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.