MA.6.NSO.2.1

Multiply and divide positive multi-digit numbers with decimals to the thousandths, including using a standard algorithm with procedural fluency.

Clarifications

Clarification 1: Multi-digit decimals are limited to no more than 5 total digits.
General Information
Subject Area: Mathematics (B.E.S.T.)
Grade: 6
Strand: Number Sense and Operations
Date Adopted or Revised: 08/20
Status: State Board Approved

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

 

Terms from the K-12 Glossary

  • Area Model
  • Commutative Property of Multiplication
  • Expression
  • Dividend
  • Divisor

 

Vertical Alignment

Previous Benchmarks

http://flbt5.floridaearlylearning.com/standards.html

Next Benchmarks

 

Purpose and Instructional Strategies

In grade 5, students multiplied and divided multi-digit whole numbers and represented remainders as fractions. They also estimated and determined the product and quotient of multi-digit numbers with decimals to the hundredths and multiplied and divided the product and quotient of multi-digit numbers with decimals to the hundredths by one-tenth and one-hundredth with procedural reliability. In grade 6, students multiply and divide positive rational numbers with procedural fluency, including dividing numerators by denominators to rewrite fractions as decimals. In grade 7, students will become fluent in all operations with positive and negative rational numbers.
  • Instruction includes representing multiplication in various ways. 
    • 3.102 × 1.1 = 3.4122 
    • (3.102)(1.1) = 3.4122 
    • 3.102(1.1) = 3.4122 
    • 3.102 · 1.1 = 3.4122
  • Students should continue demonstrating their understanding from grade 5 that division can be represented as a fraction.
  • A standard algorithm is a systematic method that students can use accurately, reliably and efficiently (no matter how many digits) depending on the content of the problem. It is not the intention to require students to use a standard algorithm all of the time. However, students are expected to become fluent with a standard algorithm by certain grade levels as stated within the benchmarks.
  • Instruction includes a variety of methods and strategies to multiply and divide multi-digit numbers with decimals.
    • Area Models

    • Partial Products

    • Multiplying as if the factors are whole numbers and applying the decimal places to the final product based on the number of decimals represented in the factors (MTR.3.1).
      Multiplying as if the factors are whole numbers
  • Students should develop fluency with and without the use of a calculator when performing operations with positive decimals.

 

Common Misconceptions or Errors

  • Students may incorrectly apply rules for adding or subtracting decimals to multiplication of decimals, believing place values must be aligned.
  • Students may confuse the lining up of place values when multiplying or dividing vertically by omitting or forgetting to include zeros as place holders in the partial products or quotients.

 

Strategies to Support Tiered Instruction

  • Instruction includes the use of estimation to ensure the proper placement of the decimal point in the final product or quotient of decimals.
    • For example, if finding the product of 12.3 and 4.8, students should estimate the product to be close to 60, by using 12 and 5 as friendly numbers, then apply the decimal to the actual product of 123 and 48, which is 5904. Based on the estimate, the decimal should be placed after 59 to produce 59.04.
  • Teacher encourages and allows for students who have a firm understanding of multiplying and dividing fractions to convert the provided decimal values to their equivalent fractional form before performing the desired operation and converting the solution back to decimal form.
  • Teacher provides graph paper to utilize while applying an algorithm for multiplying or dividing to keep numbers lined up and help students focus on place value.
  • Instruction includes providing opportunities to reinforce place values with the use of base ten blocks or hundredths grids.

 

Instructional Tasks

Instructional Task 1 (MTR.6.1)
Carlos spent $20.76 on chips when his friends came over. Each bag of chips cost $3.46 and each bag has 3 servings. What is the maximum number of friends that Carlos can have over if each person can have a single serving of chips?

Instructional Task 2 (MTR.7.1)
Samantha has 6.75 bags of candy. A full bag of candy contains 13.125 ounces of candy. How many ounces of candy does Samantha have?

Instructional Task 3 (MTR.4.1, MTR.5.1)
Part A. Complete the table below using a calculator.
  ExpressionSolutionExpressionSolution
559(5)

5.59(5)
325(25) 3.25(2.5) 
19(93) 19(9.3) 
Part B. Talk with a partner about what you notice from the table in Part A.

Part C. How are the expressions without decimals related to the expressions with decimals? Is there a relationship between the decimal placements in the expressions and the solutions?

Part D. If 2368(421) = 996,928, what would you expect 2.368(4.21) be equal to?

 

Instructional Items

Instructional Item 1
Determine the product of 23.5 and 2.3.

Instructional Item 2
The expression 13.31 ÷ 0.125 is equivalent to what number?

Instructional Item 3
Determine the quotient of 201.3 and 1.83.

Instructional Item 4
The expression 4.321 × 2.3 is equivalent to what number?


*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.
1205010: M/J Grade 6 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1205020: M/J Accelerated Mathematics Grade 6 (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 2022, 2022 and beyond (current))
1204000: M/J Foundational Skills in Mathematics 6-8 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7812015: Access M/J Grade 6 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2022, 2022 and beyond (current))

Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.6.NSO.2.AP.1: Solve one-step multiplication and division problems involving positive decimals whose place value ranges from the tens to the hundredths places.

Related Resources

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

Formative Assessments

Multiplying Multidigit Decimals:

Students are asked to multiply multidigit decimal numbers and are assessed for both accuracy and fluency.

Type: Formative Assessment

Subtracting Multidigit Decimals:

Students are asked to subtract multidigit decimal numbers and are assessed for both accuracy and fluency.

Type: Formative Assessment

Adding Multidigit Decimals:

Students are asked to add multidigit decimal numbers and are assessed for both accuracy and fluency.

Type: Formative Assessment

Dividing Multidigit Decimals:

Students are asked to divide multidigit decimal numbers and are assessed for both accuracy and fluency.

Type: Formative Assessment

Running:

Students are asked to solve a word problem involving division of a whole number by a decimal using a model or drawing, a strategy based on place value, the relationship between multiplication and division, or properties of operations.

Type: Formative Assessment

Lesson Plans

Sound Is Not The Only Place You Hear About Volume!:

This lesson introduces the idea of finding volume. Volume in sixth grade math is very "rectangular" (cubes, rectangular prisms) and this lesson brings to light that volume is simply a measure of available space, but can take on many shapes or forms (cylinders for example - graduated cylinders and beakers) in science. Students will be left to design their own data collection and organizing the data that they collect. They will apply the skill of finding volume to using fractional parts of a number (decimals) and finding the product using the volume formula.

Type: Lesson Plan

Using Nets to Find the Surface Area of Pyramids:

In this lesson, students will explore and apply the use of nets to find the surface area of pyramids.

Type: Lesson Plan

Area of a Triangle:

This lesson is primarily formative in nature and is designed to introduce students to the area of a triangle by having them derive the formula themselves using the relationship between rectangles and triangles. During the lesson the teacher will be facilitating their students as they work with their teams and shoulder partners to solve problems.

Type: Lesson Plan

Where's The POINT? What's The POINT? The Point is... a DECIMAL. "Multiply with Decimals":

Multiply efficiently and fluently with multi-digit decimals using a standard algorithm for the operation.

Type: Lesson Plan

The Price is Right:

In this activity the students will apply their knowledge of mathematical calculations to solve a real-world problem. They will analyze a collection of shipping boxes to determine which box will ship the most for the $100 allowed.

Type: Lesson Plan

Better Buy: 75 fl oz or 150 fl oz?:

The students will clip out advertisements or use the attached PowerPoint to determine the better buy between small quantities and large quantities. The students will answer the question, "Which item costs less per unit?" and demonstrate fluency in dividing with decimals.

Type: Lesson Plan

How much can it hold?:

This lesson uses a discovery approach to exploring the meaning of volume. The students will work with cubes as they construct and analyze the relationship between the length, width, and height to the total amount of cubes. Students will be able to apply this concept to real world applications of other right rectangular prisms and compare them to determine which will hold the most volume. 

 

Type: Lesson Plan

Area of a Right Triangle:

Area of a Right Triangle

Type: Lesson Plan

Rank Our Pressure Cleaners:

In this Model Eliciting Activity, MEA, students are to decide on a pressure cleaning machine that will provide the Sidewalks and Roof Cleaning Services Incorporated with the best value for their money. Students are asked to provide a "Best Value" pressure cleaner to the company owner and explain how they arrived at their solution.

Model Eliciting Activities 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

Type: Lesson Plan

The Mystery of Decimals:

This lesson reviews all four operations (adding, subtracting, multiplying, and dividing) with decimals. It is designed to easily provide differentiated instruction for students. The culmination of the lesson is a computer-based assessment which provides a fun change from a typical pencil and paper test.

Type: Lesson Plan

Dividing Fractions (Part 1) - Tackling Word Problems:

This lesson allows the students to explore the foundation for dividing fractions as well as correctly solving word problems involving division of fractions. It includes the use of the Philosophical Chairs activity and numerical solutions. Group activities are included to foster cooperative learning.

Type: Lesson Plan

Dividing Decimals Investigations:

In this introductory lesson, students test how the basic operations performed on the dividend and divisor affect the quotient of a pair of numbers. Students then conclude whether the results of their trials can be applied to solve problems with a decimal divisor.

Type: Lesson Plan

Dividing by Fractions Discovery:

This lesson allows students to derive an algorithm for dividing fractions using visual fraction models and equations to represent the problem.

Type: Lesson Plan

Enrique's Ruined Carpet:

In this activity, students use a house blueprint to find the area of carpeted floor by decomposing composite shapes into rectangles and triangles. As students critique each other's reasoning, they refine their thinking of surface area. 

Type: Lesson Plan

Perspectives Video: Experts

Fluency vs. Automaticity:

How are fluency and automaticity defined? Dr. Lawrence Gray explains fluency and automaticity in the B.E.S.T. mathematics benchmarks in this Expert Perspectives video.

Type: Perspectives Video: Expert

The Role of Procedures in Fluency:

What are the components to a good procedure? Dr. Lawrence Gray discusses the role of procedures in the path to fluency in this Expert Perspectives video.

Type: Perspectives Video: Expert

That's Not How I Learned it: Why today's math may look different:

Why do students need "a" good procedure for the arithmetic operations? Dr. Lawrence Gray explains why math may look different than in the past in this Expert Perspectives video.

Type: Perspectives Video: Expert

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. maththematics benchmarks in this Expert Perspectives video.

Type: Perspectives Video: Expert

What is Fluency?:

What is fluency? What are the ingredients required to become procedurally fluent in mathematics? Dr. Lawrence Gray explores what it means for students to be fluent in mathematics in this Expert Perspectives video.

Type: Perspectives Video: Expert

Why Isn't Getting the "Right" Answer Good Enough?:

Why is it important to look beyond whether a student gets the right answer? Dr. Lawrence Gray explores the importance of understanding why we perform certain steps or what those steps mean, and the impact this understanding can have on our ability to solve more complex problems and address them in the context of real life in this Expert Perspectives video.

Type: Perspectives Video: Expert

A Standard Algorithm:

Ever wonder why the benchmarks say, “a standard algorithm,” instead of, “the standard algorithm?" Dr. Lawrence Gray explores the role that standard algorithms can play in building and exhibiting procedural fluency through this Expert Perspectives video.

Type: Perspectives Video: Expert

Perspectives Video: Teaching Idea

Estimating Decimal Multiplication:

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

Type: Perspectives Video: Teaching Idea

Problem-Solving Tasks

Movie Tickets:

The purpose of this task is for students to solve problems involving decimals in a context involving a concept that supports financial literacy, namely inflation. Inflation is a sustained increase in the average price level. In this task, students are asked to compare the buying power of $20 in 1987 and 2012, at least with respect to movie tickets.

Type: Problem-Solving Task

Reasoning about Multiplication and Division and Place Value, Part 1:

Given the fact 13 x 17 = 221, students are asked to reason about and explain the decimal placement in multiplication and division problems where some of the numbers involved have been changed by powers of ten.

Type: Problem-Solving Task

Buying Gas:

There are two aspects to fluency with division of multi-digit numbers: knowing when it should be applied, and knowing how to compute it. While this task is very straightforward, it represents the kind of problem that sixth graders should be able to recognize and solve relatively quickly. Easily recognizing contexts that require division is a necessary conceptual prerequisite to more complex modeling problems that students will be asked to solve later in middle and high school.

This task also has a natural carryover to work with ratios and rates, so students should also be building connections between these kinds of division problems and finding unit rates.

Type: Problem-Solving Task

Gifts from Grandma, Variation 3:

Students are asked to solve problems from context by using multiplication or division of decimals.

Type: Problem-Solving Task

Jayden’s Snacks:

Students are asked to add or subtract decimals to solve problems in context.

Type: Problem-Solving Task

Tutorials

Multiplying a Decimal by a Power of 10:

This Khan Academy tutorial video explains patterns in the placement of the decimal point, when a decimal is multiplied by a power of 10.  Exponents are NOT discussed.

Type: Tutorial

Multiply and Divide Powers of 10: Zero Patterns:

This Khan Academy tutorial video presents the methodology of understanding and using patterns in the number of zeros of products that have a factor that is a power of 10. While the standard does not mention exponents, the place value understanding of multiplying or dividing by powers of ten will help students understand multiplying and dividing by decimals.  

Type: Tutorial

Dividing by a Multi-Digit Decimal:

This video demonstrates dividing two numbers that are decimals.

Type: Tutorial

Multiplying Decimals:

This video demonstrates how to multiply two decimal numbers.

Type: Tutorial

STEM Lessons - Model Eliciting Activity

Rank Our Pressure Cleaners:

In this Model Eliciting Activity, MEA, students are to decide on a pressure cleaning machine that will provide the Sidewalks and Roof Cleaning Services Incorporated with the best value for their money. Students are asked to provide a "Best Value" pressure cleaner to the company owner and explain how they arrived at their solution.

Model Eliciting Activities 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

MFAS Formative Assessments

Adding Multidigit Decimals:

Students are asked to add multidigit decimal numbers and are assessed for both accuracy and fluency.

Dividing Multidigit Decimals:

Students are asked to divide multidigit decimal numbers and are assessed for both accuracy and fluency.

Multiplying Multidigit Decimals:

Students are asked to multiply multidigit decimal numbers and are assessed for both accuracy and fluency.

Running:

Students are asked to solve a word problem involving division of a whole number by a decimal using a model or drawing, a strategy based on place value, the relationship between multiplication and division, or properties of operations.

Subtracting Multidigit Decimals:

Students are asked to subtract multidigit decimal numbers and are assessed for both accuracy and fluency.

Student Resources

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

Problem-Solving Tasks

Movie Tickets:

The purpose of this task is for students to solve problems involving decimals in a context involving a concept that supports financial literacy, namely inflation. Inflation is a sustained increase in the average price level. In this task, students are asked to compare the buying power of $20 in 1987 and 2012, at least with respect to movie tickets.

Type: Problem-Solving Task

Reasoning about Multiplication and Division and Place Value, Part 1:

Given the fact 13 x 17 = 221, students are asked to reason about and explain the decimal placement in multiplication and division problems where some of the numbers involved have been changed by powers of ten.

Type: Problem-Solving Task

Gifts from Grandma, Variation 3:

Students are asked to solve problems from context by using multiplication or division of decimals.

Type: Problem-Solving Task

Jayden’s Snacks:

Students are asked to add or subtract decimals to solve problems in context.

Type: Problem-Solving Task

Tutorials

Multiplying a Decimal by a Power of 10:

This Khan Academy tutorial video explains patterns in the placement of the decimal point, when a decimal is multiplied by a power of 10.  Exponents are NOT discussed.

Type: Tutorial

Multiply and Divide Powers of 10: Zero Patterns:

This Khan Academy tutorial video presents the methodology of understanding and using patterns in the number of zeros of products that have a factor that is a power of 10. While the standard does not mention exponents, the place value understanding of multiplying or dividing by powers of ten will help students understand multiplying and dividing by decimals.  

Type: Tutorial

Dividing by a Multi-Digit Decimal:

This video demonstrates dividing two numbers that are decimals.

Type: Tutorial

Multiplying Decimals:

This video demonstrates how to multiply two decimal numbers.

Type: Tutorial

Parent Resources

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

Problem-Solving Tasks

Movie Tickets:

The purpose of this task is for students to solve problems involving decimals in a context involving a concept that supports financial literacy, namely inflation. Inflation is a sustained increase in the average price level. In this task, students are asked to compare the buying power of $20 in 1987 and 2012, at least with respect to movie tickets.

Type: Problem-Solving Task

Reasoning about Multiplication and Division and Place Value, Part 1:

Given the fact 13 x 17 = 221, students are asked to reason about and explain the decimal placement in multiplication and division problems where some of the numbers involved have been changed by powers of ten.

Type: Problem-Solving Task

Buying Gas:

There are two aspects to fluency with division of multi-digit numbers: knowing when it should be applied, and knowing how to compute it. While this task is very straightforward, it represents the kind of problem that sixth graders should be able to recognize and solve relatively quickly. Easily recognizing contexts that require division is a necessary conceptual prerequisite to more complex modeling problems that students will be asked to solve later in middle and high school.

This task also has a natural carryover to work with ratios and rates, so students should also be building connections between these kinds of division problems and finding unit rates.

Type: Problem-Solving Task

Gifts from Grandma, Variation 3:

Students are asked to solve problems from context by using multiplication or division of decimals.

Type: Problem-Solving Task

Jayden’s Snacks:

Students are asked to add or subtract decimals to solve problems in context.

Type: Problem-Solving Task