MA.4.NSO.2.4

Divide a whole number up to four digits by a one-digit whole number with procedural reliability. Represent remainders as fractional parts of the divisor.

Clarifications

Clarification 1: Instruction focuses on helping a student choose a method they can use reliably.

Clarification 2: Instruction includes the use of models based on place value, properties of operations or the relationship between multiplication and division.

General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Number Sense and Operations
Status: State Board Approved

Benchmark Instructional Guide

• Dividend
• Divisor
• Expression
• Equation
• Quotient

Vertical Alignment

Previous Benchmarks

Next Benchmarks

Purpose and Instructional Strategies

The purpose of this benchmark is for students to choose a reliable method for dividing 4-digit numbers by 1-digit numbers. It builds on the understanding developed during exploration (MA.3.NSO.2.2) and on automaticity (MA.4.NSO.2.1), and prepares for procedural fluency (MA.5.NSO.2.2).
• This benchmark connects to previous work with division within 144. Before achieving procedural reliability it may be useful for students to engage in additional exploratory work dividing multi-digit numbers by single-digit numbers. Students should use multiple methods (MTR.2.1) such as area models or models of base-ten blocks to connect understanding to a method they will use with procedural reliability and ultimately leading to a standard algorithm.
• Base-Ten Blocks
• Long Division Algorithm
• Area Model
• Partial Quotient Division
• When students are using their preferred method they should be able to explain their thinking, connecting it to place value understanding and the relationship between division and repeated subtraction.

Common Misconceptions or Errors

• Many students are taught an algorithm for division and then tend to look at the digits within the number as single digits instead of thinking about the place value of each digit or thinking about the number as a whole. When asked if their solution is reasonable, students do not understand what is reasonable because they are unable to estimate since they do not see the number in its entirety, but rather, as individual digits. Students must have a solid understanding about place value and the properties of operations to make sense of division.
• Some students may not understand that the remainder represents a fraction with the divisor as the denominator. For example, 7 ÷ 3 = 2r1 means that 7 ÷ 3 = 2 $\frac{\text{1}}{\text{3}}$. Students should have experience with equal sharing division problems that involve remainders (MA.4.AR.1.1).

Strategies to Support Tiered Instruction

• Instruction includes connecting place value with the partial products model. Students should not view the digits as individual numbers but connect individual digits with the value of that number.
• Example: 366 is 300 + 60 + 6.
• Instruction includes problems involving division with a remainder. Students use models to understand what the remainder is. The remainder can be written as a whole number or a fraction.
• For example, Karly, Juan, and Li share 4 cookies equally. How many cookies can each person eat? Karly, Juan, and Li each can eat one whole cookie but then must split the 4th cookie into thirds so that they can each eat 1$\frac{\text{1}}{\text{3}}$ .The remainder 1 in this division problem represents the fraction $\frac{\text{1}}{\text{3}}$
• Instruction connects place value to dividing whole numbers equally. Students build the number with base ten blocks and then physically divide the number into equal groups.
• For example, when solving 366 ÷ 3, students should build the number 366 and then physically move the blocks into 3 equal groups. This will help solidify the understanding from thinking of the digit as a “3” and now thinking about it as 300.
• Instruction includes the opportunity to use models to understand what the remainder is. The remainder can be written as a whole number or a fraction. Students physically cut or break apart paper to show what is happening in problems involving remainders.
• For example, using the problem: Frank and Lisa share five brownies. How many brownies can they each eat? Students should model the problem with five pieces of paper, each representing one brownie. Students should start by labeling each brownie. Frank and Lisa each have two brownies with one brownie left over. Then, students physically cut the last brownie into two equal parts so that each person is able to eat 2 $\frac{\text{1}}{\text{2}}$ brownies. Relate this to an equation 5 ÷ 2 = 2 $\frac{\text{1}}{\text{2}}$.

Using only the number tiles 2, 3, 4, 5, 6 or 7, fill in the blanks in the division situation to find a quotient as close to 100 as possible.

Sam and Sally were given \$117 after they helped deliver groceries for a month. In order to split the money equally, Sam divides 117 by 2 and gets 58 with a remainder of 1. Explain how they should use this result to determine their equal shares in dollars and cents.

Instructional Items

Instructional Item 1

What is 1,545 divided by 5?

Instructional Item 2

What is 311 divided by 7? (Express the remainder as a fraction)

*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.
5012060: Mathematics - Grade Four (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7712050: Access Mathematics Grade 4 (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.4.NSO.2.AP.4: Explore division of two whole numbers up to two digits by one digit with and without remainders. Represent remainders as whole numbers.

Related Resources

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

Educational Software / Tool

Arithmetic Quiz:

In this activity, students solve arithmetic problems involving whole numbers, integers, addition, subtraction, multiplication, and division. This activity allows students to track their progress in learning how to perform arithmetic on whole numbers and integers. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Educational Software / Tool

Formative Assessments

Dividing Using an Area Model:

Students are asked to interpret a division problem with a one-digit divisor that has been completed using an area model. If the student is successful, he or she is asked to complete a division problem with a one-digit divisor using an area model.

Type: Formative Assessment

Book Drive:

Students are asked to solve a division problem using a strategy based on place value.

Type: Formative Assessment

Interpreting Division:

Students are asked to analyze and explain another student’s division work in terms of a partial quotients strategy and to apply this strategy to another division problem.

Type: Formative Assessment

Dividing Using Place Value:

Students are asked to complete a division problem using place value.

Type: Formative Assessment

Lesson Plans

New Puppy's Pen:

The purpose of this lesson is to help students find the missing side's length for rectangular area problems, when the total area and one side's length is given. The use of square tiles, then graph paper and equations are used throughout the lesson to help students progress from conceptual to procedural knowledge.

Type: Lesson Plan

Dividing for Equal Groups:

This lesson is meant to help solidify division understanding before teaching the standard algorithm.  Given a situational story, students will use base 10 blocks to model division in order to solve problems. It may be used for 4th or 5th grade depending on the size of the divisor.

Type: Lesson Plan

I Love Leftovers!:

In this lesson, students will explore situational problems that address the different ways to interpret the remainder.

Type: Lesson Plan

I See! Division with the Distributive Property:

In this lesson, students will use visual models to represent division using the distributive property as a strategy. Students will have an understanding of how to decompose numbers in the context of division problems using an area model.

Type: Lesson Plan

Slither Not in the Everglades! Python MEA:

This MEA will ask students to work in teams to help their client, The Florida Fish and Wildlife Conservation Commission, to decide which Burmese python traps manufacturing company to buy traps from. The traps will be placed along the Florida Keys and the Everglades to help prevent the growth of invasive Burmese Python population. The students will implement their knowledge of how plants, animals, and humans impact the environment, use mathematical and analytical problem-solving strategies, and be able report their finding in an organized, descriptive manner.

Type: Lesson Plan

Patty's Party Planning:

In this Model Eliciting Activity, MEA, students will help a party planner determine which party location is the best one to use. They will calculate the cost of the banquet hall rental based on the number of people, number of tables and hourly rental of the location by using division and multiplication.

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

Type: Lesson Plan

Fourth graders will help Cookies and Treats find cost-effective and eco-friendly packaging for its cookies. Students will organize data and compare prices using decimal notation in order to develop a procedure for choosing packaging for cookies.  Students will use multiplication and division of whole numbers to plan for how many packages to order.

Type: Lesson Plan

Share and Share Alike:

This lesson is an introduction to division and does not include a procedural recording for division. The student will be able to physically model the division of 2-, 3-, and 4-digit dividends with 1-digit divisors using objects and base ten blocks and explain the meaning of a remainder.

Type: Lesson Plan

What Are They Thinking? Understanding Division:

This lesson uses a discovery approach to exploring the meaning of division. The students will utilize math practice standards as they analyze math solutions and explain their own solutions. Since the lesson analyzes division, it is a sound lesson to use to check student understanding before introducing efficient division algorithms.

Type: Lesson Plan

Original Student Tutorials

CPALMS Aquarium: Part 3 Division with Larger Numbers:

This is part 3 of in a three-part series. Click below to learn different strategies to help you become more efficient with division.

Type: Original Student Tutorial

CPALMS Aquarium Part 2: Division Strategies:

Learn to solve division challenges using the partial quotients strategy with this interactive tutorial.

This is the second tutorial is a series on division strategies.

Type: Original Student Tutorial

CPALMS Aquarium: Connecting Multiplication and Division: Part 1:

Learn how multiplication connects to division to help understand what division is in this aquarium-themed, interactive tutorial.

This is part 1 of a two-part series. Click to open Part 2, Division Strategies.

Type: Original Student Tutorial

Perspectives Video: Teaching Idea

Representing Remainders as Fractions:

Unlock an effective teaching strategy for representing remainders as fractions in this Teacher Perspectives video for educators.

Type: Perspectives Video: Teaching Idea

Tutorials

Division: The importance of place value:

In this video tutorial from Khan Academy, learn about the importance of place value when dividing. The tutorial uses place value up to thousands to help students think about division.

Type: Tutorial

Division: Intro to remainders:

In this video tutorial from Khan Academy, you will get an introduction to the meaning of remainders.

Type: Tutorial

STEM Lessons - Model Eliciting Activity

Fourth graders will help Cookies and Treats find cost-effective and eco-friendly packaging for its cookies. Students will organize data and compare prices using decimal notation in order to develop a procedure for choosing packaging for cookies.  Students will use multiplication and division of whole numbers to plan for how many packages to order.

Patty's Party Planning:

In this Model Eliciting Activity, MEA, students will help a party planner determine which party location is the best one to use. They will calculate the cost of the banquet hall rental based on the number of people, number of tables and hourly rental of the location by using division and multiplication.

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

Slither Not in the Everglades! Python MEA:

This MEA will ask students to work in teams to help their client, The Florida Fish and Wildlife Conservation Commission, to decide which Burmese python traps manufacturing company to buy traps from. The traps will be placed along the Florida Keys and the Everglades to help prevent the growth of invasive Burmese Python population. The students will implement their knowledge of how plants, animals, and humans impact the environment, use mathematical and analytical problem-solving strategies, and be able report their finding in an organized, descriptive manner.

MFAS Formative Assessments

Book Drive:

Students are asked to solve a division problem using a strategy based on place value.

Dividing Using an Area Model:

Students are asked to interpret a division problem with a one-digit divisor that has been completed using an area model. If the student is successful, he or she is asked to complete a division problem with a one-digit divisor using an area model.

Dividing Using Place Value:

Students are asked to complete a division problem using place value.

Interpreting Division:

Students are asked to analyze and explain another student’s division work in terms of a partial quotients strategy and to apply this strategy to another division problem.

Original Student Tutorials Mathematics - Grades K-5

CPALMS Aquarium Part 2: Division Strategies:

Learn to solve division challenges using the partial quotients strategy with this interactive tutorial.

This is the second tutorial is a series on division strategies.

CPALMS Aquarium: Connecting Multiplication and Division: Part 1:

Learn how multiplication connects to division to help understand what division is in this aquarium-themed, interactive tutorial.

This is part 1 of a two-part series. Click to open Part 2, Division Strategies.

CPALMS Aquarium: Part 3 Division with Larger Numbers:

This is part 3 of in a three-part series. Click below to learn different strategies to help you become more efficient with division.

Student Resources

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

Original Student Tutorials

CPALMS Aquarium: Part 3 Division with Larger Numbers:

This is part 3 of in a three-part series. Click below to learn different strategies to help you become more efficient with division.

Type: Original Student Tutorial

CPALMS Aquarium Part 2: Division Strategies:

Learn to solve division challenges using the partial quotients strategy with this interactive tutorial.

This is the second tutorial is a series on division strategies.

Type: Original Student Tutorial

CPALMS Aquarium: Connecting Multiplication and Division: Part 1:

Learn how multiplication connects to division to help understand what division is in this aquarium-themed, interactive tutorial.

This is part 1 of a two-part series. Click to open Part 2, Division Strategies.

Type: Original Student Tutorial

Educational Software / Tool

Arithmetic Quiz:

In this activity, students solve arithmetic problems involving whole numbers, integers, addition, subtraction, multiplication, and division. This activity allows students to track their progress in learning how to perform arithmetic on whole numbers and integers. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Educational Software / Tool

Tutorials

Division: The importance of place value:

In this video tutorial from Khan Academy, learn about the importance of place value when dividing. The tutorial uses place value up to thousands to help students think about division.

Type: Tutorial

Division: Intro to remainders:

In this video tutorial from Khan Academy, you will get an introduction to the meaning of remainders.

Type: Tutorial

Parent Resources

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

Tutorials

Division: The importance of place value:

In this video tutorial from Khan Academy, learn about the importance of place value when dividing. The tutorial uses place value up to thousands to help students think about division.

Type: Tutorial

Division: Intro to remainders:

In this video tutorial from Khan Academy, you will get an introduction to the meaning of remainders.

Type: Tutorial