is a number.

Write an expression for the number that is,

Five more than One third

of

Four multiplied by Seven less than

The difference between and 10

Give your answers in correct algebraic notation.

Compare your answers with a partner’s, do you have different

but equivalent expressions? Check by substituting in values
for and comparing your answers.
is a number. Write an expression for the number that is,

Five more than One third

of

Four multiplied by Seven less than

The difference between and 10

Give your answers in correct algebraic notation.

Compare your answers with a partner’s, do you have different

but equivalent expressions? Check by substituting in values
for and comparing your answers.
On each card, a cube represents and a counter represents
Write an expression for the total of each card, giving your
answers as simply as possible.

Repeat if the cube now represents and the counter

On each card, a cube represents and a counter represents
Write an expression for the total of each card, giving your
answers as simply as possible.

Repeat if the cube now represents and the counter

Write simplified expressions for the perimeter and area of
each shape.

4
3
𝑓
2𝑔
2𝑑 2𝑑

4𝑥
3 5𝑎 6𝑝
3𝑎
4𝑎
Write simplified expressions for the perimeter and area of
each shape.

4
3
𝑓
2𝑔
Perimeter 2𝑑 2𝑑
Area
Perimeter
4𝑥 Area

3 5𝑎 6𝑝
Perimeter
3𝑎
Area 4𝑎
Perimeter Perimeter
Area Area
Work out the value of these expressions when
and
𝑎𝑏 𝑎 𝑏
𝑎+𝑏 𝑏 𝑎
𝑎− 𝑏 3𝑎𝑏
2
𝑎 +𝑏
2 2
𝑎 −𝑏
2
2𝑎− 3𝑏 3𝑎
2

Now find the values again this time using and

Which expressions give the same answer as before? Why?

Work out the value of these expressions when
and
𝑎𝑏¿ − 8 𝑎 1 𝑏
𝑎+𝑏¿ − 2 𝑏
¿−
2 𝑎
¿ − 2
𝑎− 𝑏¿ 6 3𝑎𝑏¿ − 24
2 2 2
𝑎 +𝑏 ¿ 20𝑎 − 𝑏
2¿ −12
2𝑎− 3𝑏¿ 16 3 𝑎 ¿ 12
2

Now find the values again this time using and

Which expressions give the same answer as before? Why?

2 2 2 2 2
𝑎 +𝑏 𝑎 −𝑏 3𝑎
Because when you square a negative number you get a positive result.
Mo is answering the question on the card.

2
Find the value of 𝑥 when 𝑥=−2.5
Mo enters into his calculator and is surprised to get the
answer as he thinks the answer should be positive.
Discuss why the calculator shows a negative answer.
Mo is answering the question on the card.

2
Find the value of 𝑥 when 𝑥=−2.5
Mo enters into his calculator and is surprised to get the
answer as he thinks the answer should be positive.
Discuss why the calculator shows a negative answer.
The reason Mo gets a negative result is that his calculator is
following the order of operations.
In order to work out Mo should input
Solve the equations.

𝑥−1.7=−6.8
𝑥+1.7=6.8 6.8=1.7 𝑥

6.8=2 𝑥−1.7
−6.8=1.7 𝑥 𝑥
=− 6.8
1.7
1.7 𝑥−5.1=−6.8
Solve the equations.

𝑥−1.7=−6.8
𝑥+1.7=6.8 𝑥=−5.1
6.8=1.7 𝑥
𝑥=5.1 𝑥=4
6.8=2 𝑥−1.7
−6.8=1.7 𝑥 𝑥=4.25
𝑥
=− 6.8
𝑥=− 4 1.7
𝑥=−11.56
1.7 𝑥−5.1=−6.8
𝑥=−1
Simplify the expressions on the cards.

−3 𝑝+4𝑝 −8𝑝
3 𝑝+4 𝑝− 8 𝑝 −3 𝑝− 4𝑝 −8𝑝

−3 × − 4 𝑝× −2
3 ×− 4 𝑝 −3 × − 4 𝑝

−3 × 4 𝑝
Simplify the expressions on the cards.

−3 𝑝+4𝑝 −8𝑝
3 𝑝+4 𝑝− 8 𝑝 ≡− 7 𝑝 −3 𝑝− 4𝑝 −8𝑝
≡− 𝑝 ≡− 15 𝑝
−3 × − 4 𝑝× −2
3 ×− 4 𝑝 ≡− 24 𝑝 −3 × − 4 𝑝
≡− 12𝑝 ≡ 12 𝑝
−3 × 4 𝑝
≡− 12𝑝
Annie is working out and reasons she can do the same with
any number

Compare the value of with the value of for different values

of (positive, negative, fractions, decimals)
Do they always have the same value?
Annie is working out and reasons she can do the same with
any number

Compare the value of with the value of for different values

of (positive, negative, fractions, decimals)
Do they always have the same value?

Compare results as a class.

As the value is always the same.
Explain how these representations show that

2𝑥 5
3
Explain how these representations show that

Multiplication can be shown

as repeated addition. green tile, 1 yellow tile

2𝑥 5
3

cup, 1 counter is the area

Expand these brackets.

3 (𝑥+5) 3 (𝑥 −5) −3(𝑥−5)

3 (5 − 𝑥) −3(𝑥+5) 𝑥 (𝑥 +5)

3 (5+𝑥) 2 𝑥(5−𝑥+ 𝑦)
Expand these brackets.

3 (𝑥+5) 3 (𝑥 −5) −3(𝑥−5)

≡3 𝑥 +15 ≡3 𝑥 − 15 ≡− 3 𝑥+15
3 (5 − 𝑥) −3(𝑥+5) 𝑥 (𝑥 +5)
≡15 − 3 𝑥 ≡− 3 𝑥 −15 2
≡ 𝑥 +5 𝑥

3 (5+𝑥) 2 𝑥(5−𝑥+ 𝑦)
≡15 +3 𝑥 2
≡10 𝑥 − 2 𝑥 +2 𝑥𝑦
List the factors of the numbers or expressions on each card.

10 18 3𝑥

𝑥
2
2𝑥
2
6 𝑥𝑦
 List the factors of the numbers or expressions on each card.

10 18 3𝑥

𝑥
2
2𝑥
2
6 𝑥𝑦
2
1,𝑥 ,𝑥
The area and the length of one of the sides is given for each
of the rectangles. Find the missing sides.

5𝑥 ?
? 6

? 4
3 𝑎+2 𝑏
The area and the length of one of the sides is given for each
of the rectangles. Find the missing sides.

5𝑥 4 𝑦
2 6

3 4
3 𝑎+2 𝑏 3 𝑎+2 𝑏
Complete the factorisations.
( ____ _____) ( ____ _____)
( ____ _____) ___
___ ____ ( ____
Complete the factorisations.
2 𝑥
( ____ _____) 𝑦
3 _____)
( ____ 2𝑥 3𝑦
𝑦
( ____ _____) 7
___ 𝑎
___ 3
____ 𝑞
( ____ 4 𝑑
5 𝑑
 How many ways can you find to factorise the
expression?
Fully factorise the expression.

24 𝑥𝑦+36𝑥𝑧 𝑥( ____+_____)
( ____ _____)
 How many ways can you find to factorise the
expression?
Fully factorise the expression.
3 𝑥 (8 𝑦 + 12 𝑧
6 )𝑥 ( 4 𝑦 + 6 𝑧 )

24 𝑥𝑦+36𝑥𝑧 𝑥(24
____+_____)
𝑦36 𝑧
4 𝑥𝑦
( ____ 6 𝑥𝑧
_____)
2(12 𝑥𝑦 +18 𝑥𝑧 )

Discuss other alternatives as a class.

Fully factorised:
Without expanding the brackets, decide whether you think
these expressions will be equivalent or not.

3 ( 𝑥+ 4 ) +2 2 𝑥+3 ( 𝑥 +4 )

2+3 ( 𝑥 + 4 ) 3 ( 𝑥+ 4 ) +2

Checking by building the expressions using algebra tiles.

Simplify the expressions and compare with your concrete
versions.
Without expanding the brackets, decide whether you think
these expressions will be equivalent or not.
Discuss answers as a class.

3 ( 𝑥+ 4 ) +2 2 𝑥+3 ( 𝑥 +4 )
3 𝑥 + 145 𝑥 +12
2+3 ( 𝑥 + 4 ) 3 ( 𝑥+ 4 ) +2
3 𝑥 + 143 𝑥 + 14
Checking by building the expressions using algebra tiles.
Simplify the expressions and compare with your concrete
versions.
Ron has made mistakes in both these simplifications.

Explain Ron’s errors and work out the correct answers.

Ron has made mistakes in both these simplifications.

Explain Ron’s errors and work out the correct answers.

Ron has not followed the Ron has subtracted from

order of operations. when he should have added
He should have expanded the this term.
bracket first and then added 5
as the last step. Correct answer:

Correct answer:
Expand and simplify the expressions.

3 ( 5𝑎+2 ) +4(2𝑎+3) 3 ( 5𝑎+2 ) −4(2𝑎+3)

3 ( 5𝑎+2 ) +4(2𝑎−3) 3 ( 5𝑎−2 ) −4(2𝑎+3)
3 ( 5𝑎−2 ) +4(2𝑎−3) 3 ( 5𝑎−2 ) −4(2𝑎−3)
3 ( 5𝑎−2 ) −5(3𝑎−2) 3 ( 4𝑎−2 ) −2(6 𝑎−3)
Expand and simplify the expressions.

3 ( 5𝑎+2 ) +4(2𝑎+3) ≡23 𝑎+18 3 ( 5𝑎+2 ) −4(2𝑎+3)≡ 7 𝑎 − 6

3 ( 5𝑎+2 ) +4(2𝑎−3) ≡23 𝑎 −6 3 ( 5𝑎−2 ) −4(2𝑎+3)≡7 𝑎 −18
3 ( 5𝑎−2 ) +4(2𝑎−3) ≡23 𝑎 −18 3 ( 5𝑎−2 ) −4(2𝑎−3)≡ 7 𝑎+ 6
3 ( 5𝑎−2 ) −5(3𝑎−2) ≡ 4 3 ( 4𝑎−2 ) −2(6 𝑎−3)≡ 3 𝑎
Explain how the algebra tiles show that
𝑥+ 2

𝑥+1
Use algebra tiles to expand the expressions.

(𝑥+4)(𝑥+3) (𝑥+3)(𝑥+4)

(𝑥+4)(𝑥−3) (𝑥+3)(𝑥− 4) ( 𝑥+ 3 ) 2
Explain how the algebra tiles show that
𝑥+ 2

𝑥+1
The algebra tiles show using an area
model expansion.

Use algebra tiles to expand the expressions.

(𝑥+4)(𝑥+3) (𝑥+3)(𝑥+4)
2 2
𝑥 +7 𝑥 +12 𝑥 +7 𝑥 +12
(𝑥+4)(𝑥−3) (𝑥+3)(𝑥− 4) ( 𝑥+ 3 ) 2

2 2 2
𝑥 + 𝑥 −1 2 𝑥 − 𝑥 − 12 𝑥 + 6 𝑥+ 9
Here is Tommy’s method for working out by thinking of the
calculation as

× 60 2
40 2400 80 2400+ 180+80 +6=2666
3 180 6

Complete this adaptation of Tommy’s method to work out

×
𝑎𝑏+3 𝑏 +¿
____ _____

What would be different if Tommy was working out

Here is Tommy’s method for working out by thinking of the
calculation as

× 60 2
40 2400 80 2400+ 180+80 +6=2666
3 180 6

Complete this adaptation of Tommy’s method to work out

×
𝑎𝑏+3 𝑏 +¿
4 𝑎+12
𝑎 _____
4____ 12
What would be different if Tommy was working out
Annie works out as

Show that Annie is wrong using:

substitution an area model algebra tiles
Annie works out as

Show that Annie is wrong using:

substitution an area model
algebra
Substitute for 3 tiles
into both
expressions

(2 𝑥+5) 2 ≡ 4 𝑥 2+ 20 𝑥+25
2 2
4 𝑥 + 20 𝑥 +25 ≠ 4 𝑥 +25
2 2
(2 𝑥+5) ≡ 4 𝑥 + 20 𝑥+25

4 𝑥2 + 20 𝑥 +25 ≠ 4 𝑥2 +25
Solve the equations
Solve the equations

𝑎=11 𝑏=1

𝑥=−1.2 𝑥 =9

𝑒= 2
Compare these solutions of the equation
Explain the steps in each method.
Which approach do you prefer?
Compare these solutions of the equation
Explain the steps in each method.
Which approach do you prefer?

Discuss answers as a class – right hand method is less suitable

for equations like as 19 is not divisible by 3
Compare the two number puzzles.

I think of a number. I think of a number.

I add on 6 I double it.
I double my answer. I add on 6 to my answer.
Now I’m thinking of 18 Now I’m thinking of 18
What was my original What was my original
number? number?

What’s the same and what’s different?

What were the numbers?

Represent the puzzles using concrete materials, pictures and
equations to check your answers.
Compare the two number puzzles.

I think of a number. I think of a number.

I add on 6 I double it.
I double my answer. I add on 6 to my answer.
Now I’m thinking of 18 Now I’m thinking of 18
What was my original What was my original
number? number?

What’s the same and what’s different?

Similarity – Answers of 18 Differences – The equations:

What were the numbers? and

Represent the puzzles using concrete materials, pictures and
equations to check your answers.
Compare different representations as a class
Write and solve equations to find the values of and

𝑥 × 4 +7 20

𝑦 +7 × 4 20
Write and solve equations to find the values of and

𝑥 × 4 +7 20

or

𝑦 +7 × 4 20
The area of the rectangle is 72 cm2
Work out the value of and hence find the perimeter of the
rectangle.

3 (2
𝑥+ cm
5)
The area of the rectangle is 72 cm2
Work out the value of and hence find the perimeter of the
rectangle.

3 (2
𝑥+ cm
5)

Perimeter cm
Which of the inequalities does the number satisfy?

𝑥> 7 7< 𝑥 7 ≤ 𝑥 𝑥< 8 𝑥 ≥ 8

What’s the same and what’s different about the inequalities?
Which of the inequalities does the number satisfy?

𝑥> 7 7< 𝑥 7 ≤ 𝑥 𝑥< 8 𝑥 ≥ 8

What’s the same and what’s different about the inequalities?
Compare similarities and differences as a class
Mo says “If , then ”
Explain why Mo is right.
Complete the spider diagram.

___
___

___ 𝑥>10 ___

___ ___
Mo says “If , then ”
Explain why Mo is right. is one greater than
Complete the spider diagram.

___14
___ 32

5
___ 𝑥>10 20
___

___ 32 ___ 17
If , which of the cards are true and which are false?

10< 𝑥

40< 4 𝑥
− 𝑥 >−10
If , which of the cards are true and which are false?

10< 𝑥 True

40< 4 𝑥 True

− 𝑥 >−10 False
Solve the inequalities.

𝑥+ 2>7 𝑥 −2> 7 𝑥 −2> −7

𝑥+ 2>−7 𝑥+ 2<−7

𝑥+ 2<7 𝑥 −2 ≤ 7 𝑥 −2 ≥ −7

2 𝑥+2< 7 4 𝑥+2≥ −7 3+ 5 𝑥 ≤7
Solve the inequalities.

𝑥+ 2>7 𝑥 −2> 7 𝑥 −2> −7

𝑥> 5 𝑥> 9 𝑥> − 5
𝑥+ 2>−7 𝑥+ 2<−7
𝑥> − 5 𝑥< − 9

𝑥+ 2<7 𝑥 −2 ≤ 7 𝑥 −2 ≥ −7
𝑥< 5 𝑥≤9 𝑥 ≥ −5

2 𝑥+2< 7 4 𝑥+2≥ −7 3+ 5 𝑥 ≤7
𝑥< 2.5 𝑥 ≥ −2.25 𝑥 ≤ 0.8
Write an inequality and solve it to find the possible range of
values for Whitney’s number.
What is the smallest integer Whitney could be thinking of?

Three more than double my number is

greater than 10
Whitney
Write an inequality and solve it to find the possible range of
values for Whitney’s number.
What is the smallest integer Whitney could be thinking of?

Three more than double my number is

greater than 10
Whitney

Smallest possible integer value

 Annie has £100
She wants to buy three T-shirts and a jumper.
The jumper costs £45, and she doesn’t have enough money to buy
everything she wants.
What can be worked out about the price of the T-shirts?
 Annie has £100
She wants to buy three T-shirts and a jumper.
The jumper costs £45, and she doesn’t have enough money to buy
everything she wants.
What can be worked out about the price of the T-shirts?

The T-shirts cost more than each

Explain why you cannot make a triangle with three sides of
lengths 4 cm, 5 cm and 12cm.
Explain why you cannot make a triangle with three sides of
lengths 4 cm, 5 cm and 12cm.
The sum of the two shorter sides in any triangle must be
greater than or equal to the length of the longest side.
Each square represents and each cube represents
Write expressions for each card. The first one is done for you.

𝑥
This triangle had sides + 4
and 11 cm. 𝑥
Work out the range of
possible values of .
cm
Each square represents and each cube represents
Write expressions for each card. The first one is done for you.

𝑥
This triangle had sides + 4
and 11 cm. 𝑥
Work out the range of
possible values of .
cm

cm
Write down the equation shown by the bar model.

Write down the new equation if the two leftmost s are

removed from the bars. Work out the value of .
Write down the equation shown by the bar model.

4 𝑥+12=2 𝑥+18
Write down the new equation if the two leftmost s are
removed from the bars. Work out the value of .
Use the bar model to help you complete the workings to find
the value of .

5 𝑦 =3 𝑦 +15
−𝟑 𝒚 −𝟑 𝒚
etc.
Use the bar model to help you complete the workings to find
the value of .

5 𝑦 =3 𝑦 +15
−𝟑 𝒚 −𝟑 𝒚
÷𝟐 ÷𝟐
𝑦 =7.5
Solve the equations.

5 𝑥+1=71 5 𝑥+1=7 𝑥 5 𝑥+1=2 𝑥+7

17=4 𝑥−3 2 𝑥=4 𝑥 −3 2 𝑥+1=4 𝑥−3

Solve the equations.

5 𝑥+1=71 5 𝑥+1=7 𝑥 5 𝑥+1=2 𝑥+7

𝑥=14 𝑥=0.5 𝑥=2

17=4 𝑥−3 2 𝑥=4 𝑥 −3 2 𝑥+1=4 𝑥−3

𝑥=5 𝑥=1.5 𝑥=2
The area of the parallelogram is twice the area of the
trapezium.
Work out the total area of the two shapes.
𝑥+ 4
10 12 6
The area of the parallelogram is twice the area of the
trapezium.
Work out the total area of the two shapes.
𝑥+ 4
10 12 6

Area of the parallelogram =

Area of the trapezium

Total area
Dora and Amir are both given the same starting number.

I add two to the

I triple the
number and
number and
then multiply by
add on seven Amir
Dora 4

Dora’s answer is less than Amir’s.

Is it possible that the starting number was negative?
If so, give an example.
Dora and Amir are both given the same starting number.

I add two to the

I triple the
number and
number and
then multiply by
add on seven Amir
Dora 4

Dora’s answer is less than Amir’s.

Is it possible that the starting number was negative?

leads to , so all negative numbers are possible.

If so, give an example.
e.g.
Dora’s expression Amir’s expression
Verify, by substitution, that is the solution to the equation.

7 𝑥+3 ( 2𝑥 −4 ) =3 ( 2𝑥+4 ) −3(3 𝑥−8)

Now solve the equation algebraically.
Verify, by substitution, that is the solution to the equation.

7 𝑥+3 ( 2𝑥 −4 ) =3 ( 2𝑥+4 ) −3(3 𝑥−8)

Let

Now solve the equation algebraically.

 Esther adds together three consecutive even numbers.
Her total is less than 80. Use an algebraic method to work out the
greatest of Esther’s three numbers.
 Esther adds together three consecutive even numbers.
Her total is less than 80. Use an algebraic method to work out the
greatest of Esther’s three numbers.

Greatest value
What’s the same and what’s different about the cards?

2(𝑎+𝑏) 𝑃=2(𝑎+𝑏) 2 ( 𝑎+𝑏) ≡2𝑎+2𝑏

What’s the same and what’s different about the cards?

2(𝑎+𝑏) 𝑃=2(𝑎+𝑏) 2 ( 𝑎+𝑏) ≡2𝑎+2𝑏

Expression Formula Identity
Which of these formulas do you recognise?

Explain the meaning of all the variables in the formulas.

𝐶=𝜋 𝑑 𝑃=2𝑙+2𝑤 1
𝐴= 𝑏h
2

𝐷
𝑆=
𝑃=4 𝑠 𝑇 𝐴=𝑙𝑤
Investigate how the value of the subject of the formula
changes as the other variables change.
Which of these formulas do you recognise?
Circumference of Perimeter of a Area of a
a circle rectangle triangle

𝐶=𝜋 𝑑 𝑃=2𝑙+2𝑤 1
𝐴= 𝑏h
2
Perimeter of a Speed
Area of a
square rectangle
𝐷
𝑆=
𝑃=4 𝑠 𝑇 𝐴=𝑙𝑤
Explain the meaning of all the variables in the formulas.
Discuss as a class
Investigate how the value of the subject of the formula
changes as the other variables change.
Discuss as a class
 An equation is anything with
an equals sign.
Rosie
Explain why Rosie is wrong.
Which of these cards show equations? What do the other
cards show?

3𝑎+2 ( 𝑎+5 ) =25 𝑣 =𝑢+𝑎𝑡 5 𝑥+6 𝑥≡11 𝑥

𝑏 ×𝑏 × 𝑏 ≡𝑏
3 𝑝
10= − 3 6 − 𝑚=2
2
1
2
( 𝑎 +𝑏 ) h 2
𝑎 +𝑏 =𝑐
2 2
6 − 𝑚=𝐹
 An equation is anything with
an equals sign.
Rosie
Explain why Rosie is wrong. Formulas also use an equals sign
Which of these cards show equations? What do the other
cards show?

3𝑎+2 ( 𝑎+5 ) =25 𝑣 =𝑢+𝑎𝑡 5 𝑥+6 𝑥≡11 𝑥

𝑏 ×𝑏 × 𝑏 ≡𝑏
3 𝑝
10= − 3 6 − 𝑚=2
2
1
2
( 𝑎 +𝑏 ) h 2
𝑎 +𝑏 =𝑐
2 2
6 − 𝑚=𝐹