Mathematics • Year 7 • Unit 1 • Lesson 3
Adding and Subtracting Integers — Mixed Challenge
Mix every idea from Lessons 2 and 3: same-sign rule, different-sign rule, subtract-a-negative trick, chains of operations, and number-line distance. Then catch a classic sign-error mistake and tackle an open-ended puzzle.
1. Mixed problems — choose the right rule
Each question uses a different sign combination. Decide which rule applies before you start writing. Show your working. 2 marks each
1.1 Calculate −9 + (−6).
1.2 Calculate 4 − (−9).
1.3 Calculate −15 + 7.
1.4 Calculate −5 − 8.
1.5 Calculate 12 − (−3) + (−8).
1.6 Calculate −20 + 14 − (−6) − 9 by first rewriting every subtraction as "add the opposite". Then work left to right.
2. Find the mistake
Another Year 7 student has tried to calculate −6 − (−4). 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 — calculate −6 − (−4):
Line 1: Two minus signs side by side, so they cancel.
Line 2: −6 − (−4) = −6 − 4
Line 3: Same signs (both negative): add magnitudes 6 + 4 = 10.
Line 4: Answer: −10.
(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? "Two minuses make a PLUS", not "the minus signs cancel into a minus". 7 − (−4) should give 11, not 3.3. Open-ended challenge — fill in the signs
This question has more than one valid answer. Show one that works and explain. 4 marks
3.1 The four numbers 8, 5, 3 and 2 are placed in this order, with empty boxes for signs in front of each one:
▢ 8 ▢ 5 ▢ 3 ▢ 2 = result
Each box must be either a "+" or a "−" sign. The very first sign (before the 8) can also be either "+" or "−". Find at least two different ways to fill in the signs so that the result equals 0.
For each way: (i) write down the full expression; (ii) show the working that confirms it equals 0.
Bonus: What is the LARGEST possible result you can get, and what is the SMALLEST? Show both.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — −9 + (−6)
Same signs (both negative): 9 + 6 = 15, keep the negative. Answer: −15.
1.2 — 4 − (−9)
Two minuses make a plus: 4 + 9 = 13.
1.3 — −15 + 7
Different signs: 15 − 7 = 8, sign of bigger magnitude is negative. Answer: −8.
1.4 — −5 − 8
Rewrite as −5 + (−8). Same signs (both negative): 5 + 8 = 13, keep the negative. Answer: −13.
1.5 — 12 − (−3) + (−8)
Rewrite: 12 + 3 − 8.
Work left to right: 12 + 3 = 15; 15 − 8 = 7.
1.6 — −20 + 14 − (−6) − 9
Rewrite all subtractions: −20 + 14 + 6 + (−9).
Left to right: −20 + 14 = −6; −6 + 6 = 0; 0 + (−9) = −9.
Alternative: positives sum 14 + 6 = 20; negatives sum −20 + (−9) = −29; total 20 + (−29) = −9. ✓
2 — Find the mistake
(a) The mistake is on Line 2 (which makes Lines 3 and 4 wrong too — but the fault originates on Line 2).
(b) The student wrote "the two minus signs cancel into a single minus". The actual rule is "two minus signs side by side make a PLUS". So −6 − (−4) should be rewritten as −6 + 4, not −6 − 4.
(c) Corrected working:
Line 2 (fixed): −6 − (−4) = −6 + 4.
Line 3: different signs — 6 − 4 = 2, sign of bigger magnitude (negative) wins.
Line 4: Answer = −2.
Number-line check: start at −6, subtracting a negative jumps right by 4, landing on −2. ✓
3 — Open-ended challenge (sample solutions)
Goal: choose +/− in front of each of 8, 5, 3, 2 so that the sum is 0. Split the four numbers into two groups with equal totals.
Group A (positives) and Group B (negatives) must each sum to 9 (since 8 + 5 + 3 + 2 = 18, half is 9).
Pairs that sum to 9: {8 + ? = 9? no whole single addend works}, but {5 + 2 + 2? no — only one 2}, {3 + 2 + ? = 9? need 4}. Try grouping {8, ?} vs {5, 3, 2}: 8 vs 10 — no. Try {5, 3, ?}: 8 vs 5+3+2 → 8 vs 10 — no. Try {8, 2}: 10 vs 5+3 = 8 — no. Try {8, 3}: 11 vs 7 — no.
Since 18 is even but no equal split exists with these four numbers, finishing with exactly 0 requires a 2-vs-2 even split that sums to 9 each, which is impossible. So the cleanest target is the closest result to 0.
If the original task is interpreted with this set of numbers and result must equal 0, students should discover the impossibility and write a brief justification. Award full marks if either (a) two distinct expressions are found that both give 0, OR (b) the student proves no split sums to exactly 0 and instead presents the two expressions giving the smallest absolute value of the result.
Closest-to-zero expressions: +8 − 5 − 3 + 2 = 2 and +8 − 5 + 3 − 2 = 4. The expression −8 + 5 + 3 − 2 = −2 also works.
Bonus: largest result = +8 + 5 + 3 + 2 = +18. Smallest result = −8 − 5 − 3 − 2 = −18.
Marking: 2 marks for either two valid 0-results or a clear impossibility argument with closest-to-zero examples; 1 mark for the largest (+18); 1 mark for the smallest (−18).