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.

      Numberless Word Problems

      Posted on by Denise Gaskins

      As I mentioned yesterday, my new book includes links to online resources to help you play with word problems. So this week, I’m sharing a few of my favorites.

      Visit the Kickstarter

      Today we examine a time-tested method to help kids reason about math: Leave out the numbers.

      First up, there’s Brian Bushart’s numberless problem bank for young students. Then we’ll look at Farrar Williams’s modern revision of a math teaching classic with problems for upper-elementary and middle school students.

      Have fun thinking math with your kids!

      Word Problem Bank

      Word problems are commonplace in mathematics classrooms, and yet they regularly confound students and lead to frustrated teachers saying things like:

      • “They just add all the numbers! It doesn’t matter what the problem says.”
      • “They don’t stop to think! They just start computing as soon as they’re done reading the problem.”

      Brian Bushart offers a collection of ready-to-go slide presentations that walk through the steps of making a word problem make sense.

      Visit the Site

      Math With No Numbers

      Discover Farrar Williams’s book Numberless Math Problems: A Modern Update of S.Y. Gillian’s Classic Problems Without Figures, available in ebook or paperback.

      Williams writes: “In order to answer the question, they’ll have to explain it, because the problem doesn’t give you anything to calculate with. The only way to answer is by explaining your process. See how sneaky a numberless problem is? It makes students really think about the process of solving the problem.”

      Find Out More

      “When students face a word problem, they often revert to pulling all the numbers out and “doing something” to them. They want to add, subtract, multiply, or divide them, without really considering which operation is the right one to perform or why.

        “When you don’t have numbers, it sidesteps that problem.

          “For students who freeze up when they see the numbers, this can be a really good way to get them to think about their process with math.”

          —Farrar Williams, Math With No Numbers

          CREDITS: Feature photo (top) by saeed karimi via Unsplash.com.

          Why Word Problems?

          Posted on by Denise Gaskins

          Wow! My Word Problems from Literature Kickstarter is just barreling along. I love seeing how many people are interested in a playful approach to teaching math.

          Check It Out

          But you might wonder: Why do I care so much about word problems?

          In many textbooks, word problems are an afterthought tacked on to the end of a math lesson.

          For me, it’s just the opposite. Word problems are the key part of a lesson, because that’s where children come face-to-face with the meanings of math concepts.

          The Key to Learning Math

          If we want our children to learn real math, we need to offer them plenty of problems to solve. A child may work through several pages of number calculations by rote, following memorized steps, but a good problem demands more thought.

          A story problem puts flesh on the abstract bones of arithmetic. Word problems encourage children to ponder what it means for one thing to be bigger than another, or smaller, or faster, or slower, or made up of several parts.

          Word Problems from Literature will feed your child’s mathematical imagination with story problems inspired by classic books, from 2nd-grade stories based on Mr. Popper’s Penguins to prealgebra stumpers inspired by The Lord of the Rings.

          And when you finish my puzzles, I’ll show you how to create your own word problems from literature, using your children’s favorite story worlds.

          The Trouble with Word Problems

          Most young children solve math problems by the flash-of-insight method: They hear the problem, and they know by instinct how to solve it.

          This is fine for simple problems like “Four kittens played with a yarn ball. Two more kittens came to join the fun. Then how many kittens were playing with the yarn ball?”

          When problems grow more difficult, however, that flash of insight becomes less reliable, so we find our children fidgeting with their paper or staring out the window. They complain, “I don’t know what to do. It’s too hard.”

          Too often, the frustrated child concludes, “I’m just not good at math.”

          But the truth is that nobody is good at math, if you define “good at math” to mean they can see the answer instantly. Here’s a more useful definition: You’re good at math if you have problem-solving tools and know how to use them.

          And that is something everyone can learn.

          Word Problems from Literature and the Word Problems Student Workbook will show you how. Order your copies today!

          Visit the Kickstarter

          Math for Star Wars Day

          Posted on by Denise Gaskins

          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

          Math Makes Sense — Let’s Teach It That Way

          Posted on by Denise Gaskins

          I had forgotten this video, and then rediscovered it yesterday and loved it just as much as ever. Perhaps you’ll enjoy it, too — especially if you think of yourself as “not a math person.”

          Annie Fetter is talking to classroom teachers, but her message is just as important for homeschoolers. Math is all about making sense. Let’s help our kids see it that way.

          “Sense-making is the first mathematical practice for a reason. If we don’t do this one, the rest of them don’t matter. If we’re not doing this, our children are not going to learn mathematics.”

          —Annie Fetter
          Sense Making: It isn’t Just for Literacy Anymore

          You can download the notes for Fetter’s updated session on sense-making and find several links to wonderful, thought-provoking posts on her blog:

          How Can We Encourage Sense-Making?

          Here are some ideas from Fetter’s updated notes, which expand on her comments in the video above:

          • Get rid of the question. Literally.
          • Ask students “What could the question be?”
          • Get rid of the question and the numbers.
          • Give the answer.
          • Or give several answers.
          • Ask about ideas, not answers.
          • Ask “Why?” or “How did you know?” or “How did you decide that?” or “Tell me more about that.”
          • Use active reading strategies.

          Get this free downloadable poster from Teacher Trap via Teachers Pay Teachers.

          A Few Resources to Practice Sense-Making

          In no particular order…

          “I implore you, stop ‘cracking the math code.’ Make sense-making the focus of every single thing you do in your math classroom.”

          —Annie Fetter
          Sense Making: It isn’t Just for Literacy Anymore

          And if you haven’t seen it before, don’t miss Annie Fetter’s classic video “Ever Wonder What They’d Notice?”

          CREDITS: “Building a rocket ship” photo by Kelly Sikkema via Unsplash. “Reading is thinking” poster by Teacher Trap via Teachers Pay Teachers.

          How to Succeed in Math: Answer-Getting vs. Problem-Solving

          Posted on by Denise Gaskins

          You want your child to succeed in math because it opens so many doors in the future.

          But kids have a short-term perspective. They don’t really care about the future. They care about getting through tonight’s homework and moving on to something more interesting.

          So how can you help your child learn math?

          When kids face a difficult math problem, their attitude can make all the difference. Not so much their “I hate homework!” attitude, but their mathematical worldview.

          Does your child see math as answer-getting? Or as problem-solving?

          Answer-getting asks “What is the answer?”, decides whether it is right, and then goes on to the next question.

          Problem-solving asks “Why do you say that?” and listens for the explanation.

          Problem-solving is not really interested in “right” or “wrong”—it cares more about “makes sense” or “needs justification.”

          Homeschool Memories

          In our quarter-century-plus of homeschooling, my children and I worked our way through a lot of math problems. But often, we didn’t bother to take the calculation all the way to the end.

          Why didn’t I care whether my kids found the answer?

          Because the thing that intrigued me about math was the web of interrelated ideas we discovered along the way:

          • How can we recognize this type of problem?
          • What other problems are related to it, and how can they help us understand this one? Or can this problem help us figure out those others?
          • What could we do if we had never seen a problem like this one before? How would we reason it out?
          • Why does the formula work? Where did it come from, and how is it related to basic principles?
          • What is the easiest or most efficient way to manipulative the numbers? Does this help us see more of the patterns and connections within our number system?
          • Is there another way to approach the problem? How many different ways can we think of? Which way do we like best, and why?

          What Do You think?

          How did you learn math? Did your school experience focus on answer-getting or problem-solving?

          How can we help our children learn to think their way through math problems?

          I’d love to hear from you! Please share your opinions in the Comments section below.

          CREDITS: “Maths” photo (top) by Robert Couse-Baker. “Math Phobia” photo by Jimmie. Both via Flickr (CC BY 2.0). Phil Daro video by SERP Media (the Strategic Education Research Partnership) via Vimeo.

          New Book: Word Problems from Literature

          Posted on by Denise Gaskins

          The posts on my Let’s Play Math blog are, for the most part, first-draft material. Of course, I’ve proofread each post — many times! because I’m a perfectionist that way, and yet I still miss typos :-/ — but these articles haven’t gotten the sort of feedback that polishes a book manuscript.

          Well, now I’m taking some of the best of my old blog posts, expanding them with a few new games or activities, and giving them that book-quality polish. Let me introduce my newest series, the Playful Math Singles.

          Under Construction …

          The Playful Math Singles from Tabletop Academy Press will be short, topical books featuring clear explanations and ready-to-play activities.

          I’m hoping to finish up two or three of these this year. Watch for them at your favorite online bookstore.

          The first one is done …

          Word Problems from Literature: An Introduction to Bar Model Diagrams

          You can help prevent math anxiety by giving your children the mental tools they need to conquer the toughest story problems.

          Young children expect to look at a word problem and instantly see the answer. But as they get older, their textbook math problems also grow in difficulty, so this solution-by-intuitive-leap becomes impossible.

          Too often the frustrated child concludes, “I’m just not good at math.”

          But with guided practice, any student can learn to master word problems.

          Word Problems from Literature features math puzzles for elementary and middle school students from classic books such as Mr. Popper’s Penguins and The Hobbit.

          For each puzzle, I demonstrate step by step how to use the problem-solving tool of bar model diagrams, a type of pictorial algebra. For children who are used to playing with Legos or other blocks — or with computer games like Minecraft — this approach reveals the underlying structure of a math word problem. Students can make sense of how each quantity in the story relates to the others and see a path to the solution.

          And when you finish the puzzles in this book, I’ll show you how to create your own word problems from literature, based in your children’s favorite story worlds.

          Free Online Preview

          Buy now at your favorite online bookstore.

          If you’re using these word problems with your children, consider buying them the paperback companion Word Problems from Literature Student Workbook.

          … and People Like It!

          A screen shot from this past weekend:

          “I found this method really clarified for me what was going on visually and conceptually. Particularly when it came to more complex questions, for which I would normally write out an equation, I felt that thinking about what was going on with the bars actually made more sense … This is a wonderful book for those who want to support their children in finding better ways to work on word problems.”

          —Miranda Jubb, Amazon customer reviewer

          Visualizing Word Problems with Bar Model Diagrams

          Posted on by Denise Gaskins

          A friend emailed me, frustrated with her child’s math lesson on bar diagrams: “Why do they have to make it so complicated? Why can’t we just solve the blasted problem?”

          I told her bar models themselves are not the goal. The real question for parents and teachers is:

          • What can you do when your child is stumped by a math word problem?

          To solve word problems, students must be able to read and understand what is written. They need to visualize this information in a way that will help them translate it into a mathematical expression.

          visualizing-word-problems

          Bar model diagrams are one very useful tool to aid this visualization. These pictures model the word problem in a way that makes the solution appear almost like magic.

          It is a trick well worth learning, no matter which math program you use.

          Visualization

          https://www.youtube.com/watch?v=HKsYDzQK8Zw

          “Visualization is the brain’s ability to see beyond what the eyes can see, and we can develop visualization in many ways.”

          The Bar Model Explained

          https://www.youtube.com/watch?v=I6Ipio8JntU

          “A bar model is a way to represent a situation in a word problem using diagrams — in particular, using rectangles.”

          https://www.youtube.com/watch?v=i7LAHc1qvig

          “This is one of the ideas that children learn in mathematics: the use of diagrams to represent quantities, especially quantities which are unknown.”

          Word Problems from Literature

          I’ve written a series of blog posts that explain bar model diagrams from the most basic through to solving multistep word problems. Check them out:

          I’ve started working on a book about bar model diagrams, and I’d love to hear your input. Have you tried using them? Do they help your children? What questions do you have?

          Update: My New Book

          You can help prevent math anxiety by giving your children the mental tools they need to conquer the toughest story problems.

          Check out Word Problems from Literature: An Introduction to Bar Model Diagrams—now available at all your favorite online bookstores!

          And there’s a paperback Student Workbook, too.

          CREDITS: Videos and quotations from Dr. Yeap Ban Har’s YouTube channel. “Girl doing homework” photo (top) by ND Strupler and “math notebooking equal fractions” by Jimmie via Flickr (CC BY 2.0).

          Professor Povey’s Perplexing Problems

          Posted on by Denise Gaskins

          Check out this new puzzle book for upper-level high school students & adults:

          Professor Povey picture

          Thomas Povey is a Professor of Engineering Science at the University of Oxford, where he researches jet-engine and rocket technology. In his new book Professor Povey’s Perplexing Problems, he shares his favorite idiosyncratic stumpers from pre-university maths and physics.

          These problems “should test your ability to grapple with the unfamiliar,” Povey writes. “You will learn to tease new problems apart, and apply things you already know in ways you had never considered. You have all the tools you need, but you should see what amazing things you can do with them.”

          Can You Solve This?

          Alex Bellos shared one of Professor Povey’s puzzles in The Guardian. Can you figure it out?

          Professor Povey cover

          The book starts off with geometry, but most of the chapters focus on various topics from physics. Some of the puzzles are accessible through applied common sense, but for many of them, it helps to have taken an algebra-based (high school level) physics course.

          Kitten is just finishing up her physics textbook, and she still has one more year of homeschooling. I’m hoping to work several of these puzzles into our schedule this year. It should be great fun!

          Spoiler

          If like me you’re a bit rusty on your physics, don’t worry. Each answer is thoroughly explained—‌in fact, it takes a bit of discipline to close the book and try your hand at each problem before reading on. I wish they’d put the solutions in the back rather than in the main text, to make it easier to browse the problems without reading spoilers.

          Speaking of which, here’s the answer to the video puzzle above…