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This is a copy of Class Activity 13Q on pages 457 459 of ActivitesManual to Accompany Mathematics for Elementary Teachers

(first edition)by Sybilla Beckmann, copyright Pearson Education, 2005.

Class Activity 0A: Sums of Squares

This activity is related to Class Activity 13Q on explaining why there is afactor of 1

3in the volume formula for pyramids and cones. In that activity,

3 oblique pyramids were put together to make a cube. In this activity, youwill see that 3 oblique step pyramids fit together to make a rectangularprismexcept that there is a gap in the prism. By describing the steppyramids and the gap, you can derive a formula for a sum of squares.

Before you start this activity, you may want to review how the 3 obliquepyramids of Class Activity 13Q fit together to make a cube. Observe thatthe points of the 3 pyramids come together.

1. (a) Using blocks in 3 different colors, make 3 oblique step pyramids,as pictured in Figure 1, each of a different color. Your step pyra-mids should have 2 layers: the bottom layer made of 22 = 4 blocksand the top layer made of 1 block.

Figure 1: 3 Oblique Step Pyramids

(b) Put your 3 step pyramids together, so that their points (toplayers) come together (just like the three oblique pyramids in ClassActivity ??), and so that together, they almost form a rectangular

prism.

Observe that if you add 1 block in the middle and 2 blocks atan edge, you will get a rectangular prism that is 3 blocks wide, 3blocks deep, and 2 blocks tall.

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(c) Explain how part (b) gives rise to the following equation:

3

(1 + 22) = 3

3

2

(1 + 2)

2. (a) Using blocks in 3 different colors, make 3 oblique step pyramids,as pictured in Figure 2, each of a different color. This time yourstep pyramids should have 3 layers: the bottom layer made of32 = 9 blocks, the middle layer made of 22 = 4 blocks, and thetop layer made of 1 block.

Figure 2: 3 Oblique Step Pyramids

(b) Put your 3 step pyramids together, so that their points (toplayers) come together (just like in part (b) above, and just likethe three oblique pyramids in Class Activity ??), and so thattogether, they almost form a rectangular prism. It might help tosecure your step pyramids with masking tape.

Observe that if you add 1 block, and another 2 blocks in themiddle, and 3 blocks on the edge, you will get a rectangular prismthat is 4 blocks wide, 4 blocks deep, and 3 blocks tall.

(c) Explain how part (b) gives rise to the following equation:

3 (1 + 22 + 32) = 4 4 3 (1 + 2 + 3)

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3. If you can, make 3 step pyramids that each have 4 layers: the bottom

layer made of 4

2

blocks, and the remaining layers made of 3

2

, 2

2

, and1 block/s, respectively. Put them together as before, and explain howthis give rise to the equation:

3 (1 + 22 + 32 + 42) = 5 5 4 (1 + 2 + 3 + 4)

4. Imagine that you could continue this process of putting larger andlarger step pyramids together indefinitely. Use your discoveries above,and the fact that

1 + 2 + 3 + . . . + N =N(N + 1)

2

to give a formula for

1 + 22 + 32 + . . . + N2

In other words, write an equation for which the left hand side is 1 +

22 + 32 + . . . + N2.

5. Based on your formula in (4), calculate

1 + 22 + 32 + . . . + 1002