Mathematics • Year 7 • Unit 2 • Lesson 7
Dividing Algebraic Terms — Mixed Challenge
Pull everything from Lesson 7 together: simple divisions, the power rule for same-base divisions, sign rules, expressions with multiple variables, a classic Year 7 mistake to spot, and an open-ended fraction puzzle.
1. Mixed problems — choose the right idea
Each question uses a different part of Lesson 7. Decide which idea applies before you start. Show your working. 2 marks each
1.1 Simplify each: (a) 18x ÷ 6 (b) 20y ÷ 4 (c) 15a ÷ 3.
1.2 Simplify each using the power rule: (a) x⁵ ÷ x² (b) a⁷ ÷ a³ (c) y⁴ ÷ y.
1.3 Simplify, watching signs: (a) (−12x) ÷ 4 (b) (−18a²) ÷ (−6a) (c) 20m³ ÷ (−5m).
1.4 Simplify the algebraic fractions: (a) 6x ÷ 2x (b) 10x² ÷ 5x (c) 8xy ÷ 2y.
1.5 A rectangle has area 24x² cm² and length 6x cm. Write and simplify an expression for its width.
1.6 Simplify the algebraic fraction with multiple variables: 15x³y ÷ 3xy.
2. Find the mistake
Another Year 7 student has tried to simplify x⁵ ÷ x². 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 x⁵ ÷ x²:
Line 1: The numerator has x⁵ and the denominator has x².
Line 2: Same base means I use the power rule.
Line 3: Power rule: divide the exponents, so 5 ÷ 2 = 2.5.
Line 4: Answer: x2.5.
(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? The power rule says: same base on top and bottom → SUBTRACT the exponents (not divide them). xm ÷ xn = xm − n.3. Open-ended challenge — design your own algebraic fraction
This question has more than one correct answer. Show one that works and explain. 4 marks
3.1 Find an algebraic fraction (something on top, something on bottom) that simplifies to 4x². Then find a different fraction that also simplifies to 4x².
For each fraction, write down the numerator and denominator, then show the cancelling/dividing working that confirms it simplifies to 4x².
Bonus: Now write an algebraic fraction that simplifies to 4x² AND has a y on both the top and the bottom (the y must cancel out).
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Simple number divides
(a) 18x ÷ 6 = 3x. (b) 20y ÷ 4 = 5y. (c) 15a ÷ 3 = 5a. (Variable stays — only on top.)
1.2 — Power rule
(a) x⁵ ÷ x² = x5−2 = x³.
(b) a⁷ ÷ a³ = a7−3 = a⁴.
(c) y⁴ ÷ y = y4−1 = y³.
1.3 — Signs
(a) (−12x) ÷ 4 = −3x (neg ÷ pos = neg).
(b) (−18a²) ÷ (−6a) = +3a = 3a (neg ÷ neg = pos).
(c) 20m³ ÷ (−5m) = −4m² (pos ÷ neg = neg).
1.4 — Cancelling fractions
(a) 6x ÷ 2x = (6 ÷ 2)(x ÷ x) = 3 (x cancels).
(b) 10x² ÷ 5x = (10 ÷ 5)(x² ÷ x) = 2x.
(c) 8xy ÷ 2y = (8 ÷ 2)(x)(y ÷ y) = 4x (y cancels).
1.5 — Rectangle width
Width = area ÷ length = 24x² ÷ 6x = (24 ÷ 6)(x² ÷ x) = 4x cm.
1.6 — Multiple variables
15x³y ÷ 3xy: numbers 15 ÷ 3 = 5; x³ ÷ x = x²; y ÷ y = 1 (cancels). Answer: 5x².
2 — Find the mistake
(a) The mistake is on Line 3.
(b) The power rule says SUBTRACT the exponents, not divide them. The rule is xm ÷ xn = xm − n, not xm ÷ n.
(c) Corrected Line 3: 5 − 2 = 3. Corrected answer: x³.
Quick check with x = 2: x⁵ ÷ x² = 32 ÷ 4 = 8 = 2³ = x³ ✓.
3 — Open-ended (sample solutions)
Target: simplify to 4x².
Fraction A: 12x³ ÷ 3x. Check: (12 ÷ 3)(x³ ÷ x) = 4 × x² = 4x² ✓.
Fraction B (different): 8x⁴ ÷ 2x². Check: (8 ÷ 2)(x⁴ ÷ x²) = 4 × x² = 4x² ✓.
Other valid fractions: 16x² ÷ 4, 20x³ ÷ 5x, 4x³ ÷ x, etc.
Bonus: 12x³y ÷ 3xy. Check: numbers (12 ÷ 3) = 4; x³ ÷ x = x²; y ÷ y = 1 (cancels). Answer: 4x² ✓ — the y appeared on both top and bottom and cancelled.
Marking: 1 for the first valid fraction with working; 1 for a different valid fraction with working; 1 for both pieces of working being correct; 1 for the bonus (if attempted, with y cancelling correctly).