Lesson 1 ~30 min Unit 2 · Algebra +85 XP

Review of Algebraic Expressions

Three tools, one toolbox, collect like terms, substitute values, and apply BIDMAS. You'll lean on these in every lesson this unit.

Today's hook: Every time you split a pizza bill, scale a recipe, or work out a phone plan, you're doing algebra without thinking about it. Why letters?

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Think First

warm-up

Before you read on, if $a = 3$ and $b = 5$, what is the value of $2a + 3b - a$? Try it in your head, then note which order you did the operations in.

Record your answer in your workbook.

The Big Idea

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Algebra is shorthand for "do this with a number whatever the number turns out to be." When you write $3x$ you're saying "three of whatever $x$ is." The rest of the lesson is just three small tools, see below.

Every problem in this unit is just three small moves: combine like terms, substitute letters with numbers, and follow BIDMAS for order.

$3x + 5x = 8x$

Same letter? Combine.

Add the coefficients, keep the letter. $3x + 5x = 8x$.

Got a value? Substitute.

Replace each letter, brackets around negatives. $(-3)^2 = 9$.

Many ops? BIDMAS.

Brackets → Indices → ×÷ → +−. Same priority: left to right.

What You'll Master

objectives

Know

What an expression is vs an equation
Term, coefficient, variable, constant
The BIDMAS order of operations

Understand

Why only like terms can combine
Why $(-3)^2 = 9$, not $-9$
Why brackets matter in substitution

Can Do

Simplify by collecting like terms
Substitute values, including negatives
Evaluate with correct BIDMAS order

Words You Need

vocabulary

ExpressionA math phrase with letters, numbers and operations, but NO equals sign. e.g. $3x + 7$.

TermA single building block separated by + or −. $3x + 7$ has two terms: $3x$ and $7$.

CoefficientThe number multiplying the letter. In $3x$ it's $3$; in $-x$ it's $-1$.

VariableA letter ($x$, $a$, $n$, …) that stands for an unknown or changeable number.

ConstantA plain number with no variable. In $3x + 7$ the constant is $7$.

Like termsTerms with exactly the same letters raised to exactly the same powers, only these can be added or subtracted.

Spot the Trap

heads-up

Wrong: "$3x + 5y$ simplifies to $8xy$."

Right: $3x$ and $5y$ are NOT like terms. They stay separate: $3x + 5y$ is already in simplest form.

Wrong: "If $x = -3$, then $x^2 = -9$."

Right: $(-3)^2 = (-3) \times (-3) = +9$. A negative times a negative is positive.

Parts of the Whole

+5 XP

Every algebraic expression is built from terms, joined by $+$ or $-$ signs. Each term has up to three parts you need to recognise on sight.

An expression is a math phrase with no equals sign. It describes a value rather than asserting one. Each term is a block. Each block has a coefficient, a variable, or is a plain constant.

$3x + 7$
EXPRESSION

Coefficient

The number multiplying the letter. In $3x$ it's 3; in $-x$ it's $-1$.

Variable

The letter. Stands in for an unknown number.

Constant

A plain number, no letter. Doesn't change.

Knowing the parts by name makes everything easier. When a question says "find the coefficient of $x$" or "substitute $x = 4$", you'll know exactly where to look.

Sort and Sum

+5 XP

Two terms are like terms when they share exactly the same letters raised to exactly the same powers. Coefficients don't have to match, only the letter-and-power signature does. Sort the like terms together, then sum.

Sort terms into letter groups. Inside each group, add (or subtract) the coefficients. Groups with different letters stay apart they're already in simplest form.

$ax + bx = (a+b)x$

Same letter → combine

$3x + 7x = 10x$. Add the coefficients, keep the letter.

Different letter → leave

$3x + 5y$ stays as $3x + 5y$. No simpler form exists.

Power matters

$x$ and $x^2$ aren't like. Group them separately.

$3x$ and $7x$, like ✓ (both $x^1$)
$5a^2$ and $-2a^2$, like ✓ (both $a^2$)
$3x$ and $3y$, not like ✗ (different letters)
$4x$ and $4x^2$, not like ✗ (different powers)

When collecting, simply add or subtract the coefficients and keep the letter-part the same:

$$3x + 7x = 10x \qquad 5a - 2a = 3a \qquad 4x + 3y + 2x = 6x + 3y$$

Try It Now: Simplify $6m + 4n - 2m + 5n$. (Hint: two groups.) Answer: $4m + 9n$.

Plug It In

+5 XP

Substitution means replacing each variable with a given number, then evaluating. Golden rule: always use brackets when substituting, especially around negatives.

Wherever you see a letter, write its number in brackets. Then evaluate the resulting number-only expression using BIDMAS. Brackets are your insurance policy against sign errors.

$3(4) + 2(-2) = 8$

Brackets always

Write $3(4)$ not $3 \cdot 4$. Especially around negatives.

Negatives in brackets

$(-3)^2 = 9$, not $-9$. The bracket binds the negative.

Every instance

If $x$ appears 3 times, replace it 3 times, same value each.

Worked example: if $x = 4$ and $y = -2$, find the value of $3x + 2y$.

Watch the negatives: Without brackets, $3 \times -2$ can confuse the eye. Writing $3(-2) = -6$ keeps the sign visible and stops careless errors.

Who Goes First?

+5 XP

When an expression has more than one operation, the answer depends on which one you do first. BIDMAS is the order, and when two operations share a priority, work left to right.

Climb the ladder from the top: B rackets, then I ndices, then D iv/M ult, then A dd/S ub. Same priority? Left to right. Most mistakes come from forgetting that.

$2 + 3 \times 4^2 = 50$

Brackets always first

Do every operation inside ( ) before anything outside.

Same priority → L to R

$\times$ and $\div$ share priority. So do $+$ and $-$. Work in order.

Classic slip

$10 - 5 + 2 = 7$ (NOT 3). Subtraction isn't "before" addition.

The full ladder same as the mini above, with the example operations spelt out:

Example: evaluate $2 + 3 \times 4^2$.

$$2 + 3 \times 4^2 = 2 + 3 \times 16 \;\text{(indices)} = 2 + 48 \;\text{(mult)} = 50$$

Watch Me Solve It · 3 examples

Watch Me Solve It · Collecting like terms

+15 XP per step

PROBLEM

Simplify $5a + 3b - 2a + 7b$. Show the cleanest form.

1

Group like terms

$(5a - 2a) + (3b + 7b)$

Bring the $a$-terms together and the $b$-terms together, never mix letters when combining.
2

Combine coefficients

$= 3a + 10b$

$5 - 2 = 3$ and $3 + 7 = 10$. Keep the letter part the same.
3

Check it's simplest form

$3a + 10b$ ✓

Each letter appears once, nothing left to combine.

Answer $3a + 10b$

Watch Me Solve It · Substituting negatives

+15 XP per step

PROBLEM

If $p = 5$ and $q = -3$, evaluate $2p^2 - 3q$.

1

Substitute with brackets

$2(5)^2 - 3(-3)$

Brackets around $-3$ save you from sign errors.
2

Apply indices (BIDMAS)

$2(25) - 3(-3)$

Indices come before multiplication, so $5^2$ first.
3

Multiply then add

$= 50 - (-9) = 50 + 9 = 59$

Subtracting a negative becomes adding.

Answer $59$

Watch Me Solve It · Perimeter with algebra

+15 XP per step

PROBLEM

A rectangle has length $(3x + 2)$ cm and width $(2x - 1)$ cm. (a) Write a simplified expression for the perimeter. (b) Find the perimeter when $x = 4$.

1

Set up perimeter formula

$P = 2(3x + 2) + 2(2x - 1)$

Perimeter = 2 × length + 2 × width.
2

Expand brackets

$P = 6x + 4 + 4x - 2$

Multiply each term inside by the 2 outside.
3

Collect like terms

$P = 10x + 2$

$x$-terms together: $6x + 4x = 10x$. Constants: $4 - 2 = 2$.
4

Substitute $x = 4$

$P = 10(4) + 2 = 42$ cm

Now we have a number-only expression, evaluate it.

Answer (a) $P = 10x + 2$ · (b) $P = 42$ cm

Common Pitfalls

heads-up

Combining unlike terms

$3x + 4y \rightarrow 7xy$ ✗, these are not like terms, so they cannot be combined. The expression stays as $3x + 4y$.

Fix: check the letter AND power match before combining. Treat each letter-group separately.

Squaring a negative without brackets

If $x = -3$ and the expression is $x^2$, writing $-3^2 = -9$ is wrong. BIDMAS applies the index first, ignoring the negative.

Fix: ALWAYS substitute with brackets: $(-3)^2 = 9$.

Doing addition before division

$6 + 4 \div 2$, if you do $6 + 4 = 10$ first, you'd get $5$. Division comes first: $6 + 2 = 8$.

Fix: stop and check BIDMAS order before each step.

Copy Into Your Books

Definitions

Expression, math phrase, no equals sign
Term, block separated by + or −
Coefficient, number multiplying the variable
Constant, plain number, no variable

Like Terms Rule

Same letter AND same power → combine
$3x + 7x = 10x$
$5a^2 - 2a^2 = 3a^2$
$3x + 5y$ stays as $3x + 5y$

Substitution

Use brackets around negatives
$(-3)^2 = 9$, not $-9$
Replace every instance of each letter

BIDMAS Order

Brackets → Indices → ×÷ → +−
Same priority = left to right
$10 - 5 + 2 = 7$, not $3$

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Mixed

4 problems

Four problems mixing all three tools. Work each one, then reveal the answer to check.

1 Simplify $7a + 4b - 3a + 6b - 2a$.

Group $a$-terms and $b$-terms: $(7a - 3a - 2a) + (4b + 6b)$$= 2a + 10b$
2 If $x = -2$ and $y = 5$, evaluate $x^2 - 2y$.

Substitute with brackets: $(-2)^2 - 2(5)$$= 4 - 10 = -6$
3 Evaluate $20 - 6 \div 2 + 3 \times 4$.

Division and multiplication first (left to right): $20 - 3 + 12$$= 17 + 12 = 29$
4 If $m = 3$ and $n = -4$, find $2m^2 + 3mn$.

$2(3)^2 + 3(3)(-4) = 2(9) + (-36)$$= 18 - 36 = -18$

Complete in your workbook.

Quick Check · 5 questions

Which pair are like terms?

+10 XP

Simplify $8x + 3y - 5x + 2y$.

+10 XP

If $a = 4$ and $b = -2$, what is $3a - 2b$?

+10 XP

Simplify $2x^2 + 5x - 3x^2 + 2x$.

+10 XP

If $x = -3$ and $y = 2$, evaluate $x^2 - 2xy$.

+10 XP

Show Your Working · 3 questions

Show Your Working

8 marks total

Apply Easy 2 MARKS

Q6. Simplify $9p - 2q + 4p + 7q - 5p$.

Answer in your workbook.

Apply Medium 3 MARKS

Q7. If $m = 5$ and $n = -2$, evaluate: (a) $m^2 + n^2$ (b) $3mn - 2m$.

Answer in your workbook.

Analyse Hard 3 MARKS

Q8. A rectangle has length $(3x + 2)$ cm and width $(2x - 1)$ cm. (a) Write a simplified expression for the perimeter. (b) Find the perimeter when $x = 4$.

Answer in your workbook.

Comprehensive Answers

Multiple Choice

1. C$7m$ and $-2m$ share variable $m$ to the same power.

2. A$(8x-5x)+(3y+2y) = 3x + 5y$.

3. B$3(4) - 2(-2) = 12 + 4 = 16$. Subtracting a negative adds.

4. A$(2x^2 - 3x^2) + (5x + 2x) = -x^2 + 7x$.

5. A$(-3)^2 - 2(-3)(2) = 9 + 12 = 21$.

Short Answer Model Answers

Q6 (2 marks): $(9p + 4p - 5p) + (-2q + 7q) = 8p + 5q$. [1 grouping, 1 answer]

Q7 (3 marks): (a) $5^2 + (-2)^2 = 25 + 4 = 29$ [1]. (b) $3(5)(-2) - 2(5)$ [1] $= -30 - 10 = -40$ [1].

Q8 (3 marks): (a) $P = 2(3x+2) + 2(2x-1) = 6x+4+4x-2$ [1] $= 10x + 2$ [1]. (b) $P = 10(4) + 2 = 42$ cm [1].

Stretch Challenge · +25 XP, +10 coins

The Hidden Substitution

The identity $a^2 + 2ab + b^2 = (a + b)^2$ is worth memorising. Use it to evaluate $97^2$ without a calculator. Hint: let $a = 100$ and $b = -3$.

Reveal solution

$97^2 = (100 - 3)^2 = 100^2 + 2(100)(-3) + (-3)^2 = 10\,000 - 600 + 9 = 9409$.

Quick Review

Like Terms

Same letter, same power

Collecting

Add/subtract coefficients only

Substitute

Brackets around negatives

$(-x)^2$

Always positive

BIDMAS

B · I · DM · AS, left to right

Same priority

Work L → R

Interactive: Expression Simplifier

Practise collecting like terms and substituting values until it's automatic.

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