Pythagorean Theorem Student Data

Every few months, I meet with other beginning math and science teachers to discuss teaching and what it looks like for us (though the Knowles Science Teaching Fellowship). Recently, this has meant collecting data on our classrooms, presenting it to two other fellows, then discussing it.

The lesson I discussed was an exploration of the Pythagorean Theorem. Students read a reading guide and used Pythagorean Tangrams to see if the two smaller area squares added up to the bigger area square. (photos below)

Part of our assignment was to think about the Standards for Mathematical Practices, which, admittedly, I didn’t.

A couple of thoughts:

  • Students sometimes got stuck when trying to put together the two smaller squares to make the big square and I’d have to show them a hint or first step.This felt useful in terms of keeping the kiddos moving and not letting them get stuck on anything that was probably not hugely important (though if it wasn’t hugely important…) In general, I think I need to try and push students to struggle with math without giving up before asking for help. It feels like such a fine line between getting them engaged and than letting them be independent.
  • We also established that the students may not have understood the goal of the activity. Do they fully understand what it means when they make the two smaller squares equal the big square? Do they think that putting together the two smaller squares means that this will work for all squares?
  • It’s interesting to see how students explain and justify. One student says “I don’t know how to explain” which is frustrating because they probably could explain, but at least shows that they know they need to explain. It’s also interesting to see how students explain something tricky like “why do we need to write the small 2.” (This maybe feels like a “guess what the teacher is thinking” question)
  • In thinking about standards for mathematical practices, I go back and forth about which ones are “most important” (they’re all important, which makes it difficult for me to try and focus). So I might try and get our department (four people) to pick a math practice to focus on across courses next year. We’ll see.

Photos 1 and 2: Student AStudent A - Page 1

Student A - Page 2

Photos 3 and 4: Student BStudent B - Page 1Student B  - Page 2Photos 5 and 6: Student CStudent C - Page 1Student C  - Page 2

Day 151: The One Where We Pronounce Pythagorean

Spent today doing a reading guide on the Pythagorean Theorem. Reading guides are a structure at our school where students read together. In most classes, they learn and practice common reading strategies like visualizing, making predictions and inferences, etc. In math, they do these things as well though we often walk them through math problems as we go.

I’ve felt very up and down about reading guides this year. For me, they are usually linked to a participation quiz where I narrate (and give points) for the positive behaviors students do (today it was: work in the middle, point to what you’re talking about, connect the area of the square to the side of the square). Up until recently, I’ve felt that I wasn’t able to intervene if students needed help (an ongoing struggle on my part), though I’m beginning to get a better sense of when to stand back and when to quickly step in. (I think. Knock on wood.)

Photo One: The Reading Guide and Tangramsreading about the Pythagorean TheoremThe part of me that signs all my emails “The Worst” (ie “Sorry I haven’t email back #ImTheWorst” or “Sorry I totally Second Year Teacher’ed you when I flaked out on Friday #ImTheWorst”) will also confess to not having read the Common Core progressions in-depth* (though make no mistake, I’m fond of them). So I’m excited that we actually talked a bit about how to show that the sum of the area of the squares of the legs is equal to the square of the hypotenuse (we did not at all describe it in those words). We had students cut out tangrams of the two small squares and try to put them together over the big square. Most students were able to accomplish this, with a hint or two. (In the end, a bit of struggling seemed important, though it also felt useful to show students the result if they hadn’t discovered it) Incidentally, the two constructed squares in this photo were made by two friends in different classes (One student built the pattern in the morning, which I showed to the student I have in the afternoon when they started to struggle. They seemed impressed).

Photo 2: Pythagorean PronunciationPythagorean PronunciationCurriculum partner and I occasionally talk about words that are hard to pronounce. Trigonometry (which we never even taught – we just left it at sine, cosine and tangent) would have been tough, parallelogram is tough, Pythagorean Theorem (let alone “theorem”) is tough. That being said, kiddos have been super down to try, including this student who took notes when I wasn’t looking.

Also, what has two thumbs and is rockin’ out in the kitchen to Billy Joel? Certainly not this guy…

*Just kidding, the ones for middle school geometry, where Pythagorean Theorem should be haven’t been written yet. Told you I hadn’t read them in depth yet.

Day 150: The One with Triangles and Squares

New week, new unit. Pythagorean Theorem is usually taught in middle school, but (again), it’s not a given that our students have learned it, so here we are. It also builds nicely on what we did with right triangle trigonometry.

I should probably change seats today, but I feel like it takes the kiddos a bit to warm up, so I’m leaving them in the same seats for now.

Which probably means we won’t change seats again.

Because there are two weeks of content left to go.

Photo 1: The Recording SheetTriangles and squares activityThe idea behind today’s activity is that students use squares to make right triangles. Our first set of squares was too small and students build several right triangles that looked correct (4-5-6-nope), but weren’t. We spent 2nd period making the squares bigger and removing some of the confusing ones. One class tried to cut up the squares into smaller squares, but otherwise, this helped.

This photo is from a group that worked steadily throughout the whole period. Most kiddos made the connection between the area of the hypotenuse square (as we’re calling it) and the sum of the area of squares 1 and 2.

Photo 2: Explaining Complex Area

Day 150 - Complex AreaWith both of these photos (and most of the photos I post here), I wish I could actually capture the groupwork that is happening. Kiddos who finished the triangles and squares activity worked on a practice area worksheet. We haven’t touched much on complex area, so these kiddos had to struggle their way through it (which is difficult and good at the same time). One of the high status students really struggled with this problem and a student, who would be considered a low status math student, saw how to adjust the height of the rectangle and tried to explain it. So cool to see. This photo doesn’t really do it justice.