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Swap and Balance — Making Two Groups Equal
Lesson
A common difficulty is random trial: nudge students to compute the difference between the two sums first, then look for a pair of numbers whose difference is half that amount. Slips of paper make the physical swapping concrete and keep the reasoning visible. Encourage students to say why fewer swaps are better, not just record the count.
0–8Write two unequal groups of numbers on the board (e.g. {3, 7, 12} and {5, 8, 9}). Ask students to find each group's sum and tell you which is larger. Invite a few students to suggest a swap and check whether the sums become equal.
8–28Students work in pairs on Making Sums Equal (a)–(d), p.42 ("Interchange pairs of numbers between two groups to make their sums equal using the least number...."). Each pair writes numbers on slips so they can physically move them. After each swap the pair recomputes both sums. Pairs record the number of swaps needed for each case.
28–40Pairs share their results for two of the four cases. The class discusses: how did you decide which numbers to swap? Is there a way to know in advance? Collect one key insight — swapping two numbers changes each group's sum by the same amount in opposite directions — on the board in students' own words.
Making Sums Equal (a)–(d), p.42 — Interchange pairs of numbers between two groups to make their sums equal using the least number of moves. (a)–(d) four cases.
Group A has the numbers 2, 8, 14 (sum = 24). Group B has the numbers 5, 10, 9 (sum = 24). Now consider Group A: 3, 9, 12 (sum = 24) and Group B: 7, 6, 11 (sum = 24). Suppose instead Group A has 4, 7, 13 (sum = 24) and Group B has 6, 8, 10 (sum = 24). If you swap the 13 from Group A with the 10 from Group B, what are the new sums of each group? Are they equal? If not, suggest one swap that would make the sums equal.
After swapping 13 and 10: Group A becomes 4, 7, 10 with sum 21; Group B becomes 6, 8, 13 with sum 27. The sums are not equal. The difference is 6, so the groups need a swap where the two numbers differ by 3. Swapping the 7 (Group A) with the 10 (Group B) — wait, 10 is now in Group A. Swapping the 4 from Group A with the 7 from Group B gives Group A = 7, 7, 10 — not valid (duplicates). Better: from the original groups (before any swap), swapping 13 from A with 7 from B gives A = 4, 7, 7 — still invalid. Use the original: A = 4, 7, 13 (sum 24), B = 6, 8, 10 (sum 24). Sums are already equal — no swap needed. The question leads students to verify: after swapping 13↔10, Group A = 4, 7, 10 (sum 21), Group B = 6, 8, 13 (sum 27); difference = 6; need a pair differing by 3; swap 7
- (A) with 10 — but 10 is in A now. Correct path: swap 4
- (A) with 7 (B): A = 7, 7, 10 — invalid. Swap 10
- (A) with 6 (B): A = 4, 7, 6 (sum 17), B = 8, 13, 10 (sum 31) — worse. The one-swap solution from after the first swap: swap 10
- (A) with 13
- (B) to undo; or swap 7
- (A) with the 6 (B): A = 4, 6, 10 (sum 20), B = 7, 8, 13 (sum 28) — still unequal. Correct one-swap fix: from A = 4, 7, 10
- (21) and B = 6, 8, 13 (27), difference = 6, half = 3; find a in A and b in B where b − a = 3: 7 and 10 — no (10 is in A); 4 and 7 (7 is in A); 6+3=9 not in A; 8−3=5 not in A; 13−3=10 yes — swap 10
- (A) with 13 (B): A = 4, 7, 13 (sum 24), B = 6, 8, 10 (sum 24). Equal. So the swap that fixes it is swapping the 10 (now in A) back with the 13 (now in B), restoring the original.
Also saved as 8 periods or 11 periods.
- 2Fuel Up — Adding with Regrouping in Columns
Lesson - 3Fact Families — Addition and Subtraction Are Linked
Lesson - 4Borrowing Fuel — Subtraction with Regrouping
Lesson - 5Consecutive Clues — Patterns in Running Sums
Lesson - 6North–South Corridor — Adding 4- and 5-digit Numbers
Lesson - 7Sea Routes and Savings — Subtracting Large Numbers
Lesson - 8Piku and Namita's Tricks — Mental Addition and Subtraction
Lesson - 9Pairs and Leftovers — Even, Odd, and Their Sums
Lesson - 10Palindromes and Patterns — Reasoning Challenges Part 1
Lesson - 11Coins, Mountains, and Mela — Solving Extension Problems
Lesson - 12Let Us Do — Fluency and Word-Problem Practice
Lesson - 13Travellers' Roundup — Connecting the Whole Chapter
Lesson
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