Tova Brown’s introduction to Hilbert’s Hotel Paradox, a riddle about the nature of infinity…
Once upon a time, there was a hotel with an infinite number of rooms. You might be thinking this is impossible, and if so you’re right. A hotel like this could never exist in the real world.
But fortunately we’re not talking about the real world, we’re talking about math. And when we do math we can make up whatever rules we want, just to see what happens.
[Feature photo above from the public domain, and title background (below) by frankieleon (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. Please join us!
For many years mathematicians, scientists, engineers and others interested in math have played “year games” via e-mail. We don’t always know whether it’s possible to write all the numbers from 1 to 100 using only the digits in the current year, but it’s fun to see how many you can find.
Use the digits in the year 2016 to write mathematical expressions for the counting numbers 1 through 100. The goal is adjustable: Young children can start with looking for 1-10, middle grades with 1-25.
You know you’re a math teacher when you see a penny in the parking lot, and your first thought is, “Cool! A free math manipulative.”
My homeschool co-op math students love doing math with pennies. They’re rather heavy to carry to class, but worth it for the student buy-in.
This month, I’m finishing up the nearly 150 new illustrations for the upcoming paperback edition of my Let’s Play Math book. I’m no artist, and it’s been a long slog. But a couple of the graphics involved pennies—so when I saw that penny on the ground, it made me think of my book.
And thinking of my book made me think it would be fun to share a sneak peek at coming attractions…
Real mathematics is intriguing and full of wonder, an exploration of patterns and mysterious connections. It rewards us with the joy of the “Aha!” feeling. Workbook math, on the other hand, is several pages of long division by hand followed by a rousing chorus of the fraction song: “Ours is not to reason why, just invert and multiply.”
Real math is the surprising fact that the odd numbers add up to perfect squares (1, 1 + 3, 1 + 3 + 5, etc.) and the satisfaction of seeing why it must be so.
Did your algebra teacher ever explain to you that a square number is literally a number that can be arranged to make a square? Try it for yourself:
Each new set of pennies must add an extra row and column to the current square, plus a corner penny where the new row and column meet. The row and column match exactly, making an even number, and then the extra penny at the corner makes it odd.
Can you see that the “next odd number” pattern will continue as long as there are pennies to add, and that it could keep going forever in your imagination?
The point of the penny square is not to memorize the square numbers or to get any particular “right answer,” but to see numbers in a new way—to understand that numbers are related to each other and that we can show such relationships with diagrams or physical models. The more relationships like this our children explore, the more they see numbers as familiar friends.
A large jar of assorted coins makes a wonderful math toy. Children love to play with, count, and sort coins.
Add a dollar bill to the jar, so you can play the Dollar Game: Take turns throwing a pair of dice, gathering that many pennies and trading up to bigger coins. Five pennies trade for a nickel, two nickels for a dime, etc. Whoever is the first to claim the dollar wins the game.
Or take the Penny Birthday Challenge to learn about exponential growth: Print out a calendar for your child’s birthday month. Put one penny on the first day of the month, two pennies on the second day, four pennies on the third day, etc. If you continued doubling the pennies each day until you reach your child’s birthday, how much money would you need?
Warning: Beware the Penny Birthday Challenge! Those pennies will add up to dollars much faster than most people expect. Do not promise to give the money to your child unless the birthday comes near the beginning of the month.
The first time I did pennies on a calendar with my homeschool co-op class was during December, so we called it the Penny Christmas Challenge:
I told the kids that if their grandparents asked what gift they wanted for Christmas, they could say, “Not much. Just a few pennies…”
The Penny Square, Dollar Game, and Penny Birthday Challenge are just three of the myriad math tips and activity ideas in the paperback edition of Let’s Play Math: How Families Can Learn Math Together and Enjoy It. Coming in early 2016 to your favorite online bookstore…
Here’s a new video from Jo Boaler at YouCubed.org.
There is a huge elephant standing in most math classrooms, it is the idea that only some students can do well in math. Students believe it, parents believe and teachers believe it. The myth that math is a gift that some students have and some do not, is one of the most damaging ideas that pervades education in the US and that stands in the way of students’ math achievement.
—Jo Boaler
Unlocking Children’s Math Potential
The YouCubed site is full of encouragement and help for families learning math.
— and plenty more!
Interrupt your regular math programming to try this fantastic math doodling investigation!
Anna Weltman wrote a math/art book. It’s great fun for all ages, full of fantastic mathematical explorations — including spirolateral math doodles.
To make a spirolateral, you first pick a short series of numbers (1, 2, 3 is a traditional first set) and an angle (90° for beginners). On graph paper, draw a straight line the length of your first number. Turn through your chosen angle, and draw the next line. Repeat turning and drawing lines, and when you get to the end of your number series, start again at the first number.
Some spirolaterals come back around to the beginning, making a closed loop. Others never close, spiraling out into infinity—or at least, to the edge of your graph paper.
Martin Gardner, “Worm Paths” in Knotted Doughnuts and Other Mathematical Entertainments.Articles by Robert J. Krawczyk:
Anna Weltman appeared on Let’s Play Math blog once before, with the game Snugglenumber. And she’s a regular contributor to the wonderful Math Munch blog.
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.
I’m still having fun with David Coffey’s meme, which started a couple of years ago with this blog post:
Would you like to create a math holiday, too? Try one of these sign generators:
What kind of math will you celebrate? Leave a link to your Happy Math Day post in the comments below!
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
The Internet boasts a wide-ranging assortment of math websites, and for years I maintained (or mostly neglected) a huge page of reference links. This spring I’ve been working on the paperback edition of my book—with its appendix of favorite books and internet sites—and I decided to revise my blog links to match.
So this week, I’m in Jeju, South Korea, visiting my daughter who teaches English there. In between seeing touristy sites and gorging ourselves on amazingly delicious food, she took me to a beautiful coffee shop that overlooks the beach in Aewol.
Great place to work on my blog!
The long monster list morphed into eight topical pages. I hope you find something useful.
I will try to keep these pages up to date, but the Internet is volatile. If you find a broken link, you can search for the website by name or enter the defunct URL into the Internet Wayback Machine at Archive.org.
And if you know of a fantastic website I’ve missed, please send me an email (LetsPlayMath@gmail.com, or use the comment form on my “About” page). I appreciate your help.
Feature photo above by Fractal Ken via Flickr (CC BY 2.0). Korea photos ©2015 Denise Gaskins, all rights reserved. For more math resource suggestions, check out my Math with Living Books pages. They’re not finished yet, but I’ll be working on them next.
In the land of Fantasia, where people communicate by crystal ball, Wizard Mathys has been placed in charge of keeping the crystal connections clean and clear. He decides to figure out how many different ways people might talk to each other, assuming there’s no such thing as a crystal conference call.
Mathys sketches a diagram of four Fantasian friends and their crystal balls. At the top, you can see all the possible connections, but no one is talking to anyone else because it’s naptime. Fantasians take their siesta very seriously. That’s one possible state of the 4-crystal system.
On the second line of the diagram, Joe (in the middle) wakes up from siesta and calls each of his friends in turn. Then the friends take turns calling each other, bringing the total number of possible connection-states up to seven.
Finally, Wizard Mathys imagines what would happen if one friend calls Joe at the same time as the other two are talking to each other. That’s the last line of the diagram: three more possible states. Therefore, the total number of conceivable communication configurations for a 4-crystal system is 10.
For some reason Mathys can’t figure out, mathematicians call the numbers that describe the connection pattern states in his crystal ball communication system Telephone numbers.
Hint: Don’t forget to count the state of the system when no one is on the phone crystal ball.
Feature photo at top of post by Christian Schnettelker (web designer) and wizard photo by Sean McGrath via Flickr (CC BY 2.0). This puzzle was originally featured in the Math Teachers At Play (MTaP) math education blog carnival: MTaP #76.
Denise Gaskins' Let's Play Math 
