Lesson 2.4 · 25 min

Substitution and Constructing Expressions

Putting numbers — including negatives and decimals — into formulae, and building expressions for areas and number puzzles.

In this lesson you will learn to…

Substitute positive, negative and decimal values into an expression
Follow the order of operations, especially with powers of negative numbers
Write and simplify an expression for the area of a compound shape
Use substitution to check that an expression is correct

Why this matters

A formula is only useful once you put real values into it. Physicists substitute into $s = ut + \frac{1}{2}at^2$ to find how far something travels; engineers substitute into $V = \pi r^2 h$ to find how much a tank holds. One slip with a negative sign and the answer is useless.

The two traps with negatives

Expression	Value when $m = -4$	Why
$m^2$	$(-4)^2 = 16$	negative × negative is positive
$-m^2$	$-(-4)^2 = -16$	square first, then apply the minus
$3m$	$3 \times (-4) = -12$
$m^3$	$(-4)^3 = -64$	an odd power keeps the sign

Calculator tip. Type $(-3)^2$ with the brackets. Without them, many calculators work out $-3^2 = -9$ .

Constructing an expression

Word problems become algebra one phrase at a time. “I think of a number $n$ , add 3, then square it, then multiply by 2” becomes

$n \rightarrow n + 3 \rightarrow (n + 3)^2 \rightarrow 2(n + 3)^2$

The brackets matter: without them, $2n + 3^2$ would mean something completely different.

Compound shapes

There are usually two routes:

Split the shape into rectangles and add their areas, or
Take the surrounding rectangle and subtract the piece that is missing.

Both must give the same simplified expression — which makes a great check.

Proof with algebra

Algebra can prove things are always true. For example, if $n$ is even, the next two odd numbers are $n + 1$ and $n + 3$ , and their product is

$(n + 1)(n + 3) = n^2 + 4n + 3$

$n^2$ and $4n$ are both even, so $n^2 + 4n$ is even and adding 3 makes it odd. So the product of the next two odd numbers is always odd.

Key rules and formulas

Put the multiplication signs back
When you replace letters with numbers, the hidden $\times$ signs come back: if $x = 10$ and $y = 6$ , then $2 x + 3 y = 2 \times 10 + 3 \times 6 = 38$ .
Brackets round negatives
$p = - 3 : 5 p^{2} = 5 \times (- 3)^{2} = 5 \times 9 = 45$
Always write a negative value inside brackets. The power is worked out before the multiplication, and $(- 3)^{2} = + 9$ .
Build expressions from the picture
Area of a rectangle $=$ length $\times$ width. For a compound shape, either add the parts or take the big rectangle and subtract the missing piece.
Check by substituting
To test whether two expressions are equal, put the same number into both. If they give different answers, they are not equivalent.

Worked examples

Worked example 1

Substituting a negative and a decimal

Using

A = 100 t - 5 p^{2}

, find

A

when

p = - 3

and

t = 0.5

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

Area of a compound shape

An L-shape is a big rectangle

(x + 6)

(x + 2)

with a small rectangle

(x + 3)

x

cut out of one corner. Find an expression for its area.

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

Checking an expression

Amy says the area of a square with side

(x + 6)

x^{2} + 36

. Use

x = 5

to show she is wrong.

Try it yourself first, then reveal the solution one step at a time.

Practice: substitution and constructing expressions

11 questions

Ten questions. For substitution questions, write each step down on paper with brackets round the negatives — it is where nearly every mistake happens.

Question 1 of 11Complete the table
$m = - 4$ , $x = 2$ , $y = - 5$ and $r = 3$ . Find the value of each expression.
Expression Value
10(mx + y) Row 1, Value
−3x − y + r Row 2, Value
5my − 6r Row 3, Value
5xmr ÷ y Row 4, Value
x⁴(m + r) Row 5, Value
Question 2 of 11Type the answerFoundation
Find the value of $9 + m^{2}$ when $m = - 3$ .
Your answer
Question 3 of 11Type the answer
Find the value of $10 r^{2} + 2 r - 4$ when $r = 0.2$ .
Your answer
Question 4 of 11Step by step
Substitute into this formula one step at a time.
Using $s = u t + \frac{1}{2} a t^{2}$ , find $s$ when $u = 80$ , $a = - 3$ and $t = 5$ .
1. Work out $u t$ .
  Step 1 answer
Question 5 of 11Choose all that apply
When $x = 0.5$ and $y = - 6$ , which two of these expressions have the same value?
Select every correct answer — there may be more than one.
$4 x + y^{2}$ $(x y)^{2}$ $6 x y$ $48 x^{2} - y^{2} - y$
Question 6 of 11Algebraic answer
A rectangle has width $w$ . Its length is 7 more than double the width. Write an expression for its area, expanded.
Your expression
Type powers with ^ (so n^2 is n squared). Any equivalent form is accepted, but keep it in the form the question asks for.
Question 7 of 11Algebraic answer
I think of a number $n$ , add 3, then square it, then multiply by 2. Write an expression for the result.
Your expression
Type powers with ^ (so n^2 is n squared). Any equivalent form is accepted.
Question 8 of 11Algebraic answerChallenge
An L-shape is a rectangle $(x + 8)$ by $(x + 4)$ with a rectangle $(x + 5)$ by $(x + 2)$ cut from one corner. Find its area, expanded and simplified.
Your expression
Type powers with ^ (so n^2 is n squared). Any equivalent form is accepted, but keep it in the form the question asks for.
Question 9 of 11Multiple choice
A cylinder has radius 6 cm and height 7 cm ( $V = π r^{2} h$ ). A sphere has radius 6 cm ( $V = \frac{4}{3} π r^{3}$ ). Which has the smaller volume?
Question 10 of 11True or false
Amy says the area of a square with side $(x + 6)$ is $x^{2} + 36$ . Substituting $x = 5$ proves she is wrong.
Question 11 of 11Type the answerChallenge
Heron's formula gives the area of a triangle with sides $a$ , $b$ , $c$ as $A = s (s - a) (s - b) (s - c)$ , where $s = \frac{a + b + c}{2}$ . Use it to find the area of a triangle with sides 5, 5 and 6 cm.
Your answercm²

Expression	Value
10(mx + y)	Row 1, Value
−3x − y + r	Row 2, Value
5my − 6r	Row 3, Value
5xmr ÷ y	Row 4, Value
x⁴(m + r)	Row 5, Value

In this lesson you will learn to…

Why this matters

The two traps with negatives

Constructing an expression

Compound shapes

Proof with algebra

Key rules and formulas

Put the multiplication signs back

Brackets round negatives

Build expressions from the picture

Check by substituting

Worked examples

Practice: substitution and constructing expressions