This game pushes students to consider area and perimeter beyond the basic shapes.

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!

Area Battle

Math Concepts: area, square units, perimeter, regular and irregular polygons.

Players: two to four.

Equipment: homemade game cards, pencils or markers.

The FREE 68-page printable (pdf) Prealgebra & Geometry Printables file includes Area Battle card templates, along with all the game boards for the Math You Can Play: Prealgebra & Geometry book.

Set-Up

Students cut square grid paper into 5 × 5 squares. Each student creates a personal deck of eleven cards, as follows:

• Regular or irregular polygons, one with each shaded-in area: 1, 2, 3, …, 8, and 9 square units. The area must be easy to count, so build each polygon from a combination of squares and half-squares.
• Two additional polygon cards with areas of the player’s choice.

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

How to Play

Stack your card deck face down in front of you. Draw the top two cards for your hand.

On your turn, call “high” or “low” for whether the greatest or least area wins the trick. Then each player chooses a card from their hand and holds it face down. When all players are ready, turn the cards up, and the winner captures them all. Each player keeps a pile of prisoner cards.

In case of a tie, those players reveal their other card, and the longest perimeter takes the trick. If it’s still a tie, leave the cards on the table for the winner of the next trick.

Players draw one or two new cards to replenish their hands before the next turn.

The game ends when at least one player runs out of cards. Whoever has captured the most prisoners wins the game. Give the prisoner cards back to their owner, unless you are playing the Spoils of War variation.

Variations

In between games, players can make new cards. But they must always play with a deck that fits the rules: nine cards with the areas 1–9, plus two extras.

Perimeter Battle: On each turn, the low or high perimeter takes the trick. In case of a tie, the greatest area wins the second fight.

Spoils of War: If you win the game, you may claim one card from your prisoner pile to replace the matching card from your deck. Erase the identification and add your own symbol on the back. You may offer your old card to the losing player as a consolation prize, or keep it as a backup card.

History

When John Golden created and tested this game, the students gave it “universal thumbs up as to whether they would recommend it to other teachers. They were requesting time to play in class later, which is surely a good sign.”

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 241: Number Race

Write the numbers 1–12 down the side of your page, each on its own line. Outline a row of four boxes next to each number (or more boxes for a longer race).

Roll two dice, and shade in a box for the sum.

The number that fills its boxes first wins the race.

Try another race. Does the same number always win? Which number can never win, and why? What other questions can you ask?

Do you ever take your kids’ math tests? It helps me remember what it is like to be a student. I push myself to work quickly, trying to finish in about 1/3 the allotted time, to mimic the pressure students feel. And whenever I do this, I find myself prone to the same stupid mistakes that students make.

Even teachers are human.

In this case, it was a multi-step word problem, a barrage of information to stumble through. In the middle of it all sat this statement:

“…and there were 3/4 as many dragons as gryphons…”

My eyes saw the words, but my mind heard it this way:

“…and 3/4 of them were dragons…”

What do you think — did I get the answer right? Of course not! Every little word in a math problem is important, and misreading even the smallest word can lead a student astray. My mental glitch encompassed several words, and my final tally of mythological creatures was correspondingly screwy.

But here is the more important question: Can you explain the difference between these two statements?

If Johnny Can’t Read, Then He Can’t Do Math

Few people realize that a math test is also a test of reading comprehension. In some cases the reading level required far exceeds the level of math.

Before they can solve a word problem, children must be able to understand what is written. They must hold that information in their working memory as they think about how to find a solution. They need to comprehend what they read, paraphrase it—concentrating on the relevant facts—and translate that information into a mathematical expression. Frequently they have to read between the lines and understand something that is implied, not explicitly stated.

That quote is from an article at Teen Literacy Tips blog. Does a literature teacher have anything useful to say about solving math problems? Well, the fact that word problems are also called story problems should clue us in to a significant connection.

When students struggle with word problems, more often than not it is a language issue that confuses them.

Consider, for example, the maddening confusion that may be caused by these two simple statements:

• Eight divided in half is four.
• Eight divided by one-half is sixteen.

These sentences seem to contradict each other, yet both are true. Can you explain the difference? If your students keep a math journal, you might use this question as a writing prompt.

(I’ve included one possible answer at the bottom of this post.)

My Problem with Dragons and Gryphons

In my word problem, it turned out there were fifty-six fantasy creatures in all. I got that part of the answer, but then I needed to find the number of gryphons.

This is how I did it…

“… and 3/4 of them were dragons …”

Imagine all fifty-six creatures in a zoo, split evenly into four pens. Three of the pens hold various breeds of dragon, and the fourth pen has the gryphons.

4 pens = 56 creatures

1 pen = 56 ÷ 4 = 14 gryphons

But that was not at all what the problem said. If I had been paying better attention to what I read, this is how I should have solved the problem…

“… and there were 3/4 as many dragons as gryphons …”

This zoo is wiser than the first one, because it splits the troublesome creatures into smaller groups, so there’s less chance of a fight that might endanger visitors. The gryphons are split into four pens, so each pen holds one-fourth of the gryphons. The dragons have three pens all to themselves, which makes three-fourths as many as the gryphons. Thus, we have a total of seven fenced-in areas.

7 pens = 56 creatures

1 pen = 56 ÷ 7 = 8 creatures per pen

4 pens = 4 × 8 = 32 gryphons

Math Journal Prompt

To make the reading issue more difficult, consider this: All the statements in the following list are equivalent. Compare each statement to the second diagram above, the correct one. Can you see each relationship?

• There are 25% fewer dragons than gryphons.
• For every four gryphons, there are three dragons.
• The ratio of dragons to gryphons is 3:4.
• Four out of every seven creatures is a gryphon.
• There are one-third more gryphons than dragons.
• There are one-fourth fewer dragons than gryphons.
• If you tag a creature at random from the group, the probability of choosing a dragon is three-sevenths.

Can you think of any other ways to say it? This would be another good math journal writing prompt.

The more we can work with our children on reading, paraphrasing, and translating word problems into mathematical expressions, the better prepared they will be to face the challenges they will meet in future math classes and in the textbook of life.

One Possible Answer to Dividing in/by Half

When you divide a number in half, you split it into two equal parts.

But when you divide a number by 1/2, you are finding how many halves it takes to make that number — that is, you are cutting it into half-size pieces and counting how many there are.

And in that case, because each whole thing is two halves, there will be twice as many pieces as the number you started with.

This game challenges players to make logical deductions about number permutations.

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!

Codebreaker

Math Concepts: permutations, logical deduction.

Players: two or more.

Equipment: paper and pencil, or whiteboard and marker.

How to Play

One player (the codemaker) thinks of a three-digit number between 000 and 999, writes this number on a piece of paper, and folds the paper to hide it. The other players (codebreakers) draw a two-column chart labeled Guesses and Clues. Players can each keep separate written records, or all may share a single chart on a whiteboard.

The codebreakers take turns guessing a three-digit number, which they record in the guess column. The codemaker responds with the appropriate clues:

• If one of the digits is the right number in the right place, say “There’s one number correct,” and draw a smiley face in the clues column. Do not tell which digit won the smiley.

• If one of the digits is a correct number but in the wrong place, say “There’s one number right, but it’s not where it belongs,” and draw an empty circle in the clues column.

• If more than one digit matches the secret code, tell the total number of correct and almost-right digits—and draw that many smiley faces and plain circles—but never reveal which digit matches which clue.

With two players, a round of play is two numbers, so each player has a chance to be the codemaker. Whoever breaks the code with the fewest guesses wins. For group play, whoever guesses the number gets to be codemaker for the next round.

Variations

For beginners, don’t allow repeating digits, and make it easier by using only the numbers 1–5. A secret number like 142 would be fine, but not 223. Advanced players can use four- to six-digit numbers and allow repetition.

Or try playing without paper, keeping track of guesses and clues in your head. This is a mind-bending challenge, but it can be done.

History

In the wee morning hours on a cross-country road trip, I made up the paperless version of Codebreaker to keep the driver awake. In my mind, it was a modification of the commercial board game Mastermind, but according to Wikipedia, similar guessing games with numbers or words date back at least a century.

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 282: Success

One thing I’m proud of in math is…

[Based on a problem I made up for my co-op students, once upon a time…]

Conversion factors are special fractions that contain problem-solving information. Why are they called conversion factors?

• “Conversion” means change, and conversion factors help you change the numbers and units in your problem.

• “Factors” are things you multiply with. So to use a conversion factor, you will multiply it by something.

For instance, if I am driving an average of 60 mph on the highway, I can use that rate as a conversion factor. I may use the fraction:

Or I may flip it over to make:

It all depends on what problem I want to solve.

After driving two hours, how far have I gone?

But if I am planning to go 240 more miles, how much longer will it take?

Any rate can be used as a conversion factor. You can recognize them by their form: this per that. Miles per hour, dollars per gallon, cm per meter, and many, many more.

Of course, you will need to use the rate that is relevant to the problem you are trying to solve. If I were trying to figure out how far a tank of gas would take me, it wouldn’t be any help to know that an M1A1 Abrams tank would get about 1/3 mile per gallon. I won’t be driving one of those.

Using Conversion Factors Is Multiplying by One

If I am driving a steady 60 mph on the highway, then driving for one hour is exactly the same as driving 60 miles, and:

This may be easier to see if you think of kitchen measurements. Two cups of sour cream are exactly the same as one pint of sour cream, so:

If I want to find out how many cups are in 3 pints of sour cream, I can multiply by the conversion factor:

Multiplying by one does not change the original number. In the same way, multiplying by a conversion factor does not change the original amount of stuff. It only changes the units that you measure the stuff in. When I multiplied 3 pints times the conversion factor, I did not change how much sour cream I had, only the way I was measuring it.

Conversion Factors Can Always Be Flipped Over

If there are:

Then there must also be:

If I draw house plans at a scale of:

Then I know that:

If there are:

Then there are:

Or if a single-engine airplane is burning fuel at 8 gallons/hour, then the pilot has only 1/8 hour left to fly for every gallon left in his tank.

This is true for all conversion factors, and it is an important part of what makes them so useful in solving problems. You can choose whichever form of the conversion factor seems most helpful in the problem at hand.

How can you know which form will help you solve the problem? Look at the units you have, and think about the units you need to end up with. In the sour cream measurement above, I started with pints and I wanted to end up with cups. That meant I needed a conversion factor with cups on top (so I would end up with that unit) and pints on bottom (to cancel out).

Just as numbers cancel out when the same number is on the top and bottom of a fraction (2/2 = 2 ÷ 2 = 1), so do units cancel out if you have the same unit in the numerator and denominator. So pints/pints= 1.

Important Tip 1: Some teachers don’t like to use the word “cancel” because students forget that it’s a shortcut for saying “divide by each other to equal one.” So the forgetful kids cross out numbers or units and put a zero in their place. But you would never do that, right?

You Can String Conversion Factors Together

String several conversion factors together to solve more complicated problems. How many cups of milk are there in a gallon jug?

As you write out your string of factors, you will want to draw a line through each unit as it cancels out, and then whatever is left will be the units of your answer.

Important Tip 2: Notice that only the units cancel — not the numbers. Even after I canceled out the quarts, the 4 was still part of my calculation.

Let’s Try One More

The true power of conversion factors is their ability to change one piece of information into something that at first glance seems to be unrelated to the number with which you started.

Suppose I drove for 45 minutes at 55 mph in a pickup truck that gets 18 miles to the gallon, and I wanted to know how much gas I used.

To find out, I start with a plain number that I know (in this case, the 45 minutes) and use conversion factors to cancel out units until I get the units I want for my answer (gallons of gas). How can I change minutes into gallons?

I need a string of conversion factors:

How Old Are You, Anyway?

If you want to find your exact age in nanoseconds, you need to know the exact moment at which you were born. But for a rough estimate, just knowing your birthday will do. First, find out how many days you have lived:

Once you know how many days you have lived, you can use conversion factors to find out how many nanoseconds that would be.

You know how many hours are in a day, minutes in an hour, and seconds in a minute.

And just in case you weren’t quite sure:

Have fun playing around with conversion factors. You will be surprised how many problems these mathematical wonders can solve.

This game builds conceptual understanding of multiplication or fraction models.

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!

Math Model Rummy

Math Concepts: multiplication or fraction models.

Players: two or more.

Equipment: one deck of math model cards, or a double deck for four or more players.

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, or books. 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.

Or take the quick route: download my free Multiplication & Fraction Printables file, which includes two decks of premade math model cards. Print the pages on card stock, and laminate for durability if desired.

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 includes products from 2×2 to 6×6. Each book consists of four cards — the multiplication equation or fraction, plus three pictures.

How to Play

Deal seven cards per player, and place the remaining cards face down as a draw pile. Turn up the top card of the deck to start the discard pile.

On your turn, you may either draw the top card from the deck or pick up the discard pile as far back as desired. But if you pick up more than the top discard, you have to meld the farthest-back card you take.

After drawing, you may meld — that is, place three or more matching cards face up on the table in front of you. If you have the fourth card in a set that has already been played (by any player), you may also lay that down in front of you.

Finally, put one card on the discard pile to end your turn. But if you discard a card that could have been played, any other player can call “Rummy!” and meld your discard.

If the deck runs out, take all the discards except the top one. Shuffle these cards and place them face down so the next player has a stack to draw from. Play continues until one player runs out of cards (either by laying them all down or by discarding the last one).

Scoring

Count each player’s score as follows:

• Each card played on the table is worth 5 points.
• For every card remaining in the hand, subtract 2 points.
• The player who went out gets a bonus of 15 points.

You may play a single hand, just for fun. Or play several hands, and the first player to reach 300 points wins the game. Or set a different point goal based on how long you want the game to last.

Variations

Floater: You must have a discard to end the game, and this must be a card that could not be melded. If you lay down all your cards without a discard, you become a floater. Continue to play your turn — drawing or picking from the discard pile and melding cards — until you can go out with a proper discard. (But it is illegal to pick up just the top discard and immediately discard it again.)

History

According to the Pagat website, Rummy-style games first appeared in the early twentieth century. Like many card games, Rummy picked up different flavors as it traveled from one player to another over the years. Feel free to modify the rules above to fit your family’s favorite way to play.

We talked about how math is a whole world full of big, living ideas that we can explore with our children. What it means to have a living education in math, opening our children’s minds to big ideas about the relationships of numbers, shapes, and patterns. And how to tell if our kids really understand what they are doing, or if they’re just parroting back the steps we taught them.

And lots more — listen for yourself:

“Math doesn’t have to be a drudgery, a chore that we finish so we can check it off of our list. Math can be a delight when we give children things to think about, ideas to wonder about. When we explore the world of numbers and shapes together, math can be a playful delight for the whole family.”

—Denise Gaskins

For full show notes, see Season 11, Episode 144: Charlotte Mason’s Living Math with Denise Gaskins.

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

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

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

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

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

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

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

How To Use a Math Game Prompt

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

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

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

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

Journaling Prompt 53: 2-D Nim

(two players)

Draw a rectangular grid of 10–20 squares in area. For example, a 3 × 5 grid makes a 15-square gameboard.

On your turn, cross out one or two of the grid squares — but you can mark two squares only if they share a side.

Whoever marks the last square wins.

Misère variation: Whoever marks the last square loses.

[Now available exclusively in my Playful Math Store! Ebook only at the moment, but print editions are coming soon.]

Here’s an example of such a question, an excerpt from my earlier book, 70+ Things To Do with a Hundred Chart: Number, Shape, and Logic Activities from Preschool to Middle School.

Have fun exploring math with your kids!

Column Jumping

Print your choice of hundred chart from any online resource or from my free Hundred Charts Galore booklet.

Choose any column of the hundred chart to study. For example, the threes column: 3, 13, 23, and so on.

• Pick any two numbers in that column and add them together. In which column will you find the sum?

• Try a different pair of numbers. Does their sum land in the same column?

• What if the numbers are so big they go off your chart, like 73 + 93? Will the pattern still work? How can you tell?

• What happens if you add four of the threes-column numbers together? Where do you find that sum?

• How many threes-column numbers do you have to add for the sum to land in the eights column? Does it matter which numbers you pick?

Make a chart of the column jumping pattern for the threes. If you add enough threes-column numbers, can you get to each of the other columns?

Swim in the Sea of Ideas

Examine other column jumping patterns.

• How many fours-column numbers do you have to add to land back in the fours column?

• Can you add fours-column numbers and end up in the sevens? Why not?

• What is the column jumping pattern for the tens?

Dive Deeper

Can you think of a way to predict the pattern for any numbers? If you choose a random column n, and you jump j times, could you tell how to know in advance where you’ll land?

Do you think column jumping works for subtraction?

• It’s easy to understand “add four of the numbers in the threes column.” But what might it mean to “subtract four of the numbers”?

• What happens when you subtract one threes-column number from another one? In which column will the difference land?

• What if you subtract a big number from a smaller one? Will the pattern extend into negative numbers?

What other questions can you ask?

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.