Mathematics • Year 10 • Unit 2 • Lesson 2
Expanding Brackets — Mixed Challenge
Pull together everything from Lesson 2: distributive law, negative multipliers, x as a multiplier, and expand-then-simplify when two brackets share an x-line. Choose the right tool, find another student's slip, then design a pair of expressions that are equal after expansion.
1. Mixed problems — choose the right tool
Show your working. 3 marks each
1.1 Expand 7(2x − 3).
1.2 Expand −4(x − 6).
1.3 Expand 3a(a + 5).
1.4 Expand and simplify 2(3x + 1) + 5(x − 4).
1.5 Expand and simplify 6(x + 2) − 4(2x − 1).
1.6 The area of a rectangle is given by 3x(x + 6). (a) Expand this expression. (b) Find the area when x = 5.
2. Find the mistake
Another Year 10 student has tried to expand and simplify 4(2x + 3) − 2(x − 5). Their working is shown below. Exactly one line contains a mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Student's working — expand and simplify 4(2x + 3) − 2(x − 5):
Line 1: = 4(2x) + 4(3) − 2(x) − 2(−5)
Line 2: = 8x + 12 − 2x − 10
Line 3: = (8x − 2x) + (12 − 10)
Line 4: = 6x + 2
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected working in full, including the corrected final answer.
Stuck? On Line 2 — what does −2 × (−5) actually equal?3. Open-ended challenge — design equivalent expressions
This question has many valid answers. Be creative but show every number. 4 marks
3.1 Design two different expressions involving brackets that both expand to exactly the same simplified form: 6x + 18.
In your submission, include:
(i) Expression A (with brackets) and its full expansion to 6x + 18.
(ii) Expression B (with brackets, but structured differently from A — for example, A uses one bracket and B uses two) and its full expansion to 6x + 18.
(iii) A one-sentence explanation of why two different-looking bracket expressions can give the same expansion.
Bonus: Make at least one of your expressions use a negative multiplier somewhere.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — 7(2x − 3)
= 7(2x) + 7(−3) = 14x − 21.
1.2 — −4(x − 6)
= (−4)(x) + (−4)(−6) = −4x + 24. Negative × negative = positive.
1.3 — 3a(a + 5)
= 3a · a + 3a · 5 = 3a² + 15a.
1.4 — 2(3x + 1) + 5(x − 4)
= (6x + 2) + (5x − 20) = (6x + 5x) + (2 − 20) = 11x − 18.
1.5 — 6(x + 2) − 4(2x − 1)
= (6x + 12) + (−8x + 4) = (6x − 8x) + (12 + 4) = −2x + 16.
The −4 multiplier turns the −1 inside into +4.
1.6 — Rectangle area 3x(x + 6)
(a) = 3x · x + 3x · 6 = 3x² + 18x.
(b) When x = 5: 3(5)² + 18(5) = 3(25) + 90 = 75 + 90 = 165 square units.
2 — Find the mistake
(a) The mistake is on Line 2.
(b) The last term in Line 2 is "−10", but the previous line correctly showed it as −2(−5), which equals +10 (negative times negative is positive). The student dropped the sign flip.
(c) Corrected working:
= 4(2x) + 4(3) − 2(x) − 2(−5)
= 8x + 12 − 2x + 10
= (8x − 2x) + (12 + 10)
= 6x + 22.
Lesson § "Watch the Negative" flags this exact slip.
3 — Open-ended challenge (sample solutions)
Pair 1:
Expression A: 6(x + 3) = 6x + 18 ✓.
Expression B: 3(2x + 4) + 6 = 6x + 12 + 6 = 6x + 18 ✓.
Both reach 6x + 18 even though A has one bracket and B has one bracket plus a loose constant. The expansion strips the brackets and exposes the underlying expression — different bracket arrangements just hide the same sum.
Pair 2 (with a negative multiplier — bonus):
Expression A: 2(3x + 9) = 6x + 18 ✓.
Expression B: 10(x + 3) − 4(x + 3) = (10x + 30) + (−4x − 12) = 6x + 18 ✓.
The −4 multiplier in B brings a negative through. Both still simplify to 6x + 18.
Marking: 1 for each valid expression with correct expansion (2 marks total), 1 for them being structurally different, 1 for the explanation. Full marks for any valid pair.