Mathematics • Year 7 • Unit 2 • Lesson 8
Expanding Single Brackets — Mixed Challenge
Pull everything from Lesson 8 together: positive multipliers, negative multipliers, brackets with three inside terms, variable multipliers (like x outside), checking by substitution, and a classic Year 7 mistake. Finish with an open-ended bracket puzzle.
1. Mixed problems — choose the right idea
Each question uses a different part of Lesson 8. Decide which idea applies before you start. Show your working. 2 marks each
1.1 Expand each: (a) 5(x + 4) (b) 6(2x + 1) (c) 3(x − 7).
1.2 Expand each (negative outside): (a) −3(x + 2) (b) −4(2x − 3) (c) −(x − 5).
1.3 Expand bracket with three inside terms: 2(x + 3y − 4).
1.4 Expand each (variable as the outside term): (a) x(x + 5) (b) a(2a − 3) (c) y(y + 1).
1.5 Expand 3(2x − 4) and check your answer by substituting x = 5 into both the original and the expanded form.
1.6 A rectangle has width 4 and length (3x − 2). Write and expand an expression for its area.
2. Find the mistake
Another Year 7 student has tried to expand −2(x − 3). 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 −2(x − 3):
Line 1: Multiplier = −2. Inside terms = x and −3.
Line 2: Multiply outside by first inside: −2 × x = −2x.
Line 3: Multiply outside by second inside: −2 × (−3) = −6.
Line 4: Combine: −2x − 6.
(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 = 1 in the original. −2(1 − 3) = −2 × (−2) = +4. Does the student's −2x − 6 give +4 when x = 1? −2 − 6 = −8. Doesn't match. So something is wrong.3. Open-ended challenge — design your own bracket
This question has more than one correct answer. Show one that works and explain. 4 marks
3.1 Find a bracket expression of the form a(bx + c) that expands to 6x + 12. Then find a different bracket that also expands to 6x + 12.
For each bracket, write the bracket form first, then show the working that confirms it expands to 6x + 12.
Bonus: Now find a bracket with a negative outside multiplier that expands to −6x − 12.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Positive multipliers
(a) 5(x + 4) = 5x + 20. (b) 6(2x + 1) = 12x + 6. (c) 3(x − 7) = 3x − 21.
1.2 — Negative multipliers
(a) −3(x + 2) = −3x − 6 (both signs negative).
(b) −4(2x − 3) = −8x + 12 = −8x + 12 (neg × neg = pos for the second term).
(c) −(x − 5) = −x + 5 = −x + 5 (the implicit multiplier is −1).
1.3 — Three inside terms
2(x + 3y − 4) = 2x + 6y − 8 = 2x + 6y − 8. (Multiply 2 by each of x, 3y, and −4 in turn.)
1.4 — Variable as the outside term
(a) x(x + 5) = x² + 5x = x² + 5x.
(b) a(2a − 3) = 2a² − 3a = 2a² − 3a.
(c) y(y + 1) = y² + y = y² + y.
1.5 — Expand 3(2x − 4) and check
Expanded: 3 × 2x + 3 × (−4) = 6x − 12.
Check x = 5: Original 3(2 × 5 − 4) = 3 × 6 = 18. Expanded 6 × 5 − 12 = 30 − 12 = 18 ✓.
1.6 — Rectangle area
Area = 4(3x − 2) = 4 × 3x + 4 × (−2) = 12x − 8.
2 — Find the mistake
(a) The mistake is on Line 3.
(b) −2 × (−3) is NOT −6. Negative × negative = positive, so −2 × (−3) = +6. The student dropped the sign-flip when multiplying two negatives.
(c) Corrected Line 3: −2 × (−3) = +6. Corrected Line 4: combine: −2x + 6.
Quick check with x = 1: original −2(1 − 3) = −2 × (−2) = +4. Corrected −2x + 6 = −2 + 6 = 4 ✓.
3 — Open-ended (sample solutions)
Target: expand to 6x + 12.
Bracket A: 2(3x + 6). Check: 2 × 3x + 2 × 6 = 6x + 12 ✓.
Bracket B (different): 3(2x + 4). Check: 3 × 2x + 3 × 4 = 6x + 12 ✓.
Other valid brackets: 6(x + 2), 1(6x + 12) (trivial), etc.
Bonus — negative multiplier to give −6x − 12: −2(3x + 6). Check: −2 × 3x + (−2) × 6 = −6x − 12 ✓. (Or −3(2x + 4), or −6(x + 2), etc.)
Marking: 1 for the first valid bracket with working; 1 for a different valid bracket with working; 1 for both pieces of working being correct; 1 for the negative-multiplier bonus.