But there is one resolution that I enjoy keeping—the resolve to play more math.

My favorite way to celebrate at any time of the year is by playing the Year Game. It’s a prime opportunity for players of all ages to fulfill the two most popular resolutions: spending more time with family and friends, and getting more exercise.

So grab a partner, slip into your workout clothes, and pump up those mental muscles!

Rules of the Game

Use the digits in the year 2026 to write mathematical expressions for the counting numbers 1 through 100. The goal is adjustable: Young children can start with looking for 1-10, middle grades with 1-25.

• You must use all four digits: 2, 0, 2, and 6. You may not use any other numbers.

• You may use +, -, x, ÷, sqrt (square root), ^ (raise to a power), ! (factorial), and parentheses, brackets, or other grouping symbols.

• You may use a decimal point to create numbers such as .2, or you may create multi-digit numbers such as .02 or 20 or 202.

While this year offers more options than many recent years, we’ll need plenty of arithmetic tricks to create variety in our numbers. Experiment with decimals, two-digit numbers, and factorials. Remember that dividing (or using a negative exponent) creates the reciprocal of a fraction, which can flip the denominator up where it might be more helpful.

My Special Variations on the Rules

• Challenge yourself: Keep the year digits in 2-0-2-6 order, if you can. And stick to the single-digit numbers as long as possible, leaving multi-digit numbers like .02 or 20 as a last resort.

• You may use the overhead-bar (vinculum), dots, or brackets to mark a repeating decimal.

• You may also use a double factorial, n!! = the product of all integers from 1 to n that have the same parity (odd or even) as n. But save it until you’ve tried everything else. I feel much more creative when I can wrangle a solution without invoking double factorials.

Clarifying the Do’s and Don’ts

Finally, here are a few rules that players have found confusing in past years.

These things ARE allowed:

• You must use each of the digits 2, 0, 2, 6 exactly once in each expression.

• For this game, 0! = 1 and 0^0 = 1.

• Unary negatives count. That is, you may use a “−” sign to create a negative number.

• You may use (n!)!, a nested factorial, which is a factorial of a factorial. Nested square roots are also allowed.

• You may use n!!, a double factorial, which is a factorial that uses only the numbers with the same parity (odd or even) as n.

These things are NOT allowed:

• You may not write a computer program to do the puzzle for you. Or at least, if you do, PLEASE don’t ruin our fun by telling us all the answers!

• You may not use any exponent unless you create it from the digits 2, 0, 2, 6. You may not use a square function, but you may use “^2”. You may not use a cube function, but you may use “^(2+0!)”. You may not use a reciprocal function, but you may use “^(−0!)”.

• While we do allow the square root function, you must create any other roots from the digits 2, 0, 2, 6. For example, to take the cube root of a number, use the radical symbol along with (2+0!) to mark it as cube root.

• “0!” is not a digit, so it cannot be used to create a base-10 numeral. You cannot use it with a decimal point, for instance, or put it in the tens digit of a number.

• The decimal point is not an operation that can be applied to other mathematical expressions: “.(2+0!)” does not make sense.

• You may not use the integer, floor, or ceiling functions. You must “hit” each number from 1 to 100 exactly, without rounding off or truncating decimals.

Helpful Links

• Mathematics Game Worksheet
  For keeping track of which numbers you’ve solved.

• Mathematics Game Manipulatives
  This may help visual or hands-on thinkers.

For more tips, check out this comment from the 2008 game. And Heiner Marxen has compiled hints and results for past years (and for the related Four 4’s puzzle).

Dave Rusin describes a related card game, Krypto, which is much like my Target Number game.

Alexander Bogomolny offers a great collection of similar puzzles on his Make An Identity page. And Pat Ballew takes a brief look at the history of such arithmetic puzzles.

I love the year game! It’s new every time, and such a fun way to build number skills. I do hope you give it a try.

This game challenges students to plan ahead and think strategically.

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!

Sim

Math Concepts: geometry, strategy.

Players: only two.

Equipment: colored pencils or markers, paper.

How to Play

Draw a circle. Add 6 dots spaced out around the circumference.

Each player needs a different colored pencil or marker.

Take turns drawing straight lines across the circle to connect two of the dots. The first person to complete a triangle in their own color with all three corners on the circle loses the game.

Variations

Use the game as a journaling prompt. Here are some sample questions:

• What is your strategy for winning?

• Do you think this is a fair game, or does one player have an advantage?

• How would you count score, so you could compare your performance from one game to the next?

• How would you modify the game rules? Is your version easier or harder than the original game?

• Do you prefer logical strategy games or games of chance? Or do the best games have a bit of both? Explain.

Writing to Learn Math: At its heart, geometry is all about seeing connections and relationships.

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 Geometry Prompt

At its heart, geometry is all about seeing connections and relationships. How can students break shapes apart, put them together, move them around the page, turn them, or distort them? Which properties change, and which stay the same?

Every activity has the potential to spawn hundreds of variations. Alter something in the prompt to make a fresh investigation. Tweak the size, shape, or other properties of interest. What new things can your children see in the math? What questions can they ask?

For older students, use algebra to put some teeth in the relationships they see. Give the points names. Identify the line segments. Can your students write any equations about them? Which distances are equal to other distances, or areas equal to other areas? How can they know for sure? When they add new points, lines, or circles to the diagram, what new connections do they find?

Journaling Prompt 61: Perimeter Puzzle 1

A rectangle has a perimeter of _____ grid units. [Choose any number.] What might the area be? How many different rectangles can you find with that perimeter? What if the sides don’t have to be whole unit lengths?

 

Extra challenge: Perimeter values less than 4 units force the use of at least one fraction or decimal side length.

This is a common problem, and there’s no easy answer.

You see, it’s easy for humans to convince ourselves we understand something when someone else explains it. It seems to make sense, but it doesn’t stick in our minds.

If you think of times when you’ve tried to learn something new, you can probably remember the feeling—you thought you had it, but then when you tried to do it yourself, your mind went blank.

So how can we help our kids when they can’t remember what to do?

Explanations Are Easily Forgotten

One thing that can help is to NOT explain the lesson. Just start with a problem, and ask how your son would think about it. What would he try?

For example, if you are working on times-8 strategies, how would he try to figure out 6 × 8? What does he remember that would help him? Where would he start?

Then you can build on his answer.

If he figured it out, then can he think of another way to do it? There is always more than one way to do anything in math. So, if he solved it by counting 8’s, what’s another way? What if he wasn’t allowed to count? Could he figure it out using any math facts he knows?

Talking about how he reasons things through will help it stick in memory.

Posing His Own Problems

Or if he couldn’t figure it out, then let him name a problem he can do.

Perhaps 6 × 8 is beyond him, but he does know 6 × 2. Then work from there. If two 6s are 12, then how much would four 6s be? And if four of them are 24, then how many would double-4 of them be?

And then once he’s got that answer, can he think of another problem that will help to fix it in his mind? Maybe from knowing 6 × 8, can he figure out what 6 × 9 would be?

Or let him pose a problem for you to solve.

Maybe he gives you 16 × 8. How would you think about that? Talk about your reasoning. Perhaps you already know that 8 × 8 = 64, so 16 eights would be twice that much. Or you used some other way of thinking.

Going Deeper

Push the idea of multiplication beyond what the book has in mind.

• How about fractions? If he knows what 1 × 8 is, can he use that to figure out what 1/2 times 8 would be?

• Or −1 times 8?

• Or if he knows what 3 × 8 is, can he use that to figure out 300 × 8? Or something harder, like 33 × 8?

The idea is to start from where he is and push him to think as deeply as he can.

When we ask a student to listen to our explanation and follow our instructions, we are asking them to think our thoughts. But thinking someone else’s thoughts is boring.

What we want is to have kids who think their own thoughts about the topic at hand. Because thinking their own thoughts is fun and leads to more learning.

This game lays a great foundation for your child’s understanding of multiplication and 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!

Concentration with Math Model Cards

Math Concepts: multiplication or fraction models, visual/spatial memory.

Players: any number.

Equipment: one deck of math model cards.

Set-Up

You can work with your children to create a deck of math model cards, sketching the pictures on index cards or on a stack of old business cards with blank backs. Each deck should include 10–15 sets (also called books). Each book consists of four cards — the multiplication equation or fraction, plus three pictures.

Your card deck need not include every possible fraction or every math fact in the times tables. But it should offer enough variety to cement the most common multiplication models in your children’s minds. For example, my multiplication card deck only includes products from 2×2 to 6×6, but students who master these models can extend the concepts to think about any calculation.

Take your time, making just two or three books each day while talking about real-life situations the models might symbolize. When you think the deck is finished, lay the cards out on the table in sets, to make sure each book has all its members.

The free 44-page PDF Multiplication & Fraction Printables file features two decks of mathematical model playing cards, plus hundred charts and all the game boards for the Math You Can Play: Multiplication & Fractions book.

Get Your Printables File ❱

Print either the Multiplication Models or Fraction Models card deck for this game.

How to Play

Shuffle the cards and lay them all face down on the table, spread out in a single layer. The cards may be placed in an array or arranged in a haphazard cloud, as long as no card covers any other card.

On your turn, flip two cards face up. If the cards match, representing the same product or fraction, then you get to take the pair. If they do not match, leave the cards showing for a few seconds so all players can see what they are. Then turn them face down and let the next player take a turn.

Keep the cards you capture in a personal score pile. When all the cards are claimed, whichever player has collected the most is the winner.

Variations

House Rule: How will you handle the frustrating cycle where a player turns up new cards and sees that one of them would match a previously exposed card, but the other player grabs that pair, leaving the first player to try unknown cards again next turn? At our house, if you find a pair, you get a free turn and can flip over two more cards — which means every player exposes new cards that the next player can use. Free turns expire when there are ten or fewer cards left on the table, to keep one lucky player from claiming all the last pairs.

Mixed Groups: When playing with a wide range of ages, let the younger players flip three cards per turn and keep any two that match.

Equivalent Fractions: Instead of matching the fraction and pictures exactly, players may take any two cards that name the same amount of stuff. A card labeled 3/6 can match with a picture of 2/4, since both of them are worth half of one whole thing.

History

Concentration is my favorite ice-breaker game for math club meetings because the game is quick to learn and easy to play in large groups. It is also a game that older children and adults can enjoy as much as the beginning students do. More than once, when my teenage daughter walked through the room where the younger children were playing, she asked to join in the game.

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 Leo Tolstoy

A man is like a fraction whose numerator is what he is and whose denominator is what he thinks of himself. The larger the denominator, the smaller the fraction.

—Leo Tolstoy

Welcome to the 184th edition of the Playful Math Education Blog Carnival — a smorgasbord of delectable tidbits of mathy fun. It’s like a free online magazine devoted to learning, teaching, and playing around with math from preschool to high school.

With all the links, a blog carnival can feel overwhelming. Bookmark this article, so you can take your time reading the posts.

“Living math” means bringing our children face-to-face with the big ideas of mathematics to help them develop their reasoning skills. When the ideas of math come to life for our children, their minds delight in seeing how numbers and shapes connect to each other and exploring these relationships.

Scattered between the playful math links below, you’ll find quotations from my new book Charlotte Mason’s Living Math, along with several paintings of children playing and learning which I considered for the book but ran out of room.

By tradition, we start the carnival with a puzzle/activity in honor of our 184th edition. But if you’d rather jump straight to our featured blog posts, click here to see the Table of Contents.

Puzzle: Playing with Primes

Prussian mathematician Christian Goldbach, in correspondence with Swiss polymath Leonhard Euler, posed several ideas about number theory (the study of natural numbers) which he couldn’t prove. The version we know now as Goldbach’s Conjecture is:

Every even natural number greater than 2 is the sum of two prime numbers.

Goldbach’s Conjecture is one of the oldest and best-known unsolved problems in math. It has been shown to hold for numbers less than 4 × 10^18, but remains unsettled for larger numbers.

Our carnival number can be written as the sum of primes in eight ways:

184 = 181 + 3
= 179 + 5
= 173 + 11
= 167 + 17
= 137 + 47
= 131 + 53
= 113 + 71
= 101 + 83

184 can also be written as the sum of four consecutive prime numbers:

184 = 41 + 43 + 47 + 53

Play around with Goldbach’s Conjecture and prime numbers.

• What do you notice about sums of primes?
• What do you wonder?
• What other questions can you ask?

Did you know the prime numbers come in a pattern? While it can be hard to prove for sure which large numbers are prime, there are infinitely many numbers that we can be confident are not prime.

• List some large numbers that you know are not prime. How do you know?
• Fill in this worksheet and consider the patterns.
• Now can you list more large numbers that are not prime? Can you tell some that might be?

Contents

And now, on to the main attraction: the blog posts. Some articles were submitted by their authors; others were drawn from the immense backlog in my rss reader. If you’d like to skip directly to your area of interest, click one of these links.

• Talking Math with Kids
• Exploring Elementary Arithmetic
• Adventuring into Algebra and Geometry
• Scaling the Slopes of High School Math
• Enjoying Recreational Puzzles and Math Art
• Teaching with Wisdom and Grace
• Giving Credit Where It’s Due

Talking Math with Kids

“From their birth, children have minds just like ours, hungry for knowledge and able to digest solid mental food. They enter life full of wonder, questioning, investigating the world, reasoning, and drawing conclusions.”

—Denise Gaskins

• Dan Finkel invents Trifle — a new, very small game. “I invented a small game off the cuff last night. I needed a super quick game to play with my 6.5-year-old before bedtime.”

• Christopher Danielson shares the conversations that launched his yet-unpublished book: On vehicles and the meanings of words. “These questions are simple yet deep. They can inform our current conversations in which we seek to regain our understanding of truth, meaning, and empathy.”

• Digging through Christopher Danielson’s delightful archives, I find: A circular conversation, and Doll years, and Does the Earth have an end?. “If you are new to talking math with your kids, don’t worry about getting the timing right. Just start to make a habit of asking those questions. The first few times, you may not get much. That’s OK. It can be like introducing new foods — children need multiple exposures to new things before they accept them.”

• Tom Hobson tells how to build A State-of-the-Art Preschool Playground (for under $200) that will prompt all sorts of math (and other) talk. “As educational as these kinds of spaces are for children, these wonderlands of loose parts, dirt, rocks and compost, these bastions of junkyard chic, they are often perceived as eyesores by the uninitiated. Before going too far, you might want to save up to build a fence.”

• Writing is talking on paper. Dylan Kane tries Incorporating more writing in 7th grade math. “Slowing down and writing about a topic is a great way to think deeply about it, to make connections, to consider hypotheticals, to reason about cause and effect.”

• Jenna Laib gets 4th-graders writing math: How Many Nivelsnorts: Assessing Silly Story Problems. “Perhaps this speaks to who I was as a child, but there is a playfulness and a creativity to getting to write about nivelsnorts and flying tomatoes, and I don’t see a need to deny students that.”

• Karrie E. suggests 5 Easy Ways to Add Writing into Your Math Class. “The goal isn’t to create more work for your or your students. It’s to make thinking visible.”

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Exploring Elementary Arithmetic

“That he should do sums is of comparatively small importance; but the use of those functions which ‘summing’ calls into play is a great part of education.”

—Charlotte Mason

• Patrick Vennebush explores the Mathematical Mysteries of 2026. Meanwhile, Iva Sallay collects 2026 Math Facts and Factors, including a “powerful” math joke. You may also enjoy George Sicherman’s Roman New Year puzzle.

• Sarah Dees explains one of my favorite math games: How to Play Target Number. “This is a fabulous math game that works for third grade all the way up through 6th. You can choose the complexity of the game by choosing how many dice to use.”

• Erick Lee poses A Problem Worth Solving. “I liked this problem because there are so many different ways to approach it and so many interesting patterns to see while solving it.”

• Susan Smith and Kim Montague explore Factor Puzzles: Helping Math Make Sense. “That’s the challenge we pose to students: What do you notice about the puzzle you see? What relationships pop out to you?”

• Pat Ballew plays around with The 3×3 Magic Square, More Magical than You Thought! And Greg Ross shares a magic Venn diagram: Set Piece.

• For my contribution to the carnival, I finish up my series on mental math with Advanced Multiplication, Part 1, followed by Advanced Multiplication, Part 2, and then Advanced Division to wrap things up. “The more time our children spend playing around with numbers and making sense of these relationships, the better they’ll be prepared for algebra and beyond.”

• Jenna Laib shares important information: The 8 Fast Food Chains in the US Most Likely to Get Your Drive-Thru Order Right. And some good news: Extreme Poverty Fell Sharply Worldwide – Even Excluding China. Slow Reveal Graphs “invite learners to examine trends, relationships, and possible interpretations.”

• John Golden teaches Statistics and Probability for K-8 Teachers with some fun resources: Who Wins? “Lots of great discussion about variation, mean, median and their limitations.”

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Adventuring into Algebra and Geometry

“If students find our lessons boring, that is because we are not engaging their minds. Children learn by thinking, imagining, reflecting, reasoning, arguing, justifying, and communicating, putting their thoughts into words. Learning, like digestion, is active.”

—Denise Gaskins

• Iva Sallay graphs a Cat Rotating Around a Bouncing Ball. “This year, the 9th-graders I work with at school need to know how to rotate a shape around a point that is NOT the origin. This is a topic I had never thought about before. To patch up this hole in my math knowledge, I decided to play with rotations in Desmos.”

• Ramsha Waseem reports on 14-year-old Miles Wu’s origami invention. “I was really shocked by how much [weight] these simple pieces of paper could hold,” says Wu.

• Karen Latham offers suggestions for Making the Most of Probability. “I like to have students work several counting problems by hand, so they have experience simplifying expressions with factorials. Then they can move on and let the calculator do the work.”

• David Butler plays with a geometric puzzle: When perimeter is equal to area. “One answer I discovered to this question is completely surprising and delightful to me. You may want to attempt to prove it yourself before I show you my proof…”

• Andrew Staccy categorizes Catriona Agg’s Puzzles by Topic. “The various techniques are grouped into categories and ordered roughly by complexity. Puzzles may appear more than once as they can have multiple ways of being solved.”

• Pat Ballew takes a look at Heron and his Formula(s). “Heron is also remembered for his invention of a primitive steam engine and many early automatons, and a coin operated vending machine, and one of the earliest forerunners of the thermometer.”

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Scaling the Slopes of High School Math

“Never are the operations of Reason more delightful and more perfect than in mathematics. There is great joy in standing by, as it were, and watching our own thought work out an intricate problem.”

—Charlotte Mason

• Christopher Burke posts a series of Geometry Problems of the Day from the Regents Exam, with solutions. “This is a silly question because it’s obvious that both Choices (1) and (2) cannot both be true, so one of them must be the answer.”

• Oliver Johnson explores the probability that your shopping receipt ends in .00: Time’s Arrow. “This ‘everything has the same chance’ collection of probabilities is called the uniform distribution, and it’s really important. But what might be surprising is how easily we reach it, and how few items we need to pick up in a supermarket to make it appear.”

• Nicola Rennie discusses How to create a more accessible line chart. “It’s a myth that accessibility means compromising on aesthetics. Instead, accessibility means prioritising communication.”

• Sue VanHattum needs beta-readers for Althea and the Mysteries of Calculus, version 8.2. “I’m still finding places where I need to add a bit to make the math clearer. Adding a touch of color here, removing something out of place there. I feel like a sculptor or painter.”

• Erick Lee shares A Calculus Esti-Mystery. “Instead of using basic number properties for the clues, I leveled it up. To unlock the clues, my students had to evaluate derivatives.”

• And don’t miss the 249th Carnival of Mathematics.

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Enjoying Recreational Puzzles and Math Art

“We take strong ground when we appeal to the beauty and truth of Mathematics. Mathematics are to be studied for their own sake and not as they make for general intelligence and grasp of mind. But then how profoundly worthy are these subjects of study for their own sake!”

—Charlotte Mason

• David Butler (and daughter) explore a math/art puzzle: Jenga Views. “One thing I particularly like about them is that you don’t need an answer key, because you can just look at your construction and tell if it looks right or not. There’s something empowering about being able to check it yourself.”

• John Golden shares a couple of shape puzzles: Five by Five, and his variation on the theme: Seven by Seven. “Easier to solve, but more solutions. Better for free play. If you made something cool, tangram style, I would love to see it!”

• Rick Mohr welcomes us to a world of Tiled Art. “enjoy a gallery of artworks as each emerges from an underlying grid. And try creating your own tessellation, with the tiles staying interlocked automatically.”

• I post two excerpts from my new book, Charlotte Mason’s Living Math: Discover Math in Art, and the math equation journaling prompt True-False-True. “This puzzle pushes students to consider the structure of mathematical expressions. Make it a game by letting your children challenge you, too.”

• Andrew Taylor creates things with math. I’ve been enjoying his puzzle game Celtix. “Celtic knots are a rich tradition, but for the sake of reducing them to a game mechanic you can think of them as ribbons which ‘bounce’ off ‘walls’ which you can add by clicking anywhere that the ribbons cross each other.”

• Paula Beardell Krieg demonstrates how to make a Cube with an Open Pocket. You may also enjoy her Snow Day Octahedrons. “I’ve been at my desk, working on geometric solids, doing art and math together. I find focus and comfort here.”

• Jonathan Halabi plays with a couple of Birthday Puzzles. Greg Ross finds some harder Words and Numbers. And Pat Ballew digs deeper into Some History Notes about Alphametic Puzzles (and some early versions of a Topology Gem).

• Ben Orlin seeks play-testers: Puzzle Planet. “Your collective generosity and wisdom help me to clarify confusing bits, fine-tune difficulty levels, cull inferior puzzles, spotlight superior ones, and sprinkle play-testers’ insight and wit throughout the text.”

• Laura looks at the math of bell ringing: I can hear the bells. “Each bell ringer controls a huge bell, often on the order of a tonne in weight. One by one they ring their bells, then switch to a new permutation (a change) and repeat.”

• James Propp plays around with probability: In Praise of Stupid Questions. “The way to find things out is to ask a lot of questions. Ask enough questions, and you’re likely to find a new answer: new to you, and once in a while, new to others.”

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Teaching with Wisdom and Grace

“It can be exhausting to shovel information into our child’s head. We put out a lot of effort without much return, and daily lessons become a struggle of wills. But when we treat a child as a person capable of learning for himself and meet him mind-to-mind, with a focus on understanding big ideas, lessons become a delight.”

—Denise Gaskins

• Dan Finkel concludes that Manipulatives in Math Class Are So Worth It. “When it came to mathematizing — linking equations and the rods — the kids who had previous experience building and free playing with the rods made the connections far more readily. Playing with the rods seems to prime the mind to make connections.”

• JoAnne Growney wonders, Can Poems Affect Students’ Math-Attitudes? “One useful viewpoint is that math need not be treated as an isolated subject . . . it is connected to our lives in VERY MANY ways.”

• Growney also links to two contests your students still have time to enter: Creative Writing — Including Mathematics. For inspiration, consider her poem Like Poetry, Mathematics is Beautiful.

• Cathy Yenca details how to prepare The April Fool’s Day Math Activity Students Never Forget. “I decided I wanted to prank my students, but not the kind of prank that derails a lesson. I wanted something that would make students lean into the math instead of checking out.“

• Jenna Laib discusses Measuring Growth in Mathematical Reasoning: What Mia’s Thinking Reveals. “I confess: sometimes I find wrong answers more interesting than right ones. There can be so many ways to get something wrong!”

• Craig Barton creates interactive tools and games for teachers. “End of lesson treat or whiteboard activity for the whole class. Project a game and play together!”

• The Math is FigureOutAble team highlights Leonhard Euler: The Blind Mathematician Who Saw Everything. “Every student can learn to think like a mathematician. They can persist through challenges, adjust their thinking, and experience the satisfaction of figuring something out. They do not need perfect conditions or special talent. They need instruction that invites sense making and builds confidence over time.”

• David Butler shares More wisdom from the Dodecahedron about the humanity of doing math. “I find that maths is full of emotion. Frustration, curiosity, surprise, satisfaction, pride, sadness, companionship, wonder, silliness, joy — they’re all there, sometimes in quick succession.”

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Giving Credit Where It’s Due

Quotations are from my new book Charlotte Mason’s Living Math. Public domain art is primarily from Wikimedia Commons.

And that rounds up this edition of the Playful Math Education Blog Carnival. I hope you enjoyed the ride.

The next installment of our carnival will open sometime during the 2nd quarter of 2026 at Nature Study Australia. Visit our blog carnival information page for more details.

• Feed your children’s minds with big ideas.
• Give them confidence they can learn and make sense of things.
• Develop intuition and creative thinking about math.

You’ll love how Charlotte Mason’s Living Math transforms your children’s experience of math, awakening their imagination about numbers, shapes, and patterns.

It’s a fun way to enrich any math curriculum, and great for unschoolers, too.

But you have to ACT FAST: The Kickstarter campaign ends Thursday night!

Order Your Copy Today ❯

This game helps preschool children develop counting and number sense.

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!

Dinosaur Race

Math Concepts: number symbols, counting beyond ten, number line.

Players: any number.

Equipment: subitizing cards, number line racetrack, small plastic dinosaur or other toy for each player.

Set-Up

Draw a straight path on paper or a manila file folder, either horizontal or slanted uphill (so the larger numbers will be higher). Divide the racetrack into twelve to twenty spaces large enough for small toys to rest in. Or glue squares of colored construction paper in a long line on poster board. Number the spaces in order, beginning with one.

Create subitizing cards by drawing 1, 2, or 3 dots on several index cards or cut-up squares of cardboard. Better yet, let your kids make them with dot markers (but make sure the dots don’t show through the back).

Turn the subitizing cards face down and spread them out to form a fishing pond. Do not use dice or regular playing cards. The number of squares moved each turn must be low enough to recognize at a glance, or else counting will distract the player from saying the track numbers in order.

How to Play

Each player should choose a small dinosaur or other toy and place it near the beginning of the racetrack. On your turn, draw a card and move your dinosaur that many spaces, saying each number as you land on it. Cards should be mixed back into the pond after each turn.

This is the most important rule: when moving their toys, players must say the number in each space. Repeating the numbers in order focuses the child’s attention and helps build number sense, a gut feeling for how numbers work, which is important to future learning.

The first player to reach the end of the path wins the race.

Variations

After children have played the game normally many times, try starting at the end of the path and counting down the number line.

Or use the game board for counting practice. Count pennies or dried beans onto the racetrack spaces, or write numbers on small plastic lids so children can match them to the board.

Whole-Body Counting: Draw a Dinosaur Race path outdoors with sidewalk chalk, or use colored painter’s tape along a hallway floor. Children can walk or jump along the line, saying the numbers as they go.

History

Counting up and down a number line forms a strong foundation for children’s understanding of arithmetic. Dinosaur Race is based on the research of Robert S. Siegler and Geetha B. Ramani, who studied how preschool children responded to a variety of games. Playing a number line game like Dinosaur Race for as little as an hour (in fifteen-minute segments spread out over a couple of weeks) made a dramatic difference in the children’s ability to learn and retain arithmetic facts, while similar games played on a round track or on a linear track without numbers produced no measurable change.

Writing to Learn Math: Measurement is our way of connecting numbers to the things we find in the world, in daily life.

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 Measurement & Data Prompt

Measurement is our way of connecting numbers to the things we find in the world, in daily life. Those numbers become data that students can examine, compare, and reason about.

Some measurements are clear and easy to determine, such as the length of a stick or the weight of a bunch of bananas. But other measurements are fuzzy and open to debate. For example, how can anyone measure the value of an idea or the intelligence of a puppy?

My Measurement & Data prompts give students a chance to collect and examine a variety of measurements and to practice different ways of representing data with charts or graphs.

Older students may want to examine how data shape the way people understand their society. Two websites to explore: slowrevealgraphs.com and nytimes.com/column/whats-going-on-in-this-graph.

Journaling Prompt 34: Comparison Puzzles

A melon weighs as much as 5 premium apples, or as much as 15 plums. How do plums and apples compare?

Make up a measurement-comparison question of your own.

But as I’m working on my new book, I wanted to add another valuable supplement that we often overlook. So here’s a preview excerpt from Charlotte Mason’s Living Math…

Discover Math in Art

In addition to books, supplement your child’s experience of math with art. 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 lines, curves, shapes and symmetries, exploring a wide range of mathematical structures and relationships.

While any piece of artwork has mathematical elements, geometric art and sculpture are easy places for beginners to start. Consider, for example, M. C. Escher’s tessellations, Islamic and Arabic decorative designs, Celtic knot patterns, or the playfulness of Op Art.

You can use the Notice-Wonder-Create cycle of learning: Choose an image to study, and begin by paying attention to everything you can see. What sort of math was the artist playing with?

Consider elements such as…

• Number: unit, counting, order, sequence, series, precision, approximation, and so on.

• Shape: line, angle, parallel, perpendicular, polygon, face, side, vertex, curve, arc, circle, ellipse, chord, radius, diameter, shape, space, boundary, position, depth, perspective, size, and so on.

• Pattern: symmetry, reflection, rotation, repetition, reverse, tessellation, illusion, rhythm, growth, spiral, structure, decoration, color (and relations on a color wheel), and so on.

Wonder about the things you notice, the choices the artist made, or what they might have been thinking. Wonder about relationships to mathematical concepts or to other artistic works. If you were creating this piece, what would you have changed or done differently?

Then create a math art image of your own, riffing off one or more of your noticings and wonderings.

Sources for Math Art

• John Golden’s “(Mathhombre) Miscellanea” Tumblr: mathhombre.tumblr.com, and “Math × Art” page: mathhombre.blogspot.com/p/mathart.html.

• “Mathematical Art Lessons” by Clarissa Grandi: artfulmaths.com/mathematical-art-lessons.html.

• “#MathArtChallenge” by Annie Perkins: arbitrarilyclosecom.wordpress.com/home.

• The Bridges Conference “Mathematical Art Galleries”: artgallery.bridgesmathart.org.

• “Mathematics and art” article: en.wikipedia.org/wiki/Mathematics_and_art, and “List of mathematical artists”: en.wikipedia.org/wiki/List_of_mathematical_artists.

• Islamic Art and Geometric Design: Activities for Learning by The Metropolitan Museum of Art: archive.org/details/IslamicArtandGeometricDesignActivitiesforLearning.

• Celtic Art, The Methods Of Construction by George Bain: archive.org/details/CelticArtTheMethodsOfConstruction.

• “What can you do with a page full of dots?” from my Let’s Play Math blog: denisegaskins.com/2017/03/23/dot-grid-doodling.

And Three Examples to Enjoy

“Theseus Mosaic” by artist unknown, marble and limestone pebbles, circa 350.

“The importance of knowing perspective” by William Hogarth, engraving on paper, circa 1750.

“Composition I (still life)” by Theo van Doesburg, oil on canvas, 1916.

Order Your Book Today

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Check Out the Kickstarter

This game encourages players to reason about the relationships between length dimensions and volume in a 3-D shape.

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!

Prism Power

Math Concepts: rectangular volume, cubic units.

Players: two or more.

Equipment: printed gameboard or plain paper, pencils or markers, one six-sided die, 40–50 cubic blocks per player.

Set-Up

Interlocking cubes work well for this game, as do plain wooden cubes. We’ve also played with sugar cubes when we found them cheap — but do spread a napkin or something to catch the inevitable sugar crumbs.

Print a gameboard for each player. Or make your own scoresheet with four columns labeled length, width, height, and volume (total number of cubes) and use a second sheet of paper for your building property. Draw a straight line to represent the street-facing edge of your property.

If you have enough dice, players may all take their turns at the same time.

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.

How to Play

Each player starts with two blocks, creating a rectangular prism (box shape). Place the long edge of your initial prism parallel to the street-edge of your paper property. This side is your building’s “length” throughout the game.

All players record their building’s measurements for length (2), width (1), height (1), and the initial volume (2 cubes). One side-edge of a block is one length unit, and the volume is the total number of blocks.

On your turn, roll the die and follow the instructions that match your number:

• 1 = Architect’s Choice: Add one layer to any dimension.

• 2 = Elbow Room: Make your prism one layer wider.

• 3 = Expanding Storefront: Make your prism one layer longer.

• 4 = Penthouse Apartment: Make your prism one layer taller.

• 5 = Community Investment Grant: Increase the smallest dimension by one.

• 6 = Zoning Violation: Remove one layer from the dimension of your choice.

After each turn, record your building’s new dimensions and volume on your scoresheet.

Notice that the street-facing length of your building may not always be the longest side, depending on how the dice roll. It often happens in algebra that the side we called “length” when we began a problem turns out to be shorter than the side we initially called “width.” That’s fine because these names make no difference in the final calculation of area or volume.

The game ends when any player’s building exceeds forty cubes. Finish that round, so all players have the same number of turns. Then count up your score based on the Architect’s Prizes below, and the highest score wins.

Architect’s Prizes

Award one point in each of the following prize categories:

• Forty or more cubes

• Tallest building

• Greatest volume

• Greatest perimeter around the base

• Most area on any one side

If two or more players tie for an award, each player gets a point.

Words to Know

What most people call a “box shape,” mathematicians call a right rectangular prism — a prism with a rectangle for its base.

A prism is any shape with a polygonal base — actually two identical bases, the top and bottom — and flat sides that connect the two bases. All the side edges are parallel to each other. Think of a deck of cards with each card cut into an identical polygon shape. When you stack all the cards in the deck, you create a prism with that polygon as the base.

When the deck of cards is stacked straight up, we call that a right prism because the side edges make a right angle with the table. Each side of a right prism is a rectangle, no matter what shape is the base. The common glass or plastic prism used to turn light rays into rainbows is a right triangular prism.

If the deck of cards slants sideways, making each side a parallelogram, the prism is oblique. Notice that one deck of cards can form either a right or oblique prism — or several oblique prisms at different angles — but the volume (total space enclosed) never changes. Any oblique prism has the same volume as a right prism with the same base and height.

Incidentally, the idea of imagining a shape as a stack of cards is called Cavalieri’s Principle (named after Bonaventura Cavalieri, one of Galileo’s students). Your children will use this principle in calculus to find the volume of two- and three-dimensional objects.

History

John Golden shared this game on his Math Hombre blog.

Writing to Learn Math: You can spark creative thought by removing any need to worry about spelling or punctuation rules.

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 Freewrite Prompt

You can spark creative thought by removing any need to worry about spelling or punctuation rules. During a freewriting session, students should write fast and raw until they reach the end of the page.

If students can’t think of what to write, they might consider how the journalist’s 5W1H questions apply to their prompt: who, what, where, when, why, how? Or they can reword their previous sentence, or look for a way to add extra details. It’s always valuable to rethink and revise our writing to make it better express the ideas in our heads.

If all else fails, students can keep writing anything that comes into their minds, even if it seems to have nothing to do with math. They may be surprised to find mathematical ideas pop up in the most unexpected places.

Freewriting prompts may be reused, especially if you change one or two words to make them new. Most of the questions are general enough to spawn entire books, so there will always be ideas the student didn’t have time to think about before.

Or let students propose their own topics. Make a long list of prompt sentences and cut the paper into strips, then crumple the strips into a jar. When it’s time to write, they can pull out a prompt and let the pencil run with it.

If children have trouble filling a whole page, let them write to a timer instead. Set it for five minutes or however long fits their energy level.

Journaling Prompt 229: Math Debate

Is math easy or hard? Explain.

You’ll love how Charlotte Mason’s Living Math transforms your children’s experience of math, awakening their imagination about numbers, shapes, and patterns.

But I know there are many readers who don’t know much about crowdfunding. Some people even think that a campaign like this is just online begging.

So I want to share what I totally love about Kickstarter and how much it offers you, the customer:

(1) It’s the “New World” of publishing.

In the old days, traditional publishing companies paid writers in advance to write the books the publisher wanted to see. In this new world, you get to choose and support the writers creating books you want to read.

(2) Backers get their books early.

I’ll publish the Charlotte Mason’s Living Math book anyway, even if no one supports my project. But through Kickstarter, I can send copies to backers several months before the books are available through normal bookstores.

And I can offer you books the regular bookstores won’t carry, like a special stretch-goal edition with full-color art.

Plus, because you’re buying direct, I can offer a bit of a discount over the recommended retail price.

(3) You get a bundle of cool special perks.

When you buy at a standard bookstore, or even at my publisher’s online store, you get a copy of the book. Period.

When you support my work on Kickstarter, I can send you extra treats like one of my favorite printable math activity books. What fun!

And you can order some cool math merchandise to inspire your family’s math adventures. Or pick up some of my earlier books you may have missed, like a Math Games Bundle targeted at your child’s age and skill level.

(4) You can help make a better book.

Once the project passes its relatively low funding goal and moves into Stretch Goal territory, each additional pledge improves the final book.

It’s a collaborative project. Depending on how high our funding goes, you’ll be helping to pay for an extra appendix, full-color art, and more mathy fun, at no additional cost to you.

Does all that sound good to you?

Then I hope you’ll join me as we take this book over the finish line, covering its production costs in advance of publication and spreading the idea of playful math to a whole world of new readers.

I’ll see you behind the backer wall!

Preorder Your Books Today ❯

This simple strategy game challenges players to think ahead and visualize what their opponent might do.

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!

Square Nim

Math Concepts: logic and strategy.

Players: only two.

Equipment: printed blank hundred chart or draw a 10×10 grid, pencil or pen.

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

You will need a blank hundred chart and a pencil, pen, or marker. Draw lines to divide the chart into five sets of twenty squares each, to make five games.

Each game is played in one set of twenty squares.

Take turns going first. On your turn, you must mark one square, and you may take up to four squares.

The player who marks the last square of the set wins that game. The player who wins the most games on the page is the champion.

Variations

Nim is a simple game, so it’s easy for children to think of new ways to play.

How will you modify the rules?

Teacher’s Tip: Of all the games in the world, I think the math strategy game Nim has the most variations. Traditionally, it’s a misère game, which means the player who takes the last item loses. My math club students call this the “poison” variation and enjoy acting dramatic death scenes when they lose the game.