MFM2P - Day 70: More Trig

I had to take two of my kids to a doctor's appointment this morning so a colleague was kind enough to cover my class. They started with this warm up:

And this is where they went with it:

Then they worked on the trig matching some more. They also talked about the steps involved in solving a trig word problem and did one example together.

We will on to quadratics on Monday.

MFM2P - Day 69: Quadratic Visual Pattern & Trig

As we missed doing the visual pattern warm-up yesterday, we did it today. This is what they started with:

Most showed me the pattern growing along the diagonal but were able to see it growing in other ways. They figured out that you needed to add 3 blocks, then 4 blocks, then 5 blocks... I asked if this was a linear pattern and some said yes while others said no. I asked what a graph of the number of the blocks vs. step number would look like. 

After they thought about that for a while I pulled up Desmos and plotted the points.

They could see that the points did not form a line and someone said that this was quadratic (yay!). I asked how we have found the equation to represent a quadratic pattern so far, and they said that they have used graphing calculators. I then told them that we could figure out the rule without using graphing calculators. We talked a little about how rearranging blocks to form a square plus some blocks can be helpful with quadratic patterns, but that it didn't work here. Admittedly, I was stuck on this yesterday, but thanks to Dave Lanovaz, I was able to share a fantastic strategy with my class today. He suggested doubling the pattern like this:

We now have rectangles whose side lengths we can relate to the step number:

Now that we have a rule for this pattern, we can divide by 2 to get the rule for the original pattern:

I love this! I love that I learned a new way of finding a rule for a quadratic pattern and that it is accessible to all of my students.

We spent the rest of today's class working on the trig matching activity. There was an error in my original fine - the corrected version is here. They are very slowly working through these problems and will continue tomorrow.

MFM2P - Day 68: Assembly

Today all grade 9, 10 and 11 classes had an assembly first thing this morning. We got back to class with only a few minutes to spare. There was not enough time to do the visual pattern as it was quadratic:

I can see how this pattern is growing in lots of ways. I can come up with the rule from the number of blocks. What I am having difficulty with is coming up with a rule from the pattern in such a way that my grade 10 applied students could understand it. If anyone can help me with that I would greatly appreciate it.

What we did for the 5 minutes I had them in class was multiply two binomials. We did (x - 4)(x - 3) and I had a student put his solution on the board with a lot of help from the class. A quick refresher in the short time we had. Back to trig tomorrow.

MFM2P - Day 67: Trig Matching

It seems like a long time since we have done an Estimation 180. This is today's:

And here are their estimates:

As you can see a few students changed their estimates along the way (I love that they are thinking about it!). We discussed the factors involved (oops, I misspelled marshmallow) including the number of layers of marshmallows and the number of marshmallows per layer. We talked about the "melty" factor, too. This would be a fun one to actually do with a class.

When I corrected their cycle 3 tests I noticed that many students were still struggling with trigonometry and quadratics. So this week is going to be devoted to trig. Well, yesterday was a holiday and I lose my class to an assembly tomorrow, so that's only 3 days of trig. I made up a warm-up, matching activity and extension over the weekend - you can find them here. My inspiration was this post by Tina Palmer.

I handed out the warm-up and circulated to help them get going. Here is a snapshot of what they look like:

One of the reasons for the warm-up was to get my students to differentiate between similar triangle, trig and Pythagoream theorem questions. Some of them see a triangle and immediately do <choose one from above list> regardless of whether it makes sense. The warm-up really forced them to think about what information they were given and what they were being asked to find. Many were stuck on 2 of the 3 kinds of questions but got through them with a little help. A couple of students took the entire period to get through the sheet, mostly because they were getting distracted... The others all began the matching activity. For this one I made up 10 typical right angle trig word problems along with 10 skeleton diagrams and 10 answers. None, of course, were in the same order. My intent was to immediately stop any of them from reading a question and saying "I don't know what to do." I mean, they never get away with that, but this didn't even give them the option.

I wasn't sure if they would want to cut the pieces out but they all seemed happy to number the questions and match the numbers up on the diagrams page. They didn't get very far with this part today, but will continue next class.

MFM2P - Day 66: Vroom Vroom!

Today's warm-up was the following math talk:

This is what they did:

They came up with lots of different strategies. I tried to help the ones who were stuck at the beginning by asking how much money they would have if they had 18 quarters. 

Next up, Vroom Vroom! This is a Fawn Nguyen activity so you know it's good. I decided that I wanted to leave this one more open-ended and not give students a handout including a table, graph, room for an equation... My students have done a lot of different activities where they have had to collect data and come up with a model so I thought they should be able to figure out what to do with this one without scaffolding it. This is what I showed them:

I put them in random groups, provided graph paper and let them get going. 

I think it would have gone better had the cars gone straight for more than 30 cm (sigh). The data that was collected was okay - better for some groups than others - and the work that followed was the same. Each group did hand in some form of a report. Here is a good one:

Sadly, our attempted competition at the end failed as the cars would simply not go straight. I only got 7 of my 9 cars back and 2 of those are very broken : ( So...if I do this again I will have to get better cars. The positives were that, without much prompting from me, they gathered data, looked at the rate of change, graphed, and came up with an equation (kinda-sorta). And I think that if I give them another similar task, they will run with it.

MFM2P - Day 65: In-N-Out

It was so good to be back in class today. After getting a couple of kids who have been away all week set up with the test they had missed, we started on today's warm-up: Would You Rather...

They worked in groups on the big whiteboards and really impressed me. Although some groups based their decision solely on distance along the ramp, many used trig to find the angle of each ramp. Once they had all worked on it, the class was pretty evenly divided between triangle A and triangle B. Here is some of their work:

Once a group had chosen A or B I asked them if changing the weight of the box would change their answer. It gave them something to think about and discuss. One student did say he would put wheels on the box which prompted another big discussion about friction and gravity!

Next up, In-N-Out! This is from Robert Kaplinsky's site and is a quick, fun activity. I started by showing them this picture which really grabbed the attention of some (and grossed out others):

and asked what they noticed and wondered:

I was very happy to have the questions I wanted to tackle on the list so I asked what they needed to know to be able to figure out the cost. I think the initial answer was Google, but this is where I went:

and I showed them this which helped us clarify what a double-double and 20x20 or 100x100 consists of:

Then I gave them this handout and let them get to work. There were a couple of students that came up with an answer quickly - a wrong answer - because they Googled it and quoted the price with tax. Eventually one of these students explained the mathematical solution to the class as his Googled answer didn't impress me as much as he thought it would! I circulated and asked questions to get groups going if they were stuck or had the wrong model. This was our consolidation:

I think they all could explain what the 0.90 and 0.85 represented in terms of the burger parts. We then compared to the receipt which is itemized with a double-double and 98 meat patties with cheese. We found a model for their system and were able to show that it was equivalent to ours.

The groups that finished quickly moved on to calculating the number of calories based on this information:

I loved that they didn't even need to check their answers - they were 100% confident that they were right (and they were). They left hungry, but mathematically satisfied.

MFM2P - Day 64: Toothpicks

Today is the wrap-up day for the lesson study in which I have been involved. So, again, I will blog about what should be happening in my class today.

Here is today's visual pattern:

Students need to find the pattern in the number of toothpicks at each step. They are very good at counting squares so this one might throw them off a bit. If I was there, I might extend it and ask about area, also. Next time...

Coincidentally (really!) they will then be doing this toothpick activity which I got from Alex Overwijk. This is really the 2nd part - the first has them counting toothpicks, whereas this one has them counting vertices. It's a nice activity to connect patterns, graphs and algebra.