No peeking! This post is for those of you who have given the trisection proof a good workout on your own.
If you have a question about the proof or a solution you would like to share, please post a comment here.
But if you haven’t yet worked at the puzzle, go back and give it a try.
When someone just tells you the answer, you miss out on the fun. Figure it out for yourself — and then check the answer just to prove that you got it right.
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.
On a friend’s blog, I found this delightful letter written by Charles Lutwidge Dodgson to a young man named Wilton Cox.
The pseudonym Lewis Carroll comes from an alteration of the Latinized name Carolus (for Charles) and the replacement of Lutwidge with Lewis.
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.
1-3 narrators (or more, if you have a large group)
7 friends (non-speaking parts, adjust to fit your group)
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.
There’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
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…
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.
English mathematician Henry Ernest Dudeney wrote logic puzzles and mathematical games for several newspapers and magazines, later collected into books. This poem is from Amusements in Mathematics, published by Thomas Nelson and Sons, 1917.
The numbers are simple enough that younger students can solve it by the guess-and-check method. Older students or adults may want to set up a quadratic equation.
Historical Note: In the British currency of the time, there were 12 pennies to a shilling and 20 shillings to a pound (which was also called a sovereign).
’Twas last Bank Holiday, so I’ve been told,
Some cyclists rode abroad in glorious weather.
Resting at noon within a tavern old,
They all agreed to have a feast together.
“Put it all in one bill, mine host,” they said,
“For everyone an equal share will pay.”
The bill was promptly on the table laid,
And four pounds was the reckoning that day.
But, sad to state, when they prepared to square,
’Twas found that two had sneaked outside and fled.
So, for two shillings more than his due share
Each honest friend who had remained was bled.
They settled later with those rogues, no doubt.
How many were they when they first set out?
One fun thing about math is that you really don’t need the answer book. You can always check the math for yourself: Does your answer make sense? Does it fit the story?
Would you like to write a math poem puzzle of your own? I’d love to hear it!
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For more ideas on writing math poetry, check out Math Makers: Write a Poem.
This blog is reader-supported.
If you’d like to help fund the blog on an on-going basis, then please join me on Patreon for mathy inspiration, tips, and an ever-growing archive of printable activities.
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!
“Can You Solve It? The Cyclists’ Feast” copyright © 2023 by Denise Gaskins. Image at the top of the post copyright © yanlev / Depositphotos.
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.
Denise Gaskins' Let's Play Math 