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The following questions could be used to help students make conjectures and justify their thinking as they work on The Frosted Cube Cake problem.
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Teacher: What pattern do you notice about the number of pieces with three sides frosted? How do you know this pattern will always be true?
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Student:
There will always be eight pieces with three sides frosted, because the corners of the cube are the pieces with three sides showing, and a cube has eight corners.
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Teacher: What pattern do you notice about the number of pieces with two sides frosted? How do you know this pattern will always be true?
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Student:
Each time the size of the cake increases by one, the number of these pieces increases by 12.
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Teacher (follow-up question): That will work if we continue to build the cake by ones. Can you find a pattern that will work for any size cake?
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Student:
The two-sided pieces are always along the edge of the large cube, except the corners. So the number on each edge would be the length of the edge, subtracting two for the corner pieces, or (n - 2). Since there are 12 edges on the cube, you would multiply 12 times (n - 2), and that would give you the number of pieces with two sides frosted.
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Teacher: How could I find the number of pieces with one side frosted for any size cake?
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Student:
The number of pieces with one side frosted is the number of pieces in the center of each face of the cake. That is a square with a side that is two squares smaller than the side of the cake or (n - 2). After I find the number of pieces in that square by multiplying (n - 2) by itself, I can take that answer and multiply it by 6.
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Student:
Because the original cube has six faces, and in the center of each face is a piece with one side frosted.
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Teacher: How could you find the number of pieces with no frosting for any size cake?
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Student:
The pieces with no frosting are hiding in the middle. They are really a smaller cube hiding inside the original cube. That cube would be two shorter than the cake since it is inside the top and the bottom. So the side of that cube would be (n - 2). I can find out how many unit cubes there are by taking (n - 2) and multiplying it by itself. That is one layer. I would then multiply it by itself one more time to get all of the layers.
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 Learn about connections to the other process standards
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