Lesson 14 of 15

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9 min

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Ten ideas, ten checks. The table comes first, so nothing has to be remembered cold.

Ten ideas for numbers built by squaring and by cubing
Idea	How it goes
Square number	Same count across as down. 49 tiles go 7 across, 7 down.
Odd staircase	Each new layer is the next odd number: 1, 3, 5, 7, and on.
Ruling out	Squares end in 0, 1, 4, 5, 6 or 9. Another ending rules it out.
Square root	The side the count came from. Check by multiplying back.
Pair the primes	Break, pair, keep one per pair, multiply. A lone prime means no whole side.
In between	Find the square below and the square above.
Cube number	A stack as deep and tall as it is wide. 3 × 3 × 3 = 27, written 3³.
Cube patterns	Cubes end in every digit. The n-th group of odd numbers adds up to n³.
Group in threes	Groups of three give the cube root. 2,744 gives 14.
Which count	Covering a square patch: side × side. Filling a cube-shaped space: side × side × side.

A block of 121 tiles has the same number across as down. How many are along one edge?

60, because that is about half of 121
11
12, because 12 is close to 121

A square 11 tiles across grows to 12 tiles across. How many tiles does the new layer add?

1, because the edge went up by one
23
12, because the new edge is 12

Which number can its last digit rule out as a square?

What is the square root of 100?

50
10
10,000

1,024 is ten 2s multiplied together. What is its square root?

20, because there are ten 2s
32
512, because that is half of 1,024

A square patch has an area of 700 square units. Between which two whole numbers does its side sit?

Between 25 and 26
Between 26 and 27
Between 34 and 35

A crate is packed full with unit blocks. It is 4 blocks along every edge. How many blocks is that?

12, because 4 × 3 = 12
64
16, because that is 4 rows of 4

What do 13 + 15 + 17 + 19 add up to?

16, because 4 × 4 = 16
64
52, because the numbers are all about 13

8,000 is six 2s and three 5s multiplied. What is its cube root?

7, because 2 + 5 = 7
20
40, because 2 × 2 × 2 × 5 = 40

A cubical tank has edges of 6 units. Its square base is being tiled. How many tiles does the base need?

216, because the tank is 6 × 6 × 6
36
18, because 6 × 3 = 18

Part I, Chapter 1 is A Square and A Cube. You have drawn the blocks and stacked the cubes. You have pulled numbers apart into primes. Look at what the chapter asks you to do with all this.

Next: Power Play. Short, clear ways of writing numbers too long to say out loud.

Reveal every answer to finish this lesson.

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