# MA.5.AR.1.1

Solve multi-step real-world problems involving any combination of the four operations with whole numbers, including problems in which remainders must be interpreted within the context.

### Clarifications

Clarification 1: 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.
General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Algebraic Reasoning
Status: State Board Approved

## Benchmark Instructional Guide

• Dividend
• Divisor
• Equation

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

• MA.6.NSO.2.3

### Purpose and Instructional Strategies

The purpose of this benchmark is for students to solve multistep word problems with whole numbers and whole-number answers involving any combination of the four operations. Work in this benchmark continues instruction from grade 4 where students interpreted remainders in division situations (MA.4.AR.1.1) (MTR.7.1), and prepares for solving multi-step word problems involving fractions and decimals in grade 6 (MA.6.NSO.2.3).
• To allow for an effective transition into algebraic concepts in grade 6 (MA.6.AR.1.1), it is important for students to have opportunities to connect mathematical statements and number sentences or equations.
• During instruction, teachers should allow students an opportunity to practice with word problems that require multiplication or division which can be solved by using drawings and equations, especially as the students are making sense of the context within the problem (MTR.5.1).
• Teachers should have students practice with representing an unknown number in a word problem with a variable by scaffolding from the use of only an unknown box.
• Offer word problems to students with the numbers covered up or replaced with symbols or icons and ensure to ask students to write the equation or the number sentence to show the problem type situation (MTR.6.1).
• Interpreting number pairs on a coordinate graph can provide students opportunities to solve multi-step real-world problems with the four operations (MA.5.GR.4.2).

### Common Misconceptions or Errors

• Students may 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: “There are 34 students in a class bowling tournament. They plan to have 3 students in each bowling lane. How many bowling lanes will they need so that everyone can participate?” the student response is “11 r 1 bowling lanes,” without any further understanding of how many bowling lanes are needed and how the students may be divided among the last 1 or 2 lanes. To assist students with this misconception, pose the question “What does the quotient mean?”

### Strategies to Support Tiered Instruction

• Instruction includes opportunities to engage in guided practice completing multi-step word problems with any combination of the four operations, including problems with remainders. Students use drawings and models to understand how to interpret the remainder in situations in which they will need to drop the remainder as their solution.
• For example, the teacher displays and reads the following problem aloud: “There are 58 fourth grade students and 45 fifth grade students going on a class field trip. They plan to have 20 students in each van. How many vans will they need so that everyone can participate?” Students use models or drawings to represent the problem and write an equation to represent the problem. The teacher uses 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 5r3 vans, the teacher refers to the models to prompt students that a sixth van is needed for the remaining three students. If students state that they will need 3 more vans since the remainder is 3, the teacher reminds students through guided questioning that the remainder of 3 represents 3 remaining students and only 1 more van is needed (i.e., “add 1 to the quotient”). This is repeated with similar multistep real-world problems, asking students to explain what the quotient means in problems involving remainders.
• Instruction includes opportunities to engage in practice with explicit instruction completing multi-step word problems with any combination of the four operations, including problems with remainders. Students use manipulatives to understand how to interpret the remainder in situations in which they will need to drop the remainder as their solution.
• For example, the teacher displays and reads the following problem aloud: “There are 18 red markers and 26 black markers on the art table. Ms. Williams is cleaning up and can put 10 markers in each box. How many boxes will she need so all the markers will be put into box?” The teacher uses manipulatives (e.g., base ten blocks) to represent the problem, having students write an equation to represent the problem. The teacher uses guided questioning to encourage students to identify that they will need to add 1 to the quotient as their solution. If students state that she will need 4r4 boxes, the teacher refers to the models to prompt students that a fifth box is needed for the remaining four markers. If students state that they will need 4 more boxes since the remainder is 4, the teacher reminds students through guided questioning that the remainder of 4 represents 4 remaining markers and only 1 more box is needed (i.e., “add 1 to the quotient”). This is repeated with similar multistep real-world problems, asking students to explain what the quotient means in problems involving remainders.

There are 128 girls in the Girl Scouts Troop 1653 and 154 girls in the Girl Scouts Troop 1764. Both Troops are going on a camping trip. Each bus can hold 36 girls. How many buses are needed to get all the girls to the camping site?

### Instructional Items

Instructional Item 1

A shoe store orders 17 cases each containing 142 pairs of sneakers and 12 cases each containing 89 pairs of sandals. How many more pairs of sneakers did the store order?

*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.AR.1.AP.1: Solve one- and two-step real-world problems involving any combination of the four operations with whole numbers. Explore problems in which remainders must be interpreted within the context.

## Related Resources

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

## Formative Assessment

Picking Strawberries:

Students are asked to solve a three-step word problem.

Type: Formative Assessment

## Lesson Plans

Students will investigate the relationship between mass and the force required to lift an object, as well as the impact of forces on the object’s motion. Using payload data and rocket specs, they will develop a plan with a budget to launch five modules of a new spaceport into low Earth orbit.

Type: Lesson Plan

Solving Multi-Step Word Problems With "Undecided" Colonists:

Students will be divided into groups and given a set of multi-step real-world problems to solve. The word problems will be specific to topics important to the neutral colonists during the American Revolution. The lesson will conclude with a class discussion of the word problems tying the math and civics together in this integrated lesson plan.

Type: Lesson Plan

Solving Multi-Step Word Problems with Loyalists and Patriots:

Students will be divided into groups and given a set of multi-step real-world problems to solve. The word problems will be specific to topics important to the Loyalists during the American Revolution. The lesson will  conclude with a class discussion of the word problems tying the math and civics together in this integrated lesson plan.

Type: Lesson Plan

Solving Real-World Problems P-3:

This integrated lesson includes students practicing multistep problem-solving while analyzing the mathematical thinking of others and justifying their results by explaining methods and processes.  Students will work in groups to solve a real-world problem, involving a citizenship context, by using a strategy of their choice.  As a class, the students will analyze the different ways the remainder can be interpreted based on different questions involving the same context.

Type: Lesson Plan

Solving Multi-Step Word Problems With Patriots and Loyalists:

Students will be divided into groups and given a set of multi-step real-world problems to solve. The word problems will be specific to topics important to the Patriots during the American Revolution. The lesson will conclude with a class discussion of the word problems tying the math and civics together in this integrated lesson plan.

Type: Lesson Plan

Solving Real-World Problems P-2:

This integrated lesson includes students solving multi-step real-world word problems using the four operations. Students will solve word problems involving citizen’s duties and responsibilities by using a strategy of their choice and analyzing the context to interpret remainders.

Type: Lesson Plan

Solving Real World Problems P-1:

Students will be exposed to several word problems involving citizens' duties and responsibilities, guiding them to reflect on what could happen if citizens do not fulfill their responsibilities. As the lesson progresses, the students will learn to combine addition, subtraction, multiplication, and division to solve multistep word problems in this integrated lesson plan.

Type: Lesson Plan

Gimme Two Steps!:

In this lesson, students will create representations for different multi-step word problems. One of these representations will be an expression with a variable.

Type: Lesson Plan

Blessings in a Bag!!:

In this MEA, the students will help a charitable organization select 5 snack items from a list to provide nutritious snacks for children in low-income communities.  Students will practice using the four operations to solve real-world problems and use decimal notation to make calculations involving money.  Additionally, they will be asked to compare multi-digit numbers to the thousandths.

Type: Lesson Plan

Getting the Top Mini-Fridge not a Small Deal:

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.

Type: Lesson Plan

Textbook Predicament:

In this Model Eliciting Activity, MEA, students multiply and compare information to provide the most appropriate textbook for a county.

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

Museum Dilemma:

In this MEA, students evaluate the contributions of various explorers to help a museum select the subject who provided the most impact on Western development for a new exhibit. Students will need to convert units to have the necessary information to help come up with a solution to the problem.

Type: Lesson Plan

One Step at a Time: Word Problems:

In this lesson, students will use the four operations to solve multi-step word problems composed of whole numbers. Students will be asked to estimate, write equations, decide if their answers are reasonable, and explain their decision. Several problems include explaining the meaning of the remainder in a division problem.

Type: Lesson Plan

Birthday Balloon Planner:

In this Model Eliciting Activity, MEA, students will develop a procedure for choosing a balloon company for a birthday party and rank them from best to worst.

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

Party Planners Wanted:

In this Model Eliciting Activity, MEA, students will work in collaborative groups to solve multistep problems with whole numbers and decimals by using different mathematical operations such as addition, subtraction, multiplication, and division. The students will be asked to assist a businessman who is planning a party for his employees. They will need to read several ads and decide which company offers the best deal in renting tables, chairs, and tablecloths for the client. They will need to take into consideration the number of guests attending the party and the budget allowed. A twist is added to the problem when the students are asked to consider an additional ad and the fact that the guest list is now slightly larger.

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

Cameras on Campus:

In this MEA, students will interpret data related to digital cameras to make a recommendation for a school to purchase for students to use.

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

Which Food Should I Feed My Cat?:

The students will convert fractions into decimals and multiply to determine the amount of food to feed their assigned cat. Using this amount, they will calculate (division of decimals) the number of servings and the price per serving of cat food. Students will work with three brands of food and enter the data into their charts. Finally, the students will analyze the data to determine the most affordable food.

Type: Lesson Plan

Diving deeper into division:

This lesson introduces students to dividing with 2 digit divisors.  Students are asked to apply strategies that they learned in dividing with 1 digit divisors such as partial quotients or breaking numbers apart using the distributive property.

Type: Lesson Plan

Wallpaper Woes Money Math: Lessons for Life:

Students hear a story about a middle-school student who wants to redecorate his bedroom. They measure the classroom wall dimensions, draw a scale model, and incorporate measurements for windows and doors to determine the area that could be covered by wallpaper. Students then hear more about the student's redecorating adventure and learn about expenses, budget constraints, and tradeoffs.

Type: Lesson Plan

New Coat of Paint:

In this Model Eliciting Activity, MEA, students will work in collaborative groups to solve multistep problems with whole numbers using the 4 operations. The students will be asked to assist a property owner, who is planning to repair his new property, in purchasing the right exterior paint. They will need to read a data table, rank the paints from highest to lowest, calculate the amount of gallons needed according to the surface area, and calculate the total cost of each paint. A twist is added to the problem when one of the paints is not available, but two others are added, and also the owner wants to paint the rectangular area of the dividing walls outside.

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

Rockin' Remainders:

This is a lesson designed to teach interpreting remainders in division based on the context of the word problem. Included with the lesson plan is a PowerPoint for direct instruction and word problems for small group or individual practice.

Type: Lesson Plan

Those Pesky Remainders:

This is a lesson to help students understand how to interpret the remainder in a division problem. Real world problems are presented in a PowerPoint so students may visualize situations and discover the four treatments of a remainder.

Type: Lesson Plan

## 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

Carnival Tickets:

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.

What is 23 ÷ 5?:

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.

Converting Fractions of a Unit into a Smaller Unit:

The purpose of this task is to help students gain a better understanding of fractions and the conversion of fractions into smaller units.

## STEM Lessons - Model Eliciting Activity

Birthday Balloon Planner:

In this Model Eliciting Activity, MEA, students will develop a procedure for choosing a balloon company for a birthday party and rank them from best to worst.

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

Blessings in a Bag!!:

In this MEA, the students will help a charitable organization select 5 snack items from a list to provide nutritious snacks for children in low-income communities.  Students will practice using the four operations to solve real-world problems and use decimal notation to make calculations involving money.  Additionally, they will be asked to compare multi-digit numbers to the thousandths.

Cameras on Campus:

In this MEA, students will interpret data related to digital cameras to make a recommendation for a school to purchase for students to use.

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

Getting the Top Mini-Fridge not a Small Deal:

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.

Museum Dilemma:

In this MEA, students evaluate the contributions of various explorers to help a museum select the subject who provided the most impact on Western development for a new exhibit. Students will need to convert units to have the necessary information to help come up with a solution to the problem.

New Coat of Paint:

In this Model Eliciting Activity, MEA, students will work in collaborative groups to solve multistep problems with whole numbers using the 4 operations. The students will be asked to assist a property owner, who is planning to repair his new property, in purchasing the right exterior paint. They will need to read a data table, rank the paints from highest to lowest, calculate the amount of gallons needed according to the surface area, and calculate the total cost of each paint. A twist is added to the problem when one of the paints is not available, but two others are added, and also the owner wants to paint the rectangular area of the dividing walls outside.

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

Party Planners Wanted:

In this Model Eliciting Activity, MEA, students will work in collaborative groups to solve multistep problems with whole numbers and decimals by using different mathematical operations such as addition, subtraction, multiplication, and division. The students will be asked to assist a businessman who is planning a party for his employees. They will need to read several ads and decide which company offers the best deal in renting tables, chairs, and tablecloths for the client. They will need to take into consideration the number of guests attending the party and the budget allowed. A twist is added to the problem when the students are asked to consider an additional ad and the fact that the guest list is now slightly larger.

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

Textbook Predicament:

In this Model Eliciting Activity, MEA, students multiply and compare information to provide the most appropriate textbook for a county.

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

## MFAS Formative Assessments

Picking Strawberries:

Students are asked to solve a three-step word problem.

## Student Resources

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

Carnival Tickets:

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.

What is 23 ÷ 5?:

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.

Converting Fractions of a Unit into a Smaller Unit:

The purpose of this task is to help students gain a better understanding of fractions and the conversion of fractions into smaller units.

## Parent Resources

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

Carnival Tickets:

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.