MPM2D - Day 42: Similar Triangles and Trig

We jumped back in where we left off yesterday with students investigating the relationship between perimeter of a triangle and scale factor and then area and scale factor. Here are our conclusions:

And our "official" consolidation:

Next, we moved on to trig starting with a warm-up to get them thinking of sine, cosine and tangent again. When they completed that, they started on this matching activity (the warm-up is included in this file). They will keep working on it tomorrow.

Here is today's homework set.

MPM2D - Day 23: Trig Word Problems

I can't say how many times I have changed my mind on how to teach trig in this first cycle. At some point I had thought I would start with word problems to give students the context around what we are doing. I have started with slope angle which turns into the tangent ratio before, but didn't do that. I still need to take some time to reflect on whether I chose the best path and to make notes of what might work better next time.

For our third day of trig, we worked through word problems. We started by talking about the angle of elevation and the angle of depression (I have no idea where this handout came from so I cannot give credit. If you know, please let me know.) 

At this point I realized that my entire class seemed to still be in a post-turkey haze so I made up an example on the whiteboard and got them going. We talked about how you could find an angle inside the triangle given the angle of depression and came up with a couple of options.

I had them all solve using whichever angle they preferred. This turned out to be a really good thing. I wrote both solutions on the board and many decided that the one on the left was easier as the variable was in the numerator on that first line. This led to a good discussion about whether they could always make that happen. Others didn't care that the variable was in the denominator which also made me happy.

We then did the actual examples I had planned, which turned out to be a little repetitive as many use the tangent ratio.

I purposely removed the diagrams for as many questions as I could as drawing them and correctly labelling them is a big hurdle for some students. They worked through this one and then we consolidated together.

Here is the next one, again they had to draw the diagram:

Then they worked through three practice problems. Along the way we defined "stringer", "guy wire" and "clinometer". I randomly chose names of students to put up solutions to the practice problems and we looked over them together.

We had time to look at one more example together.

At this point, their eyes glaze over because they were overwhelmed by the sheer length of the question. I asked them to read it as many times as they needed to in order to draw a diagram. They tentatively did so. I showed them that they could break it up into two triangles, each of which they could solve. They seemed to understand that they needed to take the sum of the answers to get the height of the taller building.

Here is today's homework set. This is the end of cycle 1 so we will be doing review tomorrow and Thursday, and the test will be on Friday and Monday.

MPM2D - Day 22: Trig

We started today's class by looking at the trig tables they had created along with my version of it. 

I asked if we had similar numbers for the ratios and, for the most part, they said we did. I looked at them with a furrowed brow and said "But my triangles were not the same as yours. Why would we have the same ratios?" I was surprised (in a really awesome way) at how quickly someone said that all of our triangles for a particular angle were similar (reflecting on this, I should have given more wait time and had them talk about this in their groups). We explored that a little further to emphasize that each side length is being multiplied by the scale factor and that the scale factor divided by the scale factor is 1, hence the ratios will be the same.

Then I explained that they had created a trig table which could be used to find the side length or acute angle of a right triangle. I showed them a trig table for all whole numbers from 1° to 90° and told them that, back before there were computers, people had accurately created triangles with all of these angles and calculated the ratios. 

Side note: Up until yesterday, I was going to have them use the trig table for all their trig work in cycle 1. But I changed my mind yesterday afternoon. I use trig tables in grade 10 applied very successfully. Many of those students have trouble with any solution that requires multiple steps, so giving them the trig table removes one layer of abstraction - they don't need to think about which trig ratio they are finding, just that it's opposite/hypotenuse, for example. However, my grade 10 academic students should all be able to work through that and I decided it might be doing them a disservice by waiting to introduce the names of the ratios.

Back to today's class. I then told them that all of these values had been programmed into their calculators <insert speech about needing a good scientific calculators>. Although it might have been fun to have student give the ratios names, we went with tradition instead.

We went over some basics to make sure everyone knew how to use their calculators and that none was in radian mode.

I really emphasized that they should use the brackets (parentheses to my American readers) on their calculators to help keep their answers as exact as possible. I showed them that for part (c), above, using 0.88 resulted in the answer being off by a full degree. Then we moved on to actually using trig.

They were doing really well up to this point, but got stuck on the next one.

I was so happy that no one said "cross multiply!". We looked back at what we had done to get rid of the fraction when there was a number in the denominator, and followed the same process. They got stuck again. Many wanted to subtract cos(40) from both sides. I wrote it on the whiteboard x * 0.6 = 30 (where 0.6 was my bad approximation of cos 40) and they still wanted to subtract. But when I rewrote that as 0.6x = 30 the light bulb went on and they said "We need to divide!". They don't seem to be looking for shortcuts and seem to be really buying in to understanding why we are doing what we are doing. I like that.

Here is today's homework set. We have a PD day tomorrow and Monday is Thanksgiving so my next blog post should be on Tuesday.

MPM2D - Day 21: Intro to Trig

I talked very little in today's class because my students were busy working. They started with the similar triangle word problems that I had handed out at the end of yesterday's class. I had them write the answers on the board and if there were two different answers for the same question, someone wrote their solution on the whiteboard. It all went very smoothly - they didn't have many questions.

Once most students had finished those questions, we moved on to trig. They have never encountered trig before so we are starting fresh. I like to introduce it with act I of this (Boat on the River) 3-act from Andrew Stadel. 

They quickly got to the question:

We talked about why, as the captain of this ship, this would be an important issue to figure out before attempting to fit under the bridge. In order to calculate that angle...

I like to throw in a bit about how trig is actually used in "real life".

After we discussed how to name the sides of a right triangles (opposite, adjacent & hypotenuse), I had them work in groups to make their own trig tables. With the help of protractors, they had to draw triangles, then measure and record the sides lengths. They then had to calculate the ratios of sides. Here's my version of the table they were working on, found in this handout.

There were some good questions along the way as I circulated that could be paraphrased "Does it matter what size my triangle is?". That's as far as we got today. Here is homework set 18.

MFM2P - Day 58: Pyramids & Trig

I am home with 2 sick kids today and my class is likely being covered by a non-math teacher, so seat work it is :(  Along with working on the pyramids handout from yesterday, I want them to do some trig again so I left this handout (which I know some kids already started a while back). I hope they will work collaboratively today...

MFM2P - Day 22 (Finding Angles with Trig)

Today's warm up was this Would You Rather:

They mostly worked in groups on the big whiteboards. Many groups wrote 50 x 3 = 150 and 25 x 5 = 125 and they circled the second calculation as the better option. But what did the 150 and 125 mean? They couldn't tell me. One student even told me that those numbers had no meaning. This was the big takeaway for my class - just because you have numbers doesn't mean that you should necessarily start calculating with them. We discussed how the 50 lb box would be harder to carry up that third flight of stairs than it was for the first flight of stairs. Some told me that 50 lb wasn't heavy so they were okay with carrying that instead of the 25 lb box. There were interesting discussions and they agreed that this was an opinion question (because they don't have the physics background to work with) and it depended greatly on their level of fitness and how strong they were.

Today I was bound and determined to finish Boat on the River 3-act. But we didn't. Sigh. I started by getting them to add a "find an angle" trig question to their exercise books:

and I thought this would be an excellent opportunity to review sum of squares so we did that for the same triangle:

They practices a few more trig questions (finding the angle, only) before the quiz. Open-book, again. Some similar triangle questions followed by sum of squares and trig. I was pleased that when some of my students seemed stuck, they responded really well to a quick prompt from me. I also really loved it when a student asked for highlighters for the similar triangles questions. 

No time for Boat on the River, but I will get back there!

MFM2P - Day 21 (Finding Angles with Trig)

I love Visual Pattern Wednesdays! Here was today's:

and this is what we did with it:

I found it difficult to capture student thinking, but did my best with the different colours, above. They all noticed that the pattern was growing by 6 each time, but the starting value (step 0) was harder to find (see green). In blue, a student noticed that if they did 6 times the step number it always gave them 1 more than the actual number of blocks. I found drawing the pattern at step 0 really difficult - I had many failed attempts before coming up with the one shown.

We jumped back in where we left off yesterday with the triangles on chart paper. Here is an example of what they did:

and a (different group's) data:

At least one student noticed that everyone was getting about the same numbers for the ratios, despite all having different triangles. She was surprised and looked skeptical. We talked about why this happened and I tried to tie it back to similar triangles, but did not do the best job:

A little too abstract for many... I have updated the handout for next time. I also didn't get them to notice that opp/hyp and adj/hyp (should have) had the same values for different angles and why. Sometimes it is hard to find the balance between doing everything and not overwhelming them. I will try to include that when we do trig in the next cycle.

I gave them each a trig table and had them compare their values to the ones from the table - most were pretty close. Note that the trig table is all in terms of ratios of opposite, adjacent and hypotenuse.

Next, I asked them all to draw (using a ruler) a right angle triangle. Any right angle triangle. Then measure the lengths of 2 sides. Any 2 sides. Then choose the angle they were going to determine and label it with theta. Mark the sides with O, A and H for opposite, adjacent and hypotenuse from the perspective of the marked angle. Then figure out which ratio they could find based on the sides they measured and calculate the ratio. Then look up that value in the correct column of the trig table and read off the angle. And they all did!

I was working through an example at the same time, although I did all 3 cases:

They got to try a few on their own and we will do a few more tomorrow. Yesterday, before the end of class, one student asked if the boat made it (see post here). Tomorrow, they will work out the angle and see Act III - did the captain do the math?!?

I have decided to just find the measure of angles in right triangles for cycle 1. We will find the lengths of sides in cycle 2.

MFM2P - Day 20 (Starting Trig)

We did our first non-height Estimation 180 today:

The too-low guess was 1 and the too-high ended at 56 (not 1 million, as was suggested). Their guesses were all in the 15-25 range. Someone even suggested that the measuring cup is a cylinder and we could estimate its volume. Estimating the volume of one almond made the rest of this process difficult, however the idea of figuring out approximately how many almonds would fit in the bottom of the cup, then how many "rows" there would be is a good strategy for estimating.

I decided to also do the follow up one:

There were interesting strategies here that led to very specific estimates - clearly from actual calculations. One student saw the large container as holding 3 of the 1/4 cups along the length and 3 along the width, then estimated the height and used the number of almonds in the 1/4 cup measure to come up with an estimate.

I am still happy with how the warm ups are going and really like the routine they provide at the start of each class.

I intended to jump into trigonometry, but realized that my students had not done any similar triangle work in their exercise books because I was away last Thursday and Friday. I gave them the copies of this handout I made oh-so-long ago. It was especially good to go over this as I had two new students added to my class in the past few days (another benefit of spiralling is that these students will see all the concepts they missed in the 1st cycle during the 2nd, 3rd and 4th cycles). This is what it looked like:

Again, I focused on having them make obvious which angles and which sides matched up and on making sense of our answers. The student I asked to explain how to find y said that the y side matched up with 14 side so we needed to multiply 14 by the scale factor of 1.75. We did and decided that the answer should, in fact, not be bigger than 14 so we should have divided by the scale factor instead. I want my students to reflect on their answers and adjust their strategy if their answer is not reasonable.

On to trig. I started by showing them this video called Boat on the River which I found on Andrew Stadel's 3-act catalog:

There were some silly questions that arose, but I jumped on this one: "What's the angle of the boat?" I explained that sailboats are expensive and taking the mast down is an expensive process that usually requires a crane, as does putting the mast back up. So as the captain of this boat, if they could make it under the bridge without taking the mast down, that would be a very good thing. 

I talked a little about the fact that we are still working with triangles, only they have the be right triangles now (unlike some of our similar triangles). We started by going over how we name the sides of a right triangle, based on a marked angle:

I got them to practice identifying the sides of triangles with the first page of this handout that I found on-line here. Once they seemed to have a good handle on correctly naming the sides we moved on the activity part of this handout. In groups, they had to create right triangles with an angle of 10, 20, 30, ... , 80 degrees, measure the sides and come up with ratios of the sides equivalent to sine, cosine and tangent. (I should note that I do not mention the words sine, cosine or tangent until cycle 3.) They did this on chart paper so that all the members of each group could be contributing. They started creating their own trig tables... and will finish tomorrow.

The Ambiguous Case

This song came on the radio this morning, which naturally made me think of trig (less the song, more the band - sorry if I was ambiguous). I thought that if I was teaching the ambiguous case I would start with an activity like this.

Student would be paired up and one of each pair (let's call this student A) would receive an envelope. Their job would be to describe the contents of the envelope to student B so that student B could replicate it. Student B could not ask any questions, nor could they show their work until they finished. Then the two students would compare triangles.

Inside the envelope would be something like this:

I would be curious to see how many students would create a triangle that matched.

The irony here is that I don't teach the ambiguous case. I teach trig before it and trig after it, but not that part!

Sine Law

Jennifer Wilson wrote a post about productive struggle that you can find here. I did my usual trick of sending myself the link. I have a filter on my school email that automatically labels emails from me and files them away under "Twitter" so that I never actually see these emails. Thankfully I keep vague recollections of what I have tagged and earlier this week I knew to search my "Twitter" label for something interesting relating to sine law.

I stole this from Jennifer's post, having first shown my students the picture of the riverbank and surveyor and the diagram without any numbers. Straight from Jennifer's post.

Students worked in groups on their big whiteboards to find the width of the river. There was definitely some productive struggle going on and a lot of good mistakes were made along the way. I love that students try to use other mathematics they know, in this case Pythagorean theorem and right angle trigonometry. There were some good conversations around when those apply and why they don't in this case (with the triangle as drawn here). One group split up the 107° angle into a 90° and and a 17° angle (below). They struggled a lot and did some good math along the way. When they realized that they needed to create an altitude instead, they were quick to solve the whole thing.

Groups, one by one, figured out that they needed to add the altitude from S2. Some made mistakes along the way...

But they eventually got there!

Once they had found the width of the river, they had to work with an oblique triangle that did not have any numbers on it (again stolen from Jennifer's post):

Find an expression that represents the length of side c, I said. I got a lot of, uh, interesting looks. Most did not want to work without numbers. Didn't know where to start. They struggled in their groups for a bit then I told them to finish it for homework. That was their only homework.

The next day, we worked through the development of the sine law together and it went fairly well - better than in previous years, I think. 

Working in groups on the big whiteboards has really made a difference to the culture of my classroom. Students are willing to try things and make mistakes and they persevere. That makes me happy.

As a side note, none of my students thought of solving it the way Jennifer showed on her blog here. I will have to make sure I work that in next time around.