FAQ: Playful Math for Older Students

My students are so busy that time-consuming math projects are a luxury. How is it possible for older kids to play with mathematics?

Too often, the modern American school math curriculum is a relentless treadmill driving students toward calculus. (Does this happen in other countries, too?)

But that’s definitely not the only way to learn. For most students, it’s not the best way, either.

Here are a few ideas to get your older children playing with math…

Continue reading FAQ: Playful Math for Older Students

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.

Which is NOT to say that our country always protects the rights of true minorities. Clearly, that’s an ongoing struggle.

But overall, the political parties stay relatively balanced, making for a stable government. After more than two centuries, we still have, as Ben Franklin said, “a republic, if you can keep it.”

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

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

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.

Advent Math Activity Calendars

Once again, 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.

Including an especially tough Advent meta-puzzle for truly determined problem-solvers…

Click the images below to visit the corresponding December Math Calendar pages.

For Primary Students

Easier activities for elementary and middle school.

A problem or game that uses dice for each day in the run up to Christmas.

2018 Primary Advent Calendar

When you get to the Nrich website, click a number to go to that day’s math.

For Secondary Students

Activities for middle and high school.

Most of the tasks require dotty paper or circles which you can find on the Nrich Printable Resources page.

2018 Secondary Advent Calendar

When you get to the Nrich website, click a number to go to that day’s math.

For Teens and Adults

“Because sometimes, in the midst of all the family fun, it’s good to put your headphones in and retreat into a word of your own, this year’s advent calendar brings you some of our favourite Plus podcasts. From the secrets of the Universe to the maths of football stadiums, there should be something there for everyone. Happy listening and happy Christmas from the Plus team!”

When you get to the +Plus Magazine website, you can tell which links are live because they jump to a larger size when you tap or mouse over the picture.

Plus Advent Calendar 2018

One link becomes live each day — so come back tomorrow and discover something new!

Christmas Meta-Puzzle

Or try your hand at the biggest mathematical mystery of them all — and save Christmas!

“It’s nearly Christmas and something terrible has happened: one of Santa’s five helpers — Jo Ranger, Fred Metcalfe, Kip Urples, Meg Reeny, and Bob Luey — has stolen all the presents during the North Pole’s annual Sevenstival. You need to find the culprit before Christmas is ruined for everyone.”

Christmas Logic Puzzle

If you solve all the clues and enter the answer on Christmas day, you may win a present for yourself, too.

Still More Mathy Fun

Colleen Young has collected several more holiday math calendars — enough to keep your kids playing and learning well into the New Year. Take a look!

Mathematical Advent Calendars

CREDITS: “Peanuts Christmas Panorama” photo [top] by Kevin Dooley via Flicker. (CCBY2.0)

A Beautiful Puzzle

This lovely puzzle (for upper-elementary and beyond) is from Nikolay Bogdanov-Belsky’s 1895 painting “Mental Calculation. In Public School of S. A. Rachinsky.” Pat Ballew posted it on his blog On This Day in Math, in honor of the 365th day of the year.

I love the expressions on the boys’ faces. So many different ways to manifest hard thinking!

Here’s the question:

No calculator allowed. But you can talk it over with a friend, as the boys on the right are doing.

You can even use scratch paper, if you like.

Thinking About Square Numbers

And if you’d like a hint, you can figure out square numbers using this trick. Think of a square number made from rows of pennies.

Can you see how to make the next-bigger square?

Any square number, plus one more row and one more column, plus a penny for the corner, makes the next-bigger square.

So if you know that ten squared is one hundred, then:

… and so onward to your answer. If the Russian schoolboys could figure it out, then you can, too!

Update

Simon Gregg (@Simon_Gregg) added this wonderful related puzzle for the new year:

Playful Math Education Carnival 106

Do you enjoy math? I hope so! If not, browsing this post just may change your mind.

Welcome to the 106th edition of the Math Teachers At Play math education blog carnival — a smorgasbord of links to bloggers all around the internet who have great ideas for learning, teaching, and playing around with math from preschool to pre-college. Let the mathematical fun begin!

By tradition, we start the carnival with a puzzle in honor of our 106th edition. But if you would like to jump straight to our featured blog posts, click here to see the Table of Contents.

Try This Puzzle

If you slice a pizza with a lightsaber, you’ll make straight cuts all the way across. Slice it once, and you get two pieces.

If you slice it five times, you’ll get a maximum of sixteen pieces. (And if you’re lucky you might get a star!)

  • How many times would you have to slice the pizza to get 106 pieces?

Click here for all the mathy goodness!

Dot Grid Doodling

What can you DO with a page full of dots?

Yesterday, I mentioned my new series of paperback dot grid notebooks, and I promised to share a few ideas for mathematical doodling.

But first, let me share my new dot-grid journals for kids. Aren’t they pretty?

Click Here for More Information

Doodling gives our minds a chance to relax, wander, and come back to our work refreshed. And though it goes against intuition, doodling can help us remember more of what we learn.

Math doodles let us experiment with geometric shapes and symmetries. We can feel our way into math ideas gradually, through informal play. Through doodles, our students will explore a wide range of mathematical structures and relationships.

Our own school experiences can make it hard for us to teach. What we never learned in school was the concept of playing around with math, allowing ideas to “percolate,” so to speak, before mastery occurs, and that process may take time.

—Julie Brennan

I like to doodle on dotty grid paper, like the pages in my math journals, but there’s No Purchase Necessary! You can design your own printable dot page at Incompetech’s PDF generator, or download my free coloring book (which includes several pages of printable dot and graph paper).

Patterns in Shape and Angle

To make a faceted mathematical gemstone, start with any shape you like. Then build other shapes around it. What do you notice? Does your pattern grow outward from its center? Or flow around the corner of your page? How is each layer similar, and how is it different?

Arbitrary constraints can lead to mathematically interesting doodles. For instance, create a design out of 45-45-90 triangles by coloring exactly half of every grid square. How many variations can you find?

Symmetry Challenge

Play a symmetry puzzle game. Draw a line of symmetry and fill in part of the design. Then trade with a partner to finish each other’s doodles.

Make more complex symmetry puzzles with additional reflection lines.

Math Doodle Links

  • Who can talk about mathematical doodling without mentioning Vi Hart? If you’ve never seen her “Doodling in Math Class” video series, you’re in for a treat!
  • See if you can draw a rotational-symmetry design, like Don’s “Order 4” graphs.
  • Or experiment with the more flexible rules in John’s “Knot Fun” lesson.
  • And my latest obsession: the “ultimate” tutorial series on Celtic Knotwork, which explores the link between knots and their underlying graphs.
My favorite knot doodle so far.
Inspirations: A Recreational Mathematics Journal
Reflections: A Math Teacher’s Journal
Explorations: A Math Student’s Journal
Contemplations: A Homeschooler’s Journal

Before you start doodling: How to Break In Your New Math Journal.

Feature photo (top): Sommermorgen (Alte Holzbrücke in Pretzfeld) by Curt Herrmann, via Wikimedia Commons. [Public domain]

March 2016 Math Calendars

Once again, a few of my favorite bloggers have come through with math calendars for our students to puzzle over. Check them out:

algebra calendar

Things to Do with a 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 and trim off the dates. Then challenge your students to arrange them in ascending (or descending) order.

Make up problems to fill a new calendar for next month.
And if you do, please share!