Mathematics • Year 7 • Unit 2 • Lesson 9
Expanding and Simplifying — Mixed Challenge
Pull everything from Lesson 9 together: two added brackets, two subtracted brackets, three brackets with mixed signs, brackets with multiple variables, checking your answer by substitution, and a classic Year 7 mistake. Plus an open-ended bracket puzzle.
1. Mixed problems — choose the right idea
Each question uses a different part of Lesson 9. Expand each bracket first, then collect like terms. Show your working. 2 marks each
1.1 Expand and simplify each: (a) 2(x + 3) + 3(x + 1) (b) 4(x + 1) + 2(x + 5).
1.2 Expand and simplify each (subtracted bracket): (a) 5(x + 2) − (x − 3) (b) 4(2x − 1) − 2(x + 3).
1.3 Expand and simplify the three-bracket expression: 2(x + 1) + 3(x + 2) − (x − 1).
1.4 Expand and simplify with two variables: 3(x + 2y) + 2(2x − y).
1.5 Expand and simplify 3(x − 4) + 2(x + 6), then check by substituting x = 3 into BOTH the original and your simplified answer.
1.6 The perimeter of a rectangle is 2(x + 5) + 2(x − 1). Expand, simplify, and find the perimeter when x = 6.
2. Find the mistake
Another Year 7 student has tried to expand and simplify 4(x + 2) − 3(x − 1). 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 4(x + 2) − 3(x − 1):
Line 1: Expand the first bracket: 4(x + 2) = 4x + 8.
Line 2: Expand the second bracket: −3(x − 1) = −3x − 3.
Line 3: Combine: 4x + 8 − 3x − 3.
Line 4: Collect: x + 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? Try x = 2 in the original. 4(2 + 2) − 3(2 − 1) = 16 − 3 = 13. Does the student's "x + 5" give 13 when x = 2? 2 + 5 = 7. Doesn't match — so something is wrong.3. Open-ended challenge — design your own expression
This question has more than one correct answer. Show one that works and explain. 4 marks
3.1 Write a two-bracket expression of the form a(x + p) + b(x + q) (with a, b, p, q being positive integers of your choice) that, after being expanded and simplified, equals 5x + 11.
Write your two-bracket expression, then show the expansion and simplification that confirms it.
Bonus: Now write a different two-bracket expression — but this time one of the brackets must be subtracted — that also simplifies to 5x + 11.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Two added brackets
(a) 2(x + 3) + 3(x + 1) = 2x + 6 + 3x + 3 = 5x + 9.
(b) 4(x + 1) + 2(x + 5) = 4x + 4 + 2x + 10 = 6x + 14.
1.2 — Subtracted bracket
(a) 5(x + 2) − (x − 3) = 5x + 10 − x + 3 = 4x + 13.
(b) 4(2x − 1) − 2(x + 3) = 8x − 4 − 2x − 6 = 6x − 10.
1.3 — Three brackets
2(x + 1) + 3(x + 2) − (x − 1) = 2x + 2 + 3x + 6 − x + 1. Collect: x-terms 2x + 3x − x = 4x; constants 2 + 6 + 1 = 9. Answer: 4x + 9.
1.4 — Two variables
3(x + 2y) + 2(2x − y) = 3x + 6y + 4x − 2y. Collect: x-terms 3x + 4x = 7x; y-terms 6y − 2y = 4y. Answer: 7x + 4y.
1.5 — Expand and check
3(x − 4) + 2(x + 6) = 3x − 12 + 2x + 12 = 5x (the constants cancelled!).
Check with x = 3: Original 3(3 − 4) + 2(3 + 6) = −3 + 18 = 15. Simplified 5 × 3 = 15 ✓.
1.6 — Rectangle perimeter
2(x + 5) + 2(x − 1) = 2x + 10 + 2x − 2 = 4x + 8. Substitute x = 6: 4 × 6 + 8 = 32 (so perimeter = 32 units).
2 — Find the mistake
(a) The mistake is on Line 2.
(b) −3(x − 1) is not −3x − 3. The −3 multiplies BOTH inside terms, and neg × neg = pos for the second term: −3 × (−1) = +3, not −3. So −3(x − 1) = −3x + 3.
(c) Corrected Line 2: −3(x − 1) = −3x + 3. Corrected Line 3: 4x + 8 − 3x + 3. Corrected Line 4: x-terms 4x − 3x = x; constants 8 + 3 = 11. Final answer: x + 11.
Quick check with x = 2: original 4(2 + 2) − 3(2 − 1) = 16 − 3 = 13. Corrected x + 11 = 2 + 11 = 13 ✓.
3 — Open-ended (sample solutions)
Target: 5x + 11.
Expression A: 2(x + 1) + 3(x + 3). Expand: 2x + 2 + 3x + 9. Collect: 5x + 11 ✓.
Expression B (also valid): 3(x + 1) + 2(x + 4) = 3x + 3 + 2x + 8 = 5x + 11 ✓.
Other valid: 1(x + 5) + 4(x + 1.5)? — no, p and q should be integers. Try 4(x + 2) + 1(x + 3) = 4x + 8 + x + 3 = 5x + 11 ✓.
Bonus — with a subtracted bracket: 6(x + 2) − (x − 1). Expand: 6x + 12 − x + 1 = 5x + 13 — close, but not 11. Try 6(x + 3) − (x + 7) = 6x + 18 − x − 7 = 5x + 11 ✓.
Marking: 1 for the first valid expression with working; 1 for the working being correct; 1 for a second distinct valid expression with working; 1 for the subtracted-bracket bonus.