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.
3. Challenge, find the integer pair
The conditions determine one pair. Find it, verify each rule, and explain why it is unique. 4 marks
3.1 Find the unique pair 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.
(a) Write down a and b; (b) check all four rules; (c) sketch a number line showing both points.
Bonus: Explain 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 verifying all four rules; 1 for the number-line sketch; 1 for explaining what happens when rule (iv) is dropped.