Puzzle: Patty Paper Trisection

Posted on by Denise Gaskins
student using drafting tools

One of the great unsolved problems of antiquity was to trisect any angle, to cut it into thirds with only the basic tools of Euclidean geometry: an unmarked straight-edge and a compass.

Like the alchemist’s dream of turning lead into gold, this proved to be an impossible task. If you want to trisect an angle, you have to “cheat.” A straight-edge and compass can’t do it. You have to use some sort of crutch, just as an alchemist would have to use a particle accelerator.

One “cheat” that works is to fold your paper.

I will show you how it works, and your job is to show why.

Continue reading Puzzle: Patty Paper Trisection

Skit: The Handshake Problem

Posted on by Denise Gaskins
The handshake problem

If seven people meet at a party, and each person shakes the hand of everyone else exactly once, how many handshakes are there in all?

Our homeschool co-op held an end-of-semester assembly. Each class was supposed to demonstrate something they had learned.

I threatened to hand out a ten question pop quiz on integer arithmetic, but instead my pre-algebra students presented this skit.

Download a Printable Script

Cast

1-3 narrators (or more, if you have a large group)
7 friends (non-speaking parts, adjust to fit your group)

Props

Each friend will need a sheet of paper with a number written on it big and bold enough to be read by the audience. The numbers needed are 0, 1, 2, 3, … up to one less than the number of friends. Each friend keeps his paper in a pocket until needed.

Continue reading Skit: The Handshake Problem

Math Journal: The 1-2-3 Puzzle

Posted on by Denise Gaskins
colorful numbers 1, 2, 3

Math Journaling Adventures series by Denise GaskinsThere’s still time to check out my Math Journaling Adventures project and discover how playful writing activities will help your students learn mathematics. Preorder your books today!

Meanwhile, here’s a math puzzle to share with your kids…

Write down any whole number. It can be a single-digit number, or as big as you like. For example:

64,861,287,124,425,928

Now, count up the number of even digits (including zeros), the number of odd digits, and the total number of digits your number contains. Write those counted numbers down in order, like this:

64,861,287,124,425,928
even 12, odd 5, total 17

Continue reading Math Journal: The 1-2-3 Puzzle

Playful Math 179: Our Sweet Sixteen Carnival

Posted on by Denise Gaskins

Welcome to the sweet-16 birthday edition of the Playful Math Carnival. Originally called Math Teachers at Play, our first carnival was published in February 2009.

Each Playful Math Carnival offers 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.

There’s so much playful math to enjoy!

By tradition, we start the carnival with a math activity in honor of our 179th edition. But if you’d rather jump straight to our featured blog posts, click here to see the Table of Contents.

NOTE: Our wonderful volunteer hosts have kept the Playful Math Carnival going when so many other blog carnivals died off. If you’d like to sign up to host the carnival for a month, email Denise for information.

Try These Prime Puzzles

Did you know there are 179 even-numbered days this year?

  • How many even-numbered days will there be in a leap year?
  • But there are 365 days in a standard year and 366 in a leap year. Shouldn’t there be half that many even-numbered days?

179 is a prime number, and it’s also a knockout prime. You can knock out any of the digits, and what’s left is still prime: 17, 19, or 79.

  • Can you find another knockout prime number?

179 is a twin prime. That means that one of its odd-numbered neighbors is also prime.

  • Is the other twin 177 or 181? Can you tell without looking it up?
  • Why are twin primes limited to the odd numbers? That doesn’t seem fair!

179 is also an emirp. That’s a special kind of prime that forms a different prime number when you write it backwards: 971 is also prime.

  • How many emirps can you find?

“A palindrome is a word that when written in reverse results in the same word. for example, ‘racecar’ reversed is still ‘racecar’. Related to palindromes are semordnilaps. These are words that when written in reverse result in a distinct valid word. For example, ‘stressed’ written in reverse is ‘desserts’. Not all words are palindromes or semordnilaps.

    “While certainly not all numbers are palindromes, all non-palindromic numbers when written in reverse will form semordnilaps.

      “Narrowing to primes brings back the same trichotomy as with words: some numbers are emirps, some numbers are palindromic primes, but some are neither.”

      —The On-Line Encyclopedia of Integer Sequences A006567

      Click here for all the mathy goodness!

      Memories: The Oral Story Problem Game

      Posted on by Denise Gaskins
      photo of sheep in a field

      Homeschool Memories…

      Perhaps you’ve heard me mention the oral story problem game. It was one of my favorite ways to get my children thinking about math, back in our early days of homeschooling. We played in the car on the way to soccer practice, or while we washed dishes, or sitting in the lobby waiting for a doctor’s appointment.

      The rules are simple: I’ll make up a math problem for you to solve. And then you make up one for me.

      The kids always loved trying to stump me.

      This problem from Henry Ernest Dudeney’s Amusements in Mathematics reminded me of those days. This is exactly the way my eldest loved to torture me…

      Continue reading Memories: The Oral Story Problem Game

      Puzzle: Henry Dudeney’s Pebble Game

      Posted on by Denise Gaskins
      photo of girl playing with pebbles on the beach

      English mathematician and puzzle-meister Henry Ernest Dudeney once wrote:

      “It may be said generally that a game is a contest of skill for two or more persons, into which we enter either for amusement or to win a prize. A puzzle is something to be done or solved by the individual.

        “The example that I give here is apparently a game, but, as in every case one player may win if he only play correctly, it is in reality a puzzle. The interest, therefore, lies in attempting to discover the leading method of play.”

        Below is the puzzle game as Dudeney explained it.

        Play it for fun at first, then see if you can solve the puzzle.

        Continue reading Puzzle: Henry Dudeney’s Pebble Game

        The Rhythm of Math Puzzles

        Posted on by Denise Gaskins
        Father and son homeschooling math

        Homeschooling families naturally build patterns and routines that help us keep our sanity as we go through our homeschooling day.

        No matter what resources we choose or which curriculum packages we buy, we never end up following the book exactly as it is written. So don’t worry if you find yourself wandering away from the lesson plan. You’re not getting behind; you’re just discovering your family’s natural learning style.

        If you and your children have fallen into the rut of traditional math lessons, have patience. Give yourself time to adjust to a more relaxed mindset about math.

        And when you find the rhythm that fits your family, you’ll discover that math lessons flow so much more smoothly.

        Continue reading The Rhythm of Math Puzzles

        Learning Math with Puzzles

        Posted on by Denise Gaskins
        Mother and daughter homeschooling math

        Learning mathematics begins with puzzles. As stories are to history or science, puzzles are to math.

        In ancient times, math began with puzzles like:

        • How can we keep track of our herds, or predict the cycles of the moon, or figure out how much to pay for taxes?

        • How can we draw perfect right angles, or make beautifully symmetric designs?

        • What are the ratios of harp strings that create the most beautiful sounds?

        Math puzzles touch every area of life. Puzzles inspire curiosity and solving them brings joy.

        And we can enrich any homeschool math program with puzzles that invite children to think about mathematical concepts.

        Even those monster topics that often bring homeschooling families to tears, such as fractions.

        Continue reading Learning Math with Puzzles

        New Book: Farzanah and the 17 Camels

        Posted on by Denise Gaskins
        illustration from Farzanah and the 17 Camels

        Now’s your chance to promote creative math for kids, with Sue Looney’s new book on Kickstarter: Farzanah and the 17 Camels.

        You can order the book in digital or paperback format, get multiple copies for a classroom or book club, or just donate to support the project.

        Visit the Kickstarter

        Farzanah and the 17 Camels celebrates the excitement and the rewards of solving a challenging and intriguing math problem. Set against the backdrop of the ancient Silk Road, with bustling markets, stunning carpets, fun characters, and camels, the story draws readers into the magic of Farzanah’s surroundings. 

        As Farzanah searches for an unusual approach, a way of solving the problem that no one else could think of, she follows the wise advice of her mother: 

        “My dear Farzanah, don’t be discouraged,” said Mama. “Sometimes, being stuck is exactly where you need to be. I find the best thing I can do is to step away. I free my mind to think about other things. It is in that space that the magic happens. I am able to look at things from a different perspective. With wait time and wishful thinking comes the solution.”

        Farzanah embodies the joy of productive struggle in mathematical problem-solving. She is patient, persistent, and curious, using these qualities to tackle a perplexing dilemma that has troubled everyone.

        For a bonus math puzzle you can play today, based on Farzanah’s life, download Farzanah’s Sheep Dilemma.

        Looney’s earlier book, Ying and the Magic Turtle, is available on the Natural Math website. Or check out her Same But Different discussion prompts for all ages on her website.

        Visit the Kickstarter

         
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        Illustration above copyright © 2024 Noemi Sophia James, from the book Farzanah and the 17 Camels by Sue Looney.