MPM1D1 - Day 67 Solving Systems of Equations

When I first came across today's warm-up question I thought it would be great as a measurement problem solving type question with a bit of algebra thrown in for practice. Here is the problem:

What I did not anticipate was that this was also a great problem for solving systems of equations. A number of groups realized quickly that they needed an expression for the perimeter of each rectangle and then they had to set them equal to each other. One group quickly said "We don't know how to find the length". I asked them to start with what they did know and go from there. This quickly got the group moving forward.

I was amazed to see how easy most groups were able to set the equations equal to each other and solve. For whatever reason they were far better at this than they were last week. I'm guessing it has something to do with the context here. They can see the perimeter and know that the perimeters have to be the same (since it says so in the question). I was very impressed with the results today. One group that finished early said something along the line of "You're going to ask us to find the area next, aren't you?". Too be honest I hadn't thought about that, but it seemed like a great extension for those who were done. So I asked them to find an expression for the areas and asked if they could expand their expressions. What a great way to lead them into multiplying binomials. I love using the great ideas that students have.

The goal for today was to have students solve systems of equations graphically (the course only gets as far as solving by graphing). I mentioned earlier, we did this about a week ago. The nice thing about spiralling is that you can visit some trouble areas. This was one of those areas and I wanted to extend a bit by looking at systems in different forms.

Here are the questions I had them work on:

There were so many great questions that came out of this work. I find students always have a hard time with the equations of vertical and horizontal lines so a bit of extra practice here is alway good. Some students struggled with graphing the second equation in part b). They forgot what the slope was if there was no coefficient showing in front of the x. There was lots of good practice graphing equations and finding ways to graph different forms of equations.

One girl in the class insisted on solving the equations by substitution. This is easy enough for the first five questions, but I'me guessing she'll have a hard time with the last couple.

With about 15 minutes to go we moved onto a mastery test on solving equations.

MPM1D1 - Day 18 Optimizing Area & Perimeter

For a warm-up today I had students, individually, do the following:

Once everyone had a figure I had them share with the rest of their group and discuss their thinking. Then I gave them another one to consider on their own. 

This one seemed to be a little more challenging for some. I did see lots of squares and a couple of circles. Students had a good sense that these fit the description but they had a hard time explaining why they thought it worked.

As we were finishing up, one of the students said "Next you're going to get us to draw a figure that has the same area and perimeter. Aren't you?". I actually hadn't considered that as an option, but since he brought it up, I thought it would be a great idea. Most students thought it was easy and they drew a square. I asked a few to show some dimensions so we could compare the perimeter and the area. Many of them just wrote down a length and a width at random. We discussed, as a class, the shapes and dimensions they chose. Their answers were mostly squares that were 1x1, 2x2, 3x3 and 4x4. I drew a 1x1 and asked about the area and perimeter. Students quickly began changing their minds. We realized that a 4x4 would work. I asked if there were any others. Some suggested multiples of 4, then did a calculation to see if it would work. One student suggested 40x40 only to realize that the area and perimeter differed by a factor of 10. One girl pointed out that the areas were getting bigger faster than the perimeters so there couldn't be any more squares that had equal areas and perimeters. Wow! This would have been a good time to pull out Desmos and graph the perimeter vs. side length and area vs. side length to compare the graphs. Unfortunately, I didn't.

Next, we moved onto using a certain number of toothpicks to create all possible rectangles, using all of the toothpicks (we used pieces of straws rather than toothpicks). I gave each group eight straws and I asked them to keep track of the length, width, perimeter and area of each rectangle.

 They repeated the process with twelve and sixteen straws. Then they were asked to look back at their tables to see what they noticed. They needed a push in the right direction but it didn't take too long before they were able to find what I was hoping for. Hopefully this portion gets easier for them as they do it more.

We moved onto minimizing perimeter given a certain area. I gave out linking cubes and asked them to do the same sort of thing using 9, 16 and 25 cubes. The instructions I gave can be found here.

Some groups finished and were ready for the homework a couple of minutes before the bell. Others got far enough that they could finish up at home. Here's a link to the homework I gave. I feel like I stole this from someone. If it was you let me know so I can give you credit.

MPM1D1 - Day 10 Finishing Up The Giant Toonie

The warm-up for today was this Would You Rather problem:

I had students work in groups but at their desks rather than the whiteboards since their work from Friday was still up on the board. It was much harder to see what was going on when students were seated compared to what they would do at the board.

One student said he'd prefer the 40' pool because it was longer. He seemed happy to stop their so I asked if he could find out how much water in the pools. Some groups converted the feet to yards, others the yards to feet. One student wanted to use the formula for surface area but fortunately his group convinced him that he was calculating the wrong thing. All the groups had some good success with this problem.

We then went on to find out how many toonies would fit in the giant toonie. This problem had multiple steps to it and many students struggled with those steps. This reinforced the importance of being organized and methodical. All groups were able to find the volume of the actual coin, after asking for a formula. Determining the volume of the giant coin proved to be a challenge. They had a hard time figuring out that they needed to find the scale factor, then use the scale factor to find the thickness of  the giant toonie. There were a few unit conversion errors but all of the groups knew that they needed to divide the volume of the big coin by the volume of the little coin. Of course this assumes that the coin can be packed in (melted down?) without any space for air.

This was a fun problem to watch groups struggle through. They really had to think about what the problem was asking and come up with a plan. Some students were quite frustrated by it, but hopefully this process will get easier for them as we do it more often.

Once we were done we did a quick not on calculating area and perimeter and I gave them some questions to practice.