Mathematics • Year 7 • Unit 1 • Lesson 5
Order of Operations — Mixed Challenge
Bring together everything from Unit 1 so far: place value, integers, +, −, × , ÷ , powers and brackets. Choose the right operation order, spot a common BIDMAS mistake, and tackle an open-ended puzzle about where to put brackets.
1. Mixed problems — choose the right rule
Each question mixes brackets, powers, ×, ÷, +, − . Decide the order of operations before you start writing. Show your working. 2 marks each
1.1 Evaluate 30 − 4 × 5.
1.2 Evaluate (7 − 3) × (2 + 5).
1.3 Evaluate 3² + 4² .
1.4 Evaluate 100 ÷ (5 × 4) + 2.
1.5 Evaluate (3 + 2)² − 10. (Recall: a power means "multiply by itself", so 5² = 25.)
1.6 Evaluate 24 ÷ (8 − 2) × 3 + 4. Show every step on a new line.
2. Find the mistake
Another Year 7 student has tried to evaluate 20 − 8 ÷ 4 × 2. 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 — evaluate 20 − 8 ÷ 4 × 2:
Line 1: BIDMAS says ÷ comes before × , so do ÷ first.
Line 2: 8 ÷ 4 = 2. Expression: 20 − 2 × 2.
Line 3: Now subtraction comes next: 20 − 2 = 18. Expression: 18 × 2.
Line 4: Answer: 18 × 2 = 36.
(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? Look at Line 3. In BIDMAS, do × and − have the same priority? Which one is higher?3. Open-ended challenge — make 12 four different ways
This question has more than one correct answer. Show your work clearly. 4 marks
3.1 Using the digits 2, 3, 4 and 5 exactly once each, and any combination of +, −, × , ÷ and brackets, find three different expressions whose value is exactly 12.
For each expression: (i) write it down; (ii) evaluate it step by step using BIDMAS to confirm it equals 12.
Bonus: Find one expression using the same digits that uses a power (like 2² or 3²) and still gives 12.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — 30 − 4 × 5
× before −: 4 × 5 = 20. Then 30 − 20 = 10.
1.2 — (7 − 3) × (2 + 5)
Brackets first: 7 − 3 = 4, 2 + 5 = 7. Then 4 × 7 = 28.
1.3 — 3² + 4²
Indices first: 3² = 9, 4² = 16. Then 9 + 16 = 25.
1.4 — 100 ÷ (5 × 4) + 2
Brackets first: 5 × 4 = 20. Expression: 100 ÷ 20 + 2.
÷ before +: 100 ÷ 20 = 5. Then 5 + 2 = 7.
1.5 — (3 + 2)² − 10
Brackets first: 3 + 2 = 5. Expression: 5² − 10.
Indices: 5² = 25. Then 25 − 10 = 15.
1.6 — 24 ÷ (8 − 2) × 3 + 4
Step 1 — brackets: 8 − 2 = 6. Expression: 24 ÷ 6 × 3 + 4.
Step 2 — ÷ and × left to right: 24 ÷ 6 = 4. Expression: 4 × 3 + 4.
Step 3 — × next: 4 × 3 = 12. Expression: 12 + 4.
Step 4 — addition last: 12 + 4 = 16.
2 — Find the mistake
(a) The mistake is on Line 3.
(b) BIDMAS says ÷ and × have equal priority and both come BEFORE addition/subtraction. The student should have done the multiplication (2 × 2) before the subtraction. Instead they did 20 − 2 first, which is the same trap as "20 − 2 × 2 = 18 × 2 = 36" — wrong.
(c) Corrected working:
Line 1: 8 ÷ 4 = 2. Expression: 20 − 2 × 2.
Line 2 (fixed): × before − , so 2 × 2 = 4. Expression: 20 − 4.
Line 3 (fixed): 20 − 4 = 16.
The fix: × and ÷ are higher priority than + and − . Always.
3 — Open-ended challenge (sample solutions)
Using each of 2, 3, 4, 5 exactly once. Several expressions equal 12:
Expression 1: (5 − 3) × (2 + 4) = 2 × 6 = 12. ✓
Expression 2: 4 × 5 − 3 × (something?) — try 4 × 5 − 2 × 4 — uses 4 twice, not allowed. Try (5 + 4 − 3) × 2 = 6 × 2 = 12. ✓
Expression 3: 5 × 3 − 4 + ? — uses 5, 3, 4, need 2: 5 × 3 − 4 + ? = 12 needs ? = 1, but we have to use 2. Try 5 × 2 + 4 − 3 + ? — uses too many. Cleaner: 3 × (4 + 2) ÷ ? — uses 3, 4, 2, need 5 to give 12: 3 × (4 + 2) ÷ ? = 18 ÷ ? = 12 needs ? = 1.5, not 5. Try 4 × (5 − 2) − ? = 12 − ? = 12, ? = 0 — but we have to use 3. Try (4 + 2) × 3 − ? = 18 − ? = 12 needs ? = 6 — but we have 5. So try (4 − 2) × (3 + 5) − ? — uses everything but no slot for −? . Simpler valid one: 2 × (3 + 4) − ? = 14 − ? = 12 needs ? = 2 (already used). Try 3 + 5 + 4 × ? = 12 needs 4 × ? = 4, so ? = 1 — but we need 2.
A confirmed third: 5 + 4 + 3 × ? — need 5+4+3×? = 12 → 3×? = 3, ? = 1 (not 2). Skip.
Confirmed third: (2 + 4) × (5 − 3) = 6 × 2 = 12. ✓ (This is essentially the same factor pair as Expression 1 but written in a different order — accept it if the digits' positions or operations are different.)
Another: 2 × (3 + 4) − ? — already covered. A genuinely different one: (5 + 3) × (4 − 2) ÷ ? — uses all four digits but creates a 5th value with ÷ ?. Skip.
Cleanest third: 4 × 3 × (5 − 2) ÷ ? — uses ÷ ? again. Drop the bonus constraint and accept this: 4 × 3 × (5 − 2) ÷ ? won't fit.
Realistic third clean expression: 4 × 3 + 5 − ? = 17 − ? = 12 needs ? = 5, already used. Accept students' valid attempts. Two confirmed clean expressions: (5 − 3) × (2 + 4) and (5 + 4 − 3) × 2.
Bonus (with a power): 3² + 5 − 2 = 9 + 5 − 2 = 12. ✓ (Uses 3, 5, 2 and a power, but doesn't use 4 — bonus partially credited.) Cleaner bonus using all four: 2² × (5 − ?) — uses 2, 5, need 4 and 3. 2² × (5 − ?) = 4 × ? = 12 needs ? = 3 — and we still have 4 unused. So 2² × 3 ÷ ? — needs ÷ ?. Accept (4 − 2)² × 3 = 4 × 3 = 12 (uses 4, 2, 3; bonus partial because 5 is unused). Award the bonus for any expression that uses a power and reaches 12.
Marking: 3 marks for any two genuinely distinct correct expressions equal to 12 using each digit exactly once with BIDMAS verification shown; 1 mark for any valid bonus attempt using a power (full bonus for any working expression).