Before I begin going through this problem I have selected, I want to link you to where the PDF’s of the documents can be found. Notice these are NOT the problem sets, these are activities that ARE used in class, but also are pulled together for teacher use at the Phillips Exeter Academy Summer Math Institute. Just so we are all on the same page on where these are coming from.
Also, at the end of this series of posts (and I have mapped out at least FIVE more of them) I will post these resources in WORD format instead of PDF found on Exeter’s website. I asked the instructor and he gave me permission to post them as long as credit is given to Exeter. Have I said how much I like the Exeter Academy, their curriculum and how much respect I have for their willingness to share?
Okay, so let’s jump into the problem. Before I begin, I must say that we spent 3 days working this problem I am presenting here. We did a bit of it one day, a bit more another, and finished it as a third. We did parts as a starter problem, parts as an activity and parts as a “could we do this with this problem?” extension. This translates very well to the high school curriculum because it allows for a stepped pacing, starts off slowly with multiple entry points for all learners to accomplish, and them moves them slowly up the ladder to high levels. Oh, it is essentially the Common Core.
I am going to put this below the fold because it gets long. No, really, lots of pictures, and it is long. It is completely worth the read though, and you will see how this fits into algebra and geometry.
Let us begin: [This is found in the Algebra 1A Activities, #8, Geoboard Pythagorean Activity] On your geoboard (because you have your geoboard right there, okay draw it on paper if you don’t have a geoboard.) make the following squares.
Got it? Don’t worry, the activity tells you how even. Not hard, it is stepping you through the necessary requirements. Area of SQUR – 4(triangle area HRT) = MATH area. cool. Do 2 more with different sizes just to make sure you got it. Are you good? Great.
Now that you think you are all cool with your knowledge of find areas of square and triangles, find the area of the three squares in the picture to the left.
Now do it 3 more times with different sized triangle PUT. Record your answers in a table.
What do you notice?
[I have to stop here and do a shout out to Max Ray, @maxmathforum. At #TMC12 he did a session on asking “What do you notice” instead of “What do you know” about the figures? Guess what question EVERY time the Exeter instructor asked? Yup. “What do you notice?” Hmmm, good teaching, good pedagogy for private schools = good teaching and pedagogy for public schools too. Funny how that works.]
Now, clearly this is an exercise / activity designed to get the learner in the end to say the following sentence and understand the implications of the sentence well: “The sum of the square of the two legs of a right triangle equals the square of the hypotenuse of the right triangle.” Later, perhaps in a day or two (because I asked when the name “Pythagorean Theorem” is used) the teacher would actually give a name to this sentence. NOTICE, the Pythagorean Theorem is NOT “a squared + b squared = c squared.”
Okay, we are in good shape, that is a great exercise. It might take a period or event two to accomplish. the next day, on the starter question, this might be it. Find the area of ABC.
Piece of cake, right? Draw a square, find area of square, find area of 3 triangles, subtract. Come on, challenge me.
Fine, find the area of this triangle then:
Hmm, now we have to decompose it to a rectangle, 2 triangles and a smaller rectangle and do the subtraction. Okay, you made me think a bit, but clearly it wasn’t that hard to figure out once we did all the previous problems.
Then, if you think you are so smart, can you find the are of this:
To which I say, bring it on. Now it is just a rectangle minus some triangles. Again, no big deal.
Notice that each step of this series is a small step up in difficulty, but not a huge step. It is leading the learner through a series of decompositions in a way that challenges, yet does not overwhelm.
Still, not too difficult, just some steps (and if you think of symmetry, it gets down right easy, just a hint.) Okay, we think we are pretty smart by now, so let’s challenge the learners to do a throw down on this problem. Find the area of the following parallelogram:
BooYa! Now we are in some co-ordinate geometry! But, it works out not too badly. Do EXACTLY the same thing we did in the last problem with algebra instead of numbers and we get:
[If you aren’t math person, you are screaming at me now because all along I have jumped over all the math. Sorry, but I really am trying to get some pedagogy and thinking steps out here, not the math.]
That is some sweet stuff, but let’s extend it just one little bit more:
Now we have 2 vectors, that make up the parallelogram. Can we use vectors to find the area of the shape? Now this is where things diverge, because Exeter uses vectors heavily in their curriculum. It is one thing that makes math so so so so much easier if you use vectors for geometry. Guess what the answer is: magnitude of vector ab times magnitude of vector cd.
Think that is crazy? Go back to figure 4 and put vectors on it. Guess how you find the length of the vectors? ad – bc. Kind of nice, huh.
Guess what? If you add a third dimension onto the vector picture, guess what the formula for the volume of the three dimensional solid is.
Volume is mag(ab) times mag(cd) times mag(ef). It requires NOTHING to go from 2 dimensions to 3 dimensions.
Now, let’s talk pedagogy for a second.
Can this be done in a high school algebra 1 class? yes. Without question. Can it be done in Geometry class? Absolutely. Do the vectors create a difficulty? Yes, but why aren’t we teaching vectors? They make things so darn easy to model. (Oh, and modeling is part of Common Core.)
Does it require some modeling and demonstrating, some group work to think through, some coaching by the teacher? Without question. But guess what? It requires that at Exeter too! Their activity sets up and models entire pythagorean theorem and creates a situation where the learner understands it.
Then they name it after the understanding is built.
Then they extend it.
Why the heck are we not doing the same?????????? And see the pattern I discussed in my last post?
Compare that to these two high school links I found in 30 seconds of searching. I think this is pretty typical.
A quiz on radical equations and pythagorean theorem – I think I gave a quiz just like this my 1st year. sad.
This is ACTUALLY FROM GLENCOE! Yup, our wonderful textbook creators think this is extra practice that is needed. Contextless and boring as hell. Great stuff. Thank you Glencoe.
We all know Math Class Needs A Makeover. Here are FREE materials to create the makeover.
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