Sunday, July 29, 2012

Draw a Picture, You Idiot!

I'd like to make my lesson on writing algebraic expressions more concrete for students. My hope is that students' understanding of simple expressions will lead to better interpretation of equation word problems. I was inspired by Steve Leinwand's book, Accessible Mathematics, where he argues that diagrams should be drawn as much as possible to help students conceptualize material. Here is an excerpt:

"Without question, one of the most common responses I have when sitting in the back of a mathematics class is screaming under my breath, 'Draw a picture!' or 'Use a number line!' or 'Ask them what it looks like!'" (page something or other...I don't know, it's on my Kindle app, how am I supposed to cite these things?)

I think he really meant to say, "Draw a picture, you idiot!" (hence, my post title). So I figured, why not apply it to this lesson:
What does everyone think about teaching this way? Will this help students gain a better understanding of variables and operations? Should I preface this with diagramming of numerical expressions first? What about expressions that don't lend themselves well to a diagram (such as 5 divided by n)? Any suggestions?

Nathan Kraft

Saturday, July 28, 2012

Gang Violence and Adding Integers

This is my borderline inappropriate way to teach integer addition to students. It was inspired by a presentation I saw by Dr Kadhir Rajagopal on solving equations (NCTM 2009, DC). Check out his website here.

Algebra tiles are a great way to teach integer addition. But I like to represent positives and negatives as members of two rival gangs.
The yellow gang member is basically the same as a yellow algebra tile (+1) and the red gang member is the same as a red algebra tile (-1). Yellow gang members get along with other yellow gang members. Same goes for the reds. But when a yellow and red meet up, bad things happen.
To a 13 year old child, this explanation is much more satisfying than some nonsense about "zero-pairs". And when one of my students is confused about an addition problem, all I have to say is "gang violence", and they know exactly what to do.

Say a student is presented with a problem like this: -4 + 7. I have to ask two questions: Which gang will win? (the positives) How many will be left? (three)

To me, this is much better than some silly rule that students have to memorize. They are visualizing the numbers. And they are seeing positives and negatives as opposites that cancel one another out. And best of all, they remember it.

Credits: Graphics are from the Smart Notebook software. I'm sure the people at Smart Technologies appreciate that I've found this use for them.

Nathan Kraft

Sunday, July 22, 2012

Measuring S'more Gooeyness

One day I was making S'mores in the microwave.

I noticed that the directions on the package of chocolate bars and package of marshmallows disagreed with each other. Chocolate: 10 to 15 seconds on medium. Marshmallows: 15 to 20 seconds on high. This was especially odd since both packages suggested using the other product to create the S'mores. (Honey Maid Graham Crackers remained neutral with no suggestions for how to make your S'more.)


I saw some potential for this in a math lesson. So I made S'mores with varying times in the microwave (10 seconds on high, 20 seconds on high, and 1 minute on high).

I took height measurements for each scenario. First 0 seconds...

Then 10 seconds...


Then 20 seconds...


Then 1 minute...


But none of these measurements seemed like a satisfying way to measure gooeyness, especially since 10 seconds and 20 seconds appeared to have the same height. This would have to be a qualitative experiment. So I ate them.


I suppose it comes down to personal preference. For me, 10 seconds (top) was perfect. 20 seconds (bottom left) was too messy, and 1 minute destroyed my tongue. Unfortunately, I don't see this as something I can use in math class. Any suggestions?

Nathan Kraft