Writing to Learn Math: Games are the ultimate re-playable activity prompts.

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 Game Prompt

Games are the ultimate re-playable activity prompts. As children repeat a game, they try variations on their previous moves to gain extra advantage. This sort of experiment mirrors the approach a mathematician may take when faced with a problem. What if we try this, or that? How do things change, and what stays the same?

After your child masters the ordinary version of a game, try a misère variation. In a misère game, the move that otherwise would win now makes you the loser. Students must reconsider their strategy and think more deeply about the game.

Encourage children to modify the game rules. What if they changed the number of cards to draw, or how many dice to throw? If the game uses dice, can they figure out a way to play it with cards or dominoes? Or transfer it to a gameboard? Or is there a way to use money in the game? Or can they change it into a whole-body action game? Perhaps using sidewalk chalk?

Older students may want to analyze a game. Does one player have the advantage, or do both players have an equal chance of winning? What’s the best move? Can they find a strategy to increase their odds? How are fairness and randomness linked?

Journaling Prompt 262: Ben Orlin’s Row Call

(two players)

Draw a tic-tac-toe grid. Each player chooses a symbol: X, O, your initial, a small star, etc. On your turn, your opponent names a row or column that has at least one open square. You choose where in that row or column to make your mark.

The player who gets three in a row in any direction wins the game.

Variations: Play on a 5 × 5 grid and try to get four in a row. Or use a 6 × 6 grid, trying for five in a row.

When two non-mathematicians meet on the street and one says, “I’ve got problems,” the other answers, “I’m so sorry for you.”

When two mathematicians meet and one says, “I’ve got problems,” the other says, “Oh, goody!”

—Herta Taussig Freitag

Whenever you get a group of mathematicians together, at least one of them will suggest a puzzle to play with. This is how mathematics grows, puzzle upon puzzle, generation after generation.

And since human nature hasn’t changed, we know that the same thing happened in the ancient Ptolemaic Egyptian of Alexandria.

So let’s explore a few of the math stumpers that enchanted those old Greek scholars…

Euclid and Straight Lines

Before the days of mass-produced paper, Euclid’s students drew diagrams in the sand. Almost any flat section of ground served as a classroom, and sketches could be easily erased by smoothing out the sand.

Take a ruler or other flat object outdoors, rake up a bit of dirt until it’s loose, then smooth it out. What geometric principles can you discover?

• Press your straightedge into the ground to make a straight line. How far can you make that line extend?

• Pick a point somewhere near the middle of your line and center your straightedge on that point. Lift the straightedge so it barely touches the soil. Rotate it a partial turn around that point, and then press your straightedge into the ground to make a new line.

What can you notice about the two lines, and about the angles swept out by your rotation?

What else can you notice? Don’t stop with one idea, but milk the diagram for everything you can get.

• Now hold your straightedge in the air over your second line.

• Keeping it in the same orientation, parallel to that line, slide it up or down along the first line.

• At some point of your choice, press the straightedge into the ground to make a new line.

What can you notice about the parallel lines, and the places where they intersect the original line? How are they the same? How are they different?

Look for intersecting and parallel lines in the world around you. Can you find the same relationships you noticed in your geometry sketches?

Pappus’s Invariant

Pappus of Alexandria lived sometime around AD 300. He wrote a summary of all the mathematics that was known in his day. He called it the Collection. Many works of the ancient Greek mathematicians survive only because Pappus wrote them down in his book.

He also wrote about his own studies in geometry and other subjects. Here is a math-art puzzle that Pappus enjoyed, which demonstrates what mathematicians call an invariant—something that never changes, no matter how you shift the things around it.

Smooth out your geometric ground to make a new diagram. Create this pattern:

• Make a line and mark on it any three points of your choice: A, B, and C from left to right.

• Somewhere not too far away (so you have room to draw connections) make another line. It go wherever you like, but the rest of the diagram will be easier if the slant isn’t too extreme compared to your first line. Mark three points on your new line, and label them from left to right: D, E, and F.

• Now construct the pattern: Connect each point to the points on the other line that are not in the same left-to-right position. That is, connect point A to E and F, and connect B to D and F, and so on.

What can you notice about the diagram you constructed? What shapes and angles can you find?

Go indoors and draw the pattern on paper. Make several designs, each with different points and slants. Color them, if you like. What can you notice about the patterns? What is the same, and what is different?

Pappus noticed something about his diagram of intersecting triangles. In particular, he looked at the points where corresponding lines crossed.

• Mark where the two left-to-middle lines meet, and where the right-to-middle lines meet, and where the left-to-right lines cross each other.

Can you see what’s special—that is, what’s invariant—about these three points in every design you constructed?

The Epitaph of Diophantus

Diophantus, a mathematician who lived in Alexandria around AD 250. He wrote Arithmetica, the first known book about solving problems with algebra. He is famous for his study of Diophantine equations, where the variables must always represent whole numbers.

Can you solve the epitaph of Diophantus?

“This tomb holds Diophantus. Ah, what a marvel! And the tomb tells scientifically the measure of his life.

“God vouchsafed that he should be a boy for the sixth part of his life; when a twelfth was added, his cheeks acquired a beard; He kindled for him the light of marriage after a seventh; and in the fifth year after his marriage, He granted him a son.

“Alas! late-begotten and miserable child, when he had reached the measure of half his father’s life, the chill grave took him.

“After consoling his grief by this science of numbers for four years, he reached the end of his life.”

The Math Adventurer’s Rule: Figure It Out for Yourself!

Whenever I give a problem in an Alexandria Jones story or companion post, I will try to post the answer soon afterwards.

But don’t peek!

If I tell you the answer, you miss out on the fun of solving the puzzle. Figure it out for yourself—and then check the answer just to prove that you got it right. But never read the answers until you’ve given the problem a good workout on your own.

Euclid and Straight Lines

Things you might notice about intersecting lines:

• The lines form four angles around the point of rotation.
• The angles directly opposite each other (called vertical angles) grow or shrink by the same amount as the line rotates. They are always equal to each other.
• Any two angles that touch each other add up to a full straight line (180 degrees).
• If we rotate the line until all the angles are the same, we form four right angles. Each right angle is half of a straight line (90 degrees).

Things you might notice about parallel lines:

• The third line is an exact copy of the second, except in a new position.
• All the angles around the new intersection are exact copies of the corresponding angles around the first intersection.
• Therefore, we can mark angles that are equivalent to each other. Can you find them all?
• Assuming the intersections are not perpendicular, then all four of the smaller angles match, and all four of the larger angles also match.
• Any smaller angle of either intersection could add up to any of the larger angles to make the equivalent (in imagination) of one straight line, 180 degrees.

In Euclid’s Elements, Book 1, these ideas are counted as propositions 13–15 and 27–29.

Pappus’s Invariant

Pappus discovered that no matter how you draw your original lines or where you make the points on them, if you follow his instructions, the three points where the corresponding lines cross will always be collinear. That is, you can always connect them to make a new straight line.

The Epitaph of Diophantus

Let’s make the letter D stand for the number of years that Diophantus lived.

• He was a boy for 1/6 of his life, which would be D/6 years.
• Then he was a youth for 1/12 of his life, until he got a beard—that’s D/12 years.
• After another seventh, he got married, so that’s D/7 more years as a bachelor.
• Then, when he had been married 5 years, his son was born. That’s just a plain 5 years, not a fraction.

This is where the riddle gets tricky. The phrase “when he had reached the measure of half his father’s life” could mean the son was half as old as Diophantus was when the son died, or that the son lived half as long as Diophantus’s total life.

You can try both options, but the first will lead to Diophantus living a fractional age, some number of years plus part of the next year. But Diophantine problems must always have whole number solutions.

• So, the son lived D/2 years.
• Finally, Diophantus lived 4 more years after his son died. Again, that’s a plain 4 years, not a fraction of anything.

Put all of these parts together to make an algebraic equation for how long Diophantus lived:

D = D/6 + D/12 + D/7 + 5 + D/2 + 4

And then imagine solving that with Roman numerals!

Diophantus lived for 84 years.

This game helps players grow comfortable with all the attributes of linear graphs.

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!

Linear War

Math Concepts: coordinate graphing, linear equations, slope, intercepts.
Players: two to four.
Equipment: printed or homemade game cards, pencils or markers, ruler or other straightedge.

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

Players each create a personal deck of eleven game cards. The printable download file includes coordinate grid templates for the game cards. Print on card stock or sturdy paper for best results. Or cut regular grid paper into 10 × 10 squares.

Draw coordinate axes with the origin at the center of each card. Number the points from −5 to +5 on each axis.

On each card, draw a line that passes through at least two points with integer coordinates, such as (−3,1) and (0,5). Mark these points with dots on the line.

Claim your deck: Mark the back of each card with your insignia (initials, emoticon, math symbol, etc. — your choice). Decorate as desired, so long as it doesn’t bleed through the paper and obscure your lines.

How to Play

Place your game card deck face down in front of you and draw the top two cards for your hand. The player whose turn it is names the trump:

• Least slope or greatest slope
• Least or greatest x-intercept
• Least or greatest y-intercept

Each player chooses a card from their hand and holds it face down. When all players are ready, turn the cards up, and the card that best fits the trump captures them all. Each player keeps a pile of prisoner cards.

In case of a tie, leave the cards on the table for the winner of the next trick.

Players draw a new card to replenish their hands. When the decks run out, continue playing until all the cards are taken or the last round ends in a tie. Whoever captures the most prisoners wins the game.

Give the prisoner cards back to their owner, unless you are playing the “Spoils of War” variation.

Variations

In between games, players can make new cards. But they must always choose a deck of eleven cards for play.

House Rule 1: Are players allowed to draw lines with undefined slopes? If so, how will you treat those cards when the trump is “greatest slope”?

House Rule 2: How will you handle cards whose x- or y-intercept is off the chart? Are they allowed to compete for the intercept trumps categories?

Spoils of War: If you win the game, you may claim one card from your prisoner pile to replace a card in your deck. Erase the identification and add your own symbol on the back. Offer your old card to the losing player as a consolation prize, if you wish.

History

This is yet another game from the inventive mind of John Golden, the Math Hombre. He suggests using it throughout a unit on linear graphing, having students make different game cards as you introduce each vocabulary term. At the end of the unit of study, they can choose their eleven-card decks and play the game for review.

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 Gian-Carlo Rota

A mathematician’s work is mostly a tangle of guesswork, analogy, wishful thinking, and frustration. And proof, far from being the core of discovery, is more often than not a way of making sure that our minds are not playing tricks.

—Gian-Carlo Rota

“It is the supreme art of the teacher to awaken joy in creative expression and knowledge.”

—Albert Einstein

Whenever a new set of test results comes in, society goes through a new cycle of hand- wringing and blame-casting. The most current PISA results are no different. The best analysis I’ve seen: The Sky Is Always Falling by Bill McCallum.

Rather than join that fray, let’s step back and ask a bigger question: What do we want for our children?

Why do we school our kids? What does it mean to be “educated”? And in particular, why do we teach math? What do we hope to achieve?

We can’t think rationally about how to teach until we are clear about our goals.

What’s Your Priority?

• Do you want to produce good test-takers, children who know how to follow instructions and work efficiently?
  
• Or do you care less about efficiency and more about curiosity? Do you want students who enjoy wondering about concepts and are eager to learn more?
  
• Or maybe you want problem-solvers, students who can step into an unfamiliar situation and figure it out?

Can we choose option D, all of the above? Is it possible to work toward all these goals at once, to build solid knowledge and teach mathematical skills while at the same time encouraging curiosity and developing flexible thinking?

Like a toddler throwing a tantrum, I want it all, and I want it right now.

I don’t know how to solve the problems of classroom schooling, but here’s some advice for my fellow home schoolers.

This school year…

(1) Build Problem-Solving Skills

Whatever math curriculum you use, focus math lessons on your child’s own thinking.

Rather than telling them what to do, ask about what they know. Help children find a way to use what they’ve already learned to reason about the new topic.

Consider doing your math lessons buddy-style:

• Buddy-Style Math: Doing Homework Without Tears
  In a way, it’s like Charlotte Mason-style narration applied to math.

And when your children get stuck, follow the tips in my “If Not Methods” blog series:

• If Not Methods, Then What?
  As always, real math is not about the answers but the thinking.

(2) Practice for Fluency

Reinforce learning with creative practice.

Don’t just memorize facts, but use them in a variety of ways. Build mental math skills by playing math games, especially games that require strategic use of math skills.

Check out my blog for ideas:

• My Best (Free) Math Games for All Ages
  Games put children in exactly the right frame of mind for learning difficult things.
• Math Game Monday
  Learn a new game every week, for free!
• Mental Math Series
  The more our children practice these structures in mental math, the better prepared they will be to recognize the same principles in algebra.

(3) Explore Non-Textbook Math

Encourage curiosity through puzzles, enrichment activities, reading good books, playing with math art, and journaling.

For example:

• Learning Math with Puzzles
  As stories are to history or science, puzzles are to math.
• Make Your Own Nim Games
  Mathematicians enjoy studying patterns, whether they are patterns within our system of numbers or patterns of shapes or of abstract ideas.
• Living Books for Math
  Here are a few of my favorite books of playful, living math, both old and new.
• Discover Math in Art
  Art lets children experiment with lines, curves, shapes and symmetries, exploring a wide range of mathematical structures and relationships.
• The Creative Way To Help Your Kids Learn Math
  When you embrace this adventure of learning math through playful exploration, you’ll be surprised how much fun thinking hard can be.

“How to Teach Math” copyright © 2026 by Denise Gaskins. Image at the top of the blog copyright © Wavebreakmedia / Depositphotos.

Are you looking for more creative ways to play math with your kids? Check out all my books, printable activities, and cool mathy merch at Denise Gaskins’ Playful Math Store. Or join my free email newsletter on Substack.

This blog is reader-supported. If you’d like to help fund the blog on an ongoing basis, then please join me on Patreon (or choose the paid level on Substack) for mathy inspiration, tips, and an ever-growing archive of printable activities.

This game reinforces the most important model for understanding numerical and algebraic multiplication.

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!

How Close to 100?

Math Concepts: area model for multiplication.

Players: two or three.

Equipment: blank hundred chart or graph paper, two six-sided dice, pencil or marker.

The free 50-page PDF Hundred Charts Galore! file features printable 1–100 charts, 0–99 charts, bottom’s-up versions, multiple-chart pages, blank charts, game boards, and more.

How to Play

Print a blank hundred chart for players to share, or mark off 100 squares on your graph paper. The goal of this cooperative game is to color in all the squares on the hundred chart.

On your turn, roll the dice. Color a rectangle with sides that match the numbers rolled.

For example, if you rolled a two and a four, you might color a rectangle with two rows and four squares in each row. Or you could make it four rows tall with two squares in each row. [Can you see how either rectangle shows the product 2 × 4? Can you explain why we say, “To find the area of a rectangle, multiply the length times the width”?]

If you can’t fit your rectangle anywhere on the chart, you have to pass.

The game ends when all players have rolled the dice without being able to color a rectangle.

How close can you get to coloring the whole game board?

Variation

Rectangle Face-Off: Play as above, but players each color rectangles on their own printed chart. Whoever gets closest to filling all their squares wins the game. If you have enough dice to go around, players may all roll at once on each turn.

Writing to Learn Math: Number play doesn’t have to follow school math methods.

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 Number Play Prompt

Number play doesn’t have to follow school math methods. Remember the Math Rebel rule: A student may write anything that is true or that makes sense.

Most number play prompts offer nearly infinite variation. Change the numbers in the description, and wherever there is a blank you may put in any number you like. Each time you revisit the puzzle, it’s new again.

Older students may experiment with fractions, decimals, or exponential notation. Or try numbers in another base — do the patterns they found hold up when they change the way they count? Can they express these patterns with algebra?

Journaling Prompt 59: Make a Million

Draw six boxes in a row, each big enough to write a digit from 0 to 9. Below that, draw six more boxes:

[_] [_] [_], [_] [_] [_]
[_] [_] [_], [_] [_] [_]

Draw cards or roll dice to get random numbers. Write each number into one of the boxes. (For 10s and face cards, write a zero.) Once a number is written, you can’t move it. When all the boxes are full, add your two six-digit numbers.

How close did you get to making one million? Compete with a friend to see who can get closest.

“In most sciences, one generation tears down what another has built, and what one has established another undoes. In mathematics alone, each generation adds a new story to the old structure.”

—Hermann Hankel

In my old snail-mail Mathematical Adventures newsletter, I used to squeeze bits and snippets into the margins. Quotations, historical tidbits, and small puzzles tangentially related to the main story.

These marginal notes were not simply decoration. Rather, they served the same purpose as the small bumps on a Lego brick, connections on which to build a greater understanding of the world of math.

Thinking about how other people have explored the world of mathematics in the past can offer us new ways of looking at, the things we’re learning today.

Besides, math history is just plain interesting. After all, it’s the story of people just like us, people who wondered, fought, loved, grieved, and lived rich lives—and who played around with math puzzles that we can still enjoy today.

The internet flows like water, however, adjusting content to fit your screen size, which means it doesn’t really have margins to fill. But that’s fine, because it lets me expand the tidbits into a post of their own and include links for further research. You can enjoy my bits and bobs, and then click through to explore on your own.

So let’s turn the clock back a couple thousand years and see what we can find…

About the Names

Everyone in Alexandria Jones’s family takes his or her name from someone (or in Alex’s case, something) important in the history of mathematics.

Alexandria Jones is named after the Egyptian city of Alexandria, founded in 332 BC by Alexander III of Macedon, commonly called “the Great.” As his army marched out from Macedonia and rampaged across the world, Alexander founded many cities, and he named most of them after himself.

(But he did name at least one city in honor of his horse, Bucephalus.)

The Egyptian Alexandria, near the mouth of the Nile River, became a great center of learning. Among the scholars of Alexandria were…

Euclid

Euclid of Alexandria, the most famous mathematician of antiquity, lived around 300 BC, and little is known of his life. That he was an influential teacher seems clear, for he put together a treatise he called The Elements, which became the best-selling math textbook of all time.

The Elements consists of thirteen books, covering plane (flat) geometry, number theory, and three-dimensional geometry. Euclid’s work is admired for the clarity with which the theorems are stated and proved, setting a standard of rigor for later mathematicians to follow.

For further study:

• Play the interactive introduction to Euclidean geometry at sciencevsmagic.net. Start by connecting the two points to draw a line.

• Learn How to Study Euclid’s Elements and Actually Understand It.

• Read The Elements online.

“There is no royal road to geometry.”

—Euclid of Alexandria

Eratostenes

In the late 200’s BC, Eratosthenes of Cyrene served as librarian at a temple of the Muses called the Mouseion. This famous library in Alexandria may have contained hundreds of thousands of papyrus and vellum scrolls.

But Eratosthenes was also a scholar in his own right, known for his work in mathematics, philosophy, astronomy, geography, poetry, and music. He worked on challenge of doubling a cube, developed a system for identifying prime numbers, and is best remembered for his accurate measurement of the Earth’s circumference.

For further study:

• Read Julia Diggins’ chapter on Eratosthenes.

• Play with the Sieve of Eratosthenes to identify prime numbers. Why does it work?

• Try my printable activity to Measure the Earth (a mathematical experiment for the spring or autumn equinox).

“Eratosthenes was, indeed, recognised by his contemporaries as a man of great distinction in all branches of knowledge, though in each subject he just fell short of the highest place. On the latter ground he was called Beta, and another nickname applied to him, Pentathlos, has the same implication, representing as it does an all-round athlete who was not the first runner or wrestler but took the second prize in these contests as well as others.”

—Thomas Little Heath

Ptolemy

The most influential astronomer of ancient Greece, Claudius Ptolemy lived in the early AD 100’s and invented an Earth-centered mathematical theory of the solar system that scientists relied on for more than a thousand years.

He compiled the results of his research in a thirteen -book treatise now called The Almagest, summarizing everything the Greeks know about how the sun, moon, and planets moved around the Earth. This second-most-famous math textbook was studied up through the late Renaissance, but was finally replaced by Copernicus’s sun-centric theory.

Ptolemy also wrote works about geography, optics, and the mathematics of musical harmony.

For further study:

• Read about Ptolemy’s model of the solar system.

• See Ptolemy’s epicycles and equants in action, to scale.

• Find out how mistakes in Ptolemy’s Almagest led to Christopher Columbus’s famous journey.

“Well do I know that I am mortal, a creature of one day.
But if my mind follows the winding paths of the stars
Then my feet no longer rest on earth, but standing by
Zeus himself I take my fill of ambrosia, the divine dish.”

—Claudius Ptolemy

Fibonacci

Leonardo of Pisa, nicknamed Fibonacci (from whom Alexandria Jones’s father takes his name), didn’t live in ancient Greece, but he traveled all around the Mediterranean world while working for his merchant father, so he almost certainly visited the famous Egyptian port city of Alexandria. Though by that time, the Mouseion and its wonderful library were long gone.

When he returned to his hometown of Pisa, Italy, in AD 1200, he brought back “new” ideas about math. He rediscovered concepts that had been lost (to the Europeans) since the fall of Rome.

Even more important, Fibonacci brought to Europe the Hindu-Arabic numbering system—including the notion of zero. This efficient way of dealing with numbers led to revolutions in banking and in science.

But most people know him because of a story problem about breeding rabbits

For further study:

• Read about Fibonacci’s life and mathematics.

• Discover how the Fibonacci numbers appear in nature.

• But don’t fall for a Golden Sales Pitch.

“When I had been introduced to the art of the Indians’ nine symbols through remarkable teaching, knowledge of the art very soon pleased me above all else.”

—Leonardo Pisano Fibonacci

This game offers young children a chance to practice counting and early addition skills while racing to beat their parents.

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!

Push the Penny

Math Concepts: addition to one hundred, thinking ahead.

Players: two or more.

Equipment: playing cards (remove face cards and jokers), a hundred chart, and a penny or other small token.

Set-Up

Print a hundred chart for players to share. You may choose any style you like: 1–100, 0–99, top-down, or bottoms up.

The 23-page printable (pdf) Number Game Printables Pack file includes hundred charts, graph paper, and game boards from the first two Math You Can Play books.

How to Play

Deal three cards per player and set the rest of the deck face down as a draw pile. Place the chart in the middle of the table, with the penny on the first square.

On your turn, lay one card from your hand face up on the discard pile. Move the penny forward that many squares, and then draw to replenish your hand.

If adding the number on your card would move the penny past one hundred, then you must subtract your number and move backward instead. Whoever pushes the penny to one hundred by exact count wins the game.

Variations

Play two cards from your hand and choose whether you want to add or subtract the numbers to find out how far the penny moves.

Or start at one hundred and push the penny down to number one.

History

My math club students have enjoyed playing a variety of count-to-one-hundred games. This version is a modification of the 99 Game from Julie Reulbach’s collection “Math Games for Classrooms.”

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.

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

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.”