Writing to Learn Math: People learn math by playing with ideas. A math journal can be like a science lab book.

Do you want your children to develop the ability to reason creatively and figure out things on their own?

Help kids practice slowing down and taking the time to fully comprehend a math topic or problem-solving situation with these classic tools of learning: Notice. Wonder. Create.

Notice: Look carefully at the details of the numbers, shapes, or patterns you see. What are their attributes? How do they relate to each other? Also notice the details of your own mathematical thinking. How do you respond to a tough problem? Which responses are most helpful? Where did you get confused, or what makes you feel discouraged?

Wonder: Ask the journalist’s questions: who, what, where, when, why, and how? Who might need to know about this topic? Where might we see it in the real world? When would things happen this way? What other way might they happen? Why? What if we changed the situation? How might we change it? What would happen then? How might we figure it out?

Create: Create a description, summary, or explanation of what you learned. Make your own related math puzzle, problem, art, poetry, story, game, etc. Or create something totally unrelated, whatever idea may have sparked in your mind.

Math journaling may seem to focus on this third tool, creation. But even with artistic design prompts, we need the first two tools because they lay a solid groundwork to support the child’s imagination.

How To Use an Experiment Prompt

Many people know it’s important for students to do hands-on experiments in science. But did you know the same is true for mathematics? People learn math by playing with ideas.

A math journal can be like a science lab book. Not the pre-digested, fill-in-the-blank lab books that some curricula provide. But the real lab books scientists write to keep track of their data, and what they’ve tried so far, and what went wrong, and what finally worked. Children may draw pictures of their investigations, write explanations, or play with equations.

When students find a solution to their prompt question, that’s when the fun begins. The point of a math experiment is to change something in the problem and explore how that changes the answer. For older students, the bonus challenge of a math puzzle is to generalize the solution: Can they discover a method that works for any starting conditions?

Besides these prompts, let students pose research topics of their own. What do they wonder? What questions can they ask? Can they expand one of the previous activity prompts into a more general investigation?

Any math topic or prompt offers an overwhelming variety of paths for exploration. Pick your rabbit hole, dive in, and discover the crazy Wonderland of mathematics.

Be patient with math experiments. Allow plenty of time for students to be scientists, doing their own research. Work for a while and then let the investigation rest. Come back to it later to see if they can discover anything new.

Journaling Prompt 41: Linear Crossings

Draw straight lines. Each line must cross all the others. How many lines can you draw?

Try the experiment again. Can you get more lines crossing this time?

What do you think is the maximum number of straight lines that all cross each other?

This week, let’s explore how students learn and retain new information.

Because if we want our children to be able use what we teach them, the first step is to help them remember it.

How Do Students Learn and Retain New Information?

Principles from The Science of Learning:

• To learn, students must transfer information from working memory (where it is consciously processed) to long-term memory (where it can be stored and later retrieved). For long-term knowledge to be useful, students have to think about meaning.

• Practice is essential to long-term learning, but not all practice is equivalent.

• Effective feedback is essential for learning.

Tips for Teachers

• Assign tasks that focus on meaning, such as asking students to explain or organize material.

• Focus lessons on sense-making, while making sure that students’ background knowledge is solid.

• Use mnemonics for hard-to-remember details, not for foundational concepts. Mnemonics can’t replace reasoning.

• Practice should require thinking, not simple recall. Focus on using the material, not just repeating it.

• Feedback should not focus on grading performance, but encourage growth and offer suggestions for improvement.

My Comments

We connect information (facts, data) to the ideas we have made sense of. We retain the information that we use, that we focus our thinking on, and that reinforces those ideas.

I appreciate the caution that practice needs to focus on sense-making, not simple recall.

Games are great for practice because they have low stakes, no pressure to perform, and they offer immediate feedback. But we need the right kind of games, the ones that rely on using meaning, not just answer-getting.

Knowledge becomes solid in our minds when we use it to do something else, when our focus is not on the answer to this particular problem but on a goal that this problem will help us reach. The best practice is not an end in itself, but a tool we use to achieve our strategic goal.

This reminds me of the common math-teacher observation that students never really learn the last math course they take. Rather, they learn whatever they studied previously as they put that knowledge to use in the new course.

But we can activate the power of reasoning practice for our students by avoiding the “insufferably dull” math-fact-recall games and instead playing games that require strategic thinking.

Try This Today: The Substitution Game

The Substitution Game features low-floor, high-ceiling cooperative play that works with any age (or with a mixed-age group). It’s a great way to build skills through practice that involves creative thinking, using math and enjoying it.

The first player writes a simple equation at the top of the paper, such as “1 + 1 = 2.” Then all players take turns complexifying this equation.

On your turn, copy the equation to the next line, replacing one number with an equivalent expression. For instance, replace the number 2 with:

5 − 3
or 50 ÷ 25
or (1/3) × 6

… or any other calculation that equals two.

Use parentheses or brackets as needed to make your expression perfectly clear. For example, I put parentheses around my fraction above so people can tell I didn’t mean “1/(3×6),” which is definitely not a substitute for two.

If you have colored pencils or markers, circle the number you plan to substitute. Then write the substitution below in the same color. Finally, fill out the rest of the equation using a neutral-colored pencil or marker.

The other players should check to make sure they agree with your math.

After you change part of the equation, it is no longer available for anyone else to use. If you substitute “5 − 3” for the 2, the other players cannot replace your creation with their own version of that number. But they can alter individual numbers within your creation. So the next player may decide to substitute for the 5, writing a new expression in its place.

For young players, an older child or an adult may take dictation. This lets the young one focus on thinking about the numbers without struggling to write an ever-growing equation.

Continue until the paper is full, or until the equation looks satisfyingly complex, or until you run out of time.

Targeted Practice

As students grow comfortable with the basic game, pose additional challenges. Perhaps each substitution must use multiplication, or a fraction, or contain a specific number.

For example, if your challenge is to play division, you could replace the number 3 in an existing equation with:

15 ÷ 5
or (70 ÷ 7) − 7
or (1/4) × 300 ÷ 25

… or any other calculation that uses division. Remember to use parentheses or brackets as needed to make your expression perfectly clear.

If you’re playing with a mixed-ability group, each player may have a different challenge. An elementary student may be practicing the times-7 number family while a teenager practices with trigonometric functions.

You can also play the Substitution Game as solitaire, just for the fun of complexifying equations.

How crazy can you make the math?

This game features the Cuisenaire rods, but you may play with any math manipulative based on length: Montessori bead chains, Mortensen or Math-U-See blocks, etc. Or draw pictures on graph paper.

Many parents remember struggling to learn math. We hope to provide a better experience for our children.

And one of the best ways for children to enjoy learning is through hands-on play.

The Partitions Game

Math Concepts: partitions, part/whole, addition, multiplication, algebra.

Players: two or more.

Equipment: Cuisenaire rods (or other math manipulatives) or graph paper, pencil and paper (optional).

Set-Up

Dump the rods on the table, where all players can reach them. Clear a space in the middle for building your game.

Players must agree in advance whether the same rods in a different order will count as a new partition.

How to Play

The first player chooses any rod (yellow or longer) and sets it in the middle of the table. Or if using graph paper, the first player outlines a row of squares (five or longer).

The second player adds a new row of two or more rods that make the same length. Or draws a new same-length row of squares, split into two or more sections.

Players continue to add rows in turn. Each row must be different from all those already played.

Whoever plays the last new partition, leaving the other players stumped, wins that round and gets to choose a new starting rod.

Variation

Depending on the age of your child, you may want to keep “score” by writing an expression to represent each row. You may use numbers, but I like to give children a head-start on understanding algebra by using the color names or initials.

For example, you might write the game above as:

• orange = 2 × yellow = green + black, etc.
• o = 2y = g + k, etc.
  [We use k for black, t for tan (brown), and b for blue.]
• 10 = 2 × 5 = 3 + 7, etc.
  [Assuming you treat the white rod as one unit.]

Writing everything out does not increase the mathematical content of the game. The written symbols are not mathematics, any more than the lines and dots written on a staff are music. Music only happens when we play the notes, just as math only happens when we think the ideas.

Your children will be thinking like a mathematician, even if they write nothing down. But writing helps them grow comfortable with math notation (the way we write math on paper), which will be useful when they want to communicate their own mathematical thoughts.

Writing to Learn Math: When students create their own math, they forge a personal connection to mathematical concepts and relationships. And it’s fun!

Do you want your children to develop the ability to reason creatively and figure out things on their own?

Help kids practice slowing down and taking the time to fully comprehend a math topic or problem-solving situation with these classic tools of learning: Notice. Wonder. Create.

Notice: Look carefully at the details of the numbers, shapes, or patterns you see. What are their attributes? How do they relate to each other? Also notice the details of your own mathematical thinking. How do you respond to a tough problem? Which responses are most helpful? Where did you get confused, or what makes you feel discouraged?

Wonder: Ask the journalist’s questions: who, what, where, when, why, and how? Who might need to know about this topic? Where might we see it in the real world? When would things happen this way? What other way might they happen? Why? What if we changed the situation? How might we change it? What would happen then? How might we figure it out?

Create: Create a description, summary, or explanation of what you learned. Make your own related math puzzle, problem, art, poetry, story, game, etc. Or create something totally unrelated, whatever idea may have sparked in your mind.

Math journaling may seem to focus on this third tool, creation. But even with artistic design prompts, we need the first two tools because they lay a solid groundwork to support the child’s imagination.

How To Use a Create-Your-Own-Math Prompt

When students create their own math, they forge a personal connection to mathematical concepts and relationships. And it’s fun!

Children might make up a math game, write a story or poem, draw a comic, or pose a problem. Create math art, think up a challenging question, or write a puzzle. Since earlier chapters focused on writing and math art, most of these prompts involve creating puzzles or problems.

The “Story Problem Challenge” is one of my favorite math club activities. My students invent their own word problems in any style they like. They don’t have to know how to solve the problems they create. We read the stories aloud, and everyone works together to find the solutions.

For puzzles where the child already knows the answers (for example, “Two Truths and a Lie”), let them trade with a friend. Can they each solve the other’s puzzle? Can they stump each other? Or save the child’s work and let them come back to it another day, after they’ve forgotten the answers.

And when students create something they’re proud of, let them share it with the world. Visit the Student Math Makers Gallery at tabletopacademy.net/math-makers to share your children’s math creations.

Journaling Prompt 49: Math Quilt

Draw a rectangular grid of squares on your page, like a quilt made of large, square pieces. For each piece of the quilt, color a different fraction of the square.

Or color the same fraction in every square, but make each one look different. For example, one square might represent 1/2, but drawn creatively (not just a line down the middle).

Cognitive science focuses on the “learning” part of life, studying how people acquire and retain information and then incorporate it into their understanding of the world. Earlier this year, Deans for Impact published a new edition of their white paper, “The Science of Learning,” summarizing cognitive science research and its implications for teachers.

Download a copy and read it for yourself:

• The Science of Learning

In this series, we’ll go through Deans for Impact’s six key questions and learn how to apply the principles of science in teaching our children. First, let’s explore how students understand new ideas.

Because if we want our children’s learning to stick, our focus must always be on helping them make sense of things.

How Do Students Understand New Ideas?

Principles from The Science of Learning:

• Students learn by connecting new knowledge to what they already know.

• Students have limited working memory capacity that can be overwhelmed if given too much information at once or if tasks are cognitively too demanding.

• Cognitive development does not progress in a fixed sequence at age-related stages. Understanding does not happen all at once or just one time; the mastery of new concepts happens in fits and starts.

Tips for Teachers

• Encourage students to remember what they’ve already learned, and help them connect new knowledge to old.

• Use stories, activities, and visuals that clearly relate to the concept being studied. Avoid distracting clutter that is merely decorative.

• When working example problems, discuss the reasoning behind each step and help students make connections.

• Don’t limit students to “grade level” ideas, but wonder about big concepts.

• Don’t assume that learning something once is enough. Repetition and reinforcement make learning solid.

My Comments

We make sense of new ideas by connecting them to things we already know, by recognizing relationships.

I appreciate the reminder of how important background knowledge is to new learning. It’s easy for me to jump straight into the new material, thinking that’s the interesting part of a lesson, but I need to let my students orient themselves by remembering what we’ve done before.

Just as I begin a history read-aloud with, “Tell me what happened last time,” I can begin a math lesson with, “Remind me of what you know about division.” My goal is not to make connections for my students, but to open their eyes so they can see the relationships for themselves.

Try This Today: Number Webs

For each post in this Science of Learning series, I’ll share an activity idea that gives you and your children a way to put it into practice. Today, let’s play with relationships and see how many connections we can find between math ideas by creating Number Webs.

For best results, get a large whiteboard or a sheet of poster-size paper.

Write a number or math expression in the center of your space. Then take turns writing equivalent expressions, other ways to say the same thing. Draw lines to show how these new expressions branch out from the original to form families of related math.

For example, if you begin with the number five, branches might include:

• 1+4, 2+3, etc.
• 10÷2, 15÷3, and so on
• 20/4, 50/10, etc.
• What else can you think of?

As you build your number web, each transformation will spark new ideas. You can create branches off the branches. For example, you could shoot off from 15÷3 with an expression like (5+5+5)÷3, or (7+8)÷(9−6).

For each bit of math you add, be prepared to explain why it fits in the family you chose, or how it’s different from the others and needs to start a new branch. Often an expression could fit in several places, and players may argue for their own point of view. But the player whose turn it is has the final say where to place his or her creation.

Instead of (or in addition to) writing in a new expression, players may draw in a line that connects it to an expression on a different branch. Numbers relate to each other in various ways, and there is almost always more than one way to think about linking them. For example, you might join the division expression 15÷3 with the fraction 15/3 and the multiplication (⅓)×15.

Don’t worry about keeping your Number Web neat and systematic. Explore and let the connections get a bit wild. There’s always another relationship to discover.

This game helps students master multiplying and dividing fractions.

Many parents remember struggling to learn math. We hope to provide a better experience for our children. And one of the best ways for children to enjoy learning is through hands-on play.

So what are you waiting for? Let’s play some math!

Fraction Product Game

Math Concepts: multiplying fractions, equivalent fractions.
Players: two, or two teams.
Equipment: printed game board, colored markers or a set of matching tokens for each player, two glass gemstones or other small tokens to mark the factors.

The free 44-page PDF Multiplication & Fraction Printables file includes traditional and spiral forms for each of the Product Games.

How to Play

The first player places a stone on any one of the factors at the bottom of the board. The second player places the other stone on a factor — the same or different — and then marks the product of those two numbers by coloring the square or placing a token.

On each succeeding turn, a player moves just one stone to a new number and then marks the product of those two factors. If both players agree that all possible moves have already been colored in, the player whose turn it is may make a fresh start by moving both stones.

Whichever player marks four (or more) squares in a row — horizontally, vertically, or diagonally — wins the game. The squares must touch each other at edges or corners, with no gaps. If neither player can make four in a row, then the player who has the most sets of three in a row wins.

History

Have you figured out yet how much I love John Golden’s blog? About this game, Golden writes:

“I launched the game by playing me vs. the class. Re-emphasizing that you only get to change one factor at a time, the goal is to get four in a row, and the new idea that there are equivalent fractions. If you multiply and get 6/12, you can cover 1/2, and vice versa.

“Then the students played pair vs. pair. At the end we summarized by discussing what they noticed about the game, and what they thought made for a good strategy.

“I did point out to them that someone would tell them fraction division was hard, but they’ve already done it when they’re figuring what to multiply 3/4 by to get 6/12.”

Sue VanHattum put together a delightful collection of mathematical fiction in honor of Math Storytelling Day:

• Playful Math Blog Carnival 186: Math Stories

Every Playful Math Blog Carnival brings you a great new collection of puzzles, math conversations, teaching tips, and all sorts of mathy fun. It’s like a free online magazine of mathematical adventures, helpful and inspiring no matter when you read them.

What are you waiting for? Come join the fun!

Click Here to Read the Carnival Blog

CREDITS: Feature photo (top) copyright © David Butler.

Writing to Learn Math: What did the author mean? Put the thought in your own words. Do you agree or disagree? Why?

Do you want your children to develop the ability to reason creatively and figure out things on their own?

Help kids practice slowing down and taking the time to fully comprehend a math topic or problem-solving situation with these classic tools of learning: Notice. Wonder. Create.

Notice: Look carefully at the details of the numbers, shapes, or patterns you see. What are their attributes? How do they relate to each other? Also notice the details of your own mathematical thinking. How do you respond to a tough problem? Which responses are most helpful? Where did you get confused, or what makes you feel discouraged?

Wonder: Ask the journalist’s questions: who, what, where, when, why, and how? Who might need to know about this topic? Where might we see it in the real world? When would things happen this way? What other way might they happen? Why? What if we changed the situation? How might we change it? What would happen then? How might we figure it out?

Create: Create a description, summary, or explanation of what you learned. Make your own related math puzzle, problem, art, poetry, story, game, etc. Or create something totally unrelated, whatever idea may have sparked in your mind.

Math journaling may seem to focus on this third tool, creation. But even with artistic design prompts, we need the first two tools because they lay a solid groundwork to support the child’s imagination.

How To Use a Quotation Prompt

Let students choose how they want to react to the quotation. Or offer one of the following questions:

• What did the author mean? Put the thought in your own words.
• Do you agree or disagree? If you agree, can you think of someone who would disagree? Why?
• Is this quote a general principle, or only for specific situations? Describe a time when it might apply, or when it might not.
• Tell a time in your life when you lived up to the quotation — or when you wish you had.
• How does the quote relate to math, science, history, or another subject?

Short exercises are great writing practice. But occasionally you’ll want to assign deeper essay topics, such as:

• Look up the author’s name online. Who are/were they, and why do people care what they said?
• Quotes are often misattributed. Did the author really say this?
• What have others said about the same topic? Search out a variety of quotes related to this one. How are they similar? How are they different?
• Does thinking about the quotation make you want to change anything, in yourself or in the world? How could you put that idea into action?

Quotation from Eleanor Roosevelt

Yesterday is history. Tomorrow is a mystery. Today is a gift. That’s why we call it ‘The Present’.

—Eleanor Roosevelt

“Puzzles are made of the things that the mathematician, no less than the child, plays with, and dreams and wonders about, for they are made of things and circumstances of the world he [or she] lives in.”

Long ages ago, one of the greatest mathematicians of all time played with a children’s puzzle.

The puzzle was called “The Ivory Challenge”—or something like that, translations vary. Ostomachion in Greek. Fourteen ivory polygons (flat shapes with straight sides) were supposed to fit into a square box.

Archimedes solved that puzzle. But like any good mathematician, he didn’t stop there. The solution to any puzzle is only the beginning, a doorway to further adventures.

So Archimedes asked himself one of the big questions of mathematics: “How many ways?”

The Ostomachion is a rather complex puzzle, and our students may struggle to figure out where to start. One of the classic tools in any mathematician’s toolbox is to make it simpler. So let’s begin our own investigation with a simpler challenge:

• How many ways can you bisect a square?

Divide the Square

Twelve is a wonderful number for mathematical play because of its abundant factors. There are so many ways to split twelve up into equal pieces.

And when we want to divide a square, a 12×12 grid offers plenty of options.

How can you divide it into halves? Quarters? Thirds? Can you make pieces of equal area that are not the same shape?

What designs can you make? What math do you notice in your designs.

This download includes twelve 12×12 squares for your investigation.

Ostomachion

The puzzle Archimedes played with had 14 pieces: 11 triangles, 2 scalene quadrilaterals, and one pentagon. E) so that the area of each piece is a rational fraction of the area of the square.

Download a PDF version. Print the ostomachion on cardstock and cut out the pieces.

• Can you fit the square back together?

(There are 536 ways to make an ostomachion square.)

• What other shapes can you make?

This game builds mental math skills in young students, and is fun for all ages.

Many parents remember struggling to learn math. We hope to provide a better experience for our children. And one of the best ways for children to enjoy learning is through hands-on play.

So what are you waiting for? Let’s play some math!

Shut the Box (Cards)

Math Concepts: addition, number bonds up to nine.
Players: two to four.
Equipment: one deck of playing cards, two six-sided dice.

How to Play

Each player lays out the ace through ten of one suit, face up.

On your turn, roll the dice and add the numbers together. Turn face down one or more of your cards whose numbers add up to make that sum.

For instance, if you roll a six and a four, you could choose the 8 and 2, or the 6 and 4, or the 7 and 2 and 1, or any other combination that makes a sum of ten. If you cannot turn down the full amount of your roll, you lose that turn.

Pass the dice to the next player.

If all the higher numbers are already covered, a player may choose to roll only one die.

The first player to turn down all their cards wins the game.

Variations

Traditionally, each player takes a single, long turn. You keep playing until you roll a sum you can’t make, and then you add up the uncovered numbers to make your score. Pass the cards and dice to the next player. Whoever gets the lowest score wins the game.

Older students can add the jack (=11) and queen (=12) for a greater challenge.

Children may also enjoy playing Shut the Box as a solitaire game.

Plus or Minus: Players may choose to turn down either the sum or difference of the numbers on their dice. Subtraction offers more options near the end of the game, when there are only a few numbers left.

History

Shut the Box is a traditional English pub game, also known as Canoga, which may have been played as early as the twelfth century in Normandy. The “box” is a wooden tray with a row of numbers one to nine along the top length, each with a cover that can either slide or swing to hide the number. As a gambling game, each player would ante an agreed amount into a pool, which would be awarded to the winner.

Writing to Learn Math: Research prompts help students view math as a human endeavor.

Do you want your children to develop the ability to reason creatively and figure out things on their own?

Help kids practice slowing down and taking the time to fully comprehend a math topic or problem-solving situation with these classic tools of learning: Notice. Wonder. Create.

Notice: Look carefully at the details of the numbers, shapes, or patterns you see. What are their attributes? How do they relate to each other? Also notice the details of your own mathematical thinking. How do you respond to a tough problem? Which responses are most helpful? Where did you get confused, or what makes you feel discouraged?

Wonder: Ask the journalist’s questions: who, what, where, when, why, and how? Who might need to know about this topic? Where might we see it in the real world? When would things happen this way? What other way might they happen? Why? What if we changed the situation? How might we change it? What would happen then? How might we figure it out?

Create: Create a description, summary, or explanation of what you learned. Make your own related math puzzle, problem, art, poetry, story, game, etc. Or create something totally unrelated, whatever idea may have sparked in your mind.

Math journaling may seem to focus on this third tool, creation. But even with artistic design prompts, we need the first two tools because they lay a solid groundwork to support the child’s imagination.

How To Use a Research Prompt

Where did math come from? Who thought up the rules? How are the ideas and methods we study used in actual life?

Research prompts help students view math as a human endeavor, something that grew bit by bit as people in many countries across many centuries struggled to understand their world.

While most research journaling is informal, older students may wish to venture into writing more formal reports. Here they can practice essay skills like formulating a thesis statement (a concise summary of the student’s main idea) and organizing facts and information to prove their point.

Be sure to allow plenty of time for students to respond to research-based journaling prompts.

Journaling Prompt 82: Mini-Biography 2

Write about a female mathematician who is NOT Ada Lovelace.

Look at mathigon.org/timeline or arbitrarilyclose.com/mathematician-project or mathshistory.st-andrews.ac.uk/biographies for ideas.

“The value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would-be solver.”

—I. N. Herstein

Whenever I give a problem or puzzle in an Alexandria Jones story, I’ll try to post the answer soon afterward.

But don’t peek!

In the old days of my snail mail newsletters, the answers always came in the next issue, so students had two months to let the puzzle ruminate without temptation.

After all, if you only read the answer I give, you miss out on the fun of solving the puzzle. Follow the Math Adventurer’s Rule:

• Figure it out for yourself!

Give the problem a good workout on your own. See what you can notice about the puzzle. Wonder about how it connects to the larger world of mathematics.

Play with the ideas and find out where they may lead.

Then check the answer just to prove you got it right, and maybe to see whether we did it the same way. There’s always more than one way to approach any math problem, and each method offers its own insights.

Surveying the Egyptian Farm

Did you try to find the area of the Egyptian farm?

Divide the property into rectangles and right triangles, calculate the area of each piece, and then add the areas all together to find the total area of the farm. Your answer will depend on what size the map came out when you printed it, because each printer has its own idiosyncrasies.

Also, there will be round-off error and slight differences in the way people measure—did you read the inside of the dark line, or the outside, or try to estimate the center?—so don’t expect your answer to match mine exactly. But the order of magnitude should be the same.

My map had the long line at the bottom almost exactly 7 cm long. I was able to divide the farm into one large rectangle and five smaller right triangles, and I calculated the entire area as about 45 square cm.

Since one cm stands for 60 cubits, each square cm represents 60 × 60 = 3600 square cubits.

So the total area of the farm on my print-out was approximately 162,000 square cubits.

Proportional Reasoning

But your map might have come out a very different size, depending on your printer settings. Then how can we compare answers?

You have to understand ratios and proportions.

A ratio is a fraction that compares one number to another.

A direct proportion says one ratio is equal to another ratio.

The key in a proportion is to make sure that the numbers match each other in type. For example, it would be nonsense to compare dollars/gallon to cups/quart.

But if you are buying gas, the dollars/gallon should be equal whether you buy a lot or a little. So we could set up a proportion like this:

Proportional Measurements

In the case of our farm puzzle, we want to compare the ratio of our measurements:

But what measurements shall we compare?

Area is directly proportional to length squared. That means that for any similar shapes—like your map and mine—we can make a ratio comparing the overall area to the square of one side’s length measurement. As long as we both measure the same side, your ratio should be equal to mine.

And the coolest thing about proportions is that the units don’t matter, as long as we are consistent. We can compare cm to cubits, as long as we do the same thing in each ratio.

In the case of the farm plot, the following proportion should be true despite our different printer settings:

So to check whether our answers agree:

• Measure (in cm) the long line at the bottom of your map. Call that length your baseline, b.
• Multiply that number times itself to find b^2.
• Calculate your answer (in square cubits) for the area of the farm. Call it A.
• Calculate your ratio: A ÷ b^2.

That should be the same as my ratio, within a reasonable margin for differences in measurement or rounding off numbers.

A Research Project

Would you like to know how big the Egyptian farm is in “real life”?

There is enough information given in the problem. You will need to use conversion factors to switch between a hodgepodge of units: cm, cubits, feet, and square kilometers or acres (your choice).

If you don’t remember how conversion factors work, check out my blog post “How Old Are You in Nanoseconds?”

I will give you this hint: Our hypothetical Egyptian farmer was not a wealthy man.

Playing with Ratios

If you’d like to have some more fun with ratio math, try this puzzle from my Let’s Play Math blog:

• Rate Puzzle: How Fast Does She Read?

(Rates are just ratios where the two numbers are different types, like dollars/gallon or for my voracious daughter, books/day.)

This game gives students a fun reason to think about calculating with exponents.

Many parents remember struggling to learn math. We hope to provide a better experience for our children. And one of the best ways for children to enjoy learning is through hands-on play.

So what are you waiting for? Let’s play some math!

Power Up

Math Concepts: powers (exponents).
Players: only two.
Equipment: one printed character sheet for each player, two six-sided dice, pencils or markers. Calculator optional.

The FREE 68-page printable (pdf) Prealgebra & Geometry Printables file features hundred charts, coordinate grids, assorted graph paper, and all the game boards for the Math You Can Play: Prealgebra & Geometry book.

Set-Up

Students create their own superheroes or supervillains using character sheets from the printables file or on regular notebook paper. Give your character a name and three superpowers, magical abilities, or technological gadgets. Describe the character’s powers:

• Greater Ability = …
• Lesser Ability = …
• Random Ability = …

Briefly explain your character’s origin story and draw a picture (optional).

How to Play

Each battle consists of three rounds, one for each of the superpowers. In the first round, players may choose which ability they want to use. For the second round, choose either unused power. Finally, the players call upon their remaining abilities.

Each player rolls one or two dice, as needed for their chosen superpower:

• Greater Ability: Roll one die. Use 4 as the base, your roll as the exponent.
• Lesser Ability: Roll one die. Use 3 as the base, your roll as the exponent.
• Random Ability: Roll two dice. Choose which is the base and which is the exponent.

Calculate how much damage your attack inflicts upon your opponent. Whoever does the most damage wins that round.

The player who wins two of the three rounds wins the battle. If there is a tie, fight one more round, with both players using their random ability.

Variations

Players get twelve base points to split between their character’s superpowers. For example, you might give one ability base 6 and both of the others base 3. A random ability (rolling two dice) costs four base points, so you could make all three superpowers random.

Power Team Assemble: Two teams of players battle each other. For each round, teammates multiply their individual scores to calculate the total damage they inflict.

History

Game creator John Golden writes:

“Note that it’s okay for heroes to fight heroes, as those kinds of misunderstandings happen all the time. Villains might fight another villain because they’re just so darn evil. (Or misunderstood.) Also, it’s okay to end in a tie: ‘We may have fought to a standstill, but I’ll get you next time, Snake Lady!’”

Read the delightful origin story for this game — including a cartoon battle between Captain Victory and the Snake Lady — at Golden’s Math Hombre blog.

Writing to Learn Math: When students learn to visualize shapes, designs, and patterns, it makes them better at math.

Do you want your children to develop the ability to reason creatively and figure out things on their own?

Help kids practice slowing down and taking the time to fully comprehend a math topic or problem-solving situation with these classic tools of learning: Notice. Wonder. Create.

Notice: Look carefully at the details of the numbers, shapes, or patterns you see. What are their attributes? How do they relate to each other? Also notice the details of your own mathematical thinking. How do you respond to a tough problem? Which responses are most helpful? Where did you get confused, or what makes you feel discouraged?

Wonder: Ask the journalist’s questions: who, what, where, when, why, and how? Who might need to know about this topic? Where might we see it in the real world? When would things happen this way? What other way might they happen? Why? What if we changed the situation? How might we change it? What would happen then? How might we figure it out?

Create: Create a description, summary, or explanation of what you learned. Make your own related math puzzle, problem, art, poetry, story, game, etc. Or create something totally unrelated, whatever idea may have sparked in your mind.

Math journaling may seem to focus on this third tool, creation. But even with artistic design prompts, we need the first two tools because they lay a solid groundwork to support the child’s imagination.

How To Use a Math Art Prompt

When students learn to visualize shapes, designs, and patterns, it makes them better at math. Even topics like algebra can be surprisingly visual.

Art lets children experiment with geometric shapes and symmetries. They can feel their way into math ideas through informal play. As they draw, students explore a wide range of mathematical structures and relationships.

Math doodles allow a student’s mind to relax, wander, and come back to its work refreshed. And though it goes against intuition, doodling helps people remember more of what they learn.

And art is one of the most replayable of all types of journal prompts. Have you noticed how professional artists love to tinker with their creations? Even a slight change produces delightful new variations. Come back to each math art prompt and enjoy the adventure of exploring possibilities.

Journaling Prompt 221: Make a Maze

Use dotty graph paper. Connect the dots on your page to draw a maze or labyrinth with horizontal and vertical passages. Include a few diagonal passages, too, if you like.

Make your design as complex as you like, but be careful to leave at least one path through your maze.

“One factor that has remained constant through all the twists and turns of the history of physical science is the decisive importance of the mathematical imagination.”

—Freeman Dyson

In the previous episode, Alexandria Jones discovered a mysterious treasure: three wooden sticks, like tent pegs, and a long loop of rope with 12 evenly spaced knots. Her father explained that it was an ancient Egyptian surveyor’s tool, used to mark right angles.

Back at the camp, Fibonacci Jones stacked multi-layer sandwiches while Alexandria poured milk and set the table for supper.

“Geometry,” Fibonacci said.

“What?”

“Geo means earth, and metry means to measure. So geometry means to measure the earth. That is what the Egyptian rope stretches did.”

Alex thought for a moment. “So in the beginning, math was just surveying?”

“And taxes…”

Measuring the Land

Professor Jones sliced a sandwich diagonally into two triangles. “Every farmer in Egypt had to pay a property tax to the Pharaoh each year, based on the size of his land, so the rope stretchers spent most of their time measuring farm land. The scribes could easily calculate the area of a rectangular plot of land.”

Area = length × width

He waved the knife as he talked, drawing imaginary diagrams in the air.

“Dad, would you please put the knife down?” Alex said. “You’re making Rammy nervous.”

“Oh, yes, of course. And since a right triangle made exactly half of a rectangle, the area of a right triangle was simple, too.”

“I know that one,” Alex said. “Area = ½ (base × height), where the base and height are the two legs of the triangle—the two sides that form the right angle. But what if the farmer’s property had some really weird shape?”

Her father began to lay the pieces of sandwiches on the tabletop in an asymmetrical design. “The Egyptians discovered that they could divide any piece of property with straight borders into right triangles.”

“They could cut any property into triangles?”

“If it had straight sides.” Fibonacci picked up the last sandwich and took a big bite.

Alex grabbed a sandwich half and held it up, tracing its edges with her finger. “Then when they measured the sides of each triangle, they could calculate its area. What a neat system!”

Her father nodded. “Adding the areas of all the triangles together gave them the area of the entire property—”

“And the proper amount for the farmer’s taxes!” Alex laughed.

Try It for Yourself

Of course, Egyptian rope stretchers, like modern surveyors, laid out land in rectangles whenever possible. But some properties came out with irregular shapes no matter what the rope stretchers did, and the fact that the Nile flooded every year made their work particularly difficult.

On a blank sheet of paper, draw a few large polygons: closed shapes made of straight line segments. Start with relatively simple shapes, using only four or five lines, then work your way up to a complex shape that fills half the page. Use a ruler to keep your lines straight.

Try to divide your shape into right triangles. If you don’t have a drafting triangle, you can use the corner of a sheet of paper to help you draw right angles. The more complicated your original shape, the more lines you will need to cut it up, but try to find the fewest lines you can. Like everyone else, Egyptian rope stretchers tried to make their work as easy as possible.

Finally, try this challenge from my old newsletter:

• Egyptian farm puzzle

Can you find the area of this farmer’s property, so that he will know how much of his crop to send to Pharaoh for taxes?

A Puzzle for Older Students

Given the equation for the area of a rectangle:

Area (rectangle) = length × width

Show that Alexandria Jones’s equation for the area of a right triangle is true, where the base and height are the two legs that meet at the right angle:

Area (right triangle) = ½ base × height

Then can you show how that area formula applies to any triangle, and why it’s true even for wildly slanty ones?

You don’t have to do a formal proof or write it in the two-column format often taught in geometry class. But make sure your explanations contain enough information that a reader can follow your reasoning.

The toughest part of any geometry proof is to make sure your logic will stand up to scrutiny. How do you know that everything you said is true?

HINT: You may find the following tips from Euclid useful.

Or not, depending on how you approach the problem. Even when there’s only one right answer (the formula you’re trying to prove), there are always many ways to get there. But these are some of the principles I find helpful in thinking about triangles.

• Book 1, proposition 29
  What happens when you draw a diagonal through a rectangle?

• Book 1, proposition 4
  How can you show that two triangles are congruent?

• Book 1, common notions
  If you know that this equals that, what else can you do with the information?

And I also like to use Cavalieri’s principle, which says you can imagine any geometric shape sliced into thin pieces like a deck of cards or a stack of pennies. Then even when the stack is pushed slantwise into different shapes, it will always have the same area or volume.

Myin36, CC BY-SA 4.0, via Wikimedia Commons

How might that relate to triangles?

CHALLENGE: Can you show where geometry’s other area formulas come from? Try to prove the area of a parallelogram or trapezoid, or to approximate the area of a circle.

I Lied to You!

I confess: I lied—or rather, I helped to propagate a math-history legend.

Scholars tell us that the Egyptian rope stretchers did not use a 3-4-5 triangle for right-angled corners. They say it is a myth, like the corny old story of George Washington and the cherry tree, which bounces from one storyteller to the next—as I got it from a book I bought as a library discard.

None of the Egyptian papyri that have been found show any indication that the Egyptians knew of the Pythagorean Theorem, one of the great theorems of mathematics, which is the basis for the 3-4-5 triangle. Unless a real archaeologist finds a rope like Alexandria Jones discovered in my story, or a papyrus describing how to use one, we must assume the Pharaoh’s 3-4-5 rope triangle is an unfounded rumor.

Then why did I tell the story like that?

Unlike Egyptian surveyors, students today do need to know the Pythagorean Theorem and to be familiar with at least a few specific examples. As you go through your high school and college math classes, you will find 3-4-5 (and 5-12-13, 7-24-25, 8-15-17, and more) triangles popping up in all sorts of problems.

I hope that reading about and working with “the Pharaoh’s treasure” will help you remember a few of these useful math facts.

Any multiples of 3-4-5 form similar triangles, which means that sides of 6-8-10 or 9-12-15 also make right angles.

Can you see why this is true?

The Egyptian surveyor could tie an extra knot in the middle of each space of his rope without changing the overall shape of the triangle. That would double the number of spaces on each side, turning 3-4-5 into 6-8-10. This is true for any triangles:

If the sides are proportional, the triangles must be similar.

Another Way to Form Right Angles

But if they did not use the 3-4-5 rope triangle, how did the Egyptian surveyors measure right angles? They still used their knotted ropes, and they used another important fact of geometry:

If you cut an isosceles triangle in half, you get two right triangles.

An isosceles triangle is one in which two of the sides are equal. Here is one way the Egyptian rope-stretchers might have used an isosceles triangle to make a right angle:

• Get four pegs and three ropes. Two of the ropes should have an odd number of knots, evenly spaced. (The reason for an odd number is to make it easy to find the center.) One of the knotted ropes must be much longer than the other.

• Peg the shorter knotted rope along the line of your wall or the edge of the farmer’s property, with the center knot where you want the right angle to be and one knot exactly at each peg. This will form the base, or bottom line, of your isosceles triangle.

• Tie the longer knotted rope to the same pegs, so that it also has one knot at each peg. Stretch it tight to form a triangle, and put the third peg at the exact center knot to hold it in place.

• Tie the third rope to this last peg and stretch it out until it passes the center knot of the first rope. Use your last peg to hold it in place. This rope will be the altitude of your triangle, and it will form a right angle with the first rope.

If you made a “rope” loop out of string, as described in The Secret of the Pharaoh’s Treasure, Part 1, you can use it this way:

• Make the base of your isosceles triangle with either 3 or 5 knots, putting the center knot where you want the right angle and one knot exactly at each corner pin.

• Then pin the center of the rest of the string as far away as it will stretch, making two equal sides for your triangle.

• Stretch an extra piece of string from this top corner down past the center of the base, which will form your right angle.