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.
Sue VanHattum and I were chatting about her young adult math books.
[Sue would love to get your help with beta-reading her books. Scroll down to the bottom of this post for details.]
In the first book of the series, Althea and the Mystery of the Imaginary Numbers, Althea learns that Tartaglia came up with a formula to solve cubic equations and wrote about it in a poem.
Sue had discovered an English translation of that poem and shared it with me. (You can read it on JoAnne Growney’s blog.) Then we wondered whether we could come up with a simpler poem, something an algebra student might be able to follow.
Perhaps you and your kids would enjoy making up poems, too. An algebra proof-poem might be too difficult for now, but check out my blog for math poetry ideas.
The two ideas that Mason considered important in math — rightness and reason — are connected. It is our reasoning that convinces us an answer is right or wrong. How do we know we got a sum correct? We can take the numbers apart and add them another way, to see if we get the same answer. Or we can subtract one of the numbers from the sum and see if we get the other number. Or … well, how would you prove it?
More than anything else, Mason wanted her students to discover in math a sense of immutable truth, a truth that stands on its own, apart from anything we say or do, a truth we can explore and reason about but can never change.
This sense of rightness, of solid, unalterable truth, inspires a feeling of wonder and awe — she calls it “Sursum corda,” a call to worship — that delights our minds. It’s that “Aha!” feeling we get when something we’ve been struggling with suddenly fits together and makes sense.
From the very beginning, children should be doing this sort of informal proof, explaining how they figured things out. Don’t wait until high school geometry to let your children wrestle with ideas.
Kitten (my daughter) and I sat on the couch sharing a whiteboard, passing it back and forth as we took turns working through our prealgebra book together.
The chapter on number theory began with some puzzles about multiples and divisibility rules.
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.
One of the most persistent math myths in popular culture is the idea that mathematics is primarily about getting right answers.
The truth is, the answer doesn’t matter that much in math. What really matters is how you explain that answer. An answer is “right” if the explanation makes sense.
And if you don’t give an explanation, then you really aren’t doing mathematics at all.
Here is a short sequence of numbers. Can you figure out the rule and fill in the next three blanks?
2, 3, 5, 7, ___, ___, ___, …
Remember, what’s important is not which numbers you pick, but rather how you explain your answer.
Perhaps the sequence is the prime numbers?
2, 3, 5, 7, 11, 13, 17, …
The prime numbers make a wonderful sequence, though it isn’t the one I was thinking of.
Math Concepts: division as equal sharing, naming fractions, adding fractions, infinitesimals, iteration, limits
Prerequisite: able to identify fractions as part of a whole
This is how I tell the story:
Continue reading Infinite Cake: Don Cohen’s Infinite Series for Kids
[Feature photo above by Michael Cory via Flickr (CC BY 2.0).]
I hear so many people say they hated geometry because of the proofs, but I’ve always loved a challenging puzzle. I found the following puzzle at a blog carnival during my first year of blogging. Don’t worry about the arbitrary two-column format you learned in high school — just think about what is true and how you know it must be so.
I hope you enjoy this “Throw-back Thursday” blast from the Let’s Play Math! blog archives:
One of the great unsolved problems of antiquity was to trisect any angle using 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 or something.
One “cheat” that works is to fold your paper. I will show you how it works, and your job is to show why …
[Feature photo above by hom26 via Flickr.]
My free time lately has gone to local events and to book editing. I hope to put up a series of blog posts sometime soon, based on the Homeschool Math FAQs chapter I’m adding to the paperback version of Let’s Play Math. [And of course, I’ll update the ebook whenever I finally publish the paperback, so those of you who already bought a copy should be able to get the new version without paying extra.]
But in the meantime, as I was browsing my blog archives for an interesting “Throw-Back Thursday” post, I stumbled across this old geometry puzzle from Dave Marain over at MathNotations blog:
Jake shows Jack a piece of wood he cut out in the machine shop: a circular arc bounded by a chord. Jake claimed that the arc was not a semicircle. In fact, he claimed it was shorter than a semicircle, i.e., segment AB was not a diameter and arc ACB was less than 180 degrees.
Jack knew this was impossible and argued: “Don’t you see, Jake, that O must be the center of the circle and that OA, OB and OC are radii.”
Jake wasn’t buying this, since he had measured everything precisely. He argued that just because they could be radii didn’t prove they had to be.
Which boy do you agree with?
If you have a student in geometry or higher math, print out the original post (but not the comments — it’s no fun when someone gives you the answer!) and see what he or she can do with it.
Dave offers many other puzzles to challenge your math students. While you are at his blog, do take some time to browse past articles.
Denise Gaskins' Let's Play Math 