## Playful Math: Getting Students To Write Their Own

To wrap up our week of exploring the resources from Word Problems from Literature, let’s talk about getting students to write their own math.

First up, I’m sharing an excerpt from the Word Problems Student Workbook. The “Story Problem Challenge” is one of my favorite math club activities.

Following that, you’ll find an amazing online mathemagical adventure for middle school: The Arithmetiquities. It’s great fun, and a great inspiration for students to create their own math stories.

Have fun writing math with your kids!

### The Story Problem Challenge

What do you get when you cross a library book or favorite movie with a math worksheet? A great alternative to math homework!

The rules are simple:

(1) Choose a worksheet calculation to be the basis for your word problem.

(2) Solve the calculation.

(3) Consider where these numbers could make sense in your book or movie universe. How might the characters use math? What sort of things would they count or measure? Do they use money? Do they build things, or cook meals, or make crafts? Do they need to keep track of how far they have traveled? Or how long it takes to get there?

To make the game easier, you may change the numbers to make a more realistic problem. But you must keep the same type of calculation. For example, if your worksheet problem was 18÷3, you could change it to 18÷6 or 24÷3 or even 119÷17 to fit your story, but you can’t make it something like 18−3.

Remember that some quantities are discrete and countable, such as hobbits and fireworks. Other quantities are continuous, such as a barrel of wine or a length of fabric. Be sure to consider both types when you are deciding what to use in your problem.

Then share your problem with friends, and you try their problems. Can you stump each other?

Old books are in the public domain, so you can always use characters like Robin Hood, Sherlock Holmes, or Winnie-the-Pooh (but not the newer Disney version with the red jacket). But most books and movies are the protected intellectual property of their authors or estates, or of the company who bought those rights.

When you write problems for your own private use, feel free to use your favorite characters from any story. That’s like fan fiction, secret, just for your own pleasure.

But if you decide to share your creation beyond your own home or classroom, then be sure to “genericize” it first. Change or remove the proper names, using general descriptions instead.

For example, if you love the Harry Potter series, you might want to use Harry or Hermione in your story problems. Instead, write about “the boy wizard destined to fight an evil sorcerer.” Or “the bright young witch who can master any spell.”

Or if you like the Star Wars movies, you might write about “an interstellar justice warrior with an energy sword.” Or “an alien master of martial arts training a cocky but inexperienced apprentice.”

We’d love to add your story to the Student Math Makers Gallery.

### The Arithmetiquities

When the world of Sfera is threatened by the machinations of a malevolent sorcerer, it will be up to a band of unlikely heroes to become the brightest light in the darkness.

The adventurers fan out across the land to find and retrieve the Arithmetiquities, a set of ancient mathemagical artifacts.

The Arithmetiquities is a fantasy adventure story told through a sequence of 36 mathematical puzzles.

“Though it is still before sunrise, Lumparland Harbor is already bustling. Sailing ships moor at the misty docks, bringing travelers and goods to the seaside town. Three dwarves disembark from different ships, each adventurer returning home from some faraway locale. The three women gather at the end of the pier.

“The strangers discover that they all live along the main road that leads from the harbor, so they decide to split the cost of a wagon. Egga lives 10 miles away, Floora lives 20 miles away, and Greeta lives 30 miles away. The wagon ride costs \$1.50 per mile regardless of the number of passengers.

“How much should each of the adventurers pay so that each one has a fair fare?”

—Jason Ermer, “Lumparland Harbor,” The Arithmetiquities Chapter I

CREDITS: Feature photo (top) by Hannah Olinger via Unsplash.com.

## Playful Math Education Carnival 147

Welcome to the 147th edition of the Playful Math Education Blog Carnival — a smorgasbord of delectable tidbits of mathy fun. It’s like a free online magazine devoted to learning, teaching, and playing around with math from preschool to high school.

Bookmark this post, so you can take your time browsing. There’s so much playful math to enjoy!

By tradition, we start the carnival with a puzzle in honor of our 147th edition. But if you’d rather jump straight to our featured blog posts, click here to see the Table of Contents.

## New! Your Student Can Be a Math Maker

When children create their own math, they build a deep understanding of mathematical concepts and relationships.

And it’s fun!

So take a break from your normal math program to play with creative math. Students can:

### Check Out the Gallery

We have a few entries already in the Student Math Makers Gallery.

### Join the Student Math Makers

We’d love to add your students’ math to our collection and share it with viewers all around the world!

To submit a math creation, download a Math Makers Invitation and Submission Form below:

CREDITS: “Creating Math Puzzles by Sian Zelbo, the author of Camp Logic, via NaturalMath.com.

## Math Activity: Polite Numbers

Did you know that numbers can be polite? In math, a polite number is any number we can write as the sum of two or more consecutive positive whole numbers.

(Consecutive means numbers that come one right after another in the counting sequence.)

For example, five is a polite number, because we can write it as the sum of two consecutive numbers:
5 = 2 + 3

Nine is a doubly polite number, because we can write it two ways:
9 = 4 + 5
9 = 2 + 3 + 4

And fifteen is an amazingly polite number. We can write fifteen as the sum of consecutive numbers in three ways:
15 = 7 + 8
15 = 4 + 5 + 6
15 = 1 + 2 + 3 + 4 + 5

How many other polite numbers can you find?

### What Do You Notice?

Are all numbers polite?

Or can you find an impolite number?

Can you make a collection of polite and impolite numbers? Find as many as you can.

How many different ways can you write each polite number as a sum of consecutive numbers?

What do you notice about your collection of polite and impolite numbers?

Can you think of a way to organize your collection so you can look for patterns?

### What Do You Wonder?

Make a conjecture about polite or impolite numbers. A conjecture is a statement that you think might be true.

For example, you might make a conjecture that “All odd numbers are…” — How would you finish that sentence?

Make another conjecture.

And another.

Can you make at least five conjectures about polite and impolite numbers?

Can you think of any way to test your conjectures, to know whether they will always be true or not?

### Real Life Math Is Social

This is how mathematics works. Mathematicians play with numbers, shapes, or ideas and explore how those relate to other ideas.

After collecting a set of interesting things, they think about ways to organize them, so they can look for patterns and connections. They make conjectures and try to imagine ways to test them.

And mathematicians compare their ideas with each other. In real life, math is a very social game.

So play with polite and impolite numbers. Compare your conjectures with a friend.

And check out the list of student conjectures at the Ramblings of a Math Mom blog.

CREDITS: Numbers photo (top) by James Cridland via Flickr (CC BY 2.0). I first saw this activity at Dave Marain’s Math Notations blog, and it’s also available as a cute printable Nrich poster. For a detailed analysis, check out Wai Yan Pong’s “Sums of Consecutive Integers” article.

## Math Journals for Elementary and Middle School

This fall, my homeschool co-op math class will play with math journaling.

But my earlier dot-grid notebooks were designed for adults. Too thick, too many pages. And the half-cm dot grid made lines too narrow for young writers.

So I created a new series of paperback dot-grid journals for my elementary and middle school students.

I hope you enjoy them, too!

## Math Journaling Prompts

So, what can your kids do with a math journal?

Here are a few ideas:

I’m sure we’ll use several of these activities in my homeschool co-op math class this fall.

### Noticing and Wondering

Learning math requires more than mastering number facts and memorizing rules. At its heart, math is a way of thinking.

So more than anything else, we need to teach our kids to think mathematically — to make sense of math problems and persevere in figuring them out.

Whenever your children need to learn a new idea in math, or whenever they get stuck on a tough homework problem, that’s a good time to step back and make sense of the math.

Kids can write their noticings and wonderings in the math journal. Or you can act as the scribe, writing down (without comment) everything child says.

For more tips on teaching students to brainstorm about math, check out these online resources from The Math Forum:

Problem-solving is a habit of mind that you and your children can learn and grow in. Help your kids practice slowing down and taking the time to fully understand a problem situation.

### Puzzles Are Math Experiments

Almost anything your child notices or wonders can lead to a math experiment.

For example, one day my daughter played an online math game…

A math journal can be like a science lab book. Not the pre-digested, fill-in-the-blank lab books that some curricula provide. But the real lab books that scientists write to keep track of their data, and what they’ve tried so far, and what went wrong, and what finally worked.

Here are a few open-ended math experiments you might try:

##### Explore Shapes
• Pick out a 3×3 set of dots. How many different shapes can you make by connecting those dots? Which shapes have symmetry? Which ones do you like the best?
• What if you make shapes on isometric grid paper? How many different ways can you connect those dots?
• Limit your investigation to a specific type of shape. How many different triangles can you make on a 3×3 set of dots? How many different quadrilaterals? What if you used a bigger set of dots?
##### Explore Angles

• On your grid paper, let one dot “hold hands” with two others. How many different angles can you make? Can you figure out their degree without measuring?
• Are there any angles you can’t make on your dot grid? If your paper extended forever, would there be any angles you couldn’t make?
• Does it make a difference whether you try the angle experiments on square or isometric grid paper?
##### Explore Squares
• How many different squares can you draw on your grid paper? (Don’t forget the squares that sit on a slant!) How can you be sure that they are perfectly square?
• Number the rows and columns of dots. Can you find a pattern in the corner positions for your squares? If someone drew a secret square, what’s the minimum information you would need to duplicate it?
• Does it make a difference whether you try the square experiments on square or isometric grid paper?

### Or Try Some Math Doodles

Create math art. Check out my math doodling collection on Pinterest and my Dot Grid Doodling blog post. Can you draw an impossible shape?

## How Would YOU Use a Math Journal?

I’d love to hear your favorite math explorations or journaling tips!

P.S.: Do you have a blog? If you’d like to feature a math journal review and giveaway, I’ll provide the prize. Send a message through my contact form or leave a comment below, and we’ll work out the details.

## My Favorite Math Games

Take a break from textbook math and enjoy yourself!

I like to use games as a warm-up with my co-op math classes. Some homeschoolers make every Friday a game day, and some turn gaming into a family lifestyle.

“Playing games with your kids offers a host of educational benefits, plus you build relationships and make memories. I am constantly amazed by the amount of learning that happens when I sit down to play games with my children.”

—Caitlin Fitzpatrick Curley
Gameschool Challenge

### Family Games for All Ages

“Games put children in exactly the right frame of mind for learning difficult things. Children relax when they play — and they concentrate. They don’t mind repeating certain facts or procedures over and over, if repetition is part of the game.”

### Accessible to Young Children

“Coming back from winter break can be hard. Everyone is sleepy, unfocused, and daydreaming of the holiday gifts that await them at home after school. And that’s just the teachers!”

—Andrew Gael
Beat the Back to School Blues…Play a Math Game

### For Elementary Students

“If you play these games and your child learns only that hard mental effort can be fun, you will have taught something invaluable.”

“Mathematics is mental play, the essence of creative problem solving. This is the truth we need to impart to our children, more important than fractions or decimals or even the times tables. Math is a game, playing with ideas.”

They don’t have to be math! Please share in the comment section below!

## Puzzle: Exploding Dots

I’m planning ahead for my fall semester homeschool co-op math class. Definitely going to try this with the kids…

Encourage your children to have some fun this week with this Exploding Dots math puzzle from The Global Math Project. What do they notice? Does it make them wonder?

### More Explosive Math

You may recognize the connection between Exploding Dots and binary numbers. Or not — the puzzle is accessible to people at almost any age and level of mathematical sophistication.

But what I find amazing is that this puzzle can help us understand all sorts of topics in elementary arithmetic and algebra. So cool!

If you’d like to investigate Exploding Dots in depth, check out James Tanton’s free G’Day Math online course.

## 2016 Mathematics Game

[Feature photo above from the public domain, and title background (below) by frankieleon (CC BY 2.0) via Flickr.]

Have you made a New Year’s resolution to spend more time with your family this year, and to get more exercise? Problem-solvers of all ages can pump up their (mental) muscles with the Annual Mathematics Year Game Extravaganza. Please join us!

For many years mathematicians, scientists, engineers and others interested in math have played “year games” via e-mail. We don’t always know whether it’s possible to write all the numbers from 1 to 100 using only the digits in the current year, but it’s fun to see how many you can find.

## Rules of the Game

Use the digits in the year 2016 to write mathematical expressions for the counting numbers 1 through 100. The goal is adjustable: Young children can start with looking for 1-10, middle grades with 1-25.

• You must use all four digits. You may not use any other numbers.
• Solutions that keep the year digits in 2-0-1-6 order are preferred, but not required.
• You may use +, -, x, ÷, sqrt (square root), ^ (raise to a power), ! (factorial), and parentheses, brackets, or other grouping symbols.
• You may use a decimal point to create numbers such as .2, .02, etc., but you cannot write 0.02 because we only have one zero in this year’s number.
• You may create multi-digit numbers such as 10 or 201 or .01, but we prefer solutions that avoid them.

#### My Special Variations on the Rules

• You MAY use the overhead-bar (vinculum), dots, or brackets to mark a repeating decimal. But students and teachers beware: you can’t submit answers with repeating decimals to Math Forum.
• You MAY use a double factorial, n!! = the product of all integers from 1 to n that have the same parity (odd or even) as n. I’m including these because Math Forum allows them, but I personally try to avoid the beasts. I feel much more creative when I can wrangle a solution without invoking them.

## Infinite Cake: Don Cohen’s Infinite Series for Kids

Math Concepts: division as equal sharing, naming fractions, adding fractions, infinitesimals, iteration, limits
Prerequisite: able to identify fractions as part of a whole

This is how I tell the story:

• We have a cake to share, just the two of us. It’s not TOO big a cake, ‘cuz we don’t want to get sick. An 8 × 8 or 16 × 16 square on the graph paper should be just right. Can you cut the cake so we each get a fair share? Color in your part.

• How big is your piece compared to the whole, original cake?
• But you know, I’m on a diet, and I just don’t think I can eat my whole piece. Half the cake is too much for me. Is it okay if I share my piece with you? How can we divide it evenly, so we each get a fair share? How big is your new piece? Color it in.
• How much of the whole, original cake do you have now? How can you tell?
• I keep thinking of my diet, and I really don’t want all my piece of cake. It looks good, but it’s still just a bit too big for me. Will you take half of it? How big is that piece?
• Now how much of the whole, original cake do you have? How could we figure it out?
[Teaching tip: Don’t make kids do the calculation on paper. In the early stages, they can visualize and count up the fourths or maybe the eighths. As the pieces get smaller, the easiest way to find the sum is what Cohen does in the video below‌—‌identify how much of the cake is left out.]
• Even for being on a diet, I still don’t feel very hungry…

## April 2015 Math Calendar

Feature photo above by Kelly Sikkema via Flickr (CC BY 2.0).

Six years ago, my homeschool co-op classes had fun creating this April calendar to hand out at our end-of-semester party. Looking at my regular calendar today, I noticed that April this year starts on Wednesday, just like it did back then. I wonder when’s the next time that will happen?

A math calendar is not as easy to read as a traditional calendar — it is more like a puzzle. The expression in each square simplifies to that day’s date, so your family can treat each day like a mini-review quiz: “Do you remember how to calculate this?”

The calendar my students made is appropriate for middle school and beyond, but you can make a math calendar with puzzles for any age or skill level. Better yet, encourage the kids to make puzzles of their own.

### How to Use the Math Calendar

At home:
Post the calendar on your refrigerator. Use each math puzzle as a daily review “mini-quiz” for your children (or yourself).

In the classroom:
Post today’s calculation on the board as a warm-up puzzle. Encourage your students to make up “Today is…” puzzles of their own.

As a puzzle:
Cut the calendar squares apart, then challenge your students to arrange them in ascending (or descending) order.

### Help Us Make the Next Math Calendar

If you like, you may use the following worksheet:

Submission details here: Kids’ Project — More Math Calendars?