Optimizing the Learning....

In my view, optimization is not always the most exciting or engaging topics in the Grade 9 math curriculum.  I do try to make it interesting, but I always feel something seems to be missing. We investigate, and after much discussion about different rectangle dimensions and reminders that a square is in fact a rectangle, students finally conclude that the "best shape" is a square. And then it happens - I give them a question on optimizing perimeter and they have no idea what to do.  UGH! I have failed to reach them again....

Fast forward to our most recent activity using Ozobots.  Students were asked to draw a rectangle that has an area of 60 cm2  and then time how long it takes for the Ozobot to travel around the rectangle.  The rectangles that groups started with varied where some had 6 cm x 10 cm, others 5x12, one even 20x3.

Then we asked them to draw another rectangle, also with an area of 60 cm2 , but now it should take less time for the Ozobot to travel around it.  It was interesting that some groups didn't take the time to think through what property of the rectangle determined how long it took for the Ozobot to travel around it, they just picked another rectangle that had an area of 60 cm2  and drew it. It wasn't long before they realized that it was taking longer for the Ozobot to travel one length of the rectangle than it did the entire perimeter of the rectangle. One discussion I heard:

Student 1: "Well why would that be?"

Student 2: "I don't know, do you think it has something to do with the length of the sides?"

Student 1: "Well I guess so, but not sure.  Well... wouldn't the side lengths have to be smaller for it to go around faster?"

Student 2: "Makes sense to me."

These students tried a rectangle with a smaller perimeter and realized that yes, the Ozobot took less time to travel around it.

Then it was draw a rectangle that would take the Ozobot the least amount of time to travel around. Students got stuck on the dimensions of 6 x 10 cm. They said that 32 cm was the smallest perimeter.  Discussion:

Me: "Are you sure?"  

Student(s): "Yup"

Me: "What makes you say that?"

Student(s): "Because there are no other factors of 60."

Me: "Are you sure?"

Student(s): "We did a factor tree"

Me:  "Really, there are no other factors of 60, just the ones you have listed?"

Student(s): "Yup!" 
.......
This exchange goes back and forth a few more times
........
Student(s): Well are we able to use decimals...?"

Me: "What is stopping you from using decimals?"

Student(s): "Really? We can use decimals?"  

Me:  "Yup!"

Off they went calculating again. They started to realize that the perimeter would get smaller as the dimensions became closer to the same value. Then again, many groups stopped at 8 x 7.5 cm.

Me: "Is this the smallest perimeter?"  

Student(s): "Yes"

Me: "What makes you say that?"

Student(s): "If you go any closer together it becomes a square"

Me: "What is wrong with a square?"

Student(s): "A square isn't a rectangle"

Me: "What makes you say that?"

This lead to another interesting discussion on the properties of rectangles and squares and remembering that squares are rectangles. Finally, they tested the square and it turned out to be the fastest time.

Next we asked them to do the following:

What are the dimensions of a rectangle with area of 40 cm2 which will take the least amount of time for the Ozobot to travel?

1. Make a prediction
2. Test the prediction
3. Reflect on your prediction

The groups immediately predicted a square.  What was interesting is that they drew the square, timed it and then stopped.  Students didn't think of what "test" meant.  They didn't realize they needed to check other rectangle dimensions to prove or disprove their prediction.  We had a discussion that although they found that the square had the minimum perimeter for an area of 60 cm2, was that the case for all rectangles?  We have one piece of evidence.  Can we make a conclusion based on one piece of evidence?  For example a square with a side lengths of 4 cm, the area and perimeter both have a value of 16.  Is this the case for all squares?  

In the case of our investigation, how could they test their prediction that a square will provide the fastest time?  We talked about testing other dimensions to confirm the prediction.  

Then we did the same for an area of 80 cm2.  Again, a lot of them predicted a square, timed the square and then stopped.  The idea of testing didn't include looking at other rectangles to prove their prediction was true, it was just time how long it took the Ozobot to get around the square. We had the discussion again about testing predictions.  Fascinating....

When we consolidated the activity students did understand that the square would give the fastest time.  Then, for the first time ever, when I asked a question about minimizing perimeter students knew how answer the question.

SUCCESS!!! 

Reflecting on the lesson as a whole:

1. What made this lesson better than other optimization lessons?  Is it the technology?  Is it the design of the lesson? Is it both?
2. Why do students get stuck on only working in integer values?  
3. Why do students think that squares aren't rectangles?
4. Why do students not understand what it means to test a prediction?

Reflecting on the first and last points:

The technology:  this was the first time I felt students had an understanding of why minimizing the perimeter was important -  what dimensions allowed for the Ozobot to travel around the rectangle in the least amount of time.  They could see the relationship in the drawings as well as in the recorded times.  Most importantly students were engaged.  

After this lesson, Dianne and I were trying to figure out how to use the Ozobots for maximizing area, but we realized it wouldn't work.  This was one of those situations of trying to fit the technology to the lesson instead of designing the lesson to use technology.  Instead of forcing the lesson Debbie and I went back to using one of our previous investigations for maximizing area.  

In this next investigation we were maximizing the area of a skating rink.  We used Desmos on the Chromebooks for the investigation to record our data and make conclusions.  Today in our TLLP meeting I had a moment where I realized I was getting stuck on what the word "technology" meant.  Really,  Desmos is using technology - of course it is, it just wasn't the "new" technology we were trying to incorporate with our TLLP.  The most important part is having the technology readily available. 

Lesson design: predict, test, reflect.  Predict, test, reflect.  I think this is one of the few times we have really incorporated predict, test, reflect into our lessons.  Usually, we do an investigation, consolidate the investigation, make conclusions and then have students answer questions. This time we had students make conclusions based on an initial investigation, use those conclusions to predict the outcome of another similar situation and then test the prediction.  Then do this again. This lead me to thinking that we need to do this in other investigations.  It also made me think how often do we set up an investigation where we don't give students an opportunity to do predict, test and reflect?  How does this impact their future learning?

For the last 2 classes, we incorporated predict, test, reflect in optimizing three sided enclosures.  They used Desmos to organize their data and test their predictions.  As the students worked through the predict, test, reflect cycle, they were the ones that developed formulae for determining the dimensions based on their knowledge of the relationship between the length and the width for three sided enclosures.  Under these circumstances, the length is twice the width.  For example, to minimize perimeter for a given area, they knew that l=2w, therefore A=2w(w) and finally A=2w2 .  Then because they had an equation with one unknown and could solve for w.  In previous optimization investigations, we would develop this as a group during consolidation, this time, I didn't need to, each group had found it themselves.  Most importantly, I felt students had a much better understanding of optimizing three sided enclosures.

Optimizing Area Given 72 m Perimeter

The last three points:

The learning in the lesson with Ozobots went far beyond optimizing rectangles.  Students were questioning their thinking on: restricting shape dimensions to integer values; properties of quadrilaterals where squares are rectangles, but rectangles aren't necessarily squares; and what does testing a prediction really mean?  Then the subsequent optimization lessons that incorporated predict, test, reflect saw students, on their own, using their understanding of variables in shapes to modify an existing formula and then manipulate that formula to determine the dimensions. 

I don't think I will ever see optimization the same way again...  not only was my student's learning optimized, but so was mine.