Standard #: MA.5.NSO.2.4


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Explore the multiplication and division of multi-digit numbers with decimals to the hundredths using estimation, rounding and place value.


Examples


The quotient of 23 and 0.42 can be estimated as a little bigger than 46 because 0.42 is less than one-half and 23 times 2 is 46.

Clarifications


Clarification 1: Estimating quotients builds the foundation for division using a standard algorithm.

Clarification 2: Instruction includes the use of models based on place value and the properties of operations.



General Information

Subject Area: Mathematics (B.E.S.T.)
Grade: 5
Strand: Number Sense and Operations
Standard: Add, subtract, multiply and divide multi-digit numbers.
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

The purpose of this benchmark is for students to explore multiplication and division of multi- digit numbers with decimals using estimation, rounding, place value, and exploring the relationship between multiplication and division. This benchmark connects to the work students did in grade 4 with addition and subtraction of decimals (MA.4.NSO.2.7). Students achieve procedural fluency with multiplying and dividing multi-digit numbers with decimals in grade 6 (MA.6.NSO.2.1)
  • Instruction of this benchmark focuses on number sense to help students develop procedural reliability while multiplying and dividing multi-digit numbers with decimals. 
  • During instruction, students should explore how the products and quotients of whole numbers relate to decimals. 
    • For example, if students know the product of 8 × 7 and the quotient of 56 ÷ 4, then they can reason through 0.08 × 7 or 5.6 ÷ 0.4 through place value relationships. Classroom discussions should allow students to explore these patterns and use them to estimate products and quotients (MTR.4.1, MTR.6.1). 
  • Teachers should connect what students know about place value and fractions. o For example, because students know that multiplying a number by one-fourth will result in a product that is smaller, multiplying a number by 0.25 (its decimal equivalence) will also result in a smaller product. In division, dividing a number by one-fourth and 0.25 will result in a larger quotient. Continued work in this benchmark will help students to generalize patterns in multiplication and division of whole numbers and fractions (MTR.5.1). 
  • Models that help students explore the multiplication and division of multi-digit numbers with decimals include base ten representations (e.g., blocks) and place value mats.

 

Common Misconceptions or Errors

  • Students may not understand the reasoning behind the placement of the decimal point in the product. Modeling and exploring the relationships between place value will help students gain understanding.
  • Students can confuse that multiplication always results in a larger product, and that division always results in a smaller quotient. Through classroom discussion, estimation and modeling, classroom work should address this misconception.

 

Strategies to Support Tiered Instruction

  • Instruction includes opportunities to explore place value of decimals with concrete models and objects.
    • For example, students use place value understanding and a place value chart to compare 0.14 and 0.2. The teacher explains that when comparing decimals, we start with the digit to the far left because we want to compare the greatest place values first. Both values have a 0 in the ones place, so we will move to the tenths place. One-tenth is less than two-tenths, so 0.14 < 0.2. 

  • For example, students compare 0.3 and 0.03 using decimal grids and represent each value and explain that 0.3 covers a greater are of the decimal grid than 0.03, so 0.3 is greater than 0.03. 

decimal grids

  • Instruction includes opportunities to predict and explain the relative size of the product of two decimals. Students use models to check their prediction and solve. The teacher guides students to connect that multiplying a given number by a number less than one will result in a smaller number, and that multiplying a given number by a number greater than one will result in a larger number. 
    • For example, students solve the following problem 0.2 × 0.5. Students should reason about the size of the decimals and connect it back to their fraction understanding and think about the multiplication sign signaling “groups of.” This expression could be interpreted as 0.2 “of” 0.5. This will help with the misconception of multiplying equals a larger product. The picture below illustrates the product of 0.2 and 0.5. If the entire square is 1 unit, the gray region represents 0.5 units, and the red region represents 0.2 units. The overlap in purple contains 10 small squares, each of which represents 0.01 units. Therefore, the overlap portion contains 10 × 0.01 = 0.10 units. The overlap portions show a 0.2 by 0.5 rectangle, so the number of units it contains is the product 0.2 and 0.5.

Grid

 

Instructional Tasks

Instructional Task 1 (MTR.4.1) 

What is the same about the products of these expressions? What is different? Explain. 
14 × 5         0.14 × 0.05 


Instructional Task 2 (MTR.4.1) 

What is the same about the quotients of these expressions? What is different? Explain. 
50 ÷ 25         50 ÷ 0.25

 

Instructional Task 3 (MTR.5.1)

How can you use 2 × 12 = 24 to help you find the product of 2 × 1.2? Explain.

 

Instructional Items

Instructional Item 1 

Raul reasons that the product of 82 × 0.56 will be greater than 41 and less than 82. Explain whether or not his conclusion is reasonable. 

 

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



Related Courses

Course Number1111 Course Title222
5012070: Grade Five Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 - 2024, 2024 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 - 2024, 2024 and beyond (current))
5012015: Foundational Skills in Mathematics 3-5 (Specifically in versions: 2019 - 2022, 2022 - 2024, 2024 and beyond (current))


Related Access Points

Access Point Number Access Point Title
MA.5.NSO.2.AP.4 Explore the estimation of products and quotients of two multi-digit numbers with decimals to the tenths (e.g., 8.9 × 2.3 becomes 9 × 2 by rounding both factors to the nearest whole number). Multi-digit numbers not to exceed 9.9.


Related Resources

Educational Game

Name Description
Ice Ice Maybe: An Operations Estimation 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.

Lesson Plans

Name Description
Catch Me If You Can: Engineering Design Challenge

In this lesson, 5th grade students work in small groups on a STEM challenge that involves science and math standards related to the water cycle, as well as learning the engineering design process.

How Much Did I Earn? Division with Decimals

This lesson will introduce division of decimals using place value decomposition. Students will use base 10 blocks, division strategies and place value knowledge to divide decimals by whole numbers.

Multiplying Decimals: Finding Part of a Whole

This is an introductory lesson for this concept.   Students will use area models to show that multiplying a whole number by a decimal creates a product that is only part of the original whole.

Deft Drawings for Decimal Division

In this lesson, students divide decimals to hundredths in real-world word problems by drawing illustrations based on place value and explaining the reasoning used.

Getting the Top Mini-Fridge not a Small Deal

In this MEA, students will create a procedure to rank five mini-refrigerators to determine which one should be purchased for the school by the PTA based on size, type, features, energy usage, and cost.  In the process, students will solve real-world problems involving the multiplication of multi-digit numbers with decimals to the hundredths, including using money.  Students will also determine the volume of a rectangular prism using a formula.

Workouts That Work

Students will create a rating system for workout DVD's according to weight loss, muscle toning, and increased physical condition.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

Kudos for Kicks - MEA

In this Model Eliciting Activity, MEA, the students will be asked to assist a discount shoe store owner, who is planning a one day sale promotion, to choose a famous brand shoe to feature for the one day sale. Students will determine which one will bring in more customers, as well as provide the most profit. Students will need to read a data table, calculate the total profit margin per pair, and the total sales potential profit margin determined by the number of sneakers in stock. Students will also need to consider comfort, durability, and specific details about each brand. A twist is added to the problem when additional stock items are added, plus one of the brands is removed and two new brands are added.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem, while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought process. MEAs follow a problem-based, student-centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEA’s visit: https://www.cpalms.org/cpalms/mea.aspx

To Oregon by Wagon

Students work in teams to plan the contents of a covered wagon for a family relocating from Missouri to Oregon. Students must calculate the weight and cost of the wagon by adding, subtracting, and multiplying with decimals.

Out to Lunch: Decimal Operations with a Menu

In this lesson students work toward fluency with decimal operations by using a snack bar menu and going "Out to Lunch" with a friend.

Party Planners Wanted

In this Model Eliciting Activity, MEA, students will work in collaborative groups to solve multistep problems with whole numbers and decimals by using different mathematical operations such as addition, subtraction, multiplication, and division. The students will be asked to assist a businessman who is planning a party for his employees. They will need to read several ads and decide which company offers the best deal in renting tables, chairs, and tablecloths for the client. They will need to take into consideration the number of guests attending the party and the budget allowed. A twist is added to the problem when the students are asked to consider an additional ad and the fact that the guest list is now slightly larger.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem, while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought process. MEAs follow a problem-based, student-centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEA’s visit: https://www.cpalms.org/cpalms/mea.aspx

Vacation Time

The students will create a package list for a travel company. They must use all operations with decimals as well as compare decimals.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought processes. MEAs follow a problem-based, student-centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEAs visit: https://www.cpalms.org/cpalms/mea.aspx

Pick A Pet

This Model Eliciting Activity (MEA) asks students to develop a procedure for choosing a reptile or amphibian to place in a school reception area. Students will need to consider safety, price of animal, cost by week to feed animal, size and cost of the enclosure, and the life span of the animals they are considering. In the second portion of the problem statement, the students will need to prepare a budget and cost analysis for the year to consider if they have still made the correct choices while adding three more animals for consideration. The culminating activity for this MEA will have the student write a proposal for the Principal to state their choice of animal, give a year's budget for cost and care for the animal.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem, while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought process. MEAs follow a problem-based, student centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEA’s visit: https://www.cpalms.org/cpalms/mea.aspx

Estimating Decimal Products

In this lesson, students will learn to estimate decimal products using different strategies to arrive at compatible numbers. They will learn that estimates will vary depending on the strategy chosen and that the closer the compatible numbers are to the actual factors, the closer the estimate will be to the exact answer. Students will also learn that estimation is used to solve problems that don't require exact answers and to check exact answers for reasonableness.

Banana County Public School-Painters MEA

This Model Eliciting Activity (MEA) is written at a 4th grade level.

This activity allows students to think critically using information provided. Students will write a procedure on how they determined which painting company would be suitable for the client.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

The 20 Second Game

This is a game students will love to play to improve their understanding of estimating decimal products and increase speed when finding these estimations. The game can be modified to practice estimating products of whole numbers, quotients of whole numbers, and quotients of decimals.

Vending Machine Snacks

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem, while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought process. MEAs follow a problem-based, student centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEA’s visit: https://www.cpalms.org/cpalms/mea.aspx

In this MEA, students are challenged to choose the snacks that will be in a vending machine in a school. Students will need to divide whole numbers and decimal numbers by whole numbers. Students will work in groups to solve the problem and write a letter to the client explaining their thinking.

Original Student Tutorial

Name Description
Designing Dog Playgrounds: Multiplying Decimals

Help your town build a dog park by multiplying whole numbers by decimals to the tenths place in this interactive tutorial. 

Note: this is an introductory tutorial on multiplying whole numbers by decimals before students move on to multiplying decimals by decimals.

Perspectives Video: Expert

Name Description
B.E.S.T. Journey

What roles do exploration, procedural reliability, automaticity, and procedural fluency play in a student's journey through the B.E.S.T. benchmarks? Dr. Lawrence Gray explains the path through the B.E.S.T. mathematics benchmarks in this Expert Perspectives video.

Perspectives Video: Teaching Ideas

Name Description
Multiplying Multi-digit Numbers

Unlock an effective teaching strategy for teaching multiplying multi-digit numbers using ten frames in this Teacher Perspectives video for educators.

Estimating Decimal Multiplication

Unlock an effective teaching strategy for teaching decimal multiplication in this Teacher Perspectives video for educators.

Precision of Measurement <p>Classroom activities that teach students precision of measurement.</p>

Problem-Solving Tasks

Name Description
Reasoning about Multiplication and Division and Place Value, Part 2

Students are asked to reason about and explain the placement of decimals in quotients.

What is 23 ÷ 5?

When a division problem involving whole numbers does not result in a whole number quotient, it is important for students to be able to decide whether the context requires the result to be reported as a whole number with remainder (as with Part (b)) or a mixed number/decimal (as with Part (c)). Part (a) presents two variations on a context that require these two different responses to highlight the distinction between them.

Teaching Idea

Name Description
Guide for Providers: Best Practices For K–5 Mathematics Support During Out-Of-School Time

This guide offers five practical strategies to help after-school providers support meaningful, low-pressure math experiences that lead to deeper understanding and better long-term outcomes for K-5 students.

Student Resources

Original Student Tutorial

Name Description
Designing Dog Playgrounds: Multiplying Decimals:

Help your town build a dog park by multiplying whole numbers by decimals to the tenths place in this interactive tutorial. 

Note: this is an introductory tutorial on multiplying whole numbers by decimals before students move on to multiplying decimals by decimals.

Educational Game

Name Description
Ice Ice Maybe: An Operations Estimation 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.

Problem-Solving Tasks

Name Description
Reasoning about Multiplication and Division and Place Value, Part 2:

Students are asked to reason about and explain the placement of decimals in quotients.

What is 23 ÷ 5?:

When a division problem involving whole numbers does not result in a whole number quotient, it is important for students to be able to decide whether the context requires the result to be reported as a whole number with remainder (as with Part (b)) or a mixed number/decimal (as with Part (c)). Part (a) presents two variations on a context that require these two different responses to highlight the distinction between them.



Parent Resources

Problem-Solving Tasks

Name Description
Reasoning about Multiplication and Division and Place Value, Part 2:

Students are asked to reason about and explain the placement of decimals in quotients.

What is 23 ÷ 5?:

When a division problem involving whole numbers does not result in a whole number quotient, it is important for students to be able to decide whether the context requires the result to be reported as a whole number with remainder (as with Part (b)) or a mixed number/decimal (as with Part (c)). Part (a) presents two variations on a context that require these two different responses to highlight the distinction between them.

Teaching Idea

Name Description
Guide for Providers: Best Practices For K–5 Mathematics Support During Out-Of-School Time:

This guide offers five practical strategies to help after-school providers support meaningful, low-pressure math experiences that lead to deeper understanding and better long-term outcomes for K-5 students.



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