A Penny for Your Math

You know you’re a math teacher when you see a penny in the parking lot, and your first thought is, “Cool! A free math manipulative.”


My homeschool co-op math students love doing math with pennies. They’re rather heavy to carry to class, but worth it for the student buy-in.

This month, I’m finishing up the nearly 150 new illustrations for the upcoming paperback edition of my Let’s Play Math book. I’m no artist, and it’s been a long slog. But a couple of the graphics involved pennies‌—‌so when I saw that penny on the ground, it made me think of my book.

And thinking of my book made me think it would be fun to share a sneak peek at coming attractions…

The Penny Square: An Example of Real Mathematics

Real mathematics is intriguing and full of wonder, an exploration of patterns and mysterious connections. It rewards us with the joy of the “Aha!” feeling. Workbook math, on the other hand, is several pages of long division by hand followed by a rousing chorus of the fraction song: “Ours is not to reason why, just invert and multiply.”

Real math is the surprising fact that the odd numbers add up to perfect squares (1, 1 + 3, 1 + 3 + 5, etc.) and the satisfaction of seeing why it must be so.

Did your algebra teacher ever explain to you that a square number is literally a number that can be arranged to make a square? Try it for yourself:

  • Gather a bunch of pennies‌—‌or any small items that will not roll away when you set them out in rows‌—‌and place one of them in front of you on the table. Imagine drawing a frame around it: one penny makes a (very small) square. One row, with one item in each row.
  • Now, put out three more pennies. How will you add them to the first one in order to form a new, bigger square? Arrange them in a small L-shape around the original penny to make two rows with two pennies in each row.
  • Set out five additional pennies. Without moving the current four pennies, how can you place these five to form the next square? Three rows of three.
  • Then how many will you have to add to make four rows of four?
Twenty-five is a square number, because we can arrange twenty-five items to make a square: five rows with five items in each row.

Each new set of pennies must add an extra row and column to the current square, plus a corner penny where the new row and column meet. The row and column match exactly, making an even number, and then the extra penny at the corner makes it odd.

Can you see that the “next odd number” pattern will continue as long as there are pennies to add, and that it could keep going forever in your imagination?

The point of the penny square is not to memorize the square numbers or to get any particular “right answer,” but to see numbers in a new way‌—‌to understand that numbers are related to each other and that we can show such relationships with diagrams or physical models. The more relationships like this our children explore, the more they see numbers as familiar friends.

The Penny Birthday Challenge: Exponential Growth


A large jar of assorted coins makes a wonderful math toy. Children love to play with, count, and sort coins.

Add a dollar bill to the jar, so you can play the Dollar Game: Take turns throwing a pair of dice, gathering that many pennies and trading up to bigger coins. Five pennies trade for a nickel, two nickels for a dime, etc. Whoever is the first to claim the dollar wins the game.

Or take the Penny Birthday Challenge to learn about exponential growth: Print out a calendar for your child’s birthday month. Put one penny on the first day of the month, two pennies on the second day, four pennies on the third day, etc. If you continued doubling the pennies each day until you reach your child’s birthday, how much money would you need?

Warning: Beware the Penny Birthday Challenge! Those pennies will add up to dollars much faster than most people expect. Do not promise to give the money to your child unless the birthday comes near the beginning of the month.

A Penny Holiday Challenge

The first time I did pennies on a calendar with my homeschool co-op class was during December, so we called it the Penny Christmas Challenge:

  • How many pennies would you need to cover all the days up to the 25th?

I told the kids that if their grandparents asked what gift they wanted for Christmas, they could say, “Not much. Just a few pennies…”

LPM-ebook-300The Penny Square, Dollar Game, and Penny Birthday Challenge are just three of the myriad math tips and activity ideas in the paperback edition of Let’s Play Math: How Families Can Learn Math Together and Enjoy It. Coming in early 2016 to your favorite online bookstore…

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Happy Math Equation Day!

math equation day

Every Day Is Mathematics Day!

I’m still having fun with David Coffey’s meme, which started a couple of years ago with this blog post:

Make Your Own

Would you like to create a math holiday, too? Try one of these sign generators:

What kind of math will you celebrate? Leave a link to your Happy Math Day post in the comments below!

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Review Game: Once Through the Deck

Ace of Spades

[Feature photo above by Shannon (shikiro famu) via Flicker (CC BY 2.0).]

Math Concepts: basic facts of addition, multiplication.
Players: one.
Equipment: one deck of math cards (poker- or bridge-style playing cards with the face cards and jokers removed).

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.

Continue reading Review Game: Once Through the Deck

Math Games with Factors, Multiples, and Prime Numbers

factor game

Students can explore prime and non-prime numbers with two free favorite classroom games: The Factor Game (pdf lesson download) or Tax Collector. For $15-20 you can buy a downloadable file of the beautiful, colorful, mathematical board game Prime Climb. Or try the following game by retired Canadian math professor Jerry Ameis:

Factor Finding Game


Math Concepts: multiples, factors, composite numbers, and primes.
Players: only two.
Equipment: pair of 6-sided dice, 10 squares each of two different colors construction paper, and the game board (click the image to print it, or copy by hand).

On your turn, roll the dice and make a 2-digit number. Use one of your colored squares to mark a position on the game board. You can only mark one square per turn.

  • If your 2-digit number is prime, cover a PRIME square.
  • If any of the numbers showing are factors of your 2-digit number, cover one of them.
  • BUT if there’s no square available that matches your number, you lose your turn.

The first player to get three squares in a row (horizontal, vertical, or diagonal) wins. Or for a harder challenge, try for four in a row.

Feature photo at top of post by Jimmie via flickr (CC BY 2.0). This game was featured in the Math Teachers At Play (MTaP) math education blog carnival: MTaP #79. Hat tip: Jimmie Lanley.

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Math Game: Thirty-One

Canterbury Puzzles

Math Concepts: addition to thirty-one, thinking ahead.
Players: best for two.
Equipment: one deck of math cards.

How to Play

Lay out the ace to six of each suit in a row, face up and not overlapping, one suit above another. You will have one column of four aces, a column of four twos, and so on‌—‌six columns in all.

The first player flips a card upside down and says its number value. Players alternate, each time turning down one card, mentally adding its value to the running total, and saying the new sum out loud. The player who exactly reaches thirty-one, or who forces the next player to go over that sum, wins the game.


Continue reading Math Game: Thirty-One

Teaching the Standard Algorithms

Peter Pan Arithmetic Quiz

[Feature photo above by Samuel Mann, Analytical Engine photo below by Roͬͬ͠͠͡͠͠͠͠͠͠͠͠sͬͬ͠͠͠͠͠͠͠͠͠aͬͬ͠͠͠͠͠͠͠ Menkman, both (CC BY 2.0) via Flickr.]

Babbage's Analytical Engine

An algorithm is a set of steps to follow that produce a certain result. Follow the rules carefully, and you will automatically get the correct answer. No thinking required — even a machine can do it.

This photo shows one section of the first true computer, Charles Babbage’s Analytical Engine. Using a clever arrangement of gears, levers, and switches, the machine could crank out the answer to almost any arithmetic problem. Rather, it would have been able to do so, if Babbage had ever finished building the monster.

One of the biggest arguments surrounding the Common Core State Standards in math is when and how to teach the standard algorithms. But this argument is not new. It goes back at least to the late 19th century.

Here is a passage from a book that helped shape my teaching style, way back when I began homeschooling in the 1980s…

Ruth Beechick on Teaching Abstract Notation

Understanding this item is the key to choosing your strategy for the early years of arithmetic teaching. The question is: Should you teach abstract notation as early as the child can learn it, or should you use the time, instead, to teach in greater depth in the mental image mode?


Abstract notation includes writing out a column of numbers to add, and writing one number under another before subtracting it. The digits and signs used are symbols. The position of the numbers is an arbitrary decision of society. They are conventions that adult, abstract thinkers use as a kind of shorthand to speed up our thinking.

When we teach these to children, we must realize that we simply are introducing them to our abstract tools. We are not suddenly turning children into abstract thinkers. And the danger of starting too early and pushing this kind of work is that we will spend an inordinate amount of time with it. We will be teaching the importance of making straight columns, writing numbers in certain places, and other trivial matters. By calling them trivial, we don’t mean that they are unnecessary. But they are small matters compared to real arithmetic thinking.

If you stay with meaningful mental arithmetic longer, you will find that your child, if she is average, can do problems much more advanced than the level listed for her grade. You will find that she likes arithmetic more. And when she does get to abstractions, she will understand them better. She will not need two or three years of work in primary grades to learn how to write out something like a subtraction problem with two-digit numbers. She can learn that in a few moments of time, if you just wait.

— Ruth Beechick
An Easy Start in Arithmetic (Grades K-3)
(emphasis mine)

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2015 Mathematics Game

happy new year 2015

[Feature photo above by Scott Lewis and title background (right) by Carol VanHook, both (CC BY 2.0) via Flickr.]


Did you know that playing games is one of the Top 10 Ways To Improve Your Brain Fitness? So slip into your workout clothes and pump up those mental muscles with the Annual Mathematics Year Game Extravaganza!

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.

Math Forum Year Game Site

Rules of the Game

Use the digits in the year 2015 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 2-0-1-5 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 NOT use a double factorial, n!! = the product of all integers from 1 to n that have the same parity (odd or even) as n. Math Forum allows these, but I’ve decided I prefer my arithmetic straight.

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