Green

Book 11

The wee Maths

Book of Big

Brain

Growth

Expressions and Formulae, Patterns and

Gradient

Grow your brain

Guaranteed to make

your brain grow, just

add some effort and

hard work

Don’t be afraid if

you don’t know how

to do it, yet!

It’s not how fast you

finish, but that you

finish.

It’s always better to

try something than

to try nothing.

Don’t be worried

about getting it

wrong, getting it

wrong is just part of

the process known

better as learning.

Page | 2

Expressions and Formulae

N31 I can use algebraic shorthand and gather like terms.

1. Ben was thinking of a number, the last

digit of the password for the laptop, and

Eva hasn’t been told what it is.

At this point, as far as Eva is concerned,

the number can be anything from zero to

nine.

It is unknown and variable.

Eva says to Ben, I’m going to call the

number 𝑃 just like we did in Maths when

you were off with the flu.

(a) Ben says he will tell Eva what double the number is, so Eva wrote

down an expression for this in algebraic shorthand.

Write down, in algebraic shorthand, an expression for double the

variable 𝑃.

(b) Ben changes his mind and says he will tell Eva the answer to five

times the number.

Write down, in algebraic shorthand, an expression for five times

the variable 𝑃.

(c) In your own words write down what you think 7𝑃 means.

(d) Ben eventually writes this down for Eva

10𝑃 = 60

What is the last digit of the password?

Page | 3

2. Ben missed some lessons on algebra, so he has asked Eva to help him.

Eva says that 5𝑝 is an expression which means 5 groups of an

unknown, or variable, number 𝑝.

She also says that if we add a known, or constant, number to 𝑥 we

can also show this in the expression.

Eva showed Ben this note in her jotter

Complete the following which Eva used to test Ben’s understanding.

(a) Write down an expression for five groups of a variable number 𝑥.

(b) Write down an expression for 7 groups of the unknown number 𝑡.

(c) Write down an expression for 6 groups of the variable number 𝑎

plus the constant 10.

(d) Write down an expression for 2 groups of the variable 𝑝 plus 3

groups of the variable 𝑞.

(e) Write down an expression for 5 groups of the variable number 𝑟

plus the constant 8.

(f) Write down an expression for 4 groups of the variable 𝑢 plus 3

groups of the variable 𝑣.

Page | 4

3. Write down an expression which

contains two different variables.

4. Write down an expression which

contains one variable and one

constant.

5. Write down an expression which contains three different variables.

6. Write down an expression which contains one unknown and a one

constant.

7. Rewrite each of the following products without the multiplication

sign.

(a) 4×𝑝 (b) 10×𝑦 (c) 𝑝×3 (d) 𝑧×5

(e) 𝑝×𝑞 (f) 𝑏×𝑐×𝑑 (g) 5×𝑟×𝑠 (h) 𝑝×𝑝

8. Simplify each of the following expressions.

(a) 4𝑦 + 3𝑦 (b) 5𝑝 + 5𝑝 (c) 3𝑥 + 𝑥

(d) 6𝑦 − 2𝑦 (e) 5𝑑 − 𝑑 (f) 4𝑝 + 2𝑞

9. Simplify each of the following expressions.

(a) 2𝑝 + 5𝑝 + 3𝑞 + 4𝑞 (b) 6𝑦 + 7𝑦 + 4𝑣 + 5𝑣

(c) 6𝑥 − 2𝑥 + 3𝑦 + 4𝑦 (d) 8𝑐 − 3𝑐 + 5𝑑 − 𝑑

(e) 5𝑥 + 2𝑥 + 4 + 3 (f) 8𝑦 − 𝑦 + 7 − 4

(g) 3𝑥 + 2𝑦 + 2𝑥 + 5𝑦 (h) 6𝑥 + 7 + 5𝑥 + 3

Page | 5

10. Write these out in algebraic shorthand

(a) 𝑥 + 𝑥 + 𝑥 + 𝑥 + 𝑥 + 𝑥 (b) 𝑎 + 𝑎 + 𝑎 + 𝑎 + 𝑎

(c) 𝑝 + 𝑝 + 𝑝 + 𝑝 + 𝑝 + 𝑝 (d) 𝑡 + 𝑡 + 𝑡 + 𝑡

(e) 𝑟 + 𝑟 + 𝑟 + 𝑟 + 𝑟 + 𝑟 + 𝑟 (f) 3×𝑥

(g) 5×𝑝 (h) 7×𝑟

(i) 8×𝑡 (j) 4×𝑎

11. Gather up these like terms

(a) 5𝑥 + 3𝑥 (b) 6𝑟 + 2𝑟 (c) 5𝑝 + 12𝑝

(d) 4𝑡 + 𝑡 (e) 𝑠 + 12𝑠 (f) 9𝑥 − 5𝑥

(g) 35𝑟 − 13𝑟 (h) 19𝑝 − 11𝑝 (i) 44𝑠 − 33𝑠

(j) 5𝑡 − 5𝑡 (k) 12𝑠 − 3 + 7𝑠 (l) 12𝑟 + 3𝑟 + 12

(m) 7𝑥 − 4𝑥 + 9 (n) 𝑠 + 8 − 𝑠 (o) 4𝑡 + 9 + 𝑡

(p) 11 + 2𝑝 − 𝑝 (q) 4𝑟 + 6𝑟 − 2 (r) 4𝑡 + 9𝑦 − 𝑡

(s) 9𝑠 + 6𝑡 − 7𝑠 (t) 12𝑝 − 3 − 5𝑝 (u) 2𝑥 + 3 − 2𝑥 + 9

12. Simplify by gathering like terms

(a) 3𝑥 + 5𝑥 + 2𝑦 + 8𝑦 (b) 4𝑠 + 3𝑠 + 7𝑡 + 8𝑡

(c) 4𝑝 + 9𝑝 + 3𝑝 + 6𝑝 (d) 7𝑢 + 5𝑢 + 3𝑣 + 6𝑣

(e) 5𝑥 − 2𝑥 + 6𝑦 + 7𝑦 (f) 9𝑝 − 3𝑝 + 8𝑞 − 3𝑞

(g) 4𝑥 + 3𝑥 + 7 + 9 (h) 9𝑦 − 2𝑦 + 8 − 3

(i) 4𝑥 + 3𝑦 + 6𝑥 + 2𝑦 (j) 5𝑥 + 7 + 8𝑥 + 1

(k) 5𝑥 + 3𝑦 − 3𝑥 − 2𝑦 (l) 7𝑥 + 12 + 3𝑥 − 1

Page | 6

13. Eva asked Ben to simplify 7𝑥 + 3 − 4𝑥 by gathering like terms.

Ben’s working is shown below.

(a) Can you spot Ben’s mistake?

Write out the correct solution showing all working.

(b) What advice would you give Ben so that he could learn from this

mistake?

14. Define a variable and write down expressions to illustrate the

following.

(a) Eva is a baker and bakes some cakes. Each cake needs 3 eggs.

How many eggs did she use?

(b) Ben caught 10 fish and gave some to Morven.

How many fish does Ben have left?

(c) Cara has some money and is going to share it equally amongst

five charities.

How much will each charity get?

(d) Mark had some plectrums for his guitar but then bought four

more.

How many plectrums does Mark now have?

Page | 7

N32 I can use the Distributive Law to multiply out brackets and

gather Like Terms

15. Expand the brackets:

(a) 2(4𝑎 − 3) (b) 6(4𝑦 + 3) (c) 3(2𝑥 − 5)

(d) 4(5𝑐 + 6) (e) 7(2𝑎 + 1) (f) 2(8𝑥 + 3)

(g) 5(6𝑥 − 7𝑦) (h) 3(8𝑡 5𝑢) (i) 3(9𝑥 − 4𝑦)

(j) 8(7𝑥 + 5𝑦) (k) 7(2𝑏 + 9𝑐) (l) 2(12𝑥 + 7𝑦)

16. Expand and simplify:

(a) 2𝑎 + 3(𝑎 + 5) (b) 3𝑥 + 2(𝑥 + 3)

(c) 4𝑏 + 8(𝑏 + 2) (d) 5ℎ + 4(2ℎ + 1)

(e) 11𝑥 + 5(3𝑥 + 4) (f) 10𝑐 + 3(2𝑐 + 1)

(g) 2(4𝑡 + 3) + 10𝑡 (h) 3(5𝑝 + 4) + 7𝑝

(i) 7(1 + 3𝑐) + 10 (j) 3(3𝑎 1) + 2𝑎

(k) 2(5𝑥 + 3) − 3𝑥 (l) 8(𝑏 + 2) − 9

17. Expand and simplify:

(a) 6 + 2(𝑎 + 7) (b) 10 + 2(𝑥 + 3)

18. Expand and simplify:

(a) 2(𝑎 + 1) + 7(𝑎 + 3) (b) 3(𝑥 + 2) + 3(𝑥 + 1)

(c) 5(𝑏 + 3) + 2(𝑏 − 1) (d) 2(ℎ + 9) + 3(2ℎ − 1)

(e) 7(𝑥 + 5𝑦) + 2(3𝑥 + 2𝑦) (f) 6(𝑐 + 5𝑑) + 4(𝑐 + 2𝑑)

Page | 8

N33 I can use the correct order of operations (BODMAS).

19. Evaluate the following

(a) 2×3 + 5×4 (b) 6 − 2×2 (c) 3 + 20 ÷ 4

(d) 3×(5 + 1) (e) 30

5− (4 + 2) (f) (3 + 2)×(6 − 4)

20. Evaluate the following

(a) 15 ÷ 3 + 6×2 − 10 (b) 8×3 + 12×2 − 1

(c) 7×3 + 3 − 2×4 (d) 10×2 + 3 − 4 ÷ 2

21. Evaluate the following

(a) 3+9

6 (b)

6+12

5−1 (c)

4×(6−1)

18−4×2

22. In each of the examples below, spot the mistake and write out the

correct solution.

(a) 3 + 4×(5 − 3) (b) 5 − 2×2

= 7×2 = 3×2

= 14 = 6

(c) 6×3 − 4×3 (d) 8 ÷ 2 + 5×2

= 18 − 4×3 = 4 + 5×2

= 14×3 = 9×2

= 42 = 18

Page | 9

N34 Evaluating an expression or a formula which may include

more than one variable.

23. Evaluate the following expressions

(a) 𝑥 + 𝑦 when 𝑥 = 5 and 𝑦 = 3.

(b) 𝑎 + 4 when 𝑎 = 7.

(c) 5𝑥 when 𝑥 = 8.

(d) 3𝑦 when 𝑦 = 12.

24. Evaluate the following expressions

(a) 𝑡 − 𝑠 when 𝑡 = 8 and 𝑠 = 3.

(b) 17 − 𝑏 when 𝑏 = 9.

(c) 3𝑎 + 2𝑏 when 𝑎 = 7 and 𝑏 = 5.

(d) 4𝑥 + 3𝑦 when 𝑥 = 9 and 𝑦 = 2.

25. Evaluate the following expressions

(a) 7𝑝 + 5𝑞 when 𝑝 = 1 and 𝑞 = 3.

(b) 8𝑠 + 6𝑡 when 𝑠 = 0 and 𝑡 = 8.

(c) 𝑎

𝑏 when 𝑎 = 12 and 𝑏 = 3.

26. Evaluate the following expressions

(a) 48

𝑏 when 𝑏 = 6. (b)

3𝑟

𝑠 when 𝑟 = 8 and 𝑠 = 4.

Page | 10

27. The cost of hiring a taxi can be found using the formula

𝑐 = 2𝑑 + 1

Where 𝑐 is the cost of the hire in pounds and 𝑑 is the distance

travelled in miles.

(a) Use the formula to find the cost of a 3 mile hire.

(b) John is charged £11 for a 5 mile journey.

Was John charged the correct fee?

Use a calculation, involving the formula, to justify your answer.

28. The perimeter of a rectangle can

be found using the formula

𝑃 = 2𝑙 + 2𝑏

Where 𝑃 is the perimeter, 𝑙 is the length of the rectangle and 𝑏 is

the breadth of the rectangle.

(a) Use the formula to find the perimeter of a rectangle of length 14

cm and breadth 8 cm.

(b) Does a rectangle with breath 5m and length 7m have a perimeter

of 22m?

Use a calculation, involving the formula, to justify your

answer.

Page | 11

N35 I can factorise expressions using common factor

29. Factorise each of the following expressions.

Always check your answer by “multiplying out”.

(a) 3𝑥 + 27 (b) 10𝑒 – 50𝑘 (c) 30 – 5𝑏

(d) 16𝑐 + 4𝑟 (e) 4𝑝 – 6 (f) 6𝑔 + 9𝑎

(g) 25 + 10𝑑 (h) 28𝑘 – 16𝑚 (i) 8𝑡 + 16

30. Factorise each of the following expressions

Always check your answer by “multiplying out”.

(a) 𝑝𝑞 + 4𝑝 (b) 5𝑎𝑏 – 2𝑏𝑑 (c) 10𝑔ℎ + 3ℎ

(d) 𝑥𝑦 – 2𝑦 (e) 8𝑡𝑣 + 7𝑡 (f) 𝑚𝑛 – 6𝑚

(g) 5𝑒𝑔 + 𝑔 (h) 9𝑠𝑡 – 11𝑠 (i) 8𝑝𝑡 + 16𝑡

31. Factorise each of the following expressions

Always check your answer by “multiplying out”.

(a) 7𝑥𝑦 + 21𝑥 (b) 16𝑎𝑏 – 12𝑏 (c) 10𝑚 + 25𝑚𝑛

(d) 30𝑔 + 2𝑔ℎ (e) 6𝑠𝑡 – 27𝑠 (f) 8𝑝𝑞𝑟 + 32𝑞𝑟𝑠

(g) 12𝑏𝑧 + 18𝑏𝑞 (h) 30𝑣𝑤 – 20𝑒𝑤 (i) 4𝑝𝑡 + 12𝑝𝑟

Page | 12

Patterns

N36 I can extend a straightforward sequence.

1. List the next five terms in each sequence.

(a) 5, 10, 15, 20, …… (b) 40, 37, 34, 31, ……….

(c) 1, 2, 4, 8, 16, …… (d) 4, 9, 16, 25, ……….

(e) 1, 3, 6, 10, …… (f) 0, 1, 1, 2, 3, 5, ……….

2. List the next two terms in each sequence.

(a) 512, 256, 128, 64, 32, ….. (b) 10, 13, 16, 19, 22, 25, , ,

(c) 129, 116, 103, 90, 77, ….. (d) 101, 90, 79, 68, 57, 46, , ,

(e) 2, 4, 8, 16, 32, 64,….. (f) 34, 31, 28, 25, 22, 19, , ,

(g) 7, 7, 10, 13, 13, 16,….. (h) 21, 19, 17, 15, 13, 11, , ,

3. The first 5 terms of some sequences are shown below.

In each case what is the value of the twentieth term.

(a) 4, 8, 12, 16, 20 ………. (b) 1, 3, 5, 7, 9 ……….

(c) 4, 9, 14, 19, 24, ………. (d) 3, 7, 11, 15, 19 ……….

(e) 1, 4, 9, 16, 25 ………. (f) 1, 1, 2, 3, 5, ……….

Page | 13

Pattern 1 Pattern 2 Pattern 3

N37 I can extend a pattern in diagram format.

4. Draw the next two patterns in the sequence of dots below.

5. Draw the next two patterns in this sequence of dots.

6. Draw the next two patterns in the sequence of dots and squares

below.

Page | 14

N38 I can complete a table from a pattern or formula.

7. A pattern, using matchsticks to make triangle, is shown below.

(a) Draw the next set of triangles in the sequence.

(b) Copy and complete the table below:

8. Look at the pattern of trapezium shaped tables and customers in a

restaurant.

(a) Draw the next set of tables in the sequence.

(b) Copy and complete the table below:

No. of Triangles 1 2 3 4 5 6

No. of Matchsticks 3 5

No. of Tables 1 2 3 4 5 6

No. of customers

Page | 15

N39 I can find a formula for a linear Patterns and can evaluate

the formula for a given value.

9. The diagrams below show the number of people sitting at desks.

(a) Draw the next diagram in the sequence.

(b) Copy and complete the table below for this pattern.

Number of Desks (𝑑) 1 2 3 4 5 6

Number of People (𝑝) 4 6

(c) Write down a formula connecting 𝑝 and 𝑑.

(d) Use the formula to find the number of people who can be sat at:

(i) 35 desks. (ii) 78 desks. (iii) 105 desks.

(e) How many desks would be needed for 78 people?

Page | 16

10. A manufacturer makes necklaces in various sizes.

(a) Draw the next size in the sequence.

(b) Complete this table to show how the pattern is built up.

Number of Rings (𝑟) 1 2 3 4 5 6

Number of Beads (𝑏) 4 7

(c) Write down a formula connecting 𝑏 and 𝑟.

(d) Use the formula to find the number of beads needed for:

(i) 24 rings. (ii) 54 rings. (iii) 98 rings.

(e) How many rings can be made from 76 beads.

2 rings 7 beads 1 ring 4 beads

Page | 17

Pattern 1 Pattern 2 Pattern 3

11. These patterns are made with circles.

(a) Draw pattern 4.

(b) Copy and complete the table below.

Pattern Number (𝑃) 1 2 3 4 5 6

Number of Circles (𝑐) 1 4

(c) Write down a formula connecting 𝑐 and 𝑃.

(d) Use the formula to find the number of circles in:

(i) Pattern 12. (ii) Pattern 45. (iii) Pattern 86.

(e) What pattern number would contain 28 circles?

Page | 18

12. A design consists of rectangles and triangles.

The first three patterns are shown.

(a) Draw the 4th Pattern.

(b) Copy and complete the table below.

(c) Write down a formula connecting 𝑡 and 𝑟.

(d) Use the formula to find the number of triangles in a pattern with:

(i) 33 rectangles. (ii) 75 rectangles. (iii) 112 rectangles.

(e) How many rectangles in a pattern with 42 triangles?

Number of Rectangles (𝑟) 1 2 3 4 5 6

Number of Triangles (𝑡) 6 8

Page | 19

13. Fences are built using posts and boards as shown below.

(a) Draw the next fence in the sequence.

(b) To help plan how many boards are needed to build a fence copy

and complete the table below:

(c) Write down a formula connecting 𝐵 and 𝑝.

(d) Use the formula to find the number of boards required for:

(i) 47 posts. (ii) 86 posts. (iii) 150 posts.

(e) How many posts are required for a fence with 106 boards?

No. of Posts (𝑝) 2 3 4 5 6 7 15

Number of Boards (𝐵) 4 7

Page | 20

Gradient

M12 I can calculate the gradient of a slope

1. Express each fraction in its simplest form, as a ratio, a decimal and a

percentage

(a) 5

10 (b)

3

9 (c)

2

8 (d)

10

2

(e) 6

9 (f)

8

12 (g)

15

20 (h)

30

20

(I) 14

35 (j)

16

12 (k)

24

40 (l)

50

42

For each of the following diagrams, find the gradient and give your answer

as a fraction in its simplest form, as a ratio, as a decimal fraction and as a

percentage.

Example: 2

5= 2: 5 (or 2 in 5) = 0 ∙ 4 = 40%

2. 3.

12m

3m

35cm

7cm

Page | 21

4. 5.

6. 7.

8. 9.

20mm

18mm

20cm

26cm

14m

35m

30m

18m

18m

45m

12m

3m

Page | 22

M13 I can communicate a decision which involved gradient.

10. The cross section of a doorstep is shown below.

(a) Calculate the vertical height, ℎ.

(b) A satisfactory doorstop has a gradient less than 0∙3. Is the

doorstop shown satisfactory?

Give a reason for your answer

11. A mountain railway tunnel

rises 150m over a horizontal

distance of 500m.

(a) Calculate the gradient of the railway line going through the

tunnel.

(b) The rail company are investing in new trains.

The new trains cannot operate if the gradient is greater

than 0∙15.

Will the new trains be able to operate through the tunnel?

Give a reason for your answer

22cm

20cm ℎ

500m

150m

Page | 23

12. A new regulation states that the gradient of all ramps into a building

must be less than 0∙26.

An existing ramp is 410cm long and has a horizontal distance of

400cm.

Does this ramp satisfy the new regulation?

Show all your working and give a reason for your answer.

13. To meet safety regulations, the gradient of a children’s slide must be

between 2

5 and

3

4.

A company designs the slide shown.

Does this slide fit the safety regulations?

Use your working to explain your answer.

5m

530cm

Page | 24

14. The ‘Accessibility Guidelines for Buildings and Facilities for

Wheelchair Access’ give two recommendations.

The drawing below shows the design of a new ramp.

(a) Does the new ramp meet Recommendation 1?

Give a reason.

(b) Does the new ramp meet Recommendation 2?

Give a reason.