Math Storytelling Day: The Hospital Floor

Posted on by Denise Gaskins

[Feature photo above by Christiaan Triebert via flickr (CC BY 2.0).]

Have you ever heard of Math Storytelling Day? On September 25, people around the world celebrate mathematics by telling stories together. The stories can be real — like my story below — or fictional like the tale of Wizard Mathys from Fantasia and his crystal ball communication system.

Check out these posts for more information:

My Math Story

tiles2

My story begins with an unexpected adventure in pain. Appendicitis sidewhacked my life last week, but that’s not the story. It’s just the setting. During my recovery, I spent a lot of time in the smaller room of my hospital suite. I noticed this semi-random pattern in the floor tile, which made me wonder:

  • Did they choose the pattern to keep their customers from getting bored while they were … occupied?
  • Is the randomness real? Or can I find a line of symmetry or a set of tiles that repeat?
  • If I take pictures from enough different angles, could I transfer the whole floor to graph paper for further study?
  • And if the nurse finds me doing this, will she send me to a different ward of the hospital? Do hospitals have psychiatric wards, or is that only in the movies?
  • What is the biggest chunk of squares I could “break out” from this pattern that would create the illusion of a regular, repeating tessellation?

I gave up on the graph paper idea (for now) and printed the pictures to play with. By my definition, “broken” pattern chunks need to be contiguous along the sides of the tiles, like pentominoes. Also, the edge of the chunk must be a clean break along the mortar lines. The piece can zigzag all over the place, but it isn’t allowed to come back and touch itself anywhere, even at a corner. No holes allowed.

I’m counting the plain squares as the unit and each of the smaller rectangles as a half square. So far, the biggest chunk of repeating tiles I’ve managed to break out is 283 squares.

tiles1

What Math Stories Will You Tell?

Have you and your children created any mathematical stories this year? I’d love to hear them! Please share your links in the comments section below.

 
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Which I am going to say right now. Thank you!

“Math Storytelling Day: The Hospital Floor” copyright © 2014 by Denise Gaskins.

Fractions: 1/5 = 1/10 = 1/80 = 1?

Posted on by Denise Gaskins

[Feature photo is a screen shot from the video “the sausages sharing episode,” see below.]

Fractions: 1/5 = 1/10 = 1/80 = 1?

How in the world can 1/5 be the same as 1/10? Or 1/80 be the same as one whole thing? Such nonsense!

No, not nonsense. This is real-world common sense from a couple of boys faced with a problem just inside the edge of their ability — a problem that stretches them, but that they successfully solve, with a bit of gentle help on vocabulary.

Here’s the problem:

  • How can you divide eight sausages evenly among five people?

Think for a moment about how you (or your child) might solve this puzzle, and then watch the video below.

What Do You Notice?

Continue reading Fractions: 1/5 = 1/10 = 1/80 = 1?

Reblog: Patty Paper Trisection

Posted on by Denise Gaskins

[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.]

More Than One Way To Find the Center of a Circle

Posted on by Denise Gaskins

[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.

The Linear Inequality Adventures of Ohio Jones

Posted on by Denise Gaskins

Ohio Jones 1

Last week, Kitten and I reached her textbook’s chapter on graphing linear equations, and a minor mistake with negative numbers threw her into an “I can’t do it!” funk. It’s not easy teaching a perfectionist kid.

Usually her mood improves if we switch to a slightly more advanced topic, and luckily I had saved these worksheets on my desktop, waiting for just such an opportunity. Today’s lesson:

  • Some fun(ish) worksheets
    “For tomorrow, students will be graphing systems of inequalities, so I decided to create a little Ohio Jones adventure (Indiana’s lesser known brother)…”

I offered to give her a hint, but she wanted to try it totally on her own. It took her about 40 minutes to work through the first few rooms of the Lost Templo de los Dulces and explain her solutions to me. I’m sure she’ll speed up with experience.

So far, she’s enjoying it much more than the textbook lesson. It’s fascinating to me how the mere hint of fantasy adventure can change graphing equations from boring to cool. Thanks, Dan!

2014 Mathematics Game

Posted on by Denise Gaskins

2014-Calendar

[Feature photo above by Artis Rams (CC BY 2.0) via flickr. Title background (right) by Dan Moyle (CC BY 2.0) via flickr]

Have you made a New Year’s resolution to spend more time with your family this year, and to get more exercise? Problem-solvers of all ages can pump up their (mental) muscles with the Annual Mathematics Year Game Extravaganza!

For many years mathematicians, scientists, engineers and others interested in mathematics have played “year games” via e-mail and in newsgroups. We don’t always know whether it is possible to write expressions for all the numbers from 1 to 100 using only the digits in the current year, but it is fun to try to see how many you can find.

— Math Forum Year Game Site

Rules of the Game

Use the digits in the year 2014 to write mathematical expressions for the counting numbers 1 through 100. The goal is adjustable by age: Young children can start with looking for 1-10, or 1-25.

  • You must use all four digits. You may not use any other numbers.
  • Solutions that keep the year digits in 2-0-1-4 order are preferred, but not required.
  • You may use +, -, x, ÷, sqrt (square root), ^ (raise to a power), ! (factorial), and parentheses, brackets, or other grouping symbols.
  • You may use a decimal point to create numbers such as .2, .02, etc., but you cannot write 0.02 because we only have one zero in this year’s number.
  • You may use the overhead-bar (vinculum), dots, or brackets to mark a repeating decimal.
  • You may create multi-digit numbers such as 10 or 201 or .01, but we prefer solutions that avoid them.
  • You may use a double factorial, but we prefer solutions that avoid them. n!! = the product of all integers from 1 to n that have the same parity (odd or even) as n.

[Note to students and teachers: If you want to take part in the Math Forum Year Game, be warned that they do not allow repeating decimals.]

Click here to continue reading.

Algebra for (Almost) Any Age

Posted on by Denise Gaskins

VisualPatterns-org2

Fawn Nguyen’s Visual Patterns website just keeps getting better and better. Check it out:

In addition to the 115 puzzle patterns (as of this writing), the site features a Gallery page of patterns submitted by students. And under the “Teachers” tab, Fawn shares a form to guide students in thinking their way through to the algebraic formula for a pattern.

How can you use these patterns to develop algebraic thinking with younger students? Mike Lawler and sons demonstrate Pattern #1 in the YouTube video below.

Things To Do with a Hundred Chart #30

Posted on by Denise Gaskins

100chartpuzzle

Here’s one more entry for my 20+ Things to Do with a Hundred Chart post, thanks to David Radcliffe in the comments on Monday’s post:

(30) Can you mark ten squares Sudoku-style, so that no two squares share the same row or column? Add up the numbers to get your score. Then try to find a different set of ten Sudoku-style squares. What do you notice? What do you wonder?
[Suggested by David Radcliffe.]

Share Your Ideas

Can you think of anything else we might do with a hundred chart? Add your ideas in the Comments section below, and I’ll add the best ones to our master list.

Puzzle: Algebra on Rectangles

Posted on by Denise Gaskins

Gordon Hamilton of Math Pickle recently posted these videos on how to make algebra 1 puzzles on rectangles. As I was watching, Kitten came in and looked over my shoulder. She said, “Those look like fun!”

They look like fun to me, too, and I bet your beginning algebra students will enjoy them:

Continue reading Puzzle: Algebra on Rectangles