# To Accumulate a Rate — Integrate!

Teaching High School Mathematics

# Farmer Allinson!

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 remember 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.

Here is how I group students: I take a deck of cards and select 3 Kings, 3 Queens, 3 etc.. for as many sets needed for the class. Then I shuffle the deck in front of the students and randomly hand out the cards. Because, it is random, and they know that I will not switch them even if they are randomly with a friend, they are way more likely to work well and with someone they don’t know or would rather not work with. I call out their cards as I hand them out and tell them they cannot switch cards.

This group work 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 after teaching a lesson and see if they could figure it out for themselves in their group. I figured that if it didn’t work we could regroup after winter break.

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? (See diagram above.)

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 (not Calculus) 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 many examples they showed me that were less than the one they got… 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 they chose. 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 maximum of the function. The immediately started 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 fencing(say 120ft or 200ft). 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. I honestly was pretty excited.

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 for any amount of fencing.

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:

New day, new random groups.

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. One group in first period did start with trial and error again 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 and then told them there were more than 2 equal enclosures (3, 5, 8…). 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 to make the sections have equal area.

Afterwards:

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.

Farmer Allinson (Part 2)