The Pythagorean theorem. To many, a formula memorized in middle or high school used to calculate the side lengths of right triangles in order to answer questions on tests.
But an idea to play with? That is what 7th grade math teacher Liz Caffrey wants her students to do -- play with Pythagoras’ idea in order to have a better understanding of the concept. Liz’s 45-minute zoom lesson in mid-May follows a format familiar to her students from the time they were physically together. They begin with “Math from the world." Liz explains that, “today’s offering, in the spirit of fun and randomness, is from the New York Times’ puzzle mania.” Liz shares her screen and encourages the students to count how many squares they see.
As class verbally discusses where they see squares, students also type into the chat section:
Max: I don’t see any
Mira : I see them!!
Avery : Me too!!!
Rachel: me too!!
Camilo: I see the squares too
Sophia : OH MY
Sophia : I SEE THEM
Sophia : AND I CANT UNSEE THEM
As the chat unfolds, Liz highlights some of the squares on the screen. She draws the conversation to an end with a reference to The Hitchhikers Guide to the Galaxy: “There is a total of 42 total squares, which is the meaning of life, which is important for you to know.”
The next part of the class is a deep dive into the main topic of the day, a better understanding of the Pythagorean Theorem and its proofs. Liz reviews the class’s previous explorations where they had “discovered” that the area of figures created from sides of a right triangle allows for calculations of the length of sides of those triangles.
Picking up on the previous day’s conversation, Liz then introduces a game called Is it a Triangle. She asks, “Will these three sides make a triangle: three, four and five? Write yes or no in the chat.” At first, there is little response from the students so Liz adds:
Ten seconds to stake your claim. I really recommend you do stake your claim because then you are really on the hook for the answer. Once you put yourself out there your brain is primed to learn something. If you are averse to the risk you’re less engaged. Type in an answer. It’s OK to be wrong. I won’t even remember what you wrote.
Most children type yes. Liz then asks, “How about three, four and ninety-nine?” There is laughter, some very adamant noes in the chat (e.g., Rosa: N!!!). Camilo comments that it is “a triangle wannabe.”
After some further explanation, it is time to play. Liz explains that in small groups they will be exploring different geometric proofs of the theorem which can be manipulated on the computer screen. Each proof has a different right triangle whose sides are part of another shape—in some cases triangles and in other cases squares or hexagons (see diagram below). The shapes that border sides a and b of the triangle are split into colored “puzzle pieces” that can be moved into the shape bordering side c. That the pieces fit neatly into the third shape is proof that a2 (the area of the first shape) plus b2 (the area of the third shape) is equal to c2 (the area of the third shape).
Liz asks the students to play with the proofs and share what they are noticing.
In breakout room three, there is some initial frustration on how the proofs work. After a few minutes, the students have sorted this out and the following conversation unfolds:
Rosa: I like this second one
Hannah: They all fit together, nice.
Rachel: This is fun
Hannah: The second one is pretty
Rosa: So is the fourth one
Rachel: Are we discussing what we notice?
Rosa: Yeah, I think
Rachel: Okay, what do you guys notice?
Hannah: I notice that we never had to rotate
Rachel: You can change the shape on this
Hannah: What did you say Rachel?
Rachel: That is cool
Hannah: Woo, ten is crazy
Rachel: Woo, seven is crazy
Rosa: Woo, seven is really pretty
Rachel: Ten is super cool
Hannah: Woooooooo. This is cool
Rachel: Ten is super cool. What is it showing?
Rosa: There is kind of a pattern. One of the squares is broken into four triangles and another square
goes in the center.
Rachel: Can I share my screen for a minute?
Rosa: Sure, sure.
Rachel: If you do this it will make two squares
Rosa: Oh, that’s cool
Hannah: My dad is blasting Twist and Shout downstairs and I don’t know why
Rachel: You can make it into two squares and I don’t know why.
As students return from the breakout rooms Liz asks, “how was that for an amount of playtime? Too much? Not enough? Just right goldilocks amount?” A few call out yes, while there are a couple of noes recorded in the chat. Liz will take this input into consideration for the next session.
To end class, Liz walks the students through an extension of the Pythagorean Theorem, a proof involving hexagons rather than squares. It turns out that the sums of the areas of any shape built off of the two legs of a right triangle will always equal the area of the same shape built off of the hypotenuse! Liz adds that the proofs they explored were visual proofs; the next day they will do an algebraic proof together, which is “both rewarding and challenging.”
During distance learning, it was important to Liz to preserve as much of a sense of normalcy to her class as possible. So she maintained the routines of class over Zoom, and kept math learning playful and exploratory. As distance learning “fatigue” set in after spring break, Liz changed the warmups to make them lighter and more fun, altering the theme each week to promote student enjoyment and also to gauge how students were feeling that day before diving into math. A surprising side effect of distance learning was how much students utilized the chat feature to share their thinking, allowing quieter students to have more of a voice in class. A particularly exciting achievement during distance learning was that Liz and her middle school team were able to make their month-long interdisciplinary, end of year project go entirely online. Students were able to use collaborative whiteboards, simulators, and 3D modeling software to make scale models of community centers situated in different areas of the world.
Liz Caffrey is the middle school math teacher at Atrium. She will be sharing more about her teaching practices, in particular how she creates projects that connect math and social justice, at Atrium’s summer math institute, July 20-24. Come play!