This game makes children think about multiplication and notice how prime numbers are special.

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!

Tax Collector

Math Concepts: multiplication, prime numbers, factors and multiples.

Players: two or more players or two teams.

Equipment: blank paper or a printed hundred chart, pencils or markers. Calculator optional.

Set-Up

Choose a set of numbers (the more numbers, the longer the game) such as 1–30 or 1–100. Write the numbers at the top of your paper, so you can mark them out as they are used. Or use a printed hundred chart to mark numbers, keeping score in the margins or on scratch paper.

How to Play

One player is the tax collector, and all the others work together as a team of shop owners. Or the players on one team are tax collectors, and the other team entrepreneurs.

First, a shop owner marks any available number by crossing it out or coloring that square on the chart. Add this number to the running total of shop owner profits.

Once a number is marked by any player, it cannot be used again.

After each shop-owner turn, the tax collector marks all the factors of that number that have not been previously scored. Add these numbers to the running total of taxes.

Continue turns with shop owners claiming a profit amount and the tax collector looking for factors to tax. The game ends when the shop owners have no legal play — that is, there are no numbers with factors left. The tax collector claims all remaining numbers.

Whoever collects the most money (profit or tax) wins.

Warning: You must always pay the tax collector! No shop owner may mark a number that doesn’t have any factors remaining. If you try to claim a number with no factors, the tax collector takes that whole amount as a penalty.

Variations

The Factor Game: Alternate roles on each turn. The first player chooses a number, and the second collects the tax. Then the second player chooses a number, and the first one collects the tax. At the end of the game, neither player gets the unclaimed numbers.

Factor Blaster: Players alternate roles, as in the Factor Game, and they may choose any unmarked number — including numbers without factors for the other player to claim. In Tax Collector or the Factor Game, you may choose only one prime number (you’ll find out why as you play), but the lack of penalties in Factor Blaster makes the prime numbers into prime targets.

History

In Simply Great Math Activities: Number and Operations, authors Bill Lombard and Brad Fulton write:

“The tax collector usually wins the first game so handily that the students are left believing there is no way for them to win this game. However, assure them there are many ways for the taxpayer to win, and ask them to play another game.”

Teacher coach Terry Kawas created the Factor Blaster variation that emphasizes the importance of prime numbers, to prepare for teaching prime factoring.

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 William James

The union of the mathematician with the poet, fervor with measure, passion with correctness, this surely is the ideal.

—William James

Game options might include:

• How many stones to take from a pile.
• Which position to claim on a gameboard.
• How far to count in a given sequence.

The rules can vary at the players’ whim (as long as both players agree). How many possibilities do you start with, what are the rules for removing options, and how do you win or lose the game? Everything is open to change. And with every tweak, players must reanalyze their strategy.

The History of Nim

For centuries, people all around the world have played Nim-like folk games, though the rules are rarely written down. Some say Nim originated in China because the rules are similar to the Chinese game Jian-shízi, or “picking stones.”

The first version of the game in print dates to about 1500 in a book of mathematical recreations by Luca Pacioli, a Franciscan friar who also collaborated with Leonardo da Vinci on a geometry book.

Charles Bouton coined the modern name (perhaps based on the German word for “to take”) and brought mathematical attention to the game with his 1901 article “Nim, A Game with a Complete Mathematical Theory.” But it was Martin Gardner who made the game famous when he wrote about Nim in his Scientific American column “Mathematical Games” in 1958.

How Is This Math?

Mathematicians enjoy studying patterns, whether they are patterns within our system of numbers or patterns of shapes or of abstract ideas.

Did you know there’s a branch of mathematics called game theory? Game theorists study patterns of logical decision-making.

They began by studying the pattern of wins and losses in 2-player strategy games like Nim. Now the science has grown into a study of much larger and more complex “games” like politics and economics.

First, Play Several Nim Games

When we reason about game strategy, that’s mathematical thinking. We are dealing logically with the things we see and the facts we know.

So try one of the Nim games on my blog:

• Math Club Nim
• Hundred Chart Nim

Or play some of the sample games (some with printable gameboards) in my activity guide Creative Nim: Make Your Own Math & Logic Games.

And then take a break and talk about the things you noticed. Wonder together about what might happen the next time you play.

• Did you figure out a winning strategy?
• How might your opponent try to block you?
• Would you rather play first or second, or does it matter?

Then, Make Your Own Math

When you create your own games, you do mathematical reasoning at a deeper level, considering the types of choices to give your players and how limiting those options will affect the game.

First, think about a theme or story behind your game. Perhaps your players are dragons trying to hide their gold, or ninja warriors trying to sneak into an enemy camp, or birds building nests on the branches of a tree. Or would you rather make an abstract game of counting or picking up stones?

Will you stick to the tradition of a 2-player strategy game, or will you make a way for more people to join in the fun?

Decide what options players will have in your game:

• Will they be taking physical items like stones or game pieces?
• Will they be moving or claiming positions on a gameboard?
• Or will they be choosing among abstract ideas like numbers or shapes?

In a Nim game, players remove options on each turn until no choices remain. How will players remove options in your game? What choices can they make? What are the limits? How will they win (or lose) the game?

Write out the instructions for your game. Copy what you need from the games in this book, modifying the rules to fit the way you want to play. Draw your gameboard, if needed, and collect whatever game pieces you desire.

Test your new game by playing with a friend. How did it go? Do you want to tweak the rules?

A Puzzle for You

Think about all the games you know. How many can you find that have the features of a Nim game?

Buy the Printable Guide

Did you know there’s a branch of mathematics called game theory? Game theorists study patterns of logical decision-making.

Mathematicians began by studying the pattern of wins and losses in 2-player strategy games like Nim. Now the science has grown into a study of much larger and more complex “games” like politics and economics.

Encourage your children to enter the fascinating world of game theory by creating their own Nim-style math and logic games.

Creative Nim includes instructions and teaching tips, several sample games, and make-your-own-game worksheets in full color and ink-saving black-and-white.

For ages 8 and up.

FORMAT: 47-page printable PDF file with your choice of 8.5″×11″ (letter size) or A4 pages.

GET IT TODAY

And start making math with your kids!

This game helps students make sense of those confusing rules about multiplication with inequalities.

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!

Greater Than

Math Concepts: integer addition, integer multiplication, working with inequalities.

Players: two players or two teams.

Equipment: printed gameboard or sheet of paper, one deck of playing cards, pencils or markers.

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

Agree on which color represents negative numbers. Face cards count as ten, aces as eleven.

The suits also represent operations:

• Spades and hearts = add the number.
• Clubs and diamonds = multiply by the number.

Print a gameboard for players to share. Or prepare a scoresheet with three columns: one for each player’s running score, and a column in the middle for the inequality sign. Choose a player to deal the first hand.

How to Play

Deal four cards to each player or team. Players each choose one card as their initial value, holding the card face down in front of them. At the dealer’s signal, both players reveal their cards.

If the two cards played have the same value, players return them to their hands and choose a different starter card.

Write the players’ initial numbers in their columns. In the middle column, draw an inequality symbol (> or <) with its open end toward the greater value.

The non-dealer plays first. Choose one card from your hand and lay it on the table. Write the operation represented by that card in the first box of the next line on the gameboard (or beside the next line of the scoresheet). Then do the indicated calculation:

• If your card is a spade or heart, add the value to each player’s current score.
• If your card is a club or diamond, multiply each player’s score by that number.

Write the new sum or product in each player’s column on the scoresheet. Finally, draw the correct inequality symbol in the middle column.

A hand consists of two turns for each player, beginning with the non-dealer. So the dealer takes the next turn, playing a card, writing it at the start of the third line, and doing the calculations based on the values in the previous line. The non-dealer fills the next line, and the dealer plays the final line of that hand. Players do not draw new cards after each turn, so plan ahead how best to use the cards you have.

After the dealer’s second turn, whoever has the greater value wins that hand. Circle the winning score. Mix all cards back into the deck and pass it to the other player to deal the next hand.

The first player to win three hands wins the game.

Sample Game

Tony challenges Steve to a game of Greater Than. Tony deals the first hand, and the players reveal their initial cards. Tony plays the queen of hearts, for a value of −10. Steve has the eight of spades, for a value of +8.

As the non-dealer, Steve goes first. He plays the three of clubs and writes “×3” on the next row. Then he multiplies both scores by three and writes each product in that player’s column. He writes the less-than sign “<” to show that he’s in the lead.

Tony lays down the ten of spades, adding 10 to each score. Steve chooses the five of hearts, adding −5 to the scores. Steve still has the greater value, but Tony gets one more turn.

Tony plays the two of diamonds, which multiplies both scores by −2. Multiplying by a negative number changes the sign of the scores, and Tony wins the hand with 50 points.

Variations

House Rule: Does having the last play offer too great an advantage? Give the dealer one less card at the start of each hand.

Or add a level of risk to the game by having both players reveal their operation cards at the same time, as they did with their initial numbers. On each turn, choose a card and hold it face down until both players are ready. The card with the lesser value does its operation first. (And remember that −9 is less than +2.) This means the dealer may play before the non-dealer, depending on which cards they choose.

History

John Golden, who created this game, writes:

“I think this is a good educational game, but only close to a good strategy game. I can’t quite figure out what’s missing, so if you have a variation or adaptation to try, please let me know.”

If you think of a variation to share, post it in the comments on his original Math Hombre blog post:

• mathhombre.blogspot.com/2012/09/greater-than.html

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 46: Age Puzzles

Jewel’s father is 3 times as old as Jewel. In 10 years, he will be twice her age. How old is Jewel? Make up some age puzzles of your own.

All my children have figured out ways to do things in math that I would never have expected, and I’ve learned quite a bit from listening to their explanations.

But what if the child’s creative method makes it hard for them to learn what our textbook wants to teach?

A Reader Asks

My seven year old son is learning how to subtract with “regrouping.” We used base-ten blocks to learn the traditional method, but today, he showed me how he works the problems.

Instead of regrouping, he subtracts the ones column and gets a negative number. Then he subtracts the rest of the numbers (tens and hundreds columns), and then he adds the negative number back in.

So for 342 − 225, he does:

2 − 5 = −3,

then 340 − 220 = 120,

and adding back the −3 is 117.

I know this method works and I’m blown away that he came up with it. (Not sure how he knows about negative numbers!)

But what do I do — do I continue reviewing the regrouping method? Do I let him do his own thing and eventually go back to the traditional method? Do I leave it alone and see what happens? Do I try to combine the two methods in his learning?

Great Questions

And congratulations to your son for developing such a solid understanding of numbers!

There is nothing magic in the standard algorithms (the rules for pencil-and-paper arithmetic). Your son’s method is perfectly fine and logical — and also much better suited for working in his head on the fly, which is how we often do math in real life.

The main thing you want to teach your son is that math makes sense. That it’s logical, something he can figure out on his own, not just “rules from on high.”

So let him use the method that makes sense to him.

There may come a day (maybe in third or fourth grade) when the numbers get too big for him to handle with his method. At that point, he may adapt his own algorithm to include written notes about the negative numbers in each place value column, or he may be ready to learn the standard algorithm.

Either way, if you follow his lead, you will strengthen his ability to reason things through.

Develop Mathematical Thinking

In adult life, when we have big numbers to deal with, we have calculators on our phones and computers, and we do our accounting in spreadsheets that do the calculations for us.

The standard pencil-and-paper algorithms were designed for a different world, a place where clerks had to do hundreds of manual calculations every day with as little thought as possible because thinking about the numbers slowed them down.

But cognitive scientists tell us that children only learn what they think about, and the deeper they think, the more the concept links itself to other ideas in their mind. That means we want our children to think about the numbers, not just follow rules to get answers.

The math that was perfect for Bob Cratchit really doesn’t help our kids today. Because the thinking is what’s valuable — it’s the thing computers and phones can’t do for us.

So enjoy your son’s insight, and continue your adventure of exploring the mathematical world together.

This classic family game is a great way to play with numbers and logical strategy.

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!

Dominoes

Math Concepts: subitizing dot patterns, thinking ahead.

Players: two or more.

Equipment: one set of double-six or double-nine dominoes, or two sets for five or more players.

Set-Up

Turn all domino tiles (also called bones) face down on the table and mix them around. Each player draws several tiles, as follows:

• Two players draw seven tiles each.
• Three or four players draw five tiles each.
• Five or more players use a double set of dominoes and draw five each.

Set your tiles upright on their sides, so the other players cannot see them. These tiles are your hand, and the ones left on the table are called the wood pile (also known as the bone yard).

How to Play

Whoever has the highest double goes first, placing that tile face up on the table. If no doubles are available, turn everything face down, reshuffle, and draw again.

Play proceeds to the left around the table. Domino tiles are played end-to-end in a long row (the train), with only the outer ends available for adding new tiles. On your turn, you may play one of the tiles in your hand to either open end of the train if the dots (called pips) on one side of your tile match the tile on that end.

Doubles are placed crosswise to the direction of the train, with the middle of the domino touching its neighboring tiles. Players may not match tiles to the ends of the doubles, however, only to the middle of the other side, continuing the train in whichever direction it was growing.

If you have no tile that will play, draw one tile from the wood pile. If you can play that one, do so. Otherwise, add it to your hand. This marks the end of your turn. If you cannot play from your hand and there are no tiles left to draw, you must pass.

The first player to run out of tiles wins the game (or in Muggins, wins that round). If no player is able to go out, then all players add up the pips on their remaining tiles. The player with the smallest sum wins.

Variations

Domino games vary tremendously around the world, and even from one family to another within the same town. Whenever you play with friends, be sure to agree on the rules before you draw the first tile.

Cross Dominoes: Make a double train. After the first double tile is played, the next four tiles must match it. Two of these are placed normally, at the middle of each side of the first tile, and the other two connect to the ends. If players do not have a matching tile, they must pass. After the four matching tiles have made a cross shape, play continues normally, except in four directions instead of two.

Muggins: Play as for Cross Dominoes, but keep score as you go along. At the end of each turn, the player adds up the pips showing on the live tiles at all four ends of the train, including both halves of any exposed doubles. If these make a multiple of five, the player adds that number to his or her score. At the end of each round, players add up the pips remaining in their hands, round to the nearest five, and subtract those points from their total so far. The first player to reach 200 points (or some other agreed-upon total) wins the game.

Fives and Threes: Play as for Muggins, but score all multiples of three or five.

History (and a Puzzle)

Domino-like tile games seem to have originated in China, and they came to Europe through the great trading cities of Venice and Naples. Some game historians claim the European game was invented independently, because European domino sets are different from Chinese sets in several ways. (For instance, Chinese tiles come in suits, like a set of playing cards.) Dominoes spread across France and reached England in the late eighteenth century, where the game became a favorite pastime in British pubs.

Puzzle: Encourage your children to examine a set of domino tiles and describe what they notice. For example, every possible combination (double-0, 0|1, 0|2, etc.) is a single tile, but there are no duplicates: 0|1 is the same tile as 1|0.

Ask them, “If you bought a set of dominoes at a garage sale, how could you tell whether any of the tiles were missing? Can you figure out how many tiles there should be?”

Spoiler: To find the answer, make a systematic list, and be careful not to count any of the combinations twice. A double-six set should have twenty-eight tiles, and a double-nine set will have fifty-five. A new set from the store may contain extra blank tiles, which can be decorated with paint or white nail polish to replace lost pieces.

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 145: Collatz Hailstones

Imagine the whole numbers as tiny ice particles flying in the wind, bouncing up and down inside a storm cloud. Pick any number.

If it’s even, cut it in half. If it’s odd, multiply by 3 and then add 1.

Follow the same rule with your new number, and with each answer after that. If you get down to 1, your original number formed a hailstone that hit the ground.

Try several different starting numbers. What patterns do you see? Can you think of any questions to ask?

But lately, I’ve added a new teaser to wake up my brain for the day.

While Set was a visual-logic puzzle, this one is straightforward (though not simple) deduction. I think you’ll enjoy it.

Clues by Sam

I first discovered Clues by Sam back in March, as I was putting together Playful Math Carnival 184. I can’t remember why I didn’t include it in the carnival, but I did save it to my phone and add it to my wake-up routine.

The puzzle consists of a 4 × 5 array of characters labeled by their career. Each character is either a criminal — not really a murderer, but a thief with unusual aspirations (think of Vector in Despicable Me) — or innocent. Your job is to identify which is which, based on what they say.

Even if a character is a cop or a judge, you can’t assume they’re innocent. Make your decisions based on pure logic. But this isn’t a “truth-teller or liar” puzzle. You can trust that each statement is valid, though not always helpful.

Try It for Yourself

Clues by Sam offers a tutorial game to kick-start your little gray cells. Or you can try the partial puzzle below…

Each day begins with a single clue that lets you identify at least one character. If you’re on a small screen (like my phone), you may have to hit the “Inspect” button to read the last bit of a longer clue.

Whenever you label someone “innocent” or “criminal,” they offer a new clue.

The game won’t let you cheat. If there’s not enough information to prove a character’s status, trying to label them counts as an error.

I Make a Lot of Mistakes

The game starts each week with an “easy” level and gets harder day by day.

I’ve rarely made it through a puzzle without committing at least one mistake. Usually it’s just a mental glitch: I switch “row” for “column” when interpreting a clue, or I hit “innocent” when I really meant the other.

But I’ve made actual logic errors, too. The clues keep coming, and it can be a lot of information to remember.

As you go through the puzzle, some of the “clues” turn out to be snarky comments, like Frida’s in the image above. This was a Saturday puzzle, and while it started out relatively simple, it quickly turned into a challenge.

Vince was my nemesis this time. I misread one of the clues, and when the app counted me wrong, it took several minutes to figure out why.

Give Clues by Sam a try, and have fun playing with logic!

This game pushes students to think more deeply about the meaning of place value.

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!

Double Digit

Math Concepts: place value, addition.

Players: any number.

Equipment: six-sided die, pencil and paper for keeping score.

How to Play

Take turns rolling a die. Add that number to your score as either “ones” or “tens.” For example, if you roll 3, you can choose to add 3 or 30.

After everyone has seven turns, the player whose score is closest to 100 wins.

Optional challenge: Anyone who goes past 100 loses.

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: Problem-solving cares less about whether an answer is right and more about whether a solution makes sense.

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 Problem-Solving Prompt

When children face a tough math problem, their attitude can make all the difference — not so much their “I hate homework!” attitude, but their mathematical worldview. Does your child see math as answer-getting or as problem-solving?

Answer-getting asks “What is the answer?”, decides whether it is right, then forgets it and goes on to the next question. Problem-solving cares less about whether an answer is right and more about whether a solution makes sense.

Students who care about problem-solving want to explore the web of interrelated ideas they discovered along the way: How can they recognize this type of problem? Can this one help them figure out others?

What could they do if they had never seen a problem like this one before? How would they reason it out?

Why does the formula work? Where did it come from, and how is it related to basic principles?

What is the easiest or most efficient way to manipulate the numbers? Does this help the problem-solver see more of the patterns and connections within our number system?

Is there another way to approach the problem? How many ways can they think of? Which do they like best, and why?

Journaling Prompt 299: Farmer’s Market

Mr. Tam had 480 more pears than apples at his fruit stand. Then he sold half of each type of fruit. Now he has four times as many pears as apples.

 

How many apples did he start with?

 

Make up a farmer’s market puzzle of your own.

Today I’m sharing a few treats from The Moscow Puzzles by Boris Kordemsky, which mixes classic brainteasers and original stumpers.

Recreational math expert Martin Gardner called Kordemsky’s book “the outstanding puzzle collection in the history of Russian mathematics.”

Have fun playing logic with your kids!

A Weather Forecast

It is raining at midnight—will we have sunny weather in 72 hours?

Shoes and Socks

I went to the closet while my sister was asleep, so I left the light off.

I found my shoes and socks, but I must confess they were in no kind of order—just a jumbled pile of 6 shoes of three brands, and a heap of 24 socks, black and brown.

How many shoes and socks did I have to take with me to be sure I had a pair of matching shoes and a pair of matching socks?

Apples

Three kinds of apples are mixed in a box. How many apples must you take to be sure of at least 2 apples of one kind? At least 3 apples of one kind?

Arbor Day

On Arbor Day the Young Pioneers of the fourth grade started early and planted 5 trees before the sixth-graders came. But they planted them on the side assigned to the sixth grade.

The fourth-graders had to cross the street and start over, so the sixth-graders finished first. To pay their debt, they crossed the street and planted 5 trees. They planted 5 more trees, and all the work was finished.

Were the sixth-graders ahead by 5 or 10 trees?

[Both rows of trees were the same length.]

A Purchase

“Your pencils, notebooks, and colored paper cost $1.70.”

“I bought 2 pencils at 2 cents each and 5 pencils at 4 cents each—and 8 notebooks and 12 sheets of colored paper, I don’t remember the prices. But the bill can’t be $1.70.”

Why not?

A Crime Story

An elementary school teacher in New York State had her purse stolen. The thief had to be Lillian, Judy, David, Theo, or Margaret.

When questioned, each child made three statements:

Lillian:

(1) I didn’t take the purse.

(2) I have never in my life stolen anything.

(3) Theo did it.

Judy:

(4) I didn’t take the purse.

(5) My daddy is rich enough, and I have a purse of my own.

(6) Margaret knows who did it.

David:

(7) I didn’t take the purse.

(8) I didn’t know Margaret before I enrolled in this school.

(9) Theo did it.

Theo:

(10) I am not guilty.

(11) Margaret did it.

(12) Lillian is lying when she says I stole the purse.

Margaret:

(13) I didn’t take the teacher’s purse.

(14) Judy is guilty.

(15) David can vouch for me because he knows me since I was born.

Later, each child admitted that two of his statements were true and one was false. Assuming this is true, who stole the purse?

A great classic game for groups or family play.

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!

Farkle

Math Concepts: addition to ten thousand, probability with dice.

Players: any number.

Equipment: six six-sided dice, pencil and paper for keeping score.

Set-Up

Optional: The Number Game Printables Pack includes a Farkle scoring guide. Cut the page in half and give one sheet to each player.

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

On your turn, throw all six dice. If you like your score (see below for scoring), write it down and pass the dice to the player on your left. Or keep going. But beware: every time you roll the dice, you must set aside at least one scoring die. If none of the dice score, you have farkled, and you lose all the points you have accumulated that turn.

If you get all six dice to score, you have hot dice, and you may roll them all again to continue building up points. If you don’t farkle, then whenever you decide to stop rolling, add the points you have collected to your running total for the game. Writing down this score marks the end of your turn.

The first player to reach 10,000 points might win the game. But all the other players get one last chance to try and pass that score without farkling. Highest final score wins.

Scoring the Dice

There is often more than one way to score your dice, and you do not have to set aside every die that could score. Try to keep the highest score you can, while still leaving yourself as many dice as possible to throw again, thus decreasing your risk of a farkle.

Every roll is scored separately. If you set aside two ones and then get another one on your next throw, that does not count for 1,000 points.

Each 1 = 100 points
Each 5 = 50
Three 1s = 1,000
Three 2s = 200
Three 3s = 300
Three 4s = 400
Three 5s = 500
Three 6s = 600
Four of a kind = Double the score for three of them
Five of a kind = Double the score for four of them
Six of a kind = Double the score for five of them
Three pairs = 500
Straight (1-2-3-4-5-6) = 1,000
Two triples = 2,500

History

Farkle-style games have been played around the world for centuries, probably since dice were invented. Because of this, rule variants are common, so be sure that all players agree on which version you’re playing before the game begins.

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 Lao-Tzu

Knowing others is intelligence; knowing yourself is true wisdom. Mastering others is strength, mastering yourself is true power.

—Lao-Tzu

Here are some dreams shared the authors of Avoid Hard Work!

For our children, we dream that mathematics…

• … makes sense.
• … is more than just arithmetic.
• … is joyous.
• … makes them strong.
• … is meaningful.
• … is creative.
• … is full of fascinating questions.
• … opens up many paths to solutions.
• … is friendly.
• … solves big problems and makes the world better.
• … is a powerful tool they can master.
• … is beautiful.
• … lets them learn in their own ways.
• … is connected to their lives.
• … asks “why” and not just “how.”
• … opens the world.

Cool Math for Older Kids

James Tanton (one of the above-mentioned authors) used to have a website full of cool math puzzles. Unfortunately, that’s gone the way of too many sites I used to love. But I was doing research on something else and stumbled upon the Internet Archive’s web backup of his “MAA AMC Curriculum Inspirations” puzzles, so I decided to rescue a few of them.

I’ve got two puzzles for upper-elementary to middle school, and two for middle-to-high school.

These are challenges taken from the AMC competitions, along with Tanton’s wonderfully encouraging, common-sense tips for figuring them out—Including how to get started when you have absolutely no idea what to do.

Have fun playing math with your older kids!

But First…

Tanton also wrote a short-lived blog on the Medium website. This was one of my favorite articles, “Two Key — but ignored— Steps to Solving Any Math Problem.”

Great tips to help you tackle the challenge problems below!

“Every challenge or problem we encounter in mathematics (or life!) elicits a human response. The dryness of textbooks and worksheets in the school world might suggest otherwise, but connecting with one’s emotions is fundamental and vital for success — and of course, joy — in doing mathematics.

“So… Experience mathematics as a human! Help your students do so too!”

—James Tanton

Now, let’s play some math…

Two Trees

The top of one tree is 16 feet higher than the top of another tree.

The heights of the two trees are in the ratio 3 : 4.

In feet, how tall is the taller tree?

Puzzle over the problem for yourself.

For tips or to check your answer download the worked-out solution here.

Angles in a Star

The degree measure of angle A is…?

No peeking!

For tips or to check your answer download the worked-out solution here.

Areas in Triangles

The area of ∆EBD is one third of the area of 3–4–5 ∆ABC.

Segment DE is perpendicular to segment AB.

What is BD?

What fun is a puzzle if someone just gives it away?

For tips or to check your answer download the worked-out solution here.

Differences of Four Numbers

Brian writes down four integers whose sum is 44:

w > x > y > z

The pairwise positive differences of these numbers are 1, 3, 4, 5, 6, and 9.

What is the sum of the possible values for w?

Thinking hard is a challenge—but that’s what makes it fun!

For tips or to check your answer download the worked-out solution here.