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Solution Page 156:
My son's drawing of one of the cubes is different than the one your answer depicts (the left drawing in the second row on the solution page). Your top line is a straight line which he says only gives that face three edges. He says each face must have four edges. His line goes from the same left and right vertexes as your line does, but he drew those to the vertex of the light grey lines (top center above your straight line.) This gives four edges. Is he correct?
Another one of his cubes (depicted by the right drawing in the second row) shows a sliver of a face on the left, thereby demonstrating three faces. Your drawing shows a straight line on the left, thereby demonstrating only two faces total. Again he drew his line to the vertex of the light grey lines on the left. I cannot erase what he drew to see the light grey lines as it will erase them as well (plus he used colored pencils). He said your drawing cannot demonstrate the sliver of a face because the thickness of your line obscures it. Anyway, is he correct?
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I also have two questions from Lesson 156.
Question 6 asks you to find the volume of the cube that fits inside the dodecahedron. It says that if you can find angle b, you can use trig (or the Pythagorean theorem) to find line segment RS...
My son figured out the answer using trig without any trouble, but we don't see how you can use the Pythagorean Theorem unless you know two sides. We know RQ is 1 dm. We're trying to find line segment RS. But we don't see how to find line segment QS (unless we use trig: sin angle R (36 degrees) = QS / 1 dm). In that case, we may as well just use the sin of angle b.
If we're supposed to be able to use the Pythagorean Theorem, we don't see how (but we'd like to!).
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Lesson 156 Re: Section labeled Worksheet:
The first sentence asks, "Can more than one cube fit in the dodecahedron?
The third sentence states, "That's the number of different cubes that will fit in the dodecahedron."
When my son was explaining this lesson to me, I read these sentences differently than he did. I take it to mean that more than one cube might fit inside the dodecahedron, not how many "ways" the cube can fit inside the docecahedron which is what you mean, I believe.
Evidently, my son also looked at the diagram and knew exactly what you meant. He proceeded to complete the worksheet by drawing five different "orientations" of one cube inside the dodecahedron. He had to explain what you meant to me.
I can now see that what he is saying is correct, but I wanted to suggest to you a slightly different wording in the lesson for these two sentences (see below).
"Can a cube fit more than one way inside the dodecahedron?"
"That's the number of different ways a cube will fit inside the dodecahedron."
I assume everyone knows that the dodecahedron is to remain stationary as you are looking at the five different cube orientations. When I read the rest of the sentences in that paragraph it did help to clear up for me what you were saying (that you only want one cube per dodecahedron and not to repeat cubes), but it took me some time to reconcile because I got hung up on those first two sentences.
Thanks,
Julia
Aren't you glad that we only have nine lessons left!
