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Examples
A group of 243 students is taking a field trip and traveling in vans. If each van can hold 8 students, then the group would need 31 vans for their field trip because 243 divided by 8 equals 30 with a remainder of 3.Clarifications
Clarification 1: Problems involving multiplication include multiplicative comparisons. Refer to Situations Involving Operations with Numbers (Appendix A).Clarification 2: Depending on the context, the solution of a division problem with a remainder may be the whole number part of the quotient, the whole number part of the quotient with the remainder, the whole number part of the quotient plus 1, or the remainder.
Clarification 3: Multiplication is limited to products of up to 3 digits by 2 digits. Division is limited to up to 4 digits divided by 1 digit.
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 to have students solve problems involving multiplication and division by using and discussing various approaches. This work builds on problem-solving using the four operations from grade 3 (MA.3.AR.1.2).- Students should use estimation, and this can include using compatible numbers (numbers that sum to 10 or 100) and rounding.
- Instruction should include allowing students many opportunities to solve multiplicative comparison situations.
- Students should have experience solving problems that require students to interpret the remainder to fit the situation. Students may have to round up to the next whole number, drop the remainder, use the remainder as a fraction or decimal, or use only the remainder as determined.
- Add 1 to the quotient
- Thirty students are going on a field trip. They want to put 4 people in each car so that people can sit comfortably. How many cars will be needed?
- Solution: Divide 30 by 4. The answer is 7 r2.
- The answer shows that 7 cars will be needed, but 2 people still need to go to a car. Therefore, they will need 8 cars.
- Use only the remainder
- Gerardo has 19 dollars in his pocket. He wants to give the same amount of money to 4 friends. The rest of the money, if any, will go to his sister to buy toys. How much money will go to his sister if Gerardo wants to give away everything he has?
- Solution: Divide 19 by 4. The answer is 4 r3.
- The remainder is 3, so 3 dollars will go to his sister.
- Drop the remainder
- Alicia has 48 dollars in her pocket. She wants to buy meals for 5 friends. If each meal costs 10 dollars, will Darlene be able to keep all her friends happy?
- Solution: Divide 48 by 10. The answer is 4 r8.
- Alicia can only buy 4 complete meals. Therefore, she cannot buy one for each of her 5 friends.
Common Misconceptions or Errors
- Students apply a procedure that results in remainders that are expressed as r for all situations, even for those in which the result does not make sense.
- For example, when a student is asked to solve the following problem, the student responds to the problem— there are 52 students in a class field trip. They plan to have 10 students in each van. How many vans will they need so that everyone can participate? And the student answers “5r2 vans.” The student does not understand that the two remaining students need another van to go on the field trip.
- Students may not understand that the remainder represents a portion of something, rather than a whole number. Referring back to the previous example students may think r2 means two additional vans rather than a portion of an additional van.
- Students may have trouble seeing a remainder as a fraction.
- For example, 7 ÷ 3 = 2r1 means that 7 ÷ 3 = 2 . If 7 cupcakes are divided among 3 people, then each person will get 2 and cupcakes.
Strategies to Support Tiered Instruction
- Instruction includes opportunities to engage in guided practice completing real-world problems involving remainders. Students use models to understand how to interpret the remainder in situations in which they will need to “ add 1 to the quotient,” “ use only the remainder,” “ drop the remainder” or “ treat the remainder as a fraction.”
- For example, the teacher displays and reads the following problem aloud: “ There are 55 pencils that need to be sorted into boxes. 10 pencils can go into each box. How many boxes are needed so all the pencils can be put into boxes?” The teacher uses models or drawings to represent the problem and guided questioning to encourage students to identify that they will need to add one to the quotient as their solution. If students state that they will need 5r5 boxes, the teacher refers to the models to prompt students that a sixth box is needed for the remaining five pencils. If students state that they will need 5 more boxes since the remainder is 5, the teacher reminds students through guided questioning that the remainder of 5 represents 5 remaining pencils and only 1 more box is needed (i.e., “ add 1 to the quotient”). This is repeated with similar real-world problems.
- Instruction includes opportunities to complete real-world problems involving remainders
using pictorial representations to understand what the remainder is, including interpreting the remainder as a fraction.
- For example, the teacher displays and reads the following problem aloud: “ Karly, Juan, and Li share 4 cookies equally. How many cookies can each person eat?” The teacher uses drawings to represent the problem and guided questioning to encourage students to identify that Karly, Juan, and Li are able to eat 1 whole cookie but then must split the 4th cookie into thirds so that they can each eat 1 cookies, therefore 4 ÷ 3 = 1 . This is repeated with similar real-world problems.
- Instruction includes opportunities to complete real-world problems involving remainders using manipulatives. Students use hands-on models to interpret the remainder in situations in which they will need to “ add 1 to the quotient,” “ use only the remainder,” “ drop the remainder” or “ treat the remainder as a fraction.”
- For example, the teacher displays and reads the following problem aloud: “ There are 24 students in a class field trip. They plan to have 10 students in each van. How many vans will they need so that everyone can participate?” The teacher has students use counters or base-ten blocks to build a model of the problem and guided questioning to encourage students to identify that they will need to add 1 to the quotient as their solution. If students state that they will need 2r4 vans, the teacher refers to the models to prompt students that a third van is needed for the remaining four students. If students state that they will need 4 more vans since the remainder is 4, the teacher reminds students through guided questioning that the remainder of 4 represents 4 remaining students and only 1 more van is needed (i.e., “ add 1 to the quotient”). This is repeated with similar real-world problems.
Instructional Tasks
Instructional Task 1 (MTR.5.1)
Write an example of a word problem that will require the person solving the problem to “ Add 1 to the quotient” as their solution.Instructional Task 2 (MTR.5.1)
Write an example of a word problem that will require the person solving the problem to “ Use only the remainder” as their solution.Instructional Task 3 (MTR.5.1)
Write an example of a word problem that will require the person solving the problem to “ Drop the remainder” as their solution.Instructional Task 4 (MTR.7.1)
- If completing 6 benchmarks takes an entire work day, how much of Friday will be needed to complete the remaining benchmarks?
Instructional Items
Instructional Item 1
Sam has $50 to spend on video games. He buys one video game for $26. With the money he has left over, how many $9 games can Sam buy?- a. 2 games
- b. 3 games
- c. 5 games
- d. 6 games
Related Courses
Related Access Points
Related Resources
Formative Assessments
Lesson Plans
Original Student Tutorials
Perspectives Video: Teaching Idea
Problem-Solving Tasks
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.
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.
Students are asked to plan a playground for a new park within a given budget and area limit. They will analyze the best use of playground equipment using a data table of area requirements and cost. Students will convert units within a single measurement system, calculate the area of a rectangle, and perform addition/subtraction calculations involving money using decimal notation.
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
In this open-ended problem, students will work in teams to determine a procedure for ranking shoe closet styles for a person’s dream closet. Students will need to calculate the perimeter and cost for the closet, make decisions based on a table of data, and write a letter to the client providing evidence for their decisions.
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.
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
The student is asked to solve a multiplicative comparison word problem comparing 6 inches of string to 24 inches of string.
Students are given a multi-step word problem to solve that requires interpreting remainders.
Original Student Tutorials Mathematics - Grades K-5
Take a field trip while learning how to interpret remainders in multi-step division word problems.
This is part 1 of a four-part series of interactive tutorials. Click below to open the other tutorials in this series.
Learn how to interpret remainders in multi-step division problems related to a field trip in this interactive tutorial.
This tutorial is Part 2 in a four-part series about remainders. Click below to open the other tutorials in this series.
Learn how to interpret remainders in multi-step division problems in this interactive tutorial
This is the third tutorial in the Field Trip Frenzy Series about remainders. Click below to open the other tutorials in this series.
Learn when to write the remainder of a multi-step division process as a fraction or decimal in this interactive tutorial.
This is the final tutorial in the Field Trip Frenzy Series about remainders. Click below to open the other tutorials in this series.
Note: This tutorial extends beyond whole number quotients with whole number remainders to whole number quotients with fractional or decimal remainders.
Learn how to write multiplication equations based on multiplication comparisons and story problems in this magical math online tutorial!
Discover how multiplicative comparison problems, from outer space, can be solved using division in this online tutorial.
Launch into solving word problems that use multiplicative comparisons, drawings, and symbols in this space-themed interactive tutorial.
Learn how Maritza uses multiplication to stuff school supplies into backpacks with this interactive tutorial.
Student Resources
Original Student Tutorials
Learn how Maritza uses multiplication to stuff school supplies into backpacks with this interactive tutorial.
Type: Original Student Tutorial
Discover how multiplicative comparison problems, from outer space, can be solved using division in this online tutorial.
Type: Original Student Tutorial
Launch into solving word problems that use multiplicative comparisons, drawings, and symbols in this space-themed interactive tutorial.
Type: Original Student Tutorial
Learn when to write the remainder of a multi-step division process as a fraction or decimal in this interactive tutorial.
This is the final tutorial in the Field Trip Frenzy Series about remainders. Click below to open the other tutorials in this series.
Note: This tutorial extends beyond whole number quotients with whole number remainders to whole number quotients with fractional or decimal remainders.
Type: Original Student Tutorial
Learn how to interpret remainders in multi-step division problems in this interactive tutorial
This is the third tutorial in the Field Trip Frenzy Series about remainders. Click below to open the other tutorials in this series.
Type: Original Student Tutorial
Learn how to interpret remainders in multi-step division problems related to a field trip in this interactive tutorial.
This tutorial is Part 2 in a four-part series about remainders. Click below to open the other tutorials in this series.
Type: Original Student Tutorial
Take a field trip while learning how to interpret remainders in multi-step division word problems.
This is part 1 of a four-part series of interactive tutorials. Click below to open the other tutorials in this series.
Type: Original Student Tutorial
Learn how to write multiplication equations based on multiplication comparisons and story problems in this magical math online tutorial!
Type: Original Student Tutorial
Problem-Solving Tasks
The purpose of this task is to assess students’ understanding of multiplicative and additive reasoning. We would hope that students would be able to identify that Student A is just looking at how many feet are being added on, while Student B is comparing how much the snakes grew in comparison to how long they were to begin with.
Type: Problem-Solving Task
The purpose of this task is to foster a classroom discussion that will highlight the difference between multiplicative and additive reasoning. Some students will argue that they grew the same amount (an example of "additive thinking"). Students who are studying multiplicative comparison problems might argue that Jewel grew more since it grew more with respect to its original length (an example of "multiplicative thinking").
Type: Problem-Solving Task
The purpose of this task is for students to solve multi-step problems 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 can see that if the price level increases and people’s incomes do not increase, they aren’t able to purchase as many goods and services; in other words, their purchasing power decreases.
Type: Problem-Solving Task
The purpose of this task is to give students a better understanding of multiplicative comparison word problems with money.
Type: Problem-Solving Task
The purpose of the task is for students to solve a multi-step multiplication problem in a context that involves area. In addition, the numbers were chosen to determine if students have a common misconception related to multiplication. Since addition is both commutative and associative, we can reorder or regroup addends any way we like. Students often believe the same is true for multiplication.
Type: Problem-Solving Task
The purpose of this task is to generate a classroom discussion that helps students synthesize what they have learned about multiplication in previous grades. It builds on applying properties of operations as strategies to multiply and divide and interpreting a multiplication equation as a comparison.
Type: Problem-Solving Task
Tutorial
In this video tutorial from Khan Academy, you will get an introduction to the meaning of remainders.
Type: Tutorial
Parent Resources
Problem-Solving Tasks
The purpose of this task is to assess students’ understanding of multiplicative and additive reasoning. We would hope that students would be able to identify that Student A is just looking at how many feet are being added on, while Student B is comparing how much the snakes grew in comparison to how long they were to begin with.
Type: Problem-Solving Task
The purpose of this task is to foster a classroom discussion that will highlight the difference between multiplicative and additive reasoning. Some students will argue that they grew the same amount (an example of "additive thinking"). Students who are studying multiplicative comparison problems might argue that Jewel grew more since it grew more with respect to its original length (an example of "multiplicative thinking").
Type: Problem-Solving Task
The purpose of this task is for students to solve multi-step problems 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 can see that if the price level increases and people’s incomes do not increase, they aren’t able to purchase as many goods and services; in other words, their purchasing power decreases.
Type: Problem-Solving Task
The purpose of this task is to give students a better understanding of multiplicative comparison word problems with money.
Type: Problem-Solving Task
The purpose of the task is for students to solve a multi-step multiplication problem in a context that involves area. In addition, the numbers were chosen to determine if students have a common misconception related to multiplication. Since addition is both commutative and associative, we can reorder or regroup addends any way we like. Students often believe the same is true for multiplication.
Type: Problem-Solving Task
The purpose of this task is to generate a classroom discussion that helps students synthesize what they have learned about multiplication in previous grades. It builds on applying properties of operations as strategies to multiply and divide and interpreting a multiplication equation as a comparison.
Type: Problem-Solving Task
Tutorial
In this video tutorial from Khan Academy, you will get an introduction to the meaning of remainders.
Type: Tutorial
