50-minute lesson
Year 7 Mathematics: Algebraic expressions
Lesson objective
- 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 criteria:
- 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.
Materials
- Paper and pencil
- Small cups or envelopes and counters (to model unknown amounts)
- Sticky notes
Introduction
5 minIntroduce today's 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.
Ask your child what they already know about algebraic expressions.
Explanation
10 minIn 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
8 minWriting an expression
Write an expression for “five more than three times a number n”.
Step 1: “Three times a number n” is 3n.
Step 2: “Five more than” means add 5.
Answer: 3n + 5
Collecting like terms
Simplify 4a + 3b + 2a − b.
Step 1: Group the like terms: 4a + 2a and 3b − b.
Step 2: 4a + 2a = 6a.
Step 3: 3b − b = 3b − 1b = 2b.
Answer: 6a + 2b
Substituting values
If x = 4, find the value of 3x − 2 and of 2(x + 5).
Step 1: 3x − 2 = 3 × 4 − 2 = 12 − 2 = 10.
Step 2: 2(x + 5) means 2 × (x + 5). Work out the brackets first: 4 + 5 = 9.
Step 3: 2 × 9 = 18.
Answer: 3x − 2 = 10 and 2(x + 5) = 18.
Guided practice (do together)
10 min1. Which expression means “7 less than m”?
(a) 7 − m (b) m − 7 (c) 7m (d) m + 7
2. Simplify 5x + 2x − 3x.
(a) 10x (b) 4 (c) 4x (d) 7x − 3
3. Which pair are like terms?
(a) 3a and 3b (b) 4x and 4x² (c) 2y and 7y (d) 5 and 5p
4. What is the value of 2p + 3 when p = 6?
(a) 11 (b) 26 (c) 29 (d) 15
Independent practice
12 min5. Find the value of 4k − 5 when k = 3.
6. Simplify 6m + 4 − 2m + 1.
7. Find the value of n² + 1 when n = 5.
8. 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?
Questions to check understanding
- Can you write an expression such as 3n + 5 from a description in words?
- Can you identify and collect like terms?
- Can you substitute a number for a pronumeral and work out the value of an expression?
- What was the trickiest part today?
Answer guide
1. m − 7 — Start with m and take 7 away: m − 7.
2. 4x — All three are like terms: 5 + 2 − 3 = 4, so the answer is 4x.
3. 2y and 7y — 2y and 7y have exactly the same pronumeral part (y), so they are like terms.
4. 15 — 2 × 6 + 3 = 12 + 3 = 15.
5. 7 — 4 × 3 − 5 = 12 − 5 = 7.
6. 4m + 5 — Collect the m terms: 6m − 2m = 4m. Collect the numbers: 4 + 1 = 5. The answer is 4m + 5.
7. 26 — n² means n × n, so 5 × 5 + 1 = 25 + 1 = 26.
8. $22 — 4 + 2 × 9 = 4 + 18 = 22, so the trip costs $22.
Review
5 minRecap the success criteria together. Watch for these common misconceptions:
- 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.
Extension activities
- 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.
Suggested follow-up
- Revisit algebraic expressions tomorrow with two or three quick questions from memory.
- Try the online practice check for this topic and look at any questions that need another go.
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.