Teaching High School Mathematics
Winter break is over and it’s always pretty funny hearing what kids say. My favorite of the first day back: “What are we doing today? All my other teachers are acting like we didn’t have a break.”
I put students into visibly random groups and gave them this problem:
The post office will allow packages to have a combined girth plus length of 108 inches. What are the dimensions of the package with the largest volume?
I told it as a story where (as a farmer) I needed to send my goods in the mail. I changed the item each period. Their goal was to help me get the dimensions. Students quickly got on board with the task. Like in my previous farming adventures, I was again forgetful and kept changing the numbers on them. The 108 became 100 or 120 as either the post office changed it or I remembered it incorrectly and now had the right number. This time more groups jumped on board to use a variable for the 108 and solved the problem in general. I was able to have a group come up at the end of both classes to show their solutions.
It went great! The transition back into learning was smooth and they remembered what we did right before break two weeks before.
On the second day back I gave them the classic pipeline offshore to a town problem except, of course, I had to change it to reflect farming.
Here is the problem:
A water pipe needs to be laid from my farmhouse to an established well. Pipe laid along the road costs $2 per foot to install and pipe laid not along the road costs $5 per foot to install. The well is 6 miles away from my farmhouse along a straight road, and 2 miles off the road.
I also drew them a diagram showing the house and the well and the two distances 6 and 2.
Wow! We hit some serious road blocks. Don’t get me wrong, there was some good math done, but this problem was obviously harder to set up than the previous ones. I’m not even sure if any group got a final correct answer by the end of the period. Most groups were on the right track. I sent them home with the goal of making some progress to share with their group the next day. It was interesting because I didn’t consider this problem any harder than the previous one.
I made an adjustment after first period. I had students figure the cost to lay a straight line(off road), then the cost along the road and straight up, and then the cost at one random spot, say 5 miles in before taking a straight path off road to the well. I had them all compute these before I split them into groups. It definitely helped all students get progress and feel connected as groups started working on the problem. The also might have all engaged more since they knew that what they were doing would contribute to their groups.
I sent them home with the assignment of finding the ideal location to take the pipe off road and minimize cost.
After class I sat and worked out the problem with variables for the different costs and distances to in essence solve every problem of this type. It was pretty cool and it got me excited at the prospect of any of my students coming up with a general solution.
It’s quite rewarding when my students are solving these problems together without an example to follow!