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 Eight 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 SKILLS QUESTIONS 1. List the first five multiples of 7
5. Which is larger 0.3 or
9. What is
1 6
1 3
of 30 ?
13. Solve 5x = 45
2. What is the lowest common multiple of 7 and 5 ?
3. Evaluate 72
4. Write 0.25 as a fraction
6. Calculate -18 + 5
7. Calculate 4.26 + 31.5
8. Calculate 4.2 × 3
10. Find 15% of 80
11.
14. Find the missing numbers in the sequence
15. Find the nth term of 4, 7, 10, 13, 16, …
16. What is the area of a rectangle with length 8 cm and width 6 cm ?
19. Find the mode of 5, 3, 5, 4, 3, 4, 5, 5, 2
20. A letter is chosen at random from the word MATHEMATICS. What is the probability of choosing a letter T?
17. Find the perimeter of a square with side 5 cm
2
+
1 3
12, 16, ___, ___, 28
18. Two angles in a triangle are 46 and 24 . What is the third angle?
2
1
YOU R S U MMER BO O KL ET
Delta Academies Trust
12.
2 7
×
4 5
Mathematics
WEEK 1 INFORMATION FOR TOPIC: PERCENTAGES To write a number as a percentage of another without a calculator, there are two steps:
To express a change as a percentage without a calculator, again there are two steps:
1) Write it as a fraction
1) Write the difference over the original.
2) Make the denominator 100
2) Make the denominator 100.
Example – Julie scored 17 out of 20 in a test. What is this as a percentage? x5
Example – The price of a jumper increased from £50 to £70. What is this as a percentage change? x2
17 20
=
85 100
= 85%
20 50
x5
=
40 100
= 40%
x2
QUESTIONS FOR TOPIC: PERCENTAGES 1) Write these test scores as a percentage: a) 13 out of 20 b) 19 out of 25 c) 43 out of 50
d) 7 out of 10 e) 38 out of 40 2) Find the percentage change in these prices:
f) 18 out of 30
a) £20 to £27 b) £10 to £18 c) £25 to £18
d) £50 to £42 e) £40 to £42
f) £150 to £180
MATHS PUZZLE
MATHS CHALLENGE
Can you arrange the numerals 1 to 9 (1, 2, 3, 4, 5, 6, 7, 8 and 9) in a single fraction that equals exactly An example that doesn’t work:
1 3
7192 38456
The folding challenge:
(one third)?
= 0.187
See if you can complete the folding challenge with Matt Parker – you just need a piece of paper.
Y O U R S U M M E R B O O K LE T
Delta Academies Trust
3
Mathematics
WEEK 2 SKILLS QUESTIONS 1. List the factors of 12
2. What is the highest common factor of 12 and 18 ?
3. Evaluate 53
4. Write 0.2 as a fraction.
5. Which is larger 0.15
6. Calculate 16 – -7
7. Calculate 15.26 – 11.5
8. Calculate 6 × 2.8
10. Find 35% of 120
11.
13. Solve 6x + 2 = 20
14. Find the missing numbers in the sequence 11, ___, 21, 26, ___
15. Find the nth term of 5, 10, 15, 20, 25, …
16. What is the area of a triangle with base 6 cm and height 4 cm?
17. A square has perimeter 36 cm. What is its side length?
18. What is the size of each angle in an equilateral triangle?
19. Find the median of 5, 3, 5, 4, 3, 4, 5, 5, 2
20. A letter is chosen at random from the word GOOGLE. What is the probability of choosing a letter O?
or
1 8
9. What is
4
1 7
of 42 ?
YOU R S U MMER BO O KL ET
Delta Academies Trust
1 4
+
2 5
12.
6 11
×
3 7
Mathematics
WEEK 2 INFORMATION FOR TOPIC: RATIO Simplifying – To write a ratio in its simplest form we divide both parts by the highest common factor of each number.
Share in a ratio – To share an amount in a given ratio, we draw a bar model.
Example – Write 18:60 in its simplest form
There are 7 boxes to share 42 between.
2
6
6
=12
42 ÷ 7 = 6
5
6
6
6
The highest common factor of 18 and 60 is 6. So we divide by 6.
:6
18 : 60
Example – Share 42 in the ratio 2 : 5.
:6
3 : 10
6
6
=30
So 6 goes in each box.
QUESTIONS FOR TOPIC: RATIO 1) Write each ratio in its simplest form a) 4 : 6 b) 10 : 25 c) 42 : 14 d) 60 : 12
e) 14 : 64 f) 8cm : 24mm
2) Share these amounts in the given ratio: a) £40 in the ratio 7 : 3 b) $64 in the ratio 3 : 5
c) 80 in the ratio 10 : 6
d) £35 in the ratio 1 : 4 e) $140 in the ratio 9 : 5 f) 400 in the ratio 2 : 3 : 5
MATHS PUZZLE
MATHS FACTS
You are at an unmarked intersection. One way is the City of Lies and the other way is the City of Truth.
How to make a sundial Don’t forget most iPhones have a compass on them
Citizens of the City of Lies always lie.
Citizens of the City of Truth always tell the truth.
A citizen of one of those cities (you don’t know which) is at the intersection. What question could you ask to them to find the way to the City of Truth?
Y O U R S U M M E R B O O K LE T
Delta Academies Trust
5
Mathematics
WEEK 3 SKILLS QUESTIONS 1. List the factors of 18
2. What is the highest common factor of 24 and 18 ?
3. Evaluate √81.
4. Write 0.75 as a fraction
5. Which is larger 0.1
6. Calculate -14 + -7
7. Calculate 16.5 – 10.41
8. Calculate 6.1 × 2.4
10. Find 2.5% of 120
11.
14. Find the missing numbers in the sequence
15. Find the nth term of 12, 15, 18, 21, 24, …
or
1 9
9. What is
2 5
of 45 ?
13. Solve 4 + 3x = 25
17. A rectangle has a perimeter of 14 cm. If its width is 2 cm, what is its length?
6
1 5
–
1 10
YOU R S U MMER BO O KL ET
Delta Academies Trust
4 5
÷
1 10
16. A square has an area of 144 cm2. What is its side length?
___, ___, 13, 17, 21
18. Three angles in a quadrilateral are 45, 120 and 60. What is the fourth angle?
12.
19. Find the median of 7, 8, 12, 13, 15, 4, 4, 12
20. A fair six sided dice is rolled. What is the probability of getting a square number?
Mathematics
WEEK 3 INFORMATION FOR TOPIC: ALGEBRA – SIMPLIFY AND MANIPULATE Simplifying
Expand brackets – Multiply the term outside the bracket by each term inside the bracket
To simplify an algebraic expression that is made up of terms which are added or subtracted we collect like terms. Example – Simplify 4a + 2b – a – 6b
Examples
4a + 2b – a – 6b
5(4t – 6) = 20t – 30
3a – 4b
6e(2e + f) = 12e2 + 6ef
Sometimes we need to simplify an algebraic expression where the terms are multiplied together, see the examples below: 4 × x × y × 2 = 8xy
a × b × c = abc
y × y × y × 3 = 3y
3
QUESTIONS FOR TOPIC: ALGEBRA – SIMPLIFY AND MANIPULATE a) 1) Simplify: 4a + a + 2a b) 3t – t + 2t c) 5a + 2b + a + 3b d) 4t – t + 2p – 5p
e) a × b f) b × b × b g) 2 × b × c × d h) p×2×p×5
2) Expand: a) 7 (2r + 5) b) 4 (y – 9) c) 2a (a + 7) d) 3p (4p – 5)
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.
Y O U R S U M M E R B O O K LE T
Delta Academies Trust
7
Mathematics
WEEK 4 SKILLS QUESTIONS 2. What is the highest common factor of 32 and 24 ?
3. Evaluate √27
4. Write 0.3 as a fraction
5. Which is larger 0.9
6. Calculate -5 × -12
7. Calculate
8. Calculate 4.26 × 2.3
or
4
312.4 – 284.36
5
5
9. What is
13. Solve
9
x 7
of 63 ?
=4
17. A rectangle has a perimeter of 36 cm. If its width is 8 cm, what is its length?
2
–
3
4
2
10. Find 12.5% of 80
11.
14. Find the missing numbers in the sequence
15. Find the nth term of 10, 8, 6, 4, 2, …
16. A square has an area of 121 cm2. What is its side length?
19. Find the mean of
20. A fair ten sided dice is rolled. What is the probability of getting a number greater than 4?
8
3
1. List the factors of 24
3
7
20, 17, ___, ___, 8
18. One angle in a parallelogram is 60. What are the other three angles?
YOU R S U MMER BO O KL ET
Delta Academies Trust
10, 8, 9, 8, 5
12.
9
÷
3
Mathematics
WEEK 4 INFORMATION FOR TOPIC: 3D SHAPES AND VOLUME 3D shapes and volume
Below are the formulas for volumes of simple shapes.
We need to know how to identify the number of faces, edges and vertices of a shape. Triangular prism:
Prism
Cuboid height
- 5 faces, three rectangles and two triangles - 9 edges.
len
gt
- 6 vertices (corners)
h
h idt
w
Cuboid volume = l × w × h
cross section
length
Prism volume = area of cross section × length
QUESTIONS FOR TOPIC 1) Complete the table: Shape
Shape
Faces
Edges
Vertices
Cube Cuboid Tetrahedron Square based pyramid Cylinder 8
2) Find the volume: 5
8 2 4
8 cm
Area 12 cm2
14 cm
7 cm
8
5 cm
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. • Liverpool were not third. • There were two teams between Manchester City and Derby County List the order the teams finished in.
Y O U R S U M M E R B O O K LE T
Delta Academies Trust
9
Mathematics
WEEK 5 SKILLS QUESTIONS 3
1. List the first 5 multiples of 42
2. What is the lowest common multiples of 14 and 22 ?
3. Evaluate √125
4. Write 0.65 as a fraction in its simplest form
5. Which is larger 0.62
6. Calculate -60 ÷ 12.
7. Calculate
8. Calculate 26.4 × 5.3
or
2
14.28 – 7.864
3
9. What is
13. Solve
5 12
x 2
of 72 ?
+5=9
7
8
14. Find the missing numbers in the sequence
15. Find the nth term of 30, 25, 20, 15, 10, ...
16. A square has an area of 100 cm2. What is its perimeter?
19. Find the mean of 7, 10, 8, 8, 14, 13
20. What is the probability of choosing a spade, from an ordinary pack of cards?
+
9
___, ___, 16, 13, 10
17. What is the area of a triangle with base 10 cm and height 4 cm?
10
7
11.
11
18. One angle in an isosceles triangle is 30. What could the other two angles be?
YOU R S U MMER BO O KL ET
Delta Academies Trust
12.
8
10. Find 15% of £16
11
×
12
Mathematics
WEEK 5 INFORMATION FOR TOPIC: PROBABILTY Probabilities can be shown on a scale.
Even Chance
Impossible
Certain
Unlikely 0 0% 0
Likely
0.25
0.5
0.75
25% 1 4
50% 1 2
75% 3 4
We can calculate probabilities using: Probability =
1 100% 1
number of successful outcomes number of possible outcomes
QUESTIONS FOR TOPIC: PROBABILTY When rolling a fair dice, find the probability of A. rolling a 2 C. rolling a 7 E. rolling a number less than 4 B. rolling an odd number D. rolling a number bigger than 4
1
When picking a counter from this tub, find the probability of C. picking a black counter A. picking a white counter D. picking a grey or white counter B. picking a grey counter From this bag of numbered balls, find the probability of
2
1
6
3
5
70
0
0
70 300
50
10
4
0
7 4
2
3
20
1000
50
MATHS PUZZLE The Monty Hall Problem This is a really popular maths problem – all about a game show
A. picking the number 1 B. picking the number 7 C. picking a number less than 5
D. picking an odd number E. picking an even number F. picking a number less than 8
When spinning this spinner, find the probability of getting A. a score of 1000 B. more than 100 C. less than 100
MATHS FACTS Do you want to be able to multiply any number by 11 in your head? You should be able to do any single digit number by learning your times tables. To multiply a two digit number e.g. 18 × 11 1) Write the first digit 2) Add the two digits together and write it next 3) Write the second digit The answer would be 198
1 1+ 8 = 9 8
If we did 48 × 11 4 1) Write the first digit 4+ 8 = 12 2) Add the two digits together and write it next 8 3) Write the second digit The answer would be 5 (4 + 1 from the 11) 28 = 528 To multiply a three digit number e.g. 124 × 11 1) Write the first digit 1 2) Add the next two digits together 1+ 2 = 3 2+ 4 = 6 3) Move across and add the next two digits together 4 4) Write the last digit The answer would be 1364
Y O U R S U M M E R B O O K LE T
Delta Academies Trust
11
Mathematics
WEEK 6 SKILLS QUESTIONS 1. List all of the prime numbers that are below 30
2. What is the highest common factor of 120 and 210 ?
3. Evaluate 23 + 42
4. Write 0.82 as a fraction in its simplest form
5. Which is larger 0.7
6. Calculate -80 ÷ -16
7. Calculate 0.94 – 0.485
8. Calculate 0.24 × 5.1
10. Find 12.5% of £30
11.
14. Find the missing numbers in the sequence
15. Find the nth term of 5, 3, 1, -1, -3, ...
16. A square has an area of 49 cm2. What is its perimeter?
19. Find the mean of
20. What is the probability of choosing an ace, from an ordinary pack of cards?
or
5 7
3
9. What is
13. Solve
8
x 3
of 48 ?
– 5 = 15
17. What is the area of a triangle with base 15 cm and height 8 cm?
12
7 12
–
3 8
12. 1
3 4
×
6 7
-6, -3, ___, ___, 6
18. One angle in an isosceles triangle is 50. What could the other two angles be?
YOU R S U MMER BO O KL ET
Delta Academies Trust
14, 20, 16, 15, 20
Mathematics
WEEK 6 INFORMATION FOR TOPIC: CIRCLES We need to know the names of the parts of a circle. Circumference
We also need to know:
ra
di
us
Sector
= 3.14159…….
Chord
Area of a circle =
diameter Segment
Circumference of a circle =
t
en
g Tan
× r2 ×d
Arc
QUESTIONS FOR TOPIC: CIRCLES 1) Find the area and circumference of: a)
b)
c)
6 cm
2.4 mm
1.5 m
2) Find the area and perimeter of these semi-circles: a)
b)
4 cm
8m
MATHS PUZZLE Complete the Sudoku. The digits 1 to 9 must appear in each column, row and box only once.
MATHS FACTS
6 9
7 9 4 1 3
5
3 4 5
8 6 3 5 1 2 4
How to measure very tall things you can’t reach to the top of!
4 7 1 5 4 8 1 4 6 7 4 2 6 8 9
Y O U R S U M M E R B O O K LE T
Delta Academies Trust
13
Mathematics
ANSWERS WEEK 1
WEEK 2
WEEK 3
WEEK 4
WEEK 5
WEEK 6
1
7, 14, 21, 28, 35
1, 2, 3, 4, 6, 12
1, 2, 3, 6, 9, 18
1, 2, 3, 4, 6, 8, 12, 24
42, 84, 126, 168, 210
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
2
35
6
6
8
154
30
3
49
125
9
3
5
24
1
1
3
3
13
41
4
5
4
10
20
50
2
5
3
7
4 5
3
0.15
6
-13
7
1 9
0.9
23
-21
60
-5
5
35.76
3.76
6.09
28.04
6.416
0.455
8
12.6
16.8
14.64
9.798
139.92
1.224
9
5
6
18
35
30
18
10
12
42
3
10
£2.40
£3.75
5
13
1
5
6
20
10
21
8
18
35
77
13
x=9
x=3
x=7
x = 28
x=8
x = 60
14
20, 24
16, 31
5, 9
14, 11
22, 19
0, 3
15
3n + 1
5n
3n + 9
-2n + 12
-5n + 35
-2n + 7
16
48 cm2
12 cm2
12 cm
11 cm
40 cm
28 cm
17
20 cm
9 cm
5 cm
10 cm
20 cm2
60 cm2
18
110
60
135
60 , 120 , 120
30 , 120 , or 75 , 75
50 , 80 or 65 , 65
19
5
4
10
8
10
17
2
1
1
3
1
1
11
3
3
5
4
13
11 12
20
14
1
YOU R S U MMER BO O KL ET
8
Delta Academies Trust
1
52
5
99
24
2
14
3
33
1
1 2
Mathematics
ANSWERS Week 1 Questions for topic 1) a) 65% b) 76% 2) a) 35% b) 80%
c) 86% c) 28%
Week 2 Questions for topic 1) a) 2 : 3 b) 2 : 5 c) 3 : 1 2) a) £28 : £12 b) $24 : $40 c) 50 : 30 Week 3 Questions for topic 1) a) 7a b) 4t e) ab f) b3 2) a) 14r + 35 b) 4y – 36 Week 4 Questions for topic Shape Shape 1)
d) 70% d) 16%
e) 95% e) 5%
f) 60% f) 20%
d) 5 : 1 d) £7 : £28
e) 7 : 32 f) 10 : 3 e) $90 : $50 f) 80 : 120 : 200
c) 6a + 5b d) 3t – 3p g) 2bcd h) 10p2 2 c) 2a + 14a d) 12p2 – 15p
Faces
Edges
Vertices
Cube
6
12
8
Cuboid
6
12
8
Tetrahedron
4
6
4
Square based pyramid
5
8
5
Cylinder
3
2
0
2) a) 40 b) 512 c) 84 cm3
d) 280 cm3
Week 5 Questions for topic 1
1) A. 6 1
1
B. 2 3
C. 0 3
1
D. 3 5
2) A.
B.
C.
D.
3) A.
B.
C.
D.
4) A.
B.
C.
4 1
7 1 8
8 1
7 1 2
8 4
7 3
8 4 7
E.
1 2
3 E. 7
F. 1
8
Week 6 Questions for topic 1) a) Area = 28.3 cm2, Circumference = 18.8 cm b) Area = 7.1 m2, Circumference = 9.4 m c) Area = 4.5 mm2, Circumference = 7.5 m 2) a) Area = 6.3 cm2, Perimeter = 10.3 cm b) Area = 100.5 m2, Perimeter = 41.1 m
Y O U R S U M M E R B O O K LE T
Delta Academies Trust
15
Academies Trust Education House, Spawd Bone Lane, Knottingley, WF11 0EP T: 0345 196 0033 | info@deltatrust.org.uk | www.deltatrust.org.uk Summer 2020