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Hi! In the absence of a Geometry Progression, I have been reading Wu’s Geometry Progression (https://math.berkeley.edu/~wu/Progressions_Geometry.pdf) and I have a question about an interpretation he takes which is not evident in the standards. I am curious if the highly anticipated Geometry Progression will take this position and if this position is the intended reading of 7.G.4.
On page 55, Wu defines pi as the area of the unit circle. He does this in order to lead into the relationship between the circumference and area of a circle and derive the formula for the area of the circle. Shifting the definition from pi as the ratio of the circumference to the diameter to the area of the unit circle will be a big task. Do you think this is a worthwhile battle, in the grand scheme of all things CCSS which must be shifted to new understandings? Is it a necessary shift for the high school standards but not the middle school standards? I guess, in general, I am curious about your thoughts about defining pi as the area of the unit of a unit circle.
From here, Wu takes a polygon and decomposes it into triangles from the center of the polygon (pg 56-59). He creates a general formula for the area of a polygon based on the area of each triangle. Then he does an informal limit, as the polygon increases the number of sides, it gets more circular. Therefore the formula created can be used to find the relationship between the area and circumference of a circle and then he continues this logic to find the standard formula for the area of a circle. My questions are:
1) Is this line of reasoning, decomposing polygons and informal limits, the intended line of reasoning for the part of the standard asking for the informal derivation of the relationship between the circumference and area of a circle?
2) Should students be deriving the formula for the area of a circle in 7th grade? The standard only says “know and use” so I was wondering if this interpretation is an extension of the intent of this standard.
3) Would this explanation also work for (or be more appropriate for) the high school G.GMD.1 standard? Is this the intent of that standard?Finally, any word on when we will get an official Geometry Progression? 🙂
Thanks!
Hi! I am just following up and wondering if anyone has any thoughts on my question. I am planning for some Geometry PD this summer and would like to know if these perspectives are above the scope of the standards.
Thanks!
SarahSorry it is taken me so long to reply to this. First, there is no official definition of π in the standards, so no, you don’t have to take Wu’s approach. I quite like it myself but I have talked to others who disagree. Mathematically, you can do it either way; the miracle is that the two numbers (area of unit circle and constant of proportionality between circumference and diameter) turn out to be the same.
As to your other questions, I think Wu’s limit argument is a bit too much for Grade 7. I would use the argument that rearranges the triangles into an approximate rectangle with length equal to half the diameter and height equal to the radius if I were going to give any argument at all. From this you can get that the area is 1/2 the product of the circumference and the diameter, A = 1/2 Cr. As you point out, the standards do not technically require that you justify the individual formulas C = πd (or C = 2πr) and A = πr^2. But you are almost there at this point. If you have defined π as a constant of proportionality, you may have given an informal justification of why that constant of proportionality exists. Doing so would amount to a justification of the formula for the circumference. And once you have that you can get the area formula by substituting the circumference formula into A = 1/2 Cr.
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