3A · Chapter 2 · Lesson 8
Word Problems — Part 1
Distinguish a total from one of its parts before choosing an operation.
Today's challenge
A school library has 781 books. 67 are on the front shelf. How many are elsewhere? Try your own method, then compare it with a model.
What you'll learn
- Distinguish a total from one of its parts before choosing an operation.
- Explain the connection between a diagram and a number sentence.
- Solve a fresh application without step-by-step hints.
Before this lesson
Addition and subtraction.
Prerequisite competency: 3A-C02-L05
Entry · 2 questions
Warm-up: prerequisite check
Try on paper before continuing. If the questions are hard, review the prerequisite concepts.
- E1
What must be added to 68 to make 100?
Show answer and reasoning
Answer: 32
The missing part is 100 − 68 = 32; check 68 + 32 = 100.
- E2
What must be added to 280 to make 1,000?
Show answer and reasoning
Answer: 720
The missing part is 1,000 − 280 = 720; check 280 + 720 = 1,000.
Learn the concept
Words to know
- whole
- the entire known quantity
- part
- a portion of the whole
Distinguish a total from one of its parts before choosing an operation.
Draw one bar for the total. Partition it into a known part and an unknown part. If the unknown is the remaining amount, subtract the known part from the whole.
Why this works
Every part-whole question has a whole and smaller parts. If a class has 28 students and 11 go on a trip, the other part is 28 − 11 = 17 students. A model with two labeled sections, 11 and 17, makes 28 as the whole. The number 17 is not an extra group beyond the 28.
Check the meaning
Always identify the referent of “altogether,” “left,” and “some”; do not choose operations only from keywords.
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 school library has 781 books. 67 are on the front shelf. How many are elsewhere?
- Draw one bar for the total. Partition it into a known part and an unknown part. If the unknown is the remaining amount, subtract the known part from the whole.
- The 781 books are the whole, 67 is a part. Subtract: 781 − 67 = 714.
- Always identify the referent of “altogether,” “left,” and “some”; do not choose operations only from keywords.
Result: 714
Understanding check 1 — your turn
A school library has 197 books. 179 are on the front shelf. How many are elsewhere?
Reveal reasoning
Answer: 18 — The 197 books are the whole, 179 is a part. Subtract: 197 − 179 = 18.
Worked example 2
Reason and verify
A school library has 202 books. 84 are on the front shelf. How many are elsewhere?
- Draw one bar for the total. Partition it into a known part and an unknown part. If the unknown is the remaining amount, subtract the known part from the whole.
- The 202 books are the whole, 84 is a part. Subtract: 202 − 84 = 118.
- Always identify the referent of “altogether,” “left,” and “some”; do not choose operations only from keywords.
Result: 118
Understanding check 2 — your turn
A school library has 238 books. 196 are on the front shelf. How many are elsewhere?
Reveal reasoning
Answer: 42 — The 238 books are the whole, 196 is a part. Subtract: 238 − 196 = 42.
Worked example 3
Reason and verify
A school library has 243 books. 101 are on the front shelf. How many are elsewhere?
- Draw one bar for the total. Partition it into a known part and an unknown part. If the unknown is the remaining amount, subtract the known part from the whole.
- The 243 books are the whole, 101 is a part. Subtract: 243 − 101 = 142.
- Always identify the referent of “altogether,” “left,” and “some”; do not choose operations only from keywords.
Result: 142
Understanding check 3 — your turn
A school library has 279 books. 213 are on the front shelf. How many are elsewhere?
Reveal reasoning
Answer: 66 — The 279 books are the whole, 213 is a part. Subtract: 279 − 213 = 66.
Guided · 4 questions
Guided practice
Practice with the model visible. Explain the units at each step.
- G1
A school library has 460 books. 70 are on the front shelf. How many are elsewhere?
Show answer and reasoning
Answer: 390
The 460 books are the whole, 70 is a part. Subtract: 460 − 70 = 390.
- G2
A school library has 501 books. 87 are on the front shelf. How many are elsewhere?
Show answer and reasoning
Answer: 414
The 501 books are the whole, 87 is a part. Subtract: 501 − 87 = 414.
- G3
A school library has 542 books. 104 are on the front shelf. How many are elsewhere?
Show answer and reasoning
Answer: 438
The 542 books are the whole, 104 is a part. Subtract: 542 − 104 = 438.
- G4
A school library has 583 books. 121 are on the front shelf. How many are elsewhere?
Show answer and reasoning
Answer: 462
The 583 books are the whole, 121 is a part. Subtract: 583 − 121 = 462.
Independent · 8 questions
Independent practice
Work without relying on the model or checking answers prematurely.
- I1
A school library has 620 books. 60 are on the front shelf. How many are elsewhere?
Show answer and reasoning
Answer: 560
The 620 books are the whole, 60 is a part. Subtract: 620 − 60 = 560.
- I2
A school library has 661 books. 77 are on the front shelf. How many are elsewhere?
Show answer and reasoning
Answer: 584
The 661 books are the whole, 77 is a part. Subtract: 661 − 77 = 584.
- I3
A school library has 702 books. 94 are on the front shelf. How many are elsewhere?
Show answer and reasoning
Answer: 608
The 702 books are the whole, 94 is a part. Subtract: 702 − 94 = 608.
- I4
A school library has 743 books. 111 are on the front shelf. How many are elsewhere?
Show answer and reasoning
Answer: 632
The 743 books are the whole, 111 is a part. Subtract: 743 − 111 = 632.
- I5
A school library has 784 books. 128 are on the front shelf. How many are elsewhere?
Show answer and reasoning
Answer: 656
The 784 books are the whole, 128 is a part. Subtract: 784 − 128 = 656.
- I6
A school library has 205 books. 145 are on the front shelf. How many are elsewhere?
Show answer and reasoning
Answer: 60
The 205 books are the whole, 145 is a part. Subtract: 205 − 145 = 60.
- I7
A school library has 246 books. 162 are on the front shelf. How many are elsewhere?
Show answer and reasoning
Answer: 84
The 246 books are the whole, 162 is a part. Subtract: 246 − 162 = 84.
- I8
A school library has 287 books. 179 are on the front shelf. How many are elsewhere?
Show answer and reasoning
Answer: 108
The 287 books are the whole, 179 is a part. Subtract: 287 − 179 = 108.
Challenge · 2 questions
Challenge practice
Solve original application and explanation problems. Draw and defend a useful model.
- C1
A class has 28 students. 11 are away on a trip. What does the number 17 represent in the situation?
Show answer and reasoning
Answer: Students who did not go on the trip
28−11=17 is the other part of the class, not 17 additional students.
- C2
There are 63 seedlings in three trays. The first holds 18 and the second 21. How many are in the third? Sketch the whole and its three parts.
Show answer and reasoning
Answer: 24 seedlings
Known parts total 18+21=39; third part 63−39=24. Three-part whole model: 18, 21, 24.
Exit · 3 questions
Exit check
Solve independently; revisit the skill if an important explanation is missing.
- E1
A school library has 320 books. 40 are on the front shelf. How many are elsewhere?
Show answer and reasoning
Answer: 280
The 320 books are the whole, 40 is a part. Subtract: 320 − 40 = 280.
- E2
A school library has 361 books. 57 are on the front shelf. How many are elsewhere?
Show answer and reasoning
Answer: 304
The 361 books are the whole, 57 is a part. Subtract: 361 − 57 = 304.
- E3
A school library has 402 books. 74 are on the front shelf. How many are elsewhere?
Show answer and reasoning
Answer: 328
The 402 books are the whole, 74 is a part. Subtract: 402 − 74 = 328.