Mathematics • Year 7 • Unit 2 • Lesson 2
Writing Algebraic Expressions
Build the basics: use the operation keywords (sum, difference, product, quotient) to translate English phrases into algebra, and watch out for the "less than" / "subtracted from" order traps.
1. I do — fully worked example
Read every line. Each step has a short reason on the right so you can see why, not just what.
Problem. Write an algebraic expression for: "The product of 4 and the difference between a number and 3, increased by 7."
Step 1 — Choose a variable for "a number".
Let n = the number.
Reason: every time you see "a number" or "some number", pick a letter (n or x) to stand for it.
Step 2 — Translate the innermost phrase first.
"the difference between a number and 3" → n − 3
Reason: "difference between" means subtract. The number comes first because it's the one being talked about.
Step 3 — Wrap it in "the product of 4 and …".
"product of 4 and (n − 3)" → 4(n − 3)
Reason: "product" means multiply. Brackets are MANDATORY because the 4 multiplies the WHOLE difference, not just the n.
Step 4 — Add "increased by 7" at the end.
→ 4(n − 3) + 7
Reason: "increased by" means + . The +7 sits outside the brackets because it's added to the whole product.
Step 5 — Check with a number.
If n = 5: difference = 5 − 3 = 2, product = 4 × 2 = 8, plus 7 = 15.
Expression: 4(5 − 3) + 7 = 4(2) + 7 = 8 + 7 = 15 ✓
Answer: 4(n − 3) + 7.
2. We do — fill in the missing steps
Same structure as Section 1, but with the working faded. Fill in each blank line. 4 marks
Problem. Write an algebraic expression for: "5 less than 3 times a number, divided by 2."
Step 1 — Choose a variable:
Let n = ______________________________________.
Step 2 — Translate "3 times a number":
→ __________
Step 3 — Apply "5 less than" (watch the order trap!):
"5 less than 3n" → __________ (NOT 5 − 3n)
Step 4 — Apply "divided by 2" (the whole thing goes on top):
→ __________
3. You do — independent practice
Translate each phrase into an algebraic expression. Use n (or x) for "a number" unless told otherwise. Show the steps if it has more than one operation.
Foundation — single step
3.1 "A number plus 8". 1 mark
3.2 "The product of 6 and a number". 1 mark
3.3 "A number divided by 4". 1 mark
3.4 "7 less than a number". (Watch the order!) 1 mark
Standard — combine two ideas
3.5 "3 more than double a number". 2 marks
3.6 "The sum of a number and 5, divided by 2". 2 marks
Extension — push your thinking
3.7 "The product of 5 and (a number minus 2), then increased by 8". 3 marks
3.8 Write the algebra for both, then explain in one sentence why they are different: (a) "4 subtracted from a number" (b) "a number subtracted from 4". 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do ("5 less than 3 times a number, divided by 2")
Step 1: Let n = a number.
Step 2: "3 times a number" → 3n.
Step 3: "5 less than 3n" → 3n − 5 (the 3n comes first, then we subtract 5).
Step 4: "divided by 2" → (3n − 5) ⁄ 2 or written as a fraction with 3n − 5 on top and 2 on the bottom.
3.1 — "A number plus 8"
n + 8. "Plus" means + ; normal order.
3.2 — "The product of 6 and a number"
6n. "Product" means multiply; the coefficient (6) always goes in front of the letter.
3.3 — "A number divided by 4"
n ⁄ 4 (or n ÷ 4). The number being divided goes on TOP.
3.4 — "7 less than a number"
n − 7. "Less than" reverses the order — start with n, then subtract 7. (NOT 7 − n. Test with n = 10: "7 less than 10" should give 3, and n − 7 = 10 − 7 = 3 ✓.)
3.5 — "3 more than double a number"
"Double a number" = 2n. "3 more than 2n" = 2n + 3. Answer: 2n + 3. (Check with n = 4: double = 8, plus 3 = 11. 2(4) + 3 = 11 ✓.)
3.6 — "The sum of a number and 5, divided by 2"
"Sum of n and 5" = n + 5. Then divided by 2: (n + 5) ⁄ 2. The brackets (or fraction bar) are essential — they group the sum so that only the whole sum gets divided.
3.7 — "Product of 5 and (a number minus 2), then increased by 8"
"A number minus 2" = n − 2. "Product of 5 and (n − 2)" = 5(n − 2). "Increased by 8" = + 8 at the end. Answer: 5(n − 2) + 8. (Check with n = 4: 5 × 2 + 8 = 18, and 5(4 − 2) + 8 = 5(2) + 8 = 18 ✓.)
3.8 — Two "subtracted" phrases
(a) "4 subtracted from a number" → n − 4. The 4 is taken AWAY from n, so n comes first.
(b) "A number subtracted from 4" → 4 − n. The n is taken AWAY from 4, so 4 comes first.
Why different: the word "from" tells you which number is the starting point. The number AFTER "from" comes first in the algebra. Test with n = 10: (a) gives 10 − 4 = 6; (b) gives 4 − 10 = −6. Very different answers!