Year 10 Summer Workbook - Maths by Delta Academies Trust

D E LTA Academies Trust

AR LY Y E ARS FO U NDATION S TA G E BO O KL ET

MATHEMATICS YOU R S UM MER BO O KL ET Dear Year Ten We know you will have missed being with your friends and teachers over the last few weeks and we know 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 secondary maths teachers would love to see your answers in September. Enjoy Maths Team at Delta

Mathematics

WEEK 1 SKILLS QUESTIONS 3

2. 2m² + 3m² + 5m²

3. Work out

5. Ben and Lisa share £40 in the ratio 3: 5. How much do they each get?

6. Work out the value of 8-¹

7. Simplify a4 × a5

8. Simplify 3b × 5c

10. F = ma. Work out F when m = 6 and a=3

11. Write 12:28 in its simplest form

12. Work out 1

15. Write 0.54 as a fraction in its simplest form.

16. Solve 3(2x + 5) = 27

19. Write 18000 in standard form

20. Find the HCF of 36 and 42

9. Work out

1. Round 0.00602 to 2 significant figures

1 3

3 10

15x4

13. Work out 6.4 × 2.7

14. Simplify

17. Find the size of one interior angle in an octagon.

18. A factory makes T-Shirts in sizes small, medium and large. The ratio of small : medium : large is 3:5:4. What fraction of the T-Shirts are small?

YOU R S U MMER BO O KL ET

Delta Academies Trust

4. Solve 3a + 4 = 19

4 5

×1

1 3

Mathematics

WEEK 1 INFORMATION FOR TOPIC: EXPANDING SINGLE AND DOUBLE BRACKETS Expand simple brackets:

Expand double brackets:

Expand and simplify where appropriate

There are different methods you can use for this. Choose the one it suits you

1) 4(2 + m) = 8 + 4m

1) Expand and simplify (x + 4) (x - 3)

× 2) 2(7 + b) + 3 (2 - b)

Method 1:

(x + 4)(x – 3) ×

Method 2:

x × x = x² x × -3 = -3x =+x 4 × x = 4x 4 × -3 = -12

}

× × = 14 + 2b + 6 – 3b = 20 – b

}

= x² + x-12

x²

-3

-3x

-12

= x² + 4x-3x -12 = x² + x-12

QUESTIONS FOR TOPIC: EXPANDING SINGLE AND DOUBLE BRACKETS A) Expand and simplify where appropriate 1) 5(9 + b)

2) 2(4b + 6)

3) 3(8 – c)

4) 6(2 + 3a) + 4(2a – 5)

5) a(a + 7)

6) 4(2d – 6) + 8(3d – 5)

1) (x + 3)(x + 2)

2) (w + 1)(w + 3)

B) Expand and simplify 4) (2m – 2)(3m + 7)

3) (x + 3)(x – 5)

5) (5n – 3)(2n – 1)

MATHS PUZZLE

MATHS FACTS

Put in some missing operation signs to make the following sum work:

Did you know….

1 2 3 4 5 6 7 8 9 = 100 (Keep the numbers in numerical order)

Algebra was invented by the mathematician Al-Khwarizmi in the book he wrote in 820. Algebra is the Arabic word (aljabr) for “equation”, and the word “algorithm” comes from the author’s name, Al-Khwarizmi. Go to this link to find out more about Al-Khwarizmi “The father of Algebra” https://www.youtube.com/ watch?v=FTL5A0ejA-s

Y O U R S U M M E R B O O K LE T

Delta Academies Trust

Mathematics

WEEK 2 SKILLS QUESTIONS

1. Expand the brackets 5(2x + 3y)

2. Work out

5. Work out the best buy: 5 donuts for £3.25 or 3 donuts for £1.86

3. Work out the size of the exterior angle in a decagon.

4. Solve 7a + 3 = 3a + 23

6. Write 3.5 × 105 in ordinary form

7. Find the LCM of 8 and 12

8. Expand and simplify: 4(3x + 2) + 6(4x + 3)

9. Work out 35% of 400

10. Solve 4x – 2 > 10

11. Alice and Stuart share some money in the ratio of 3:5. Stuart gets £12 more than Alice. How much money do they each get?

12. Write the exact value of sin 0°

13. Work out (3 × 104) × (4 × 10-²)

14. Write 72 as a product of prime factors

15. Write an expression to represent: “Fred thinks of a number, n, adds 3, and then doubles the answer.”

16. Write 24 as a percentage of 50.

17. Write the nth term for the sequence 5, 9, 13, 17, …

18. Convert 2m² into cm²

19. Work out 4 + 3 × 5

20. Find the area of a circle with radius 5 cm

YOU R S U MMER BO O KL ET

Delta Academies Trust

Mathematics

WEEK 2 INFORMATION FOR TOPIC: FACTORISING INTO SINGLE AND DOUBLE BRACKETS Factorise into single brackets:

Factorise into double brackets:

Factorise 6ab – 4ac

Factorise x2 + 8x + 15

Find the highest common factor of both terms, this will We need two numbers that multiply together to make then go outside the bracket. 15 and when added make 8. 2a × 3b = 6ab

Add to make 8

6ab – 4ac = 2a(3b – 2c) Highest common factor

2a × -2c = -4ac

x2 + 8x + 15 (x + 3)(x + 5)

Multiply to make +15 1 × 15 3×5

Multiply to make 15

Add to make +8 1 + 15 = 16 3+5=8

5 and 3 work for both rules

QUESTIONS FOR TOPIC: FACTORISING INTO SINGLE AND DOUBLE BRACKETS A) Expand and simplify where appropriate 1) 15a + 10

2) 12 – 21t

3) 6y – 3

4) ab + 5b

5) 2mp + 2mk

6) 3ab – 3b

1) x2 + 8x + 7

2) x2 + 6x + 5

3) x2 + 9x + 8

4) x2 + 7x + 10

5) x2 + 4x + 4

B) Factorise

MATHS PUZZLE

MATHS FACTS

When my sister was 36, my father was 63.

Did you know….

How old was she when she was exactly one quarter of his age?

Around 700AD the general solution for the quadratic equation, this time using numbers, was devised by a Hindu mathematician called Brahmagupta, who, among other things, used irrational numbers; he also recognised two roots in the solution.

How old will he have to be for her to reach three quarters of his age? (Hint: it’s over 100!)

Go to this link to find out more about the history of the quadratic formula. www.mathnasium.com/ the-history-behind-thequadratic-formula

Y O U R S U M M E R B O O K LE T

Delta Academies Trust

Mathematics

WEEK 3 SKILLS QUESTIONS 2. Simplify (54)3

5. Write the following in order, starting with the smallest 3 73% 0.072 0.7

6. Represent x ≤ 4 on a number line

7. A recipe for 4 people uses 150g of sugar. How much sugar is needed for 7 people?

8. Use a calculator to work out (6.42 × 10-5) ÷ (3.21 × 10-3)

9. Expand 3(2a – 7)

10. Work out the value of

11. Solve 5x + 3 = 23

12. Write 16:24 in its simplest form

0.52

13. Write 0.00024 in standard form

17. Work out

2 7

1 3

3. Write an estimate for 451 × 34

4. Factorise 3xy + 4x2

1. 3 painters take 12 hours to a house. How long will it take 4 painters to paint the house?

1 3

3 7

14. Work out 3.6 × 7.85

15. Work out 1

18. Solve 7x + 3 = 4x + 9

19. Find the sum of the interior angles in an octagon

YOU R S U MMER BO O KL ET

Delta Academies Trust

–

16. Simplify 4ab × 2a3b

20. Given that a : b = 8 : 5 and b : c = 10 : 3, find the ratio a : b : c in its simplest form

Mathematics

WEEK 3 INFORMATION FOR TOPIC: PYTHAGORAS’ THEOREM Examples: Calculate the length of the missing side

A formula which helps us calculate any length on a right angled triangle if the two other lengths are given.

a2 + b2 = c2 Shorter sides

Hypotenuse

a2 + b2 = c2 62 + 82 = x2 100 = x2 √100 = x 10 = x

2. 12

a2 + b2 = c2 y2 + 82 = 122 y2 = 122 – 82 y = √80 y = 8.9

QUESTIONS FOR TOPIC: PYTHAGORAS’ THEOREM Calculate the length of the hypotenuse, x, in each right-angled tringle. 12 mm

5 mm

7 cm

24 cm

3.5 cm

20 cm

16 cm

2.9 cm

4.6 cm

5.8 cm

7 cm

x 25 cm

5.2 cm

5.6 cm

7.8 cm

4.9 cm

MATHS PUZZLE

MATHS FACTS

A rat was crawling up a ship’s rope which was 30 feet long. It climbed up 3 feet a minute but then slipped back 2 feet with exhaustion!

Did you know….

How long did it take to reach the ship?

7.1cm

30 mm

Pythagoras (say “pie-thag-or-as”) of Samos was a Greek philosopher who lived from about 580 BC to about 500 BC. He made important developments in mathematics, astronomy and the theory of music. Go to this link to find out more about Pythagoras’ and his work https://nrich.maths. org/2721

Y O U R S U M M E R B O O K LE T

Delta Academies Trust

Mathematics

WEEK 4 SKILLS QUESTIONS

1. Write the exact value of sin 60°

2. Calculate the perimeter of a semi-circle with diameter 10 cm

5. Expand and simplify 3a(2a – 5) – a(2 – a)

3. Write the reciprocal of 5

4. Work out the value of 1 25 2

as a 6. Write 5 mixed number

7. Share £36 in the ratio 4:5

8. Write 5.82 × 104 in ordinary form

9. Write 72 as a product of prime factors

10. Factorise fully 6ab³ – 15a²b

11. Find the percentage increase of an antique which changed in price from £45 to £60

12. Make x the subject of a = 4x + 7

13. Tony and Luke share some money in the ratio 3:7. Luke receives £20 more than Tony. How much do they each receive?

14. Work out the value of 90

15. Find the HCF of 36 and 48

16. Factorise fully 3x² + 9x

17. Change 5m into cm

18. It takes 5 workers to paint a house in 8 hours. How long will it take 4 workers to do the same job?

19. Write the exact value of cos 45°

20. Work out 1

YOU R S U MMER BO O KL ET

Delta Academies Trust

2 7

3 5

Mathematics

WEEK 4 INFORMATION FOR TOPIC: PERCENTAGES USING MULTIPLIERS TO FIND A PERCENTAGE OF AN AMOUNT

INCREASE/DECREASE AN AMOUNT USING THE MULTIPLIER

To find a percentage of an amount, change the percentage into a decimal and multiply by the amount

To increase/decrease an amount, multiply the amount by the multiplier 100 + percentage New Amount = × Amount

New Amount =

percentage 100

× Amount

Example: Calculate 24% of £600 24% as a decimal is 0.24 (24 ÷100 = 0.24) 0.24 × 600 = 144

REVERSE PERCENTAGES The original amount = New amount ÷ multiplier Examples

add the percentage for increase, subtract for a decrease

1) A pair of trainers cost £36 in a sale. If there was 20% off, what was the original price of the trainers? Multiplier = 100 – 20 = 0.8 (Sale means reduced so subtract from 100%)

100 Example: 1) Increase £40 by 6% Multiplier = 100 + 6 = 1.06 New amount = 40 × 1.06 100 = £42.40 2) Decrease 350g by 12%. Multiplier = 100 – 12 = 0.88 New amount = 350 × 0.88 100 = 308 g

100

New amount = 36 ÷ 0.8 = £45 2) A vintage car has increased in value by 5%, it is now worth £15750. What was it worth originally? Multiplier =

100 + 5 100

= 1.05 (increased means add 100%)

New amount = 15750 ÷ 1.05 = £15000

QUESTIONS FOR TOPIC: PERCENTAGES a) Work out:

1. 20% of 90 2. 30% of 50 3. 90% of 20 4. 5% of 40

5. 65% of 60

6. 33% of 660 ml 7. 63% of 8 m 8. 37% of £128 9. 19% of 253 m

10. 64% of 13 km

b) Work out using the multiplier:

1. Increase £120 by 30%

2. Increase £500 by 5%

Increase 70 by 8.5%

4. Decrease £200 by 40% 5. Decrease £500 by 6% 6. Decrease 130 by 1.2%

c) Calculate: 1. Jacob answered 80% of the questions in a test correctly. He answered 32 of the questions correctly. Work out the total number of questions in the test. 2. A camera costs £180 in a 10% sale. What was the pre-sale price? 3. The cost of a holiday, including VAT at 20% is £540. What is the pre-VAT price? 4. After fuel prices increased by 15%, a family’s annual heating bill was £1654. What would the bill have been without the price increase? 5. The sale price of a television is £420 after a 15% reduction. What was the price before the sale?

MATHS PUZZLE

MATHS FACTS

Can you find the missing number?

Did you know….

15,

34,

152

Originally the % sign had a horizontal line instead of a diagonal one. Go to this link to find out more about the percentage symbol https://en.wikipedia.org/ wiki/Percent_sign

Y O U R S U M M E R B O O K LE T

Delta Academies Trust

Mathematics

WEEK 5 SKILLS QUESTIONS 2

4. Solve 3x + 4 < 16

2. Work out the value of y in y = ma + b when a = 5, b = -3 and m = -2

3. Work out

5. Work out the size of one exterior angle in a pentagon.

6. James invested £2000 in a Savings Account. He is paid 2% per annum compound interest. Find the amount after the 3 years.

7. Find the volume of a sphere with radius 6 cm

10. Simplify 3a² × 4a³

11. Starting with the smallest, put the following in order:

9. Simplify (a3)4

(Volume =

2 3

1. Find the LCM of 24 and 36

4 3

πr 3)

0.606 60% 0.63

31 50

8. Work out (4.5 × 105) × (3 × 10-2)

12. Find the area of a semi-circle with diameter 14cm. Give your answer to 2 decimal places

13. Find the nth term of 5, 8, 11, 14, 17, …

14. Find the number of sides in a regular polygon with exterior angle of 45°

15. In a class the ratio of boys to girls is 6:5. What fraction of the class are boys?

16. Write 1.32 × 105 in ordinary form

17. Work out 8.9 × 7.36

18. Simplify 12b4 ÷ 3b5

19. Increase £600 by 35%

20. Find the 5th number in the sequence with nth term 3n² + 1

YOU R S U MMER BO O KL ET

Delta Academies Trust

Mathematics

WEEK 5 INFORMATION FOR TOPIC: ANGLES IN POLYGONS EXAMPLES

REGULAR POLYGONS

have equal lengths of sides and equal angles

ANGLES IN POLYGONS

Sum of interior angles = (number of sides – 2)×180 One interior angle = sum of interior angles Exterior angles of regular polygons =

360

Sum of interior angles in a pentagon = (5-2)×180 = 540° One interior angle = 540 = 108°

3. Find the size of one exterior angle in a regular hexagon. A

regular hexagon has 6 sides and one exterior ang =

number of sides

Number of sides in a regular polygon =

= (5 – 2)×180 = 540°

2. Find the size of one interior angle in a regular pentagon 5

Exterior angle

number of sides

1. Find the sum of the interior angles in a pentagon.

360 6

= 60°

4. The interior angle of a regular polygon is 135°.

360

Find the number of sides. Exterior angle = 180 – 135 = 45° Number of sides = 360 = 8 An 8 sided polygon is an octagon

exterior angle

Exterior angle + Interior angle = 180

Interior angle

QUESTIONS FOR TOPIC: ANGLES IN POLYGONS A) Find the size of each interior angle in a regular polygon that has: 1) 10 sides

2) 6 sides

3) 5 sides

4) 8 sides

5) 12 sides

B) Find the size of each exterior angle in a regular polygon that has: 1) 4 sides

2) 9 sides

3) 10 sides

4) 20 sides

5) 18 sides

C) Find the number of sides in a regular polygon that has an exterior angle of: 1) 120°

2) 40°

3) 72°

MATHS PUZZLE If there are 5 Mondays, 5 Tuesdays and 5 Wednesdays in January, on what day of the week will February 1st fall?

4) 45°

5) 18°

MATHS FACTS Did you know…. In 1936 a clay tablet was excavated at Shush (Khuzistan region of Iran) some 350km from the ancient city of Babylon on which was inscribed a script that was only translated as late as 1950. The text provided confirmation that the Babylonians’ measured angles using the figure of 360 to form a circle. The inscription on the tablet shows the ratio of a perimeter of a regular hexagon to the circumscribed circle i.e. Six sides of a hexagon times their base of 60 = 360. Go to this link to find out more about the history of angles https://www.fig.net/ resources/proceedings/fig_proceedings/ cairo/papers/wshs_01/wshs01_02_wallis.pdf

Y O U R S U M M E R B O O K LE T

Delta Academies Trust

Mathematics

WEEK 6 SKILLS QUESTIONS 1. 4 tins of tuna cost £5.20. How much would 9 tins of tuna cost?

2. Work out (3.2 × 104) ÷ (4 × 10-2)

3. Round 4.5672 to 2 decimal places

4. Find the next term in the sequence 1, 7, 15, 25, …

5. Calculate the area of a circle with radius 7 cm

6. Write the reciprocal of 7

7. Simplify 6a × 5ab

8. Work out

9. Work out the value of v in v = u + at when u = 12, a = 5 and t = 2

10. Calculate the circumference of a circle with diameter 10cm

11. Work out the value of 2 4 1 ×

12. Find an estimate

13. Expand the brackets and simplify m (2m+3) – 4m

14. Work out

17. Order these numbers, starting with the smallest

18. Make x the subject of the formula F = 3x + 7

3 8

0.352 0.335

3 7

2 3

35%

YOU R S U MMER BO O KL ET

Delta Academies Trust

for

6 7

–

3 5

32.6 x 38 0.48

15. Work out 2.3 × 5.7

16. Factorise 4n² – 6n

19. Adam and Beth share some money in the ratio 5 : 8. Beth gets £24 more that Adam. How much do they each get?

20. Simplify (72)4

Mathematics

WEEK 6 INFORMATION FOR TOPIC: SOLVING LINEAR EQUATIONS For each step in solving an equation we must do the inverse operation Examples: 2) Solve 5(x – 3) = 20 1) Solve 12 = 3x – 18 Expand 5x – 15 = 20 +18 +18 +15 +15 30 = 3x 5x = 35 ÷3 ÷3 ÷5 ÷5 x =10 x=7

3) Solve 7p – 5 = 3p + 3 – 3p – 3p 4p – 5 = 3 +5 +5 4p = 8 ÷4 ÷4 p=2

QUESTIONS FOR TOPIC: SOLVING LINEAR EQUATIONS A) (1) 15 = 4f + 3

(2) 19 = 7 + 3t

(4) 4 = 3y – 2

(5) 11 = 3x – 7

(3) 18 = 3 + 5y

B) (1) 2(x + 5) = 16

(2) 5(x – 3) = 20

(4) 44 = 4(2x + 5)

(5) 14 = 2(3y – 5)

(3) 3(2t + 1) = 21

C) (1) 5k = 2k + 3 (5) 2t + 7 = 4t – 3 (8) 3(2y + 3) = 5(2y + 1)

(2) 2x + 3 = x + 5 (6) 2(d + 3) = d + 12 (9) 8(m – 2) = 4(m + 9)

(3) 4a – 3 = 3a + 4 (4) 7p – 5 = 3p + 3 (7) 5(x – 2) = 3x + 12 (10) 6(2y + 7) = 3(5y + 6)

MATHS PUZZLE

MATHS FACTS

Find two numbers which when multiplied together make 1,000,000. Neither of the two numbers must use any 0’s

Did you know….

Sir William Rowan Hamilton invented the linear equation in 1843. He was an Irish mathematician of the early 1800’s contributing significant work to quantum mechanics, physics and specifically linear equations. Go to this link to find out more about Sir William Rowan Hamilton https://mathshistory. st-andrews.ac.uk/ Biographies/Hamilton/

Y O U R S U M M E R B O O K LE T

Delta Academies Trust

Mathematics

ANSWERS Week 1 1

0.0060

10 m²

4 5

Week 3

Week 4

10x +15 xy

9 hours

5¹²

36°

30000

a=5

x(3y + 4x)

£15 and £25

3 donuts

£11.70

25.7cm

y = -13

800000

4.57

x<4

7a²–17a

72°

153.9cm2

£2122.42

v = 22

350000

3 1

Week 6

0.072, 0.7, 73%,

Week 5

262.5 g

£16 and £20

904.78cm³

30a²b

15bc

36x + 26

0.02 or 2 × 10¯²

58200

13500

140

6a – 21

23×32

a12

3ab(2b² – 5a)

12a5

31.4cm or 10π cm

33.3%

10 11 12 13 14 15

16 17 18 19 20

F = 18

x>3

3:7

£18 and £30

x=4

2:3

17.28

1200 or 1.2 × 10³

2.4 × 10–4

5x³

2³ × 3²

28.26

2 5

27 50

Week 2

2(n + 3)

x=2

48%

135°

4n + 1

2 21

x=2

1.8 × 104

1080°

25π cm2 or 78.5 cm2

16:10:3

YOU R S U MMER BO O KL ET

Delta Academies Trust

2400

£15 £35

3n + 2

2m2 – m

8 sides

611

13.11

3x(x + 3)

132000

2n(2n – 3)

500cm

65.504

35%, 0.335, 0.352, 3

8a4b²

a–7

2 , 0.63,

76.97cm2

20000 cm²

60%,0.606,

2321 or 1221

10 hours 2 2 135 35

31 35

4b-1 = 4b

F–7 3

£810

£40 £64

Mathematics

ANSWERS Week 1

Week 2

Week 3

A) 1) 45 + 5b 2) 8b + 12 3) 24 – 3c 4) 26a – 8 5) a² + 7a 6) 32d – 64

A) 1) 5(3a + 2) 2) 3(4 – 7t) 3) 3(2y – 1) 4) b(a + 5) 5) 2m(p + k) 6) 3b(a – 1)

B) 1) x² + 5x + 6 2) w² + 4w + 3 3) x² – 2x – 15 4) 6m² + 8m – 14 5) 10n² – 11n + 3

B) 1) (x + 7)(x + 1) 2) (x + 5)(x + 1) 3) (x + 8)(x + 1) 4) (x + 5)(x + 2) 5) (x + 2)(x + 2)

A) 1) x = 13 mm 2) x = 25 cm 3) x = 6.77 cm 4) x = 5.44 cm 5) x = 7.07 cm 6) x = 12 cm 7) x = 24 cm 8) x = 2.71 cm 9) x = 5.81 cm

Puzzle 123 – 45 – 67 + 89 = 100

Puzzle She was 9 years old when she was a quarter of his age He will have to be 108 for her to reach three quarters of his age.

Week 4 A) 1) 18 2) 15 3) 18 4) 2 5) 39 6) 217.8 ml 7) 5.04m 8) £47.36 9) 48.07m 10) 8.32km B) 1) £156 2) £525 3) £75.95 4) £120 5) £470 6) £128.44

c) 1) 40 2) £200 3) £450 4) £1438.26 5) £494.12 MATHS PUZZLE 73 – the difference of the difference is doubling each time.

Puzzle 30 minutes

Week 5

Week 6

A) 1) 144° 2) 120° 3) 108° 4) 135° 5) 150° B) 1) 90° 2) 40° 3) 36° 4) 18° 5) 20° C) 1) 3 2) 9 3) 5 4) 8 5) 20

A) 1) f = 3 2) t = 4 3) y = 3 4) y = 2 5) x = 6 B) 1) x = 3 2) x = 7 3) t = 3 4) x = 3 5) y = 4

C) 1) k = 1 2) x = 2 3) a = 7 4) p = 2 5) t = 5 6) d = 6 7) x = 11 8) y = 1 9) m = 5 10) y = 8

MATH PUZZLE 64 x 15625 – Prime factorisation give 26×56

MATHS PUZZLE Thursday

Y O U R S U M M E R B O O K LE T

Delta Academies Trust

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