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

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

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

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

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?

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?

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

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

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

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.

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

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

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

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

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

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.

Milestone (1) Grade C 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?

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

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

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?

Milestone (1) Grade A* 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?

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.

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 = 1 + 2 + 3 4 cans high = 1 + 2 + 3 + 4 15 cans high = 1 + 2 + 3 ………+ 13 + 14 + 15 Pair up numbers 15 + 1 = 16 14 + 2 = 16 13 + 3 = 16 How many pairs have you made? Back to question 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

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

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

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) 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

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