Algebra - PBworksthomaswhitham.pbworks.com/f/Algebra+booklet+year+7[1].pdf · Thomas Whitham Sixth Form | 3 Algebra (3) Fill in the missing gaps In each of the following questions

THOMAS WHITHAM SIXTH FORM

Year 7

Algebra Mathematics

S. J. Cooper

B U R N L E Y C @ M P U S , B U R N L E Y , L A N C A S H I R E , B B 1 0 1 J D . T E L . 6 8 2 2 7 2

Thomas Whitham Sixth Form |

1

Algebra (1) Collection of like terms

Simplify each of the following terms:

1. aaa 243

2. bbb 38

3. pp 76

4. nnnn 473

5. ccc 95

6. mmm 410

7. dddd 63

8. eee 736

9. ff 1512

10. xxx 61321

11. ggg 67

12. yyy 1048

13. rrr 36

14. ttt 758

15. iii 854

16. kkk 4312

17. wwww 738

18. aaa 459

19. qqqq 223

20. yyyy 57

21. baba 238

22. dcdc 345

23. pqpq 437

24. yxyx 8429

25. tsts 355

26. fefe 627

27. hghg 243

28. yxyy 726

29. nmnm 4268

30. jiji 36410

31. yxyx 364

32. pkpk 475

33. cbcb 3879

34. yxyx 6688

35. baba 7794

36. vuvu 26

37. wwzz 486

38. mnnnm 34235

39. cbacba 2537

40. gfegfe 45538

41. jihjih 3679

42. pnmpnm 644827

43. tsrsrt 9779

44. zyxzyx 348105

45. gfegfe 913111715

46. bababa 5232

47. yxyxyx 223

48. rqprqp 7251058

49. wvuwvu 5877

50. xxyx 437

Find a simplified expression for the perimeters of the following:

51. 52. 53. 54.

a

a

a a 3y

2x 4t

b 3y

2x

b

b

4t

4t

4t

4t


2

Algebra (2) Simplifying Expressions

Simplify each of the following terms:

1. x7

2. ba 43

3. yx 35

4. dc 26

5. qp47

6. xy 82

7. nm 75

8. de 54

9. af 98

10. rs3

11. ba 43

12. nm 67

13. pq 412

14. jk 9

15. xy 37

16. vu 115

17. xx

18. pp

19. aa 92

20. rs 65

21. rr 4

22. ih 77

23. rqp 32

24. cba 326

25. gfe 74

26. zyx 64

27. tt 154

28. rpq 36

29. yy 85

30. nm 27


3

Algebra (3) Fill in the missing gaps

In each of the following questions fill in the missing gaps:

1. What number should be written in the circles in the following:

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

7 +5

7 –3

7 X 2

7 + x

7 –y

a + b

p X r

y + 2 x

3d –2d

5h X 3i


4

2. What should be written in the rectangles in the following?

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

–2

X 5

X 2

+ a

– b

X f

+ 2 x

–5d

X 3a

+ 3

7

9

20

2a

2 + a

x - b

ef

7x

2g –5d

9ab


5

3. What should be written next to the sign in the following:

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

4 +

12 –

3 +

p X

5s +

b X

2y –

9d –

5h X

4 x

12

12

4

3 + c

8p

5s + r

4bc

2y – 3x

2d

30hg


6

4. What should be written in the circles or rectangles for each of the following:

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

5 Double, and add 3

Double, and subtract 1

11

3 Multiply by 3 and add x

a Multiply by 3 and add x

2 Multiply by a and subtract b

d Double, and subtract 1

Double, and add 5

2x + 5

Multiply by 4 and subtract 2

4a –2

3p Multiply by 4 and subtract 3q

Double, and add 2k

8 + 2k


7

Algebra (4) Simple Equations

Exercise 1

Find the number which must be placed in the box to make both sides equal.

1. • + 2 = 5 6. 2 + • = 6 11. 7 – • = 5 16. • – 6 = 4

2. • + 4 = 6 7. 4 + • = 7 12. 9 – • = 6 17. • – 2 = 9

3. • + 6 = 9 8. 3 + • = 6 13. • – 2 = 3 18. • + 7 = 13

4. • + 3 = 7 9. 8 – • = 6 14. • – 3 = 5 19. • – 7 = 18

5. • + 7 = 9 10. 6 – • = 2 15. • – 4 = 5 20. • + 14 =

27

Exercise 2

Find the number which has to replace the letter to keep both sides equal.

1. x + 3 = 5 6. y + 4 = 8 11. a + 3 = 11 16. 9 – p = 8

2. y + 3 = 9 7. x + 9 = 12 12. p – 7 = 2 17. 7 – y = 2

3. x + 5 = 9 8. d + 4 = 10 13. x – 4 = 4 18. 12 – r = 8

4. x + 2 = 8 9. 2 + n = 7 14. u – 7 = 8 19. y – 3 = 2

5. x + 3 = 5 10. 1 + p = 8 15. b – 5 = 6 20. 6 – n = 3

Exercise 3

Find the value of the letter in each of the following:

1. 8x = 24 5. 6c = 48 9. 12m = 36 13. 8x = 72

2. 7x = 35 6. 2n = 40 10. 25q = 50 14. 5d = 40

3. 9a = 54 7. 30r = 90 11. 15t = 60 15. 12f = 84

4. 4b = 28 8. 3y = 27 12. 7e = 28 16. 20q = 100

Exercise 4

Find the number which has to replace the letter to keep both sides equal:

1. 2 3 9x 6. 3 4 16b 11. 4 3 5q

2. 2 6 8x 7. 3 3 18m 12. 4 8 10r

3. 2 4 12x 8. 3 7 7n 13. 6 2 10t

4. 2 1 13y 9. 2 4 2y 14. 5 9 14x

5. 3 2 8a 10. 2 7 3y 15. 5 6 6v


8

Algebra (5) Simple Equations

Exercise 5

Solve the following equations:

1. 2 4 8x 2. 2 3 13x 3. 3 2 10x 4. 3 5 4x 5. 4 1 23x

6. 5710 x 7. 232 x 8. 712 x 9. 3847 x 10. 576 x

11. 5 4 3 x 12. 19 7 2 x 13. 68 8 12 x 14. 7 4 3 x 15. 20 6 7 x

16. 49 9 14 x 17. 6 1 20x 18. 2 4 10x 19. 2 3 13x 20. 3 1 10x

21. 3 5 11x 22. 1934 x 23. 10 7 57x 24. 1032 x 25. 2 8 13x

26. 5 7 32x 27. 7 4 25x 28. 23 4 7 x 29. 7 2 1 x 30.

68 6 8 x

Exercise 6

Solve each of the following equations:

1. 6235 xx

2. 26354 yy

3. 23247 qq

4. 40675 aa

5. 31342 xx

6. 5276 mm

7. 234110 nn

8. 604613 rr

9. 53379 pp

10. 182124 cc

11. 14587 xx

12. 6398 tt

13. 4636 bb

14. 34323 dd

15. 231124 gg

16. 532105 ff

17. 4425 xx

18. 87238 yy

19. 7841211 ee

20. 56347 hh

Exercise 7

Solve the following:

1. 14226 xx

2. 8215 aa

3. 25357 yy

4. gg 4765

5. 32328 ee

6. pp 2153

7. rr 38917

8. tt 2184

9. uu 6348

10. yy 54154

11. 374911 ee

12. 681211 ww

13. 147912 ff

14. 79410 kk

15. hh 4419

16. vv 64532

17. 225411 ss

18. 208815 zz

19. 4881720 yy

20. xx 21992


9

Algebra (6) Construction of equations

1. Asif is 2 years older than Kamran. If Kamran is x years old, how old is Asif?

2. The weight of a bag of bananas is 3 kg lass than a bag of apples. If the bag of apples weigh W

kg what weight is the bag of bananas?

3. A taxi travelling between two towns has a charge of £3 plus a further £2 per person.

What is the cost for a taxi with

(a) one passenger? (b) three passengers? (c) x passengers?

4. An end of year examination consists of three sheets of A4 paper.

How many sheets of paper will be needed for

(i) one pupil? (ii) two pupils? (iii) five pupils? (iv) y pupils?

5. To cook a turkey properly it must spend 40 minutes per pound in the oven on gas mark 7.

What is the length of time required to cook a turkey of mass

(a) 2 pounds? (b) 10 pounds? (c) 15 pounds? (d) m pounds?

6. “Garden Gro” advertisers their plant food as 5 ml required per litre of water.

How many millilitres will you require for (a) 3 litres? (b) 8 litres? (c) p litres? (d) f litres?

7. A tube of smarties contains, on average, 40 smarties per tube.

How many smarties are you likely to eat in

(i) 2 tubes? (ii) 4 tubes? (iii) 10 tubes? (iv) s tubes?

8. A number N is equal to the sum of 8 plus another number a.

Write down a formula connecting N and a.

9. A second number S is equal to the sum of two numbers p and q.

Write down a formula connecting S, p and q.

10. The number A is equal to twice the number b plus 3.

Using algebra express this as a formula.

11. The number x is equal to three times the number y minus z.

What formula can be written for this statement.


10

12. Paul’s bag contains p books, q pens and r pencils. If the total number of items in his bag is T

write down a formula connecting T, p, q and r.

13. At school pens are sold at x pence each and pencils are sold at y pence each.

Write down the total cost, C, for a) 2 pens and 1 pencil.

b) Three pens and four pencils

c) m pens and n pencils.

14. Write down an expression for the perimeter of

this rectangle.

15. Danielle can run at an average speed of v metres per second.

How many metres will she cover in

(a) 2 seconds? (b) 5 seconds? (c) 1 minute? (d) g seconds?

x + 3

x


11

Algebra (7) Construction of equations

1. I think of a number, double it and then add 3. The answer is 15. What was my number?

2. I think of a number, double it and subtract 5. The answer is 7. What was my number?

3. I think of a number, treble it and add 4. The answer is 22. What was my number?

4. I think of a number, multiply it by 3 and subtract 5. The answer is 10. What was my number?

5. I think of a number, multiply it by 4 and subtract 1. The answer is 15. What was my number?

6. I think of a number, multiply it by 7 and add 4. The answer is 39. What was my number?

7. Given the perimeter of the rectangle opposite is 30

cm find its length and width

8. If the perimeter of the triangle is 45 cm find the sides of the triangle.

9. (a) Work out an expression for the perimeter of the rectangle opposite.

(b) Given the perimeter is 54 cm find the value of h.

10. Given the area of the rectangle above is equal to 80 cm2 work out the value of h.

11. I think of a number, double it and add 5. The result is exactly the same as adding 6 to the

number. What is the number?

12. I think of a number, multiply it by 3 and subtract 5. The result is equal to the same number plus

9. What is the number?

13. I think of a number, multiply it by 3 and add 1. The answer is equal to the same number

doubled add 13. What is the number?

x + 3

x

y

2y 2y

h

20 cm


12

Algebra (8) Substituting into formulae

1. If a p 4 find a when p 7

2. If m n 7 find m when n 3

3. Given t s 6 work out t when s 9

4. Given a c b what is the value of a when b 14and c 13?

5. If x y z what is the value of x when y 34and z 19?

6. If y x 2 1 work out the value of y given x 5 .

7. If y x 4 7 what is the value of y when x 2 ?

8. Given d e 6 2 work out the value of d when e 3 .

9. If q p 3 4 what is the value of q when p 12?

10. If g h 8 4 find the value of g when h 5.

11. If m nr 7 work out the value of m when n 8 and r 2 .

12. Given y mx c what is the valuw of y when m 2 , x 6 and c 1?

13. Given v u at find v when u 7 , a 3 and t 10.

14.Evaluate S T U 3 when T 9 and U 5.

15. If VM

D work out V when M 144 and D 3.

16. Given SD

T work out S for D 240 and T 12 .

17. If h i 6 3 work out h for i 1.


13

18. Given m n 4 2 work out the value of m when n 9 .

19. Evaluate P q r 5 when q 5 and r 2 .

20.Given y x 2 what is the value of y when x 6?


14

Algebra (9) Coordinates

1. Write down the coordinates for each letter in the diagram below.

2. (a) Draw a set of axes labelled from 0 to 8 on both the x and y axis.

(b) Plot the following coordinates on the set of axes

(i) (3, 1)

(ii) (8, 7)

(iii) (0, 4)

(iv) (5, 2)

(v) (1, 0)

(vi) (6, 3)

(vii) (2, 7)

(viii) (4, 4)

(ix) (8, 0)

(x) (7, 5)

3. (a) Draw a set of axes labelled from 0 to 8 on both the x and y axis

(b) Plot the coordinates for each of the following and join each coordinate with the next

coordinate, using a ruler.

(i) (2, 1), (7, 1) , (5, 6)

(ii) (0, 3) , (8, 3), (6, 7) (2, 7)

(iii) (0, 4) , (2, 0), (5, 1), (5, 4)

(iv) (8, 8) , (3, 1), (4, 6)

(v) (5, 0) , (2, 7), (0, 1)

1

1

2

3

4

5

6

7

8

9

10

2 3 4 5 6 7 8 9 10 0

x

y

N Q

R

H

B

A

F

K

I

L

J

M

U

P

T

S

G

E

D

C


15

Algebra (10) Coordinates

4. Write down the coordinates for each letter in the diagram below.

5. (a) Draw a set of axes labelled from –8 to 8 on both the x and y axis.

(c) Plot the following coordinates on the set of axes.

(i) (2, 7)

(ii) (3, –1)

(iii) (4, 2)

(iv) (–7,1)

(v) (4, –6)

(vi) (–5, 5)

(vii) (–3, –8)

(viii) (0, 4)

(ix) (2, –2)

(x) (–1, 0)

(xi) (6, 0)

(xii) (–2, 3)

(xiii) (8, –4)

(xiv) (–5, 3)

(xv) (–1, –5)

6. (a) Draw a set of axes labelled from –10 to 10 on both the x and y axis

(c) Plot the coordinates for each of the following and join each coordinate with the next

coordinate, using a ruler.

(vi) (–2, 1), (7, 1) , (–5, 6)

(vii) (0, –3) , (8, –3), (6, 7) (2, 7)

(viii) (0, –4) , (2, 0), (–5, –1), (–5, 4)

(ix) (8, –8) , (–3, 1), (4, –6)

(x) (–5, 0) , (–2, 7), (0, –1)

–4 –3 –2 –1 1 2 3 4 5 6 x –5 –6 0

–4

–3

–2

–1

1

2

3

4

5

6

y

–5

–6

N

Q

R

H

B

A

F

K

I

L

J M

U

P

T

S

G

E

D

C


16

Algebra (11) Sequences

1. Write down the next two terms for each of the following sequences and a rule in words.

(a) 2 5 8 11

(b) 1 8 15 22

(c) 8 11 14 17

(d) 5 11 17 23

(e) 6 10 14 18

(f) 4 12 20 28

(g) 9 11 13 15

(h) 7 12 17 22

(i) 7 18 29 40

(j) 13 21 29 37

2. Write down the next two terms for each of the following and give a rule for continuing the

sequence.

(a) 20 18 16 14

(b) 37 33 29 25

(c) 99 88 77 66

(d) 47 44 41 38

(e) 56 50 44 38

(f) 80 71 62 53

(g) 63 56 49 42

3. Write down the first five terms for each of the following described sequences.

a) Add 2 to the previous term: 5, …….

b) Add five to the previous term: 9, …….

c) Subtract 1 from the last term: 10, ……

d) Subtract 8 from the previous term: 65, ……

e) Multiply the previous term by 2: 1, …..

f) Multiply the previous term by 3: 3, ……

g) Divide the last term by 2: 64, ……

h) Add the next even number each time: 1, 3, ……

i) Add a number that increases by 2 each time: 3, 4,


17

4. Adding together the previous two terms generates the Fibonacci sequence. The first six terms

of the Fibonacci sequence starting with 1, 1 are 1, 1, 2, 3, 5, 8.

(a) Write down the next three terms in this sequence.

(b) Form a new Fibonacci sequence starting with 2, 3 and write down the next five terms.

5. Fill in the missing gaps in the following sequences:

a 4, ... , 12, 16, ..., ... b 3, ..., 13, 18, ..., ... c 1, ..., 5, ..., 9, ...

d 2, 4, ..., 16, ...., ... e 20, ..., 14, ..., 8, ... f 9, 18, ...., ...., 45, ...

6. For each of the following sequences give the first four terms:

a Add 3 to the previous term: 5, 8, ...

b Divide the previous term by 2: 8, 4, ...

c Subtract 4 from the previous term: 15, 11, ...

d Multiply the previous term by 3: 2, 6, ...

e Add 9 to the previous term: 1, 10, ....

7. The coordinates below form a sequence of pairs of numbers

(1, 3) , (2, 5) , (4, 7) , (8, 9)

a Find the next two sets of coordinates in the sequence.

b Find the tenth set of coordinates in the sequence.

c Describe the rule for finding (i) the x coordinates (ii) the y coordinates.

8. For each of the following sequences below give the next three terms and a rule for continuing

the sequence.

a 1, 2, 3, 4, 5, 6, ... , ... , ... b 1, 3, 5, 7, 9, ..., ..., ... c 4, 8, 12, 16, ..., ... , ...

d 7, 14, 21, 28, ..., ..., ... e 17, 15, 13, 11, ..., ..., ... f 1, 4, 9, 16, ..., ..., ...

g 1, 3, 6, 10, ..., ..., ... h 4, 2, 1, 1

2 ,

1

4, ..., ..., ... i 2, 6, 12, 20, ..., ..., ...

9. For each of the following sequences find the next two terms and a rule for continuing the

sequence.

a 21, 17, 13, ..., ... b 8, 16, 24, ..., ... c 2, 4, 8, 16, ..., ...

d 1, 3, 9, 27, ..., ... e 1, 8, 27, ..., ... f 6, 12, 18, ... , ...


18

Algebra (12) Sequences

1. The following collections of dots suggest another sequence of numbers.

The sequence is started below.

1, 4, 9, 16, etc.

a) Draw the next two patterns in the sequence.

b) Write down the next seven terms in this sequence

c) Write down the values of (i) the 2nd term (ii) the 10th term.

d) What sorts of numbers belong to this sequence?

2. Using the squares in your books draw the next two arrangements for each of the following

patterns:

a

b

c

d


19

e

f

3. Triangular numbers can be represented by dots arranged as triangles:

Draw and write down the next four in the sequence.

4. John has constructed a triangle using three pencils, as shown in diagram 1.

Elizabeth then added some pencils to John’s and constructed diagram 2.

Brian finally added some to Elizabeth’s and constructed diagram 3.

(a) Draw the next two patterns in the sequence.

(b) Copy and complete the table below.

Diagram number 1 2 3 4 5 6 7

Number of pencils 3 5

(c) State a rule in words for continuing the pattern.

5. Using a new set of pencils a series of diagrams is constructed forming squares.

1 3 6 10


20

(a) Draw the next two patterns in the sequence.


Diagram number 1 2 3 4 5 6 7

Number of pencils 4 12

(c) State a rule in words for continuing the pattern.

6. Ian is going to build a garden path in the shape of a cross. He starts by laying the centre

paving stone which is in a dark colour, as shown in figure one. He then places several lighter

paving stones around the centre stone as shown in figure 2. Next he continues to alternate the

colour of the paving stones as shown in figures 3 and 4.

Figure1 Figure2 Figure 3 Figure4

(a) Draw the next two diagrams in the sequence.


Figure number Number of dark

paving stones

Number of light

paving stones

1

2

3

4

5

6

7

(c) State a rule in words for connecting the figure number and

(i) the number of dark paving stones.

(ii) the number of light paving stones.


21

Algebra (13) The nth term

1. Give the first four terms of the sequence for which

a) 53 nUn

b) 15 nUn

c) nU n 4

d) 12 nUn

e) 58 nUn

f) 45 nUn

g) 2nU n

h) 22 nUn

i) 3nU n

j) 12 nUn

k) nUn 315

l) 39 nUn

2. Find the first 5 terms for the sequence 37 nVn

3. Find the first 4 terms for the sequence 2 nnVn

4. Find the first 5 terms for the sequence 1

n

nVn

5. Find the first 5 terms for the sequence 13

12

n

nVn

6. What nth term (formula) would give you the sequence

2, 5, 8, 11, 14, 17


22

Algebra(14) Straight line graphs

1. a) Using the formula 𝑦 = 2𝑥 + 5 complete the table below for different values of x.

x 0 1 2 3 4

y

b) Plot each of the points found above on a set of axes labeling the x axis from 0 to 5 and

the y axis from 0 to 15

2. Given 𝑦 = 3𝑥 − 1, find the different values of y for the values of x stated in the table below.

x 1 2 3 4 5

y

Copying the axes drawn below, draw the graph of 𝑦 = 3𝑥 − 1 labelling your axes carefully.

3. By completing the table below, plot the graph of 𝑦 = 4𝑥 + 2

x 0 1 2 3 4

y

4. (a) By completing the table below, plot the graph of 𝑦 = 3𝑥 − 4

x 2 3 4 5

y

(b) Using your graph find the value of y when 𝑥 = 1.

2

4

6

8

10

-2

-4

-6

2 4 -2 -4 0 x

y


23

5. (a) By completing the table below, plot the graph of 𝑦 = 4𝑥 − 7, labelling your x axis – 5 to 5

and the y axis from –10 to 15

x 2 3 4 5

y

(b) Using your graph find the value of y when (i) 𝑥 = 1. (ii) 𝑥 = −1

6. For each of the following

(i) Copy and complete the table

(ii) draw a set of axes labeling the x axis from – 5 to 5 and the y axis from –5 to 16

(iii) draw the graph of the straight line found.

a) 𝑦 = 𝑥 + 7

x 0 1 2 3 4

y

b) 𝑦 = 2𝑥 − 1

x 1 2 3 4

y

c) 𝑦 = 3𝑥 + 2

x 0 1 2 3 4

y

d) 𝑦 = 5𝑥 − 4

x 1 2 3 4

y

7. Using your own table, draw each of the following graphs.

a) 𝑦 = 𝑥 + 5

b) 𝑦 = 2𝑥 + 8

c) 𝑦 = 3𝑥 + 7

d) 𝑦 = 4𝑥 − 5

e) 𝑦 = 10 − 2𝑥


24

Algebra(15) removal of brackets

Remove the brackets for each of the following expressions

1. 3 𝑎 + 7

2. 5 𝑏 + 8

3. 9 𝑥 − 3

4. 2 2𝑎 + 5

5. 6 4𝑐 − 7

6. 4 3𝑦 − 1

7. 9 5𝑑 + 6

8. 7 2𝑚 + 9

9. 10 2𝑒 − 3

10. 3 3𝑛 + 8

11. 𝑥 𝑥 − 2

12. 𝑎 2𝑎 + 7

13. 3𝑏 𝑏 − 3

14. 4𝑚 3𝑚 + 2

15. 5ℎ ℎ − 4

16. 4 2𝑎 + 7𝑏

17. 3 6𝑥 − 7𝑦

18. 4 2𝑝 + 3𝑞 − 1

19. 3𝑥 3− 5𝑥

20. 7𝑚 3𝑚 − 8

Algebra - PBworksthomaswhitham.pbworks.com/f/Algebra+booklet+year+7[1].pdf · Thomas Whitham Sixth Form | 3 Algebra (3) Fill in the missing gaps In each of the following questions

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