### 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.

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

**Grade:**5

**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

- 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 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

- 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.

- 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.

### Instructional Tasks

*Instructional Task 1* (MTR.4.1)

*Instructional Task 2* (MTR.4.1)

*Instructional Task 3* (MTR.5.1)

### Instructional Items

*Instructional Item 1 *

**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

## Lesson Plans

## Original Student Tutorial

## Perspectives Video: Expert

## Perspectives Video: Teaching Ideas

## Problem-Solving Task

## STEM Lessons - Model Eliciting Activity

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.

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.

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.

## Original Student Tutorials Mathematics - Grades K-5

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.

## Student Resources

## Original Student Tutorial

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.

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 Task

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.

Type: Problem-Solving Task

## Parent Resources

## Problem-Solving Task

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.

Type: Problem-Solving Task