Math Detours · Grade 5 · Connections · 7

Building with blocks

How many blocks to build a wall, a tower, a hollow house? Volume, layers, and counting what you cannot see.

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Tonight's idea

You are building with cube blocks, on the floor or in a game. How many will you need? Counting one by one is slow. In a finished building many blocks are hidden, too.

A wall. This wall is 8 blocks long and 5 high. Each row holds 8 blocks and there are 5 rows. So it needs \(8 \times 5 = 40\) blocks.

8 long 5 wall: 8 × 5 = 40 empty one layer: 16 − 4 = 12 4 by 4, seen from above
A wall from the front, and one layer of a hollow tower from above.

A hollow tower. Look down on one layer. A full 4 by 4 square would be 16 blocks. The middle 2 by 2 is empty, so take away 4. One layer is \(16 - 4 = 12\) blocks. A tower 10 layers high needs \(12 \times 10 = 120\).

Count one layer, then multiply by the number of layers. For a hollow shape: full − empty.

Watch one: a small house

A house is 6 blocks long and 5 wide on the outside. Its walls are one block thick and 4 blocks high.

A full layer is \(6 \times 5 = 30\). There is a wall at both ends, so the inside is 2 shorter each way. That is 4 by 3, or 12. A layer of wall is \(30 - 12 = 18\) blocks. Four layers are 72.

Take out 2 blocks for a door: 70. Add a flat roof, 6 by 5: \(70 + 30 = 100\) blocks.

Your turn

  1. A wall is 12 blocks long, 6 blocks high and one block thick. (a) How many blocks does it need? (b) You leave a window 3 blocks wide and 2 high. How many blocks now?
  2. Each layer of a hollow tower is a 5 by 5 square with the middle 3 by 3 left empty. (a) How many blocks are in one layer? (b) How many blocks build a tower 9 layers high?
  3. From the side, a staircase is five columns of 1, 2, 3, 4 and 5 blocks. It is 3 blocks wide. (a) How many blocks does it need? (b) Turn a second staircase upside down and fit it on top of the first. What size of solid box do they make? Use the box to check (a).
    side view,3 blocks wide
  4. A treasure chest is 6 blocks long, 4 wide and 3 high on the outside. It is hollow. Its walls, floor and lid are all one block thick. (a) What size is the empty space inside? (b) How many blocks make the chest?

Stretch: a pyramid

A pyramid is built in square layers. The top layer is 1 block. The next is 2 by 2, then 3 by 3, and so on down. (a) How many blocks make a pyramid of 6 layers? (b) You have 200 blocks. How many layers can you finish?

Answers

  1. (a) \(12 \times 6 = 72\). (b) The window is \(3 \times 2 = 6\) blocks, so \(72 - 6 = 66\).
  2. (a) Full is \(5 \times 5 = 25\), empty is \(3 \times 3 = 9\), so \(25 - 9 = 16\). (b) \(16 \times 9 = 144\).
  3. (a) The side view shows \(1 + 2 + 3 + 4 + 5 = 15\) blocks. The staircase is 3 wide, so \(15 \times 3 = 45\). (b) Each column becomes 6 high: 1 + 5, 2 + 4, and so on. The box is 5 long, 6 high and 3 wide, which is \(5 \times 6 \times 3 = 90\) blocks. Half of 90 is 45, which agrees.
  4. (a) Take one block off each end, in every direction. \(6 - 2 = 4\), \(4 - 2 = 2\) and \(3 - 2 = 1\). The space is 4 by 2 by 1, which is 8 blocks. (b) Full is \(6 \times 4 \times 3 = 72\). So the chest needs \(72 - 8 = 64\) blocks.

Stretch. (a) \(1 + 4 + 9 + 16 + 25 + 36 = 91\) blocks. (b) A seventh layer is 7 by 7, which is 49 more, making 140. An eighth layer needs 64 more, making 204. That is over 200. So you can finish 7 layers, with 60 blocks left over.

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