Writing to Learn Math: Research prompts help students view math as a human endeavor.
Do you want your children to develop the ability to reason creatively and figure out things on their own?
Help kids practice slowing down and taking the time to fully comprehend a math topic or problem-solving situation with these classic tools of learning: Notice. Wonder. Create.
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.
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!
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
Here are a few pi-related links you may find interesting:
Or for pure silliness:
Have fun playing math with your kids!
Publication Day!
Prealgebra & Geometry: Math Games for Middle School hits the online bookstores today.
You can prepare your children for high school math by playing with positive and negative integers, number properties, mixed operations, algebraic functions, coordinate geometry, and more. Prealgebra & Geometry features 41 kid-tested games, offering a variety of challenges for students in 4–9th grades and beyond.
A true understanding of mathematics requires more than the ability to memorize procedures. This book helps your children learn to think mathematically, giving them a strong foundation for future learning.
And don’t worry if you’ve forgotten all the math you learned in school. I’ve included plenty of definitions and explanations throughout the book. It’s like having a painless math refresher course as you play.
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 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.)
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.
We call that biggest-inside square “inscribed” in the circle. Any polygon whose corners all sit on the circle is an inscribed polygon.
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.
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.
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:
Cimorene would love to hear about your children’s experiences playing with math! Please share your story in the comments below.
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If you’d like to help fund the blog on an on-going basis, then please head to my Patreon page.
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Which I am going to say right now. Thank you!
“Math Puzzle from the Ancient Kingdom of Cats” copyright © 2021by Denise Gaskins. Image at the top of the post copyright © Denise Gaskins.
Patrick Vennebush shared this puzzle from his new book, One-Hundred Problems Involving the Number 100:
It’s easy to cover a hundred chart with 100 small squares: 10 rows of 10 squares = 100.
It’s easy to cover a hundred chart with one big square: one 10×10 square = 100.
But can you cover the chart with 20 squares? Or with 57 squares? The squares do NOT have to be all the same size.
If we only consider squares with whole-number sides, so they exactly fit on the grid, then:
- What numbers of squares work to cover the chart?
- What numbers don’t work — and can you prove it?
Click to read the original puzzle along with some teaching tips at Patrick’s blog:
If you’d like some printable hundred charts for coloring in squares, download my free Hundred Charts Galore! file from my publisher’s online store:
And discover more ways to play with these printables in my classic blog post: 30+ Things to Do with a Hundred Chart.
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!
One of the sections in my book Let’s Play Math: How Families Can Learn Math Together — and Enjoy It encourages parents to make beautiful math with their children.
Do you have trouble believing that math can be beautiful?
In “Inspirations,” artist Cristóbal Vila creates a wonderful, imaginary work studio for the amazing M.C. Escher. You’ll want to view it in full-screen mode.
How many mathematical objects could you identify?
Vila offers a brief explanation of the history and significance of each item on his page Inspirations: A short movie inspired on Escher’s works.
Read about the inspirations, and then try making some math of your own.
“I looked into that enormous and inexhaustible source of inspiration that is Escher and tried to imagine how it could be his workplace, what things would surround an artist like him, so deeply interested in science in general and mathematics in particular. I imagined that these things could be his travel souvenirs, gifts from friends, sources of inspiration…”
—Cristóbal Vila
Inspirations: A short movie inspired on Escher’s works
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.
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!)
Have you ever wondered what mathematicians mean when they talk about a “beautiful” math proof?
“Beauty in mathematics is seeing the truth without effort.”
“There’s something striking about the economy of the counselor’s construction. He drew a single line, and that totally changed one’s vision of the geometry involved.
“Very often, there’s a simple introduction of something that’s not logically within the framework of the question — and it can be very simple — and it utterly changes your view of what the question really is about.”
CREDITS: Castle photo (top) by Rachel Davis via Unsplash. “A Mathematical Fable” via YouTube. Story told by Barry Mazur. Animation by Pete McPartlan. Video by Brady Haran for Numberphile.
Since I’ve been posting new Alexandria Jones stories this week (beginning here), I’ve gone back and re-read the old Christmas posts. I noticed that the original Graph-It Game included a religious design, but nothing for those who don’t celebrate Christmas.
So I updated the post with a new, non-religious puzzle. Here it is, if you want to play…
For this design, you will need graph paper with coordinates from −8 to +8 on both the x- and y-axis. Connect the points in each line. Stop at the periods, and then start a new line at the next point.
(-8,8) – (-8,0) – (0,8) – (-8,8) – (-4,4) – (0,4) – (0,8) – (8,8) – (4,4) – (0,8).
(8,8) – (8,0) – (4,0) – (4,-4) – (8,0) – (8,-8) – (0,-8) – (4,-4) – (0,-4) – (0,-8) – (-8,0) – (-8, -8) – (0,-8).
(-8,-8) – (4,4) – (0,4) – (4,0) – (4,4) – (8,0).
(8,-8) – (-4,4) – (-4,-4) – (0,-4) – (-4,0) – (-8,0).
(0,-2) – (0,-4) – (4,0) – (2,0) – (2,-2) – (-2,-2) – (-2,2) – (2,2) – (2,0) – (1,1) – (1,0) – (2,0) – (0,-2) – (-2,0) – (0,2) – (1,1) – (-1,1) – (-1,-1) – (1,-1) – (1,0) – (-4,0) – (0,4) – (0,-1) – (-1,0) – (0,1) – (1,0) – (0,-1) – (0,-2).
Color in your design and hang it up for the whole family to enjoy!
Of course, the fun of the Graph-It Game is to make up your own graphing puzzle. Can you create a coordinate design for your friends to draw?
You can see all the Alexandria Jones Christmas posts at a glance here:
CREDITS: “Love Christmas Lights” photo by Kristen Brasil via Flickr (CC BY 2.0).
Denise Gaskins' Let's Play Math 