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.
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These lessons were written around Australian Curriculum strands and matched to UK year groups by age (Year 1 is the year a child turns 6 in England). They are not mapped to the National Curriculum in England or the curricula of Scotland, Wales or Northern Ireland, so check them against your own plans.
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.
About 50 minutes
You'll need
- Paper and pencil
- Small cups or envelopes and counters (to model unknown amounts)
- Sticky notes
Key words
- Pronumeral (variable)
- A letter that stands for a number, such as x or n.
- Term
- A single part of an expression, such as 4a, 7 or 3xy. Terms are separated by + and − signs.
- Coefficient
- The number in front of a pronumeral. In 5x, the coefficient is 5.
- Expression
- A group of terms joined by operations, without an equals sign, such as 2x + 3.
- Like terms
- Terms with exactly the same pronumeral part, such as 3a and 7a.
- Substitute
- To replace a pronumeral with a given number.
Still tricky?
For parents
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