Mathematics • Year 7 • Unit 1 • Lesson 2
Integers, Mixed Challenge
Combine every idea from Lesson 2: plot integers on the number line, find opposites and absolute values, compare and order, count distances, and spot a classic negative-number error.
1. Mixed problems, choose the right idea
Each question uses a different idea from Lesson 2. Decide which idea applies before you start writing. Show your working. 2 marks each
1.1 Arrange from smallest to largest: 6, −9, 0, −1, 4, −3.
1.2 Find the value of: (a) |−14| (b) the opposite of +11 (c) |0| + |−7|.
1.3 What integer is 6 units to the right of −4 on the number line?
1.4 Place >, < or = between each pair:
(a) −7 ____ −12 (b) |−4| ____ |−6| (c) −3 ____ |−3|
1.5 List all the integers n for which |n| ≤ 3. How many are there?
1.6 The temperature in the morning was −4 °C. By the afternoon it had risen to 12 °C. By how many degrees did the temperature rise? Show the number line distance.
2. Find the mistake
Another Year 7 student has tried to order four integers from smallest to largest. 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, order from smallest to largest: 2, −8, −3, 5.
Line 1: "I'll separate them by sign: negatives are −8 and −3; positives are 2 and 5."
Line 2: "Positives go in order: 2, 5."
Line 3: "Negatives go in order: −3, −8 (because 3 < 8 makes −3 < −8)."
Line 4: "Final order from smallest to largest: −3, −8, 2, 5."
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected ordering in full.
Stuck? On the number line, −8 is further LEFT than −3, so −8 is the smaller of the two.3. Open-ended challenge, find two integers
This question has more than one correct answer. Show one that works and explain. 4 marks
3.1 Find two different pairs of integers (call them a and b) such that:
(i) a < 0 (a is negative)
(ii) b > 0 (b is positive)
(iii) |a| = |b|
(iv) The distance between a and b on the number line is exactly 10.
For each pair you find: (a) write down a and b; (b) check that all four rules are satisfied; (c) sketch a small number line showing both points.
Bonus: Explain why there's actually only one pair that works, and what changes if rule (iv) is dropped.
How did this worksheet feel?
What I'll revisit before next class:
1.1, Order from smallest to largest
Plot on the number line and read left to right: −9, −3, −1, 0, 4, 6.
1.2, Absolute value and opposites
(a) |−14| = 14 (distance from zero).
(b) The opposite of +11 is −11.
(c) |0| + |−7| = 0 + 7 = 7.
1.3-6 units right of −4
Each step right adds 1. Start at −4, add 6: −4 + 6 = +2. Quick check: from −4 you pass −3, −2, −1, 0, 1, 2, that's 6 jumps. ✓
1.4, Compare
(a) −7 > −12 (−7 is further right on the number line).
(b) |−4| < |−6| (4 < 6).
(c) −3 < |−3| (−3 vs 3).
1.5, Integers with |n| ≤ 3
|n| ≤ 3 means "n is at most 3 units from zero". The integers are −3, −2, −1, 0, 1, 2, 3. That is 7 integers.
1.6, Temperature rise from −4 to 12
Distance on the number line = 12 − (−4) = 12 + 4 = 16 °C. From −4 you climb 4 degrees to reach 0, then 12 more to reach +12, total 16 degrees.
2, Find the mistake
(a) The mistake is on Line 3.
(b) The student flipped the rule for negatives. The rule "3 < 8" applies to positives only; for negatives, the bigger absolute value gives the smaller number. So −8 < −3, not the other way around.
(c) Corrected ordering: negatives in order are −8, −3 (because −8 is further left on the number line). Combining: −8, −3, 2, 5. This is the "Comparing Integers" trap from the lesson.
3, Open-ended challenge (sample solution)
Rules (i)–(iii) force a and b to be opposites: |a| = |b| with one negative and one positive means b = −a. So the pair is (−k, +k) for some positive integer k.
Rule (iv): distance on the number line = k − (−k) = 2k. We need 2k = 10, so k = 5.
The only pair that works is a = −5, b = +5.
Number line: dots at −5 and +5, symmetric about 0, exactly 10 apart.
Bonus: there's only one pair because rules (iii) and (iv) together pin down k uniquely. If rule (iv) is dropped, ANY pair of opposites (−1, +1), (−2, +2), (−7, +7) … would work.
Marking: 2 marks for finding (−5, +5) and showing it satisfies all four rules; 1 for the sketch; 1 for the bonus explanation.