3A · Chapter 2 · Lesson 10
2-Step Word Problems
Track the quantity after each change, or combine changes before solving.
Today's challenge
A shop begins with 367 pencils, receives 23, then sells 67. How many pencils remain? Try your own method, then compare it with a model.
What you'll learn
- Track the quantity after each change, or combine changes before solving.
- Explain the connection between a diagram and a number sentence.
- Solve a fresh application without step-by-step hints.
Before this lesson
Part–whole and comparison problems.
Prerequisite competency: 3A-C02-L09
Entry · 2 questions
Warm-up: prerequisite check
Try on paper before continuing. If the questions are hard, review the prerequisite concepts.
- E1
Leo collected 445 stamps. Mina collected 40 more than Leo. How many does Mina have?
Show answer and reasoning
Answer: 485
Draw Leo’s bar of 445; add the extra segment 40. Mina’s total is 445+40=485.
- E2
Leo collected 476 stamps. Mina collected 49 more than Leo. How many does Mina have?
Show answer and reasoning
Answer: 525
Draw Leo’s bar of 476; add the extra segment 49. Mina’s total is 476+49=525.
Learn the concept
Words to know
- starting amount
- the value before any changes
- ending amount
- the value after all changes
Track the quantity after each change, or combine changes before solving.
Use a three-stage before/after diagram: starting quantity, after the first change, after the second. Label the direction of each change with plus or minus.
Why this works
Keep track of what the total is after each change. Start with 120 books, add 45 to get 165, and then lend out 38 to leave 127. Draw three states or write two connected equations: 120 + 45 = 165 and 165 − 38 = 127. The second change acts on the new total, not on the original 120.
Check the meaning
Check that the answer is reasonable compared with the original. Never use the final change against the wrong starting amount.
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
A shop begins with 367 pencils, receives 23, then sells 67. How many pencils remain?
- Use a three-stage before/after diagram: starting quantity, after the first change, after the second. Label the direction of each change with plus or minus.
- First 367+23=390; then take away 67. Result: 323.
- Check that the answer is reasonable compared with the original. Never use the final change against the wrong starting amount.
Result: 323
Understanding check 1 — your turn
A shop begins with 117 pencils, receives 73, then sells 17. How many pencils remain?
Reveal reasoning
Answer: 173 — First 117+73=190; then take away 17. Result: 173.
Worked example 2
Reason and verify
A shop begins with 424 pencils, receives 36, then sells 24. How many pencils remain?
- Use a three-stage before/after diagram: starting quantity, after the first change, after the second. Label the direction of each change with plus or minus.
- First 424+36=460; then take away 24. Result: 436.
- Check that the answer is reasonable compared with the original. Never use the final change against the wrong starting amount.
Result: 436
Understanding check 2 — your turn
A shop begins with 174 pencils, receives 86, then sells 34. How many pencils remain?
Reveal reasoning
Answer: 226 — First 174+86=260; then take away 34. Result: 226.
Worked example 3
Reason and verify
A shop begins with 161 pencils, receives 49, then sells 41. How many pencils remain?
- Use a three-stage before/after diagram: starting quantity, after the first change, after the second. Label the direction of each change with plus or minus.
- First 161+49=210; then take away 41. Result: 169.
- Check that the answer is reasonable compared with the original. Never use the final change against the wrong starting amount.
Result: 169
Understanding check 3 — your turn
A shop begins with 231 pencils, receives 99, then sells 51. How many pencils remain?
Reveal reasoning
Answer: 279 — First 231+99=330; then take away 51. Result: 279.
Guided · 4 questions
Guided practice
Practice with the model visible. Explain the units at each step.
- G1
A shop begins with 330 pencils, receives 40, then sells 50. How many pencils remain?
Show answer and reasoning
Answer: 320
First 330+40=370; then take away 50. Result: 320.
- G2
A shop begins with 387 pencils, receives 53, then sells 67. How many pencils remain?
Show answer and reasoning
Answer: 373
First 387+53=440; then take away 67. Result: 373.
- G3
A shop begins with 124 pencils, receives 66, then sells 24. How many pencils remain?
Show answer and reasoning
Answer: 166
First 124+66=190; then take away 24. Result: 166.
- G4
A shop begins with 181 pencils, receives 79, then sells 41. How many pencils remain?
Show answer and reasoning
Answer: 219
First 181+79=260; then take away 41. Result: 219.
Independent · 8 questions
Independent practice
Work without relying on the model or checking answers prematurely.
- I1
A shop begins with 410 pencils, receives 30, then sells 30. How many pencils remain?
Show answer and reasoning
Answer: 410
First 410+30=440; then take away 30. Result: 410.
- I2
A shop begins with 147 pencils, receives 43, then sells 47. How many pencils remain?
Show answer and reasoning
Answer: 143
First 147+43=190; then take away 47. Result: 143.
- I3
A shop begins with 204 pencils, receives 56, then sells 64. How many pencils remain?
Show answer and reasoning
Answer: 196
First 204+56=260; then take away 64. Result: 196.
- I4
A shop begins with 261 pencils, receives 69, then sells 21. How many pencils remain?
Show answer and reasoning
Answer: 309
First 261+69=330; then take away 21. Result: 309.
- I5
A shop begins with 318 pencils, receives 82, then sells 38. How many pencils remain?
Show answer and reasoning
Answer: 362
First 318+82=400; then take away 38. Result: 362.
- I6
A shop begins with 375 pencils, receives 95, then sells 55. How many pencils remain?
Show answer and reasoning
Answer: 415
First 375+95=470; then take away 55. Result: 415.
- I7
A shop begins with 112 pencils, receives 108, then sells 12. How many pencils remain?
Show answer and reasoning
Answer: 208
First 112+108=220; then take away 12. Result: 208.
- I8
A shop begins with 169 pencils, receives 31, then sells 29. How many pencils remain?
Show answer and reasoning
Answer: 171
First 169+31=200; then take away 29. Result: 171.
Challenge · 2 questions
Challenge practice
Solve original application and explanation problems. Draw and defend a useful model.
- C1
A shelf starts with 120 books, receives 45, and lends out 38. What is the ending amount, and why do you subtract from the new total?
Show answer and reasoning
Answer: 127
120+45=165 before lending. 165−38=127 because the final change occurs after receipt.
- C2
A warehouse has 280 boxes. It ships 65 in the morning and later receives 90. Would switching the two events change the final quantity? Explain.
Show answer and reasoning
Answer: 305 boxes; no
280−65+90=305 and 280+90−65=305. Both orders work here because only additions/subtractions of fixed quantities occur, with no further restrictions.
Exit · 3 questions
Exit check
Solve independently; revisit the skill if an important explanation is missing.
- E1
A shop begins with 250 pencils, receives 100, then sells 50. How many pencils remain?
Show answer and reasoning
Answer: 300
First 250+100=350; then take away 50. Result: 300.
- E2
A shop begins with 307 pencils, receives 23, then sells 67. How many pencils remain?
Show answer and reasoning
Answer: 263
First 307+23=330; then take away 67. Result: 263.
- E3
A shop begins with 364 pencils, receives 36, then sells 24. How many pencils remain?
Show answer and reasoning
Answer: 376
First 364+36=400; then take away 24. Result: 376.