3A · Chapter 4 · Lesson 4
Multiplying and Dividing with 0 and 1
Explain identities involving zero and one, including why dividing by zero is undefined.
Today's challenge
Calculate 87 × 0, and explain why. Try your own method, then compare it with a model.
What you'll learn
- Explain identities involving zero and one, including why dividing by zero is undefined.
- Explain the connection between a diagram and a number sentence.
- Solve a fresh application without step-by-step hints.
Before this lesson
Multiplication and division meaning.
Prerequisite competency: 3A-C04-L03
Entry · 2 questions
Warm-up: prerequisite check
Try on paper before continuing. If the questions are hard, review the prerequisite concepts.
- E1
12 counters are shared equally among 2 children. How many does each receive?
Show answer and reasoning
Answer: 6
2 equal groups contain 12 counters: 12÷2=6; verify 6×2=12.
- E2
27 counters are shared equally among 3 children. How many does each receive?
Show answer and reasoning
Answer: 9
3 equal groups contain 27 counters: 27÷3=9; verify 9×3=27.
Learn the concept
Words to know
- identity
- an operation that leaves the quantity unchanged
- undefined
- an expression without an ordinary numerical value
Explain identities involving zero and one, including why dividing by zero is undefined.
One group of a number has that amount; zero groups have none. Zero items shared into a positive number of groups gives zero in each.
Why this works
Zero groups of eight contain zero items: 0 × 8 = 0. Eight groups of zero also contain zero: 8 × 0 = 0. Zero items shared among four children gives zero each: 0 ÷ 4 = 0. But 8 ÷ 0 has no ordinary numerical answer, because zero times any number cannot equal 8.
Check the meaning
Division by zero cannot be assigned an ordinary quotient: no unique answer satisfies the inverse multiplication equation.
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
Calculate 87 × 0, and explain why.
- One group of a number has that amount; zero groups have none. Zero items shared into a positive number of groups gives zero in each.
- The meaning of equal groups gives 0. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
- Division by zero cannot be assigned an ordinary quotient: no unique answer satisfies the inverse multiplication equation.
Result: 0
Understanding check 1 — your turn
Calculate 43 × 0, and explain why.
Reveal reasoning
Answer: 0 — The meaning of equal groups gives 0. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
Worked example 2
Reason and verify
Calculate 0 ÷ 5, and explain why.
- One group of a number has that amount; zero groups have none. Zero items shared into a positive number of groups gives zero in each.
- The meaning of equal groups gives 0. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
- Division by zero cannot be assigned an ordinary quotient: no unique answer satisfies the inverse multiplication equation.
Result: 0
Understanding check 2 — your turn
Calculate 0 ÷ 50, and explain why.
Reveal reasoning
Answer: 0 — The meaning of equal groups gives 0. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
Worked example 3
Reason and verify
Calculate 12 ÷ 1, and explain why.
- One group of a number has that amount; zero groups have none. Zero items shared into a positive number of groups gives zero in each.
- The meaning of equal groups gives 12. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
- Division by zero cannot be assigned an ordinary quotient: no unique answer satisfies the inverse multiplication equation.
Result: 12
Understanding check 3 — your turn
Calculate 57 ÷ 1, and explain why.
Reveal reasoning
Answer: 57 — The meaning of equal groups gives 57. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
Guided · 4 questions
Guided practice
Practice with the model visible. Explain the units at each step.
- G1
Calculate 31 × 1, and explain why.
Show answer and reasoning
Answer: 31
The meaning of equal groups gives 31. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
- G2
Calculate 38 × 0, and explain why.
Show answer and reasoning
Answer: 0
The meaning of equal groups gives 0. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
- G3
Calculate 0 ÷ 45, and explain why.
Show answer and reasoning
Answer: 0
The meaning of equal groups gives 0. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
- G4
Calculate 52 ÷ 1, and explain why.
Show answer and reasoning
Answer: 52
The meaning of equal groups gives 52. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
Independent · 8 questions
Independent practice
Work without relying on the model or checking answers prematurely.
- I1
Calculate 58 × 1, and explain why.
Show answer and reasoning
Answer: 58
The meaning of equal groups gives 58. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
- I2
Calculate 65 × 0, and explain why.
Show answer and reasoning
Answer: 0
The meaning of equal groups gives 0. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
- I3
Calculate 0 ÷ 72, and explain why.
Show answer and reasoning
Answer: 0
The meaning of equal groups gives 0. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
- I4
Calculate 79 ÷ 1, and explain why.
Show answer and reasoning
Answer: 79
The meaning of equal groups gives 79. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
- I5
Calculate 86 × 1, and explain why.
Show answer and reasoning
Answer: 86
The meaning of equal groups gives 86. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
- I6
Calculate 4 × 0, and explain why.
Show answer and reasoning
Answer: 0
The meaning of equal groups gives 0. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
- I7
Calculate 0 ÷ 11, and explain why.
Show answer and reasoning
Answer: 0
The meaning of equal groups gives 0. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
- I8
Calculate 18 ÷ 1, and explain why.
Show answer and reasoning
Answer: 18
The meaning of equal groups gives 18. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
Challenge · 2 questions
Challenge practice
Solve original application and explanation problems. Draw and defend a useful model.
- C1
Why is 8 ÷ 0 not 0?
Show answer and reasoning
Answer: Undefined
If 8÷0=q, then 0×q would have to equal 8. No number q satisfies that.
- C2
Which is valid: 0÷7, 7÷0, 7×0? Give results or explain.
Show answer and reasoning
Answer: 0, undefined, 0
0÷7=0 because seven equal groups can each receive 0; 7÷0 has no defined quotient; seven groups of 0 contain 0.
Exit · 3 questions
Exit check
Solve independently; revisit the skill if an important explanation is missing.
- E1
Calculate 23 × 1, and explain why.
Show answer and reasoning
Answer: 23
The meaning of equal groups gives 23. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
- E2
Calculate 30 × 0, and explain why.
Show answer and reasoning
Answer: 0
The meaning of equal groups gives 0. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.
- E3
Calculate 0 ÷ 37, and explain why.
Show answer and reasoning
Answer: 0
The meaning of equal groups gives 0. One group preserves a quantity; zero groups give zero; zero split among nonzero groups is zero.