Tag Archives: Proofs

Infinite Cake: Don Cohen’s Infinite Series for Kids

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:

  • We have a cake to share, just the two of us. It’s not TOO big a cake, ‘cuz we don’t want to get sick. A 8 × 8 or 16 × 16 square on the graph paper should be just right. Can you cut the cake so we each get a fair share? Color in your part.

Bobby Flay German Chocolate Cake

  • How big is your piece compared to the whole, original cake?
  • But you know, I’m on a diet, and I just don’t think I can eat my whole piece. Half the cake is too much for me. Is it okay if I share my piece with you? How can we divide it evenly, so we each get a fair share? How big is your piece?
  • How much of the whole, original cake do you have now? How can you tell?
  • I keep thinking of my diet, and I really don’t want all my piece of cake. It looks good, but it’s still just a bit too big for me. Will you take half of it? How big is that piece?
  • Now how much of the whole, original cake do you have? How could we figure it out?
    [Teaching tip: Don’t make kids do the calculation on paper. In the early stages, they can visualize and count up the fourths or maybe the eighths. As the pieces get smaller, the easiest way to find the sum is what Cohen does in the video below‌—‌identify how much of the cake is left out.]
  • Even for being on a diet, I still don’t feel very hungry…

For more precalculus fun, explore Don Cohen’s Map of Calculus for Young People: hands-on activities featuring advanced ideas, for students of any age.

The Next Day

  • Your best friend comes over to visit, and we share a new cake. If you, me, and the friend all get a fair share, how much of the cake do you get?
  • But you know, I’m still on that diet. My piece of cake looks too big for me. I’ll share it with the two of you. Let’s cut my piece so each of us can have a share. How big are those pieces?
  • How much of the whole, original cake do you have now? …

Can Young Kids Really Understand This?

how tall is triangle
We did infinite cakes in Princess Kitten’s fifth-grade year, if I remember right. Three years later, I gave my middle-school math club kids this geometry puzzle from James Tanton:

  • Two circles are tangent to each other and to an isosceles triangle, as shown. The large circle has a radius of 2, and the smaller circle’s radius is 1. How tall is the triangle?

I really didn’t expect my then-8th-grade-prealgebra daughter to solve this. But I thought it might launch an interesting discussion along the lines of “What do you notice? What do you wonder?

She stared at the diagram for a minute or two, while I bit my tongue to keep from breaking her concentration.

Then she said, “Oh, I see! It’s an infinite cake.”

It took me much longer to understand what she had seen so quickly: Imagine stacking up smaller and smaller circles in the top part of the triangle. Because all the proportions stay the same, each circle is exactly half as wide as the one below it. To find the height of the triangle, we can just add up all the diameters of the circles.

The puzzle is adapted from an AMC 10/12 Practice Quiz and is available here, with Tanton’s problem-solving tips for high school students. Tanton used similar triangles to find the height, but Princess Kitten’s infinite series approach is quicker and doesn’t require algebra.

Infinite Cake


Cake photos by Kimberly Vardeman via Flickr (CC BY 2.0): Strawberry Cream Cake and Bobby Flay German Chocolate Cake.

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Reblog: Patty Paper Trisection

[Feature photo above by Michael Cory via Flickr (CC BY 2.0).]

trisection2

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:


trisection

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 …

[Click here to go read Puzzle: Patty Paper Trisection.]



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More Than One Way To Find the Center of a Circle

[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:

Is it possible that AB is a chord but NOT a diameter? That is, could circle ABC have a center that is NOT point O?

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?

  • Pick one side of the debate, and try to find at least three different ways to prove your point.

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.


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Related posts on Let’s Play Math! blog:

Math Playtime With Blocks

Feature video by Stuart Jeckel via youtube.

DO Try This at Home

And ask questions!


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Logic: The Centauri Challenge

Another fun discovery from the #MTBoS Challenge: Brian Miller (@TheMillerMath) posted this interstellar puzzle on his blog today.

centaurichallenge
[Right-click image to download a pdf you can print for your students.]

More Logic Puzzles

If you liked the Centauri Challenge, you may also enjoy the following blog posts:


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Math That Is Fun: Infinite Primes

Oh, my! Ben Orlin over at Math with Bad Drawings just published my new favorite math proof ever:

I had a fight with Euclid on the nature of the primes.
It got a little heated – you know how the tension climbs.

It started out most civil, with a honeyed cup of tea;
we traded tales of scholars, like Descartes and Ptolemy.
But as the tea began to cool, our chatter did as well.
We’d had our fill of gossip. We sat silent for a spell.
That’s when Euclid turned to me, and said, “Hear this, my friend:
did you know the primes go on forever, with no end?” …

15-eu-must-be-clidding

Click here to read the whole post at Math with Bad Drawings.


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What Is a Proof?

I’ve been enjoying the Introduction to Mathematical Thinking course by Keith Devlin. For the first few weeks, we mostly talked about language, especially the language of logical thinking. This week, we started working on proofs.

For a bit of fun, the professor emailed a link to this video. My daughter Kitten enjoyed it, and I hope you do, too.

Full lesson available at Ted-Ed.


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