### Examples

*Example:*Shanice thinks about finding the product by imagining having 8 pizzas that she wants to split equally with three of her friends. She and each of her friends will get 2 pizzas since .

*Example:* Lacey thinks about finding the product by imagining having 8 pizza boxes each with one-quarter slice of a pizza left. If she put them all together, she would have a total of 2 whole pizzas since which is equivalent to 2.

### Clarifications

*Clarification 1:*Instruction includes the use of visual models or number lines and the connection to the commutative property of multiplication. Refer to Properties of Operation, Equality and Inequality (Appendix D).

*Clarification 2:* Within this benchmark, the expectation is not to simplify or use lowest terms.

*Clarification 3: *Fractions multiplied by a whole number are limited to less than 1. All denominators are limited to 2, 3, 4, 5, 6, 8, 10, 12, 16, 100.

**Subject Area:**Mathematics (B.E.S.T.)

**Grade:**4

**Strand:**Fractions

**Date Adopted or Revised:**08/20

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

### Terms from the K-12 Glossary

- Equation
- Expression

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

- Contexts involving multiplying whole numbers and fractions lend themselves to modeling and examining patterns.
- This benchmark builds on students’ work of adding fractions with like denominators (MA.4.FR.2.2) and extending that work into the relationship between addition and multiplication.
- Students should use fraction models and drawings to show their understanding. Fraction models may include area models, number lines, set models, or equations.
- During instruction, teachers should relate “ total group size” language that was used to introduce whole number multiplication- possibly changing from “ total group size” to “ total size” or “ total amount” (see Appendix A). Using such language, the expression
5 ×$\frac{\text{3}}{\text{4}}$ can be described as the total size or amount of 5 objects, each of which has size or amount of $\frac{\text{3}}{\text{4}}$.
- For example, the weight of 5 slabs of chocolate that each weigh $\frac{\text{3}}{\text{4}}$ of a pound is 5 × $\frac{\text{3}}{\text{4}}$ pounds. Students need to understand that when multiplying a whole number by a fraction, the most important idea is that the whole number describes the number of objects and the fraction describes the size of each object.

- Instruction should include representing a whole number times a fraction as repeated addition: 5×$\frac{\text{3}}{\text{4}}$ = $\frac{\text{3}}{\text{4}}$ + $\frac{\text{3}}{\text{4}}$ + $\frac{\text{3}}{\text{4}}$ + $\frac{\text{3}}{\text{4}}$ + $\frac{\text{3}}{\text{4}}$
- When multiplying a fraction by a whole number, teachers can use language like “portion of” to convey that the fraction represents the “portion of” the whole number.
- For example, the $\frac{\text{3}}{\text{4}}$ portion of a 5 mile run is $\frac{\text{3}}{\text{4}}$ × 5 miles.

- Instruction includes fractions equal to and greater than one.
- Exploring patterns of what happens to the numerator when a whole number is multiplied by a fraction will help students make sense of multiplying fractions by fractions in Grade 5. When multiplying whole numbers by mixed numbers, students can use the distributive property or write the mixed number as a fraction greater than one. During instruction, students should compare both strategies. Using the distributive property to multiply a whole number by a mixed number could look like this.

### Common Misconceptions or Errors

- Students may multiply both the numerator and the denominator by the whole number. It is important to provide students with visual models, manipulatives or context to model the computation.
- Students may be confused by the fact that a fraction times a whole number often represents something quite different than a whole number times a fraction, even though the commutative property says the order does not affect the value.
- Without a conceptual understanding of how fraction multiplication is modeled, students can be confused regarding why the denominator remains the same when multiplying a whole number by a fraction. During instruction, teachers should relate fraction multiplication to repeated addition to explain why only the numerator changes.

### Strategies to Support Tiered Instruction

- Instruction includes connecting repeated addition and models to understand the meaning of the factors in a multiplication equation.
- For example, 8 × $\frac{\text{1}}{\text{4}}$ can be shown using repeated addition with 8 groups of $\frac{\text{1}}{\text{4}}$.
- $\frac{\text{1}}{\text{4}}$ + $\frac{\text{1}}{\text{4}}$ + $\frac{\text{1}}{\text{4}}$ + $\frac{\text{1}}{\text{4}}$ + $\frac{\text{1}}{\text{4}}$ + $\frac{\text{1}}{\text{4}}$ + $\frac{\text{1}}{\text{4}}$ + $\frac{\text{1}}{\text{4}}$ = $\frac{\text{8}}{\text{4}}$ or 2 wholes

- For example, $\frac{\text{1}}{\text{4}}$ × 8 can be shown as $\frac{\text{1}}{\text{4}}$ of 8 using a number line.

- For example, 8 × $\frac{\text{1}}{\text{4}}$ can be shown using repeated addition with 8 groups of $\frac{\text{1}}{\text{4}}$.

- Instruction includes real-world situations aligned with the content. The teacher provides a multiplication expression with real-world context and items to represent the situation to make connections.
- For example, the teacher provides the student with the following situation: “The theater class has to make costumes that require $\frac{\text{1}}{\text{4}}$ foot of ribbon. They need to make 8 costumes. How many feet of ribbon will they need total?” The teacher provides ribbon already cut into $\frac{\text{1}}{\text{4}}$ foot pieces and has students model the problem. Students then write out the repeated addition problem to match $\frac{\text{1}}{\text{4}}$ + $\frac{\text{1}}{\text{4}}$ + $\frac{\text{1}}{\text{4}}$ + $\frac{\text{1}}{\text{4}}$ + $\frac{\text{1}}{\text{4}}$ + $\frac{\text{1}}{\text{4}}$ + $\frac{\text{1}}{\text{4}}$+ $\frac{\text{1}}{\text{4}}$ = $\frac{\text{8}}{\text{4}}$ or 2 feet.
- For example, the teacher provides the student with the following situation: “The theater class has 8 feet of ribbon. They need to make 4 costumes and they want to use up all of the ribbon. How many feet of ribbon does each costume use?” The teacher will provide the student with a piece of ribbon 8 feet long and have them mark it off into 4 equal sections and find the length of 1 section (2 feet).

- Teacher provides models that represent multiplication expressions to represent the Commutative Property of Multiplication and has students explain the difference in meaning.
- For example, the teacher provides the student with the following models.

- The teacher provides models and asks students which expression it models.
- For example, the teacher provides students with the following model and asks if it represents $\frac{\text{1}}{\text{6}}$ x 4 or 4 x $\frac{\text{1}}{\text{6}}$

### Instructional Tasks

*Instructional Task 1* (MTR.2.1)

### Instructional Items

*Instructional Item 1 *

*Instructional Item 2 *

**The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.*

## Related Courses

## Related Access Points

## Related Resources

## Educational Game

## Formative Assessments

## Image/Photograph

## Lesson Plans

## Original Student Tutorial

## Perspectives Video: Teaching Idea

## Problem-Solving Tasks

## Tutorial

## MFAS Formative Assessments

Students use a visual fraction model to explain how many one sixths are in and record their work with an equation.

Students are asked to multiply a fraction by a whole number and to represent the product with a visual fraction model.

Students are asked to multiply a fraction by a whole number to solve a word problem and to represent the product with a visual fraction model.

Students are asked to consider an equation involving multiplication of a fraction by a whole number and create a visual fraction model. Additionally, the student is asked to interpret multiplying the number of parts by the whole number.

Students are asked to multiply an improper fraction by a whole number to solve a word problem and use a visual model or equation to represent the problem.

## Original Student Tutorials Mathematics - Grades K-5

Find the total amounts of repeated fraction quantities by multiplying a fraction by a whole number using visual models that represent real-world problems and cookies in this interactive tutorial.

## Student Resources

## Original Student Tutorial

Find the total amounts of repeated fraction quantities by multiplying a fraction by a whole number using visual models that represent real-world problems and cookies in this interactive tutorial.

Type: Original Student Tutorial

## Educational Game

This fun and interactive game helps practice estimation skills, using various operations of choice, including addition, subtraction, multiplication, division, using decimals, fractions, and percents.

Various levels of difficulty make this game appropriate for multiple age and ability levels.

*Addition/**Subtraction:* The addition and subtraction of whole numbers, the addition and subtraction of decimals.

*Multiplication/Division: *The multiplication and addition of whole numbers.

*Percentages: *Identify the percentage of a whole number.

*Fractions: *Multiply and divide a whole number by a fraction, as well as apply properties of operations.

Type: Educational Game

## Problem-Solving Tasks

This task provides a familiar context allowing students to visualize multiplication of a fraction by a whole number. This task could form part of a very rich activity which includes studying soda can labels.

Type: Problem-Solving Task

The purpose of this task is to have students think about the meaning of multiplying a number by a fraction, and use this burgeoning understanding of fraction multiplication to make sense of the commutative property of multiplication in the case of fractions.

Type: Problem-Solving Task

## Tutorial

In this Khan Academy video visual fraction models are used to represent the multiplication of a whole number times a fraction.

Type: Tutorial

## Parent Resources

## Image/Photograph

Illustrations that can be used for teaching and demonstrating fractions. Fractional representations are modeled in wedges of circles ("pieces of pie") and parts of polygons. There are also clipart images of numerical fractions, both proper and improper, from halves to twelfths. Fraction charts and fraction strips found in this collection can be used as manipulatives and are ready to print for classroom use.

Type: Image/Photograph

## Problem-Solving Tasks

This task provides a familiar context allowing students to visualize multiplication of a fraction by a whole number. This task could form part of a very rich activity which includes studying soda can labels.

Type: Problem-Solving Task

The purpose of this task is to have students think about the meaning of multiplying a number by a fraction, and use this burgeoning understanding of fraction multiplication to make sense of the commutative property of multiplication in the case of fractions.

Type: Problem-Solving Task