Writing to Learn Math: Problem-solving cares less about whether an answer is right and more about whether a solution makes sense.
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.
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.
Writing to Learn Math: Math journal explanations avoid the formality that turns so many students away from geometry proofs.
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!
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.
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!
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?
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.
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.
“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
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This blog is reader-supported.
If you’d like to help fund the blog on an on-going basis, then please head to my Patreon page.
If you liked this post, and want to show your one-time appreciation, the place to do that is PayPal: paypal.me/DeniseGaskinsMath. If you go that route, please include your email address in the notes section, so I can say thank you.
Which I am going to say right now. Thank you!
“Playful Math: Getting Students To Write Their Own” copyright © 2022 by Denise Gaskins. Image at the top of the post copyright © Hannah Olinger via Unsplash.com.
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.
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 problems are commonplace in mathematics classrooms, and yet they regularly confound students and lead to frustrated teachers saying things like:
Brian Bushart offers a collection of ready-to-go slide presentations that walk through the steps of making a word problem make sense.
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.”
“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.
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.
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.
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.
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!
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This blog is reader-supported.
If you’d like to help fund the blog on an on-going basis, then please head to my Patreon page.
If you liked this post, and want to show your one-time appreciation, the place to do that is PayPal: paypal.me/DeniseGaskinsMath. If you go that route, please include your email address in the notes section, so I can say thank you.
Which I am going to say right now. Thank you!
“Why Word Problems?” copyright © 2022 by Denise Gaskins. Photos copyright © TarasMalyarevich, ArturVerkhovetskiy, Wavebreakmedia / Depositphotos.
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.
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:
Here are some ideas from Fetter’s updated notes, which expand on her comments in the video above:
Get this free downloadable poster from Teacher Trap via Teachers Pay Teachers.
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.
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.”
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 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: “Math Phobia” photo by Jimmie (blog post title added) via Flickr (CC BY 2.0). Phil Daro video by SERP Media (the Strategic Education Research Partnership) via Vimeo.
Denise Gaskins' Let's Play Math 