I blogged last year about providing students the opportunity to make sense of special right triangles. This year I just want to make a few observations. Over the last year, I have gone through and thrown out old binders of notes and transparencies that I used to use. I took a picture of how I used to make sure students could solve special right triangles, just as a reminder of how far we’ve come.

In our lesson on 45°-45°-90° triangles, students use the Pythagorean Theorem to **look for regularity in repeated reasoning**, **reason abstractly and quantitatively**, and **look for and make use of structure**.

Students did well recognizing that a 45°-45°-90° triangle is half of a square divided by a diagonal. And they did well on the formative assessment Quick Polls that I sent to assess their progress.

Note: These two questions were not on the same side of the page on the student handout.

But something happened when I asked them to calculate the perimeter of a non-familiar polygon.

I monitored their progress after I sent the poll. After two minutes, I saw the following results. 4 students had a correct response, and 15 students had an incorrect response.

After another minute, 5 students had a correct response, and 21 students had an incorrect response.

I stopped the poll.

What would you do next?

I unchecked “Show Correct Answer” when I showed the results. And I asked a student who got 15 to explain his thinking (**construct a viable argument**). He counted 15 “pieces of segments” in the figure. Then I asked the class to **critique his reasoning**. Is every piece congruent? Another student asked to come to the board so that she could show how she used the practice **look for and make use of structure**.

Of course everyone understood the mistake in measuring after the second student drew the auxiliary line. But I wonder why more students didn’t connect what we had been learning to this diagram on their own?

What would have happened if we had started class with the perimeter of the polygon? Whether I had asked them to answer it then or not, would it have made a different in what they saw later?

Not unrelated, I recently asked my 9-year old daughter to load the dishwasher. Several hours later, I opened the dishwasher and found a big surprise.

I’m not sure if you can tell or not, but among other problems, the coffee mugs still have on their lids. It’s hard to believe that AKW has ever unloaded the dishwasher, much less on many occasions.

We talked about how the dishwasher works – and why we should turn dishes towards the water. This was take 2, a week later.

I’m still a bit flabbergasted at how difficult learning how to load the dishwasher has been without direct instruction. But I wonder what would have happened if I had asked her a different question to get her to think about how the dishwasher works. I wonder what would have happened if I had specifically asked her what she noticed when she was unloading the dishwasher, causing her to **look for regularity in repeated reasoning**. My daughter is still learning how to think and problem solve.

As all learners are.

And I am still learning the questions to ask, to promote thinking and problem solving, to uncover misconceptions.

As all Learners are.

And so the journey continues …

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Special Right Triangles”