# MA.7.NSO.2.3

Solve real-world problems involving any of the four operations with rational numbers.

### Clarifications

Clarification 1: Instruction includes using one or more operations to solve problems.
General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Number Sense and Operations
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Rational Number

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

Students solve real-world problems involving any of the four operations with positive multi-digit decimals or positive fractions, including mixed numbers in grade 6, with all rational numbers in grade 7, and with rational numbers including exponents and radicals in grade 8.
• This benchmark applies the procedural fluency skills of the previous benchmark to real-world problems (MTR.3.1).
• Students should develop fluency with and without the use of a calculator when performing operations with rational numbers.
• Instruction includes the use of technology to help emphasize the proper use of grouping symbols for order of operations.
• With the completion of operations with rational numbers in grade 7, students should have experience using technology with decimals and fractions as they occur in the real world. This experience will help to prepare students working with irrational numbers in grade 8.
• Open-ended tasks with real-world contexts (MTR.7.1) will allow students to practice multiple pathways for solutions as well as to make comparisons with their peers (MTR.4.1) to refine their problem-solving methods.
• Instruction includes support in vocabulary development as related to the context of the real-world problems when necessary.

### Common Misconceptions or Errors

• Students may incorrectly perform operations with the numbers in the problem based on what has recently been taught, rather than what is most appropriate for a solution. To overcome this misconception, have students estimate or predict solutions prior to solving and then compare those predictions to their actual solution to see if it is reasonable (MTR.6.1).
• Students may incorrectly oversimplify a problem by circling the numbers, underlining the question, boxing in key words, and eliminating context information that is needed for the solution. This process can cause students to not be able to comprehend the context or the situation (MTR.2.1, MTR.4.1, MTR.5.1, MTR.7.1).

### Strategies to Support Tiered Instruction

• Instruction includes the use of visual representations and manipulatives to represent the given situation and use the chosen representation to help find the solution.
• The teacher provides opportunities for students to comprehend the context or situation by engaging in questions like the ones below.
• What do you know from the problem?
• What is the problem asking you to find?
• Can you create a visual model to help you understand or see patterns in your problem?
• Teacher co-creates a graphic organizer with students to review operations with positive fractions and operations with integers to assist when applying operations with rational numbers.
• Instruction includes the use of a three-read strategy. Students read the problem three different times, each with a different purpose. Laminating these questions on a printed card for students to utilize as a resource in and out of the classroom would be helpful.
• First, read the problem with the purpose of answering the question: What is the problem, context, or story about?
• Second, read the problem with the purpose of answering the question: What are we trying to find out?
• Third, read the problem with the purpose of answering the question: What information is important in the problem?
• Instruction includes having students estimate or predict solutions prior to solving and then compare those predictions to their actual solution to see if it is reasonable.

All of the 7th grade homeroom classes collected recycling, with the top three classes splitting the grand prize of \$800 toward building their own gardens. Mr. Brogle’s class turned in 237 pounds of recycling, Mrs. Abiola’s class turned in 192 pounds and Mr. Wheeler’s class turned in 179 pounds. How should these top three divide the money so that each class gets the same fraction of the prize money as the fraction of recycling they collected?

Kari and Natalia went to the Fun Warehouse with \$20 each to spend. There is a \$3 entry fee each and the menu of activities is shown below. What are some possible combinations of activities Kari and Natalia can enjoy before they each run out of money?

Anjeanette is making cupcakes for her sister’s birthday. Among other ingredients, her recipe calls for 2 cups of flour, $\frac{\text{1}}{\text{2}}$ cup of butter and $\frac{\text{3}}{\text{4}}$ cup sugar in one batch. In the kitchen, she has 8 cups of flour, 2 cups of butter and 2 cups of sugar.
• Part A. How many batches of cupcakes can Anjeanette make?
• Part B. What should Anjeanette ask for if she wants to borrow from her neighbor to make one more batch?

### Instructional Items

Instructional Item 1
Kari and Natalia went to the Fun Warehouse with \$20 each to spend. They paid the \$3 entry fee each and then decided they would both play laser tag and mini-bowling. If they finished the day with playing 4 video games each, how much money will be left?

*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.
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1205020: M/J Accelerated Mathematics Grade 6 (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 2022, 2022 and beyond (current))
1205040: M/J Grade 7 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1204000: M/J Foundational Skills in Mathematics 6-8 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7812020: Access M/J Grade 7 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.7.NSO.2.AP.3: Using tools or models, solve real-world problems involving any of the four operations with rational numbers.

## Related Resources

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

## Formative Assessments

Trail Mix Munchies:

Students are asked to solve a word problem involving division of fractions.

Type: Formative Assessment

Monitoring Water Temperatures:

Students are asked to solve a word problem that involves finding the average of positive and negative decimal numbers.

Type: Formative Assessment

Using Estimation:

Students are asked to assess the reasonableness of answers using estimation strategies.

Type: Formative Assessment

Alexa’s Account:

Students are asked to assess the reasonableness of an answer using mental computation and estimation strategies.

Type: Formative Assessment

## Lesson Plans

Who's in the House? Part 3:

Students will use percentages and states' apportionment of representatives in the House to determine how much funding should be allocated to each state, in this integrated lesson plan.

Type: Lesson Plan

Who's in the House? Part 2:

Use data from U.S. Census Bureau that shows Apportionment Population, Resident Population, and Overseas Population for 2020 & 2010 Census to create and compare ratios in this integrated lesson plan.

Type: Lesson Plan

Students will work collaboratively to rank civic duties and responsibilities needed to keep a constitutional republic. They will utilize mathematical strategies to convert measurements of time as they calculate costs using the four operations with decimals and create an effective schedule for the ads within a budget in this model eliciting activity.

Type: Lesson Plan

Bias in Media:

Students will analyze the mathematical accuracy of fictitious political messages to explain bias in media.

Type: Lesson Plan

Budget Committee:

In this MEA, students will take on the role as a member of the Sunshine County Budget Committee. Members will collaborate to determine the optimal sales tax rate, use that rate to calculate how much money can be used for special projects, then decide which special projects to include in the budget proposal. Students will use percentages to problem-solve in context while considering citizen input and constraints on spending.

Type: Lesson Plan

Radioactive Dating: Half-Life & Geologic Time:

In this Model Eliciting Activity (MEA), students must use their knowledge of radioactive dating and geologic time to select an effective elemental isotope to be used to date three rare specimens. This decision requires an understanding of the concept of a half-life and the benefits and limitations of radiometric dating. Students must complete mathematical calculations involving equations and operations with fractions and percentages. Students completing this MEA must develop two essays that respond in a professional manner to a client in the scientific industry.

Type: Lesson Plan

How Fast Can One Travel on a Bicycle?:

Students investigate how the pedal and rear wheel gears affect the speed of a bicycle. A GeoGebra sketch is included that allows a simulation of the turning of the pedal and the rear wheel. A key goal is to provide an experience for the students to apply and integrate the key concepts in seventh-grade mathematics in a familiar context.

Type: Lesson Plan

Students will apply skills from the Geometry Domain to build graduation caps for themselves using heavyweight poster paper. They will also apply some basic mathematical skills to determine dimensions and to determine minimum cost. Some of the Geometric skills reinforced in Building Graduation Caps: Cooperative Assignment are finding area, applying the concept of similarity, and the application of the properties of parallelograms. Other skills also involved in this application are measuring, and statistical calculations, such as finding the mean and the range. In addition to the hands-on group project that takes place during the lesson, there is the Prerequisite Skills Assessment: Area that should be administered before the group activity and a home-learning activity. Building Graduation Caps: Individual Assignment is the home-learning assignment; it is designed to reinforce the skills learned in the group activity.

Type: Lesson Plan

NASA Salaries:

This is a NASA-themed, MEA (Model Eliciting Activity) lesson that challenges students to solve a real world open ended problem, while promoting collaboration through teamwork. This lesson asks each group of students to choose five positions and assign salaries to the positions with a given budget of \$500,000. The students' original decision (and "twist") will be based on information from the client's letter(s) and data set(s). Groups are to write a detailed letter to the client of the procedure used.

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

Family Restaurant:

In this Model Eliciting Activity, MEA, students will use unit rates and scoring systems to analyze and interpret data to recommend the best store from which a family restaurant should purchase its weekly non-frozen food items.

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 processes. 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 MEAs visit: https://www.cpalms.org/cpalms/mea.aspx

Type: Lesson Plan

Car Shopping:

In this Model Eliciting Activity, MEA, students will analyze and interpret data to recommend the best vehicle purchases for a school district. Students will work collaboratively to perform calculations that can be used to make comparisons and create composite scores for each vehicle.

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 processes. 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 MEAs visit: https://www.cpalms.org/cpalms/mea.aspx

Type: Lesson Plan

Sticks and Stones May Break My Bones:

In this Model Eliciting Activity, MEA, "Sticks and Stones May Break My Bones", teams of students work as forensic anthropologists and use equations to determine the height and gender of persons to whom a series of newly discovered bones may belong.

Type: Lesson Plan

Say Cheese!:

In this Model Eliciting Activity, MEA, students will apply their knowledge of rational numbers and order of operations to analyze and compare data to provide recommendations on the best camera to use in an introductory photography class.

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 processes. 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 MEAs visit: https://www.cpalms.org/cpalms/mea.aspx

Type: Lesson Plan

Along for the Ride!:

In this Model Eliciting Activity, MEA, students will use the order of operations to develop and apply a scoring system to evaluate different lawn tractors for a company. Students will justify their rankings using their analysis, calculations, and scoring system.

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 processes. 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 MEAs visit: https://www.cpalms.org/cpalms/mea.aspx

Type: Lesson Plan

Whirl Wind:

The Whirl Wind Corporation would like to install Wind Turbines in the Mojave Desert. The company produces various models of these turbines and is looking for help in selecting the best one for the job.

Type: Lesson Plan

Math in Mishaps:

Students will explore how percentages, proportions, and solving for unknowns are used in important jobs. This interactive activity will open their minds and address the question, "When is this ever used in real life?"

Type: Lesson Plan

## Perspectives Video: Teaching Idea

KROS Pacific Ocean Kayak Journey: Overview:

Why did the math teacher KROS the ocean? Because it made for a wonderful way to engage students and do something unique.

Related Resources:
KROS Pacific Ocean Kayak Journey: GPS Data Set [.XLSX]
KROS Pacific Ocean Kayak Journey: Path Visualization for Google Earth [.KML]

Type: Perspectives Video: Teaching Idea

Comparing Freezing Points:

In this task, students answer a question about the difference between two temperatures that are negative numbers.

Sharing Prize Money:

Students are asked to determine how to distribute prize money among three classes based on the contribution of each class.

## STEM Lessons - Model Eliciting Activity

Along for the Ride!:

In this Model Eliciting Activity, MEA, students will use the order of operations to develop and apply a scoring system to evaluate different lawn tractors for a company. Students will justify their rankings using their analysis, calculations, and scoring system.

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 processes. 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 MEAs visit: https://www.cpalms.org/cpalms/mea.aspx

Car Shopping:

In this Model Eliciting Activity, MEA, students will analyze and interpret data to recommend the best vehicle purchases for a school district. Students will work collaboratively to perform calculations that can be used to make comparisons and create composite scores for each vehicle.

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 processes. 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 MEAs visit: https://www.cpalms.org/cpalms/mea.aspx

Family Restaurant:

In this Model Eliciting Activity, MEA, students will use unit rates and scoring systems to analyze and interpret data to recommend the best store from which a family restaurant should purchase its weekly non-frozen food items.

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 processes. 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 MEAs visit: https://www.cpalms.org/cpalms/mea.aspx

NASA Salaries:

This is a NASA-themed, MEA (Model Eliciting Activity) lesson that challenges students to solve a real world open ended problem, while promoting collaboration through teamwork. This lesson asks each group of students to choose five positions and assign salaries to the positions with a given budget of \$500,000. The students' original decision (and "twist") will be based on information from the client's letter(s) and data set(s). Groups are to write a detailed letter to the client of the procedure used.

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

Radioactive Dating: Half-Life & Geologic Time:

In this Model Eliciting Activity (MEA), students must use their knowledge of radioactive dating and geologic time to select an effective elemental isotope to be used to date three rare specimens. This decision requires an understanding of the concept of a half-life and the benefits and limitations of radiometric dating. Students must complete mathematical calculations involving equations and operations with fractions and percentages. Students completing this MEA must develop two essays that respond in a professional manner to a client in the scientific industry.

Say Cheese!:

In this Model Eliciting Activity, MEA, students will apply their knowledge of rational numbers and order of operations to analyze and compare data to provide recommendations on the best camera to use in an introductory photography class.

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 processes. 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 MEAs visit: https://www.cpalms.org/cpalms/mea.aspx

Sticks and Stones May Break My Bones:

In this Model Eliciting Activity, MEA, "Sticks and Stones May Break My Bones", teams of students work as forensic anthropologists and use equations to determine the height and gender of persons to whom a series of newly discovered bones may belong.

Whirl Wind:

The Whirl Wind Corporation would like to install Wind Turbines in the Mojave Desert. The company produces various models of these turbines and is looking for help in selecting the best one for the job.

## MFAS Formative Assessments

Alexa’s Account:

Students are asked to assess the reasonableness of an answer using mental computation and estimation strategies.

Monitoring Water Temperatures:

Students are asked to solve a word problem that involves finding the average of positive and negative decimal numbers.

Trail Mix Munchies:

Students are asked to solve a word problem involving division of fractions.

Using Estimation:

Students are asked to assess the reasonableness of answers using estimation strategies.

## Student Resources

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

Comparing Freezing Points:

In this task, students answer a question about the difference between two temperatures that are negative numbers.

Sharing Prize Money:

Students are asked to determine how to distribute prize money among three classes based on the contribution of each class.

## Parent Resources

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

## Perspectives Video: Teaching Idea

KROS Pacific Ocean Kayak Journey: Overview:

Why did the math teacher KROS the ocean? Because it made for a wonderful way to engage students and do something unique.

Related Resources:
KROS Pacific Ocean Kayak Journey: GPS Data Set [.XLSX]
KROS Pacific Ocean Kayak Journey: Path Visualization for Google Earth [.KML]

Type: Perspectives Video: Teaching Idea