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.

## Thursday, December 14, 2017

### 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.

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.

Labels:
equations of lines,
line of best fit,
mpm1d,
visual pattern

## Tuesday, December 12, 2017

### 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:

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.

Labels:
mpm1d,
perimeter,
system of equations

## Monday, December 11, 2017

### 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.

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.

Labels:
equations,
fraction talks,
mpm1d

## Sunday, December 10, 2017

### 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.

## Thursday, December 7, 2017

### 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.

## Wednesday, December 6, 2017

### 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.

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.

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