MPM1D1 - Day 75 Snow Day

The buses were cancelled today and as a result I had very few students show up. I did have one girl email me to see if she could write her test today. She came, got some help with a few things and wrote her test. Good for her.

I did have a number of struggling students request some extra work to do over the holidays. I was impressed with their desire to improve. So I sent them some work.

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 73 Finishing Up 3D Printing

Today we picked up right where we left off yesterday with the pentomino activity. We fired up the 3D printer and started printing right away. Groups that hadn't finished the calculations from yesterday kept working away. They stumbled a little with the different units but eventually most of the groups figured things out. We printed one figure after another but eventually ran out of time. I'll get the others printed at some point.

Once groups were done I gave a review for them to work on to prepare for this week's test. I was hoping to do the test Thursday, but after things didn't go well yesterday I figured they would need an extra day. So, yes we are having a test on the last day before the holidays. It's not ideal, but I figure it's better than doing the test Thursday or after the holidays.

Generally, most students worked well today, either on the pentominoes assignment or the review. There was also a fair bit of excitement when a group's pentomino began printing.

During the printing process today I remembered that prints were not solid plastic. They are infilled with either a hexagonal pattern or a rectilinear pattern. I'll have to do some research to see what percentage we were infilling. This actually adds another layer to the assignment. The more I think about it, the more I think this activity would make a great culminating activity.

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 71 Rearranging Formulas

We started today practicing solving equations with fractions as groups at the whiteboards. Here we the warm-up questions.

There weren't any problems with the first question. There were a couple of problems with the second one and more problems with the third. We spent some time working through the issues. Some groups wanted more questions to practice with so I had them make up their own and go from there.

Once everyone seemed comfortable solving these equations we moved on to rearranging formulas. I posted a few on the board and let them get to work. Here are the questions they started with.

The most challenging one here seemed to be part c. For groups that struggled I gave them an example where y, m and b were given and asked them to solve for x. They had no trouble doing so, so I asked them to replace the numbers with variables. That seemed to be enough to get them going.

Once groups were done I gave them some questions to practice.

MPM1D1 - Day 70 Equations of Parallel and Perpendicular Lines

During yesterday's warm-up, one group was convinced that two of the lines were perpendicular. Based on that comment I thought today's warm-up should be about perpendicular lines.

Once groups found equations for the lines I asked what they knew about how the lines intersected. They responded with the point of intersection and I asked if there was anything else. When they told me that the lines were perpendicular I asked how they knew the lines were perpendicular. Some groups could justify their claim immediately, while others need time to formulate their ideas.  

After the warm-up we moved right into finding equations of parallel and perpendicular lines. We had a conversation to remind them about how to find the equation of a line given two points and what it means for lines to be parallel or perpendicular. They worked on these problems at the board:

None of the groups had any trouble with the first two questions. A couple of groups struggled with the third and all groups needed some reminders about rearranging equations for the last question. When they were finished they went to work on some practice questions.

MPM1D1 - Day 69 Properties of Quadrilaterals

The warm-up for today was to find the equation of the line segments shown below.

I figured this would be a good opportunity to practice dealing with horizontal lines, as well as others. One group tried using the formula for slope but when I asked if there was an easier way they told me that they could use the graph. Another group was convinced that the two segments on the right were perpendicular. When I asked if they could explain how they knew this was true, they began doubting themselves and then verified that they were in fact wrong by looking at the slopes of each.

The goal for today was to investigate properties of quadrilaterals. I found this Geogebra activity. I had originally thought that I would make something up but finding this saved me some time.. Students worked through the activity at their own pace and took notes about what they observed. Some notes were better than others. I had students working individually on their own computers. I'm thinking it may have been better to have them working in pairs.

Once most students were done I summarized with this graphic:

Then it was time for some practice. I gave these questions from this page. Some students chose to work in groups at the board, some chose to work in groups at their desks and some chose to work individually. The two strongest students, who hated working with each other earlier in the semester, decided to team up along with a third person because they realized they could get the work done faster if they worked together. They stayed in past the bell and got it done.

MPM1D1 - Day 68 Speedy Lines

We started with a visual pattern today:

It was interesting to see how students counted the number of watermelons in each step. It seemed as though most groups had a couple of different ways of counting which made for some interesting discussions. I think the hardest part here was generating the table of values. Once they had that, groups quickly came up with the equation and the number of watermelons in the forty-third step.

We took up the equations mastery test from yesterday and hopefully cleared up some misconceptions.

Today's main event was practicing coming up with the equation for a line of best fit. Everyone can draw a line of best fit but when I ask for the equation many students go to their tables to find the slope. When they do this they don't always choose points that are on the line. We need to work on realizing that we want to use points that are on the line to find the equation.

Today we timed to see how long it takes to assemble 5,6,7,8,9 and 10 linking cubes. This is an idea that I modified from Mary Bourassa's Speedy Squares. Rather than making squares we just connected cubes to form a line. Groups worked to collect data. Some groups needed to work on being consistent but got it sorted out pretty quickly. Then they plotted the data and worked to find an equation of the line of best fit. They then practiced using their equation. Here's the handout.

With about fifteen minutes to go we tried the mastery test on solving equations again.

It occurred to me at the end of the period that I don't take enough pictures of students working or of their work. Something to work on. Sorry about the lack of photos.

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 66 More Equations With Fractions

We started with the following Fraction Talks, where students had to determine which fraction of the picture was red.

It was interesting to hear all the different ways students did these. Their were some good discussions about adding and multiplying fractions, which was a good reminder for some.

I then put the following equations with fractions on the board and had students work individually to solve them.

A couple of people asked for a refresher on how to solve equations with fractions so we worked through a question as a class, then they began working away. It was slow going for some, but everyone was moving along and getting a little better. Lots of students were those in their groups who were struggling.

Once they were done, students continued the handout from Friday. It was a good day of individual work.

MPM1D1 - Day 65 Solving Equations With Fractions

We started with this Which One Doesn't Belong:

It was great to hear all of the terminology that came out of the discussions.

After the warm-up we moved into solving equations with fractions up at the board in groups. I didn't give any instructions. I gave some equations for groups to solve and they did a great job.

Here are the questions they worked on:

I was amazed at how well the groups worked. They required little to no assistance from me. They were able to apply what they had learned about solving equations without fractions and things worked out great. It was also great to see the stronger students really working with those that struggled to bring them along. I feel that we've really developed a community of learners in the class and I'm really happy about that. One of the groups consisted of two students who pretty much refused to work with each other at the beginning of the semester. Today they worked as though they were good buddies.

Once the groups were done I brought the class together and asked what was different about the equations today. Many students said that these equations were more difficult. When I asked why they were more difficult the response was because of the fractions. We then talked about how we could eliminate the fractions by multiplying both sides of the equation by a common denominator. We did a couple so they could see how it was done then they did the first part of this handout.

MPM1D1 - Day 64 Test Day

We had our test today. Students worked away diligently and asked for clarification as needed. I'm hoping for some good results. There are a few topics that we'll revisit in the coming weeks to solidify understanding.

MPM1D1 - Day 63 Another Day of Review

The plan for today was to have students write their individual tests. But, based on what I saw yesterday during the group test, many students would not have been prepared to write today. I decided to give the class another day to work on the review questions.

At the beginning of class I made the rounds and discovered that very few students had done much of the review at all. I was a little disappointed.

Most students used their time wisely today. It was great to see some of the stronger students helping those that were struggling.

Here's hoping for good results tomorrow.

MPM1D1 - Day 62 Group Test

Today was our group test. I let students choose their own groups. I'm not sure whether this was a good idea or not. It seemed that some students weren't really sure who to work with. Others had a good sense but I had said that the group must be three people. This meant that those hoping to be a group of four had to decide who was going to leave. Not doing random groups today made me really appreciate doing random groups.

Each group received the test and started working on their whiteboards. I had each group start at a different question. Some groups were very efficient and others really seemed to have a hard time making headway. I went around and did lots of listening and asked if students could clarify their thinking for me.

It was pretty obvious that many students had not studied for the test. There were lots of incorrect assumptions being made along with lots of simple errors. I really like the group test because of all the discussion and learning that is taking place. I also like the fact that I have some conversations (which I recorded today) that I can use to supplement (one way or the other) a student's test results. What I don't like about this extra information is that I don't have a clean way to count it.

I struggle with how to assess the group test (and maybe it just needs to be formative). I don't want to penalize a student who learned a ton between the group test and the individual test. I also don't want students to feel like they did well on the group test so they don't need to prepare for the individual test. In any case, I do have a record of the conversations along with photos of all the work that I can use to help in making an overall judgement.

The plan for tomorrow was to do the individual test but I don't think my students are ready for that yet.

MPM1D1 - Day 61 Solving a System of Equations Algebraically

We started the class solving some equations. I wanted students to really focus on showing their work and being methodical in their approach.

After the warm-up we looked at solving a system of equations algebraically. We looked at the Knotted Rope work we did last week and discussed the ways we could determine when the ropes would be the same length. Students mentioned that they could graph it or look at the table. I asked how many of them wanted to fill in the table to get up to 60 knots. We decided that in this case a table would take too long and one student suggested we should use an equation. We talked about how if the lengths of the ropes were equal then the equations representing their lengths must also be equal. I helped them setup the equation and let them solve it.

For the remainder of the period students could work on their own or in groups on the review for the upcoming test.

MPM1D1 - Day 60 More Knotted Ropes

We started with another Which One Doesn't Belong.

The goal here was to connect the different representations of a linear relation. Students did a great job of discussing slope, y-intercepts, x-intercepts, continuous, discrete and the equations for these relations.

They then had some time to finish up the Knotted Ropes work that they started yesterday.  The groups were much more focused today. They graphed the relationship between rope length and number of knots for each roe. They generated an equation and made connections between the slope and y-intercept and what they meant in terms of the rope. I asked them to use there equations to find how long will each rope be if it had 25 knots? How many knots are needed to get each rope to a length of 60 cm? Then they plotted their data in Desmos and performed a linear regression to get the equation of the lines best fit. They then checked on the graph to see where the point of intersection was located.

I had hoped to look at how to find the point of intersection algebraically but one or two of the groups weren't quite ready for it yet. I'll pick up here on Monday.

Some groups finished early so I gave them some more parallel and perpendicular lines practice. We're having a test next week so I also gave out the test review in case anyone wanted to get started on it over the weekend.