Mathematics • Year 7 • Unit 2 • Lesson 6
Multiplying Algebraic Terms — Mixed Challenge
Pull everything from Lesson 6 together: simple products, products with the same variable twice, products with three terms, sign rules with negatives, and an open-ended puzzle. Plus one classic Year 7 mistake to spot.
1. Mixed problems — choose the right idea
Each question uses a different part of Lesson 6. Decide which idea applies before you start. Show your working. 2 marks each
1.1 Simplify each product: (a) 4x × 3y (b) 6a × 2 (c) 5m × n.
1.2 Simplify: (a) 3x × 4x (b) 2a × 5a (c) x² × x³.
1.3 Simplify each, watching the signs: (a) (−3x)(2y) (b) (−4a)(−5b) (c) (−2x)(3x).
1.4 Simplify the three-term product 2a × 3b × 4a. Write the variables in alphabetical order.
1.5 A rectangle has width 2x cm and length 3x cm. Write and simplify an expression for its area.
1.6 A rectangular cuboid (box) has dimensions 2x, 3y and 4. Write and simplify an expression for its volume (volume = length × width × height).
2. Find the mistake
Another Year 7 student has tried to simplify the product 3x × 4x. Their working is shown below. Exactly one line contains a mistake. Spot it, explain why it's wrong, then redo the working correctly. 3 marks
Student's working — for 3x × 4x:
Line 1: Split into coefficients and variables: (3 × 4) × (x × x).
Line 2: Multiply coefficients: 3 × 4 = 12.
Line 3: Multiply variables: x × x = 2x.
Line 4: Put together: 12 × 2x = 24x.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write the corrected working in full, including the corrected final answer.
Stuck? Try x = 5: 3 × 5 = 15 and 4 × 5 = 20. So 3x × 4x = 15 × 20 = 300. Does the student's answer 24x give 24 × 5 = 120? No — so something's wrong.3. Open-ended challenge — design your own product
This question has more than one correct answer. Show one that works and explain. 4 marks
3.1 Find two algebraic terms (each with at least one variable) whose product is 12x²y. Then find a different pair that also gives 12x²y.
For each pair, write down the two terms, then show the working that confirms their product is 12x²y.
Bonus: Now find a product of three algebraic terms (each with at least one variable) whose product is 12x²y.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Simple products
(a) 4x × 3y = 12xy. (b) 6a × 2 = 12a. (c) 5m × n = 5mn.
1.2 — Same variable twice
(a) 3x × 4x = (3 × 4)(x × x) = 12x².
(b) 2a × 5a = (2 × 5)(a × a) = 10a².
(c) x² × x³ = x2+3 = x⁵ (same base, add the powers).
1.3 — Signs
(a) (−3x)(2y) = −6xy (neg × pos = neg).
(b) (−4a)(−5b) = 20ab (neg × neg = pos).
(c) (−2x)(3x) = (−6)(x²) = −6x².
1.4 — Three-term product
2a × 3b × 4a: coefficients 2 × 3 × 4 = 24; variables a × b × a = a²b. Answer: 24a²b (alphabetical order — a before b).
1.5 — Rectangle area
Area = width × length = 2x × 3x = (2 × 3)(x × x) = 6x² cm².
1.6 — Volume of a box
Volume = 2x × 3y × 4 = (2 × 3 × 4)(xy) = 24xy cubic units.
2 — Find the mistake
(a) The mistake is on Line 3.
(b) x × x is NOT 2x — that would be x + x. x times x means "x multiplied by x", which adds the powers: x¹ × x¹ = x1+1 = x².
(c) Corrected working: Line 3 (fixed): x × x = x². Line 4 (fixed): 12 × x² = 12x².
Quick check with x = 5: 3x × 4x = 15 × 20 = 300, and 12x² = 12 × 25 = 300. ✓
3 — Open-ended (sample solutions)
Target: 12x²y. We need coefficients to multiply to 12, the variable parts to multiply to x²y.
Pair A: 3xy × 4x. Check: (3 × 4)(xy × x) = 12 × x²y = 12x²y ✓.
Pair B (different): 6x × 2xy. Check: (6 × 2)(x × xy) = 12 × x²y = 12x²y ✓.
Other valid pairs: 12x × xy, 2x² × 6y, 4x² × 3y, etc.
Bonus — three terms: 2x × 3x × 2y. Check: (2 × 3 × 2)(x × x × y) = 12 × x²y = 12x²y ✓.
Marking: 1 for the first valid pair with working; 1 for a different valid pair with working; 1 for the working being correct in both cases; 1 for the three-term bonus (if attempted).