Here's some of what we've been up to in math recently.

Card Tricks

I start with a deck of cards. I make 6 piles of cards, face up. Then I turn each pile over, so they're face down now. The kids choose 3 piles to keep and give me back the other cards. Then they turn over the top cards on two of the piles. I perform some magic, and tell them the top card on the remaining pile. Of course this trick has to do with math. Maybe I'll reveal the details in a future post.

Measuring Slope

I made a slanted line, and the kids measured my slope. See Renata's post for the story and pictures.

Russell's Paradox

We talked about Russell's Paradox, the short version of which is

"The barber shaves every man who does not shave himself.

Who shaves the barber?"

A more detailed version, with fewer loopholes, is presented here. The kids were great at finding loopholes in the version I gave ("Maybe the barber's a woman!", "The barber is bald!").

Set Theory and Infinity

We talked about sets and about union, intersection, and set difference. Start with two sets,

{1,2,3,4,5} and {2,4,6,8}.

Their union is the set of stuff that's in one or both of the sets:

{1,2,3,4,5,6,8}

Their intersection is the set of stuff that's in BOTH sets:

{2,4}

The set difference {1,2,3,4,5}-{2,4,6,8} is everything in the first set but not the second set:

{1,3,5}

Union, intersection, and set difference can be represented with Venn Diagrams (at least when we're only working with 2 or 3 sets).

We revisited the idea of defining the numbers as sets, only this time we went further.

0={}

1={0}

2={0,1}

10={0,1,...,9}

100={0,1,...,99}

Any whole number can be defined as a set in this way. We can also define infinity as the set of all whole numbers, called w (the Greek letter omega):

w={0,1,2,...,100,...1000,...}

We can add 1 to any whole number, and get "the next" whole number (or set). We can also subtract 1 from any whole number (except 0) and get "the previous" whole number (or set). "Infinity minus 1" doesn't make sense, though, because there is no "previous" set. There is no set that comes immediately before w.

Clocks, origami, and sets

The first week in October, we did clock arithmetic. On a clock, if you count up to twelve you get back to the beginning, so 12 is basically the same thing as zero. Weird things happen in clock arithmetic. For example, it's possible to multiply two non-zero numbers (2 and 6, for example) and get zero. This doesn't happen with regular integers and real numbers!

The last couple of weeks we've been doing "magic tricks" with a compass and straightedge. We cut a line segment in half without even knowing how long it was, and we cut an angle in half without knowing the measure of the angle. These magic tricks are done in high school geometry. Magic trick 3, however, usually isn't done in school at all: while it's provably impossible to cut an angle into thirds using only a compass and straightedge, it can be done with origami paper!

Today we did some set theory. We created all whole numbers using only the symbols {, }, and a comma. Here's how:

0={} is the empty set

1={{}}

2={{},{{}}}

3={{},{{}},{{},{{}}}}

and so on. Since it's hard to make sense out of all those brackets, we can write the numbers they represent instead. Then we get something prettier:

0={}

1={0}

2={0,1}

3={0,1,2}

and so on, and so on...

Areas, volumes, and dimensions

This has been the week of finding areas. Renata's been working with the kids on finding the areas of funny shapes by decomposing them into rectangles and triangles and squares. I decided to approach areas from the point of view of units and dimension. If a square has side length 1 ft, what's the area of the square in square inches? Most of the kids said 144 very quickly, which made me happy (having had lots of college students who would have said 12). Then we went to volume, finding volumes in cubic feet and cubic inches.

Today we had a little time left at the end for more fun stuff, so we drew 4-dimensional cubes (and, in the case of one of the kids, a 3-dimensional pair of pants!).

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.