The Gerrymander Math Project

With a big election on the horizon, now is a great time to talk about the math of politics.

Does “One person, one vote” make a fair democracy?

Or does it give the majority license to trample a minority?

How can planners arrange voting districts to give everyone the best representation? And is that really what politicians would do, if they had the choice?

Try the Gerrymander Project with your students to investigate these questions and spark real-world mathematical discussion.

First, Create a Map

[Or buy a copy of my printable activity guide, The Gerrymander Project: Math in the World of Politics, which includes a prepared city map with more detailed instructions, answers, and journaling prompts. My publisher has extended the 10% discount code TBLTOP10 through to Election Day, 3 November 2020.]

  • Print a blank hundred chart or outline a 10×10 square on grid paper. This represents your city. Give it a name.
  • Pull out your colored pencils. Choose one color for your city’s Majority Party and another for the Minority Party.
  • Color 10 squares in a neutral color for non-voting areas. These might be malls or parks or the downtown business district — your choice.
  • Color the remaining 90 blocks in a random distribution so that 60% are the Majority color and 40% the Minority. How will you choose which squares to make which colors? Can you think of a way to use dice or playing cards to make your choices random, yet still get the right proportion?

Slip your finished map into a clear page protector, so you can mark on it with dry-erase markers. Or make several copies, so you can write on them without destroying the original.

Then Gerrymander Your City

“Gerrymandering” is the American political tradition of adjusting the voting district boundaries to favor one’s own party at the expense of one’s opponents.

The city has hired you to mark out 10 new voting districts of 9 squares each (not counting the neutral squares, which can go in any district). The squares in each district must touch side-to-side, not just meet at a corner.

So now you get to play “political hack.”

First, see how fair you can make the map:

  • What happens if you ignore the party colors and make your districts as compact as possible, so the people living nearest to each other vote together? Will the Majority Party always win?
  • Can you give all your voters a proportional representation? Both parties should win the number of districts that most closely matches their percentage of the voting population.

Next, try your hand at gerrymandering, but make sure all the squares in each district stay connected. Can you create ten voting districts that will guarantee:

  • A come-from-behind triumph for the Minority Party? They need to carry at least six districts to wrest control of the City Council from their opponents.
  • The greatest possible margin of victory for the Majority Party? Can you keep the Minority from winning any districts at all?

Share Your Thoughts

I’d love to hear your students’ reaction to this project. Please share in the comments section below.

For myself, the more I play with this project, the more I admire the work done by the framers of the U.S. Constitution. Our Electoral College divides the country into “districts” based on state boundaries, giving each a vote roughly proportional to its population — but in a way that slightly strengthens the Minority Party. The system may not be perfect, but it’s done an amazing job through the centuries of maintaining a balance of power, making sure that neither major political party can destroy the other.

How to Draw Minecraft Blocks

Running out of time on my Math You Can Play Kickstarter, so I better get to work on that Kickstarter Special Edition math-art book I promised to all the backers as a bonus reward.

Today I’m working on the Isometric Drawing and Impossible Figures section, because my co-op math classes had so much fun learning how to draw those.

Here’s a starter image on how to draw Minecraft blocks. At first I called them “isometric blocks” — but changing the name to “Minecraft” made the students really excited to learn. I’m not sure whether I like the pencil sketch, or if I should remake the illustrations on the computer…

Key steps:

  1. Make a Y.
  2. Turn it into an M.
  3. Slant down for the bottom.
  4. Slant up for the top.
Student drawings from my co-op classes.

The most common problem for beginners is that they try to make the base straight. They know a block can sit on a table, so the bottom has to be flat, right? But once students get a feel for how it goes, they can really take off and have fun.

UPDATE: The Kickstarter deals have ended, but my playful math books are still available through your favorite online store or by special order at your local bookshop. (Except for the Prealgebra & Geometry Games book, scheduled for publication in early 2021. Sign up for my email list to get the latest news.)

Playing with Math: My Favorite Posts

The articles that attract a browsing reader aren’t the same as the ones that pull on my heart. Yesterday, I shared my most-visited blog posts of 2019. Today, I give you my most-loved posts of the year.

#12

A fun challenge, based on an old family-favorite game.

Math Game: Six Hundred

Players around the world have played poker-style dice games for ages. Reiner Knizia included this mathematical version in his book Dice Games Properly Explained

#11

Math art — what a delightful way to learn!

Updated Geometric Coloring Designs

I created these coloring pages for my homeschool co-op math kids, and then collected them into a downloadable 42-page PDF coloring book for your family to enjoy…

#10

I could watch this video every day.

Math That Is Beautiful

Do you have trouble believing that math can be beautiful?

In “Inspirations,” artist Cristóbal Vila creates a wonderful, imaginary work studio for the amazing M.C. Escher…

#9

One of my favorite hobbies is collecting inspirational quotes as I read.

Find the Sweetness in Math

Good problems can help us fall in love with math and make a delicious meal of it, sinking our teeth into tough problems, tenderized by their intrigue…

#8

From the beginning of this blog, more than a decade ago, my desire has been to help homeschooling parents (and other teachers) change how they look at math.

Math Makes Sense — Let’s Teach It That Way

Annie Fetter is talking to classroom teachers, but her message is just as important for homeschoolers. Math is all about making sense. Let’s help our kids see it that way…

#7

An excerpt from the draft of my coming-someday-it-always-takes-longer-than-I-expect Prealgebra & Geometry Games book.

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…

#6

A sweet little game that was a big hit with my K–2nd grade homeschool co-op math class this year.

Playing with a Hundred Chart #35: The Number Grid Game

You’ll need a 6-sided die, a hundred chart (printables here), and a small token to mark each player’s square. A crumpled bit of colored construction paper works well as a token…

#5

This number-pattern activity was so much fun with my upper-elementary class.

A Puzzle for Palindromes

Here’s a math puzzle for palindrome week — or any time you want to play with math…

#4

For years, I’ve been meaning to create an email series about learning math through play. This year I finally did.

8 Weeks of Playful Math for Families

Yes, your kids CAN learn to love math. Keep your children’s math skills fresh with my 8-week email series of math games and activities…

#3

Not a single blog post but a whole new feature, which I hope to continue in the new year.

Morning Coffee

One of the best ways we can help our children learn mathematics (or anything else) is to always be learning ourselves…

#2

The Playful Math Blog Carnival is a labor of love for all our volunteer hosts. This year, August was my month to play.

Playful Math Education Carnival 130

The Playful Math Carnival is like a free online magazine devoted to learning, teaching, and playing around with math. It’s back-to-school time in the U.S., so this month’s edition focuses on establishing a creative math mindset from preschool to high school…

#1

But the work closest to my heart this year wasn’t math at all. My daughter finally concluded her fantasy novel series, the major project of her teen homeschooling years. When she gave me the manuscript to edit, I didn’t get any sleep that night. She had me hooked — I had to find out how it ended!

The Final Books from a Homeschooled Teen Author

Do you enjoy binge reading tales of epic fantasy? Love exploring magical worlds wracked by the struggle of good against evil? Then don’t miss Teresa Gaskins’s four-book serial adventure, The Riddled Stone

How About You?

Do you play math with kids? Please tell us one (or a few) of your favorite things. What touched your heart this year?

CREDITS: “Sparkling heart” photo (top) by Jamie Street on Unsplash.

Top Playful Math Posts of 2019

Here are my most-visited posts and pages in 2019. So many ways to play with math!

#12

I love books, don’t you?

Math with Living Books

Do you want to enrich your mind with the great ideas of mathematics? Are you looking for a good book to whet your child’s appetite? Then the following pages of “living” math books are for you…

#11

A logic challenge that doubles as addition practice. Or is it the other way around?

Math Game: Thirty-One

Thirty-One comes from British mathematician Henry Dudeney’s classic book, The Canterbury Puzzles

#10

Turn a regular deck of cards into math flashcards. Adaptable to any operation.

Review Game: Once Through the Deck

The best way to practice the math facts is through the give-and-take of conversation, orally quizzing each other and talking about how you might figure the answers out. But occasionally your child may want a simple, solitaire method for review…

#9

Seasonally popular enough to make the list every year. You’ll find even more mathy fun in my updated Holiday Math Carnival.

Christmas Math Puzzles and Activities

We interrupt our regularly scheduled math program to bring you the following Christmas links…

#8

A counting game for all ages.

Math Game: Fan Tan (Sevens)

Fan Tan may also be called Crazy Sevens. Like any folk game, it is played by a variety of rules around the world…

#7

The updated post (which ranked at #18 for the year) is better: My Favorite Math Games. Eventually I hope it will surpass this old one.

20 Best Math Games and Puzzles

Over the years, Let’s Play Math blog has grown into a sprawling mess, which can make it very hard to find the specific math tip you’re looking for…

#6

What a wonderful, inspiring movie! You may also enjoy the related Women of Mathematics Carnival.

Hidden Figures Teaching Resources

Before computers were machines, computers were people who computed things. This complicated task often fell to women because it was considered basically clerical. That’s right: computing triple integrals all day long qualified as clerical…

#5

One of my all-time favorites, still helpful after all these years.

Number Bonds = Better Understanding

Number bonds let children see the inverse relationship between addition and subtraction. Subtraction is not a totally different thing from addition; they are mirror images…

#4

I intended to write a follow-up series based on this post. Maybe in 2020?

How to Read a Fraction

Fraction notation and operations may be the most abstract math monsters our students meet until they get to algebra. Before we can explain those frustrating fractions, we teachers need to go back to the basics for ourselves…

#3

A dark horse in third place! I never expected this post to draw much interest.

Puzzle: Factoring Trinomials

My high school class ended the year with a review of multiplying and factoring simple polynomials. We played a matching game, and then I gave them this puzzle worksheet…

#2

A perennial favorite: widely adaptable, easy to learn, and kids enjoy it.

The Game That Is Worth 1,000 Worksheets

Have you and your children been struggling to learn the math facts? The game of Math Card War is worth more than a thousand math drill worksheets, letting you build your children’s calculating speed in a no-stress, no-test way…

#1

A well-deserving winner, with activities for preschool through middle school.

30+ Things to Do with a Hundred Chart

Are you looking for creative ways to help your children study math? Even without a workbook or teacher’s manual, your kids can learn a lot about numbers. Just spend an afternoon playing around with a hundred chart…

CREDITS: “Sparkling 2019” photo by NordWood Themes on Unsplash.

A Puzzle for Palindromes

If you haven’t seen the meme going around, this is a palindrome week because the dates (written American style and with the year shortened to ’19) are the same when reversed.

Here’s a math puzzle for palindrome week — or any time you want to play with math:

  • Print a 100 chart.
  • Choose a color code.
  • Play!

What do you think: Will all numbers eventually turn into palindromes?

Links

You can find all sorts of hundred charts on my Free Math Printable Files page.

Read about the history of palindromes on Nrich Math’s Palindromes page.

Find out more about the Palindromic Number Conjecture in Mark Chubb’s article An Unsolved Problem your Students Should Attempt.

Or play with Manan Shah’s advanced palindromic number questions.

Playful Math Education Carnival 130

Play. Learn. Enjoy!

Welcome to the 130th edition of the Playful Math Education Blog Carnival, a feast of delectable tidbits of mathy fun.

The Playful Math Carnival is like a free online magazine devoted to learning, teaching, and playing around with math. It’s back-to-school time in the U.S., so this month’s edition focuses on establishing a creative math mindset from preschool to high school.

You’re sure to find something that will delight both you and your child.

By tradition, we start the carnival with a puzzle in honor of our 130th edition. But if you would like to jump straight to our featured blog posts, use our handy Table of Contents.

Click here for all the mathy goodness!

Updated Geometric Coloring Designs (Free)

I created these coloring pages for my homeschool co-op math kids, and then collected them into a downloadable 42-page PDF coloring book for your family to enjoy.

The booklet includes several ready-to-color designs, a wide assortment of graph paper, advanced create-your-own tessellation pages, and links to all sorts of online math art resources.

But when I posted the link to Twitter, a friend noticed that several of the resource links were broken. (Thanks, Mark!)

While I was fixing those, I added some new links (to the wonderful School of Islamic Geometric Design pattern templates and classroom resources).

So even if you’ve downloaded the file before, you may want to pick up this new-and-improved edition.

Click to Visit My Free Printables Page

More Ways to Play with Math Art

8 Weeks of Playful Math for Families

Yes, your kids CAN learn to love math. Keep your children’s math skills fresh with my 8-week email series of math games and activities.

No purchase necessary! Just sign up for my email newsletter, and every week for the next two months you’ll automatically receive one of my favorite math club activities or an excerpt from my series of math game books.

Continue reading 8 Weeks of Playful Math for Families

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?

You can build polite numbers (like fifteen) with a staircase of blocks.

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?

What is your favorite conjecture? Does thinking about it make you wonder about 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.

Share your ideas in the comments section below.

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.

Holiday Math Puzzles and Activities for Christmas, Winter Break

Hapollonian Holidays from my Math Circle kids, and best wishes for a grace-filled holiday season.
Hapollonian Holiday Greetings from my co-op class kids, and best wishes for a grace-filled holiday season.

Do you know of any great math-related seasonal games, crafts, or activities I missed? Please add them to the comments section below.

As you scroll through the links below, you find several puzzle graphics from the wonderful Visual Patterns website.

Use them as conversation-starters with your kids: What do you notice? How does each pattern grow?

For older students: Can you write a formula to describe how each pattern? What will it look at stage 43?

Pattern #7, Trees

A BIT OF FUN

Setting the mood: Enjoy this bit of seasonal fidgeting from Vi Hart. If you don’t understand some of the references, that’s normal! Pick a phrase, Google it, and enjoy the fun of learning something new.

ADVENT MATH ACTIVITY CALENDARS

Every year, some of my favorite websites offer a seasonal selection of activities to encourage your children’s (and your own!) mathematical creativity, one for each day in the run-up to Christmas.

Pattern #9, Snowflakes

LET IT SNOW! LET IT SNOW! LET IT SNOW!

  • Clarissa (@c0mplexnumber) demonstrates how to make beautiful, challenging origami snowflakes. She recommends beginners try the first few folds — which create a pretty cool design on their own. Let it Snow…
Pattern #20, Helmets

HAPPY CHANUKAH

Pattern #30, from John Golden, Squares

HANDS-ON HOLIDAYS

Pattern #197, from Stephanie Bowyer, Symbols

FOLLOWING YONDER STAR

Pattern #132, from Math Curmudgeon, Diagonals

MATHY CHRISTMAS CARDS

Pattern #98, Centers are collinear, Fraction of the original circle shaded

SANTA CLAUS IS COMING

Pattern #8, Penguins

ROCKIN’ AROUND THE CHRISTMAS TREE

Pattern #152, from John Golden, Circles

PUZZLES UNDER THE TREE

  • Unfortunately, the holidays come smack in the middle of flu season. Did you come down with The Grinch Bug?
Pattern #52, Cubes
  • Speaking of Christmas carols, the Christmas Price Index shows the current cost for one set of each of the gifts given in the song “The Twelve Days of Christmas.” I wonder what’s the cumulative cost of all the gifts, when you count each repetition in the song?

CHRISTMAS ADVENTURES WITH ALEXANDRIA JONES

Alexandria Jones and her family are fictional characters from my old Mathematical Adventures newsletter. Their stories appear sporadically as I find time to transcribe them from the back-issues. You can find them all on this blog page.

Here are all the Alexandria Jones stories Christmas stories, with activity and craft ideas…

Pattern #174, from Katie Gates, Squares

WHAT ABOUT WORKSHEETS?

Do you need to keep your kids busy and work in a bit of math practice? Try these Christmas word problems:

Or visit the sites below for worksheets to cover all ages:

Pattern #28, Surface area

CREDITS: “Circle Packing” feature graphic (top) by fdecomite via Flickr (CC BY 2.0). Picture pattern puzzles from Visual Patterns website.