Tilings and Symmetry

This week we've been making tilings, also known as tessellations. A tiling is what you get when you lay a bunch of shapes edge-to-edge to cover a flat space. The shapes aren't allowed to overlap, and you aren't allowed to leave gaps.

I gave the kids stencils for squares and equilateral triangles whose sides were 1 inch long, and told them to either trace them onto paper to make a tiling, or cut out tiles from construction paper and glue them down to make a tiling. They could use the squares and triangles to make different shapes, like bigger triangles, or "house-shaped" pieces.

We got some very nice tilings, and also some nice pictures made of tiles that weren't tilings because they had overlapping tiles or gaps.

Today we used the tilings to talk about reflectional and rotational symmetry. Reflectional symmetry is when you can draw a line (called the "line of symmetry") through the middle of a picture and the two sides of the picture look the same. Rotational symmetry is when you can rotate the picture and it looks the same after rotation as it did before (for example, take a square and turn it one-quarter turn). We drew some lines of symmetry for squares and triangles, then looked at the types of symmetry in our tilings.

We also looked at pictures from this website, which has pictures of ancient tilings including ones that are Egyptian, Persian, Arabian, and Moresque. When we look at symmetries, all those tilings can be clumped into 17 groups. We talked about some of those groups and the funny names they have.

Discussion: What is Math?

Today I said I wanted to talk about math.  That didn't go over particularly well - the kids responded with things like "math is boring," "I don't like assessments," and "no fractions!"

We digressed to talking about our favorite numbers, which went over a little better.  A couple of people liked the number "2" because it's a fun number to write.  "-12" was a favorite, as was "1".  The discussion got so animated that Renata had to get out a feather (for a talking stick) to keep order.

Then we went around the circle, with (almost) every kid giving an answer to the question "what is math?"  Some fun thoughts that came out of this:

• mathematicians don't even know that 1+1=2
  
• math is "random," and the only order there is the order we put on it
  
• math is about asking questions and answering them
  
• the word that comes to mind is "progress," because without math we couldn't have computers or cars
  
• do trees have anything to do with math?
• in ancient times, didn't people have to know "how far" they could throw a rock, so they could successfully hunt animals?

Along the way we talked about the complex numbers (mathematicians wanted a number that you could square to get -1, so they made one up and called it i) and about mathematical logic (using the notation x arrow y to abbreviate the statement "if it is raining, then there are clouds in the sky").

Introduction

Hello, world!

My name is Jessica Metcalf-Burton, but I go by Jesse (no "i").  I teach math to a bunch of kids at Summers-Knoll School.  This blog is intended for parents to follow along and see what their kids are doing.  Since I haven't met most of the parents yet, I figure I should introduce myself first.

I have a Ph.D. in mathematics from the University of Michigan, and undergraduate degrees in math and computer science from the University of Maryland. My first experience as a teaching assistant was my sophomore year of college, and I've pretty much been teaching ever since.  At University of Michigan I've taught a simply ridiculous number of calculus courses, and over the last several years I've tutored everything from high school algebra and geometry through calculus.  

When I'm not teaching math, I have a side business teaching swing dancing.  I learned how to swing dance my first year of college, and it stuck.  I also like baking, reading, traveling, traveling to Spain, attempting to speak Spanish, writing poetry, listening to music, playing music (on piano, djembe, recorder, bones, bodhran, or zills), making beaded jewelry, crocheting, trying other forms of dance besides swing, doing photography (preferably while traveling), and eating ice cream.  

As the semester gets a little more underway, I'll be introducing some of the kids at Summers-Knoll to advanced math ideas that students usually don't get to until high school (if they're lucky) or college.  I have a pretty long list of ideas already, and if anyone has fun math activities or topics they'd like to suggest, I welcome comments.