3A · Chapter 2 · Lesson 7
Sum and Difference
Represent two unknown quantities with equal bars and an extra segment.
Today's challenge
Two collections contain 180 items altogether. One has 64 more than the other. How many are in EACH collection? Try your own method, then compare it with a model.
What you'll learn
- Represent two unknown quantities with equal bars and an extra segment.
- Explain the connection between a diagram and a number sentence.
- Solve a fresh application without step-by-step hints.
Before this lesson
Part–whole models and division by 2.
Prerequisite competency: 3A-C02-L03
Entry · 2 questions
Warm-up: prerequisite check
Try on paper before continuing. If the questions are hard, review the prerequisite concepts.
- E1
Find 282 − 59 by breaking apart the amount subtracted.
Show answer and reasoning
Answer: 223
Decompose 59 into tens and ones, subtract each part from 282, and check the difference 223.
- E2
Find 342 − 82 by breaking apart the amount subtracted.
Show answer and reasoning
Answer: 260
Decompose 82 into tens and ones, subtract each part from 342, and check the difference 260.
Learn the concept
Words to know
- total
- the sum of the two amounts
- difference
- how much larger one amount is
Represent two unknown quantities with equal bars and an extra segment.
Draw two bars that share a common length; give the longer bar an extra segment representing the difference. Subtract that segment from the total before dividing the two equal parts.
Why this works
Suppose two amounts total 73 and one exceeds the other by 15. Draw two equal-length bars and attach an extra 15 to the longer bar. Remove that extra from the total: 73 − 15 = 58. Split the remaining 58 into two equal parts of 29, then restore the 15 to find 44. Check: 29 + 44 = 73 and 44 − 29 = 15.
Check the meaning
Restore the removed segment to the larger amount, then check BOTH total and difference.
Concrete → Pictorial → Abstract
- Build: Use counters, grid paper, or labeled bundles to represent the story or quantity.
- Draw: Sketch an array, place-value exchange, or equal-unit bar; label what is known.
- Calculate: Write an equation matching the representation and solve.
- Verify: Use an inverse operation or a second strategy to check.
For advanced students: if you can already explain this concept, start with the independent or challenge set.
Worked examples and immediate checks
Explain each step aloud or in writing. Work the fresh question before revealing its solution.
Worked example 1
Reason and verify
Two collections contain 180 items altogether. One has 64 more than the other. How many are in EACH collection?
- Draw two bars that share a common length; give the longer bar an extra segment representing the difference. Subtract that segment from the total before dividing the two equal parts.
- Remove the extra 64; 180 − 64 = 116. Two equal parts are 58 each. Larger part is 58 + 64 = 122.
- Restore the removed segment to the larger amount, then check BOTH total and difference.
Result: 58 and 122
Understanding check 1 — your turn
Two collections contain 130 items altogether. One has 50 more than the other. How many are in EACH collection?
Reveal reasoning
Answer: 40 and 90 — Remove the extra 50; 130 − 50 = 80. Two equal parts are 40 each. Larger part is 40 + 50 = 90.
Worked example 2
Reason and verify
Two collections contain 160 items altogether. One has 18 more than the other. How many are in EACH collection?
- Draw two bars that share a common length; give the longer bar an extra segment representing the difference. Subtract that segment from the total before dividing the two equal parts.
- Remove the extra 18; 160 − 18 = 142. Two equal parts are 71 each. Larger part is 71 + 18 = 89.
- Restore the removed segment to the larger amount, then check BOTH total and difference.
Result: 71 and 89
Understanding check 2 — your turn
Two collections contain 170 items altogether. One has 64 more than the other. How many are in EACH collection?
Reveal reasoning
Answer: 53 and 117 — Remove the extra 64; 170 − 64 = 106. Two equal parts are 53 each. Larger part is 53 + 64 = 117.
Worked example 3
Reason and verify
Two collections contain 200 items altogether. One has 32 more than the other. How many are in EACH collection?
- Draw two bars that share a common length; give the longer bar an extra segment representing the difference. Subtract that segment from the total before dividing the two equal parts.
- Remove the extra 32; 200 − 32 = 168. Two equal parts are 84 each. Larger part is 84 + 32 = 116.
- Restore the removed segment to the larger amount, then check BOTH total and difference.
Result: 84 and 116
Understanding check 3 — your turn
Two collections contain 150 items altogether. One has 18 more than the other. How many are in EACH collection?
Reveal reasoning
Answer: 66 and 84 — Remove the extra 18; 150 − 18 = 132. Two equal parts are 66 each. Larger part is 66 + 18 = 84.
Guided · 4 questions
Guided practice
Practice with the model visible. Explain the units at each step.
- G1
Two collections contain 190 items altogether. One has 50 more than the other. How many are in EACH collection?
Show answer and reasoning
Answer: 70 and 120
Remove the extra 50; 190 − 50 = 140. Two equal parts are 70 each. Larger part is 70 + 50 = 120.
- G2
Two collections contain 230 items altogether. One has 64 more than the other. How many are in EACH collection?
Show answer and reasoning
Answer: 83 and 147
Remove the extra 64; 230 − 64 = 166. Two equal parts are 83 each. Larger part is 83 + 64 = 147.
- G3
Two collections contain 210 items altogether. One has 18 more than the other. How many are in EACH collection?
Show answer and reasoning
Answer: 96 and 114
Remove the extra 18; 210 − 18 = 192. Two equal parts are 96 each. Larger part is 96 + 18 = 114.
- G4
Two collections contain 250 items altogether. One has 32 more than the other. How many are in EACH collection?
Show answer and reasoning
Answer: 109 and 141
Remove the extra 32; 250 − 32 = 218. Two equal parts are 109 each. Larger part is 109 + 32 = 141.
Independent · 8 questions
Independent practice
Work without relying on the model or checking answers prematurely.
- I1
Two collections contain 110 items altogether. One has 30 more than the other. How many are in EACH collection?
Show answer and reasoning
Answer: 40 and 70
Remove the extra 30; 110 − 30 = 80. Two equal parts are 40 each. Larger part is 40 + 30 = 70.
- I2
Two collections contain 150 items altogether. One has 44 more than the other. How many are in EACH collection?
Show answer and reasoning
Answer: 53 and 97
Remove the extra 44; 150 − 44 = 106. Two equal parts are 53 each. Larger part is 53 + 44 = 97.
- I3
Two collections contain 190 items altogether. One has 58 more than the other. How many are in EACH collection?
Show answer and reasoning
Answer: 66 and 124
Remove the extra 58; 190 − 58 = 132. Two equal parts are 66 each. Larger part is 66 + 58 = 124.
- I4
Two collections contain 170 items altogether. One has 12 more than the other. How many are in EACH collection?
Show answer and reasoning
Answer: 79 and 91
Remove the extra 12; 170 − 12 = 158. Two equal parts are 79 each. Larger part is 79 + 12 = 91.
- I5
Two collections contain 210 items altogether. One has 26 more than the other. How many are in EACH collection?
Show answer and reasoning
Answer: 92 and 118
Remove the extra 26; 210 − 26 = 184. Two equal parts are 92 each. Larger part is 92 + 26 = 118.
- I6
Two collections contain 250 items altogether. One has 40 more than the other. How many are in EACH collection?
Show answer and reasoning
Answer: 105 and 145
Remove the extra 40; 250 − 40 = 210. Two equal parts are 105 each. Larger part is 105 + 40 = 145.
- I7
Two collections contain 100 items altogether. One has 54 more than the other. How many are in EACH collection?
Show answer and reasoning
Answer: 23 and 77
Remove the extra 54; 100 − 54 = 46. Two equal parts are 23 each. Larger part is 23 + 54 = 77.
- I8
Two collections contain 140 items altogether. One has 68 more than the other. How many are in EACH collection?
Show answer and reasoning
Answer: 36 and 104
Remove the extra 68; 140 − 68 = 72. Two equal parts are 36 each. Larger part is 36 + 68 = 104.
Challenge · 2 questions
Challenge practice
Solve original application and explanation problems. Draw and defend a useful model.
- C1
Two whole-number quantities total 73 and differ by 15. Find them and explain why the shared part is an integer.
Show answer and reasoning
Answer: 29 and 44
Remove difference 15: 73−15=58; split in two equal integer parts of 29, then add 15 to find 44.
- C2
Two children have 84 marbles altogether. One has 18 more. What would happen to the difference if the older child gives 9 marbles to the younger?
Show answer and reasoning
Answer: Difference becomes 0
The transfer decreases one amount by 9 and increases the other by 9, decreasing the difference by 18.
Exit · 3 questions
Exit check
Solve independently; revisit the skill if an important explanation is missing.
- E1
Two collections contain 200 items altogether. One has 50 more than the other. How many are in EACH collection?
Show answer and reasoning
Answer: 75 and 125
Remove the extra 50; 200 − 50 = 150. Two equal parts are 75 each. Larger part is 75 + 50 = 125.
- E2
Two collections contain 240 items altogether. One has 64 more than the other. How many are in EACH collection?
Show answer and reasoning
Answer: 88 and 152
Remove the extra 64; 240 − 64 = 176. Two equal parts are 88 each. Larger part is 88 + 64 = 152.
- E3
Two collections contain 220 items altogether. One has 18 more than the other. How many are in EACH collection?
Show answer and reasoning
Answer: 101 and 119
Remove the extra 18; 220 − 18 = 202. Two equal parts are 101 each. Larger part is 101 + 18 = 119.