# MA.6.AR.3.4

Apply ratio relationships to solve mathematical and real-world problems involving percentages using the relationship between two quantities.

### Examples

Gerald is trying to gain muscle and needs to consume more protein every day. If he has a protein shake that contain 32 grams and the entire shake is 340 grams, what percentage of the entire shake is protein? What is the ratio between grams of protein and grams of non-protein?

### Clarifications

Clarification 1: Instruction includes the comparison of  to  in order to determine the percent, the part or the whole.
General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Algebraic Reasoning
Status: State Board Approved

## Benchmark Instructional Guide

• Rate

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In elementary grades, students worked with creating equivalent fractions, as well as operations with whole numbers and fractions. In grade 6, students use ratio relationships to solve problems involving percentages. In grade 7, students will solve multi-step problems and proportional relationships involving percentages.
• Students should understand that percent (%) represents a part to whole relationship.
• Instruction includes the connection to ratio relationships to determine the part, the whole or the percentage (MTR.5.1).
• For example, when determining the how much 40% is of 24, students should compare the ratio $\frac{\text{x}}{\text{24}}$ to the ratio $\frac{\text{40}}{\text{100}}$.
• Instruction does not include the use of proportions or cross multiplication to solve problems involving percentages.
• Instruction includes the use of models to represent percentages such as bar models, number lines or ratio tables to help visually represent the relationship (MTR.2.1).
• Bar Models
70% $o$$f$ 120 = 84

• Number Lines
70% $o$$f$ 120 = 84

• Ratio Tables
70% $o$$f$ 120 = 84

### Common Misconceptions or Errors

• Students may not understand the difference between an additive relationship and a multiplicative relationship.
• Students may incorrectly set up ratios because of a misunderstanding of the part and the whole addressed in the situation.
• Students may not recognize simplified forms of ratios in order to find equivalent ratios to determine the percentage, the whole or the part.

### Strategies to Support Tiered Instruction

• Instruction includes finding an equivalent unit rate (either part or whole) then multiplying to find the desired equivalent ratio.
• For example: Steve wants to determine how much a 15% tip is if the bill is \$80.00.
Use a visual representation to show 80 represents 100%

Divide the ratio by 80

Multiply the resulting ratio by 15

Using the equivalent ratios, 12 is 15% of 80, so the tip is \$12.00.

Instructional Task 1 (MTR.4.1, MTR.6.1, MTR.7.1)
Carlos predicts that his math homework will take him 60 of the total of 75 minutes he has available for homework tonight.
• Part A. At this rate, how many minutes would Carlos spend on math homework out of a total of 100 available minutes?
• Part B. What percentage of the available homework time does Carlos predict he will spend doing math? Explain how the answer to this question is related to the answer in Part A.

### Instructional Items

Instructional Item 1
Find the percent equivalent to $\frac{\text{60}}{\text{115}}$. Round to the nearest tenth percent.

Instructional Item 2
15% of 80 is what value?

Instructional Item 3
Sami is keeping track of the amount of salt she consumes each day. According to the label on her macaroni and cheese box, one serving contains 470 mg of sodium (salt). If 470 mg is 20% of the recommended daily amount, how many milligrams of sodium are recommended for the whole day?

*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.
1205010: M/J Grade 6 Mathematics (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))
1204000: M/J Foundational Skills in Mathematics 6-8 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7812015: Access M/J Grade 6 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.6.AR.3.AP.4: Calculate a percentage of quantity as rate per 100 using models (e.g., percent bars or 10 × 10 grids).

## Related Resources

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

## Educational Game

Estimator Quiz:

In this activity, students are quizzed on their ability to estimate sums, products, and percentages. The student can adjust the difficulty of the problems and how close they have to be to the actual answer. This activity allows students to practice estimating addition, multiplication, or percentages of large numbers. 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

Finding the Whole:

Students are asked to find the whole given a part and a percent.

Type: Formative Assessment

Homework Time:

Students are asked to convert a given rate to an equivalent rate out of 100.

Type: Formative Assessment

## Lesson Plans

Real Life Tax, Tip, and Discount!:

Students calculate the tax, tip, and discount in real-world situations.

Type: Lesson Plan

All “Tired” Up:

In this Model Eliciting Activity, MEA, students will utilize mathematical computation skills involving percentages and critical thinking skills to select the best tire deals advertised.

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

Fast Food Frenzy:

In this activity, students will engage critically with nutritional information and macronutrient content of several fast food meals. This is an MEA that requires students to build on prior knowledge of nutrition and working with percentages.

Type: Lesson Plan

In this Model Eliciting Activity, MEA, students will create a procedure for ranking high school basketball players. Students are given statistics for each player and are asked to recommend the best player to play for an all-star team after determining the free throw, three-point, and field goal percentages. Students write about the procedure used to make their decisions. In a twist, students are given additional data to determine the mean points per game.

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

Best Day Care Center for William:

This MEA requires students to formulate a comparison-based solution to a problem involving choosing the BEST daycare based upon safety, playground equipment, meals, teacher to student ratio, cost, holiday availability and toilet training availability. 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. Students will receive practice on calculating a discount, finding the sum of the discounts, working with ratios and ranking day cares based on the data given.

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

Students will learn how to calculate markup, markdown, percent increase, and percent decrease. Using sales "ad" inserts from the internet, newspapers, and store flyers, students will understand how these concepts apply to real-world situations.

Type: Lesson Plan

Running and Rising:

In this lesson students will graph and compare two proportional relationships from different representations in contextual problems and be introduced to the constant of proportionality as the unit rate.

Type: Lesson Plan

Pricing Twelve Days of Celebration:

Students will discover how much items would cost if they were to give gifts for 12 days. They will learn how to calculate and add sales tax to find a total.

Type: Lesson Plan

Have you ever heard students ask the question, "Why do I have to learn this?" This lesson answers that question because it requires the students to apply their knowledge in real world scenarios but does not teach a basic conceptual understanding of percentages. The teacher may use the whole lesson or select specific problems.

Type: Lesson Plan

The Dazzling Painting Co.:

Students will read a letter from a painting company from New York who are planning to expand to Florida. They need help deciding on which paint sprayers to purchase. Students will use their understanding of rate and percentages to analyze data and make suggestions.

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

Makeover, Home Edition Part II:

This is the second part of the lesson, "Makeover, Home Edition." This lesson will continue focusing on unit prices, but also incorporates area and volume. Part I (Makeover, Home Edition #48705) is based on creating backyard dimensions for fencing. Part III (Makeover, Home Edition #49025) will deal with creating a scale drawing of this backyard. Part IV (Makeover, Home Edition Final #49090) will focus on inserting a window and painting walls inside the house.

Type: Lesson Plan

This MEA asks the students to decide which company would be the “best and the worst” to use to purchase scuba diving masks for Tino’s Scuba Diving School to provide to their diving certification students. Furthermore, the students are asked to suggest which type of scuba diving masks should be purchased in term of multiple panes – single pane mask, double pane mask, full face mask, skirt color, fit, durability, and price. Students must provide a "top choice" scuba diving mask to the company owner and explain how they arrived at their solution.

Type: Lesson Plan

Makeover, Home Edition Part I:

This is the first part of the lesson, "Makeover Home Edition." This lesson is designed to increase student engagement. Students must think critically about fencing in their new "dream" backyard by calculating the total fencing needed. They will choose the most cost-effective method of purchasing their fencing by comparing unit rates mathematically and graphically. CPALMS Lesson Part II (#48967) will concentrate on inserting a pool and patio into this backyard. Part III (#49025) will include the creation of a scale drawing of this backyard. Part IV (#49090) focuses on inserting a window and painting walls inside the house.

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

## Perspectives Video: Professional/Enthusiasts

Mathematical Thinking for Ceramic 3D Printing:

In this video, Matthew Lawrence describes how mathematical thinking is important for 3D printing with ceramic materials.

Type: Perspectives Video: Professional/Enthusiast

Coffee Mathematics: Ratios and Total Dissolvable Solids:

Math - the secret ingredient for an excellent cup of coffee!

Type: Perspectives Video: Professional/Enthusiast

Kendall's Vase - Tax:

This problem asks the student to find a 3% sales tax on a vase valued at \$450.

Security Camera:

Students are asked to determine the percent of the area of a store covered by a security camera. Then, students are asked to determine the "best" place to position the camera and support their answer.

Shirt Sale:

Use the information provided to find out the original price of Selina's shirt. There are several different ways to reason through this problem; two approaches are shown.

Dana's House:

Use the information provided to find out what percentage of Dana's lot won't be covered by the house.

Overlapping Squares:

This problem provides an interesting geometric context to work on the notion of percent. Two different methods for analyzing the geometry are provided: the first places the two squares next to one another and then moves one so that they overlap. The second solution sets up an equation to find the overlap in terms of given information which reflects the mathematical ideas reason about and solve one-variable equations and inequalities.

## Teaching Idea

Calculating Sharks-SeaWorld Classroom Activity:

• Given data about sharks and the amount of food they eat, students will be able to solve for the unknown in percentage problems.
• Given information about a shark's growth, students will be able to graph coordinates and interpret a linear graph.
• Given the conversion factor, students will be able to convert from metric to English units.

Type: Teaching Idea

## Tutorials

Finding a Percent:

In the video, we find the percent when given the part and the whole.

Type: Tutorial

Percent of a Whole Number:

This video demonstrates how to find percent of a whole number.

Type: Tutorial

Percent Word Problem:

You're asked to find the whole when given the part and the percent.

Type: Tutorial

Percent Word Problem:

Use long division to find the percent in this tutorial.

Type: Tutorial

Converting Decimals to Percents:

This video demonstrates how to write a decimal as a percent.

Type: Tutorial

The Meaning of Percent:

This video deals with what percent really means by looking at a 10 by 10 grid.

Type: Tutorial

The Meaning of Percent over 100:

This video demonstrates a visual model of a percent greater than 100.

Type: Tutorial

## Video/Audio/Animation

Understanding Percentages:

Percentages are one method of describing a fraction of a quantity. the percent is the numerator of a fraction whose denominator is understood to be one-hundred.

Type: Video/Audio/Animation

## STEM Lessons - Model Eliciting Activity

All “Tired” Up:

In this Model Eliciting Activity, MEA, students will utilize mathematical computation skills involving percentages and critical thinking skills to select the best tire deals advertised.

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 Model Eliciting Activity, MEA, students will create a procedure for ranking high school basketball players. Students are given statistics for each player and are asked to recommend the best player to play for an all-star team after determining the free throw, three-point, and field goal percentages. Students write about the procedure used to make their decisions. In a twist, students are given additional data to determine the mean points per game.

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

Best Day Care Center for William:

This MEA requires students to formulate a comparison-based solution to a problem involving choosing the BEST daycare based upon safety, playground equipment, meals, teacher to student ratio, cost, holiday availability and toilet training availability. 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. Students will receive practice on calculating a discount, finding the sum of the discounts, working with ratios and ranking day cares based on the data given.

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

Fast Food Frenzy:

In this activity, students will engage critically with nutritional information and macronutrient content of several fast food meals. This is an MEA that requires students to build on prior knowledge of nutrition and working with percentages.

This MEA asks the students to decide which company would be the “best and the worst” to use to purchase scuba diving masks for Tino’s Scuba Diving School to provide to their diving certification students. Furthermore, the students are asked to suggest which type of scuba diving masks should be purchased in term of multiple panes – single pane mask, double pane mask, full face mask, skirt color, fit, durability, and price. Students must provide a "top choice" scuba diving mask to the company owner and explain how they arrived at their solution.

The Dazzling Painting Co.:

Students will read a letter from a painting company from New York who are planning to expand to Florida. They need help deciding on which paint sprayers to purchase. Students will use their understanding of rate and percentages to analyze data and make suggestions.

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

## MFAS Formative Assessments

Finding the Whole:

Students are asked to find the whole given a part and a percent.

Homework Time:

Students are asked to convert a given rate to an equivalent rate out of 100.

## Student Resources

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

## Educational Game

Estimator Quiz:

In this activity, students are quizzed on their ability to estimate sums, products, and percentages. The student can adjust the difficulty of the problems and how close they have to be to the actual answer. This activity allows students to practice estimating addition, multiplication, or percentages of large numbers. 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

Kendall's Vase - Tax:

This problem asks the student to find a 3% sales tax on a vase valued at \$450.

Security Camera:

Students are asked to determine the percent of the area of a store covered by a security camera. Then, students are asked to determine the "best" place to position the camera and support their answer.

Shirt Sale:

Use the information provided to find out the original price of Selina's shirt. There are several different ways to reason through this problem; two approaches are shown.

Dana's House:

Use the information provided to find out what percentage of Dana's lot won't be covered by the house.

Overlapping Squares:

This problem provides an interesting geometric context to work on the notion of percent. Two different methods for analyzing the geometry are provided: the first places the two squares next to one another and then moves one so that they overlap. The second solution sets up an equation to find the overlap in terms of given information which reflects the mathematical ideas reason about and solve one-variable equations and inequalities.

## Tutorials

Finding a Percent:

In the video, we find the percent when given the part and the whole.

Type: Tutorial

Percent of a Whole Number:

This video demonstrates how to find percent of a whole number.

Type: Tutorial

Percent Word Problem:

You're asked to find the whole when given the part and the percent.

Type: Tutorial

Percent Word Problem:

Use long division to find the percent in this tutorial.

Type: Tutorial

Converting Decimals to Percents:

This video demonstrates how to write a decimal as a percent.

Type: Tutorial

The Meaning of Percent:

This video deals with what percent really means by looking at a 10 by 10 grid.

Type: Tutorial

The Meaning of Percent over 100:

This video demonstrates a visual model of a percent greater than 100.

Type: Tutorial

## Video/Audio/Animation

Understanding Percentages:

Percentages are one method of describing a fraction of a quantity. the percent is the numerator of a fraction whose denominator is understood to be one-hundred.

Type: Video/Audio/Animation

## Parent Resources

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

Kendall's Vase - Tax:

This problem asks the student to find a 3% sales tax on a vase valued at \$450.

Security Camera:

Students are asked to determine the percent of the area of a store covered by a security camera. Then, students are asked to determine the "best" place to position the camera and support their answer.

Shirt Sale:

Use the information provided to find out the original price of Selina's shirt. There are several different ways to reason through this problem; two approaches are shown.