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!
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 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.
I made an adjustment after first period. I had students figure the cost to lay a straight line, then along the road and straight up, and then at one random spot, say 5 miles in before angling up to the well, before I split them into groups. It definitely helped all students get progress and feel connected as groups started working on the problem.
After class I 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 to see if any of my students will come up with a general solution.
My solution was x = (A*W) / Sqrt(B^2 – A^2).
It’s cool seeing my students doing all the fun math!
In my calculus class we had just finished a test on a Tuesday before Christmas break. 3 days left and the next topic was optimization. Hmmm… I’m not into showing movies, or having parties or wasting the days… I wondered if they would remembered anything we did when we started back up after the break. I recently have been intrigued by the idea of a “thinking classroom,” Visibly Random Grouping, and Vertical Non-Permanent Surfaces. I’ve been experimenting with random groupings in all my classes because that is the easiest of the three parts to manage. It has gone well and I was ready to try a great problem for students to work on.
My goal was to start with a standard problem I would expect them to solve and see if they could figure it out for themselves.
I chose this problem: What dimensions of fencing would create the largest area rectangle next to a barn if you have 100 ft of fencing?
Day 1:
I drew the picture above without the x and y. I told the problem as a story… Farmer Allinson needed to create the largest enclosure possible with his 100 ft of fencing.
Students in all groups proceeded to do trial and error and fairly quickly found the ideal dimensions. As groups finished I would ask them to convince me that their answer had to be the max area. I had to tell some groups that no matter how close they got to a number on either side, how did they know for sure that there wasn’t a better answer somewhere else.
Then, one by one the groups started coming up with equations that modeled the situation. 100 = 2x + y and A = x * y. They also used substitution and got an area equation with only either x or y. Some groups used l and w. One group used some random variables. I hadn’t done anything to prep them to do this and I was flabbergasted. I thought that I would have to strongly guide them towards using equations. Some of the groups then realized they needed to find the max of the function using the first or second derivative test that we had just finished testing on the day before. For some groups I had to ask them what their function represented(Area) and how they would find the largest area. With these “hints” all groups were able to move along.
Once groups got the correct solution using calculus, I would tell them I made a mistake in the problem and give them a different amount of wire. I just made up a different number for each group. Two of the groups decided to put F into the equation instead of the 100 ft or fencing amount and came up with a formula for the answer regardless of the amount of fencing I gave them.
At the end of day one I chose one of those groups to come up to the board and present their solution. A couple of the groups were in awe as they explained their generalized solution.
As my second class came in I figured that it was a fluke that first period figured it all out without much help… I was wrong. Second period also rose to the challenge.
Day 2:
On day two I put the same drawing on the board. I told them that Farmer Allinson just realized that he needed to keep the boy sheep away from the girl sheep and I added a line straight down from the barn. A fence in the middle so that there were two equal enclosures. It was crazy… I could hardly finish talking about the problem and they were scribbling equations on their papers. One group in first period did start with trial and error on the second day.
As groups got the correct answers I would say that I made a mistake and give them a different amount of fencing or more than 2 equal enclosures. Most groups just kept figuring out the new problem. One group used F for the amount of fence to generalize, and one group used F for fencing and s for number of interior fences. Both classes had one group that did this and generalized the entire problem. In each class, they came up at the end of the period and explained. One student in particular was fabulous in their explanations tying each part of their solution to the specific solution to the original problem. The picture below shows the second half of their work.
Day 3:
This was a 25 min. period. I gave students the same drawing as the day before with a fence down the middle and told them that Farmer Allinson needed two 500 ft^2 enclosure and wanted to use the least amount of fencing possible. I asked them to think about what I might ask them and take the problem as far as they could. They worked on the problem, but since it was a short day on Dec. 21st. I didn’t get the closure I wanted on the day. Some groups had full generalized solutions and some found just the first answer.
One group bailed on the problem and stared working on the previous days problem using a semi-circle and two partitions down the middle. They spent their time trying to figure out where the partitions would go.
Conclusion:
One comment from a student was that the student explaining their math so well was going to take my job. I loved it!
I felt like such a great teacher. My students were doing all the interesting math that I normally get to do.
I ran into a student after the girls basketball game that night and they started asking me about partitioning a circle with vertical lines and showed me a picture of their computations on their phone.
I wrote emails home to 3 students bragging on their skills to their parents.
I’m so excited to find more problems that will engage the class this well and draw out their creativity and mathematical ideas.
If I’m honest, one of the most difficult parts of teaching is having patience when students struggle on an easy concept. After 3 attempts to impart understanding using different perspectives on the problem I sometimes wonder if they will ever understand. Part of that comes from teaching for so many years that every topic has become easy.
When this happens… It helps me best when I can understand that struggle. So… I pick up a challenging math book for me and start to work through it. There is always something that I don’t quite understand. Right now I’m reading both a Number Theory and Graph theory book. I’m stumped on a proof right now and although it sounds crazy, it feels good to be stuck. I enjoy personally being reminded of the struggle involved in a math problem. Additionally, if I solve it, I know I did something hard and learned something. I also enjoy solving puzzles which can have the same effect.