Year 8 · Mathematics · Algebra
Algebraic expressions
Use letters to stand for unknown numbers, write expressions from words, simplify expressions by collecting like terms and substitute values to evaluate them.
Awaiting educator review. This material was drafted by our content team with AI assistance and is waiting for review by a qualified educator. Please check it suits your child before using it. Created 10 October 2026.
These lessons were written around Australian Curriculum strands and matched to New Zealand year levels by age (NZ Year 1 is the first year of school). They are not yet mapped to The New Zealand Curriculum, so check them against your own learning programme.
Learning goals
Learners will
- Understand that a pronumeral (variable) represents a number.
- Write algebraic expressions from descriptions in words.
- Simplify expressions by collecting like terms.
- Evaluate expressions by substituting values.
Success looks like
- I can write an expression such as 3n + 5 from a description in words.
- I can identify and collect like terms.
- I can substitute a number for a pronumeral and work out the value of an expression.
The big idea
In algebra, a letter such as n stands for a number we do not know yet, or a number that can change. If one box holds n biscuits, then 3 boxes hold 3 × n biscuits. In algebra we leave out the multiplication sign and write 3n.
Some conventions to remember:
- 3 × n is written 3n, with the number first.
- 1 × n is written just n.
- n × n is written n² (n squared).
Words can be turned into expressions. “Five more than three times a number” becomes 3n + 5. “Seven less than m” becomes m − 7. Order matters with subtraction: m − 7 is not the same as 7 − m.
Like terms have exactly the same pronumeral part, so they can be added or subtracted, just as 3 apples + 2 apples = 5 apples. So 4a + 2a = 6a. But 4a + 3b cannot be combined, because a and b stand for different numbers. To simplify 4a + 3b + 2a − b, collect the a terms (4a + 2a = 6a) and the b terms (3b − b = 2b) to get 6a + 2b.
To substitute, replace each pronumeral with its value, then calculate using the order of operations. If x = 4, then 3x − 2 = 3 × 4 − 2 = 12 − 2 = 10.
Worked examples
Writing an expression
Write an expression for “five more than three times a number n”.
- “Three times a number n” is 3n.
- “Five more than” means add 5.
Answer: 3n + 5
Collecting like terms
Simplify 4a + 3b + 2a − b.
- Group the like terms: 4a + 2a and 3b − b.
- 4a + 2a = 6a.
- 3b − b = 3b − 1b = 2b.
Answer: 6a + 2b
Substituting values
If x = 4, find the value of 3x − 2 and of 2(x + 5).
- 3x − 2 = 3 × 4 − 2 = 12 − 2 = 10.
- 2(x + 5) means 2 × (x + 5). Work out the brackets first: 4 + 5 = 9.
- 2 × 9 = 18.
Answer: 3x − 2 = 10 and 2(x + 5) = 18.
Practice check
Have a go, then check your answers. Each answer comes with an explanation.
Answer guide for parents
Which expression means “7 less than m”?
m − 7 — Start with m and take 7 away: m − 7.
Simplify 5x + 2x − 3x.
4x — All three are like terms: 5 + 2 − 3 = 4, so the answer is 4x.
Which pair are like terms?
2y and 7y — 2y and 7y have exactly the same pronumeral part (y), so they are like terms.
What is the value of 2p + 3 when p = 6?
15 — 2 × 6 + 3 = 12 + 3 = 15.
Find the value of 4k − 5 when k = 3.
7 — 4 × 3 − 5 = 12 − 5 = 7.
Simplify 6m + 4 − 2m + 1.
4m + 5 — Collect the m terms: 6m − 2m = 4m. Collect the numbers: 4 + 1 = 5. The answer is 4m + 5.
Find the value of n² + 1 when n = 5.
26 — n² means n × n, so 5 × 5 + 1 = 25 + 1 = 26.
A taxi charges a $4 flag fall plus $2 for each kilometre, so the cost in dollars for k kilometres is 4 + 2k. How many dollars does a 9 km trip cost?
$22 — 4 + 2 × 9 = 4 + 18 = 22, so the trip costs $22.
Watch out for
- Combining unlike terms, for example writing 3a + 2b = 5ab. Use the fruit idea: 3 apples and 2 bananas are not 5 “apple-bananas”.
- Reading 3n as 30-something when n is a digit, such as thinking 3n with n = 4 is 34. Remind the learner that 3n means 3 × n = 12.
- Thinking a letter always stands for its position in the alphabet (a = 1, b = 2). A pronumeral can stand for any number.
- Writing m − 7 and 7 − m as if they are the same. Test with a number, such as m = 10, to show they give different results.
Tips for parents
- Model unknowns with cups: put the same hidden number of counters in each cup and ask how many there are in 3 cups plus 2 loose counters (3n + 2).
- Whenever the learner simplifies, ask them to check by substituting a number into both the original and the simplified expression.
- Keep the language consistent: “term”, “like terms” and “substitute” will be used throughout high school maths.
Go further
- Write expressions for real situations, such as the cost of n movie tickets at $16 each plus a $5 popcorn, then evaluate them for different values of n.
- Make a matchstick pattern of squares in a row (4, 7, 10, … matches) and find an expression for the number of matches needed for n squares.
- Play “think of a number”: double it, add 6, halve it, subtract the original number. Use algebra to explain why the answer is always 3.