Lesson 4 | Multiplication and Division, Part 2 | 3rd Grade Mathematics

Objective

Introduce the distributive property of multiplication.

Common Core Standards

Core Standards

The core standards covered in this lesson

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.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.NBT.A.2

Number and Operations in Base Ten

3.NBT.A.2 — Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.
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.

Criteria for Success

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

Look for structure to understand that when multiplying two numbers, either number can be decomposed into a sum of values and multiplied by the other factor, then added together (e.g., $$5\times4=(3\times4)+(2\times4)$$) (MP.7).
Make use of structure by using the distributive property to decompose a factor in a multiplication problem into a sum of values, multiplying those values separately by the other factor, then adding those products together to get the overall product (MP.7).

Tips for Teachers

Suggestions for teachers to help them teach this lesson

Students will be informally introduced to distributive notation in this lesson, but students will not be expected to use it or understand it yet, and thus it is not on the Problem Set.
Students need not use formal terms for the properties of operations, including the terms "distributive" or "distributive property." Thus, the term is used throughout teacher-facing materials but not the student-facing materials. Thus, this objective would require edits in order to be student-facing.

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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

Each box of tissues costs $2.

a. What is the cost for 3 boxes? What is the cost for 5 boxes? What is the cost for 8 boxes?

b. What do you notice about your answers to Part (a)? What do you wonder?

Guiding Questions

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Problem 2

Shawn is finding the number of diamonds in the following array. He marks the array like this:

a. Explain what Shawn did to find the number of diamonds in the array.

b. Why do you think Shawn used this strategy?

Guiding Questions

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Problem 3

Use the following array to answer Parts (a) and (b).

a. Use Shawn’s strategy from Anchor Task 2 to decompose the array into ones whose products you know from memory. Then find the total product represented by the array.

b. Find a second way to decompose the array.

Guiding Questions

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Problem Set

Problem Set

Answer Keys

Unlock the answer keys for this lesson's problem set and extra practice problems to save time and support student learning.

Discussion of Problem Set

In #1c, I think I can find $$3\times3$$ and then double that product to find the number of objects in the whole array. Do you agree or disagree? Why?
In #1d, I think I can find $$10\times6$$ and then subtract a group of 6 to find the product of 9 and 6. Do you agree or disagree? Why?
In #2 and #3, how did you find Matt’s total cards? How was that total related to the number of cards he first had and the number of cards he added?
How did you decompose your arrays in #4? Why did you choose to decompose them in that way? Was there more than one way to decompose them?
What did you always do to the products of the decomposed arrays to find the product of the larger array?

Target Task

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

Decompose the following array in a way that will help you find its total number of objects. Then show how you find the total number of objects.

Student Response

An example response to the Target Task at the level of detail expected of the students.

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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 3

Lesson 5

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

Foundational Standards

Number and Operations in Base Ten

Operations and Algebraic Thinking

Operations and Algebraic Thinking

Criteria for Success

Tips for Teachers

Anchor Tasks

Problem 1

Guiding Questions

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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