Celebrating Math with Pi Day

Posted on by Denise Gaskins

Are your students doing anything special for Pi Day?

Back when we were homeschooling, my kids and I always felt stir-crazy after two months with no significant break. We needed a day off — and what better way could we spend it than to play math all afternoon?

I love any excuse to celebrate math!

Pi Day is March 14. If you write dates in the month/date format, then 3/14 at 1:59 is about as close as the calendar can get to 3.14159etc.

(Otherwise, you can celebrate Pi Approximation Day on July 22, or 22/7.)

Unfortunately, most of the activities on teacher blogs and Pinterest focus on the pi/pie wordplay or on memorizing the digits. With a bit of digging, however, I found a few puzzles that let us sink our metaphorical teeth into real mathematical meat.

What’s the Big Deal? Why Pi?

In math, symmetry is beautiful, and the most completely symmetric object in the (Euclidean) mathematical plane is the circle. No matter how you turn it, expand it, or shrink it, the circle remains essentially the same.

Every circle you can imagine is the exact image of every other circle there is.

This is not true of other shapes. A rectangle may be short or tall. An ellipse may be fat or slim. A triangle may be squat, or stand upright, or lean off at a drunken angle. But circles are all the same, except for magnification. A circle three inches across is a perfect, point-for-point copy of a circle three miles across, or three millimeters.

What makes a circle so special and beautiful? Any child will tell you, what makes a circle is its roundness. Perfectly smooth and plump, but not too fat.

The definition of a circle is “all the points at a certain distance from the center.” Can you see why this definition forces absolute symmetry, with no pointy sides or bumped-out curves?

One way to express that perfect roundness in numbers is to compare it to the distance across. How many times would you have to walk back and forth across the middle of the circle to make the same distance as one trip around?

The ratio is the same for every circle, no matter which direction you walk.

That’s pi!

Puzzles with Pi

For all ages:

Sarah Carter created this fun variation on the classic Four 4s puzzle for Pi Day:

Using only the digits 3, 1, 4 once in each calculation, how many numbers can you make?

You can use any math you know: add, subtract, multiply, square roots, factorials, etc. You can concatenate the digits, putting them together to make a two-digit or three-digit number.

For older students:

1. Imagine the Earth as a perfect sphere with a long rope tightly wrapped around the equator. Then increase the length of the rope by 10 feet, and magically lift it off the Earth to float above the equator. Will an ant be able to squeeze under the rope without touching it? What about a cat? A person?

2. If you ride a bicycle over a puddle of water, the wheels will leave wet marks on the road. Obviously, each wheel leaves a periodic pattern. How the two patterns are related? Do they overlap? Does their relative position depend on the length of the puddle? The bicycle? The size of the wheels?

3. Draw a semicircle. Along its diameter draw smaller semicircles (not necessarily the same size) that touch each other. Because there are no spaces in between, the sum of the diameters of the small semicircles must equal the diameter of the large one. What about their perimeter, the sum of their arc lengths?

4. Choose any smallish number N. How can you cut a circular shape into N parts of equal area with lines of equal lengths, using only a straight-edge and compass? Hint: The lines don’t have to be straight.

[Solutions at Alexander Bogomolny’s Pi Page. Scroll down to “Extras.”]

It can be of no practical use to know that Pi is irrational, but if we can know, it surely would be intolerable not to know.

— Edward Titchmarsh

For More Information

Here are a few pi-related links you may find interesting:

Or for pure silliness:

Have fun playing math with your kids!

John Reid, CC BY-SA 3.0 via Wikimedia Commons

Playful Math: Getting Students To Write Their Own

Posted on by Denise Gaskins

To wrap up our week of exploring the resources from Word Problems from Literature, let’s talk about getting students to write their own math.

Check in on the Kickstarter

First up, I’m sharing an excerpt from the Word Problems Student Workbook. The “Story Problem Challenge” is one of my favorite math club activities.

Following that, you’ll find an amazing online mathemagical adventure for middle school: The Arithmetiquities. It’s great fun, and a great inspiration for students to create their own math stories.

Have fun writing math with your kids!

The Story Problem Challenge

What do you get when you cross a library book or favorite movie with a math worksheet? A great alternative to math homework!

The rules are simple:

(1) Choose a worksheet calculation to be the basis for your word problem.

(2) Solve the calculation.

(3) Consider where these numbers could make sense in your book or movie universe. How might the characters use math? What sort of things would they count or measure? Do they use money? Do they build things, or cook meals, or make crafts? Do they need to keep track of how far they have traveled? Or how long it takes to get there?

(4) Write your story problem.

To make the game easier, you may change the numbers to make a more realistic problem. But you must keep the same type of calculation. For example, if your worksheet problem was 18÷3, you could change it to 18÷6 or 24÷3 or even 119÷17 to fit your story, but you can’t make it something like 18−3.

Remember that some quantities are discrete and countable, such as hobbits and fireworks. Other quantities are continuous, such as a barrel of wine or a length of fabric. Be sure to consider both types when you are deciding what to use in your problem.

Then share your problem with friends, and you try their problems. Can you stump each other?

A Note about Copyright and Trademarks

Old books are in the public domain, so you can always use characters like Robin Hood, Sherlock Holmes, or Winnie-the-Pooh (but not the newer Disney version with the red jacket). But most books and movies are the protected intellectual property of their authors or estates, or of the company who bought those rights.

When you write problems for your own private use, feel free to use your favorite characters from any story. That’s like fan fiction, secret, just for your own pleasure.

But if you decide to share your creation beyond your own home or classroom, then be sure to “genericize” it first. Change or remove the proper names, using general descriptions instead.

For example, if you love the Harry Potter series, you might want to use Harry or Hermione in your story problems. Instead, write about “the boy wizard destined to fight an evil sorcerer.” Or “the bright young witch who can master any spell.”

Or if you like the Star Wars movies, you might write about “an interstellar justice warrior with an energy sword.” Or “an alien master of martial arts training a cocky but inexperienced apprentice.”

We’d love to add your story to the Student Math Makers Gallery.

The Arithmetiquities

When the world of Sfera is threatened by the machinations of a malevolent sorcerer, it will be up to a band of unlikely heroes to become the brightest light in the darkness.

The adventurers fan out across the land to find and retrieve the Arithmetiquities, a set of ancient mathemagical artifacts.

The Arithmetiquities is a fantasy adventure story told through a sequence of 36 mathematical puzzles.

Join the Adventure

“Though it is still before sunrise, Lumparland Harbor is already bustling. Sailing ships moor at the misty docks, bringing travelers and goods to the seaside town. Three dwarves disembark from different ships, each adventurer returning home from some faraway locale. The three women gather at the end of the pier.

    “The strangers discover that they all live along the main road that leads from the harbor, so they decide to split the cost of a wagon. Egga lives 10 miles away, Floora lives 20 miles away, and Greeta lives 30 miles away. The wagon ride costs $1.50 per mile regardless of the number of passengers.

      “How much should each of the adventurers pay so that each one has a fair fare?”

      —Jason Ermer, “Lumparland Harbor,” The Arithmetiquities Chapter I

      CREDITS: Feature photo (top) by Hannah Olinger via Unsplash.com.

      Problem Solving with James Tanton

      Posted on by Denise Gaskins

      At the back of my new Word Problems from Literature book, I’ve included an appendix with links to recommended online resources.

      Check in on the Kickstarter

      So I thought this week, I’d share some of my favorites with you. First up: Problem Solving Tips from James Tanton.

      You may know Tanton from the popular Exploding Dots and other activities at the Global Math Project website. But he’s been busy for decades sharing the delight and the beauty of the subject. He currently serves as the Mathematician-at-Large for the Mathematical Association of America.

      Read on to discover several of Tanton’s best problem-solving tips for middle school and older students.

      Have fun exploring math with your kids!

      How to Think like a School Math Genius

      In this 4-video series, Tanton presents five key principles for brilliant mathematical thinking, along with loads and loads of examples to explain what he means by each of them. A call for parents and teachers to be mindful of the life thinking we should foster, encourage, promote, embrace and reward — even in a math class!

      Watch the Videos

      Two Key — but Ignored —Steps to Solving Any Math Problem

      How many degrees in a Martian circle?
      Every challenge or problem we encounter in mathematics (or life!) elicits a human response. The dryness of textbooks and worksheets in the school world might suggest otherwise, but connecting with one’s emotions is fundamental and vital for success — and of course, joy — in doing mathematics.

      Read the Article

      MAA AMC Curriculum Inspirations

      Essays and videos showing how to approach math puzzles in a way that a) is relevant and connected to the curriculum, and b) revels in deep, joyous, mulling and flailing, reflection, intellectual play and extension, insight, and grand mathematical delight.

      Scroll down and start with the Ten Problem-Solving Strategies.

      Download the Puzzles

      Think Puzzles and Think Cool Math

      Here are some essays illustrating astounding tidbits of mathematical delight. And here are some purely visual puzzles to surprise.

      Explore and Enjoy

      “The true joy in mathematics, the true hook that compels mathematicians to devote their careers to the subject, comes from a sense of boundless wonder induced by the subject.

        “There is transcendental beauty, there are deep and intriguing connections, there are surprises and rewards, and there is play and creativity.

          “Mathematics has very little to do with crunching numbers. Mathematics is a landscape of ideas and wonders.”

          —James Tanton

          CREDITS: Feature photo (top) by Ian Stauffer via Unsplash.com.

          Playful Math Journaling with a Cat

          Posted on by Denise Gaskins

          As queen of the house, Cimorene insists on being involved in anything that happens in her domain. This includes promoting the Playful Math Journaling Kickstarter.

          So she created a cat math journaling prompt to help your children experience the fun of playing around with math.

          But first, she encourages you to visit the Kickstarter page and download the free 16-page printable Math Journaling Sampler file. Your kids will love solving Cimorene’s puzzle on one of the parchment-style pages!


          [The free download will always be there, even after the Kickstarter project ends.]
          Visit the Kickstarter

          Here is Cimorene’s Puzzle

          “The Princess of Cats has a luxuriously soft tail about 12 inches (30 cm) long. Her tail is three times the length of her noble head. Her beautiful, furry body is as long as head and tail together. How long is the Princess from her delicate nose to the tip of her majestic tail?”

          So, how does math journaling work? What do children do with a problem like this?

          They may want to make a list of the things they know from the story. Perhaps they will draw a picture of the cat and label the proportions. Each will take their own approach to figure it out.

          And then the best part of any math journal prompt is when kids make their own math.

          • Can they write a new puzzle about their own pet?
          • Or about their favorite animal?

          Encourage your children to share their math creations with their friends and family.

          Cimorene would love to read it, too! If you share your story in the comments section below, I will be sure to show it to her.

          And remember to back the Playful Math Journaling Kickstarter so your whole family can enjoy the adventure of playing with math!

          Math Puzzle from the Ancient Kingdom of Cats

          Posted on by Denise Gaskins

          It may look like Cimorene has lain down on the job, but don’t be fooled! She’s hard at work, creating a math investigation for your students to explore.

          Cats know how important it can be for students to experiment with math and try new things. Playing with ideas is how kittens (and humans!) learn.

          Cimorene wants you to know that the Make 100 Math Rebels Kickstarter offers a great way for human children to learn math through play. She encourages you to go watch the video and read all about the project.

          Too often, school math can seem stiff and rigid. To children, it can feel like “Do what I say, whether it makes sense or not.” But cats know that kids are like kittens — they can make sense of ideas just fine if we give them time to play around.

          So Cimorene says you should download the free sample journaling pages from the Math Rebels Kickstarter page. The beautiful parchment design makes doing math an adventure.


          [The free download will always be there, even after the Kickstarter project ends.]
          Make 100 Math Rebels Kickstarter

          Cimorene’s Puzzle Challenge

          Cimorene’s math puzzle is a classic geometry problem from the ancient Kingdom of Cats: Squaring the Circle.

          Draw a circle on your journal page. Can you draw a square (or rectangle) that has the same area?

          How would you even begin such a task?

          Notice Cimorene’s hint in the photo above: Try drawing the square that just touches the edges of your circle. (We call those just-touching lines “tangents” to the circle.)

          • What do you notice? Do the square and the circle have the same area? How close are they?

          The tangent square sets an upper limit on the area of the circle. You can see that any square that exactly matches the circle would have to be smaller than the tangent square.

          • Can you find a square that sets a lower limit on the area of the circle? That is, a square that must have less area than the circle?
          • What’s the biggest square you can draw inside your circle? Can you find a square that has all four corners on the circle?

          We call that biggest-inside square “inscribed” in the circle. Any polygon whose corners all sit on the circle is an inscribed polygon.

          • Play around with circles and squares. How close can you get to matching their size?

          Further Exploration

          After you have explored for awhile on your own, Cimorene has one more twist in her puzzle.

          In the ancient Kingdom of Cats, the wise ones estimated the area of a circle this way:

          Divide the width of the circle in thirds, and then in thirds again. (That is, cut the diameter into nine parts.) Draw a square with sides measured by eight such parts.

          You can try this on your journaling page by drawing a circle that is nine squares wide. Then draw a square overlapping it, with sides that are eight squares in length.

          • How closely do the areas match?

          Playing with Pi

          Here’s a surprise: Cimorene’s puzzle isn’t really about squares, but about calculus.

          The problem of Squaring the Circle is really a much bigger question: Finding the area of a square, rectangle, or other polygon is relatively easy, but how can we discover the area of a curved shape?

          For a circle, the area is related to the number pi, which is the number of times you would have to walk across the circle to equal the distance of one time walking around it.

          graphic by John Reid (cc by-sa 3.0)

          graphic by John Reid (cc by-sa 3.0)
          So the problem of Squaring the Circle is really the same as asking, “What is the value of pi?”

          • Can you figure out what approximate value for pi matches the 8/9 square used in the ancient Kingdom of Cats?

          If you’d like to learn more about pi, get ready for a celebration: Pi Day is coming soon! Every year, millions of children celebrate math on March 14th, because if you write the date as 3/14, it’s the same as the first three digits of pi.

          Find out more about playing with pi in my Pi Day Round-Up post.

          You may also enjoy:

          Share Your Stories

          Cimorene would love to hear about your children’s experiences playing with math! Please share your story in the comments below.

          Math Journals: Save the Cat!

          Posted on by Denise Gaskins

          Puck is concerned that some people don’t understand the idea behind the Math Rebel journals. He decided to create a journaling prompt so your children can experience the joy of creative reasoning (and save cats from their mortal enemy!)

          Journaling is a great way to help children learn to see with mathematical eyes. Not just to remember what we tell them, but to create their own math.

          Many people know it’s important for students to do hands-on experiments in science. But Puck realized that most adults don’t know how to do a math experiment.

          So Puck created this Cat Escape puzzle…

          Continue reading Math Journals: Save the Cat!

          Can You Do the Math Salute?

          Posted on by Denise Gaskins

          Did your device hide the video? Find it on YouTube here.

          How Is This Math?

          The idea that math is only about numbers, calculations, and textbook exercises is one of the greatest lies we learn in school. Of course, nobody ever comes straight out and actually says that. But the whole system teaches us every day what counts for math and what doesn’t.

          James Tanton’s math salute is a physical puzzle.

          How in the world did he do that?

          Physical puzzles don’t fit into our cultural understanding of math. But the process of figuring out the puzzle is the same problem-solving process we use to figure out other puzzles — including the puzzles we call math.

          In fact, real mathematics is all about figuring out puzzles without a teacher showing you what to do. Problem-solving is a universally useful skill.

          As master teacher W. W. Sawyer said:

          “Everyone knows that it is easy to do a puzzle if someone has told you the answer. That is simply a test of memory. You can claim to be a mathematician only if you can solve puzzles that you have never studied before. That is the test of reasoning.”

          —W. W. Sawyer, Mathematician’s Delight

          So tackle the puzzle of the math salute. Show it to your kids. (And don’t be surprised if they figure it out before you do!)

          [THE FINE PRINT: I am an Amazon affiliate. If you follow the link and buy something, I’ll earn a small commission (at no cost to you). But this book is a well-known classic, so you should be able to order it through your local library.]

          New Printable Puzzle Books: Diffy Inception

          Posted on by Denise Gaskins

          The best way to practice math is to play with it—to use the patterns and connections between math concepts in your pursuit of something fun or beautiful.

          Diffy Inception puzzles have their own symmetric beauty, but mostly they are just plain fun. Students can practice subtraction and look for patterns in the difference layers.

          I just published four new activity books to our online store:

          Notes to the teacher include puzzle instructions, game variations, journaling prompts, and more. Plus answers for all puzzles.

          Available with 8 1/2 by 11 (letter size) or A4 pages.

          Click for a Preview

          My publishing company runs this online store, so you can find all my playful math books there — including an exclusive pre-publication ebook edition of my newest title, Prealgebra & Geometry: Math Games for Middle School. Click here to browse the Tabletop Academy Press store.

          The Value of Puzzles

          Posted on by Denise Gaskins

          I love puzzles. Don’t you?

          Here are several examples of river-crossing puzzles you and your kids can try. They date back at least to the time of Alcuin, the famous scholar from the court of Charlemagne.

          I wish someone would write a whole math curriculum devoted entirely to puzzles.

          W.W. Sawyer on the Value of Puzzles

          Master teacher W.W. Sawyer didn’t write a curriculum, but he often used puzzles in the classroom.

          “It is quite possible to use simultaneous equations as an introduction to algebra. Within a single lesson, pupils who previously did not know what x meant can come not merely to see what simultaneous equtions are, but to have some competence in solving them.

          “No rules need to be learnt; the work proceeds on a basis of common sense.

          “The problems the pupils solve in such a first lesson will not be of any practical value. They will be in the nature of puzzles.

          “Fortunately, nature has so arranged things that until the age of twelve years or so, children are more interested in puzzles than in realistic problems.”

          —W. W. Sawyer, Vision in Elementary Mathematics

          Then he gives this example:

          “A man has two sons. The sons are twins; they are the same height. If we add the man’s height to the height of one son, we get 10 feet. The total height of the man and the two sons is 14 feet. What are the heights of the man and his sons?”

          Try This at Home

          Not only can children solve puzzles like this, but even better — they can make up story puzzles of their own. You could spend a whole week or more making up silly height puzzles for each other to solve. By the time you were done, your kids would have a great introduction to algebra!

          Maybe I never grew up. Because I still prefer puzzles over “real world” math problems.

          What are your favorite kinds of puzzles? Please share in the comments section.

          CREDITS: “Boat puzzles” comic from xkcd.com.
          [THE FINE PRINT: I am an Amazon affiliate. If you follow the book link and buy something, I’ll earn a small commission (at no cost to you). But this book is a well-known classic, so you should be able to order it through your local library.]

          More Dover Samples

          Posted on by Denise Gaskins

          “Without mathematics you can’t do anything! Everything around you is mathematics. Everything around you is numbers.”

          —Anna Claybourne, I Can Be a Math Magician


          Dover Publications sent out a new email today with fun coloring and craft samples. And several puzzles from I Can Be a Math Magician: Fun STEM Activities for Kids by Anna Claybourne.

          Enjoy!

          If you’d like to receive future Dover Sampler emails, you can sign up here.

          THE FINE PRINT: I am an Amazon affiliate. If you follow the book link above and buy something, I’ll earn a small commission (at no cost to you).