some (quick) ways to (probably) make consolidation tasks at least a bit more interesting

• reverse the question• greater generality • seek (exhaust) all possibilities• look at/for alternative methods

solve 3x – 4 = 20 make up some equations with a solution of x = 8: make them as complicated as you can

what integer solutions can you find to 3x – 4y = 20?

what numbers must ‘n’ be for 3x – n = 20 to have an integer solution?

in trying to solve 3x – 4 = 20, what happens if you divide everything by 3 first?

what are the coordinates of the mid-point between (2 , 1) and (6, 9)?

which pairs of points have (4, 5) as a mid-point?

if (4, 5) is one third of the way along a line what could the two end-points be?

what other integer points lie on the line joining (2, 1) to (6, 9)?

can you find two (or more) ways to find the mid-point between two points?

find several two, 2-digit subtraction sums with an answer of 7

find all the (positive) options for the result of a subtraction sum using the digits 6, 7, 8 and 9 (without repeats)

what general properties do the digits in a two, 2-digit subtraction sum, AB – CD have if the result is 7?

what is 86 – 79?

how can you see that the result of the sum 86 – 79 must be 7 by relating the numbers to multiples of 7?

what is the mean average of 0 , 3 and 15?

give sets of three numbers with a mean of 6

can the mean be smaller than the median for a set of three integers?

find all integer sets of 4 numbers with a mean of 6 and a range of 6

can you think of (another) way to find the mean of 138, 142, 135, 148 and 137?

the rectangle is reflected and ends up at (1, – 2), (5, – 2), (1, – 4) and (5, – 4) what is the mirror line’s equation?

what happens to the coordinates of a shape when you reflect it in y = x?

what happens to the four corner coordinates when the shape is reflected in y = x + 1?

can you use a rotation followed by a reflection that has the same effect as reflecting in y = x?

reflect the rectangle in the line y = x

Simpsons

0

20

40

60

80

100

120

140

160

180

200

0 10 20 30 40 50 60 70 80

age years

hei

gh

t cm

Bart

Homer

Marge

Maggie

Lisa

Abraham

Mr Burns

Simpsons

Thanks to the Shell Centre

a large bag of flour weighs 24kgit costs £21.50

a sponge cake uses 150g of this flour

what are the questions that these calculations find out?

24000150

2421.50

215024

21.5024000

× 150 (a) (b) (c) (d)

3 7 6 1 5

3 2 5 ?

? H: L:26

46

? H: L:35

25

H: L:

H: L:

play your cards rightthe numbers 0 to 9 are arranged randomly probability of

H = higherL = lower

0 1 23 4 56 7 89

?

0 1 23 4 56 7 89

can you suggest

some subtraction sums with an answer – 5 ?

an experiment where the probability is ¾ ?

two coordinate pairs with a gradient of ½ between them ?

a quadrilateral whose diagonals cut at 90o ?

two numbers with a highest common factor of 15 ?

a hexagon with rotational but no line symmetry ?

numbers that might round to 350 ?

numbers with exactly five factors ?

the dimensions of a cuboid with a volume of 24 cm3 ?

a ratio question with an answer £180 and £240?

Egyptian fractions

the Ancient Egyptians only used unit numerator fractions

they turned other fractions into sums of two or more fractions, all with a numerator of 1

(apart from ⅔)

they used fractions with different denominators

112

Egyptianfractions

Ancient Egyptians turned fractions into a sum of fractions with unit numerators

can you find other ways to represent these two fractions in this ‘Egyptian’ way?

118

+5

36=

116

120

+9

80=

one way to do this

is to split up the numerator into a sum

so that the two numbers are both factors of the

denominator

so that they cancel…

436

136

+5

36=

536

19

136

+5

36=

Egyptianfractions

sometimes you cannot do this straight away

so, look at an equivalent fraction

until you can find two (or more) numbers that are both factors

of the denominator

which cancel…

728

128

+8

28=

27

14

128

+27

=

Egyptianfractions

remember that the denominators must be different

try to make this out of two unit fractions

and another way

512

512

try to make this out of two unit fractions

and another way

712

712

1112

why does this need three unit fractions?

try to find two ways to do this

Egyptianfractions

49

215

1021

59

1118

713

(1)

(2)

(3)

(4)

(5)

(6)

(7)

try to write these as the sum of two fractions with numerator 1 and different denominators, as the Ancient Egyptians did

29

25

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15) 38

find two ways for

(16) find two ways for3

10

(17) find three ways to writeas the sum of two unitary fractions, with different denominators

18

(18) find three ways to writeas the sum of two unitary fractions, with different denominators

110

find four ways to writeas the sum of two unitary fractions, with different denominators

(19)2

15

(20)

720

514

950

623

find three ways to writeas the sum of three unitary fractions, with different denominators

45

449

Egyptian fractions

1124

27 4

19

823

2135

21

harder

Egyptian fractions

try to write these as the sum of two unit fractions

they can all be done

1

4

1

28+

2

7=

1

5

1

95+

4

19=

1

3

1

69+

8

23=

1

7

1

91+

2

13=

answers

1

6

1

14+

5

21=

16

1??

+211

=

13

133

+?11

=

1?

1??

+311

=

1?

1??

+611

=

112

1???

+111

=

Egyptian fractions

some of the ‘elevenths’family

why/how do these work in this way?

15

120

+1?

=

18

156

+1?

=

1?

1??

+13

=

1?

1??

+1

10=

1?

1??

+16

=

Egyptian fractions

Egyptian fractions sum to another Egyptian fraction

what is a general rule for these?

1n

1n(n – 1)

+1?

=substitute some numbers for ‘n’what happens?try to prove that this will always work

2n + 1

2n(n + 1)

+2n

=substitute some odd numbers for ‘n’what happens?

2n + 2

4n(n + 2)

+2n

=substitute some even numbers for ‘n’what happens?

what do these last two statements establish?