Last Math Lessons
On the first day of school, I promised all my students that I'd do my best to make math relevant and challenging. I also promised to never waste their time, therefore they could expect to do math every day in my class, including the last day of class.
I did not promise them that math would be fun because doing math is always fun to me. I am not ashamed to share here that I gave out math puzzles at my kids' birthday parties when they were younger (I actually shared this bit on Twitter), also at a baby shower that I gave, and... get ready for this... at my own wedding. Straight up.
(I know I wasted my kids' time when I gave them three benchmark tests during the year and spent days reviewing for the state test.)
I called this lesson How Many Regions? (Adapted from AIMS)
I gave each student a piece of 4-by-4-inch paper and a handful of 6-inch long thin strips of construction paper. Each strip placed on the square paper would represent a "cut."
Question: What is the maximum number of regions that you can divide the square into using n number of "cuts"?
We did the first two cuts together as a class, and each student kept track of the data in their journal.
Because the goal was to get the maximum number of regions from the cuts, the kids learned quickly that the cuts needed to intersect. For example, two non-intersecting cuts created only 3 regions. A kid yelled out, "Parallel cuts!" Gotta forgive kids who blurt out academic language!
This class has been working with a pattern worksheet almost every Monday in my class, so they know to look for a recursive rule and try to find the equation for the nth term. Recursively, they saw the pattern, so it didn't take long for them to figure out that for any n (cuts), the maximum number of regions is the sum of n and the number of regions from the previous n.
With 30 seconds left of class, I told them my gift to them was to try and figure out the equation. Matt asked, "Can I email you then over the summer?"
Our 8th graders' last day of classes was yesterday (Wednesday). They'll show up this evening for their promotional ceremony, dressed to the hilt, the girls in their 7-inch heels.
I needed a quick one-period lesson, so I had them make 2012 Clocks. This is a common assignment: using all 4 digits in "2012" and each digit only once, they had to create expressions that would equal to the hours of 1 through 12 on a clock.
They were engaged and busy because if they didn't use class time wisely, then they'd have to sacrifice their hair-and-make-up time today to get it in to the mean teacher who didn't-even-give-us-the-last-class-to-sign-yearbooks-and-hug-each-other.
I've seen different versions of this problem; the first time I worked on it was when it involved 3,000 bananas and 1 camel traveling 1,000 miles. I know there's a perfectly good strategy called "solve a simpler problem," but we could also start with a simpler one!
I gave my kids Desert Crossing, also from AIMS:
You live in a desert oasis and grow miniature watermelons that are worth a great deal of money, if you can get them to the market 15 kilometers away across the desert. Your harvest this year is 45 melons, but you have no way to get them to the market, except to carry them across the desert. You have a backpack that holds up to 15 melons, the maximum number that you can carry at a time. To walk across the desert, you need a certain amount of fluid and nourishment that is supplied by the melons you carry. For each kilometer you walk (in either direction), one melon must be eaten.
Your challenge is to find a way to get as many melons as possible to market.
(From Desert Crossing, Just for the Fun of It!, AIMS Education Foundation)
As I type this, Slater does not know that I will be awarding him the Math Excellence Award at the Promotional Ceremony this evening.
Here is Slater's work that shows one way to get the correct answer of 8 watermelons.
I did this a few weeks ago and forgot all about it; so here's a little blurb on it.
Each student quickly constructed these two prisms from two same size papers. (Dan Meyer folded them into cylinders.)
Question: Pretend the two prisms have bottoms on them, which one holds more popcorn? Take a look... Okay, grab the one you think holds more, or grab both if you think they're both equal.
And this was how they grabbed:
Then they measured and calculated the volume.
My favorite student comment: The tallness didn't make up for the fatness.
Estimating volume is a funny thing.