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Reasoning and ProofSession 04 Overviewtab aTab btab ctab dtab eReference
Part B

Exploring Reasoning and Proof
  Introduction | Try It Yourself: The Frosted Cube Cake | Problem: Math Banquet | Solution: Math Banquet | Verifying the Problem Algebraically | Problem Reflection | Your Journal

 
 

Since each teacher received an equal-size piece of cake, we know that the total number of unit cubes will be equal to the number of teachers at the math banquet. Therefore, we are looking for the cake where the number of pieces with no sides frosted is exactly 8 times the number of pieces with three sides frosted.


One way to solve this is to use the model and the table in the previous activity. Notice that the number of pieces of cake with three sides frosted is 8, no matter the size of the cake. Take a moment to look at the cake and see why this is so. Reviewing the information given in the problem, we know that the teachers found that the number of no-sides-frosted-pieces was 8 times the number of three-sides-frosted pieces. But we know that there are always 8 three-sides-frosted pieces on a cube cake. So the number of no-sides-frosted-pieces must be 8 times 8, or 64.


Now let's consider what that means. What are possible sizes for the"inside" of the cake (that is, all of the pieces with no sides frosted)? Like the complete cake, this is a cube, so it is some number multiplied by itself 3 times. Guess and check is one way to proceed. If we check 5 x 5 x 5, we find this equals 125, too big. But 4 x 4 x 4 = 64; that's the size of the "inside" of the cake.


But what about the complete cake? By going back to our model, we can see that to account for the frosting, we need additional faces on each side of the cake. That means each face is six cubes wide, and the total cake size is 6 x 6 x 6, which equals 216. The cake contains 216 total pieces, and, therefore, 216 math teachers were at the banquet. Notice that if we go back to the table, 216 is also equal to the total number of pieces in each column of the table for the 6 x 6 x 6 cake; in this case, 8 + 48 + 96 + 64.


This is one solution approach. As is usually the case, there are others.


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