Here is the breakdown of the Common Core standard that has inspired this problem, CCSS.Math.Content.HSG.GMD:

CCSS.Math.Content.HSG.GMD.A.1

Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.

CCSS.Math.Content.HSG.GMD.A.2

Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

CCSS.Math.Content.HSG.GMD.A.3

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

**Extra Credit:**

(Follow-up to last week's Extra Credit question)

1. Will a student who has never heard the phrase "Cavalieri's principle" know how to proceed on this problem?

2. Should a student who has never heard the phrase "Cavalieri's principle" end up with fewer points on this problem than one who has?

3. Should a student who explains without reference to "Cavalieri's principle" why the two volumes are equal get full credit for this problem?

4. Is it acceptable to argue that the volumes are equal because they contain the same number of equal-volume disks?

5. To what extent does knowledge of "object permanence," typically attained in infancy, suffice for grasping why the two stacks built from the same number of equal-sized building blocks have equal volume?

6. To what degree does this problem test knowledge of labels rather than mastery of concepts?

## 4 comments:

I don't recall hearing of this Cavalieri guy, but I know how to roll quarters, so I'd get it right.

So much for deeper understanding. They don't want students to explain why they are equal, they want them to spew back memorized knowledge.

Object permanence doesn't make children grasp that a cup of water in a tall, thin glass is the same amount of water as in a short, wide glass, even if you pour it back and forth from one to another. That's something children figure out around kindergarten or so.

Good point, Anonymous. The Principle of Conservation (http://en.wikipedia.org/wiki/Conservation_(psychology)#Solid_quantity) is probably more applicable here than object permanence. Though it develops substantially later than object permanence, it's in place substantially earlier than 8th grade.

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