UNIT 2 HIGHER EDEXCEL GCSEMilestone (1)

ContentPrime factorsHCF, LCMSquares, cubesBIDMASLaws of Indices

Lesson

Grade DMilestone (1)

Simplify

(i) a6 × a3

..............................

(ii)

…………………..

(iii) (e4)5

..............................

2

8

c

c

1.

Grade DMilestone (1)

What are the first five prime numbers?

Express 40 as a product of its prime factors.

40

2

2.

Grade E/DMilestone (1)

(i) 42 =

(ii) 82 =

(iii) 23 =

(iv) 53 =

(v) =

(vi) =

√64

3√125

Evaluate3.

Grade E/DMilestone (1)

List the first ten multiples of 3

List the first ten multiples of 7

What is the Lowest Common Multiple of 3 and 7?

4.

Grade DMilestone (1)

List all the factors of 36.

List all the factors of 27.

What is the Highest Common Factor of 27 and 36?

5.

Grade DMilestone (1)

(6 - 2)2 - 1 5

Work out6.

Grade DMilestone (1)

7. A chocolate company wishes to produce a presentation box of 36 chocolates for Valentine’s Day.

The company decides that a rectangular shaped box is the most efficient shape, but can’t decide how to arrange the chocolates.

List the different possible arrangements are there:

a) Using one layer (eg. 1 x 36, …..)

b) Using two layers

c) Using three layers

Grade DMilestone (1)

8. Neal works part time in a local supermarket, stacking shelves. He has been asked to use this pattern to advertise a new brand of beans.

This stack is 3 cans high.

a) How many cans will be need to build a stack 10 cans high?

b) If he has been given 200 cans, how many cans high would his stack be?

Next he is asked to stack cans of tomato soup in a similar shape,

but this time it is two cans deep.

c) How many cans will he need to build a stack 10 cans high?

d) If he has been given 400 cans, how many cans high would his stack be?

HINT

Grade CMilestone (1)

9. One is a square number and a cube number. Find another number which is both a square and a cube number.

Grade CMilestone (1)

10. Use the calculation 42 x 62 = 576

To work out:

402 x 602

4002 x 62

5760 ÷ 62

42 x 602

43 x 62

HINT

Grade CMilestone (1)

11. Work out

2 + 4 ÷ 4

53 ÷ 5 + 5

(22)3 – (23)2

Grade C/BMilestone (1)

12. Simplify

a)

b)

c)

d)37

12

55

5

6

35

3

33

10

74

4

44

34 )2(

Grade CMilestone (1)

13. Express 252 as a product of its prime factors

Express 6 x 252 as a product of prime factors

Grade CMilestone (1)

14. James thinks of two numbers. He says:

‘The highest common factor (HCF) of my two numbers is 3.

The lowest common multiple (LCM) of my two numbers is 45’.

Write down the two numbers James could be thinking of

HINT

Grade CMilestone (1)

15. Write 84 as a product of its prime factors. Hence or otherwise write 1682 as a product of its prime factors.

HINT

Grade CMilestone (1)

16. A car’s service book states that the air filter must be replaced every 10 000 miles and the diesel fuel filter every 24 000 miles.

After how many miles will both need replacing at the same time?

Grade B/AMilestone (1)

17. Work out

a)

b)

c)

d)

22

43

81

23 )216(

3)49(

3

32

8

17

Grade A/A*Milestone (1)

18. Work out the value of x

x28 2 930 HINT

Grade A*Milestone (1)

19. Take a piece of A4 paper. If you fold it in half you create two equal pieces. Fold it in half again; you now have four equal pieces. It is said that no matter how large and how thin you make the paper, it cannot be folded more than seven times. Try it.

If you fold it seven times, how many equal pieces does the paper now have?

Grade A*Milestone (1)

20. In 2001, there were two rabbits left on an island. A simple growth model predicts that in 2002 there will be four rabbits and in 2003, eight rabbits.

The population of rabbits continues to double every year. Assuming no rabbits die, how long is it before there are over one million rabbits on the island?

Grade A*Milestone (1)

21. Is each statement true or false? If the statement is false, give an example to show it.

a) The sum of two prime numbers is always a prime number.

b) The sum of two square numbers is never a prime number.

c) The difference between consecutive prime numbers is never 2

d) The product of two prime numbers is always a prime number.

e) No prime number is a square number.

Milestone

How many cans would be in a stack 1 can high?

2 cans high?

3 cans high? Etc

Can you continue the pattern?

Back to question

How can we speed up the calculations?

3 cans high = 1 + 2 + 3

4 cans high = 1 + 2 + 3 + 4

15 cans high = 1 + 2 + 3 ………+ 13 + 14 + 15Pair up numbers

15 + 1 = 1614 + 2 = 16

13 + 3 = 16 How many pairs have you made?

Is there a number left in the middle?

Milestone

What is the connection between the numbers?

40 = 4 x ?

402 = 42 x ?

Back to question

Back to question

Milestone

Think of the rules of index numbers

23 x 25 =2?

(24)2 = 2?

Can you write 8 as a power of 2?

8 = 2?

Milestone

What is the connection between 84 and 168?

Back to question

ANSWERSMilestone

GRADE D

7) a) 5 (1x36,2x18,3x12,4x9,6x6)

b) 3 (1x18,2x9,3x6)

c) 3 (1x12,2x6,3x4)

8) a)100 b) 14 (with 4 spare) c) 200 d)14 (with 8 spare)

GRADE C

9) 64

10) a) 5 760 000 b) 5 760 000 c) 160 d) 57 000 e) 2304

11) a) 3 b) 30 c) 0

12) a) 32 b)4 c) 212 d) 52

13) a) 22x32x7 b) 23x33x7

14) 9,15

15) a) 22x3x7 b) 26x32x72

16) 120 000 miles

GRADE B

17) a)48 b)36 c)343 d)25

18) 3

GRADE A*

19)a) 27 = 128

20)2020

21) a)F eg 3+5=8 b)F eg 4+9=13

c)F eg 5-3=2 d)F eg 2x3=6 e) true

1)a9 c4 e20

2) 2,3,5,7,11

2 x 2 x 2 x 5

3) 16,64,8,125,8,5

4) 3,6,9,12,15,18,21,24,27,30

7,14,21,28,35,42,49,56,63,70

HCF =21

5) 1,2,3,4,6,9,12,18,36

1,3,9,27

HCF = 9

6) 3

Draw a diagram

Work backwards

Act it out

Seen it before?

Look fora pattern

Draw a graph

Make alist

Guess andcheck

Pattern Squares

1 1

2 4

3 9

Make a table

Test allpossibilities

Solve a simpler problem

Chunk the problem

Try one or all of these strategies