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

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

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

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

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

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

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

How To Use a Research Prompt

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

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

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

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

Journaling Prompt 82: Mini-Biography 2

Write about a female mathematician who is NOT Ada Lovelace.

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

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

—I. N. Herstein

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

But don’t peek!

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

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

• Figure it out for yourself!

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

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

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

Surveying the Egyptian Farm

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

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

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

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

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

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

Proportional Reasoning

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

You have to understand ratios and proportions.

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

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

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

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

Proportional Measurements

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

But what measurements shall we compare?

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

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

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

So to check whether our answers agree:

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

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

A Research Project

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

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

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

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

Playing with Ratios

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

• Rate Puzzle: How Fast Does She Read?

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

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

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

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

Power Up

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

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

Set-Up

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

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

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

How to Play

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

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

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

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

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

Variations

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

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

History

Game creator John Golden writes:

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

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

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

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

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

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

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

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

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

How To Use a Math Art Prompt

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

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

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

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

Journaling Prompt 221: Make a Maze

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

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

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

—Freeman Dyson

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

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

“Geometry,” Fibonacci said.

“What?”

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

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

“And taxes…”

Measuring the Land

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

Area = length × width

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

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

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

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

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

“They could cut any property into triangles?”

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

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

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

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

Try It for Yourself

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

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

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

Finally, try this challenge from my old newsletter:

• Egyptian farm puzzle

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

A Puzzle for Older Students

Given the equation for the area of a rectangle:

Area (rectangle) = length × width

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

Area (right triangle) = ½ base × height

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

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

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

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

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

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

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

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

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

Myin36, CC BY-SA 4.0, via Wikimedia Commons

How might that relate to triangles?

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

I Lied to You!

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

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

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

Then why did I tell the story like that?

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

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

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

Can you see why this is true?

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

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

Another Way to Form Right Angles

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

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

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

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

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

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

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

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

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

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

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

This game builds number sense and a playful approach to math.

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!

Next-Door Neighbors

Math Concepts: number properties, numbers within 100.

Players: two players, child and adult.

Equipment: printed hundred chart.

Set-Up

The free 50-page PDF Hundred Charts Galore! file offers many variations of hundred charts. Choose your favorite, or change it up each time you play.

How to Play

Ask your child to choose any two next-door neighbor numbers — that is, two numbers in side-by-side squares — but not to tell you which ones. Then the child should add the numbers and tell you the sum.

Can you show off your magical skills by identifying the original numbers? Just subtract one from the sum, and then divide that new number in half to find the smaller of your child’s two neighbor numbers.

It may take several tries for children to figure out this trick. After they get it, take turns adding neighbor numbers for the other person to guess.

Variations

Do you notice a pattern in the next-door neighbor sums? Will the total always be odd? Why, or why not?

What will happen if you add two apartment-house neighbors (one number above the other) — will the answer still be odd? Can you think of a “magic trick” pattern to identify the original numbers from their sum?

What about two numbers that sit diagonal to each other, touching at a corner?

Answers to the Puzzle

Teacher’s Tip: Don’t peek. Nobody likes a spoiler.
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Next-door neighbor sums are the same as the double of the smaller number, plus one. To find the original numbers, subtract that extra one and then divide the answer in half. That will give you the smaller of the two numbers.

The next-door neighbor sums are always odd because they are doubles-plus-one, which is the definition of an odd number.

Apartment-house neighbors are ten apart, numbers like 32 and 42. If someone tells you the sum, you can subtract that extra ten and then divide the result in half. That will give you the smaller of the two numbers — then add ten to get the larger number.

But numbers that sit diagonal to each other may be either nine or eleven apart. Since you don’t know which way the diagonal slants, there is no way to discover them just from their sum.

Writing to Learn Math: Math journal explanations avoid the formality that turns so many students away from geometry proofs.

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 Explanation Prompt

Math journal explanations avoid the formality that turns so many students away from geometry proofs. These informal “reason-poems” drive at the heart of a student’s understanding. How did they figure this out? Why does their method work? Is the pattern they found real or just a temporary coincidence? How do they know?

When you run out of creative journaling ideas, you can always go back to the basic mathematical question: “Why?”

For older students, challenge them to explain a concept so that a kindergarten student or second grader could understand. That’s more difficult than it sounds, but the attempt forces students to clarify their own ideas about the topic.

Journaling Prompt 129: Strategic Thinking

Play a game. Describe your strategy for winning.

How do you plan your moves?

How do you figure out responses to your opponent’s moves?

The intrepid Quote Investigator played around with math recently:

“You know, people think that mathematics is complicated. Mathematics is the simple bit. It’s the stuff we can understand. It’s cats that are complicated.”

—John Horton Conway

It reminded me of one of my favorite homeschooling exercises:

• I would write out a short paragraph, all lower-case with no punctuation (and usually including some misspellings and grammar errors) on a whiteboard.

• Then my children took turns marking the passage like editors correcting my work.

One of these exercises was a series of short quotations about cats, which my youngest daughter loved.

So in today’s post, I’ve put together an editing exercise you can share with your children, along with several links for playing with mathematical cats.

Enjoy!

Cat Quotations Workbook

This mini-workbook offers five pages of editing practice:

• Cat Quotes: Proofreading Practice with Purring Furballs

(No answers, since punctuation is flexible, and what’s right is often in the eye of the editor.)

If you need a handout of proofreader’s markup, Sharon Watson’s got you covered:

• Proofreading Marks Article with Practice Assignments
• Proofreading Marks Handout

Mathematical Cats

And here’s one of my favorite older playful math activity websites. Due to internet entropy, not everything still works, but there’s plenty to explore:

• Math Cats

And a few math puzzles featuring my own cats:

• The Wisdom of Cats
• Math Puzzle from the Ancient Kingdom of Cats
• Math Journals: Save the Cat!
• Playful Math Journaling with a Cat

And If You Have a Child Who Loves Cats

My daughter would have delighted in this workbook series, if it had come out when she was young. Absolutely adorable!

• Kitten Math by Kelli Pearson
  (affiliate link)

This game uses the area model to visualize and build understanding of integer multiplication.

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

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

Grid Fight

Math Concepts: integer multiplication, area model for multiplication.

Players: two players or two teams.

Equipment: printed gameboard or square grid paper, playing cards, pencils or colored 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

Print a gameboard for the players to share. Or make your own gameboard on square grid paper: two 12 × 12 squares separated by a “zero” line.

Label one square positive (+) and the other negative (−). Choose which player or team will use the positive side of the gameboard grid and who takes the negatives.

Remove the face cards from your deck. Agree on which color represents negative numbers.

How to Play

Deal three cards to each player. Place the rest of the deck face down as a draw pile.

On each turn, both players make their move at the same time. Choose one card to play and hold it face down on the table in front of you. When both players are ready, turn up the chosen cards and multiply the numbers.

If the product is greater than zero, the positive player colors a rectangle of that size on the positive side of the grid. If the product is negative, that player colors a rectangle on the negative side.

Your goal is to fill in as many complete rows as you can, as in the game Tetris. But if you can’t fit your rectangle on the grid, you lose that turn.

Both players discard their used cards and draw to replenish their hands.

Play twelve turns. Whoever fills in the most rows wins the game.

Variations

Do you want a longer game? Play on both halves of the gameboard at once. Players may color their rectangle in either square (on their own side of the zero line) wherever they can fit it in. The game continues until the grids are so full that players lose three turns in a row.

History

I owe this game (and many others!) to the amazing John Golden. Visit his Math Hombre Games webpage for more ideas.

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 91: Sci-Fi Puzzle 2

Aliens from the planet Crimbat double their population every year. They send out lots of colony spaceships to keep from overpopulating their home.

A Crimbatti settlement on Earth contained 6,400 aliens in its 6th year.

In which year was the population half that size?

How many Crimbatti settlers were there in the first year? How long until the aliens outnumber the humans on Earth?

This week, we have the first mathematical adventure of Alexandria Jones, which takes place in the early days of her father’s archaeological expedition mentioned in the introduction post.

Herein we meet the one major character who is not named after a mathematician from history: Alex’s nemesis, Simon Skulk.

While my fiction doesn’t rise close to the level of “art,” I hope it offers a glimpse into the playful nature of mathematics for you and your children.

The Secret of the Pharaoh’s Treasure

Alexandria Jones stood outside her father’s tent. The glare of the sun on the rocky desert hurt her eyes. Holding up a hand to shield her gaze, she spotted her dad (the world-famous archaeologist) arguing with the foreman.

Poor Dad, she thought. He was sure this was the right site, but so far he’s found nothing.

She looked down at her feet, where her faithful dog Ramus waited, panting. “Well, Rammy, it looks like Dad will be busy for while. What do you say? Shall we go exploring?”

Alexandria ducked into the tent for her backpack and canteen.

Thump! Something bounced against the side of the tent. Ramus barked.

Alex stepped outside and looked quickly around. No one was in sight. She saw a fist-sized rock beside the tent, with a note tied to it. She picked it up and read:

Ha! The real Pharaoh’s Treasure lies under a pyramid of stones, and it’s mine. You can’t stop me this time!
—Simon Skulk

“Find him, Rammy. Which way did he go?”

Growling, the dog ran down the trail to the valley. Alex followed him, but pulled up short when Ramus started into a dark cave. She swung her backpack off her shoulder and drew out a flashlight.

“The treasure must have been buried in a cave, as some of the later pharaohs were,” she said. “Okay, Rammy. Lead on.”

The cave ended at the foot of a head-high pyramid of small, rough-cut stones. Her father’s enemy was just setting his lantern on the ground.

“You creep!” Alex shouted. “That treasure belongs in a museum.”

The man laughed and reached for a stone. “Not if I find it first. The Pharaoh’s Treasure should bring me a pretty profit!”

Alexandria Jones ran to grab a stone from the pyramid. If she could just uncover the treasure before he did…

Can You Find the Pharaoh’s Treasure?

The Pharaoh’s Pyramid is a strategy game for two players. Print out the PDF gameboard or draw your own pyramid of stones.

• Each player will need a pencil or colored marker. Youngest player gets the choice of going first or second.

• On your turn, draw an “X” on one or two stones, to remove them from the pyramid. You do not have to work from the top down; you may mark any stones you wish. But if you want to mark two stones on the same turn, they must be touching each other.

• Take turns marking stones until they are all gone. Whoever marks the last stone uncovers the Pharaoh’s Treasure and wins the game.

If you draw your own game, experiment with different sized pyramids. How does changing the number of stones affect the strategy of the game?

The Pharaoh’s Treasure, Continued

“I can’t believe it!” Simon Skulk threw down the last stone in disgust and walked away. At the mouth of the cave, he turned back and shook his fist. “You haven’t seen the last of me, Alexandria Jones.”

Her muscles aching, Alex sank to the ground and hugged her dog. The she gave him a little push toward the front of the cave. “Rammy, go get Dad.”

Ramus barked once and took off running.

Alex turned back to look at the Pharaoh’s Treasure. Where the last stone had stood was a hole. In the hole lay three wooden sticks, like tent pegs, and a long loop of rope with twelve evenly-spaced knots.

What could it be?

It seemed as though Alex sat there forever, shivering as the desert sun went down, her muscles growing stiff.

Finally, Ramus came bounding into the cave and licked her face. Her father ran in behind the dog, giving her a hug that chased the stiffness away.

Then, slipping on a pair of vinyl gloves from his pocket, Fibonacci Jones reached into the hole and gently touched the pegs and rope.

“You found it,” he said. “The Pharaoh’s Treasure! Isn’t it beautiful?”

Alexandria frowned. “But what is it?”

The ancient Egyptians were careful builders, Fibonacci Jones explained. To keep their pyramids, temples, and other buildings squared up properly, Egyptian surveyors (called “rope stretchers”) used a powerful design tool: the right triangle.”

How to Use the Rope Triangle

You can make your own rope stretcher’s triangle with a piece of string and three straight pins.

Tie eleven evenly-spaced knots, about 2 inches (5 cm) apart, and then tie the ends of the string together to make the twelfth knot. Getting the knots right is the hardest part. Check your spacing with a ruler or by folding the string several times, to make sure the knots all line up.

Then use the straight pins as your pegs, and use the cardboard back of a spiral notebook or the side of a cardboard box as your ground:

• Pound one peg into the ground where you want the square corner. Loop the rope around the peg with one knot exactly at the peg.

• Move along the line of your wall a distance of three knots on the rope. Pound in the second peg, with the rope looped around it and with one knot exactly at the peg.

• Slip the last peg into the rope loop and pull it tight, four knots from the first peg at the corner. Pound this last peg into the ground, making sure it is exactly at its knot.

This makes a right triangle with sides of 3, 4, and 5 lengths on the rope. The corner is perfectly square.

A Puzzle for Older Students

Play around with your Pharaoh’s Treasure model.

• Can you explain why the instructions above form a right angle?

• There are other ways to form a right angle using the rope and pegs. Can you find an alternate method?

You need to find some way to make two lines that are perpendicular to each other. Feel free to use extra ropes or pegs, if needed. The Egyptian rope stretchers could use as many or as few ropes as they liked.

If you want a hint, consider Euclid’s Book 1, Proposition 11.

And if you get stumped, I will put one possible answer in a future post. But don’t give up too quickly. Remember the math adventurer’s rule.

Math Adventurer’s Rule: Figure it out for yourself!

When I tell you the answer, you miss out on the fun of solving the puzzle. Figure it out for yourself—and then check the answer just to prove that you got it right.

To be continued…

This game requires students to make logical deductions about number relationships.

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!

Digit Disguises

Math Concepts: addition, subtraction, multiplication, division, algebraic reasoning.

Players: two players or two teams. Best with teams.

Equipment: printed gameboards (optional), pencils and paper, or whiteboard and markers.

Set-Up

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.

Each player or team needs a separate gameboard.

If you’re not using printed gameboards, each player or team makes a chart with two columns labeled Mine and Theirs. List the letters A–J in each column. Next to each letter in your own column, write one of the whole numbers 0–9 in any mixed-up order. Fold your paper to keep that column hidden from your opponent. Slip a paperclip over the fold for greater security.

How to Play

On your turn, ask for the result of a basic arithmetic calculation involving two different letters. For example, you might ask:

• A + B = ?
• C − D = ?
• E × F = ?
• G ÷ H = ?

But you may not ask for something like “I − I” or “J ÷ J.”

The other player/team figures out the answer to your question, using their values for each letter, but they do not tell you the number. If the answer is on their chart, they tell you the letter for that value. Or they tell you that the result is not one of their letters.

Players never say a number in response to a question. They only say a letter or “Not a letter.”

Write notes to help you remember each clue. You’ll need these to deduce your opponent’s number code.

Then it is their turn to ask you for a clue about your secret code.

When one player/team claims to have broken the opponent’s number code, the game is over. They read aloud the number values they have figured out. If they’re entirely correct, they win the game. But if any of the answers are wrong, the opponent wins.

Variations

Don’t end the game after the first code is guessed. Whether the guessing team won or lost, the other team keeps asking questions until they also break the code. If both teams fail, the game is a tie.

For Beginners: Start with only four unknown numbers, and allow players to give number answers in response to each question.

Keeper of the Code: (Mastermind-style) One player makes the code, and the other players try to crack it. Players take turns asking for clues, but they may discuss the answers and plan a strategy together. This variation is good for younger players, if you let them be the Keeper while older students or adults are guessing.

History

Math lecturer David Butler created this game and writes:

“I was struck by how quickly the logic got complicated, and then how quickly it all cascaded into finding everything when I finally got a few different numbers.”

Butler also shared a printable gameboard that can fold to stand on the table Battleship-style, hiding your secret code.

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 John von Neumann

If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.

—John von Neumann

It began with some 4th-8th grade friends who met in my dining room to work math puzzles and play games. At first, a few of the kids wondered how anyone could have fun with math. But we did enjoy ourselves, and Math Club grew until we couldn’t fit anyone else around the table.

When one girl had to move away, I thought, “Why not send Math Club with her?” Thus was born Math Club by Mail newsletter (later renamed Mathematical Adventures), a 4-year snail-mail newsletter that brought puzzles, strategy games, art projects, historical tidbits, and more to readers as far away as Hawaii.

Never very many readers, I admit—certainly never enough to make a profit—but it was a blast while it lasted.

So I thought you might like to read Alex’s story from the beginning, as my original math club kids did. The stories will appear sporadically, scattered between my more typical math-activity and teaching tip posts.

I hope you will have as much fun reading these adventures as I had writing them.

Introducing Alexandria Jones

My newsletter stories followed the adventures of homeschooler Alexandria Jones, her archaeologist father, and the rest of their family as they discovered a triangular treasure from Ancient Egypt, puzzled over patterns in Pythagorean pebbles, or explored the applications of math in daily life.

We learned about hieroglyphic math, fractals and chaos theory, logarithms, geometric algebra, and how to multiply any numbers using only the times-two table.

As I attempt to resurrect Alexandria Jones and her family here, the stories will appear sporadically, along with snippets from history and a side dish of math puzzles.

The first story, down below, actually appeared later in the original newsletters, but it makes a nice introduction.

Alexandria’s Dog is Now a Teacher

From the beginning, Alex’s dog supported her adventures with eager energy. In the original stories, Rammy was named after a famous Indian mathematician, but a reader of Indian descent was kind enough to write to me about a difference in our cultures.

In the US, or at least in the parts with which I am familiar, it is common to name one’s pet after a famous person. In India, however, to name a dog after a human is a very deep insult.

Therefore, I have re-named Alexandria Jones‘s dog after a Westerner. I would like to keep the tradition of naming the characters in my Alex stories after people in math history, so I am hoping this one will not offend anyone.

Peter Ramus: Homeschooler, Teacher, Magnet for Trouble

Peter Ramus is also known as Petrus Ramus or as Pierre de la Ramée. Born in France in 1515, he was educated at home until the age of twelve, when he entered the Collège de Navarre in Paris. He worked his way through college, graduated with a master’s degree at the age of 21, and began teaching philosophy—until his attacks on the dominant Aristotelian curriculum of his day got him banned.

Ramus switched to the study of mathematics, was reinstated, and rose to become the head of the the Collège de Presles. Always involved in controversy, however, he converted to Calvinism and spent several years in exile due to religious wars.

In addition to being a popular lecturer and author, Ramus was determined to reform educational philosophy. He had no patience for mnemonic tricks or rote learning. Instead, he thought it was the teacher’s job to break each of the liberal arts down into its most basic principles and organize these into a systematic and practical method of study. Above all else, education should be useful.

Due to his growing conviction about the fundamental importance of mathematics, he proposed a chair of mathematics at the University of Paris. Putting his money where is mouth was, Ramus endowed the chair in his will.

In 1572, he was killed in a massacre of French Huguenots.

A New Style of Teaching

Peter Ramus promoted many educational reforms, including:

• the simplification of knowledge, using his “laws of method”
• attention to pedagogy, the science of effective teaching
• increased attention to teaching physics and other sciences
• an emphasis on experimentation and practical application
• a de-emphasis of theoretical speculation (including abstract geometry proofs)

Ramus had great respect for Aristotle and other ancient philosophers, but fought repeatedly with the Aristotelians of his day. For at least a century after he died, Ramists and Aristotelians continued to argue about what logic really was and how we should teach students.

Enough Background! Here’s a Story: The Mysterious Temporal Freeze

Alexandria Jones stepped into the huge tent that protected her father’s excavation site from the desert winds. She laughed to herself. It was like walking into a circus.

She knelt down to whisper in the ear of her faithful dog Ramus.

“In this ring, grad students carefully brush away another layer of sand. In the next ring, the artist sketches every piece as it is found.” She waved her arm. “And over there, our flashiest attraction—drum roll, please—the photographers shoot each shard of pottery from every possible angle. But where is the Master of Ceremonies?”

Alex and Rammy found Professor Jones near the back of the tent, talking to another student. While she waited for her dad, she looked through an assortment of numbered artifacts that were ready to be packed and sent to the museum.

Fibonacci Jones had discovered the tomb of a scribe, an ancient Egyptian secretary and bookkeeper. It didn’t look like much—merely a hole in the ground, way different from a pharaoh’s hoard—but Alex knew her father. To him, a roll of papyrus was a treasure more precious than jewels.

She picked up the scribe’s palette, a carved wooden block about the size of a box of watercolors. This palette had three bowl-shaped pits for ink. Two of these held dried-out remains of the scribe’s black ink, and in the last pit she could see a faint stain of red. Four thin reeds lay in a groove, ready for the scribe to dip in the ink and write.

(Styluses, she thought they were called. Or styli? Have to ask Dad.)

Alex looked up to see her father walking toward her. Strangely, he seemed to be moving in slow motion. He froze in mid-step and stood statue-still. When she opened her mouth to speak to him, no sound came out.

At her feet, Rammy whimpered, then fell silent. Alex tried to turn her head, but she couldn’t move.

What could be happening to them?

What Really Happened

The Alexandria Jones stories were originally written for my math newsletter. Since it was a one-person operation, when I got the flu, all production ground to a halt.

I recovered just before the next issue was due out, in time to send a one-page apology to my subscribers, promising them that they would still get their full number of issues. I wrote the vignette above for the apology letter.

My newsletter also included puzzles and games unrelated to the Alexandria Jones stories. Here is a word problem from the May/June 1998 issue:

Can You Solve This Story Problem Challenge?

You may estimate the pan of lasagna to be one square foot. An 11″×13″ pan is actually one square inch shy of that, but it won’t change the solution enough to matter—simply round your final answer to two significant figures.

(As any science teacher will tell you, you should never copy down all the numbers your calculator spews out.)

If you get stumped, my recent post on conversion factors might be helpful.

And in case you were wondering, InfoPlease Almanac lists the total area of Illinois as 57,915 square miles.

A Not-Quite Answer

I won’t give the solution away, but to put the answer in perspective:

• Columbus landed in the New World 5 centuries ago.
• Jesus of Nazareth lived 20 centuries ago.
• And about 46 centuries ago, the ancient Egyptians were building pyramids.

To eat one pan of lasagna in 3 seconds sounds like a speed-eater to me, but it still would take an unimaginably long time to eat a pasta the size of just one mid-sized state.

Isn’t the world an amazing place?

This game takes strategic thinking and planning ahead, and it makes a great journaling prompt, too.

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!

Walter Joris’s Sequencium

Math Concepts: strategic thinking.

Players: two players or two teams.

Equipment: blank paper, colored pencils or markers.

How to Play

Draw a 6 × 6 grid of squares as a gameboard for players to share. Each player needs a different color pencil or pen.

Players begin by each writing the number 1 in opposite corner squares. Then take turns writing numbers until the grid is full.

On your turn, choose a square that touches any of your own previously numbered squares at a side or corner. Write the number that is one greater than the square you are touching. Draw a short line to connect those two numbers.

Your number sequence may branch and may cross the opponent’s line. Each branch continues the counting sequence from whichever square it touches.

If your new square is touching more than one of your numbers, continue counting from the higher number. Whoever reaches the greatest number wins the game.

Variations

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

• What is your strategy for winning?

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

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

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

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