Lesson 2 | Multiplication and Division, Part 2 | 3rd Grade Mathematics

Objective

Understand the zero and identity properties of multiplication.

Common Core Standards

Core Standards

The core standards covered in this lesson

3.OA.A.4 — Determine the unknown whole number in a multiplication or division equation relating three whole numbers. Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = _ ÷ 3, 6 × 6 = ?

Operations and Algebraic Thinking

3.OA.A.4 — Determine the unknown whole number in a multiplication or division equation relating three whole numbers. Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = _ ÷ 3, 6 × 6 = ?
3.OA.B.5 — Apply properties of operations as strategies to multiply and divide. Students need not use formal terms for these properties. Example: Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.) Example: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.)

Operations and Algebraic Thinking

3.OA.B.5 — Apply properties of operations as strategies to multiply and divide. Students need not use formal terms for these properties. Example: Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.) Example: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.)
3.OA.C.7 — Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

Operations and Algebraic Thinking

3.OA.C.7 — Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.
3.OA.D.9 — Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.

Operations and Algebraic Thinking

3.OA.D.9 — Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.

Foundational Standards

The foundational standards covered in this lesson

3.OA.A.1

Operations and Algebraic Thinking

3.OA.A.1 — Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7.
3.OA.A.2

Operations and Algebraic Thinking

3.OA.A.2 — Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.
3.OA.B.6

Operations and Algebraic Thinking

3.OA.B.6 — Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Look for and express regularity in repeated reasoning when multiplying by $$1$$ to generalize the pattern of multiplying any number by $$1$$ results in a product that is just that number (i.e., $$1\times n = n$$) (MP.8).
Look for and express regularity in repeated reasoning when multiplying by $$0$$ to generalize the pattern of multiplying any number by $$0$$ results in a product that is $$0$$ (i.e., $$0\times n=0$$) (MP.8).
Use the two properties mentioned above as well as the relationship between multiplication and division to determine the unknown whole number in a multiplication or division equation relating three whole numbers, one of which is $$0$$ or $$1$$.
Explain why any number divided by $$0$$ results in an impossible product by rewriting the division sentence as a multiplication one to see that no such value exists (e.g., $$6 \div 0 = a \rightarrow a\times 0 = 6$$, and no such a exists since any number multiplied by $$0$$ would have a product of $$0$$) (MP.3).

Tips for Teachers

Suggestions for teachers to help them teach this lesson

Students are not expected to use the terms "zero property" or "identity property," but they are included in the objective so that teachers can understand this lesson’s connection to the topic. Teachers may choose to rewrite the objective to be more student-friendly if they use objectives in a student-facing way.

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Anchor Tasks

Tasks designed to teach criteria for success of the lesson, and guidance to help draw out student understanding

Problem 1

Rudy is waiting patiently for his new hens to lay some eggs.

a. On Monday he went to the henhouse to check their nests. He has 12 hens in the house, and each hen had laid 0 eggs. How many eggs did Rudy have in total? Write an equation to represent the situation.

b. On Tuesday when he went to the henhouse, he discovered that each of his 12 hens had laid 1 egg! How many eggs does he have now? Write an equation to represent the situation.

Guiding Questions

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References

Achievement First Grade 3, Unit 1, Lesson 16 (2018-2019) — Pose the Problem

Grade 3, Unit 1, Lesson 16 (2018-2019) is made available by Achievement First as a part of their Open Source web portal under a CC BY 4.0 license. Copyright © 1999-2017 Achievement First. Accessed Nov. 9, 2018, 3:07 p.m..

Modified by Fishtank Learning, Inc.

Problem 2

Solve.

a. $$4\times 1 = $$ _____

b. _____ $$= 1\times 7$$

c. $$5\ \times$$ _____ $$=5$$

d. $$6\div 6 = $$ _____

e. _____ $$=9\div1$$

Guiding Questions

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Problem 3

Solve.

a. $$2\times 0 = a$$

b. $$b = 0\times 3$$

c. $$7\times c = 0$$

d. $$0\div 8 = d$$

Guiding Questions

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Problem Set

Problem Set

Answer Keys

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Discussion of Problem Set

Discuss with a partner what patterns for multiplying and dividing by $$0$$ and $$1$$ helped you solve #1? #5?
What pattern for multiplying by $$1$$ does #3 represent?
Why was A incorrect in #4? What would the correct quotient be?
Why were you able to compare $$0\div43$$ with $$96\times 0 $$ even though we don’t know how to compute with numbers as large as $$43$$ and $$96$$?
Which problems show that we can’t define a single specific value when we divide by $$0$$?
How is multiplying by $$1$$ and multiplying units of $$1$$ similar to adding $$0$$ to a number?
Let’s fill out more of the multiplication table based on what we’ve learned today. Why do you think there are no squares that correspond with the $$0$$ facts?

Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Determine the value of the unknown number that makes each equation below true.

a. $$6=a \times 1$$

b. $$8 \times b = 8$$

c. $$0=c \div 5$$

d. $$6 \times d = 0$$

e. $$1 =7 \div e$$

f. $$1\times f=9$$

Student Response

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Additional Practice

The Extra Practice Problems can be used as additional practice for homework, during an intervention block, etc. Daily Word Problems and Fluency Activities are aligned to the content of the unit but not necessarily to the lesson objective, therefore feel free to use them anytime during your school day.

Extra Practice Problems

Extra Practice Problems

Answer Keys

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Word Problems and Fluency Activities

Help students strengthen their application and fluency skills with daily word problem practice and content-aligned fluency activities.

Lesson 1

Lesson 3

Lesson Map

Topic A: Introduction to The Properties of Operations

Study commutativity to find known facts of 6, 7, 8, and 9.

3.OA.B.5 3.OA.D.9

Understand the zero and identity properties of multiplication.

3.OA.A.4 3.OA.B.5 3.OA.C.7 3.OA.D.9

Understand the role of parentheses and apply to solving problems.

3.OA.B.5 3.OA.D.8

Introduce the distributive property of multiplication.

3.OA.B.5 3.OA.D.9

Introduce the associative property of multiplication.

3.OA.B.5 3.OA.D.9

Topic B: Multiplication and Division by 6 and 7

Build fluency with multiplication and division facts using units of 6.

3.OA.A.4 3.OA.C.7

Build fluency with multiplication and division facts using units of 7.

3.OA.A.4 3.OA.C.7

Use the associative property as a strategy to multiply by units of 6.

3.OA.B.5 3.OA.C.7 3.OA.D.9

Use the distributive property as a strategy to multiply by units of 6.

3.OA.B.5 3.OA.C.7 3.OA.D.9

Use the distributive property as a strategy to multiply by units of 7.

3.OA.B.5 3.OA.C.7 3.OA.D.9

Solve one- and two-step word problems involving units up to 7.

3.OA.A.3 3.OA.D.8

Topic C: Multiplication and Division by 8 and 9

Build fluency with multiplication and division facts using units of 8 and 9.

3.OA.A.4 3.OA.C.7

Use the associative property as a strategy to multiply by units of 8 and 9.

3.OA.B.5 3.OA.C.7 3.OA.D.9

Use the distributive property as a strategy to multiply by units of 8 and 9.

3.OA.B.5 3.OA.C.7 3.OA.D.9

Use the distributive property as a strategy to multiply by units of 8 and 9, including the subtractive use of the distributive property.

3.OA.B.5 3.OA.C.7 3.OA.D.9

Solve one- and two-step word problems involving units up to 9.

3.OA.A.3 3.OA.D.8

Topic D: Multiplication and Division by Values Greater than 10

Multiply one-digit whole numbers by multiples of 10.

3.NBT.A.3 3.OA.A.4 3.OA.B.5

Multiply one-digit whole numbers by two-digit whole numbers using the associative and distributive properties.

3.NBT.A.3 3.OA.A.4 3.OA.B.5

Topic E: Two-Step Word Problems and Patterns in Arithmetic

Solve two-step word problems involving all four operations.

3.OA.D.8

Solve two-step word problems involving all four operations, including multiplying one-digit whole numbers by multiples of 10.

3.NBT.A.3 3.OA.D.8

Identify patterns in single or multiple rows/columns of the multiplication table.

3.OA.C.7 3.OA.D.9

Identify other patterns in the multiplication table.

3.OA.D.9

Identify arithmetic patterns and explain them using properties of operations.

3.OA.D.9

Multiplication and Division, Part 2

Objective

Common Core Standards

Core Standards

Operations and Algebraic Thinking

Operations and Algebraic Thinking

Operations and Algebraic Thinking

Operations and Algebraic Thinking

Foundational Standards

Operations and Algebraic Thinking

Operations and Algebraic Thinking

Operations and Algebraic Thinking

Criteria for Success

Tips for Teachers

Anchor Tasks

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

Problem 3

Guiding Questions

Problem Set

Discussion of Problem Set

Target Task

Student Response

Additional Practice

Extra Practice Problems

Word Problems and Fluency Activities

Lesson Map

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