Monday, October 29, 2012

Day 39: Practice with Squares

Today was spent recapping the conclusions we made last week and giving students time to practice with U3 WS1 - Basic Properties of Squares. 

The only "new" content discussed today was the relationship between the length of a square's diagonal and its side length. 

Student minds always surprise me - when I asked the class to try determine the length of a diagonal by counting boxes (like they'd done with the sides), I'd expected them to be stymied. "But Mr. Fuller, those boxes (corner to corner) aren't the same as the horizontal/vertical ones" I imagined them saying. Nope - they'd declare an answer - 5 (for example) - which always happened to be the same as the side length for the particular square we were looking at. And then it hit me: Because it's a square, the diagonals have a slope of 1 (or -1) and they were simply counting the spaces on the grid that the diagonal passed through, just like we did for the sides. Huh. 

So let's run with it: Projecting at image of our graph onto the screen (using Geometer's Sketchpad, so it looks precise), I can stand at the screen with a meter stick and have a volunteer who's close by read the length of a side. "57 cm" they'd say. Ok, now I'll pivot the meter stick at the vertex until it aligns with the diagonal. VERY clear that the 57 cm of the meter stick that's exposed doesn't reach the far corner of the square. Conclusion: Diagonal lengths are longer than horizontal/vertical lengths. But what's the relationship?

At this point, my original idea (back in July when I was still optimistic) was to have students "discover" the scale factor of sqrt(2), but I was running short on time and we needed to get going. Instead I worked at the computer and took suggestions (using GSP) for what mathematical operation might connect side length to diagonal length. This is why GSP is awesome - I can perform a calculation and leave the result on the screen while I alter the square to see if the result changes. I couldn't ask for a better visual for testing a mathematical hypothesis. 

Anyway, we try adding the side length to the diagonal and see if the sum is a constant. Nope. Subtraction? Nada. Multiplication? Still nothing. Division? Jackpot. Out pops 1.41 as this unshakable constant. 

Here's where I finally broke down and became "lecturer" for a moment. I asked: "If that number had hypothetically come out as 3.14, what would you think?" Students immediately jumped on Pi. Awesome. So I explained that there are a handful of very famous numbers in math that you'll start to recognize as you spend more time with math. So I lead them to the idea of sqrt(2) and thought we were done. 

Nope. The number of students that have NO IDEA how to work with exponents and radicals in a 10th grade geometry (honors or regular - it made no difference) class was astounding. Worse yet - there was no shame on the part of the students for not remembering the concept. Instead, they got frustrated with me that I wouldn't simply tell them the answer, insisting instead that they should be able to figure it out or ask a neighbor for help. 

Still a solid day. 

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