# MAFS.7.EE.2.3

Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making \$25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or \$2.50, for a new salary of \$27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

### Clarifications

Fluency Expectations or Examples of Culminating Standards

Students solve multistep problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. This work is the culmination of many progressions of learning in arithmetic, problem solving and mathematical practices.

Examples of Opportunities for In-Depth Focus

This is a major capstone standard for arithmetic and its applications.
General Information
Subject Area: Mathematics
Domain-Subdomain: Expressions & Equations
Cluster: Level 2: Basic Application of Skills & Concepts
Cluster: Solve real-life and mathematical problems using numerical and algebraic expressions and equations. (Major Cluster) -

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

Date of Last Rating: 02/14
Status: State Board Approved
Assessed: Yes
Test Item Specifications

• Assessment Limits :
Items should not use variables. Items should require two or more steps
• Calculator :

Yes

• Context :

Required

Sample Test Items (5)
• Test Item #: Sample Item 1
• Question:

Rolando is 13. In five years, his age will be the age of his sister Marisa.

How old will Marisa be in three years?

• Difficulty: N/A
• Type: EE: Equation Editor

• Test Item #: Sample Item 2
• Question: A set of pencils sells for \$1.75 and costs \$0.40 to make. Twenty percent of the profit (the difference between the purchase price and the amount it costs to make) from each set of pencils goes to a school.

If 500 sets are sold, what is the amount of money that will go to the school?

• Difficulty: N/A
• Type: EE: Equation Editor

• Test Item #: Sample Item 3
• Question: A bucket holds 243.5 ounces (oz) of water when full. The bucket loses 0.3 oz of water per second.

In how many seconds will the bucket be 40% full?

• Difficulty: N/A
• Type: EE: Equation Editor

• Test Item #: Sample Item 4
• Question: A plane is flying at 31,348 feet. It needs to rise to 36,000 feet in two stages.

In stage 1, it rises 5% of its initial altitude of 31,348 feet.

In stage 2, it rises at a rate of 140.3 feet per minute.

How many minutes does it take for the plane to rise during stage 2?

• Difficulty: N/A
• Type: EE: Equation Editor

• Test Item #: Sample Item 5
• Question:

The dimensions of a rectangular pool are 24.5 feet by 13 feet. The depth of the water is 4 feet. Each cubic foot contains 7.48 gallons of water.

How many gallons of water, to the nearest tenth, are needed to fill the pool to 80% capacity?

• Difficulty: N/A
• Type: EE: Equation Editor

## Related Courses

This benchmark is part of these courses.
1205040: M/J Grade 7 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1205050: M/J Grade 7 Accelerated Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 2022 (current), 2022 and beyond)
1204000: M/J Foundational Skills in Mathematics 6-8 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond)
1200410: Mathematics for College Success (Specifically in versions: 2014 - 2015, 2015 and beyond)
7812020: Access M/J Grade 7 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 and beyond)
7912115: Fundamental Explorations in Mathematics 2 (Specifically in versions: 2013 - 2015, 2015 - 2017 (course terminated))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MAFS.7.EE.2.AP.3a: Solve real-world, multi-step problems using positive and negative rational numbers (whole numbers, fractions and decimals).

## Related Resources

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

## Assessments

Sample 3 - Seventh Grade Math State Interim Assessment:

This is a State Interim Assessment for seventh grade.

Type: Assessment

Sample 4 - Seventh Grade Math State Interim Assessment:

This is a State Interim Assessment for seventh grade.

Type: Assessment

Sample 2 - Seventh Grade Math State Interim Assessment:

This is a State Interim Assessment for seventh grade.

Type: Assessment

Sample 1 - Seventh Grade Math State Interim Assessment:

This is a State Interim Assessment for seventh grade.

Type: Assessment

## Educational Games

Timed Algebra Quiz:

In this timed activity, students solve linear equations (one- and two-step) or quadratic equations of varying difficulty depending on the initial conditions they select. This activity allows students to practice solving equations while the activity records their score, so they can track their progress. 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 Game

Algebra Four:

In this activity, two students play a simulated game of Connect Four, but in order to place a piece on the board, they must correctly solve an algebraic equation. This activity allows students to practice solving equations of varying difficulty: one-step, two-step, or quadratic equations and using the distributive property if desired. 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 Game

## Formative Assessments

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

Gas Station Equations:

Students are asked to solve a multi-step problem involving percent.

Type: Formative Assessment

Reeling in Expressions:

Students are asked to solve a multi-step problem involving rational numbers.

Type: Formative Assessment

Discount and Tax:

Students are asked to solve a multi-step problem involving percent.

Type: Formative Assessment

## Lesson Plans

Gather Data For Distribution by Programming an App:

This lesson allow students to gather, calculate, and plot data using both computer code and mathematical equations. In this lesson students will create a pedometer app to demonstrate the understanding of algorithms, components (such as buttons, textboxes, sensors, etc.), and If/Then statements. This lesson uses algebraic equations and random data to access the needed components to store data in a spreadsheet.

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

3-D Printers: Cost Effective for the Quality?:

Students will compare the cost of pre-made solar car kits to cars made from a 3-D printer. In the second part of the activity, students will research other available 3-D printers and determine what attributes are important to consider. There is also an optional solar panel car race for day 3.

Type: Lesson Plan

Steps to Solving Equations:

This lesson unit is intended to help you assess how well students are able to form and solve linear equations involving factorizing and using the distributive law. In particular, this unit aims to help you identify and assist students who have difficulties in using variables to represent quantities in a real-world or mathematical problem and solving word problems leading to equations of the form px + q = r and p(x + q) = r.

Type: Lesson Plan

Percent of Change:

Students will investigate percent of change in real-world situations and will differentiate between an increase or a decrease. The students will use a formula to find the percent of change.

Type: Lesson Plan

Car Shopping:

This MEA requires students to formulate a comparison-based solution to a problem involving finding the best decision on purchasing official vehicles for school district considering different aspects. Students are provided the context of the problem, a request letter from a client asking them to provide a recommendation, and data relevant to the situation. Students utilize the data to create a defensible model solution to present to the client.

Type: Lesson Plan

In Whose Best Interest is Interest?:

The students will explore real world examples of interest rates. Students will explore loan rates, CD rates and compare benefits of different rates versus different terms of loans. Students will use the formula for simple interest.

Type: Lesson Plan

It's All About Properties of Equality:

Students will apply properties of equality to solve multi-step equations with one variable. Students will complete a gallery walk as formative assessment, to determine students' understanding of properties of operations and equality when applied to equations. Equations increase in difficulty as the lesson progresses. Students complete an error analysis toward the end of the lesson. This lesson includes a powerpoint presentation.

Type: Lesson Plan

Pricing The Twelve Days of Christmas:

Students will discover how much the items in the classic song, "The Twelve Days of Christmas," would cost in the current year; and then they will update the list for modern times.

Type: Lesson Plan

Say Cheese!:

This resource provides a Model-Eliciting Activity where students will analyze a real-world scenario to solve a client's problem and provide the best possible solution based on a logically justified process. The students will consider a request from Simple Photography Classes to evaluate several digital cameras and help them decide which one they should purchase.

Type: Lesson Plan

Along for the Ride!:

This resource provides a Model-Eliciting Activity where students will analyze a real-world scenario to solve a client's problem and provide the best possible solution based on a logically justified process. The students will consider a request from Cut It Out Section of the Building and Grounds Maintenance Department of a school district to evaluate several lawn tractor models and help them decide which unit they should purchase.

Type: Lesson Plan

Travel Troubles:

This activity engages the students into time scheduling, budgeting, and decision making to maximize time efficiency.

Type: Lesson Plan

Batteries Included:

This resource provides a Model-Eliciting Activity where students will analyze a real-world scenario to solve a client's problem and provide the best possible solution based on a logically justified process. The students will consider a request from E-Z Go Taxi Cab Service to evaluate several batteries and help them decide which battery they should purchase.

Type: Lesson Plan

Where in the world?:

This resource provides a Model-Eliciting Activity where students will analyze a real-world scenario to solve a client's problem and provide the best possible solution based on a logically justified process. The students will consider a request from Always On Time Delivery Service to evaluate several GPS units and help them decide which unit they should purchase.

Type: Lesson Plan

Math in Mishaps:

Students will explore how percents, 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

Which Food Should I Feed My Cat?:

The students will convert fractions into decimals and determine with multiplication the amount of food per serving for their assigned cat. Using this amount, they will calculate (division of decimals) the number of servings and the price per serving of cat food. Student will work with three types of food and enter the data into the chart. Finally, the students will analyze the data to determine the most cost-effective food.

Type: Lesson Plan

Here's a tip!:

Students will solve problems involving sales tax and tips; students will apply the properties of operations with numbers in decimal, percent and fraction form. Students will convert between numbers in any form as appropriate.

Type: Lesson Plan

Uncle Henry's Dilemma:

Uncle Henry's Dilemma is a problem solving lesson to determine the global location for the reading of Uncle Henry's will. The students will interpret data sets which include temperature, rainfall, air pollution, travel cost, flight times and health issues to rank five global locations for Uncle Henry's relatives to travel to for the reading of his will. This is an engaging, fun-filled MEA lesson with twists and turns throughout. Students will learn how this procedure of selecting locations can be applied to everyday decisions by the government, a business, a family, or individuals.

Type: Lesson Plan

Bargain Town, USA:

In this lesson, students will participate in a simulated real-world exploration of the relationship between fractions, decimals, and percents, by converting number forms and calculating discounted prices.

Type: Lesson Plan

Let's Go Shopping: Calculating Percents:

In this lesson, students will participate in a simulated shopping experience where they choose items they would like to purchase from local sale advertisements. The students will be able to apply the percent formula and the percent of change formula to real world financial situations. Students will learn how to calculate percent discounts, their percent of savings, and tax. The students will analyze, compare, draw conclusions and explain in writing why specific types of discounts are the most advantageous given specific situations.

Type: Lesson Plan

Understanding Decimals:

These games and activities help students investigate the relationship between fractions and decimals, focusing on equivalence.

Type: Lesson Plan

## Original Student Tutorial

Balancing the Machine:

Use models to solve balance problems on a space station in this interactive, math and science tutorial.

Type: Original Student Tutorial

Maximizing Area: Gold Rush:

Before the lesson, students attempt the Gold Rush task individually. You then look at their responses and formulate questions for students to think about as they review their work. At the start of the lesson, students reflect on their individual responses and use the questions posed to think of ways to improve their work. Next, students work collaboratively in small groups to produce, in the form of a poster, a better solution to the Gold Rush task than they did individually. In a whole-class discussion students compare and evaluate the different methods they used. Working in small groups, students analyze sample responses to the Gold Rush task, then, in a whole-class discussion, review the methods they have seen. Finally, students reflect on their work.

Anna in D.C.:

The purpose of this task is to give students an opportunity to solve a challenging multistep percentage problem that can be approached in several different ways. Students are asked to find the cost of a meal before tax and tip when given the total cost of the meal. The task can illustrate multiple standards depending on the prior knowledge of the students and the approach used to solve the problem.

Discounted Books:

This purpose of this task is to help students see two different ways to look at percentages both as a decrease and an increase of an original amount. In addition, students have to turn a verbal description of several operations into mathematical symbols. This requires converting simple percentages to decimals as well as identifying equivalent expressions without variables.

Guess My Number:

This problem asks the students to represent a sequence of operations using an expression and then to write and solve simple equations. The problem is posed as a game and allows the students to visualize mathematical operations. It would make sense to actually play a similar game in pairs first and then ask the students to record the operations to figure out each other's numbers.

Shrinking:

Students are asked to determine the change in height in inches when given a constant rate of change in centimeters. The answer is rounded to the nearest half inch.

Gotham City Taxis:

The purpose of this task is to give students an opportunity to solve a multi-step ratio problem that can be approached in many ways. This can be done by making a table, which helps illustrate the pattern of taxi rates for different distances traveled and with a little persistence leads to a solution which uses arithmetic. It is also possible to calculate a unit rate (dollars per mile) and use this to find the distance directly without making a table.

## Tutorials

Solving Percentage Problems with Linear Equations:

Many real world problems involve involve percentages. This lecture shows how algebra is used in solving problems of percent change and profit-and-loss.

Type: Tutorial

Age word problem :

Students will learn how to set up and solve an age word problem.

Type: Tutorial

Adding and Subtracting Numbers in Different Formats:

In this example, you will work with three numbers in different formats: a percent, a decimal, and a mixed number.

Type: Tutorial

Percent Word Problem Example 1:

We're putting a little algebra to work to find the full price when you know the discount price in this percent word problem.

Type: Tutorial

## Video/Audio/Animation

Averages:

This Khan Academy video tutorial introduces averages and algebra problems involving averages.

Type: Video/Audio/Animation

## STEM Lessons - Model Eliciting Activity

3-D Printers: Cost Effective for the Quality?:

Students will compare the cost of pre-made solar car kits to cars made from a 3-D printer. In the second part of the activity, students will research other available 3-D printers and determine what attributes are important to consider. There is also an optional solar panel car race for day 3.

Along for the Ride!:

This resource provides a Model-Eliciting Activity where students will analyze a real-world scenario to solve a client's problem and provide the best possible solution based on a logically justified process. The students will consider a request from Cut It Out Section of the Building and Grounds Maintenance Department of a school district to evaluate several lawn tractor models and help them decide which unit they should purchase.

Batteries Included:

This resource provides a Model-Eliciting Activity where students will analyze a real-world scenario to solve a client's problem and provide the best possible solution based on a logically justified process. The students will consider a request from E-Z Go Taxi Cab Service to evaluate several batteries and help them decide which battery they should purchase.

Car Shopping:

This MEA requires students to formulate a comparison-based solution to a problem involving finding the best decision on purchasing official vehicles for school district considering different aspects. Students are provided the context of the problem, a request letter from a client asking them to provide a recommendation, and data relevant to the situation. Students utilize the data to create a defensible model solution to present to the client.

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!:

This resource provides a Model-Eliciting Activity where students will analyze a real-world scenario to solve a client's problem and provide the best possible solution based on a logically justified process. The students will consider a request from Simple Photography Classes to evaluate several digital cameras and help them decide which one they should purchase.

Travel Troubles:

This activity engages the students into time scheduling, budgeting, and decision making to maximize time efficiency.

Uncle Henry's Dilemma:

Uncle Henry's Dilemma is a problem solving lesson to determine the global location for the reading of Uncle Henry's will. The students will interpret data sets which include temperature, rainfall, air pollution, travel cost, flight times and health issues to rank five global locations for Uncle Henry's relatives to travel to for the reading of his will. This is an engaging, fun-filled MEA lesson with twists and turns throughout. Students will learn how this procedure of selecting locations can be applied to everyday decisions by the government, a business, a family, or individuals.

Where in the world?:

This resource provides a Model-Eliciting Activity where students will analyze a real-world scenario to solve a client's problem and provide the best possible solution based on a logically justified process. The students will consider a request from Always On Time Delivery Service to evaluate several GPS units and help them decide which unit they should purchase.

## MFAS Formative Assessments

Alexa’s Account:

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

Discount and Tax:

Students are asked to solve a multi-step problem involving percent.

Gas Station Equations:

Students are asked to solve a multi-step problem involving percent.

Reeling in Expressions:

Students are asked to solve a multi-step problem involving rational numbers.

Using Estimation:

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

## Original Student Tutorials Science - Grades K-8

Balancing the Machine:

Use models to solve balance problems on a space station in this interactive, math and science tutorial.

## Student Resources

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

## Original Student Tutorial

Balancing the Machine:

Use models to solve balance problems on a space station in this interactive, math and science tutorial.

Type: Original Student Tutorial

## Educational Games

Timed Algebra Quiz:

In this timed activity, students solve linear equations (one- and two-step) or quadratic equations of varying difficulty depending on the initial conditions they select. This activity allows students to practice solving equations while the activity records their score, so they can track their progress. 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 Game

Algebra Four:

In this activity, two students play a simulated game of Connect Four, but in order to place a piece on the board, they must correctly solve an algebraic equation. This activity allows students to practice solving equations of varying difficulty: one-step, two-step, or quadratic equations and using the distributive property if desired. 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 Game

Anna in D.C.:

The purpose of this task is to give students an opportunity to solve a challenging multistep percentage problem that can be approached in several different ways. Students are asked to find the cost of a meal before tax and tip when given the total cost of the meal. The task can illustrate multiple standards depending on the prior knowledge of the students and the approach used to solve the problem.

Discounted Books:

This purpose of this task is to help students see two different ways to look at percentages both as a decrease and an increase of an original amount. In addition, students have to turn a verbal description of several operations into mathematical symbols. This requires converting simple percentages to decimals as well as identifying equivalent expressions without variables.

Guess My Number:

This problem asks the students to represent a sequence of operations using an expression and then to write and solve simple equations. The problem is posed as a game and allows the students to visualize mathematical operations. It would make sense to actually play a similar game in pairs first and then ask the students to record the operations to figure out each other's numbers.

Shrinking:

Students are asked to determine the change in height in inches when given a constant rate of change in centimeters. The answer is rounded to the nearest half inch.

Gotham City Taxis:

The purpose of this task is to give students an opportunity to solve a multi-step ratio problem that can be approached in many ways. This can be done by making a table, which helps illustrate the pattern of taxi rates for different distances traveled and with a little persistence leads to a solution which uses arithmetic. It is also possible to calculate a unit rate (dollars per mile) and use this to find the distance directly without making a table.

## Tutorials

Solving Percentage Problems with Linear Equations:

Many real world problems involve involve percentages. This lecture shows how algebra is used in solving problems of percent change and profit-and-loss.

Type: Tutorial

Age word problem :

Students will learn how to set up and solve an age word problem.

Type: Tutorial

Adding and Subtracting Numbers in Different Formats:

In this example, you will work with three numbers in different formats: a percent, a decimal, and a mixed number.

Type: Tutorial

Percent Word Problem Example 1:

We're putting a little algebra to work to find the full price when you know the discount price in this percent word problem.

Type: Tutorial

## Video/Audio/Animation

Averages:

This Khan Academy video tutorial introduces averages and algebra problems involving averages.

Type: Video/Audio/Animation

## Parent Resources

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

Anna in D.C.:

The purpose of this task is to give students an opportunity to solve a challenging multistep percentage problem that can be approached in several different ways. Students are asked to find the cost of a meal before tax and tip when given the total cost of the meal. The task can illustrate multiple standards depending on the prior knowledge of the students and the approach used to solve the problem.

Discounted Books:

This purpose of this task is to help students see two different ways to look at percentages both as a decrease and an increase of an original amount. In addition, students have to turn a verbal description of several operations into mathematical symbols. This requires converting simple percentages to decimals as well as identifying equivalent expressions without variables.

Guess My Number:

This problem asks the students to represent a sequence of operations using an expression and then to write and solve simple equations. The problem is posed as a game and allows the students to visualize mathematical operations. It would make sense to actually play a similar game in pairs first and then ask the students to record the operations to figure out each other's numbers.

Shrinking:

Students are asked to determine the change in height in inches when given a constant rate of change in centimeters. The answer is rounded to the nearest half inch.

Gotham City Taxis:

The purpose of this task is to give students an opportunity to solve a multi-step ratio problem that can be approached in many ways. This can be done by making a table, which helps illustrate the pattern of taxi rates for different distances traveled and with a little persistence leads to a solution which uses arithmetic. It is also possible to calculate a unit rate (dollars per mile) and use this to find the distance directly without making a table.