MPM1D1 - Day 74 Test Review

We started today by looking at these two objects that were printed yesterday.

The goal was to consolidate some of yesterday's work and to reinforce one of yesterday's big ideas. Because of the work my students had done with their pentominoes, they knew that each pentomino was made up of five cubes. The cubes on the small pentominoes were 0.5 cm in all direction and the cubes on the large one were 1 cm in all directions. So, I asked how many times bigger the volume of the larger one was compared to the smaller. I received a couple of answers of 2 (which I expected), an answer of 4 (with the justification that that's what they found yesterday) and an answer of 8. I held the figures up and asked if anyone thought it would only take two of the little ones to fit into the bigger one. Strictly by intuition everyone knew that 2 couldn't be the answer. One student offered up an explanation of doubling in more than one dimension. At this point we jumped into a bit of algebra and looked at an expression for the volume of a cube that was x units long and compared that expression to one for a cube that had a length of 2x.

The large figure in the picture is a model of an original that is twice the size, in all dimensions. I asked how many of the little ones would fit in the giant one.

There was some discussion about 16 vs. 64, but eventually we settled on 64 (supported with some algebra). Once the relationship for volume was squared away we quickly touched on the relationship for surface area.

It was a good, fun discussion that I think made a lot of sense because of the manipulatives on hand, that were made by the students.

After the warm-up we wrote a mastery test on equations of lines then students continued working on the review for their test tomorrow.

MPM1D1 - Day 72 Surface Area, Volume & 3D Printing

I had seven students away yesterday so I spent some time at the beginning of the period recapping what we did yesterday. Once the recap was done we moved right into some volume and surface area.

My own kids have a pentomino based game called Katamino. It's a fun game that really stretches your spatial reasoning skills.

I thought it would be fun to make this game the basis for an assignment. If you don't have the game you could always modify the assignment to work with any pentominoes or even have students build their own figures using linking cubes.

The gist of the lesson is that students get a pentonmino and calculate its surface area and volume. Then they create a scale diagram of a pentomino that is half the size (in all dimensions) of their original pentomino. They calculate the surface area and volume of their model and make note of the relationship between the original and the half-sized model.

They were given the length of a spool of filament (in metres), the diameter of the filament (in millimetres) and the cost of the spool and they needed to determine the cost of their pentomino. Once they had done all of that they designed their pentomino in TinkerCAD. The designing was pretty simple and didn't take long at all. Once their design was complete they were able to print on the 3D printer.

This was meant to be a bit of a fun lesson but for whatever reason many students didn't seem to be into it. I had students working in pairs which is not something we normally do. They also weren't working at the board. Some students spent a lot of time fidgeting with the pentomino (I thought fidget spinners were so last year?), others watched as their partner did the work. One group took 30 minutes just to get their measurements, despite repeated calls to get going. I think next time I need to get groups to do their work at the whiteboards and maybe I need to be explicit about how they could split up their work. Maybe groups of three would have been better than pairs. I'll have to rethink the logistics of this one.

Having said that I had two groups print today. They did a great job and were pretty excited about the result. I had one more group that finished everything except for the printing at lunch. They will print first thing tomorrow. We will finish up tomorrow. I'm looking forward to trying this again to see how it goes.

Here's a description of the task. This task could also be used in an MFM2P class. I've also modified the task to fit the MBF3C course. You can find that here. Here is a link to a quick introduction to TinkerCAD.

MPM1D1 - Day 51 Optimizing Volume and Surface Area

We started with these problems:

I gave the problems orally, one at a time and groups made their way through them. It was great to watch them work. For groups that made mistakes, it was often enough for me to say "Are you sure?" for them to think a bit about what they did and find their mistakes.

We then moved onto today's work.

This problem was closely related to yesterday's Dandy Candies only this time the side lengths didn't need to be integer values. Most groups easily made the connection to yesterday's work and quickly came up with a solution. One of the groups was really struggling so I spent some time walking them through it. 

Then we moved onto these problems:

Theses problems seemed to be at just the right level. Students seemed to be in flow for the entire period. When groups would get stuck I'd ask a question or provided a hint and off they'd go. 

The last problem I gave dealt with a cylinder rather than a rectangular prism.

A couple of groups didn't get this far today, but those that did made some great headway. A couple of groups struggled with what the cylindrical equivalent to a cube would be. I asked how they would approach the problem if the top and bottom were rectangles rather than squares. That was enough to get them going. One of the groups, on their own, actually drew a cylinder inside a cube. It was a thing of beauty. I wish I'd taken a picture of it.

We were out of time so I gave a couple of rectangular prism questions to practice for homework. We'll pick up with the cylinders again on Monday and consolidate all of the optimization.

The period flew by today. Students were right into the work. It was challenging but not so much so that they couldn't overcome the challenges. What a great period.

MPM1D1 - Day 50 Dandy Candies

We started by revisiting this type of problem:

I was curious to see how much of this work that we did a while ago they would remember. Some groups had it figured out right away. Others tried making a table which was great to see. They started with a pen that was 25 by 50 then increased (and decreased) the dimensions by 10. Which meant that they missed the optimal solution. We talked about the properties of the rectangle that seemed to give the largest area and what they noticed about it compared to the others. They quickly realized that their rectangle needed to be a square.

Before we moved onto the main event for the day we consolidated the work that we did on the distributive property yesterday. I also gave a couple of questions for them to try.

The main event for the day was Dandy Candies. I asked what they noticed and what they wondered. There were lots of good observations and a few good questions. The most common thing they wondered about was what question I was going to ask them.

We had some good discussion about volume and surface area and they had some practice calculating surface areas. Some went immediately for the formula at which point we had a discussion about what surface area actually means. No formulas were needed after that.

We finished up the class with a mastery test on collecting like terms, multiplying and dividing monomials and the distributive property.

MPM1D1 - Day 40 Surface Area

We had a guest in our class today. I had a meeting with a teacher from another school this morning and he decided to stick around and check out my class. It's always nice to have visitors. Teaching is often done in isolation. It's nice when we can get together in a classroom and reflect afterwards. We need to do more of this. At some point I need to make time to get #observeMe going.

We started the day with this visual pattern:

I asked students to find an equation that represented the surface area for step n. I handed out the linking cubes and students went to the boards. Some began building the model, others quickly made a table (of values, not an actual table made of cubes). Many just made a table for the number of cubes. Others tried using a formula, without giving it much thought. Still others tried to reason their way through only to fall back on 'the formula'. When  groups got stuck with their chosen formula I asked what surface area was (we haven't really talked much about it in class). Everyone I asked was able to tell me it was the area of all the faces. I asked them to forget about the formula and find the surface area using the cubes they had in their hands and off they went. One group wanted to find the surface area of the rectangular prism then subtract the exposed surface of the cube that was missing. I asked what happens when they take out the middle cube and they realized that they would have to add in the surface area of centre. An neat approach.

Some groups finished the task quickly while others took a long time to get there, but did manage eventually. Once they were done I had them work on these problems at the board. There were lots of great discussions (especially about the Pythagorean Theorem). Today seemed to be one of those days that things just flowed smoothly. I guess having two adults in the room can do that.

With fifteen minutes to go we did a mastery test on finding the equation of a line and I handed out some practice questions for students to work on individually.

MPM1D1 - Day 36 Surface Area, Exponents & Compound Interest

We did the extensions to the Big Nickel today (interior angles, exterior angles, cost to resurface, etc.). This provided some great review on topics already covered to date. It was interesting to watch them struggle with surface area. Right away they jumped to their formula sheets and started looking at formulas for triangular prisms. The trouble was that many of them couldn't make sense of the variables on the page. Now to be fair we haven't done any work with surface area. I stopped the class and asked what surface area means. They knew exactly what it was. I asked them to show me which parts of the nickel we had to find the area of. They nailed it, so I set them to work to find the area of all the pieces. No formula required. I can't believe I thought we'd get the entire task done in a day.

Once we finished with the nickel it was time to head into exponents. I threw this question up on the board and asked them to come up with an answer individually.

It was great to see where everyone was starting from. One student said "This is easy! I just have to multiply 3 by 3 by 3..." and off he went. Others used the exponent button but got caught up on  the order of operations. Many students found the value of the numerator, then found the value of the denominator, wrote both of them down then did the division. A handful of students were able to use the exponent laws to quickly realize that the entire expression was equal to 1. After getting all of the errors corrected a student who knew the rules wanted to share, so she told us how to do it.  It was such any easy transition into what is often a boring list of rules.

As we discussed the rules one boy asked why we needed exponents. He understood that the rules were helpful in evaluating the expression above but wondered when in his life would he ever come across exponents. This sidetracked me as we discussed one of my favourite mathematical topics: compound interest.

I'm always surprised that students aren't aware that they are paid interest by their banks. In any case we talked about compound interest and how if you set aside money every month for the majority of your live you can become quite wealthy. When I was in high school my classes received these types of lessons from Bob Gunter. I had Bob as a teacher for four out of seven math courses. To me these lessons were the most memorable parts of any course. It was clearly something he was passionate about and he was able to share and spread that passion to his students. So Monday (tomorrow is a professional activity day) we'll look at some financial scenarios (at least a little bit).

I handed out some exponents practice and bid the class a happy and productive weekend.