Lesson 1 ~25 min Unit 2 · Patterns & Algebra +85 XP

What is Algebra?

Why do mathematicians use letters? Discover how variables, constants, expressions and equations work together — and why algebra is just arithmetic with mystery boxes.

Today's hook: A friend says "I'm thinking of a number. If I triple it and add 5, I get 20." Could you figure it out? That's algebra — and you already know how to do it.

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Think First

warm-up

I'm thinking of a number. If I add 7 to it, the answer is 15. What's the number? Write down how you figured it out — there's more than one way.

Record your answer in your workbook.

The Big Idea

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Algebra is a way of writing maths using letters to stand for unknown numbers. Instead of saying "some number plus 5 equals 12", we write $x + 5 = 12$. The letter $x$ is just a placeholder — a mystery box that holds a number.

Every algebra problem has three building blocks: a variable (the unknown letter), a constant (a fixed number), and either an expression (a phrase) or an equation (a sentence with an equals sign).

$x + 5 = 12$

Variable = unknown

A letter like $x$, $n$, or $a$ stands for a number we don't know yet.

Constant = fixed

A number like 3, 10, or $-7$ that never changes its value.

Expression vs Equation

$2x + 3$ is an expression (no equals). $2x + 3 = 11$ is an equation.

What You'll Master

objectives

Know

What a variable and constant are
The difference between an expression and an equation
What a coefficient is

Understand

Why letters are used in algebra
That a variable can represent any number
Why an equation needs an equals sign

Can Do

Identify variables, constants and coefficients
Tell expressions apart from equations
Write simple algebraic phrases

Words You Need

vocabulary

AlgebraA branch of maths where letters stand for unknown numbers. From Arabic "al-jabr" meaning reunion of broken parts.

VariableA letter (like $x$, $n$, $a$) that represents an unknown or changeable number.

ConstantA fixed number that doesn't change, like 5 or $-3$.

CoefficientThe number multiplying a variable. In $3x$, the coefficient is 3. In $-x$, it's $-1$.

ExpressionA maths phrase with variables, numbers and operations — but NO equals sign. e.g. $2x + 3$.

EquationA maths sentence that says two things are equal. It ALWAYS has an equals sign. e.g. $2x + 3 = 11$.

Spot the Trap

heads-up

Wrong: "$2x$ means $2 + x$"

Right: $2x$ means $2 \times x$. A number next to a letter always means multiplication.

Wrong: "$x = 5$ is an expression"

Right: $x = 5$ is an equation because it has an equals sign. Expressions never have one.

The Mystery Box

+5 XP

Think of a variable as a mystery box with a number inside. You can't see the number yet, but you can still do maths with the box. When you finally open it (solve the equation), you find the number.

A variable is just a container. We label it with a letter so we can talk about it. A constant is a number that sits by itself. Together they make an expression — a recipe for calculating a value.

$3x + 5$

Any letter works

$x$, $n$, $a$, $m$ — they all mean "some number". Pick one and stick with it.

Coefficient of 1

$x$ secretly means $1x$. We don't write the 1, but it's always there.

Number then letter

We always write $3x$, not $x3$. The coefficient goes first by convention.

Look at the diagram above. $3x + 5$ is an expression — it's just a phrase, like "three boxes plus five." But $3x + 5 = 8$ is an equation — it's a complete sentence that claims two things are equal. The equals sign is what makes the difference.

Expression or Equation?

+5 XP

The single most important distinction in beginning algebra: does it have an equals sign? If yes, it's an equation. If no, it's an expression.

An expression is like a recipe — it tells you how to calculate something. An equation is like a puzzle — it tells you two things are the same, and you need to find the missing piece.

$\text{expr} \rightarrow \text{no =} \quad \text{eq} \rightarrow \text{has =}$

Expression = phrase

$5n - 2$ is a phrase. It has a value that depends on $n$.

Equation = sentence

$5n - 2 = 18$ makes a claim. We can solve it to find $n$.

Quick test

Scan for =. No equals → expression. Has equals → equation.

Try It Now: Classify each: (a) $7m + 2$ (b) $7m + 2 = 23$ (c) $x - 5 = 0$ Answers: (a) expression (b) equation (c) equation

Picking Out the Parts

+5 XP

Every algebraic expression is built from terms. Each term has parts you need to name. Being able to point at $5x$ and say "the coefficient is 5 and the variable is $x$" is a superpower.

Look at an expression like $4x + 3y - 7$. It has three terms: $4x$, $3y$, and $-7$. In $4x$, the coefficient is 4 and the variable is $x$. The term $-7$ is just a constant — no variable at all.

$4x + 3y - 7$

Sign belongs to term

In $4x - 7$, the second term is $-7$, not just 7. The minus sign stays attached.

Hidden coefficient

$x$ means $1x$, and $-x$ means $-1x$. The 1 is invisible but real.

Constant term

A term with no variable (like $+5$ or $-3$) is called a constant term.

From Words to Algebra

+5 XP

Algebra started because people got tired of writing "some number" over and over. Here's how everyday phrases become algebra.

Look for key words. "Sum" and "more than" mean $+$. "Difference" and "less than" mean $-$. "Product" and "times" mean $\times$.

$\text{words} \rightarrow \text{algebra}$

"a number" = $n$

Whenever you see "a number" or "some number", pick a letter to represent it.

"Less than" reverses

"7 less than $x$" is $x - 7$, NOT $7 - x$. The order matters!

"Of" means multiply

"Half of a number" = $\frac{1}{2}n$ or $n \div 2$.

Words	Operation	Algebra
A number increased by 4	addition	$n + 4$
5 less than a number	subtraction	$n - 5$
Triple a number	multiplication	$3n$
A number divided by 2	division	$n \div 2$ or $\frac{n}{2}$

Watch Me Solve It · Worked example

Watch Me Solve It · Identifying parts

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PROBLEM

For the expression $5a - 3b + 7$, identify: (a) the number of terms, (b) the coefficient of $a$, (c) the constant term, (d) the variable in the second term.

1

Count the terms

$5a$, $-3b$, and $+7$ = 3 terms

Terms are separated by + and - signs. Each sign belongs to the term that follows it.
2

Find the coefficient of $a$

Coefficient of $a$ = 5

In the term $5a$, the number multiplying $a$ is 5. That's the coefficient.
3

Identify the constant term

Constant term = $+7$ (or just 7)

The term $+7$ has no variable — it's just a number. That makes it the constant term.
4

Variable in the second term

Second term is $-3b$, so the variable is $b$

The second term is $-3b$. The letter part is $b$, so that's the variable. Don't forget the minus sign belongs to the term!

Answer (a) 3 terms · (b) 5 · (c) 7 · (d) $b$

Common Pitfalls

heads-up

Confusing $2x$ with $2 + x$

$2x$ means $2 \times x$ (multiplication), NOT $2 + x$ (addition). A number next to a letter always means multiply.

Fix: remember that $2x$ is shorthand for multiplication — the operation is hidden but it's definitely $\times$, not $+$.

Forgetting that $-x$ means $-1x$

When you see just $-x$, the coefficient is $-1$, not 0 or "nothing". Students often miss the negative sign.

Fix: always ask "what number is multiplying the variable?" For $-x$, the answer is $-1$.

Thinking expressions and equations are the same

$3x + 2$ is an expression — no equals sign, so nothing to solve. $3x + 2 = 14$ is an equation — has an equals sign and can be solved.

Fix: do a quick scan for the = sign. No equals → expression. Has equals → equation.

Copy Into Your Books

Key Definitions

Variable — a letter that stands for an unknown number
Constant — a fixed number that doesn't change
Coefficient — the number multiplying a variable
Term — a single part of an expression, separated by + or -

Expression vs Equation

Expression: NO equals sign (e.g. $3x + 5$)
Equation: HAS equals sign (e.g. $3x + 5 = 20$)
Expression = phrase, Equation = sentence

Word Clues

"a number" → use $n$ or $x$
"sum", "more than", "plus" → $+$
"difference", "less than" → $-$
"product", "times" → $\times$

Hidden Coefficients

$x$ means $1x$ (coefficient is 1)
$-x$ means $-1x$ (coefficient is $-1$)
Always write the number first: $3x$, not $x3$

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Mixed

4 problems

Four problems to test your understanding. Work each one, then reveal the answer to check.

1 Is $4x + 9 = 25$ an expression or an equation? Explain why.

Equation — it has an equals sign, which makes it a statement that two things are equal.Equation
2 In the expression $7m - 2n + 4$, what is the coefficient of $n$?

The term with $n$ is $-2n$. The coefficient is $-2$. Don't forget the minus sign belongs to the term!$-2$
3 Write an algebraic expression for: "5 more than triple a number."

Let $n$ = the number. Triple = $3n$. 5 more than = $+5$. So: $3n + 5$$3n + 5$
4 How many terms are in $3x + 2y - z + 8$? Name the constant term.

4 terms: $3x$, $2y$, $-z$, and $+8$. The constant term is $8$ (the term with no variable).4 terms; constant = 8

Complete in your workbook.

Quick Check · 5 questions

Which of the following is an equation?

+10 XP

In the expression $5x - 3y + 8$, what is the coefficient of $y$?

+10 XP

Which expression means "4 less than a number $n$"?

+10 XP

How many terms are in the expression $2a + 3b - c + 7$?

+10 XP

A student writes: "$-x$ has no coefficient." What is the actual coefficient of $-x$?

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

Understand Easy 2 MARKS

Q6. Explain the difference between a variable and a constant. Give one example of each.

Answer in your workbook.

Apply Medium 3 MARKS

Q7. For the expression $6a - 2b + 9$, identify: (a) the number of terms, (b) the coefficient of $a$, (c) the constant term.

Answer in your workbook.

Create Hard 4 MARKS

Q8. Write algebraic expressions for the following: (a) Double a number increased by 3. (b) The difference between a number and 8, divided by 2.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — $3x + 7 = 22$ is an equation because it has an equals sign.

2. C — The coefficient of $y$ is $-3$ (the term is $-3y$).

3. A — "4 less than $n$" means $n - 4$.

4. D — 4 terms: $2a$, $3b$, $-c$, and $+7$.

5. B — The coefficient of $-x$ is $-1$.

Show Your Working Model Answers

Q6 (2 marks): A variable is a letter that stands for an unknown or changeable number (e.g. $x$, $n$) [1 mark]. A constant is a fixed number that does not change its value (e.g. $5$, $-3$) [1 mark].

Q7 (3 marks): (a) 3 terms: $6a$, $-2b$, and $+9$ [1 mark]. (b) The coefficient of $a$ is 6 [1 mark]. (c) The constant term is 9 [1 mark].

Q8 (4 marks): Let $n$ = the number. (a) Double = $2n$, increased by 3 = $+3$, so $2n + 3$ [2 marks]. (b) Difference between $n$ and 8 = $n - 8$, divided by 2 = $\frac{n - 8}{2}$ or $(n - 8) \div 2$ [2 marks]. Accept $\frac{n-8}{2}$.

Stretch Challenge · +25 XP, +10 coins

The Expression Puzzle

The expression $ax + b$ has value 11 when $x = 2$, and value 23 when $x = 5$. Find the values of $a$ and $b$.

Reveal solution

When $x = 2$: $2a + b = 11$ (1). When $x = 5$: $5a + b = 23$ (2). Subtract (1) from (2): $3a = 12$, so $a = 4$. Substitute into (1): $8 + b = 11$, so $b = 3$.

Quick Review

Variable

A letter that stands for an unknown number

Constant

A fixed number that never changes

Coefficient

The number multiplying a variable

Expression

A maths phrase with NO equals sign

Equation

A maths sentence WITH an equals sign

Term

A single part separated by + or -

Interactive: Algebra Machine

Substitute numbers into algebraic expressions and see them evaluate step by step.

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