12 ÷ 3 = 4

48 × 0.15 = 2.4 × 3

2.4 ÷ 0.3 = 8

48 ÷ 3 = 2.4 ÷ 0.15

1.2 ÷ 0.3 = 4

2.4 ÷ 0.15 = 16

3 apples and 4 pears cost £1.55

I know the cost of an apple

I know the cost of a pear

6 apples and 8 pears cost £3.10

I know which costs more, one apple or one pear

6 apples and 11 pears cost £3.70

Every perfect cube is a multiple of 9, or 1 more or 1 less than a multiple of 9

1 + 2 + … + 8 + 9 = 45 1 + … + 6 + 8 + 9 = 38

(3n – 1)3 ≡ 27n3 – 27n2 + 9n – 1 and (3n + 1)3 ≡ 27n3 + 27n2 + 9n + 1

Every integer is a multiple of 3, or 1 more or 1 less than a multiple of 3

12,345,689 is not a perfect cube

(3n)3 ≡ 27n3

n2 + 2n + 1 ≡ (n + 1)2

n2 + n + 1 < n2 + 2n + 1

n is positive

The value of n2 + n + 1 is never a perfect square

n2 + n + 1 > n2

The value of n2 + n + 1 is between the value of n2 and the value of (n + 1)2

a + d = ½ cm2 4a + d = 1 cm2

d 2 cm b a e c 2 cm

e = ½ cm2 a + d + e = 1 cm2 a + e + c = 1 cm2 b + c = 1 cm2

b = ⅔ cm2

3a = ½ cm2

These triangles are similar, length ratio 1:2