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.

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