# MA.5.FR.2.3

When multiplying a given number by a fraction less than 1 or a fraction greater than 1, predict and explain the relative size of the product to the given number without calculating.

### Clarifications

Clarification 1: Instruction focuses on the connection to decimals, estimation and assessing the reasonableness of an answer.
General Information
Subject Area: Mathematics (B.E.S.T.)
Grade: 5
Strand: Fractions
Date Adopted or Revised: 08/20
Status: State Board Approved

## Benchmark Instructional Guide

• NA

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

The purpose of this benchmark is for students to examine how numbers change when multiplying by fractions (MTR.2.1). Students already had experience with this idea when they multiplied a fraction by a whole number in grade 4 (MA.4.FR.2.4). Work from this benchmark will help prepare students to multiply and divide fractions and decimals with procedural fluency in grade 6 (MA.6.NSO.2.2).
• It is important for students to have experiences examining:
• when multiplying by a fraction greater than 1, the number increases;
• when multiplying by a fraction equal to 1, the number stays the same; and
• when multiplying by a fraction less the 1, the number decreases.
• Throughout instruction, encourage students to use models or drawings to assist them with a visual of the relative size. Models to consider when multiplying fractions to assist with finding relative size without calculating include, but are not limited to, area models (rectangles), linear models (fraction strips/bars and number lines) and set models (counters). Include examples with equivalent fractions and decimals (MTR.2.1).
• Have students explain how they used the model or drawing to arrive at the solution and justify reasonableness of their answers (MTR.4.1).

### Common Misconceptions or Errors

• Students may believe that multiplication always results in a larger number. This is why it is imperative to include models during instruction when multiplying fractions so students can see and experience the results and begin to make generalizations that are based on their understanding. Ultimately, allowing students to begin to understand that multiplying by a fraction less than one will result in a lesser product, but when multiplying by a fraction greater than one will result in a greater product.

### Strategies to Support Tiered Instruction

• Instruction includes opportunities to predict and explain the relative size of the product of a given number by a fraction less than one or a fraction greater than one. Students use models to check their prediction and solve. The teacher guides students to connect that multiplying a given number by a fraction less than one will result in a smaller number and that multiplying a given number by a fraction greater than one will result in a larger number.
• For example, the teacher displays the problem 7 x $\frac{\text{4}}{\text{5}}$ and asks students to predict if the product will be greater than, equal to, or less than 7 (it will be less than 7). Students use a visual model to represent the problem to determine 7 × $\frac{\text{4}}{\text{5}}$ = $\frac{\text{28}}{\text{5}}$. This is repeated with additional examples using fractions both greater than, equal to, and less than one.
• Instruction includes providing hands-on opportunities to predict and explain the relative size of the product of a given number by a fraction less than one or a fraction greater than one. Students use fraction strips/bars or counters to check their prediction and solve, connecting that multiplying a given number by a fraction less than one will result in a smaller number and that multiplying a given number by a fraction greater than one will result in a larger number.
• For example, the teacher displays the problem 4 × $\frac{\text{3}}{\text{8}}$ = ___. Then, the teacher asks students to predict if the product will be greater than, equal to, or less than 4 (it will be less than 4).
• Using fraction bars or fraction strips, the teacher models solving this problem with explicit instruction and guided questioning. Students explain how to use fraction bars or fraction strips as a model to solve this problem. This is repeated with additional examples using fractions both greater than, equal to, and less than one.

### Instructional Tasks

Instructional Task 1

Derrick is playing a computer game where he must multiply a number by a factor that increases the number’s size each time. Select all of the factors that he could multiply by to continue to increase the size of his number? Explain your thinking.
• a. $\frac{\text{3}}{\text{4}}$
• b. $\frac{\text{4}}{\text{3}}$
• c. 1 $\frac{\text{1}}{\text{9}}$
• d. 1.01
• e. $\frac{\text{5}}{\text{2}}$
• f. $\frac{\text{8}}{\text{9}}$
• g. $\frac{\text{99}}{\text{100}}$
• h. $\frac{\text{2}}{\text{2}}$

### Instructional Items

Instructional Item 1

Which of the following expressions will have a product greater than 4?
• a. 4×$\frac{\text{8}}{\text{8}}$
• b. $\frac{\text{3}}{\text{4}}$×4
• c. 4×$\frac{\text{99}}{\text{100}}$
• d. $\frac{\text{101}}{\text{100}}$×4

Instructional Item 2

• Fill in the blank. The product of the expression $\frac{\text{63}}{\text{65}}$× 20 will be _____________ 20.
• a. less than
• b. equal to
• c. greater than
• d. half of

*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.
5012070: Grade Five Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7712060: Access Mathematics Grade 5 (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2022, 2022 and beyond (current))
5012065: Grade 4 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.5.FR.2.AP.3: Explore the impact on the size of the product when multiplying a given number by a fraction less than 1 or by a whole number.

## Related Resources

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

## Formative Assessments

Multiplying by a Fraction Greater Than One:

Students are asked to describe the size of a product of a fraction greater than one and a whole number without multiplying.

Type: Formative Assessment

Multiplying by a Fraction Less than One:

Students are asked to describe the size of a product of a fraction less than one and a whole number without multiplying.

Type: Formative Assessment

More Than or Less Than Two Miles:

Students are asked to reason about the size of the product of fractions and whole numbers presented in context.

Type: Formative Assessment

Estimating Products:

Students are given three products, each involving a whole number and a fraction, and are asked to estimate the size of the product and explain their reasoning.

Type: Formative Assessment

## Lesson Plans

Real-World Fractions:

This lesson focuses on providing students with real-world experiences where they will be required to multiply fractions. Students will be required to use visual fraction models or equations to represent the problem.  This is a practice and application lesson, not an introductory lesson.

Type: Lesson Plan

Multiplying a Fraction by a Fraction:

In this lesson, students will solve problems related to training for a marathon to apply and make sense of multiplying fractions. The student will complete a function table to help illustrate patterns in the numerator/denominator relationships. This lesson utilizes the linear model as a concrete representation and moves towards the standard algorithm (a/b) x (c/d) = ac/bd.

Type: Lesson Plan

Looking for Patterns in a Sequence of Fractions:

Students generate and describe a numerical pattern using the multiplication and subtraction of fractions.

Type: Lesson Plan

## Original Student Tutorial

Scaling Up to Escape:

Try to escape from this room using multiplication as scaling in this interactive tutorial.

Note: this tutorial is an introductory lesson on multiplying a given number without calculating before working with fractions.

Type: Original Student Tutorial

## Problem-Solving Tasks

Running a Mile:

The solution uses the idea that multiplying by a fraction less than 1 results in a smaller value. The students need to explain why that is so.

Type: Problem-Solving Task

Reasoning about Multiplication:

This is a good task to work with kids to try to explain their thinking clearly and precisely, although teachers should be willing to work with many different ways of explaining the relationship between the magnitude of the factors and the magnitude of the product.

Type: Problem-Solving Task

Grass Seedlings:

The purpose of this task is to gain a better understanding of multiplying with fractions. Students should use the diagram provided to support their findings.

Type: Problem-Solving Task

Fundraising:

This problem helps students gain a better understanding of multiplying with fractions.

Type: Problem-Solving Task

Comparing a Number and a Product:

The purpose of this task is for students to compare a number and its product with other numbers that are greater than and less than one. As written, this task could be used in a summative assessment context, but it might be more useful in an instructional setting where students are asked to explain their answers either to a partner or in a whole class discussion.

Type: Problem-Solving Task

Calculator Trouble:

This particular problem deals with multiplication. Even though students can solve this problem by multiplying, it is unlikely they will. Here it is much easier to answer the question if you can think of multiplying a number by a factor as scaling the number.

Type: Problem-Solving Task

## MFAS Formative Assessments

Estimating Products:

Students are given three products, each involving a whole number and a fraction, and are asked to estimate the size of the product and explain their reasoning.

More Than or Less Than Two Miles:

Students are asked to reason about the size of the product of fractions and whole numbers presented in context.

Multiplying by a Fraction Greater Than One:

Students are asked to describe the size of a product of a fraction greater than one and a whole number without multiplying.

Multiplying by a Fraction Less than One:

Students are asked to describe the size of a product of a fraction less than one and a whole number without multiplying.

## Original Student Tutorials Mathematics - Grades K-5

Scaling Up to Escape:

Try to escape from this room using multiplication as scaling in this interactive tutorial.

Note: this tutorial is an introductory lesson on multiplying a given number without calculating before working with fractions.

## Student Resources

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

## Original Student Tutorial

Scaling Up to Escape:

Try to escape from this room using multiplication as scaling in this interactive tutorial.

Note: this tutorial is an introductory lesson on multiplying a given number without calculating before working with fractions.

Type: Original Student Tutorial

## Problem-Solving Tasks

Running a Mile:

The solution uses the idea that multiplying by a fraction less than 1 results in a smaller value. The students need to explain why that is so.

Type: Problem-Solving Task

Reasoning about Multiplication:

This is a good task to work with kids to try to explain their thinking clearly and precisely, although teachers should be willing to work with many different ways of explaining the relationship between the magnitude of the factors and the magnitude of the product.

Type: Problem-Solving Task

Grass Seedlings:

The purpose of this task is to gain a better understanding of multiplying with fractions. Students should use the diagram provided to support their findings.

Type: Problem-Solving Task

Fundraising:

This problem helps students gain a better understanding of multiplying with fractions.

Type: Problem-Solving Task

Comparing a Number and a Product:

The purpose of this task is for students to compare a number and its product with other numbers that are greater than and less than one. As written, this task could be used in a summative assessment context, but it might be more useful in an instructional setting where students are asked to explain their answers either to a partner or in a whole class discussion.

Type: Problem-Solving Task

Calculator Trouble:

This particular problem deals with multiplication. Even though students can solve this problem by multiplying, it is unlikely they will. Here it is much easier to answer the question if you can think of multiplying a number by a factor as scaling the number.

Type: Problem-Solving Task

## Parent Resources

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

## Problem-Solving Tasks

Running a Mile:

The solution uses the idea that multiplying by a fraction less than 1 results in a smaller value. The students need to explain why that is so.

Type: Problem-Solving Task

Reasoning about Multiplication:

This is a good task to work with kids to try to explain their thinking clearly and precisely, although teachers should be willing to work with many different ways of explaining the relationship between the magnitude of the factors and the magnitude of the product.

Type: Problem-Solving Task

Grass Seedlings:

The purpose of this task is to gain a better understanding of multiplying with fractions. Students should use the diagram provided to support their findings.

Type: Problem-Solving Task

Fundraising:

This problem helps students gain a better understanding of multiplying with fractions.

Type: Problem-Solving Task

Comparing a Number and a Product:

The purpose of this task is for students to compare a number and its product with other numbers that are greater than and less than one. As written, this task could be used in a summative assessment context, but it might be more useful in an instructional setting where students are asked to explain their answers either to a partner or in a whole class discussion.

Type: Problem-Solving Task

Calculator Trouble:

This particular problem deals with multiplication. Even though students can solve this problem by multiplying, it is unlikely they will. Here it is much easier to answer the question if you can think of multiplying a number by a factor as scaling the number.

Type: Problem-Solving Task