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.

FAQ: I’ve Ruined My Daughter

My daughter is only eleven, but I’m afraid I’ve ruined her chance of getting into college because she is so far behind in math. We’ve tried tutors, but she still has trouble, and standardized testing puts her three years below grade level. She was a late reader, too, so maybe school just isn’t her thing. What else can I do?

Standardized tests are not placement tests. They cannot tell you at what level your daughter should be studying. They aren’t designed that way. The “placement” they give is vague and general, not indicative of her grade level but rather a way of comparing her performance on that particular test with the performance of other students.

There can be many different reasons for a low score. I’ve listed a few of them in my post In Honor of the Standardized Testing Season.

Continue reading FAQ: I’ve Ruined My Daughter

Math Game: What Two Numbers?

Here’s a simple, conversational game you can play anywhere — no equipment necessary. It’s great for helping your children develop number fluency and algebraic thinking.

Excerpted from my upcoming book, Prealgebra & Geometry: Math Games for Middle School, scheduled for publication in early 2021. Sign up for my newsletter to get updates.

What Two Numbers?

Math Concepts: addition, multiplication, inverse operations, positive and negative numbers.

Players: two or more.

Equipment: no equipment needed.

Continue reading Math Game: What Two Numbers?

Math Game: War with Special Decks

The all-time most-visited page on this site is my post about Math War: The Game That Is Worth 1,000 Worksheets. It’s easy to adapt to almost any math topic, simple to learn, and quick to play. My homeschool co-op students love it.

But Math War isn’t just for elementary kids. Several teachers have shared special card decks to help middle and high school students practice math by playing games.

Take a look at the links below for games from prealgebra to high school trig. And try the Math War Trumps variation at the end of the post to boost your children’s strategic-thinking potential.

Have fun playing math with your kids!

Continue reading Math Game: War with Special Decks

Math for Star Wars Day

May the Fourth be with you!

Here is a math problem in honor of one of our family’s favorite movies…

Han Solo was doing much-needed maintenance on the Millennium Falcon. He spent 3/5 of his money upgrading the hyperspace motivator. He spent 3/4 of the remainder to install a new blaster cannon. If he spent 450 credits altogether, how much money did he have left?

Stop and think about how you would solve it before reading further.

Continue reading Math for Star Wars Day

Homeschooling Tip #1: Start with Play

For children, learning always begins with play. This is how they wrap their minds around new ideas and make them their own.

“There should be no element of slavery in learning. Enforced exercise does no harm to the body, but enforced learning will not stay in the mind. So avoid compulsion, and let your children’s lessons take the form of play.”

—Plato, The Republic

If we want our children to learn math, our first job is to establish an attitude of playfulness.

This is especially important for anyone working with a discouraged child or a child who is afraid of math. The best way to help a discouraged child is to put away the workbook. Try something different, fun, and challenging.

Continue reading Homeschooling Tip #1: Start with Play

How to Homeschool Math

Far too many people find themselves suddenly, unexpectedly homeschooling their children. This prompts me to consider what advice I might offer after more than three decades of teaching kids at home.

Through my decades of homeschooling five kids, we lived by two rules:

Do math. Do reading.

As long as we hit those two topics each day, I knew the kids would be fine. Do some sort of mathematical game or activity. Read something from that big stack of books we collected at the library.

Conquer the basics of math and reading, then everything else will fall into place.

Continue reading How to Homeschool Math

Play Math with Your Kids for Free

One of the most common questions I get from parents who want to help their children enjoy math is, “Where do we start?”

My favorite answer: “Play games!”

And in this time of pandemic crisis, it’s even more important for families to play together. So my publisher agreed to make my ebook Let’s Play Math Sampler: 10 Family-Favorite Games for Learning Math Through Play free for the duration.

Continue reading Play Math with Your Kids for Free

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.

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.