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 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 60: More Trig

I was away at the OAME math conference last Thursday to Saturday. I was fortunate to have the teacher who is teaching the other section of MFM2P cover my classes, and she even sent me some pictures of what they did! (Thanks, Michelle!)

They started with this Would You Rather...

And this is what they discussed:

Then they continued with trig. The day before we had gone outside and collected data for the height of the tree/lamppost/hydro pole and for the ramp in the school's foyer. They spent some time on Thursday going through the calculations to find the height of their object and the angle of the ramp.

They spent the remaining time working through more trig practice problems.

MFM2P - Day 59: Trig

Today's visual pattern asked for both the volume and surface area. They saw how the volume (number of cubes) grew, but had more trouble with the surface area. We used linking cubes to make the shapes which helped a lot. Next time I want to show them other ways of seeing it.

Next, on to trig. They have found both angles and side lengths of right angled triangles before using trig tables. Today I introduced them to SOH-CAH-TOA. I find it helpful to write it like this so that the ratios are more obvious:

Here is the handout they received. We got through example 3 together and then moved on to this. Yes! We were going to go outside! I showed them how a clinometer works.

I asked what data they would need to collect and we came up with this diagram:

I put them in random groups and told them they had 15 minutes to be back in class. Out they went (it was beautiful out) and started measuring. Some groups were finding the height of a tree, others a hydro pole and one group the light standard. They came back in the school to take measurements (as a large group) for the ramp then we all headed back to our classroom. They wrote the measurements they took for the ramp on the board for everyone to use:

And that was it. Tomorrow they will do the calculations and practice a little more trig.

MFM2P - Day 41 (Trig)

Today's warm-up got complaints because it was too easy. I will switch it out in favour of something more challenging next time.

Today was all about trig. I gave them a few minutes to work on some examples where they had to solve for an angle. We consolidated with this one, where they were given the lengths of all three sides so could choose which ratio to use. When I wrote the first student's solution on the board, some thought they had done it incorrectly until they saw the answer. It was good for them to see multiple ways of solving this question.

I let them work for about 10 more minutes on finding angles before we stopped to look at how to solve for a side. These are from their exercise books:

They seemed to understand quite well. When we got to the second example they tried multiplying the numbers together and we got about 5.4. I asked if that looked right and they said no! I loved seeing them realize that the hypotenuse had to be longer than 8 cm so we had clearly done something incorrectly. Upon their suggestion, we tried dividing and they thought the new result was much more reasonable. I have struggled in past years with this process - so many of my students have difficulty with algebra and get lost when you show how to solve these equations in a more formal, algebraic way. I have decided that I am okay with them reasoning through, possibly getting it wrong and fixing it.

They worked for the rest of the class on this handout while I circulated. Stickers were good motivators today and I was very impressed with the work I saw.

MFM2P - Day 40 (Similar Triangles & Trig)

We did today's warm-up quickly as we had a lot I wanted to get done. Here it is:

I was surprised at how poorly some students understood what the numbers meant. A number of them divided the diameter by the number of slices and compared the results. I actually had to draw a circle, then the diameter and divide it up to help them make sense of what they had just calculated. Eventually they understood that we needed to compare areas. Here is one student's solution (he figured it all out on his own):

And my consolidation:

After the warm-up we talked about similar triangles. I asked what they knew about similar triangles. They said that they were the same only different sizes. Someone mentioned scale factor. I asked what the scale factor told you about the triangles. They replied that it was how much bigger the larger one was (I know it could go the other way). We moved on to doing a couple of practice questions (links to handouts are below). Here is the first one:

We went back to colour-coding corresponding sides which really does help some students. We found the scale factor, made sure everyone understood what that number meant then used it to find the missing side. The second example involved angle of incidence and angle of reflection, which they have learned about in science class.

They worked out the height of the statue after I demonstrated the process using a mirror. I then put them into random groups and set them on their way to collect data for two objects with inaccessible heights within the school. Some groups needed more guidance to correctly collect their data. They worked out the missing heights when they came back to class, although for many, their work ethic did not seem to make it back to class (the perils of letting them "loose"). Once done, they started to work on a trig practice sheet involving finding angles. I was pleased that they remembered to start by labeling the sides with "opposite", "adjacent" and "hypotenuse".

Here is the mirror activity handout and here is the practice worksheet for solving for an angle using trig. More trig on deck for tomorrow.

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.