This performance task is hard. So hard, that it seemed unwise to have our students try it. It requires understanding of the Triangle Sum Theorem, Isosceles Triangle Base Angle Theorem (not necessary, but helpful), ability to utilize overlapping triangles, assumed knowledge for the definition of a rectangle, and most importantly, the Pythagorean Theorem. The thing is, I was sucked into solving it, all with trying to use only the knowledge of my 7th graders. It took me a couple of days, not working straight, but doing it when I could.
I gave them some "given" information based on definition of a rectangle and understanding that a window used in traditional architecture would be symmetric. I didn't say anything about my concerns that they may not have the ability to solve it - who knows - they may just find a way that I had not imagined. They started in class on Wednesday, in teams of four and were told that it was not homework, but would be completed in class on Thursday.
I decided it was a worthwhile task, because 1) worst case scenario, it was too hard, making students want to give up which would allow us to talk about the importance of struggling and using what you DO know (this did happen to a few kids). 2) Students would realize that they could figure out most of the measurements, but not all (this was my second hour class), or 3) Someone might actually know the Pythagorean Theorem and use it to solve for the height and therefore solving the problem (this was my first hour class).
It turns out, the results of this task were amazing. My first hour was sucked into this problem, working in tight huddles, discussing, challenging, questioning, grabbing supply boxes for highlighters, etc. Students were communicating with others, not in their own teams. The energy was high and the kids were not even aware of the amazing math they were doing. During this class, I was asked if a diamond contained 360 degrees. When I asked why/how?, it was explained that they noticed the diamond could be divided into two triangles and they knew each of those had a sum of 180 degrees. I was asked if an imaginary line could be drawn to split the window horizontally since that would create a bunch of equilateral triangles. I was asked if that "A squared B squared thing" could be used to solve for the height. That question was followed with "is that only useful for right triangles?" The last question was, "how do you "unsquare" a number?"
My second hour class was not quite as dramatically sucked into this performance task. They also did not have a key member who not only knew the Pythagorean Theorem, but also of how to use it. However, nearly every team of four had disagreements/discussions about the rectangle being a square and ultimately realizing that it could not be one. Students grabbed patty paper to show this, as well as rulers- to measure the hypotenuse, proving it could not be the same length as the leg of the same right triangle. Every group in this class had moments of enlightenment realizing that overlapping triangles were important as well as properties of complementary angles.
I gave them some "given" information based on definition of a rectangle and understanding that a window used in traditional architecture would be symmetric. I didn't say anything about my concerns that they may not have the ability to solve it - who knows - they may just find a way that I had not imagined. They started in class on Wednesday, in teams of four and were told that it was not homework, but would be completed in class on Thursday.
I decided it was a worthwhile task, because 1) worst case scenario, it was too hard, making students want to give up which would allow us to talk about the importance of struggling and using what you DO know (this did happen to a few kids). 2) Students would realize that they could figure out most of the measurements, but not all (this was my second hour class), or 3) Someone might actually know the Pythagorean Theorem and use it to solve for the height and therefore solving the problem (this was my first hour class).
It turns out, the results of this task were amazing. My first hour was sucked into this problem, working in tight huddles, discussing, challenging, questioning, grabbing supply boxes for highlighters, etc. Students were communicating with others, not in their own teams. The energy was high and the kids were not even aware of the amazing math they were doing. During this class, I was asked if a diamond contained 360 degrees. When I asked why/how?, it was explained that they noticed the diamond could be divided into two triangles and they knew each of those had a sum of 180 degrees. I was asked if an imaginary line could be drawn to split the window horizontally since that would create a bunch of equilateral triangles. I was asked if that "A squared B squared thing" could be used to solve for the height. That question was followed with "is that only useful for right triangles?" The last question was, "how do you "unsquare" a number?"
My second hour class was not quite as dramatically sucked into this performance task. They also did not have a key member who not only knew the Pythagorean Theorem, but also of how to use it. However, nearly every team of four had disagreements/discussions about the rectangle being a square and ultimately realizing that it could not be one. Students grabbed patty paper to show this, as well as rulers- to measure the hypotenuse, proving it could not be the same length as the leg of the same right triangle. Every group in this class had moments of enlightenment realizing that overlapping triangles were important as well as properties of complementary angles.
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