KS3 · Maths

Algebraic notation and conventions

n + n + n and n × n × n look like twins, but at n = 10 one gives 30 and the other 1,000. Spelling rules separate them.

Maths · Algebra

Sort the longhand into its short form

Which short form does each longhand card turn into? Ask yourself first: is it repeated adding or repeated multiplying?

Still to sort

3n (0)

A letter added to itself

Where the line is: Repeated ADDITION: the number goes in front and is called a coefficient.

n³ (0)

A letter multiplied by itself

Where the line is: Repeated MULTIPLICATION: a small raised number goes on top and is called an index.

ab (0)

Different letters multiplied

Where the line is: No × sign, and the letters go in alphabetical order.

a²b (0)

A repeated letter multiplied by another letter

Where the line is: The repeated letter gets an index; the other letter stays as it is.

a/b (0)

One thing divided by another

Where the line is: Division is written as a fraction: the top over the bottom.

2a (0)

Letters added, then some taken away

Where the line is: Subtracting a letter is the same as adding its negative, so the count of letters drops.

9 of 9 still to sort.

Mathematicians agreed a spelling system for algebra so that identical terms are easy to spot and anyone can read your work. Each card is written the long way. Pick a card, then pick the short form it really is.

Maths · What is a letter?

What does the letter actually stand for?

A sticker costs a pence. Zoe buys 5 stickers and writes 5a.

Which idea is closest to what you think the letter a means here?
How sure are you?

Maths · Writing a product

Spelling a product the agreed way

Write b × 5 × a × 2 × a as simply as you can, using the algebra conventions.

  1. Sort the pieces. The numbers are 5 and 2. The letters are b, a and a.
  2. missing step
Which line is step 2?

Predict, then check

Commit to an answer before you look. The minus sign is the sneaky bit.

In the expression y − 3x, what are the coefficient of y and the coefficient of x?

Maths · Four words, four meanings

Expression, equation, identity or formula?

For each item, decide which properties it really has. Build the grid, then check it.

3x − 7
3a + 4 = 10
x + x ≡ 2x
A = lw

Maths · Spot the slip

Find the line where the grouping breaks

Maya works out 5(g + 4) when g = 3, and then (a + 6)/2 when a = 4. One of her lines ignores a grouping rule. Which one?

Maya's working — which line goes wrong?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

The spelling rules that tell 3n and n³ apart — and let anyone read your algebra at a glance.

What you need to know

  • A letter stands for a number: either an unknown (one set value) or a variable (a range of values). It never stands for an object.
  • Repeated addition becomes a coefficient: n + n + n = 3 × n = 3n. Multiplication works either way round, so a × 3 is also written 3a, never a3.
  • Repeated multiplication becomes an index: n × n × n = n³. The index counts how many times the letter is multiplied, so three n's give n³. Don't confuse n³ with 3n.
  • In a product, multiply the numbers into one, write the number first, put different letters in alphabetical order and leave out the × sign: b × 5 × a × 2 = 10ab and e × d × x × 5 = 5dex. A multiplier of 1 is not written.
  • Division is a fraction: a ÷ 4 = a/4, and the fraction line groups the whole numerator and the whole denominator. Brackets group terms: 5(g + 4) means add 4 to g first, then multiply the result by 5, with no × sign between 5 and the bracket.
  • Vocabulary: a term is a number, a letter or a product like 3y or ab, and terms are separated by + and −. A coefficient is the number multiplying the letters in a term, and a constant is a term with no letter. The factors of 2c include 1, 2, c and 2c.

The big picture

Algebra has its own spelling. 3n means n added three times, n³ means n multiplied three times, and 10ab, a²b and a/b are agreed shortcuts so that anyone can read your work at a glance. Alongside the spelling you meet the vocabulary: terms, coefficients, factors, expressions, equations, formulae and identities.

Key points

1Adding a letter to itself gives a coefficient in front (3n). Multiplying it by itself gives an index on top (n³).
2Number first, letters in alphabetical order, no × sign: 10ab, 5dex, a³b².
3The fraction line and brackets are grouping marks: work out what they group first.
4Conventions are agreed by mathematicians so that expressions are easy to read and identical terms are easy to spot.
5An expression has no equals sign, an equation is true for certain values, an identity (≡) is true for all values, and a formula is a rule linking variables in context.

Worked example

Problem

Look at the expression 5x + 2y − 8. How many terms does it have? What is the coefficient of each letter? Which term is the constant?

⚠ Watch out

Mixing up what the two small numbers record. A number in FRONT of a letter (3n) records adding, and a small raised number (n³) records multiplying. Writing the number after the letter, as in a3 for a + a + a, breaks the number-first rule too.

🧠

Memory hook

Adding goes in FRONT (3n), multiplying goes on TOP (n³). Front for adds, top for times.

✓

Check yourself

Write 4n and n⁴ the long way: which is repeated addition? Then write c × 7 × a properly. (4n = n+n+n+n; n⁴ = n×n×n×n; 7ac.)

Flashcards

(15)
What does 3n mean?
n + n + n: n added to itself three times. The 3 is a coefficient.
What does n³ mean?
n × n × n: n multiplied by itself three times. The 3 is an index.
If n = 10, what are 3n and n³?
3n = 30 (10 + 10 + 10) but n³ = 1,000 (10 × 10 × 10).
How do you write a × b, and how do you write b × a?
Both are ab: no × sign, and the letters go in alphabetical order.
Write e × d × x × 5 in conventional form.
5dex: number first, then the letters in alphabetical order.
Write a × a × a × b × b using indices.
a³b²: an index shows repeated multiplication of the same letter.
How do you write a ÷ 4, and what does the fraction line do?
a/4. In fractional notation the fraction line groups the entire numerator and the entire denominator.
What does 5(g + 4) tell you to do?
Add 4 to g first, then multiply the whole result by 5. No × sign is written between the 5 and the bracket.
What does a letter stand for in algebra?
A number: an unknown (a set value) or a variable (a range of values). Never an object.
What is a term?
A single number or letter, or a product of numbers and/or variables, like x, 5, 3y or ab. Terms are separated by + and −.
What is a coefficient?
The constant multiplier of the letters in a term. In y − 3x the coefficient of y is 1 and of x is −3.
What is a constant?
A term that does not change, because it contains no variables. In 3x − 7 it is −7.
What are the factors of 2c?
1, 2, c and 2c: terms that multiply to give 2c, found just like factors of numbers.
Expression, equation, identity, formula: one line each?
Expression: terms with no equals sign. Equation: true only for certain values. Identity: true for all values, written ≡. Formula: a rule linking variables in context.
Why do mathematicians agree conventions in algebra?
So that expressions are easy to read and identical terms are easy to spot.

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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