MA.5.NSO.2.5

Multiply and divide a multi-digit number with decimals to the tenths by one-tenth and one-hundredth with procedural reliability.

Examples

The number 12.3 divided by 0.01 can be thought of as ?×0.01=12.3 to determine the quotient is 1,230.

Clarifications

Clarification 1: Instruction focuses on the place value of the digit when multiplying or dividing.
General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Number Sense and Operations
Date Adopted or Revised: 08/20
Status: State Board Approved

Benchmark Instructional Guide

• Equation
• Expression

Vertical Alignment

Previous Benchmarks

Next Benchmarks

Purpose and Instructional Strategies

The purpose of this benchmark is for students to multiply multi-digit numbers with decimals to the tenths by .1 and by .01 with procedural reliability. Procedural reliability refers to the ability for students to develop an accurate, reliable method that aligns with a student’s understanding and learning style. Fluency of multiplying and dividing multi-digit whole numbers with decimals is not expected until grade 6 (MA.6.NSO.2.1).
• When multiplying and dividing, students should continue to use the number sense strategies built in MA.5.NSO.2.4 (estimation, rounding and exploring place value relationships). Using these strategies will helps students predict reasonable solutions and determine whether their solutions make sense after solving.
• During instruction, students should see the relationship between multiplying and dividing multi-digit numbers with decimals to multiplying and dividing by whole numbers. Students extend their understanding to generalize patterns that exist when multiplying or dividing by 10 or 100 (MTR.5.1).
• Instruction includes the language that the “digits shift” relative to the position of the decimal point as long as there is an accompanying explanation. An instructional strategy that helps students see this is by putting digits on sticky notes or cards and showing how the values shift (or the decimal point moves) when multiplying by a power of ten.
• For example, a teacher could show one card with a 3 and another with a 5, and place them on the left and right of a decimal point on a blank place value chart. The teacher could then ask students to multiply by ten and shift both digits one place left to show the equation 3.5 × 10 = 35. They could ask students to multiply by $\frac{\text{1}}{\text{10}}$ and show that 3.5 × $\frac{\text{1}}{\text{10}}$ = 0.35. Instruction also includes using the language “moving the decimal point” as long as there is an explanation about what happens to a number when multiplying and dividing by 0.1 and 0.01. Moving the decimal point does not change its meaning; it always indicates the transition from the ones to the tenths place. From either point of view, when the change is made it is important to emphasize the digits have new place values (MTR.2.1, MTR.4.1, MTR.5.1).

Common Misconceptions or Errors

• 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 the use of a place value chart to demonstrate how the value of a digit changes if the digit moves one place to the left or right. Instruction includes using place 1 value understanding to make the connections between $\frac{\text{1}}{\text{10}}$ of, ten times less, and dividing by 10. Also, the place value chart can be used to demonstrate that the decimal point marks the transition between the ones place and the tenths place.
• For example, students multiply 4 by 10, then record 4 and the product of 40 in a place value chart. This process is repeated by multiplying 40 by 10. The teacher asks students to explain what happens to the digit 4 each time it is multiplied by 10. Next, the teacher explains that multiplying by $\frac{\text{1}}{\text{10}}$  is the same as dividing by 10. Students multiply 400 by $\frac{\text{1}}{\text{10}}$ and record the product in their place value chart. The Students multiply 400 by process is repeated, multiplying 40 and 4 by $\frac{\text{1}}{\text{10}}$. Students explain how the value of the 4 changed when being multiplied by 10 and $\frac{\text{1}}{\text{10}}$

• Instruction includes opportunities to use models such as place value disks to demonstrate how the value of a digit changes if the digit moves one place to the left or right. A place value chart can be used with the models to support place value understanding and demonstrate that the decimal point marks the transition between the ones place and the tenths place. Instruction includes using place value understanding to make connections between $\frac{\text{1}}{\text{10}}$ of, ten times less and dividing by 10.
• For example, the teacher uses a familiar context such as money, asking students to explain the value of each digit in \$33.33. Then, students represent 33.33 in a place value chart using place value disks. Students compare the value of the whole numbers, (3 dollars and 30 dollars), then move to comparing 0.3 and 0.03 (30 cents and 3 cents). The teacher asks, “How does the value of the three in the hundredths place compare to the value of the three in the tenths place?” and explain that the three in the hundredths place is $\frac{\text{1}}{\text{10}}$ the value of the three in the tenths place and that multiplying by $\frac{\text{1}}{\text{10}}$ is the same as dividing by 10.

Instructional Task 1 (MTR.7.1)

Part A. What is $\frac{\text{1}}{\text{10}}$ times 15?
Part B. How many dimes are in \$1.50?
Part C. Write an expression to represent how many dimes are in \$1.50.

Instructional Items

Instructional Item 1

Which compares the products of 7.8 × 0.1 and 7.8 × 10 correctly?
• a. The product of 7.8 × 0.1 is 100 times less than the product of 7.8 × 10.
• b. The product of 7.8 × 0.1 is 10 times less than the product of 7.8 × 10.
• c. The product of 7.8 × 0.1 is 100 times more than the product of 7.8 × 10.
• d. The product of 7.8 × 0.1 is 10 times more than the product of 7.8 × 10.

*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.NSO.2.AP.5: 5 Explore multiplying and dividing single-digit whole numbers by one-tenth and one-hundredth.

Related Resources

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

Lesson Plans

Weathering Weather:

This is a design challenge lesson allowing students to utilize their understanding of climate zones to design a tropical roof that would withstand the wind and water generated by a hurricane. This lesson also allows you to assess students understanding of operations with decimals to create their design under a given budget.

Type: Lesson Plan

Icky, Icky, No More Slicky:

In this lesson, 5th grade students will build an engineering device to separate oil from water in a simulated oil spill. StudentsÂ will have an opportunity to learn about the impact that humans can have on the environment, both positively and negatively.

This is an Engineering Design Challenge that is best used after a unit or lesson that is aligned to science standards on solving problems or materials which dissolve in water. This challenge provides students a means to use their knowledge of the way materials will or will not dissolve in water to create and design an oil spill removal tool while learning the Engineering Design Process and being exposed to the field of engineering. This lesson is not intended as an initial introduction to the standard and would be best utilized as a culmination lesson.

Type: Lesson Plan

The Coasta with the Mosta:

Students will create an exciting and thrilling roller coaster model. Students will use their knowledge of forces to build a model of a roller coaster using foam insulation and a marble.

Type: Lesson Plan

When the Wind Blows:

This is an engineering design process lesson that covers forces and motion. It is designed to engage students using hands-on problem solving strategies.

Type: Lesson Plan

From Trash to Treasure - An Engineering Design Challenge:

This lesson gives students hands-on experience with sorting mixtures based on their properties. The students will relate these science standards to a real-world problem of eliminating trash in land fills. They will have to purchase the tools they use to create their assembly line to sort the garbage within the budget provided.

Type: Lesson Plan

Bridge Over Troubled Waters:

In this engineering lesson, students will design and construct a bridge that will be free-standing and support a toy car.

Type: Lesson Plan

Medic Mass Landing: 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 force, motion, and measurement, as well as learning the engineering design process.

Type: Lesson Plan

Where Will We Stay?:

In this lesson, students explore lodging options for their dream family vacation. Students will plan a vacation for a family of four. With a budget of \$5,000 students will prepare a budget to include the cost of transportation, lodging, and attractions. In this lesson, students will focus on preparing the budget for hotel costs. In lesson 1 of this unit lesson, students prepared a budget for transportation. In the subsequent lesson to this one, students will prepare a budget for the entertainment/attractions costs portion of their vacation. Teachers can choose to do one, some, or all lessons as they can be completed independently of one another.

Type: Lesson Plan

A Tasty Treat:

In this lesson, students will be given a list of ingredients and prices they must use to create their own snack mix. Students will have to add, subtract, multiply, and divide decimal numbers. The closure of the lesson focuses specifically on the division strategies students used to solve the problem.

Type: Lesson Plan

Currency Craze!:

If I travel to Italy, can I buy a raspberry gelato with a \$5.00 bill? Students will understand that other countries use currencies differentÂ from the United States, and that when visiting those countries, exchange rates are used to determine the value of the United States dollar (USD). This lesson integrates Mathematics with Social Studies as the students apply their knowledge of decimals and basic operations with their passports for a real-world application.

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

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

Tutorial

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

Student Resources

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

Tutorial

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

Parent Resources

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