Module 1: Numbers and Number Sense MODULE 1 ... - CIMT

Module 1: Numbers and Number Sense

1CCSLC CXC

SenseMODULE 1: Numbers and Number

Unit 7 Algebraic Foundations

7.1 Algebraic NotationThe objective of this section is to

• understand elementary algebraic operations, including collectinglike terms.

We start by looking at some fundamental algebraic skills by examining codes andhow to use formulae.

Example 1

Use this code wheel, which codes A on the outer ring as Y on the inner ring, to:

(a) code the word M A T H S,

(b) decode Q M L G A.

Solution

(a) Look for M on the outside circle of letters; this is coded as K which is theletter on the inside circle. Coding the other letters in the same way gives:

M

↓K

A

↓Y

T

↓R

H

↓F

S

↓Q

A BC

DE

F

H

I

G

JK

LMNOP

QR

S

T

U

V

WX

Y Z

AB

CDEF

GH

IJKLMN

OP

QRST

UV

W X Y Z


2CCSLC CXC

7.1

(b) Look for Q on the inside circle. This decodes as S, which is the letter on theoutside circle. Decoding the other letters in the same way gives:

Q

↓S

M

↓O

L

↓N

G

↓I

A

↓C

Example 2

If a b c= = =4 7 3, and , calculate:

(a) 6 + b (b) 2 a b+ (c) ab (d) a b c−( )

Solution

(a) 6 + b = 6 7+

= 13

(b) 2 a b+ = 2 4 7× + since 2 2a a= ×

= 8 7+

= 15

(c) ab = 4 7× since ab a b= ×

= 28

(d) a b c−( ) = 4 7 3× −( ) since a b c a b c−( ) = × −( )

= 4 4×

= 16

Example 3

Simplify where possible:

(a) 2 4x x+ (b) 5 7 3 2p q p q+ − +

(c) y y y+ −8 5 (d) 3 4t s+

Solution

(a) 2 4x x+ = 2 4× + ×x x

= x x x x x x+( ) + + + +( )

= 6 × x

= 6 x


3CCSLC CXC

(b) 5 7 3 2p q p q+ − + = 5 3 7 2p p q q− + +

= 5 3 7 2−( ) + +( )p q

= 2 9p q+

(c) y y y+ −8 5 = 1 8 5y y y+ −

= 1 8 5+ −( ) y

= 4y

(d) 3 4t s+ cannot be simplified.

Example 4

Write down formulae for the area and perimeter of this rectangle:

x

y

Solution

Area = x y× Perimeter = x y x y+ + +

= x y = 2 2x y+

Exercises1. Use the code wheel of Example 1 to:

(a) code this message,

M E E T M E A T H O M E

(b) decode this message,

M T C P R M W M S

2. Use the code wheel opposite to:

(a) codeG O N E F I S H I N G,

(b) decode T U S T R U H Q

7.1


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7.1

3. Laura used a code wheel similar to the one above, but with the outer ring ofletters rotated. She used her code wheel to code

as

(a) Draw the code wheel that she used.

(b) Use the code wheel to decode:

G D Q J H U D K H D G

4. If a b c d= = = =2 6 10 3, , and , calculate:

(a) a b+ (b) c b− (c) d + 7

(d) 3a d+ (e) 4a (f) a d

(g) 3b (h) 2c (i) 3c b−

(j) 6a b+ (k) 3 2a b+ (l) 4a d−

A BC

DE

F

H

I

G

JK

LMNOP

QR

S

T

U

V

WX

Y Z

OP

QRST

UV

WXYZAB

CD

EFGH

IJ

K L M N

S

↓V

E

↓H

V

↓Y

E

↓H

N

↓Q

U

↓X

P

↓S


5CCSLC CXC

5. If a b c d= = − = = −3 1 2 4, , and , calculate:

(a) a b− (b) a d+ (c) b d+

(d) b d− (e) 3d (f) a b+

(g) c d− (h) 2c d+ (i) 3a d−

(j) 2 3d c+ (k) 4 2a d− (l) 5 3a d+

6. If a b c d= = = − =7 5 3 4, , and , calculate:

(a) 2 a b+( ) (b) 4 a b−( ) (c) 6 a d−( )

(d) 2 a c+( ) (e) 5 b c−( ) (f) 5 d c−( )

(g) a b c+( ) (h) d b a+( ) (i) c b a−( )

(j) a b c2 −( ) (k) d a b2 3−( ) (l) c d −( )2

7. Use the formula s u v t= +( )12

to find s, when u v t= = =10 20 4, and .

8. Use the formula v u at= + to find v, if u a t= = − =20 2 7, and .

9. Simplify, where possible:

(a) 2 3a a+ (b) 5 8b b+

(c) 6 4c c− (d) 5 4 7d d d+ +

(e) 6 9 5e e e+ − (f) 8 6 13f f f+ −

(g) 9 7 8 2 6g g g g g+ − − − (h) 5 2p h+

(i) 3 4 2a b a+ − (j) 6 3 2x y x y+ − −

(k) 8 6 7 2t t s s− + −

(l) 11 3 5 2 2 9 8 14m n p q n q m p+ − + − + − +

10. Write down formulae for the perimeter of each of these shapes:

(a) (b)

7.1

a

b

c

a a

b


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(c) (d)

(e) (f)

11. Sam asks her friend to think of a number, multiply it by 2 and then add 5. Ifthe number her friend starts with is x, write down a formula for the numberher friend gets.

12. A removal firm makes a fixed charge of $50, plus $2 for every km travelled.Write down the formula for the cost of a removal when travelling x km.

13. A delivery company charges $1 per item, plus 50 cents per km. Write downa formula for the cost of delivering 1 item for a distance of x km.

7.2 Function MachinesThe objectives of this section are to

• use function machines to determine the outputs, given the inputs

• to reverse the order of operations described above.

INPUT OUTPUTFUNCTION MACHINE

7.1

a

b b

c

2a

a

2a

a

2b 2b

b

a a

a a

b

b

b b


7CCSLC CXC

Example 1

Calculate the output of each of these function machines:

(a) 4 ?× 5

(b) 5 ?× 2 − 1

(c) – 3 + 8 × 7 ?

Solution

(a) The input is simply multiplied by 5 to give 20:

4 20× 5

(b) The input is multiplied by 2 to give 10, and then 1 is subtracted from thisto give 9:

5 910× 2 − 1

(c) Firstly, 8 is added to the input to give 5, and this is then multipliedby 7 to give 35:

– 3 355+ 8 × 7

Example 2

Calculate the input for each of these function machines:

(a) ? 8× 4

(b) ? 25+ 2 × 5

(c) ? 6− 5 × 3

Solution

The missing inputs can be found by reversing the machines and using the inverse(i.e. opposite) operations in each machine:

(a) ? × 4 8

2 8÷ 4

7.2


8CCSLC CXC

(b) ? + 2 × 5 25

3 25− 2 ÷ 55

(c) ? 6− 5 × 3

7 6+ 5 ÷ 32

Note that:

Exercises1. What is the output of each of these function machines:

(a) 4 ?+ 6

(b) 3 ?× 10

(c) 10 ?− 7

(d) 14 ?÷ 2

(e) 21 ?÷ 3

(f) 100 ?× 5

2. What is the output of each of these function machines:

(a) 3 ?× 4 − 7

(b) 10 ?− 8 × 7

(c) 8 ?− 5 × 5

7.2

Operation InverseOperation

+ –

– +

× ÷

÷ ×


9CCSLC CXC

(d) –2 ?+ 20× 6

(e) 7 ?+ 2 ÷ 3

(f) –5 ?× 9+ 8

3. What is the output of each of these function machines:

(a) ? 30× 5 (b) ? 12+ 8

(c) ? 11− 9 (d) ? 5÷ 4

(e) ? 21+ 12 (f) ? 42× 7

4. What is the input of each of these double function machines:

(a) ? 12+ 1 × 4

(b) ? 4+ 7 ÷ 6

(c) ? 37+ 9× 4

(d) ? 34× 9 − 20

(e) ? 7− 1÷ 6

(f) ? 9− 6 ÷ 7

(g) ? 24+ 8 × 4

(h) ? −3× 2 + 7

5. Here is a triple function machine:

OutputInput × 7 − 5 ÷ 2

(a) What is the output if the input is 8 ?

(b) What is the input if the output is 3 ?

(c) What is the input if the output is −11 ?

7.2

÷ 3


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6. A number is multiplied by 10, and then 6 is added to get 36.

What was the number?

7. Karen asks her teacher, Miss Sharp, how old she is. Miss Sharp replies thatif you double her age, add 7 and then divide by 3, you get 21. How old isMiss Sharp?

8. Sally is given her pocket money. She puts half in the bank and then spends$3 in one shop and $2.50 in another shop. She goes home with $1.25. Howmuch pocket money was she given?

9. A bus has its maximum number of passengers when it leaves the bus station.At the first stop, half of the passengers get off. At the next stop 7 people geton and at the next stop 16 people get off. There are now 17 people on thebus. How many passengers were on the bus when it left the bus station?

10. Paul buys a tomato plant. In the first week it doubles its height. In thesecond week it grows 8 cm. In the third week it grows 5 cm. What was theheight of the plant when Paul bought it if it is now 35 cm in height?

7.3 Linear EquationsThe objectives of this section are to

• to solve simple linear equations

• to convert written problems into linear equations and solve them.

All equations contain an 'equals' sign, which means that the numbers on each sideof the sign have the same value. In an equation such as 3 2 17x + = , we need tofind the unknown number, x.

To solve an equation, you need to reorganise it so that the unknown value is byitself on one side of the equation. This is done by performing operations on theequation. When you do this, in order to keep the equality of the sides, you mustremember that

whatever you do to one side of an equation,you must also do the same to the other side

7.2


11CCSLC CXC

Example 1

Solve these equations:

(a) x + =2 8 (b) x − =4 3 (c) 3 12x =

(d)x

27= (e) 2 5 11x + = (f) 3 2 7− =x

Solution

(a) To solve this equation, subtract 2 from each side of the equation:

x + 2 = 8

x + −2 2 = 8 2−

x = 6

(b) To solve this equation, add 4 to both sides of the equation:

x − 4 = 3

x − +4 4 = 3 4+

x = 7

(c) To solve this equation, divide both sides of the equation by 3:

3x = 12

33x=

123

x = 4

(d) To solve this equation, multiply both sides of the equation by 2:

x

2= 7

22

×x= 2 7×

x = 14

(e) This equation must be solved in 2 stages.

First, subtract 5 from both sides:

2 5x + = 11

2 5 5x + − = 11 5−

2 x = 6

7.3


12CCSLC CXC

Then, divide both sides of the equation by 2:

22x=

62

x = 3

(f) First, subtract 3 from both sides:

3 2− x = 7

3 2 3− −x = 7 3−

−2x = 4

Then divide both sides by −( )2 :

−−22x=

42−

x = −2

Example 2

Solve these equations:

(a) 3 2 4 3x x+ = −

(b) 2 7 8 11x x+ = −

Solution

These equations contain x on both sides. The first step is to change them so that xis on only one side of the equation. Choose the side which has the most x; here,the right hand side.

(a) Subtract 3x from both sides of the equation:

3 2x + = 4 3x −

3 2 3x x+ − = 4 3 3x x− −

2 = x − 3

Then add 3 to both sides of the equation:

2 = x − 3

2 3+ = x − +3 3

5 = x

so x = 5

Note: it is conventional to give the answer with the unknown value, x,on the left hand side, and its value on the right hand side.

7.3


13CCSLC CXC

(b) First, subtract 2 x from both sides of the equation:

2 7x + = 8 11x −

2 7 2x x+ − = 8 11 2x x− −

7 = 6 11x −

Next, add 11 to both sides of the equation:

7 11+ = 6 11 11x − +

18 = 6 x

Then divide both sides by 6:

186

=66x

3 = x

so x = 3

Example 3

You ask a friend to think of a number. He then multiplies it by 5 and subtracts 7.He gets the answer 43.

(a) Use this information to write down an equation for x, the unknown number.

(b) Solve your equation for x.

Solution

(a) As x = number your friend thought of, then

x × 5 − 75 x

5 7x −

So 5 7x − = 43

(b) First, add 7 to both sides of the equation to give

5 x = 50

Then divide both sides by 5 to give

x = 10

and this is the number that your friend thought of.

7.3


14CCSLC CXC

Exercises1. Solve these equations:

(a) x + =2 8 (b) x + =5 11 (c) x − =6 2

(d) x − =4 3 (e) 2 18x = (f) 3 24x =

(g)x

64= (h)

x

59= (i) 6 54x =

(j) x + =12 10 (k) x + =5 3 (l) x − = −22 4

(m)x

72= − (n) 10 0x = (o)

x

24 5+ =

2. Solve these equations:

(a) 2 4 14x + = (b) 3 7 25x + = (c) 4 2 22x + =

(d) 6 4 26x − = (e) 5 3 32x − = (f) 11 4 29x − =

(g) 3 4 25x + = (h) 5 8 37x − = (i) 6 7 31x + =

(j) 3 11 5x + = (k) 6 2 10x + = − (l) 7 44 2x + =

3. Solve these equations, giving your answers as fractions or mixed numbers:

(a) 3 4x = (b) 5 7x = (c) 2 8 13x + =

(d) 8 2 5x + = (e) 2 6 9x + = (f) 4 7 10x − =

4. The perimeter of this triangle is 31 cm.

Use this information to write down anequation for x and solve it to find x.

5. (a) Write down an expression for the length of the perimeter of thisrectangle:

x cm x cm

18 cm

18 cm

7.3

x

12 cm

11 cm


15CCSLC CXC

(b) Find x if the perimeter length is 48 cm.

(c) Find x if the perimeter length is 45 cm.

6. Tom asks each of his friends to think of their age, double it and then takeaway 10.

Here are the answers he is given:

(a) Using x to represent Ben's age, write down an equation for x andsolve it to find Ben's age.

(b) Write down and solve equations to find the ages of Ian, Adam and Sam.

7. The perimeter of this octagon is 9.6 cm.

Write down an equation and solve itto find x.


(a) x x+ = −2 2 1 (b) 8 1 4 11x x− = +

(c) 5 2 6 4x x+ = − (d) 11 4 2 23x x− = +

(e) 5 1 6 8x x+ = − (f) 3 2 5 44x x x+ + = +

(g) 6 2 2 23x x x+ − = + (h) 2 3 6 58x x x− = + −

(i) 3 2 8x x+ = − (j) 4 2 2 8x x− = −

(k) 3 82 10 12x x+ = + (l) 6 10 2 14x x− = −

9. The diagram below shows three angles on a straight line:

2 x°80°

3x°

(a) Write down an equation and use it to find x.

(b) Write down the sizes of the two unknown angles and check that thethree angles shown add up to 180 °.

7.3

Ben Ian Adam Sam

8 10 14 11

1 cm

1 cm

1 cm1 cm

x cm x cm

x cm x cm


16CCSLC CXC

10. Use an equation to find the sizes of the unknown angles in this triangle:

40 °

x° 3x°

11. Karen thinks of a number, multiplies it by 3 and then adds 10. Her answeris 11 more than the number she thought of. If x is her original number,write down an equation and solve it to find x.

7.4 Expansion of Single BracketsThe objective of this section is to

• expand brackets in algerbaic expressions.

We will consider how to expand (multiply out) brackets to give two or moreterms, as shown below:

3 6 3 18x x+( ) = +

First we revise negative numbers and order of operations.

Example 1

Evaluate:

(a) − +6 10 (b) − + −( )7 4

(c) −( ) × −( )6 5 (d) 6 4 7× −( )

(e) 4 8 3+( ) (f) 6 8 15−( )

(g) 3 5− −( ) (h)−( ) − −( )

−

2 3

1

7.3


17CCSLC CXC

Solution

(a) − + =6 10 4

(b) − + −( ) = − −7 4 7 4

= −11

(c) −( ) × −( ) =6 5 30

(d) 6 4 7 6 3× −( ) = × −( )= −18

(e) 4 8 3 4 11+( ) = ×

= 44

(f) 6 8 15 6 7−( ) = × −( ) = −42

(g) 3 5 3 5− −( ) = +

= 8

(h)−( ) − −( )

−=

−( ) +−

2 3

1

2 3

1

=−11

= −1

When a bracket is expanded, every term inside the bracket must be multiplied bythe number outside the bracket. Remember to think about whether each number ispositive or negative!

Example 2

Expand 3 6x +( ) using a table.

Solution

From the table,

3 6 3 18x x+( ) = +

7.4

× x 6

3 3x 18


18CCSLC CXC

Example 3

Expand 4 7x −( ).

Solution

4 7x −( ) = × − ×4 4 7x

= −4 28x

Example 4

Expand x x8 −( ) .

Solution

x x8 −( ) = × − ×x x x8

= −8 2x x

Example 5

Expand −( ) −( )3 4 2 x .

Solution

−( ) −( )3 4 2 x = −( ) × − −( ) ×3 4 3 2 x

= − − −( )12 6 x

= − +12 6 x

Exercises1. Calculate:

(a) − +6 17 (b) 6 14− (c) − −6 5

(d) 6 9− −( ) (e) − − −( )11 4 (f) −( ) × −( )6 4

(g) 8 7× −( ) (h) 88 4÷ −( ) (i) 6 8 10−( )

(j) 5 3 10−( ) (k) 7 11 4−( ) (l) −( ) −( )4 6 17

7.4

Remember that every term insidethe bracket must be multiplied bythe number outside the bracket.


19CCSLC CXC

2. Copy and complete the following tables, and write down each of theexpansions:

(a) (b)

4 2x +( ) = 5 7x −( ) =

(c) (d)

4 3x +( ) = 5 2 5x +( ) =

3. Expand:

(a) 4 6x +( ) (b) 3 4x −( )

(c) 5 2 6x +( ) (d) 7 3 4x −( )

(e) 3 2 4x +( ) (f) 8 3 9x −( )

(g) −( ) −( )2 4x (h) −( ) −( )3 8 2 x

(i) 5 3 4x −( ) (j) 9 2 8x +( )

4. Jordan writes 3 4 8 12 8x x−( ) = − .

Explain why his expansion is not correct.

5. Copy and complete the following tables and write down each of theexpansions:

(a) (b)

x x −( ) =2 x x y−( ) =

6. Copy the following expansions, filling in the missing terms:

(a) 4 8 4 2x x x+( ) = + ? (b) −( ) −( ) = +3 2 7 21x ?

(c) 4 9 4 2x x x−( ) = − ? (d) 6 7 6 2x x x−( ) = − ?

(e) 3 3 2x x y x−( ) = − ? (f) −( ) +( ) = −4 2 8 32x x x?

7.4

× x 2

4

× x 3

4

× x – 2

x


20CCSLC CXC

7. Expand:

(a) x x −( )7 (b) x x8 2−( )

(c) 6 2x x +( ) (d) 4 3 5x x −( )

(e) x x y+( ) (f) x y x4 3−( )

(g) 2 2 3x x y+( ) (h) 5 2 1x y −( )

8. Write down expressions for the area of each of these rectangles, and thenexpand the brackets:

(a) (b)

(c) (d)

(e) (f)

9. Write down an expression for the areaof this triangle, that:

(a) contains brackets,

(b) does not contain brackets.

10. Write down an expression for the volumeof this cuboid, that:

(a) contains brackets,

(b) does not contain brackets.

7.4

2

x + 4

2x

2x

3x – 5

x + 9

2x

x + 2

x

2x

3 x − 2

5 + x

2x

4 x

6 2− x

12

x – 5


21CCSLC CXC

7.5 Equations with BracketsThe objective of this section is to

• solve simple equations that include brackets.

Expanding a bracket will usually be the first step when solving an equation like

4 3 20x +( ) =

Example 1

Solve

5 3 35x −( ) =

Solution

5 3x −( ) = 35

Expanding brackets gives: 5 15x − = 35

Adding 15 to both sides gives: 5 x = 50

Dividing by 5 gives: x = 10

Example 2

Solve

6 7 50x +( ) =

Solution

6 7x +( ) = 50

Expanding brackets gives: 6 42x + = 50

Subtracting 42 from both sides gives: 6 x = 8

Dividing by 6 gives: x =86

= 113

Example 3

Jane thinks of a number and adds 7 to it. She then multiplies her answer by4 and gets 64.

(a) Write down an equation that can be used to calculate the number withwhich Jane started.

(b) Solve your equation to give the number.


22CCSLC CXC

Solution

(a) Start with x.

Add 7 to give x + 7

Multiply by 4 to give 4 7x +( )

This expression equals 64, so the equation is 4 7 64x +( ) =

(b) 4 7x +( ) = 64

Expanding brackets gives; 4 28x + = 64

Subtracting 28 from both sides gives: 4 x = 36

Dividing by 4 gives: x =364

= 9

Exercises1. Solve these equations:

(a) 2 6 14x +( ) = (b) 5 8 40x −( ) =

(c) 3 5 12x +( ) = (d) 7 4 42x +( ) =

(e) 2 7 19x +( ) = (f) 3 4 11x −( ) =

(g) 5 4 12x −( ) = (h) 10 7 82x +( ) =


(a) 4 2 7 8x −( ) = (b) 3 3 6 27x +( ) =

(c) 3 2 1 30x +( ) = (d) 8 2 12 24x −( ) =

3. A rectangle has sides of length

3 m and x +( )4 m.

Find the value of x, if the area of therectangle is 18 m2 .

4. George chooses a number, adds 7, multiplies the result by 5 and gets theanswer 55.

(a) If x is the number George first chose, write down an equation thatcan be used to determine the number.

(b) Solve the equation to determine the value of x.

7.5

3 m

x + 4( ) m


23CCSLC CXC

5. The following flow chart is used to form an equation:

x + 6 × 4 17

(a) Write down the equation.

(b) Solve the equation to find the value of x.

6. Solve the following equations:

(a) 4 7 20−( ) =x (b) 3 9 15−( ) =x

(c) 6 5 2 18−( ) =x (d) 5 7 3 20−( ) =x

(e) 2 10 3 17−( ) =x (f) 6 9 5 4−( ) =x

7. Alice thinks of a number, subtracts it from 11 and then multiplies heranswer by 5 to get 45. What was the number that Alice started with?

8. Solve the following equations:

(a) 2 1 6 3x x+( ) = −( ) (b) 3 4 11x x+( ) =

(c) 5 4 2 10 1x x+( ) = +( ) (d) 4 7 5 2−( ) = +( )x x

9.

(a) Write down an expression for the area of the triangle.

(b) What is x if the area is 15 m2 ?

7.5

3 m

x + 4( ) m

Module 1: Numbers and Number Sense MODULE 1 ... - CIMT

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