3A · Chapter 2 · Lesson 9
Word Problems — Part 2
Compare two quantities using equal-length and excess bar segments.
Today's challenge
Leo collected 326 stamps. Mina collected 89 more than Leo. How many does Mina have? Try your own method, then compare it with a model.
What you'll learn
- Compare two quantities using equal-length and excess bar segments.
- Explain the connection between a diagram and a number sentence.
- Solve a fresh application without step-by-step hints.
Before this lesson
Part–whole word problems.
Prerequisite competency: 3A-C02-L08
Entry · 2 questions
Warm-up: prerequisite check
Try on paper before continuing. If the questions are hard, review the prerequisite concepts.
- E1
A school library has 610 books. 120 are on the front shelf. How many are elsewhere?
Show answer and reasoning
Answer: 490
The 610 books are the whole, 120 is a part. Subtract: 610 − 120 = 490.
- E2
A school library has 651 books. 137 are on the front shelf. How many are elsewhere?
Show answer and reasoning
Answer: 514
The 651 books are the whole, 137 is a part. Subtract: 651 − 137 = 514.
Learn the concept
Words to know
- difference
- the extra amount in a comparison
- comparison bar
- a diagram aligning two related quantities
Compare two quantities using equal-length and excess bar segments.
Draw equal-length sections for the shared amount and an additional section for the extra quantity. To find the larger amount from the smaller, add the difference.
Why this works
A comparison question describes two quantities and the distance between them. If Hall A has 145 chairs and Hall B has 39 fewer, draw Hall B’s bar shorter by a 39-chair segment. The missing bar is 145 − 39 = 106. Read the labels carefully: "39 fewer" describes the difference, not the total in Hall B.
Check the meaning
Read the question again before calculating: which bar is unknown and which is given?
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
Leo collected 326 stamps. Mina collected 89 more than Leo. How many does Mina have?
- Draw equal-length sections for the shared amount and an additional section for the extra quantity. To find the larger amount from the smaller, add the difference.
- Draw Leo’s bar of 326; add the extra segment 89. Mina’s total is 326+89=415.
- Read the question again before calculating: which bar is unknown and which is given?
Result: 415
Understanding check 1 — your turn
Leo collected 409 stamps. Mina collected 66 more than Leo. How many does Mina have?
Reveal reasoning
Answer: 475 — Draw Leo’s bar of 409; add the extra segment 66. Mina’s total is 409+66=475.
Worked example 2
Reason and verify
Leo collected 357 stamps. Mina collected 98 more than Leo. How many does Mina have?
- Draw equal-length sections for the shared amount and an additional section for the extra quantity. To find the larger amount from the smaller, add the difference.
- Draw Leo’s bar of 357; add the extra segment 98. Mina’s total is 357+98=455.
- Read the question again before calculating: which bar is unknown and which is given?
Result: 455
Understanding check 2 — your turn
Leo collected 440 stamps. Mina collected 75 more than Leo. How many does Mina have?
Reveal reasoning
Answer: 515 — Draw Leo’s bar of 440; add the extra segment 75. Mina’s total is 440+75=515.
Worked example 3
Reason and verify
Leo collected 388 stamps. Mina collected 107 more than Leo. How many does Mina have?
- Draw equal-length sections for the shared amount and an additional section for the extra quantity. To find the larger amount from the smaller, add the difference.
- Draw Leo’s bar of 388; add the extra segment 107. Mina’s total is 388+107=495.
- Read the question again before calculating: which bar is unknown and which is given?
Result: 495
Understanding check 3 — your turn
Leo collected 471 stamps. Mina collected 84 more than Leo. How many does Mina have?
Reveal reasoning
Answer: 555 — Draw Leo’s bar of 471; add the extra segment 84. Mina’s total is 471+84=555.
Guided · 4 questions
Guided practice
Practice with the model visible. Explain the units at each step.
- G1
Leo collected 105 stamps. Mina collected 80 more than Leo. How many does Mina have?
Show answer and reasoning
Answer: 185
Draw Leo’s bar of 105; add the extra segment 80. Mina’s total is 105+80=185.
- G2
Leo collected 136 stamps. Mina collected 89 more than Leo. How many does Mina have?
Show answer and reasoning
Answer: 225
Draw Leo’s bar of 136; add the extra segment 89. Mina’s total is 136+89=225.
- G3
Leo collected 167 stamps. Mina collected 98 more than Leo. How many does Mina have?
Show answer and reasoning
Answer: 265
Draw Leo’s bar of 167; add the extra segment 98. Mina’s total is 167+98=265.
- G4
Leo collected 198 stamps. Mina collected 107 more than Leo. How many does Mina have?
Show answer and reasoning
Answer: 305
Draw Leo’s bar of 198; add the extra segment 107. Mina’s total is 198+107=305.
Independent · 8 questions
Independent practice
Work without relying on the model or checking answers prematurely.
- I1
Leo collected 165 stamps. Mina collected 40 more than Leo. How many does Mina have?
Show answer and reasoning
Answer: 205
Draw Leo’s bar of 165; add the extra segment 40. Mina’s total is 165+40=205.
- I2
Leo collected 196 stamps. Mina collected 49 more than Leo. How many does Mina have?
Show answer and reasoning
Answer: 245
Draw Leo’s bar of 196; add the extra segment 49. Mina’s total is 196+49=245.
- I3
Leo collected 227 stamps. Mina collected 58 more than Leo. How many does Mina have?
Show answer and reasoning
Answer: 285
Draw Leo’s bar of 227; add the extra segment 58. Mina’s total is 227+58=285.
- I4
Leo collected 258 stamps. Mina collected 67 more than Leo. How many does Mina have?
Show answer and reasoning
Answer: 325
Draw Leo’s bar of 258; add the extra segment 67. Mina’s total is 258+67=325.
- I5
Leo collected 289 stamps. Mina collected 76 more than Leo. How many does Mina have?
Show answer and reasoning
Answer: 365
Draw Leo’s bar of 289; add the extra segment 76. Mina’s total is 289+76=365.
- I6
Leo collected 320 stamps. Mina collected 85 more than Leo. How many does Mina have?
Show answer and reasoning
Answer: 405
Draw Leo’s bar of 320; add the extra segment 85. Mina’s total is 320+85=405.
- I7
Leo collected 351 stamps. Mina collected 94 more than Leo. How many does Mina have?
Show answer and reasoning
Answer: 445
Draw Leo’s bar of 351; add the extra segment 94. Mina’s total is 351+94=445.
- I8
Leo collected 382 stamps. Mina collected 103 more than Leo. How many does Mina have?
Show answer and reasoning
Answer: 485
Draw Leo’s bar of 382; add the extra segment 103. Mina’s total is 382+103=485.
Challenge · 2 questions
Challenge practice
Solve original application and explanation problems. Draw and defend a useful model.
- C1
A student says “Mira has 12 more than Leo” means Mira has 12 altogether. Explain the mistake.
Show answer and reasoning
Answer: 12 is the difference, not the total
Mira has Leo’s quantity PLUS an extra 12; Mira’s total cannot be known without Leo’s quantity.
- C2
A school has 145 chairs in Hall A and 39 fewer in Hall B. Write a number sentence for Hall B and explain which bar should be shorter.
Show answer and reasoning
Answer: 106 chairs
145−39=106. Hall B is 39 fewer, so its bar is shorter by a 39-chair segment.
Exit · 3 questions
Exit check
Solve independently; revisit the skill if an important explanation is missing.
- E1
Leo collected 285 stamps. Mina collected 70 more than Leo. How many does Mina have?
Show answer and reasoning
Answer: 355
Draw Leo’s bar of 285; add the extra segment 70. Mina’s total is 285+70=355.
- E2
Leo collected 316 stamps. Mina collected 79 more than Leo. How many does Mina have?
Show answer and reasoning
Answer: 395
Draw Leo’s bar of 316; add the extra segment 79. Mina’s total is 316+79=395.
- E3
Leo collected 347 stamps. Mina collected 88 more than Leo. How many does Mina have?
Show answer and reasoning
Answer: 435
Draw Leo’s bar of 347; add the extra segment 88. Mina’s total is 347+88=435.