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Special number families
Square numbers up to 15², cube numbers up to 5³, square roots and cube roots.
Some numbers belong to special families. Once you know them, lots of questions become quick to answer, because you just remember them.
In this lesson you will learn:
- the square numbers up to 152 = 225
- the cube numbers up to 53 = 125
- how to find a square root and a cube root
- how to tell which two whole numbers a square root sits between
1. Square numbers
- We write 5 × 5 as 52 and say “5 squared”.
- They are called square numbers because that many tiles make a square: 5 rows of 5.
Learn these by heart:
| 12 = 1 | 22 = 4 | 32 = 9 | 42 = 16 | 52 = 25 |
| 62 = 36 | 72 = 49 | 82 = 64 | 92 = 81 | 102 = 100 |
| 112 = 121 | 122 = 144 | 132 = 169 | 142 = 196 | 152 = 225 |
The square builder: add a layer at a time
- Watch the new layer each time. How many tiles does it add?
- Build up to 102. What do the new layers add up to?
- How many tiles do you need to go from 122 to 132?
2. Cube numbers
- We write 2 × 2 × 2 as 23 and say “2 cubed”.
- They are called cube numbers because that many small cubes make a big cube.
Learn these by heart:
| 13 = 1 | 23 = 8 | 33 = 27 | 43 = 64 | 53 = 125 |
A quick way: 43 = 4 × 4 × 4. Do 4 × 4 = 16 first, then 16 × 4 = 64.
The cube builder: watch the layers stack up
3. Square roots
- 7 × 7 = 49, so the square root of 49 is 7.
- We write it as √49 = 7. The sign √ means “the square root of”.
- If you know your square numbers, you know their square roots too.
Example: between which two whole numbers is √50?
- 50 is not a square number, so its square root is not a whole number.
- The square numbers either side are 49 (72) and 64 (82).
- So √50 is between 7 and 8. 50 is much closer to 49, so √50 is just over 7.
Square or not? Guess, then watch the tiles try to make a square
4. Cube roots
- 4 × 4 × 4 = 64, so the cube root of 64 is 4.
- We write it as ∛64 = 4. The small 3 tells you it is a cube root.
Match the pairs: tap a card, then tap its partner
5. Practice: pick your level
Start with Foundation if you are in Year 4 or just starting. Move up when you are getting most of them right.
The mistakes that cost marks
- Multiplying by 2 instead of squaring. 62 is not 12. square means times itself: 6 × 6 = 36.
- Multiplying by 3 instead of cubing. 43 is not 12. cube means times itself, then times itself again: 4 × 4 × 4 = 64.
- Halving instead of finding the square root. √36 is not 18. ask which number times itself makes 36: 6.
- Mixing up the square root and the cube root. √64 = 8, but ∛64 = 4. look for the small 3 on the root sign.
- Not knowing the bigger squares. 132, 142 and 152 come up a lot. learn them: 169, 196 and 225.
Quick checklist
Tick each one when you can do it without help.
All ticked. Brilliant work! You have finished Lesson 1.7.
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