I spend a lot of time trying to keep math interest high by introducing concepts in a fun way, exploring old material from new angles, and bringing hands-on learning to our “Math Lab” whenever possible. This isn’t always easy when delving into more advanced math topics – baking with fractions is one thing but information theory, combinatorics, and algebra are quite another. However, I firmly believe this is the only way to teach talented math students at the elementary level without burning them out or turning them into grinds. Fortunately I love math and computer science so the hours I spend researching and planning activities is never a chore. It does involve a lot of experimentation and dabbling though. Not every topic is brought to completion in a neat, well thought out package. A lot of math enrichment is about throwing cool ideas out there and seeing what sticks. My hope is that even if something is not a hit, I’m planting a seed for future investigations. This week I’m giving you a peek behind the curtain to see how random our weekly math explorations really are sometimes. Fortunately most of them were a hit, but that’s not always the case!
If you don’t know what a Flexagon is, you are really missing out, and even if you’ve heard of them you may not be aware of all of these awesome mathematical properties. First discovered by British graduate student Arthur Stone in 1929, Flexagons are geometric models that have hidden faces when flexed in certain ways. They are super simple to make out of paper and a fun project for home or the classroom. I was pretty dismissive of the math behind them until I started doing some research and then was blown away by their awesomeness. The famous physicist Richard Feyman even invented visual state diagrams, which are especially complex for hexa-hexaflexagons and my student has had a great time trying to make his own.
Non-Transitive Dice Paradoxes
We’ve been using two new resources this week to study probability in a novel way. The Amazing Mathematical Amusement Arcade by Brian Bolt is a fantastic book that distills famous math puzzles into a style that is more visually appealing for kids. I picked this up at the library last week, and just coincidentally a puzzle in this book called the Gambler’s Secret Strategy completely mirrored a more technical chapter I was reading on non-transitive dice paradoxes in Martin Gardner’s Colossal Book of Mathematics. The crux of the puzzle involves dice that have equal total values, but different numbers on each face, such that they allow a gambler to always have a better chance of winning in a head-to-head roll even when letting his opponent choose first. We created paper dice and built conditional probability trees to prove that the gambler always has the advantage. Definitely a topic we will be revisiting.
I put this sequence on the board one day and told my student to ponder it.
1, 5, 12, 22, 35, 51, 70, 92, 117…
The first day he told me that he didn’t know what the pattern meant but that he had figured what the next number in the sequence was (145). He then created a formula for figuring out the next number in the sequence given the two previous. Then, since he had trouble visualizing what these numbers meant, we drew pictures to illustrate why they are called pentagonal numbers. Now we’re trying to figure out what the connection is between this series and the series of triangular numbers. After that we will explore the set of integers that cannot be expressed by a sum of three pentagonal numbers. This one has been a huge hit. I love these long term math explorations!
I have a series planned on recreational math books for kids. Our latest acquisition in this category is a British series called Murderous Maths. Surprisingly dense and a whole lot of fun. Expect a detailed review soon.
Beast Academy 5c. The long awaited next book in this elementary math series from Art of Problem Solving. My student has moved beyond the math in this series but he still loves reading the comic book adventures of the little monsters as they solve math problems, and the workbook makes for great review.
The Thrilling Adventures of Lovelace and Babbage – a fun and fascinating romp about two of the founders of the computer age. Only slightly educational but a great graphic novel for any geek. I wasn’t expecting my son to get into, but he’s been dragging it around and re-reading like crazy.
We’ve also been taking our digital logic to the next level with a free circuit design tool called Logism. It’s very easy to use and allows for quick design and simulation of functional circuits. He had no problem completing simple projects of building a half-adder and full-adder, and is now working on a 7 segment display controller. So many possibilities here and an easy way to explore electronics without an expensive science kit.
That’s our math roundup for the week! We’ve been trying to enjoy our gorgeous fall weather here on the east coast by doing less school and spending more time outdoors, but we always have time to squeeze in lots of math so stay tuned.