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velvetpageThe grade five curriculum says that students are only supposed to measure angles up to 90 degrees this year. In grade six it's 180 degrees, and by grade seven it's the whole circle.
Of all the ridiculous and counter-productive divisions of concepts they could have made, this one has to be near the top of the list.
My kids were having difficulty understanding exactly what it was they were measuring with a protractor, and understanding the units was even worse. So I drew a co-ordinate grid on an overhead (no numbers, just the lines.) They told me it was a t-chart, it was an X, it was four triangles - that's when I stopped them. Where are the triangles? So one kid came up and drew the lines that connected the four points of the grid into a square, bisected diagonally twice. We measured the angles in the middle and the angles on the outside and discovered they were ninety degrees and added up to 360 degrees.
Then we talked a bit about the Babylonians, who came up with a number system with a base of sixty. They figured out that they needed a way to divide up a circle into even fractions. Since they did things in base 60, when they divided a circle their first instinct was to divide it into six parts, each of which was 60 degrees, and the 360 degree circle was born. We talked about why you might need to measure angles, and how that was used. I gave them the vocabulary words they needed: bisect, radius, diameter. At this point I'm well into the grade seven geometry curriculum, and they're eating it up.
Then I pointed out that you can draw a triangle in a circle by going through the centre point of the circle, using the radius as sides. We measured the inside angle of that triangle and found it to be 58 degrees. With one more line, I showed them the complementary triangle: the one made with the 122 degree angle.
They spent the rest of the period exploring triangles as fractions of circles and investigating the number of degrees involved. I have three different kids who realized that the sum of the outside angles would be greater than 360 because they were measuring a radius but also a line that went through two points on the circumference. I had a fourth kid who realized that if she drew the X so that the angles weren't all 90 degrees, she could figure out the measure of all the other angles if she knew the measure of any one, by doubling it for the opposite angle, subtracting that number from 360, and then halving the result. (Convoluted but it works just as well as figuring it out using 180 for the straight angle. Efficiency will come later; right now I want comprehension and wonder.)
I'm going to get those four kids to present their findings to the class tomorrow so the rest of the class can be exposed to those ideas and play around with them themselves. So basically, I can give a B to any kid who can measure an angle less than 90 degrees accurately, and an A to those who can explain how angles are related to circles and why triangles add up to 180 degrees.
I love it when lessons become kid-directed explorations of important mathematical ideas.
Of all the ridiculous and counter-productive divisions of concepts they could have made, this one has to be near the top of the list.
My kids were having difficulty understanding exactly what it was they were measuring with a protractor, and understanding the units was even worse. So I drew a co-ordinate grid on an overhead (no numbers, just the lines.) They told me it was a t-chart, it was an X, it was four triangles - that's when I stopped them. Where are the triangles? So one kid came up and drew the lines that connected the four points of the grid into a square, bisected diagonally twice. We measured the angles in the middle and the angles on the outside and discovered they were ninety degrees and added up to 360 degrees.
Then we talked a bit about the Babylonians, who came up with a number system with a base of sixty. They figured out that they needed a way to divide up a circle into even fractions. Since they did things in base 60, when they divided a circle their first instinct was to divide it into six parts, each of which was 60 degrees, and the 360 degree circle was born. We talked about why you might need to measure angles, and how that was used. I gave them the vocabulary words they needed: bisect, radius, diameter. At this point I'm well into the grade seven geometry curriculum, and they're eating it up.
Then I pointed out that you can draw a triangle in a circle by going through the centre point of the circle, using the radius as sides. We measured the inside angle of that triangle and found it to be 58 degrees. With one more line, I showed them the complementary triangle: the one made with the 122 degree angle.
They spent the rest of the period exploring triangles as fractions of circles and investigating the number of degrees involved. I have three different kids who realized that the sum of the outside angles would be greater than 360 because they were measuring a radius but also a line that went through two points on the circumference. I had a fourth kid who realized that if she drew the X so that the angles weren't all 90 degrees, she could figure out the measure of all the other angles if she knew the measure of any one, by doubling it for the opposite angle, subtracting that number from 360, and then halving the result. (Convoluted but it works just as well as figuring it out using 180 for the straight angle. Efficiency will come later; right now I want comprehension and wonder.)
I'm going to get those four kids to present their findings to the class tomorrow so the rest of the class can be exposed to those ideas and play around with them themselves. So basically, I can give a B to any kid who can measure an angle less than 90 degrees accurately, and an A to those who can explain how angles are related to circles and why triangles add up to 180 degrees.
I love it when lessons become kid-directed explorations of important mathematical ideas.