GCSE · Maths · AQA · Spec 8300 · Foundation

Vocabulary of expressions, equations, formulae, inequalities, terms, factors

3x + 2 can be five different things. The middle symbol is your first clue; what it claims settles it. Soon you'll name each one and say why.

Sort it

What is it?

Tap a card, then tap the box it belongs in. Look at the symbol in the middle first, then ask what the statement is claiming.

Still to sort

Expression (0)

No =, ≡ or inequality sign. It's just an amount.

Where the line is: Link it to something with =, ≡, <, >, ≤ or ≥ and it stops being an expression. It becomes a statement that can be true or false.

Equation (0)

An = sign, and true only for particular values of the letter.

Where the line is: An identity is true for every value. An equation is only true for particular ones.

Formula (0)

A rule linking different quantities. Each letter stands for something real.

Where the line is: An equation asks which value of the letter works. A formula tells you how to work out one quantity from the others.

Identity (0)

True for every value of the letter. Often written with ≡.

Where the line is: Test it. If even one value of the letter makes it false, it's not an identity.

Inequality (0)

Uses <, >, ≤ or ≥ to compare two sides.

Where the line is: ≤ and ≥ include 'or equal to', but they still compare. They make inequalities, not equations.

10 of 10 still to sort.

Look back at the first five cards. They're all built from 3x + 2 (the taxi fare just calls the letter m), yet each is a different kind of statement. The symbol, and what the statement claims, made the difference.

Watch out: A formula uses = too. It's a formula because its letters stand for real quantities and it works as a rule for any values you put in.

Predict, then check

The trickiest pair is equation versus identity, so let's test them. Don't solve anything. Just substitute.

Both of these use =. A: 3x + 2 = 11. B: x + 2x + 2 = 3x + 2. If you try x = 1, x = 3 and x = 10 in each, what will you find?

Check someone else's work

Spot the slip

A student named five statements and gave a reason for each. One line is wrong. Which one?

A student's answer — which line goes wrong?

8 − 3w < 2 uses <, which compares the two sides, so it's an inequality.
8 − 3w has a letter in it, so it's an equation.
8 − 3w = 2 is only true when w = 2, so it's an equation.
3w − w ≡ 2w is true for every value of w, so it's an identity.
V = lwh links the volume of a cuboid to its length, width and height, so it's a formula.

Inside an expression

Terms and factors

Now zoom inside an expression: 4x² − 3xy + 5.

Which of these is closest to how you'd split it into terms?

Three terms: 4x², −3xy and +5. The + and − signs split it up, and each sign belongs to the term after it.Lots of terms: 4, x², 3, x, y and 5 are all separate terms.Three terms: 4x², 3xy and 5. The minus sign is just a gap between them.Just one term. The whole thing is one expression.

How sure are you?

I'm sureI think soI'm guessing

Three terms: 4x², −3xy and +5. The + and − signs split it up, and each sign belongs to the term after it.

This matches the evidence

Spot on. Terms are the chunks that are added or subtracted, and each one keeps its sign, so the middle term is −3xy. Inside a term, things are multiplied: 3xy means 3 × x × y, so 3, x and y are factors of that term. The same goes for 4x², which is 4 × x × x.

Lots of terms: 4, x², 3, x, y and 5 are all separate terms.

This is a common misconception

Tempting, because you can see lots of separate symbols. But 4 and x² aren't added together. They're multiplied: 4x² means 4 × x². Things multiplied together are factors of one term. In an expression like this, terms start and stop at the + and − signs, so there are three.

Three terms: 4x², 3xy and 5. The minus sign is just a gap between them.

This is a common misconception

Nearly. You've found the right three chunks, but the minus sign isn't a gap. It belongs to the term, which is −3xy. Here's why it matters. Reorder it as 5 + 4x² − 3xy and nothing changes, because −3xy moved with its sign. Lose the sign and you get 5 + 4x² + 3xy, which is a different expression. With x = 1 and y = 1, the first is worth 6 and the second is worth 12.

Just one term. The whole thing is one expression.

This is a common misconception

An expression can be built from several terms, just as a sentence is built from several words. 4x² − 3xy + 5 is one expression made of three terms, split by the + and − signs.

Terms are added or subtracted, and each one keeps its sign. Factors are multiplied together inside a term.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Expression, equation, formula, identity, inequality. Plus terms and factors. The symbol in the middle is your first clue. What the statement claims settles it.

What you need to know

An expression has no =, ≡ or inequality sign. It's an amount, like 3x + 2, not a claim.
An equation uses = and is true only for particular values of the letter, like 3x + 2 = 11 (true when x = 3).
A formula is a rule linking different quantities, like A = lw for the area of a rectangle.
An identity is true for every value of the letter and is often written with ≡, like x + 2x ≡ 3x.
An inequality compares two sides using <, >, ≤ or ≥, like 4n ≤ 20.
Terms are the parts added or subtracted, and each keeps its sign. Factors are the parts multiplied inside a term.

The big picture

Algebra has five kinds of statement, and the symbol in the middle is your first clue. With no linking sign, it's an expression. With <, >, ≤ or ≥, it's an inequality. An = sign can start three different things, so ask what the statement claims. True only for particular values? Equation. A rule linking quantities? Formula. True for every value? Identity. The ≡ sign is a way of saying 'identity' out loud, but an identity can be written with = too. Inside an expression, terms are added or subtracted and factors are multiplied.

Key points

1Check the symbol in the middle first: nothing, =, ≡, or <, >, ≤, ≥.

2Equations, formulae and identities can all use =. An equation is true for particular values of an unknown. A formula is a rule that links quantities. An identity is true for every value.

3To tell an identity from an equation, substitute values. If any value makes it false, it's not an identity.

4≤ and ≥ still make inequalities, even though they include 'or equal to'.

5In 4x² − 3xy + 5 there are three terms: 4x², −3xy and +5. 3, x and y are factors of 3xy.

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Worked example

Problem

Name each statement and give the feature that decides it. (a) 7y − 4 (b) 7y − 4 = 10 (c) 6y − y ≡ 5y (d) 7y − 4 < 10 (e) P = 2l + 2w, where P is the perimeter of a rectangle with length l and width w.

⚠ Watch out

Calling anything with an x in it 'an equation'. 3x + 2 on its own has no = sign, so it's an expression and there's nothing to solve. An equation needs = and is only true for particular values.

🧠

Memory hook

Read the middle. Nothing there: expression. <, >, ≤ or ≥: inequality. = means ask what it claims. A value to find: equation. A rule for quantities: formula. True for every value: identity. ≡ just says 'identity' out loud. Then look inside: terms add, factors multiply.

✓

Check yourself

What kind of statement is 10 − 3k ≥ 1, and what are the terms on its left-hand side? (An inequality, because ≥ compares the sides. The terms are 10 and −3k.)

Flashcards

(6)

What makes something an expression?

It has no =, ≡ or inequality sign. It's an amount, like 5a − 1, not a claim that can be true or false.

What is an equation?

A statement with = that is true only for particular values of the letter. For example, 2n + 1 = 9 is only true when n = 4.

What is a formula?

A rule linking different quantities, such as P = 4s for the perimeter of a square with side s. Put in a value for s and it gives you P.

What is an identity, and what does ≡ mean?

An identity is true for every value of the letter, like 2m + m ≡ 3m. The sign ≡ means 'is identically equal to'.

What is an inequality?

A statement that compares two sides using <, >, ≤ or ≥. For example, t − 1 > 6 is true when t = 10 but false when t = 2.

What's the difference between a term and a factor?

Terms are added or subtracted and keep their sign: 6a − 2b has terms 6a and −2b. Factors are multiplied inside a term: 6a is 6 × a.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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What is it?

Spot the slip

Terms and factors

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