1. Content Milestone (1) UNIT 2 HIGHER EDEXCEL GCSE Prime factors
HCF, LCM
Squares, cubes
BIDMAS
Laws of Indices
Lesson

2. Milestone (1) Grade D 1. Simplify (i) a6 × a3
(ii)
…………………..
(iii) (e4)5

3. Express 40 as a product of its prime factors.
Milestone (1)
Grade D 2.
What are the first five prime numbers?
Express 40 as a product of its prime factors. 40 2

4. (i) 42 = 82 = 23 = 53 = = √64 3√125 Evaluate Milestone (1) Grade E/D3.
(i) =
82 =
23 =
53 = =
√64
3√125

5. List the first ten multiples of 3
Grade E/D
Milestone (1) 4.
List the first ten multiples of 3
List the first ten multiples of 7
What is the Lowest Common Multiple of 3 and 7?

6. Grade D
Milestone (1) 5.
List all the factors of 36.
List all the factors of 27. What is the Highest Common Factor of 27 and 36?

7. Milestone (1)
Grade D 6.
Work out
(6 - 2)2 - 1 5

8. Milestone (1)
Grade D
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:
Using one layer (eg. 1 x 36, …..)
Using two layers
Using three layers

9. Milestone (1)
Grade D
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.
How many cans will be need to build a stack 10 cans high?
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

10. Milestone (1)
Grade C
9. One is a square number and a cube number Find another number which is both a square and a cube number.

11. Milestone (1) 10. Use the calculation 42 x 62 = 576 To work out:
Grade C
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

12. Milestone (1)
Grade C
11. Work out
2 + 4 ÷ 4
53 ÷ 5 + 5
(22)3 – (23)2

13. Milestone (1)
Grade C/B
12. Simplify a) b) c) d)

14. 13. Express 252 as a product of its prime factors
Milestone (1)
Grade C
13. Express 252 as a product of its prime factors
Express 6 x 252 as a product of prime factors

15. Milestone (1) 14. James thinks of two numbers. He says:
Grade C
HINT
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

16. Milestone (1)
Grade C
HINT
15. Write 84 as a product of its prime factors. Hence or otherwise write 1682 as a product of its prime factors.

17. Milestone (1)
Grade C
16. A car’s service book states that the air filter must be replaced every miles and the diesel fuel filter every miles.
After how many miles will both need replacing at the same time?

18. Milestone (1)
Grade B/A
17. Work out a) b) c) d)

19. Milestone (1)
Grade A/A*
18. Work out the value of x
HINT

20. Milestone (1)
Grade A*
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?

21. Milestone (1)
Grade A*
20. In 2001, there were two rabbits left on an island. A simple growth model predicts that in 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?

22. The sum of two prime numbers is always a prime number.
Milestone (1)
Grade A*
21. Is each statement true or false? If the statement is false, give an example to show it.
The sum of two prime numbers is always a prime number.
The sum of two square numbers is never a prime number.
The difference between consecutive prime numbers is never 2
The product of two prime numbers is always a prime number.
No prime number is a square number.

23. Milestone How many cans would be in a stack 1 can high? 2 cans high?
3 cans high? Etc
Can you continue the pattern?
How can we speed up the calculations?
3 cans high =
4 cans high =
15 cans high = ………
Pair up numbers
= 16
= 16
= 16
How many pairs have you made?
Back to question
Is there a number left in the middle?

24. Milestone
What is the connection between the numbers?
40 = 4 x ?
402 = 42 x ?
Back to question

25. 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?
Back to question

26. What is the connection between 84 and 168?
Milestone
What is the connection between 84 and 168?
Back to question

27. Milestone ANSWERS 1)a9 c4 e20 14) 9,15 2) 2,3,5,7,11 GRADE D
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
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) b) c) 160 d) 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) 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

28. Test all possibilities
Try one or all of these strategies
Draw a diagram
Act it out
Work backwards
Seen it before?
Draw a graph
Make a list
Look for a pattern
Pattern
Squares 1 2 4 3 9
Make a table
Test all possibilities
Guess and check
Chunk the problem
Solve a simpler problem