# MA.3.AR.1.1

Apply the distributive property to multiply a one-digit number and two-digit number. Apply properties of multiplication to find a product of one-digit whole numbers.

### Examples

The product 4×72 can be found by rewriting the expression as 4×(70+2) and then using the distributive property to obtain (4×70)+(4×2) which is equivalent to 288.

### Clarifications

Clarification 1: Within this benchmark, the expectation is to apply the associative and commutative properties of multiplication, the distributive property and name the properties. Refer to K-12 Glossary (Appendix C).

Clarification 2: Within the benchmark, the expectation is to utilize parentheses.

Clarification 3: Multiplication for products of three or more numbers is limited to factors within 12. Refer to Properties of Operations, Equality and Inequality (Appendix D).

General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Algebraic Reasoning
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Expression
• Equation
• Distributive property
• Factors

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

The purpose of this benchmark is for students to apply what they have learned about the multiplication of one-digit numbers and multiples of ten to then multiply a one-digit number and a two-digit number (MA.3.NSO.2.3).
• Students are introduced to the distributive property of multiplication over addition as a strategy for using products that they know in order to solve products that they do not know. For example, if students are asked to find the product of 6 x 9, they might decompose 6 into 4 and 2 and then multiply 4 x 9 and 2 x 9 to arrive at 36 + 18, which equals 54. Because of the distributive property, students use parentheses to show how to decompose two-digit numbers by the value of their tens and ones, and then multiply the one-digit number by both the values of the two-digit number’s tens and ones values and find the sum of those products. The application of the commutative and associative properties of multiplication allows for two-digit numbers to be decomposed and multiplication expressions reorganized so that the distributive property can work (MTR.2.1).
• During instruction, teachers should model where the properties are applied while multiplying and expect students to explain how they work during explanations of their strategies and solutions. Splitting arrays can help students understand the distributive property. They can use a known fact to learn other facts that may cause difficulty (MTR.2.1, MTR.4.1).
• Building understanding of the distributive property in Grade 3 will help students decompose larger numbers as they continue to multiply multi-digit numbers with procedural reliability and procedural fluency in Grade 4. Splitting arrays can help students understand the distributive property. They can use a known fact to learn other facts that may cause difficulty.

### Common Misconceptions or Errors

• Students can be confused about how to write expressions using the distributive property. One common mistake that students make is writing an expression 4 × 72 as (4 × 70) × (4 × 2) instead of (4 × 70) + (4 × 2). Instruction should show concrete models (e.g., base ten drawings) along with equations so students can understand the relationship between multiplication and addition while applying the property and writing expressions.

### Strategies to Support Tiered Instruction

• Instruction includes opportunities to use concrete models and drawings along with equations to increase understanding of the relationship between multiplication and addition when applying the distributive property and writing equations. The teacher begins by modeling a one-digit number multiplied by a one-digit number, guiding students to decompose one of the factors, and using models or drawings to demonstrate the reorganization of the multiplication expression using parentheses. Next, the teacher models the multiplication of a one-digit number by a two-digit number, guiding students to decompose the two-digit number into the value of the tens and the ones using models or drawings. The teacher clarifies that the decomposed factor can be represented in expanded form by adding the tens and the ones, repeating with additional one-digit by two-digit multiplication equations.
• For example, the teacher uses a model or drawing to use the distributive property to solve 3 x 24.
• 3 × (20 + 4)
• (3 × 20) + (3 × 4)
• (3 × 20) + (3 × 4) = 60 + 12
• (3 × 20) + (3 × 4) = 72
so 3 × 24 = 72
• Teacher provides opportunities to apply the distributive property to solve one-digit by two-digit multiplication equations using base-ten blocks or place value disks. The teacher provides the equation and guides students to decompose the two-digit number into the value of the tens and the ones using manipulatives. If needed, the teacher prompts students to count by 10s and 1s using the base-ten blocks or place value disks.
• For example, the teacher uses base-ten blocks to solve 3 × 24 while asking guiding questions such as “How many tens are in 24?” “How many ones are in 24?” “How would we write 24 in expanded form?”

In each equation, find the missing value, n.

• Part A. 4 × 52 = (4 × 50) + (4 × $n$
• Part B. $n$ × 3 = (20 × 3) + (9 × 3)
• Part C. 8 × 36 = ($n$ × 30) + ($n$ × 6)
• Part D. 48 × 6 = $n$

• Tory tried to use the associative and commutative properties to create the following equations. Using pictures and/or words, explain why Tory is incorrect.
• 4 x (11 + 6) = (4 x 11) + 6
• 4 x (11 + 6) = 11 x (4 + 6)

### Instructional Items

Instructional Item 1

Which of the following correctly uses the distributive property to multiply 8 × 39 ?
• a. (8 × 30) × (8 × 9) = 24 + 72 = 96
• b. (8 × 30) + (8 × 9) = 240 + 56 = 296
• c. (8 × 30) + (8 × 9) = 38 + 17 = 45
• d. (8 × 30) + (8 × 9) = 240 + 72 = 312
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

## Related Courses

This benchmark is part of these courses.
5012050: Grade Three Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7712040: Access Mathematics Grade 3 (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2022, 2022 and beyond (current))
5012055: Grade 3 Accelerated Mathematics (Specifically in versions: 2019 - 2022, 2022 and beyond (current))
5012015: Foundational Skills in Mathematics 3-5 (Specifically in versions: 2019 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.3.AR.1.AP.1: Apply the commutative property of multiplication to find a product of one-digit whole numbers.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Formative Assessments

Using the Associative Property of Multiplication:

Students are asked to find the product of three numbers and are observed to see if they use the Associative Property to find the product more easily.

Type: Formative Assessment

Break Apart and Put Together:

Students are given two arrays, one representing the equation 7 x 9 = 63 and the other representing the equation (5 + 2) x 9 = 63, to see if they recognize a relationship between the two.

Type: Formative Assessment

Students are given two problems to solve, one represented by the equation 4 x 6 = 24 and the other by the equation 6 x 4 = 24, to see if they recognize the answer to the second problem based on the Commutative Property.

Type: Formative Assessment

## Lesson Plans

Representing Symbols Using Perimeter and Area:

In this integrated lesson, students will create Uncle Sam cards encouraging responsible citizenship, find the dimensions of their card, and then use measurement, addition, and multiplication to solve a real-world task requiring calculation of perimeter and area of a larger space to display all of the student-created Uncle Sam cards.

Type: Lesson Plan

Feeding the Community:

Students analyze various proposed sites to determine which site would be best for a group of volunteers to construct and maintain a community garden in this model eliciting activity.

Type: Lesson Plan

Making Sense of Multiplication to Build Fluency of 6's, 7's, 8's, and 9's:

This lesson will help students multiply numbers with factors of 6, 7, 8, or 9 through decomposing numbers in an array and applying the distributive property. Many times, these factors are difficult for students to recall from memory. Teaching students how to use an array can give them a visual representation of the final product. This visual can also help students to make the connection that multiplying whole numbers is a sum of equal groups. Decomposing the numbers and using the distributive property is a strategy for students to use who are having trouble solving these higher factor multiplication facts.

Type: Lesson Plan

Candy Apple Fun:

Students will learn how to solve one-digit by two-digit multiplication problems using the distributive property.

Type: Lesson Plan

The Distributive Property, Revealed! with a 100-dot matrix:

This lesson is designed as an introduction to the Distributive Property by using a 100-dot matrix. The lesson addresses one-digit x one-digit multiplication challenges as a precursor to one-digit x two-digit multiplication.

Type: Lesson Plan

Zero on a Hero (Exploring the Zero Property of Multiplication):

Students will explore the Zero Property of Multiplication using array and equal-group models for multiplication. Students will model story problems, translate problems into multiplication facts, and identify patterns in a set of multiplication facts to develop understanding of the Zero Property of Multiplication.

Type: Lesson Plan

One with a Bun (Exploring the Multiplicative Identity Property of 1):

In this lesson students will explore the Multiplicative Identity Property of 1, using array and equal-group models for multiplication. Students will model story problems, translate problems into multiplication equations, and identify patterns in a set of multiplication facts to develop understanding of the Multiplicative Identity Property of 1.

Type: Lesson Plan

Multiply by Multiples of 10 with Number Cubes:

In this lesson students will use various strategies to multiply one-digit numbers by multiples of 10 within the range of 10-90. The strategies will encompass the Distributive, Commutative, and Associative properties, place value, number lines, base-ten blocks, diagrams, hundreds chart. Students will play a game with number cubes to practice this multiplication.

Type: Lesson Plan

Efficient Multiplication:

Students will engage with questions to evaluate the students' abilities to select and apply multiplication strategies with fluency and efficiency. The focus of the lesson is decomposing numbers to multiply using the Distributive property and understanding and applying the Commutative property. Then, students will reinforce decomposing of factors while playing Decomposition of Factors. The lesson concludes with a real world application problem on an Exit Slip.

Type: Lesson Plan

Area Architects, Lesson 4:

In this 5-lesson unit on area, students explore geometric measurement by becoming "Area Architects" in order to learn the concepts of area and relate area to multiplication and addition. In this 4th lesson, students will use tiling to show in a concrete case that the area of a rectangle can be found using the distributive property of multiplication. This lesson is focused on single-digit x single-digit dimensions using proper units for dimensions (e.g. ft, yd, m) and square units for the area (e.g. sq. ft, sq. yd, sq. m).

Type: Lesson Plan

Fall Fun and Games! (Exploring the Commutative Property of Multiplication):

In this lesson, students will build and manipulate a variety of arrays in the context of creating games for a Fall Festival. They will practice using the Commutative Property of Multiplication to find related multiplication facts.

Type: Lesson Plan

Amazing Arrays:

This is a hands-on lesson for introducing and practicing building arrays to create models that represent the distributive property of multiplication, and then using those arrays to draw models of the equations they represent.

Type: Lesson Plan

## Original Student Tutorial

Monkeying Around with Multiplication: Commutative Property:

Learn strategies, like the commutative property, to help you become better at multiplying in this interactive tutorial.

Type: Original Student Tutorial

## Perspectives Video: Teaching Ideas

Making Connections with the Area Model:

Unlock an effective teaching strategy for making connections in area models in this Teacher Perspectives video for educators.

Type: Perspectives Video: Teaching Idea

The Van de Walle Dot Matrix: A tool to support concepts from counting to multiplying polynomials:

Unlock an effective teaching tool that can help students all the way from basic counting principles to multiplying polynomials.

Type: Perspectives Video: Teaching Idea

## MFAS Formative Assessments

Break Apart and Put Together:

Students are given two arrays, one representing the equation 7 x 9 = 63 and the other representing the equation (5 + 2) x 9 = 63, to see if they recognize a relationship between the two.

Students are given two problems to solve, one represented by the equation 4 x 6 = 24 and the other by the equation 6 x 4 = 24, to see if they recognize the answer to the second problem based on the Commutative Property.

Using the Associative Property of Multiplication:

Students are asked to find the product of three numbers and are observed to see if they use the Associative Property to find the product more easily.

## Original Student Tutorials Mathematics - Grades K-5

Monkeying Around with Multiplication: Commutative Property:

Learn strategies, like the commutative property, to help you become better at multiplying in this interactive tutorial.

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

## Original Student Tutorial

Monkeying Around with Multiplication: Commutative Property:

Learn strategies, like the commutative property, to help you become better at multiplying in this interactive tutorial.

Type: Original Student Tutorial

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.