Year 9 Summer Workbook - Maths by Delta Academies Trust

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MATHEMATICS YOU R S UM MER BO O KL ET Dear Year Nine We know you will have missed being with your friends and teachers over the last few weeks and that you have worked really hard with home learning tasks. We want you to enjoy this summer holiday but we also want to make sure you do not forget all the amazing maths you have learnt. This booklet is just something to keep your brain active over the summer and we hope you enjoy trying some of the questions. There is a page for each week of the holiday and we know your new maths teachers would love to see your answers in September. Enjoy Maths Team at Delta

Mathematics

WEEK 1

WEEK 1 SKILLS QUESTIONS

INFORMATION FOR TOPIC: ROUNDING AND ESTIMATION Rounding to 1 decimal place

1. Calculate 3.6 × 7

2. Solve

x 5

= 12

3. Convert 0.64 to a fraction in its simplest form

6. Calculate 2 × (3 + 7)

7. Evaluate 33

4. Write 12:30 in its simplest form

8. Convert 1 into an 4 improper fraction

The number that may change

If this number is between 5 and 9 the 6 will increase by one

5 . 6 7

The number that may change

Round to 1 decimal place:

Estimate the answer to each:

13. Convert 2m into cm

14.

12. Simplify x3 × x2

1.87 8 × 30

324 will round to 300

a) 294 b) 6410 a)

240

= 120

6.5 × 9.8 11

c) 8.35 d) 0.53

3.4 × 21.6 b) 4.89

7.8 × 12.8 38

65.3 × 12.4 d) 0.54

e) Sheila has £145 to spend at the garden centre. She wants to buy a BBQ costing £65.85 and six bags of compost costing £12.95. Using estimation to support your answer, does Sheila have enough money?

1 5

1 6

15. Calculate the area of a rectangle with length 12cm and height 10cm

16. If a car travels at 60mph, how far will it have travelled in three hours?

MATHS PUZZLE

MATHS FACTS

In the Great British Pie Eating Contest the winner, Greedy Greg, ate a total of 105 pies in 5 hours. Each hour he ate six less than he did during the previous hour.

Did you know….

How many did he eat each hour? 17. Round 5.64 to 1 decimal place

Round each number to 1 significant figure and solve.

8.2 × 34.1

a) 1.371 b) 2.58 c) 104.237 d) 2.99

11. Simplify 2a + 5b + 7a + 9b

If this number is between 5 and 9 the 3 will increase by one

QUESTIONS FOR TOPIC: ROUNDING AND ESTIMATION

Round to 1 significant figure:

10. If x = 2 find the value of 4x + 3

3 2 4

If this number is between 0 and 4 the 3 will stay the same

5.67 will round to 5.7

9. Fill in the next two terms of the sequence 7, 10, 13, …, …

Estimating using rounding

We round to the first non zero number

If this number is between 0 and 4 the 6 will stay the same

5. Calculate 10% of 70

Rounding to 1 significant figure

18.

YOU R S U MMER BO O KL ET

2 7

3 4

19. Expand 3(x + 9)

20. Calculate -4 × 6

The Google search engine term is derived from the word “googol” which is a mathematical term for the number 1 followed by 100 zeros. It is supposed to reflect an infinite amount of searches on the internet. Scan the QR code to find out more about the googol….

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Mathematics

WEEK 2

INFORMATION FOR TOPIC: nth TERM

SKILLS QUESTIONS

Sequences can be arithmetic, these are sequences which increase or decrease by the same amount each time.

1. Calculate 4.2 × 6.5

5. Calculate 5% of 90

2. Solve

3x 10

= 21

6. Calculate 32 × 5 + 2

9. Fill in the next two terms of the sequence 12, 7, 2 …, …

10. If x = 8 find the value of 9x – 14

13. Convert 65cm into mm

14.

1 3

2 7

3. Convert 0.03 to a fraction in its simplest form

7. Evaluate 42

4. Write 45:35 in its simplest form

8. Convert 2 into an 11 improper fraction

11. Simplify 3a + 5b – 7a + 12b

12. Simplify x3 ÷ x2

15. Calculate the area of a triangle with base 14cm and height 8cm

16. If a car travels at 40mph for 2 and a half hours how far will it have driven?

Example 1: Find the nth term of an arithmetic sequence

3, 8, 13, 18, 23

Step 1:

Find the common difference between the values

3, 8, 13, 18, 23

The common difference means the nth term begins with

Step 2:

Write out your 5 times table and find out how far has the sequence been ‘shifted’ from it

Our nth term is therefore

4 9

2 13

19. Expand x(x – 3)

20. Calculate -12 × -5

5 Times Table: 5, 10, 15, 20, 25 3, 8, 13, 18, 23

5n – 2

-2

Step 1: Substitute the term value into where ‘n’ is (4 × 25) + 15 Step 2: Solve the calculation (4 × 25) + 15 = 115

1. Here are the first five terms in a number sequence. 6, 13, 20, 27, 34, … a) Find the next two terms in this sequence. b) Find the 10th term in this number sequence. c) Write an expression, in terms of n, for the nth term of this number sequence. 2. Here are the first five terms of an arithmetic sequence. -5, -2, 1, 4, 7, … a) i. Write down the next two terms of this sequence. ii. Explain how you found your answer. b) Find the 12th term of this sequence. c) Write an expression, in terms of n, for the nth term of this number sequence. 3. Write down the nth term of the sequence 8, 13, 18, 23, ... 4. Write down the nth term of the sequence -6, -2, 2, 6, ... 5. Write down the nth term of the sequence 29, 26, 23, 20, ... 6. If the nth term of a sequence is 5n + 6, what is the 20th term? 7. If the nth term of a sequence is 12 – 3n, what is the 3rd term?

MATHS PUZZLE

MATHS FACTS

Do you want to be able to multiply any number by 11 in your head?

The Fibonacci sequence

You should be able to do any single digit number by learning your times tables

1. Write the first digit

18.

QUESTIONS FOR TOPIC: nth TERM

To multiply a two digit number e.g.

17. Round 6.496 to 2 decimal places

+5 +5

Example 2: If the nth term of a sequence is 4n + 15, what is the 25th term?

18 × 11 1

To multiply a three digit number e.g. 124 × 11 1. Write the first digit

2. Add the two digits together and write it next 1+8=9

2. Add the next two digits together

1+2=3

3. Write the second digit

3. Move across and add the next two digits together

2+4=6

The answer would be If we did 1. Write the first digit

8 198 48 × 11 4

The Fibonacci sequence is often found in nature but find out more...

4. Write the last digit

The answer would be

1364

2. Add the two digits together and write it next 4 + 8 = 12 3. Write the second digit The answer would be 5 (4 + 1 from the 12) 28 =

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Mathematics

WEEK 3

WEEK 3 SKILLS QUESTIONS 1. Calculate 9.8 × 3.7

INFORMATION FOR TOPIC: INDEX LAWS 2. Solve 5x + 4 = 39

3. Convert 0.96 to a fraction in its simplest form

4. Write 32:64:96 in its simplest form

For higher and foundation students

For higher students only

x3 × x4 = x7

1 1 4-3 = 43 = 64

Add the powers

2 83

2x4 × 5x6 = 10x10 Add the powers and multiply the coefficients x3 ÷ x4 = x-1

Subtract the powers

12x9 ÷ 2x5 = 6x4

Subtract the powers and divide the coefficients

(x3)4 = x12

Multiply the powers

3 2 8

Use the reciprocal when there is a negative power 4 With fractional powers, the denominator is the root, and the numerator becomes the power

Multiply the powers and complete (2x3)4 = 16x12 24 = 2 × 2 × 2 × 2 = 16

5. Calculate 15% of 120

6. Calculate (9 – 2) × 42

7. Evaluate 54

8. Convert 3 into an 7 improper fraction

QUESTIONS FOR TOPIC: INDEX LAWS Higher and foundation students Simplify a) x5 × x6 b) x8 ÷ x6 c) 3x2 × 5x6 d) 12x5 ÷ 3x4

9. Fill in the next two terms of the sequence 2, 4, 8, 16, …, …

10. If x = 5 find the value of x2 + 20

11. Simplify 3a + 9b – 8a – 6b

12. Simplify 2x3 × 4x4

e) (x5)3 f) (2x4)3

x ×x 9

4 8

5 7

15. Calculate the area of a parallelogram with base 15cm and height 9cm

16. If a car travels for 3 hours and goes a distance of 210 miles, what speed in mph was it travelling?

18.

7 10

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19. Expand 2x(4x + 8)

3x7 × 4x3

2x4

e) 25

3 2

f) 27

4 3

MATHS PUZZLE

MATHS FACTS

A rat was crawling up a ship’s rope which was 30 feet long. It climbed up 6 feet every minute. Exhausted, the rat would always slip back 2 feet which took 15 seconds.

The folding challenge

How long did it take for the rat climb the 30 feet rope? 17. Round 365 to 1 significant figure

a) 3-4 b) 5-2 c) x-6 d) 16 2

14.

Higher students only Simplify

13. Convert 89mm into cm

See if you can complete the folding challenge with Matt Parker – you just need a piece of paper.

20. Calculate 15 × -8

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Mathematics

WEEK 4

WEEK 4 SKILLS QUESTIONS

INFORMATION FOR TOPIC: STANDARD FORM Standard form is used to write very large numbers or very small numbers in a shortened version.

1. Calculate 12.3 × 4.6

2. Solve 45 = 8x – 35

3. Convert 0.006 to a fraction in its simplest form

4. Write 14:63:91 in its simplest form

Can be greater than or equal to 1 but less than 10

a × 10 b

Always a multiplication

5. Calculate 35% of 90

6. Calculate (9 + 11) × (32 – 1)

7. Evaluate 63

8. Convert into a 9 mixed number

Example 1: Write 30000 into standard form 3 × 104 Write 4800 into standard form 4.8 × 103 Write 0.0007 into standard form 7 × 10-4 Write 0.0032 into standard form 3.2 × 10-3

Always a ten The power of 10 can be a positive or negative integer

Example 2: Calculate (3 × 104) × (2 × 103) Multiply the initial numbers together: 3×2=6 Multiply the powers of 10 together: 104 × 103 = 107 6 × 107 Example 3: Calculate (6 × 105) ÷ (2 × 103) Divide the initial numbers: 6÷2=3 Divide the powers of 10 together: 105 ÷ 103 = 102 3 × 102

QUESTIONS FOR TOPIC: STANDARD FORM Write the following in standard form: a) 90000 b) 450 c) 0.003 d) 0.018

Write the following as an ordinary number:

9. Fill in the next two terms of the sequence 3, 4, 6, 9, 13, 18, …, …

10. If x = 4 find the value of x2 + x – 11

11. Simplify 8a – 3b – 6a – 7b

12. Simplify 8x5 ÷ 4x3

5.7 × 102 a) 3 × 106 b)

c) 8 × 10-3 d) 9.7 × 10-5

Calculate the following:

13. Convert 2 litres into ml

14.

9 10

–

3 7

15. Calculate the area of a circle with radius 4cm

16. If a car travels for 1 and a half hours and goes a distance of 75 miles, what speed in mph was it travelling?

a) (4 × 107) × (2 × 102) b) (2 × 106) × (2.5 × 108) c) (5 × 103) × (6 × 105)

Calculate the following:

a) (9 × 109) ÷ (3 × 102) b) (10 × 107) ÷ (4 × 103) c) (26 × 109) ÷ (2 × 105)

MATHS PUZZLE

MATHS FACTS

Four English football teams compete against each other to see who is the best – Manchester City, Liverpool, Derby County and Chelsea. After every team has played every other team in a mini league competition, they finish as follows.

Can you solve the Prisoner Hats riddle?

• Chelsea finished above Manchester City.

17. Round 2.78 to 1 significant figure

18.

7 10

3 4

19. Expand 3x(5 – 4x)

20. Calculate -9 × -8

• Liverpool were not third. • There were two teams between Manchester City and Derby County.

List the order the teams finished in.

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Mathematics

WEEK 5

WEEK 5 SKILLS QUESTIONS 1. Calculate 5.9 × 13.3

INFORMATION FOR TOPIC: DIVIDING INTO A GIVEN RATIO 2. Solve

6x + 8 10

= 80

3. Convert 0.504 to a fraction in its simplest form

4. Simplify 9:54 in the ratio 1:n

Example 1: Keith and Graham share £105 in the ratio 4:3. Work out how much Keith gets. Keith Graham Draw the correct number of boxes for each person. Divide the full amount into the number of boxes. 105 ÷ 7 = 15 per box

5. Calculate 3% of 90

6. Calculate (62 + 1) – 12

7. Evaluate 25

Example 2: Keith and Graham share some money in the ratio 4:3. Keith receives £120. How much money was there all together?

Keith Graham Multiply the amount per box by the number of boxes Keith has. Keith = 15 × 4 = £60

Draw the correct number of boxes for each person. Divide Keith's amount into the number of boxes he has. 120 ÷ 4 = 30 per box

Multiply the amount per box by the total number of boxes. Total money = 30 × 7 = £210

QUESTIONS FOR TOPIC: DIVIDING INTO A GIVEN RATIO

8. Convert into a 7 mixed number

1) Share £650 in the ratio 14:11 2) Divide 180 miles in the ratio 22:14 3) Divide 220g in the ratio 4:7 4) The ratio of adults to children at a football match is 7:5. There were 240 people at the match. How many children attended the football match? 5) The angles in a triangle are in the ratio 9:2:7. Find the size of each angle.

9. Fill in the next two terms of the sequence 48, 24, 12, 6, …, …

10. If x = 8 find the value of 2x2

11. Simplify -3a + 5b – 9a – 12b

12. Simplify 15x ÷ 9x 6

6) The ratio of apple sweets to orange sweets in a tub is 5:3. There are 120 apple sweets in the tub. How many

orange sweets are in the tub?

7) Abi, Jake and Sadie share a sum of money in the ratio 3:4:2. Jake receives £150. Work out how much money

Sadie receives.

8) There are red and green peppers in a crate. There are 90 green peppers in the crate. The ratio of the number of red peppers to green peppers is 4:5. How many peppers are there altogether?

13. Convert 345m into km

14.

12 15

–

2 3

15. Calculate the area of a circle with diameter of 4cm

16. A cuboid has a mass of 100g and a volume of 50cm2. What is its density?

MATHS PUZZLE

MATHS FACTS

Can you draw this shape without taking your pencil off the paper and without going over any line more than once?

This puzzle comes from a very famous maths problem – The Seven Bridges of Königsberg Find out more from the QR code

17. Round 9628 to 2 significant figures

18.

9 14

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19. Expand and simplify 3(2x + 5) + 9(2x + 7)

20. Calculate -28 ÷ 4

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Mathematics

WEEK 6

PRIME FACTORISATION, HIGHEST COMMON FACTOR INFORMATION FOR TOPIC: AND LOWEST COMMON MULTIPLE

SKILLS QUESTIONS 1. Calculate 20.4 × 15.2

5. Calculate 2.5% of 160

2. Solve 5 =

3x + 9

3. Convert 0.0004 to a fraction in its simplest form

6. Calculate (12 + 3 ) × 5 2

7. Evaluate 3

4. Simplify 8:52 in the ratio 1:n

8. Convert into a 8 mixed number in its simplest form

Example 1: Write 48 as a product of its Example 2: Use prime factorisation to find the HCF and LCM of prime factors. 48 and 60 Complete prime Divide your starting 48 48 60 factorisation for number into two factors. 48 60 both numbers. 2 24 2 24 2 30 If either factor is prime, 2 circle it. Place prime factors 2 2 12 2 12 2 15 2 5 into a Venn diagram. 2 Continue to divide the 3 2 6 2 6 3 5 Those that are in both factors that are not go into the centre of prime until you can go 2 3 2 3 the Venn. no further. 2 × 2 × 2 × 2 × 3 = 24 × 3 = 48

Write each of the following as a product of its prime factors: a) 80 b) 126 c) 96

9. Fill in the next two terms of the sequence 1, 4, 9, 16, …, …

13. Convert 3.64km into m

17. Round 815 to 2 significant figures

10. If x = 4 find the value of 3x2 + 2x

14. 1

18. 2

1 4

–

2 3

2 5

11. Simplify 9a – 8b – 3a – 15b

15. Calculate the area of a semi-circle with radius of 10cm

19. Expand and simplify 4(8x + 3) + 4(5x – 7)

16. A cuboid has a density of 200g/cm2 and a volume of 10cm2. What is its mass?

a) 84 and 180 b) 56 and 140 c) 64 and 224

Complete the Sudoku. The digits 1 to 9 must appear in each column, row and box only once.

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MATHS PUZZLE

MATHS FACTS 6 9

20. Calculate -120 ÷ -12

6 3 3

How to measure very tall things you can’t reach to the top of!

4 7

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Find the highest common factor and lowest common multiple of:

1 4

Lowest Common Multiple Found from multiplying all values that are in the Venn diagram 2 × 2 × 3 × 2 × 2 × 5 = 240

FACTORISATION, HIGHEST COMMON FACTOR QUESTIONS FOR TOPIC: PRIME AND LOWEST COMMON MULTIPLE

x6 × x9 12. Simplify x4

Highest Common Factor Found from multiplying the values that are in the intersection 2 × 2 × 3 = 12

1 4 6

6 8 9

7 4

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Mathematics

ANSWERS Week 1

ANSWERS

Week 2

Week 3

Week 4

Week 5

Week 6

25.2

27.3

36.26

56.58

78.47

310.08

x = 60

x = 70

x=7

x = 10

x = 132

x = 17

16 25

3 100

24 25

3 500

63 125

1 2500

2:5

9:7

1:2:3

2:9:13

1:6

1:6.5

4.5

31.5

2.7

112

160

105

625

216

7 4

25 11

23 7

16, 19

-3, -8

32, 64

24, 31

3, 1.5

25, 36

128

9a + 14b

-4a + 17b

-5a + 3b

2a – 10b

-12a – 7b

6a – 23b

x or (x )

5 4 x 3

200cm

650mm

8.9cm

2000ml

0.345km

3640m

11 30

13 21

17 14

33 70

2 15

7 12

120cm2

56cm2

135cm2

50.3cm2

12.6cm2

157.1cm2

180 miles

100 miles

70mph

50mph

2g/cm2

2000g

5.6

6.50

400

9600

820

6 28

8 117

7 34

14 15

27 28

3x + 27

x2 – 3x

8x2 + 16x

15x – 12x2

24x + 78

52x -16

-24

-120

-7

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2 3

2 7

1 4

5 8

Week 1

Week 2

1. a) 41, 48 b) 69 c) 7n –1

a) 1.4 b) 2.6 c) 104.2 d) 3.0 a) 300 b) 6000 c) 8 d) 0.5

2. a) i. 10, 13 ii. Added three b) 28 c) 3n – 8

a) 7 b) 12 c) 2 d) 1400 e) 13 × 6 = 78 78 + 70 = 148

3) 4) 5) 6) 7)

5n + 3 4n – 10 -3n + 32 106 3

MATHS PUZZLE 33, 27, 21, 15, 9

Week 4 a) b) c) d)

9 × 104 4.5 × 102 3 × 10-3 1.8 × 10-2

a) 3000000 b) 570 c) 0.008 d) 0.000097

Week 3 a) x11 b) x2 c) 15x8 d) 4x e) x15 f) 8x12 g) x11 h) 6x6 1 81 1 b) 25 1 c) x6 d) 4 e) 125 1 f) 81 a)

MATHS PUZZLE 9 minutes 45 seconds MATHS PUZZLE Derby County Liverpool Chelsea Manchester City

Week 5

Week 6

1) £364:£286

a) 24 × 5

2) 110miles:70 miles

b) 2 × 32 × 7

3) 80g:140g

c) 25× 3

4) 100 children 5) 90°:20°:70°

a) HCF 12 LCM 1260

6) 72 orange sweets

b) HCF 28 LCM 280

7) Sadie gets £75

c) HCF 32 LCM 448

8) 162

a) 8 × 109 b) 5 × 1014 c) 3 × 109 a) 3 × 107 b) 2.5 × 104 c) 1.3 × 105

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Academies Trust Education House, Spawd Bone Lane, Knottingley, WF11 0EP T: 0345 196 0033 | info@deltatrust.org.uk | www.deltatrust.org.uk Summer 2020