3A · Chapter 4 · Lesson 8
Word Problems — Part 2
Solve multiplicative comparisons with repeated equal bar units.
Today's challenge
Sam has 7 beads. Nora has 3 times as many. How many beads does Nora have? Try your own method, then compare it with a model.
What you'll learn
- Solve multiplicative comparisons with repeated equal bar units.
- Explain the connection between a diagram and a number sentence.
- Solve a fresh application without step-by-step hints.
Before this lesson
Comparison bars and multiplication.
Prerequisite competency: 3A-C04-L07
Entry · 2 questions
Warm-up: prerequisite check
Try on paper before continuing. If the questions are hard, review the prerequisite concepts.
- E1
Each of 10 shelves holds 19 equally sized boxes. How many boxes altogether?
Show answer and reasoning
Answer: 190
The total is 10 equal groups of 19: 10 × 19 = 190.
- E2
Each of 12 shelves holds 26 equally sized boxes. How many boxes altogether?
Show answer and reasoning
Answer: 312
The total is 12 equal groups of 26: 12 × 26 = 312.
Learn the concept
Words to know
- times as many
- a multiplicative comparison with equal copies
Solve multiplicative comparisons with repeated equal bar units.
A comparison of “four times as many” means four equal copies of the reference amount. The larger bar contains four equal segments, not the reference plus four.
Why this works
"Three times as many" means three copies of the starting amount, not three extra objects. If Sam has eight cards and Lila has three times as many, draw three equal bars of eight for Lila: 3 × 8 = 24. But three more than eight would be 11. Bar lengths make this difference visible.
Check the meaning
Check the language: four MORE is additive; four TIMES is multiplicative.
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
Sam has 7 beads. Nora has 3 times as many. How many beads does Nora have?
- A comparison of “four times as many” means four equal copies of the reference amount. The larger bar contains four equal segments, not the reference plus four.
- Draw one unit 7 for Sam and 3 matching units for Nora: 7 × 3 = 21.
- Check the language: four MORE is additive; four TIMES is multiplicative.
Result: 21
Understanding check 1 — your turn
Sam has 16 beads. Nora has 4 times as many. How many beads does Nora have?
Reveal reasoning
Answer: 64 — Draw one unit 16 for Sam and 4 matching units for Nora: 16 × 4 = 64.
Worked example 2
Reason and verify
Sam has 10 beads. Nora has 4 times as many. How many beads does Nora have?
- A comparison of “four times as many” means four equal copies of the reference amount. The larger bar contains four equal segments, not the reference plus four.
- Draw one unit 10 for Sam and 4 matching units for Nora: 10 × 4 = 40.
- Check the language: four MORE is additive; four TIMES is multiplicative.
Result: 40
Understanding check 2 — your turn
Sam has 19 beads. Nora has 5 times as many. How many beads does Nora have?
Reveal reasoning
Answer: 95 — Draw one unit 19 for Sam and 5 matching units for Nora: 19 × 5 = 95.
Worked example 3
Reason and verify
Sam has 13 beads. Nora has 5 times as many. How many beads does Nora have?
- A comparison of “four times as many” means four equal copies of the reference amount. The larger bar contains four equal segments, not the reference plus four.
- Draw one unit 13 for Sam and 5 matching units for Nora: 13 × 5 = 65.
- Check the language: four MORE is additive; four TIMES is multiplicative.
Result: 65
Understanding check 3 — your turn
Sam has 22 beads. Nora has 6 times as many. How many beads does Nora have?
Reveal reasoning
Answer: 132 — Draw one unit 22 for Sam and 6 matching units for Nora: 22 × 6 = 132.
Guided · 4 questions
Guided practice
Practice with the model visible. Explain the units at each step.
- G1
Sam has 9 beads. Nora has 2 times as many. How many beads does Nora have?
Show answer and reasoning
Answer: 18
Draw one unit 9 for Sam and 2 matching units for Nora: 9 × 2 = 18.
- G2
Sam has 12 beads. Nora has 3 times as many. How many beads does Nora have?
Show answer and reasoning
Answer: 36
Draw one unit 12 for Sam and 3 matching units for Nora: 12 × 3 = 36.
- G3
Sam has 15 beads. Nora has 4 times as many. How many beads does Nora have?
Show answer and reasoning
Answer: 60
Draw one unit 15 for Sam and 4 matching units for Nora: 15 × 4 = 60.
- G4
Sam has 18 beads. Nora has 5 times as many. How many beads does Nora have?
Show answer and reasoning
Answer: 90
Draw one unit 18 for Sam and 5 matching units for Nora: 18 × 5 = 90.
Independent · 8 questions
Independent practice
Work without relying on the model or checking answers prematurely.
- I1
Sam has 6 beads. Nora has 2 times as many. How many beads does Nora have?
Show answer and reasoning
Answer: 12
Draw one unit 6 for Sam and 2 matching units for Nora: 6 × 2 = 12.
- I2
Sam has 9 beads. Nora has 3 times as many. How many beads does Nora have?
Show answer and reasoning
Answer: 27
Draw one unit 9 for Sam and 3 matching units for Nora: 9 × 3 = 27.
- I3
Sam has 12 beads. Nora has 4 times as many. How many beads does Nora have?
Show answer and reasoning
Answer: 48
Draw one unit 12 for Sam and 4 matching units for Nora: 12 × 4 = 48.
- I4
Sam has 15 beads. Nora has 5 times as many. How many beads does Nora have?
Show answer and reasoning
Answer: 75
Draw one unit 15 for Sam and 5 matching units for Nora: 15 × 5 = 75.
- I5
Sam has 18 beads. Nora has 6 times as many. How many beads does Nora have?
Show answer and reasoning
Answer: 108
Draw one unit 18 for Sam and 6 matching units for Nora: 18 × 6 = 108.
- I6
Sam has 21 beads. Nora has 2 times as many. How many beads does Nora have?
Show answer and reasoning
Answer: 42
Draw one unit 21 for Sam and 2 matching units for Nora: 21 × 2 = 42.
- I7
Sam has 24 beads. Nora has 3 times as many. How many beads does Nora have?
Show answer and reasoning
Answer: 72
Draw one unit 24 for Sam and 3 matching units for Nora: 24 × 3 = 72.
- I8
Sam has 27 beads. Nora has 4 times as many. How many beads does Nora have?
Show answer and reasoning
Answer: 108
Draw one unit 27 for Sam and 4 matching units for Nora: 27 × 4 = 108.
Challenge · 2 questions
Challenge practice
Solve original application and explanation problems. Draw and defend a useful model.
- C1
Explain the difference between “Lila has 3 more” and “Lila has 3 times as many” when Sam has 8.
Show answer and reasoning
Answer: 11 versus 24
Three more gives 8+3=11; three times gives 3 copies of 8, or 24.
- C2
Maya has 7 beads, and Eva has five times as many. How much larger is Eva’s collection than Maya’s?
Show answer and reasoning
Answer: 28 beads
Eva has 5×7=35. Difference is 35−7=28. The difference is four equal seven-bead units.
Exit · 3 questions
Exit check
Solve independently; revisit the skill if an important explanation is missing.
- E1
Sam has 29 beads. Nora has 2 times as many. How many beads does Nora have?
Show answer and reasoning
Answer: 58
Draw one unit 29 for Sam and 2 matching units for Nora: 29 × 2 = 58.
- E2
Sam has 32 beads. Nora has 3 times as many. How many beads does Nora have?
Show answer and reasoning
Answer: 96
Draw one unit 32 for Sam and 3 matching units for Nora: 32 × 3 = 96.
- E3
Sam has 6 beads. Nora has 4 times as many. How many beads does Nora have?
Show answer and reasoning
Answer: 24
Draw one unit 6 for Sam and 4 matching units for Nora: 6 × 4 = 24.