Tuesday, October 17, 2017

MPM1D1 - Day 29 Interior & Exterior Angles

I started by having groups up at the board working on this Would You Rather problem.

This was the first time we've worked at the board in a little while. There were a number of people who were quite off task. Others got to work right away but the group dynamics were not what they should have been. Once they were done the activity I sent them back to their seats and we talked about contributing effectively to a group and how it benefits everyone in the group.

The main event for today was some geometry, specifically interior and exterior angles of polygons. I put the image below up on the board and asked students to work at the whiteboards to see how they would do without any instruction. 

The group work was much better this time around and all groups were able to answer all of the questions. A few groups needed some reminders about supplementary angles and a couple asked about opposite angles. They were doing great.

I brought them back together as a group and we summarized the different types of triangles, supplementary angles, opposite angles and began exploring the sum of interior angles. Everyone knew that the interior angles in a triangle sum to 180° so we began looking at other polygons. We did this by looking at the number of triangles in each polygon:

From this they were able to determine the sum of the interior angles. Next I had them fill out the table below to come up with an equation.

We've done enough visual patterns that to many this process came easily. They had no trouble finding the rate but had to think a bit about the initial value.  I had a couple of different results which was neat. The most common was that the sum of the interior angles = 180n-360 and the other was that the sum of the interior angles = 180(n - 2). It was exciting to see these different results.

We had just enough time to look at exterior angles. To do so I showed this video:

 We had a few minutes left which was enough time for me to handout the work for the day.

Monday, October 16, 2017

MPM1D1 - Day 28 WODB, Tables of Values & Equations of Lines

We started with this Which One Doesn't Belong:

I figured that this would be a bit of a challenge. I knew that students would be able to look at the equations and pick some characteristics out (which they did). I also knew that my students see these four equations differently than I do. When I see these equations I picture the graphs. My students can't do that yet. We've done a lot of graphing of equations but most of that graphing was with scenarios that are concrete and have some meaning. These equations are very abstract.

In any case, I put this up to see what would happen. Right away one student asked for graph paper. Others then started asking if they had to graph it. I told them that they didn't have to. Many said "Wait! What? I don't know how to graph that." I gave them a bit of time to think this one through. I helped a few students who really wanted to be able to graph. We took it up and had some great discussions. I had some superficial type answers (the first one doesn't have a number added or subtracted). I thought I would be disappointed with these types of answers but I really wasn't because each of them led to some talk (led by other students) about what those parts of the equation are, what that means about the relationship and what it means about the graph. The discussions were fantastic. Some of the students were having a hard time connecting all the pieces but it's a conversation we can revisit throughout the semester.

This seemed to be a good time to discuss how we can graph a relationship using a table of values. This was a bit of a leap for some students, which surprised me given the number of visual patterns that we've done and the number of times we've graphed those patterns. I think had I told them that each equation came from a certain pattern, and given the pattern, they would have been fine. We took a step up the ladder of abstraction and talked about how we can create a table of values and plot those point. We did this, together, for the equation in the upper left and then they worked on the one in the bottom right. Did I mention this was just the warm-up? #longestWarmUpEver?

With the warm-up behind us we could move onto connecting slopes, y-intercepts and equations of lines. As it turns out we already had some equations, tables and graphs on the board. We talked about how to find the slope and y-intercept from the graph, from the table and finally they told me how to find them from the equation. My favourite comment of the day: "You mean we can just look at the equation and get slope and y-intercept? We don't have to graph it or make a table?" I think there was some incentive to understand y=mx+b.

I gave them the first two pages of  this handout (thanks @MrHoggsClass). Once that was done I handed out some practice on creating tables of values and equations of lines.

Sunday, October 15, 2017

MPM1D1-Day 27 Desmos Linear Activities

The goal for today was to talk about slope. We started with the Polygraph: Lines activity from Desmos. The idea is that they get paired up with another student in the class. One of them chooses a line from a list, the other asks questions that can be answered with a yes or no to help pick which graph their partner chose. It's basically Guess Who with lines. I wanted to start with this to see if students would use some of the vocabulary we've talked about.

They started right away and got right into it. I saw the use of lots of terminology but not much about what we've talked about. I heard comments about corners rather than quadrants. I heard some reference to the origin. And more than once I saw "Is your line straight?". This one drove me crazy! When I asked "Isn't every line straight?" these students would reply with something along the lines of "Yes, but I mean like this", indicating that they were talking about a vertical or horizontal line. We'll keep plugging away at the terminology.

  I let them play a round or two then brought them back together as a class. I asked which types of questions they found helpful. I then reminded them of some terminology (slope (positive and negative), quadrants) then introduced some new terms for some (x and y-intercepts). They played again and their questions were much better. There were a couple of math fights about wrong answers to questions such as "You said it had a negative slope. That slope is positive."

Once we'd had a bit of experience with the activity we moved onto Polygraph: Lines Part 2. They worked through the activity, hopefully improving their vocabulary and understanding of lines. Some students we motoring through the work, others needed a little encouragaement.

The last activity for the day was Put the Point on the Line, where students have to determine where a third point needs to go in order to be on a line with the other two. The best part about these activities is the teacher dashboard that allows me to see all the work my students have done, even after the fact. I can look the work over and see where the gaps are and then look at providing some assistance in those areas and I have a record that will allow me to see a student's growth over time.

There are lots of other Desmos activities involving linear relations here.

Once they were done the activities we talked about finding the slope between two point on a graph. We've done this before but this was a good reminder. Then we moved into finding the slope without a graph. I gave them this handout to practice with.

Thursday, October 12, 2017

MPM1D1 - Day 26 Solving One & Two Step Equations

I was away today so here's what I hope happened.

As a lead-in to solving equations students were to work through some Solve Me Mobiles as a class. I left about 10 of them to try. I really like using these because the principles are the same as those used to solve equations but students see them as non-threatening puzzles. I guess part of the reason for this could be that the puzzles are very visual and there can be some trial and error. I guess they're not quite as abstract as an equation with letters as unknowns.

After the warm-up students watched a video of me working through some examples of solving one and two step equations. After seeing the examples they had some equations to solve. I imagine that a good number of students were done early. If they finished early I'm hoping that they tried more Solve Me Mobiles on their phones.

Wednesday, October 11, 2017

MPM1D1 - Day 25 Graphing Stories & Collecting Like Terms

As part of Global Math Week I thought I'd start today with the International Math Salute. I had students stand up, put their arms out from, cross their right arm over their left, put their palms together then untwist.

The look on their faces as their hands ended in a twisted mess was priceless. I did it a couple of times. Some students said that they had figured it out but they couldn't seem to reproduce it. There were some great conversations going on.

Next, I handed out individual whiteboards so that students could sketch graphs of some Graphing Stories. We did the first two stories. The first one was a bit of a challenge given that students were graphing time vs. time. They struggled with this idea but after the first few seconds of the video they figured out what was going on. I let them create a graph then we reiterated some of the terminology. We talked about continuous vs. discrete data, partial vs. direct variation, interpolation vs. extrapolation, slope. These were all terms that we've seen before but based on last week's test many students still need practice with the terminology. We'll do a few more of these at some point.

The goal for today's lesson was to look at adding and subtracting like terms. We started by watching this video:

I was kind of surprised at how entertaining they found the video. After watching the video I asked if there would have been an easier way to place the order? Someone suggested that we group all of the similar items together and add them up. What a great idea!

We talked a little about some terminology: terms, polynomials, monomials, binomials, like terms. We took a note on adding and subtracting like terms then I gave them a handout for some practice.  I think the 'you can't add a burger and a Coke together' idea really helped some of them understand the idea of only adding (and subtracting) like terms.

Tuesday, October 10, 2017

MPM1D1 Day 24 Water Line & Distance-Time Graphs

Today we started our second cycle. For the warm-up we looked at a non-linear pattern for the first time (Visual Pattern #1).

The goal was to find out how many square were in the forty-third step and to come up with a general equation for the number of squares in the nth step. It was interesting to see the approach given that we've done so many linear patterns. Most groups created a table of values and found the pattern. They realized that the values weren't going up by the same amount. They were so accustomed to finding the first difference (though we haven't called it that yet), using that as the multiplier in the equation then finding the initial value. Some groups abandoned the idea of using the differences and instead starting looking at how the pattern actually grows from step to step (using the dimensions of the squares). Most groups that did this had no trouble finding an equation. For those that finished early I asked them to determine a rule for the number of toothpicks in each step. For the groups that didn't look at the dimension of the squares, things started to get difficult. They knew that they needed to add two more to what they added in the previous step but they couldn't figure out a way to do that in an equation. We'll do a few more of the quadratic patterns and I'm sure they will get better at them.

Once the warm-up was complete I meant to talk about distance-time graphs with motion sensors but I forgot. Instead I moved right into Water Line.

It's a great activity that allows students to graph the height of water in a glass over time. Immediate feedback is built right in as students click the play button to see if their graph matches the real life situation. The activity couldn't have gone any better. Students were working hard and some expressed how much fun they were having. Imagine, having fun in a math class! The best part seemed to be making their own glasses and trying to create a graph for their classmates' glasses.

After the Water Line activity we moved onto discussing distance-time graphs. What does it look like when you move towards a sensor, away from it, at a constant rate, speeding up, slowing down, etc. Then they practiced with this handout.

Monday, October 9, 2017

MPM1D1 - Day 23 Test Day

We wrote our first test on Friday. Many students seemed quite nervous as they started. I get excited on test days. I'm excited for students to show me what they have learned. If you're looking for a sports analogy, test day is like game day. We've spent all this time practicing and improving and I'm looking forward to seeing how much my students have improved.

As the test progressed I had questions such as "What does this mean?" and "How do I do this?". I told students to try their best. We had talked before the test about how it was important to write something down for every question. I was happy that for the most part they did. Some students need to work a little on presenting our solutions in a manner that's easy to follow.

I think that by the end of the test students had a good idea of what they needed work on before the next test. I'm hopeful that this will provide them with some focus for our upcoming work.