 181°?
The excerpt notes that testing many cases is not enough to prove it for every possible triangle.

📜 Euclid's fifth axiom

Axiom: a given fact about how geometry works, taken as a starting assumption.

Around 300 BC, Euclid wrote down five axioms for geometry.
The first four seemed obvious (e.g., you can connect two points with a line segment; all right angles are congruent).
The fifth axiom was more controversial: "The sum of the angles in a triangle is 180°."
Mathematicians felt uneasy assuming this without proof, and many tried to prove it from the other four axioms—but they could not.
Reason: there are other geometries (non-Euclidean) where the first four axioms hold but the fifth does not.

🌍 Geometry on a sphere

On a sphere (like the surface of the Earth), the first four axioms are still true.
But triangles on a sphere have angle sums greater than 180°.
Example: you can draw a triangle on Earth with three right angles, giving an angle sum of 270°.
On a sphere, bigger triangles have bigger angle sums; even tiny triangles have sums greater than 180°.
Don't confuse: Euclidean geometry (flat plane) vs. spherical geometry (curved surface)—the angle sum rule differs.

📏 Triangle Inequality

🔗 The rule

Triangle Inequality: The sum of the lengths of two sides in a triangle is greater than the length of the third side.

The excerpt describes an activity: pick three strips of paper and try to form a triangle; record which combinations work and which do not.
Goal: discover a rule that predicts when three lengths will form a triangle.
The rule is: for any two sides, their sum must be greater than the third side.

🧭 Why it makes sense

The Triangle Inequality is the same as saying "the shortest distance between two points is a straight line."
If you try to connect two endpoints with two segments that are too short, they won't reach to close the triangle.
Example: sides of 40, 40, and 100 units—40 + 40 = 80, which is less than 100, so no triangle is possible.
Example: sides of 2.5, 2.6, and 5 units—2.5 + 2.6 = 5.1, which is greater than 5, so a triangle is possible (just barely).

🔺 SSS Congruence

🔺 Congruence for triangles

Congruent: two geometric objects are congruent if they are exactly the same shape and the same size (they would exactly overlap if you moved one onto the other).

SSS (side-side-side) Congruence: If two triangles have the same side lengths, then the triangles are congruent.

The excerpt describes an activity: pick three strips that form a triangle, then try to make two different (non-congruent) triangles with the same three strips.
Result: you cannot make two different triangles with the same three side lengths.
Three side lengths uniquely determine the triangle's shape.

🔄 Why triangles can't "squish"

Imagine two of your three lengths hinged at one corner.
As you open the hinge, the two non-hinged endpoints move farther apart.
For your third length, there is exactly one position of the hinge where it will fit to close off the triangle.
No other position works, so the triangle's shape is locked in.
Don't confuse: this rigidity is unique to triangles; it does not apply to quadrilaterals or other polygons.

🔲 Quadrilaterals are different

🔲 Four sides do not determine shape

The excerpt describes an activity: pick four strips that form a quadrilateral, then try to make two different (non-congruent) quadrilaterals with the same four strips.
Result: you can make different shapes with the same four side lengths.
Example: four strips of equal length can form a square, but you can also squish that square into a non-square rhombus.

🔄 Squishing and rearranging

Quadrilaterals can be "squished" into different shapes without changing side lengths.
You can also rearrange the order of the four sides to make entirely different (non-congruent) shapes.
Example: the excerpt shows two quadrilaterals with the same four side lengths in the same order, and two more with the same lengths in a different order—all four are non-congruent.
Key difference from triangles: with three sides, you cannot rearrange them (that's just rotating or flipping the triangle); with four or more sides, rearrangement creates genuinely new shapes.

Polygons

🧭 Overview

🧠 One-sentence thesis

Polygons exhibit unique properties—such as angle sums that depend on the number of sides and the impossibility of SSS congruence for quadrilaterals—that distinguish them from triangles and lead to exactly five regular three-dimensional solids (Platonic solids).

📌 Key points (3–5)

SSS congruence fails for quadrilaterals: Unlike triangles, quadrilaterals with the same four side lengths can be "squished" or rearranged into non-congruent shapes.
Angle sum formula: Any n-gon can be divided into triangles from one vertex, yielding a predictable interior angle sum.
Regular polygons vs. regular polyhedra: Infinitely many regular polygons exist, but only exactly five Platonic solids (regular polyhedra) are possible.
Common confusion: Don't assume triangle properties (like SSS congruence) automatically apply to other polygons—triangles are special because their shape is rigid once side lengths are fixed.
The 360° constraint: For a polyhedron to close properly, the angles meeting at each vertex must sum to less than 360°, which limits the possible Platonic solids.

🔺 Why triangles are rigid but quadrilaterals are not

🔺 SSS congruence works only for triangles

For triangles: If two triangles have the same three side lengths, they are congruent (SSS congruence).
For quadrilaterals: This is "most certainly not true."
- Four strips of equal length can form a square or a non-square rhombus by "squishing."
- Four strips of different lengths can be rearranged into different non-congruent shapes.

🔧 The hinge argument

The excerpt explains why triangles cannot "squish":

Imagine two of the three sides hinged at a corner.
As the hinge opens, the distance between the free endpoints increases.
For any given third side length, there is exactly one hinge position where that third side fits to close the triangle.
No other position works, so the triangle's shape is locked.

Don't confuse: Rotating or flipping a triangle does not create a new shape; rearranging sides of a quadrilateral can.

📐 Polygon definitions and classification

📐 What is a polygon?

Polygon: (1) a plane figure (2) bounded by a finite number of straight line segments (3) in which each segment meets exactly two others, one at each endpoint.

The line segments are called edges.
The meeting points are called vertices (singular: vertex).
Polygons do not self-intersect (because of properties 2 and 3).

🏷️ Naming polygons by sides

Name	Number of sides
Triangle	3
Quadrilateral	4
Pentagon	5
Hexagon	6
Heptagon	7
Octagon	8
Nonagon	9
Decagon	10
n-gon	n

📏 Angle sums in polygons

📏 Quadrilaterals have 360° total

Any quadrilateral can be split into two triangles by drawing a diagonal.
Each triangle has an interior angle sum of 180°.
Therefore, all quadrilaterals have an angle sum of 2 × 180° = 360°.
Example: Rectangles have four 90° angles, and 4 × 90° = 360°.

Don't assume: Just because triangles have a constant angle sum doesn't automatically mean other polygons do—but the triangle-splitting method proves it.

📏 General angle sum formula

The excerpt guides readers to discover:

A pentagon splits into three triangles → angle sum = 3 × 180° = 540°.
A hexagon splits into four triangles → angle sum = 4 × 180° = 720°.
An n-gon splits into (n − 2) triangles → angle sum = (n − 2) × 180°.

🔲 Regular polygons

Regular polygon: all sides the same length and all angles the same measure.

A square is a regular quadrilateral (all sides equal, all angles 90°).
A non-square rectangle is not regular (angles equal, but sides differ).
A non-square rhombus is not regular (sides equal, but angles differ).

Finding each angle in a regular n-gon:

Total angle sum = (n − 2) × 180°.
Each angle = [(n − 2) × 180°] ÷ n.

Example: A regular triangle (equilateral) has each angle = (3 − 2) × 180° ÷ 3 = 60°.

🧊 Platonic solids (regular polyhedra)

🧊 What is a polyhedron?

Polyhedron: a solid (3-dimensional) figure bounded by polygons, with flat faces, straight edges where faces meet in pairs, and vertices where three or more edges meet.

Regular polyhedron (Platonic solid): all faces are identical (congruent) regular polygons, and all vertices are identical (the same number of faces meet at each vertex).

🔢 There are exactly five Platonic solids

Key constraint: At each vertex, the angles of the meeting faces must sum to less than 360° for the solid to close up (not lie flat or fold over).

Face shape	Faces per vertex	Works?	Result
Equilateral triangle	3	Yes	Tetrahedron
Equilateral triangle	4	Yes	Octahedron
Equilateral triangle	5	Yes	Icosahedron
Equilateral triangle	6 or more	No	6 × 60° = 360° (lies flat)
Square	3	Yes	Cube
Square	4 or more	No	4 × 90° = 360° (lies flat)
Regular pentagon	3	Yes	Dodecahedron
Regular pentagon	4 or more	No	4 × 108° > 360°
Regular hexagon	3 or more	No	3 × 120° = 360° (lies flat)
Regular n-gon (n ≥ 7)	Any	No	Angles too large

🎯 Why only five?

Triangular faces: 3, 4, or 5 triangles can meet at a vertex (each triangle angle = 60°). Six would give 360°.
Square faces: Only 3 squares can meet (3 × 90° = 270° < 360°). Four would give 360°.
Pentagonal faces: Only 3 pentagons can meet (each angle = 108°, so 3 × 108° = 324° < 360°).
Hexagons and beyond: Even 3 hexagons give 3 × 120° = 360°, which lies flat, so no polyhedron forms.

Don't confuse: Unlike regular polygons (infinitely many), there are only five regular polyhedra.

🎨 Painted cubes problem

🎨 Counting unit cubes by painted faces

The excerpt presents a problem: build an n × n × n cube from unit cubes, dip it in paint, then count how many unit cubes have 0, 1, 2, or 3 painted faces.

For a 3 × 3 × 3 cube (27 unit cubes total):

0 faces painted: The single cube at the very center (not touching any outer face).
1 face painted: Cubes in the center of each face (not on any edge).
2 faces painted: Cubes along each edge (but not at corners).
3 faces painted: The 8 corner cubes.
More than 3 faces: Impossible—a cube has only 6 faces, but no unit cube inside an n × n × n cube touches more than 3 outer faces.

🎨 Generalizing to n × n × n

The excerpt asks readers to extend the reasoning:

Total unit cubes: n × n × n = n³.
Interior (0 painted): (n − 2)³ cubes (remove one layer from each side).
Face centers (1 painted): 6 faces, each with (n − 2)² interior squares.
Edge cubes (2 painted): 12 edges, each with (n − 2) cubes (excluding corners).
Corner cubes (3 painted): Always 8 corners.

Example reasoning: The problem encourages spatial visualization and systematic counting by position.

Platonic Solids

🧭 Overview

🧠 One-sentence thesis

There are exactly five Platonic solids—a finite, surprising result that contrasts sharply with the infinite variety of regular polygons—because closing up a three-dimensional polyhedron requires less than 360° at each vertex.

📌 Key points (3–5)

What a Platonic solid is: a regular polyhedron whose faces are all identical (congruent) regular polygons and whose vertices are all identical (the same number of faces meet at each vertex).
The key constraint: for a solid to close up in three dimensions, the angles meeting at each vertex must sum to less than 360°; otherwise it lies flat or folds over.
The surprising result: unlike regular polygons (which exist for every n > 2), there are exactly five Platonic solids—no more, no fewer.
Common confusion: regular polygons vs. regular polyhedra—polygons are infinite in variety, but polyhedra are limited by the 360° vertex constraint in three dimensions.
How to find them all: systematically try each regular polygon (triangles, squares, pentagons, etc.) and count how many can meet at a vertex before exceeding 360°.

🧊 What polyhedra and Platonic solids are

🧊 Polyhedron definition

A polyhedron is a solid (3-dimensional) figure bounded by polygons. A polyhedron has faces that are flat polygons, straight edges where the faces meet in pairs, and vertices where three or more edges meet.

The plural is "polyhedra."
Key parts: faces (flat polygons), edges (where two faces meet), vertices (where three or more edges meet).
Not all 3D shapes are polyhedra—only those bounded by flat polygons.

⭐ Regular polyhedron (Platonic solid) definition

A regular polyhedron has faces that are all identical (congruent) regular polygons. All vertices are also identical (the same number of faces meet at each vertex). Regular polyhedra are also called Platonic solids (named for Plato).

"Identical faces" means all faces are congruent regular polygons (e.g., all equilateral triangles or all squares).
"Identical vertices" means the same number of faces meet at every vertex.
Example: if three triangles meet at one vertex, then three triangles must meet at every vertex.

🔄 Contrast with regular polygons

For polygons: fix the number of sides and their length, and there is one and only one regular polygon with that number of sides (e.g., every regular quadrilateral is a square, every regular octagon looks like a stop sign, just scaled).
For polyhedra: the situation is "quite different"—there are infinitely many regular polygons (one for every n > 2), but only five Platonic solids.

🔨 How to discover the Platonic solids

🔨 The hands-on method

The excerpt describes a systematic construction process:

Cut out many copies of regular polygons: equilateral triangles, squares, pentagons, hexagons, heptagons, octagons.
At each vertex, at least three polygons must meet (in any polyhedron).
Start with three polygons meeting at a vertex, tape them together, and try to close them up into a solid shape.
Then try four polygons at each vertex, then five, and so on.
Repeat for each type of regular polygon.

Example: Start with three equilateral triangles meeting at a vertex, tape them, and close them up. Check that at every vertex exactly three triangles meet. Then try four triangles at each vertex, then five, etc.

🔍 What you discover

With triangular faces: some configurations work, some don't.
With square faces: some configurations work, some don't.
With pentagonal faces: some configurations work, some don't.
With hexagons and beyond: none work.
The excerpt emphasizes working "systematically" until you "can make a definitive statement" about each type of face.

🔑 The key constraint: the 360° rule

🔑 Why there are only five

The key fact: for a three-dimensional solid to close up and form a polyhedron, there must be less than 360° around each vertex. Otherwise, it either lies flat (if there is exactly 360°) or folds over on itself (if there is more than 360°).

This is the geometric constraint that limits Platonic solids to exactly five.
If the angles at a vertex sum to exactly 360°, the polygons lie flat (no 3D solid forms).
If the angles sum to more than 360°, the polygons fold over themselves (impossible to close up properly).
Only when the sum is less than 360° can the solid close up in three dimensions.

🧮 Applying the constraint to each polygon type

Polygon type	Angle per face	How many can meet at a vertex?	Why?
Equilateral triangle	60°	Fewer than 6	6 × 60° = 360° (lies flat); so at most 5 can meet
Square	90°	At most 3	4 × 90° = 360° (lies flat); so at most 3 can meet
Regular pentagon	108°	At most 3	4 × 108° = 432° > 360° (folds over); so at most 3 can meet
Regular hexagon	120°	Cannot be used	3 × 120° = 360° (lies flat); no room for a solid
Regular n-gon (n > 6)	More than 120°	Cannot be used	Even 3 faces exceed or equal 360°; no solid possible

Triangles: fewer than 6 faces can meet at each vertex (so 3, 4, or 5 are possible).
Squares: three faces can meet, but not more (4 would be 360°).
Pentagons: three faces can meet, but not more (4 would exceed 360°).
Hexagons: three hexagons already sum to 360°, so they lie flat—no Platonic solid.
Heptagons, octagons, etc.: their angles are even larger, so even three faces exceed 360°—no Platonic solid.

✅ The theorem

Theorem: There are exactly five Platonic solids.

This is a "perhaps surprising" result.
The excerpt provides a sketch of the justification based on the 360° constraint applied systematically to each regular polygon type.
The five solids correspond to the valid combinations: three, four, or five triangles at each vertex; three squares at each vertex; three pentagons at each vertex.

🧩 Common confusions

🧩 Regular polygons vs. regular polyhedra

Don't confuse: the infinite variety of regular polygons (one for every n > 2) with the finite set of Platonic solids (exactly five).
Reason: polygons are 2D and have no vertex angle constraint; polyhedra are 3D and must satisfy the "less than 360° at each vertex" rule.
Example: you can have a regular 100-gon, but you cannot have a Platonic solid with 100-sided faces (the angles would be too large).

🧩 Flat vs. closed vs. folded

Exactly 360° at a vertex → the polygons lie flat (no 3D solid).
Less than 360° at a vertex → the polygons can close up into a 3D solid.
More than 360° at a vertex → the polygons fold over themselves (impossible to form a proper polyhedron).
This distinction is the heart of why only five Platonic solids exist.

Painted Cubes

🧭 Overview

🧠 One-sentence thesis

The painted-cube problem reveals how the position of unit cubes within a larger cube determines how many faces get painted, and this pattern can be generalized to any n × n × n cube.

📌 Key points (3–5)

Building cubes from unit cubes: An n × n × n cube is built from n³ unit cubes (1 × 1 × 1 cubes).
The painting scenario: When you dip a 3 × 3 × 3 cube into paint, only the outer surface gets painted; then you take it apart to count how many faces of each unit cube have paint.
Position determines paint: Unit cubes in different positions (interior, face centers, edges, corners) have different numbers of painted faces (0, 1, 2, or 3).
Common confusion: Don't confuse the total number of unit cubes with the number in each category—most unit cubes are not fully painted; some have no paint at all.
Generalization: The same counting logic applies to any n × n × n cube, not just 3 × 3 × 3.

🧱 Building cubes from unit cubes

🧱 What is a unit cube

A unit cube is a 1 × 1 × 1 cube.

This is the smallest building block.
Larger cubes are built by stacking unit cubes in three dimensions.

🔢 Counting unit cubes in an n × n × n cube

A 2 × 2 × 2 cube contains 2 × 2 × 2 = 8 unit cubes.
A 3 × 3 × 3 cube contains 3 × 3 × 3 = 27 unit cubes.
In general, an n × n × n cube contains n³ unit cubes.
Why: you multiply the number of cubes along each of the three dimensions (length, width, height).

🎨 The painting problem

🎨 The scenario

You build a 3 × 3 × 3 cube from 27 white unit cubes.
You dip the entire cube into blue paint.
Only the outer surface of the large cube gets painted.
Then you take the cube apart and examine each unit cube.

🔍 What the problem asks

The problem asks you to count how many unit cubes have:

0 painted faces (still all white)
1 painted face
2 painted faces
3 painted faces
More than 3 painted faces

🧩 Why position matters

Unit cubes on the interior (not touching any outer face) have 0 painted faces.
Unit cubes on a face (but not on an edge or corner) have 1 painted face.
Unit cubes on an edge (but not at a corner) have 2 painted faces.
Unit cubes at a corner have 3 painted faces.
No unit cube can have more than 3 painted faces, because each unit cube has 6 faces, but at most 3 can be on the outer surface of the large cube.

🧮 Counting painted faces for a 3 × 3 × 3 cube

🟦 Zero painted faces (interior cubes)

These are the unit cubes completely inside, not touching any outer face.
For a 3 × 3 × 3 cube, there is a 1 × 1 × 1 cube in the center.
So there is 1 unit cube with 0 painted faces.
How to know: remove one layer from each side of the 3 × 3 × 3 cube; what remains is (3 − 2) × (3 − 2) × (3 − 2) = 1 × 1 × 1.

🟦 One painted face (face centers)

These are unit cubes in the middle of a face, not on any edge.
Each face of the 3 × 3 × 3 cube is a 3 × 3 grid; the center of each face is 1 unit cube.
A cube has 6 faces, so there are 6 unit cubes with 1 painted face.

🟦 Two painted faces (edge centers)

These are unit cubes along an edge, but not at a corner.
Each edge of the 3 × 3 × 3 cube has 3 unit cubes; the two at the ends are corners, so 3 − 2 = 1 is in the middle.
A cube has 12 edges, so there are 12 unit cubes with 2 painted faces.

🟦 Three painted faces (corners)

These are unit cubes at the corners of the large cube.
A cube has 8 corners, so there are 8 unit cubes with 3 painted faces.

🟦 More than three painted faces

No unit cube can have more than 3 painted faces.
Why: a unit cube has 6 faces, but when it is part of the large cube, at most 3 of its faces can be on the outer surface (at a corner).

🔁 Generalizing to n × n × n cubes

🔁 The generalization task

Problem 11 asks you to apply the same reasoning to a 2 × 2 × 2 cube, a 4 × 4 × 4 cube, and any n × n × n cube.
The key is to use the same logic: count interior, face centers, edge centers, and corners.

🔁 General formulas (implied by the excerpt)

For an n × n × n cube:

Interior (0 painted faces): (n − 2)³ unit cubes (remove one layer from each side).
Face centers (1 painted face): 6 × (n − 2)² unit cubes (each face is an n × n grid; remove the edges to get the center).
Edge centers (2 painted faces): 12 × (n − 2) unit cubes (each edge has n unit cubes; remove the 2 corners).
Corners (3 painted faces): 8 unit cubes (a cube always has 8 corners).
More than 3 painted faces: 0 unit cubes (impossible).

🔁 Example: 2 × 2 × 2 cube

Interior: (2 − 2)³ = 0 (no interior).
Face centers: 6 × (2 − 2)² = 0 (no face centers).
Edge centers: 12 × (2 − 2) = 0 (no edge centers).
Corners: 8 (all 8 unit cubes are at corners).
Don't confuse: a 2 × 2 × 2 cube has only corner cubes; every unit cube has 3 painted faces.

🔁 Justification requirement

The excerpt emphasizes "How do you know you are right?" and "Be sure to justify what you say."
You must explain the reasoning behind each count, not just give a number.

Symmetry

🧭 Overview

🧠 One-sentence thesis

Symmetry in geometric designs comes in three main types—reflection, rotational, and translational—each defined by a specific transformation that leaves the figure appearing unchanged.

📌 Key points (3–5)

Three types of symmetry: reflection (line), rotational, and translational symmetry, each based on a different geometric transformation.
Reflection symmetry: a line divides a figure into two mirror-image halves; flipping over this line leaves the figure unchanged.
Rotational symmetry: turning a figure around a center point by less than a full circle leaves it unchanged; defined by center and angle of rotation.
Translational symmetry: sliding a figure in a specific direction and distance leaves it unchanged; common in tessellations and repeating patterns.
Common confusion: not every line that cuts a figure in half is a line of symmetry—it must create mirror-image halves.

🪞 Reflection symmetry

🪞 What line symmetry means

Reflection symmetry (or line symmetry): when you can flip a figure over a line and it appears unchanged.

The line over which you flip is called a line of symmetry.
A line of symmetry divides an object into two mirror-image halves.
The transformation is called reflecting the figure.

✂️ How to identify lines of symmetry

A line of symmetry must create mirror-image halves, not just cut the figure in half.
Don't confuse: A line that divides a figure into two equal parts is not necessarily a line of symmetry—the two halves must be mirror images.
Example: Some dashed lines cut figures in half but don't create mirror images, so they are not lines of symmetry.

🔍 Finding all lines of symmetry

A figure may have one, multiple, or no lines of symmetry.
To find them, look for all possible lines that would create mirror-image halves.
The excerpt includes exercises where you complete half a design given the line of symmetry, reinforcing the mirror-image concept.

🔄 Rotational symmetry

🔄 What rotational symmetry means

Rotational symmetry: when you can turn a figure around a center point less than a full circle and the figure appears unchanged.

The transformation is called a rotation.
Two key elements define rotational symmetry:
- Center of rotation: the point around which you rotate
- Angle of rotation: the smallest angle you need to turn

⭐ Understanding the angle of rotation

The angle of rotation is the smallest angle that makes the figure look unchanged.
Example: A star has rotational symmetry of 72°, with the center of the star as the center of rotation.
The excerpt asks how you can be certain the angle is exactly 72°, prompting reasoning about the relationship between the angle and the figure's structure.

🎯 Identifying rotational symmetry

Look for figures that appear the same after turning them partway around.
The center of rotation is typically (but not always) at the geometric center of the figure.
Exercises include completing designs given a center of rotation and an angle, reinforcing understanding of how rotation works.

➡️ Translational symmetry

➡️ What translational symmetry means

Translation (or slide): moving a figure in a specific direction for a specific distance.

Translational symmetry: when you can perform a translation on a design and the figure appears unchanged.

A vector (a line segment with an arrow) describes a translation:
- The length of the segment = distance
- The direction the arrow points = direction

🧱 Where translational symmetry appears

The excerpt shows translational symmetry in multiple contexts:

Context	Example
Architecture	Brick walls (symmetry in multiple directions)
Design	Tile patterns, mosque decorations
Art	M.C. Escher's work
Traditional art	Hawaiian and Polynesian tattoo designs

🔲 Connection to tessellations

A brick wall is an example of a tessellation.
Tessellations are designs using geometric shapes with no overlaps and no gaps.
Many (but not all) tessellations have translational symmetry.
Don't confuse: Not all tessellations have translational symmetry—the Penrose tiling is "aperiodic" (no translational symmetry).

🎨 Creating translational symmetry

To show translational symmetry, sketch a vector indicating the direction and distance of the translation.
The design repeats when you slide it along this vector.
Example: In a brick wall, you can slide the pattern horizontally or vertically and it appears unchanged.

Geometry in Art and Science

🧭 Overview

🧠 One-sentence thesis

Tessellations—patterns of shapes that fit together with no gaps or overlaps—appear throughout art, architecture, and design, and can be created systematically using geometric principles like angle sums and symmetry transformations.

📌 Key points (3–5)

What a tessellation is: a design using one or more geometric shapes with no overlaps and no gaps, conceptually extendable infinitely across the plane.
Translation symmetry vs aperiodic: most tessellations have translational symmetry (a repeating pattern shifted by a vector), but some rare "aperiodic" tessellations (like Penrose tiling) do not repeat.
Which shapes tessellate: any triangle and any quadrilateral will tessellate; the reason comes from angle sums (triangles sum to 180°, quadrilaterals to 360°).
Common confusion: not all regular polygons tessellate—regular hexagons do, but regular pentagons do not, based on whether angles fit evenly around a point (360°).
Creating Escher-like art: you can modify a tessellating tile by cutting a squiggle from one edge and moving it (via translation or rotation) to another edge; the modified shape still tessellates.

🔲 What tessellations are

🔲 Definition and concept

Tessellation: a design using one or more geometric shapes with no overlaps and no gaps.

The idea is that the pattern could continue infinitely to cover the whole plane, though we can only draw a finite portion.
Think of it as tiling a floor: if you had tiles in this shape, could you cover the floor completely without gaps or overlaps?
A single shape used in a tessellation is called a tile.
Tessellations are also called tilings.

🔄 Translational symmetry in tessellations

Translation symmetry means the pattern repeats when shifted by a fixed vector (direction and distance).
The excerpt shows translation symmetry in architecture, design, art (especially M.C. Escher's work), and traditional Hawaiian and Polynesian tattoo designs.
You can identify translational symmetry by sketching a vector that shows the direction and distance of the repeat.
Example: a row of identical tiles shifted horizontally by the same amount each time.

🌀 Aperiodic tessellations

Not all tessellations have translational symmetry.
Penrose tiling is an example of an "aperiodic" tessellation—it has no translational symmetry.
The excerpt notes that aperiodic tessellations are "much harder to come up with" than symmetric ones.
Don't confuse: a tessellation can still cover the plane without repeating in a regular pattern.

🧮 Why shapes tessellate

🔺 Any triangle tessellates

Theorem: Any triangle will tessellate.

Why: The sum of the angles in a triangle is 180°.
If you make six copies of a single triangle and arrange them at a point so each angle appears twice, the total around the point is 2 × 180° = 360°.
This means the triangles fit together perfectly with no gaps and no overlaps.
You can repeat this arrangement at every vertex using more copies of the same triangle.
Example: take any triangle (scalene, isosceles, equilateral)—all will tessellate using this method.

🔶 Any quadrilateral tessellates

Theorem: Any quadrilateral will tessellate.

Why: The sum of the angles in any quadrilateral is 360°.
If you arrange copies of a quadrilateral so that all four angles meet at a single point, they total exactly 360°, fitting together perfectly.
This works for any quadrilateral, not just regular ones.
Example: squares, rectangles, trapezoids, and irregular quadrilaterals all tessellate.

⬡ Regular polygons and angle sums

Regular hexagons tessellate because their interior angles fit evenly around a point.
Regular pentagons do not tessellate because their angles do not sum to 360° when arranged around a point.
The key test: can you arrange copies of the shape so angles around a vertex total exactly 360°?

Shape	Tessellates?	Reason
Any triangle	Yes	Six copies (each angle twice) = 360°
Any quadrilateral	Yes	Four angles meet at a point = 360°
Regular hexagon	Yes	Angles fit evenly around a point
Regular pentagon	No	Angles do not sum to 360° at a vertex

🎨 Creating Escher-like tessellations

✂️ The squiggle method

The excerpt describes a step-by-step process for creating your own Escher-like drawings:

Start with a basic tile that tessellates (e.g., equilateral triangle, square, or regular hexagon).
Draw a squiggle on one side of the tile.
Cut out the squiggle and move it to another side of the shape.
You can either translate it straight across or rotate it.
Line up the cut-out along the new edge in the same place it appeared on the original edge.
Tape the squiggle into its new location—this is your modified tile.
Trace and repeat: the modified shape will still tessellate because the squiggle was moved systematically.
Add creativity: color the shape to look like something (animal, flower, etc.) and fill the page.

🔄 Why the modified shape still tessellates

The key is that the squiggle is moved by the same transformation (translation or rotation) that the tessellation uses.
When you place tiles next to each other, the cut-out squiggle fits exactly into the space where it was removed.
Example: if you translate a squiggle from the left edge to the right edge, when you place two tiles side by side, the right edge of one tile matches the left edge of the next.
Don't confuse: you cannot move the squiggle arbitrarily—it must line up in the same position on the new edge.

🎨 M.C. Escher's work

The artist M.C. Escher created many tessellations inspired by mathematics.
His work often features recognizable figures (birds, lizards, etc.) that fit together in repeating patterns.
The excerpt references his "Symmetry" gallery and provides examples of bird and lizard tessellations inspired by his style.

🏗️ Structural applications

🏗️ Triangles in architecture

The excerpt mentions that triangles are "exceptionally sturdy" because of SSS (side-side-side) congruence.
This makes them useful in architecture and design for supports in buildings, bridges, and other structures.
The activity suggests finding and photographing triangular supports in your neighborhood.

🗼 Building towers activity

The excerpt describes a hands-on activity:

Materials: toothpicks and mini marshmallows (or clay, gummy candies) as connectors.
Warm-up problem: use exactly six toothpicks to make four triangles (without breaking toothpicks).
Main challenge: build the tallest free-standing structure in ten minutes.
Reflection questions: What design choices led to taller structures? What would you do differently?
The activity emphasizes structural stability and geometric design principles.

🔍 Multiple symmetries

🔍 Combining symmetry types

The excerpt notes that designs can have multiple types of symmetry simultaneously.
Possible combinations:
- Reflection symmetry and rotational symmetry
- Reflection symmetry and translational symmetry
- Rotational symmetry and translational symmetry
Example: a tessellation might repeat via translation and also have rotational symmetry at certain points.

Introduction to Polynesian Voyaging and Hōkūle'a

Introduction

🧭 Overview

🧠 One-sentence thesis

The Polynesian Voyaging Society built and sailed Hōkūle'a using traditional navigation to prove that Pacific Islanders intentionally settled the Polynesian islands through skilled voyaging, not accidental drift.

📌 Key points (3–5)

Historical debate: historians disagreed whether Polynesian islands were settled by intentional navigation or accidental storms washing canoes ashore.
How the debate could be resolved: building a replica canoe and successfully navigating it using only traditional techniques would demonstrate feasibility.
The Hōkūle'a experiment: PVS built a double-hulled canoe replica in 1973–1975 and sailed it from Hawai'i to Tahiti (2500+ miles) without modern instruments.
Common confusion: the debate was not about whether Polynesians could sail short distances, but whether they could deliberately navigate thousands of miles across open ocean before European sailors did.
Why it matters: the successful voyage provided evidence for intentional settlement and validated Polynesian navigational knowledge and heritage.

🌊 The historical debate

🤔 Two competing theories

The excerpt describes a mid-20th-century disagreement among historians about how Polynesian islands were populated:

Theory	What historians believed	Implication
Intentional voyages	Pacific Islanders deliberately sailed and navigated, settling islands with purpose and planning	Advanced navigational skills existed thousands of years ago
Accidental drift	Canoes were caught in storms, tossed around, and accidentally washed up on distant shores	Settlement was random, not planned

⚖️ Why the debate existed

The voyaging feat seemed impossible to some historians because it happened "thousands of years ago, before European sailors would leave the sight of land and sail into the open ocean."
Don't confuse: the question was not whether Polynesians had boats, but whether they had the navigational capability to cross 2500+ miles of open ocean intentionally.

🔍 Types of evidence needed

The excerpt prompts readers to consider:

What evidence would support intentional voyages vs accidental drift?
How could the debate be settled without time travel?
Example approach: demonstrating that the voyage can be done using only ancient techniques would show it was feasible.

🛶 The Hōkūle'a project

🎯 Purpose and founding

The Polynesian Voyaging Society (PVS) was founded in 1973 for scientific inquiry into the history and heritage of Hawai'i.

The society aimed to answer two core questions:

How did Polynesians discover and settle the Hawaiian islands?
How did they navigate without instruments across ocean distances of 2500 miles or more?

🔨 Building the replica

Timeline: 1973–1975
Design: a replica of an ancient double-hulled voyaging canoe
Designer: founder Herb Kawainui Kāne
Name: Hōkūle'a, meaning "Star of Gladness"
Launch date: March 8th, 1975

🧭 The experimental voyage

Route: Hawai'i to Tahiti
Distance: 2500 miles or more
Navigator: Mau Piailug, a master navigator from Satawal island in Micronesia
Method: traditional navigation techniques with no modern instruments at all
Result: successful navigation to Tahiti

This demonstrated that the intentional-voyage theory was feasible—skilled navigators could guide a canoe across thousands of miles of open ocean using only traditional knowledge.

🧮 Mathematical dimensions

📐 Questions the voyage raised

The excerpt prompts consideration of mathematical problems the crew had to solve:

What mathematical questions can be asked about voyaging on Hōkūle'a?
What kinds of problems (especially mathematics problems) did the crew solve before setting off?

Example areas (implied by the context):

Distance and route planning across 2500+ miles
Navigation without instruments (using stars, waves, wind patterns)
Design and construction of a stable double-hulled canoe

🎓 Cross-curricular teaching connection

The excerpt notes that elementary teachers will "mostly likely be teaching all subjects," suggesting that the Hōkūle'a story integrates:

History and heritage
Scientific inquiry and experimental method
Mathematics (navigation, measurement, design)
Cultural knowledge and traditional techniques

Hōkūle`a

🧭 Overview

🧠 One-sentence thesis

The Polynesian Voyaging Society built Hōkūle`a, a replica double-hulled canoe, to experimentally prove that ancient Polynesians could have intentionally navigated and settled Pacific islands using traditional techniques, not just drifted accidentally.

📌 Key points (3–5)

The historical debate: some historians believed Polynesians intentionally sailed and settled islands with planning; others thought canoes were accidentally swept by storms to faraway shores.
The experimental approach: PVS built Hōkūle`a (1973–1975) to test whether traditional navigation without modern instruments could achieve long-distance voyaging (2500+ miles).
Successful demonstration: master navigator Mau Piailug navigated Hōkūlea from Hawaii to Tahiti using only traditional techniques, launched March 8, 1975.
Common confusion: the debate wasn't about whether Polynesians reached the islands, but how—intentional navigation vs. accidental drift.
Mathematical applications: voyaging requires geometry (scale drawings, star compass), operations, and algebraic thinking (provisioning supplies).

🌊 The historical controversy

🗺️ Two competing views

The excerpt describes a debate about how Pacific islands, including the Hawaiian islands, were settled:

View	Claim	Mechanism
Intentional voyages	Polynesians sailed deliberately around the Pacific Ocean	Relocating as necessary, settling islands with purpose and planning
Accidental drift	Navigation feat was impossible before Europeans sailed into open ocean	Canoes caught in storms, tossed and turned, eventually washed up on faraway shores

🔍 Why the debate mattered

The disagreement centered on whether ancient Polynesians possessed the navigational skill to cross ocean distances of 2500 miles or more without instruments.
Some historians believed such voyaging was impossible "thousands of years ago, before European sailors would leave the sight of land."
Don't confuse: both sides agreed Polynesians reached the islands; the question was whether they navigated intentionally or drifted accidentally.

🛶 The experimental solution

🏗️ Building Hōkūle`a

The Polynesian Voyaging Society (PVS) was founded in 1973 for scientific inquiry into the history and heritage of Hawai`i.

Research questions: How did Polynesians discover and settle these islands? How did they navigate without instruments?
Method: Build a replica of an ancient double-hulled voyaging canoe to conduct an experimental voyage from Hawai`i to Tahiti.
Design: Founder Herb Kawainui Kāne designed the canoe; it was named Hōkūle`a ("Star of Gladness").
Timeline: Built 1973–1975; launched March 8, 1975.

🧭 The test voyage

Navigator: Mau Piailug, a master navigator from the island of Satawal in Micronesia.
Navigation method: Traditional navigation techniques with no modern instruments at all.
Route: Hawai`i to Tahiti.
Implication: Successfully completing this voyage using only traditional techniques would demonstrate that intentional, planned voyaging was feasible for ancient Polynesians.

📐 Physical specifications

📏 Dimensions of Hōkūle`a

The excerpt provides specific measurements for creating scale models:

Length overall (LOA): 62 feet 4 inches (maximum length measured parallel to the waterline).
Width at beam: 17 feet 6 inches (at the widest point).
Structure: Two hulls connected by a rectangular deck.
Deck dimensions: About 40 feet long and 10 feet wide.

👥 Crew capacity

Crew size: Usually 12–16 people for a voyage.
Deck usage: During meal times, the whole crew is on the deck together.
Example calculation task: With a 40-foot by 10-foot deck (400 square feet) and 12–16 people, each person gets approximately 25–33 square feet of space when all together.

🧮 Mathematical applications in voyaging

📊 Geometry applications

The excerpt emphasizes applying mathematical knowledge to voyaging situations:

Scale drawings: Creating a floor-plan view of the canoe from above, showing hulls and deck in proper proportion.
Star compass: Using geometry to create navigation tools (mentioned as an application).
Note: Some information may be missing; navigators must make reasonable estimates based on available data.

🍽️ Provisioning calculations

The quartermaster is responsible for provisioning the canoe — loading food, water and all needed supplies, and for maintaining Hōkūle`a's inventory.

Operations and algebraic thinking are needed to plan supplies:

Variables to consider: Which leg of the trip? How long will it take? How much food and water per person per day?
Crew size: 12–16 people affects total supply needs.
Critical importance: The excerpt notes this is "not an on board job, it is critical to the safe and efficient sailing of the canoe."
Example planning: For a voyage leg lasting X days with Y crew members, calculate total food/water = (daily amount per person) × Y × X.

🔗 Cross-curricular connections

The excerpt addresses elementary teachers specifically:

Teaching approach: "How can you connect the different subjects together?"
Goal: See mathematics in other fields of study and draw out mathematical content.
Focus: Applying mathematical knowledge to new situations (voyaging context).
Don't confuse: this is not just about learning navigation history; it's about recognizing and extracting the mathematical problems embedded in real-world contexts.

🌍 The Worldwide Voyage

🗺️ Extended mission

The excerpt mentions a "Worldwide Voyage" as a later development:

Preparation: Crew members read about daily life on Hōkūle`a and watch videos about the voyage.
Planning tasks: Crew helps the quartermaster and captain with provisioning for specific legs of the trip.
Report requirements: Document which leg, estimated duration, and supply calculations with explanations.
Research component: "The rest of this section contains pointers to information that may or may not be helpful" — navigators must evaluate sources and determine relevance.

Worldwide Voyage

🧭 Overview

🧠 One-sentence thesis

The Worldwide Voyage provisioning problem requires calculating trip duration and supply needs for a chosen leg of Hōkūle`a's journey, accounting for crew size, daily consumption, weight limits, and safety margins.

📌 Key points (3–5)

Core task: Select a leg of the voyage, estimate its duration, and calculate food and water requirements for the crew.
Weight constraint: Hōkūle`a can carry about 11,000 pounds total, including crew, provisions, supplies (3,500 pounds), and personal gear.
Daily consumption rates: Each crew member needs 0.8 gallons of water per day plus three meals.
Safety margin: The quartermaster plans for 40 days' worth of supplies even when a trip is expected to take 30 days (about 33% extra).
Common confusion: Don't confuse expected trip time with provisioning time—always add buffer for delays like bad weather.

🗺️ Route planning and duration

🗺️ Selecting a voyage leg

A "leg" means a dot-to-dot route on the Worldwide Voyage map.
Different colors on the map correspond to different years of the voyage.
The excerpt provides a reference: the first trip from Hawai`i to Tahiti in 1976 took 34 days total.
Students must pick one leg and calculate its distance (the excerpt suggests using a distance calculation tool).

⏱️ Estimating trip duration

Use the historical reference (Hawai`i to Tahiti = 34 days) as a baseline.
Calculate the distance of the chosen leg in nautical miles.
Compare the chosen leg's distance to the Hawai`i–Tahiti distance to estimate duration.
Example: If the chosen leg is twice as long as the Hawai`i–Tahiti route, it might take roughly twice as long (adjusting for conditions).

🍽️ Provisioning calculations

🍽️ Food and water requirements

Crew size: The excerpt mentions "about 16 people" during meal times when the whole crew is on deck together.
Water: Each crew member gets 0.8 gallons per day.
Meals: Three meals per day per crew member.
Example: For a 16-person crew on a 30-day expected trip:
- Water = 16 people × 0.8 gallons/day × 40 days (with buffer) = 512 gallons
- Meals = 16 people × 3 meals/day × 40 days = 1,920 meals total

🛡️ Safety buffer

For a trip expected to take 30 days, the quartermaster plans for 40 days' worth of supplies, in case of bad weather and other delays.

This is a 33% safety margin (10 extra days for a 30-day trip).
The buffer accounts for unpredictable conditions like bad weather.
Don't confuse: the 40 days is not the expected trip time; it's the provisioning target.

⚖️ Weight constraints

⚖️ Total capacity and allocation

Category	Weight (pounds)	Notes
Total capacity	11,000	Maximum Hōkūle`a can carry
Supplies	3,500	Sails, cooking equipment, safety equipment, communications equipment, etc.
Remaining	7,500	Available for crew, provisions, and personal gear

🧮 Calculating available weight for provisions

Start with total capacity: 11,000 pounds.
Subtract fixed supplies: 11,000 − 3,500 = 7,500 pounds remaining.
From the 7,500 pounds, subtract crew weight and personal gear to find how much is available for food and water.
Example: If crew + personal gear = 3,000 pounds, then provisions can weigh up to 4,500 pounds.

🧭 Navigation context

🧭 Course strategy

A voyage undertaken using modern wayfinding has three components: Design a course strategy, which includes a reference course for reaching the vicinity of one's destination, hopefully upwind, so that the canoe can sail downwind to the destination rather than having to tack into the wind to get there.

The preferred strategy is to approach the destination from upwind.
Sailing downwind to the destination is easier than tacking into the wind.
Tacking: sailing back and forth as closely as possible into the wind to make progress against the wind; it is very arduous.
This navigation detail affects trip duration—tacking takes longer and is more difficult.

📝 Report requirements

📝 What the quartermaster needs

The preliminary report must document:

Which leg of the trip you are focused on (specify the route segment from the map).
How long that leg will take, with an explanation of the calculation method.
How much food and water you will need, with an explanation of the calculation method.

📝 Level of detail

Include enough detail so the quartermaster can understand your reasoning.
Show your work: distance calculations, duration estimates, consumption rates, and weight checks.
The excerpt emphasizes that students should "do the relevant research" and explain their conclusions clearly.

Navigation

🧭 Overview

🧠 One-sentence thesis

Polynesian wayfinding uses a star compass with 32 directional houses to track position and course across thousands of miles of open ocean by memorizing star positions, wind directions, and ocean swells.

📌 Key points (3–5)

Three components of modern wayfinding: design a reference course (preferably upwind of destination), hold to that course while tracking position and distance, and find land when entering the target area.
Day vs night navigation: by day, navigators use ocean knowledge (winds, currents, clouds, animal behavior); by night, they use stars and the star compass.
The star compass structure: 32 equidistant points ("houses") around a circle with the canoe at center, organized with cardinal directions, quadrant names, and symmetrical house labels.
How stars and swells move on the compass: stars rise in the east and set in the west, staying in their hemisphere and house; winds and swells move straight across the compass (north-south or vice versa).
Common confusion: the star compass is not a physical tool carried separately—it is memorized and painted permanently on the canoe's rails at the navigator's seat.

🌊 The three components of wayfinding

🗺️ Course strategy

A reference course for reaching the vicinity of one's destination, hopefully upwind, so that the canoe can sail downwind to the destination rather than having to tack into the wind.

Why upwind matters: tacking (sailing back and forth into the wind) is very arduous and time-consuming, especially at the end of a long voyage.
The goal is to avoid tacking by positioning the canoe upwind of the target.

🧭 Holding the course and tracking position

During the voyage, navigators must:

Hold as closely as possible to the reference course.
Keep track of three things:
1. Distance and direction traveled
2. Position north/south and east/west of the reference course
3. Distance and direction to the destination

Don't confuse: "holding the course" does not mean sailing in a perfectly straight line—it means continuously adjusting based on tracking data to stay near the reference course.

🎯 Finding land (the target screen)

Called "the box" or target screen.
After entering the vicinity of the destination, navigators use specific techniques to locate land.
Example: Rapa Nui is so small and low that you must be within 30 miles to see it; a one-degree error would put the canoe 60 miles off course.

🌅 Day and night navigation methods

☀️ Navigation by day

Navigators use deep knowledge of the oceans:

Wind direction: which way do the winds blow?
Ocean currents: which way do prevailing currents move?
Environmental clues: clouds in the sky, flotsam in the water, and animal behaviors give insight into where land might be and the canoe's position relative to it.

⭐ Navigation by night

Navigators use the stars.
Stars move through the night sky in predictable patterns, rising in the east and setting in the west.
The time-lapse picture shows stars appearing to rotate around a fixed point, moving along arcs from horizon to horizon.

🧭 The star compass structure

🏠 What the star compass represents

A fundamental tool for navigators: the object in the center represents the canoe, and the shells (or points) along the outside represent directional points.

The idea: imagine stars rising from the horizon in the east, traveling through the night sky, and setting past the horizon in the west, as if on a sphere surrounding the Earth (the celestial sphere).
Nainoa Thompson's version has 32 equidistant points around a circle.
The arcs between these points are called "houses."

🧭 Cardinal directions and quadrants

Direction	Hawaiian name
North	`Ākau
South	Hema
East	Hikina
West	Komohana

Quadrant	Hawaiian name
Northeast	Ko`olau
Southeast	Malani
Southwest	Kona
Northwest	Ho`olua

🏘️ The house names and symmetry

Moving clockwise from `Ākau (North) to Hikina (East), there are seven houses in order:

Haka: "empty," describing the skies in this house
Nā Leo: "the voices" of the stars speaking to the navigator
Nālani: "the heavens"
Manu: "bird," the Polynesian metaphor for a canoe
Noio: the Hawaiian tern (a bird)
`Āina: "land"
Lā: "sun," which stays in this house most of the year

Symmetry rules:

Vertical line of symmetry: the same seven houses appear in the same order moving counterclockwise from `Ākau to Komohana (West).
Horizontal line of symmetry: use this to label houses from Hema (South) to Hikina and from Hema to Komohana.

🌟 How the star compass is used

🌊 Winds and ocean swells

Winds and swells move directly across the star compass from north to south or vice versa.
They cross through the center of the circle.
Example: if swells come from Āina Koolau, they head toward `Āina Kona (trace a straight line across the compass).
Example: if wind comes from Nālani Malani, it heads toward Nālani Ho`olua.

⭐ Star movement

Stars stay in their houses and in their hemisphere.
They do not move across the center of the circle.
Like the sun, they rise in the east and set in the west.
Example: Aā (Sirius) rises in Lā Malanai and sets in Lā Kona.
Example: Hōkūlea (the star) rises in Āina Koolau and sets in Āina Ho`olua.

Don't confuse: winds/swells cross the center; stars do not.

🧠 Memorization and use

A navigator memorizes the houses of over 200 stars.
At sunrise and sunset (when the sun or stars are rising), the navigator uses the star compass to memorize:
- Which way the wind is moving
- Which way the currents are moving
The navigator then uses that information throughout the day or night to ensure the canoe stays on course.

🎨 Drawing and positioning the star compass

✏️ Creating a precise star compass

The rough sketch is fine for teaching on land, but real navigation demands extreme precision:

Why precision matters: on the final leg to Rapa Nui, even a one-degree error would have led to being 60 miles off course; you must be within 30 miles to see the island.
What is needed:
1. A perfect circle (as perfect as possible)
2. Thirty-two points exactly evenly spaced around the circle
3. Proper labeling with all house names

Tools: the excerpt asks what tools can be used and what tools ancient Polynesian navigators would have had.

🛶 Painting the compass on the canoe

A paper star compass is not useful on a canoe (it can get lost, damaged, or soaking wet).
Solution: paint it permanently on the rails of the canoe.
Navigator's seat (kilo): located in the rear (aft) of the canoe; there is one seat on either side of the deck (port and starboard).
Positioning:
- The kilo (seat) becomes the center of the star compass.
- The navigator looks to the right and sees the star compass markings on the rails.
- Because the canoe is not circular and the navigator does not sit at the geometric center, careful planning is needed to place markings in the right positions.
- The process is repeated for the seat on the opposite side.