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Given a real-world context, determine a rate for a ratio of quantities with different units. Calculate and interpret the corresponding unit rate.
Standard #: MA.6.AR.3.2
Standard Information
Standard Examples
Tamika can read 500 words in 3 minutes. Her reading rate can be described as begin mathsize 12px style fraction numerator 500 space w o r d s over denominator 3 space m i n u t e s end fraction end style which is equivalent to the unit rate of begin mathsize 12px style 166 2 over 3 end style words per minute.
Standard Clarifications
Clarification 1: Instruction includes using manipulatives, drawings, models and words and making connections between ratios, rates and unit rates.

Clarification 2: Problems will not include conversions between customary and metric systems.

General Information
Subject Area: Mathematics (B.E.S.T.)
Grade: 6
Strand: Algebraic Reasoning
Standard: Understand ratio and unit rate concepts and use them to solve problems.
Date Adopted or Revised: 08/20
Status: State Board Approved
Standard Instructional Guide

Connecting Benchmarks/Horizontal Alignment

 

Terms from the K-12 Glossary

  • Rate
  • Unit Rate

 

Vertical Alignment

Previous Benchmarks

Next Benchmarks

 

Purpose and Instructional Strategies

In grade 5, students represented the division of two whole numbers as a fraction. In doing this, students started to work with a ratio relationship that relates parts to wholes. In grade 6, students extend this concept to include rates, which are ratios between quantities that are most often in different units. Students use ratio relationships to describe unit rates and percentage relationship and use the division of positive rational numbers to calculate unit rates from rates. In grade 7, students learn that a unit rate is the same as a constant of proportionality in a proportional relationship between two variables. 
  • Instruction connects rate and unit rate to student understanding of equivalent fractions from elementary school in both numeric and picture or model forms. Students can use the models to represent the situations in different ways (MTR.5.1).
  • Allow student flexibility in accepting both simplified and non-simplified responses for rates unless unit rate is the specified or expected form.

 

Common Misconceptions or Errors

  • Students may incorrectly identify what is being compared or the order of quantities being compared by the rate.
  • Students may have difficulty connecting a unit rate, which is represented by a single number, to a ratio or non-unit rate, which may be represented by two numbers.

 

Strategies to Support Tiered Instruction

  • Instruction includes the use of manipulatives and models to represent the provided rates and then to use multiplicative reasoning to determine the rate of one unit. Manipulatives and models include snap cubes, marbles, bar models, number lines or rate tables to help visually represent the relationship.
  • Instruction includes the use of manipulatives to allow for students to explore the meaning of a unit rate. The teacher should provide two different counters to represent a rate equivalent to a whole number unit rate and then co-model the division of the counters into equal groups to determine how many counters of one color are needed to represent a single counter of the other color.
    • For example: At the grocery store, you paid $9.00 for 3 pounds of apples. What is the unit price paid per pound of apples?
      9 red colored circles and 3 yellow colored circles

 

Instructional Tasks

Instructional Task 1 (MTR.6.1, MTR.7.1) 

When buying ground beef for hamburgers, there are several packages from which to choose as shown in the table below.

Table with data related to pounds of ground beef and its cost

  • Part A. Predict how much it would cost for a pound of ground beef. Explain why your prediction is reasonable. 
  • Part B. What is the unit cost of ground beef? Does the unit cost differ by the package size at this store?

Instructional Task 2 (MTR.4.1)
The Jones family is planning on expanding their garden so that they can plant more vegetables. The ratio of the area of the old garden to the area of the new garden is 4¼:8 ¾. Convert this ratio to a unit rate and explain what it means in this context.

Instructional Task 3 (MTR.2.1, MTR.4.1, MTR.5.1) 
Stephen is tracking the number of miles traveled to the number of minutes passed while traveling. He discovered that in 8 minutes he traveled 6 miles. 
  • Part A. In your group, use the chart below to determine the rate and unit rate in miles per minute. 
  • Part B. Which form would be most efficient for this context? Why?
    Tables with columns Ratio, Rate and Unit Rate

 

Instructional Items

Instructional Item 1
At the grocery store, you paid $9.87 for 3.3 pounds of apples. What is the unit price paid per pound of apples? (Round to the nearest cent.)

Instructional Item 2
Brenda wants to buy one of the three cereals listed below. Determine which box is the best buy. Show and explain how you determined this.
  • 16 ounces of Frosted Flurries for $3.50
  • 12.4 ounces of Chocolate O’s for $2.42
  • 11.5 ounces of Cinnamon Grahams for $2.35

 

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.
Related Courses
Related Access Points
  • MA.6.AR.3.AP.2 # Given a rate, calculate the unit rate for a ratio with different units.
Related Resources
Formative Assessments
  • Unit Rate Length # Students are asked to write ratios and unit rates from fractional values.
  • Unit Rate Area # Students are asked to convert a ratio of mixed numbers to a unit rate and explain its contextual meaning.
  • Comparing Unit Rates # Students are asked to compute unit rates from values that include fractions.
  • Computing Unit Rates # Students are asked to compute and interpret unit rates in two different ways from values that include fractions and mixed numbers.
  • Writing Unit Rates # Students are given verbal descriptions of rates and asked to write them as unit rates.
  • Identifying Unit Rates # Students are asked to decide if given statements express unit rates.
  • Explaining Rates # Students are asked to explain the meaning of given rates and identify any that are unit rates.
  • Book Rates # Students write and explain the meaning of a ratio and corresponding unit rate in the context of a word problem.
Lesson Plans
  • Cool Special Effects # In this MEA, students will apply the concepts of heat transfer, especially convection. Students will analyze factors such as temperature that affect the behavior of fluids as they form convection currents. Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.
  • Rate Your Local Produce Market # In this Model Eliciting Activity, MEA, the students will rank the local produce markets by using qualitative and quantitative data. The students will have to calculate unit rates of produce prices and then compare and order them. 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
  • Lightyear Rockets # Students are asked to evaluate and test several rocket fin designs to determine the most effective design. After launch, the students are asked to test an additional design and also design their own rocket fin. Additionally, students will record and graph their results. Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.
  • Real Estate Rental Agency # In this Model Eliciting Activity, MEA, students will choose the best location for a family relocating and will find the monthly costs per month to make the best decision. Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.
  • 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
  • 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.
  • For Students by Students # Students are presented with the task of evaluating several types of fabric based on each of its characteristics. They need to analyze their current uniform needs and decide by choosing which type of fabric will best fit their uniform needs. Then they have to write a report explaining the procedure they used to analyze their choices, reasoning for their ranking and make the requested recommendations. Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.
  • Travel Troubles # This activity engages the students into time scheduling, budgeting, and decision making to maximize time efficiency. 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.
  • "Analyzing Wordless Stories" An Introduction to Solving Unit Rates # In this lesson, students will apply their understanding of ratios and prior knowledge of division to determine the unit rate for a given ratio. After some initial instruction on unit rates, students will determine unit rates from diagrams with teacher guidance, and they will determine unit rates from narrative descriptions independently.
  • Makeover, Home Edition Final Part IV # This is the final part of the lesson "Makeover, Home Edition." This lesson is designed to teach students the applicability of finding the area of composite figures as well as understanding the importance of ratios in the real world. Part I (#48705) dealt with determining backyard dimensions for fence installation. Part II (#48967) concentrated on inserting a pool and patio into this backyard. Part III (#49025) dealt with creating a scale drawing of the backyard.
  • The Best Domestic Car # In this MEA students will use problem-solving strategies to determine which car to recommend to Americans living in India. 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
  • 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.
  • Savvy Shopper # This unit rate culminating activity has students apply knowledge to purchasing groceries. Specifically, knowledge of how unit rates can help save money over time.
  • 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.
  • Better Buy: 75 fl oz or 150 fl oz? # The students will clip out advertisements or use the attached PowerPoint to determine the better buy between small quantities and large quantities. The students will answer the question, "Which item costs less per unit?" and demonstrate fluency in dividing with decimals.
  • Is It Fair? # In this lesson students will use their understanding of ratios and unit rate to solve problems where they must decide whether various situations are fair.
  • Happy Lawns: Lawn Care Service MEA # This Model Eliciting Activity (MEA) is written at a 6th grade level.
    This MEA asks the students to decide on a lawn mower that will provide the Happy Lawns: Lawn Care Service with the best value for their money. Students are asked to rank order the lawn mowers in term of gas tank capacity, customer rating, speed, amount of time the mower takes to cut an acre of grass, shipping, and cost of the lawn mower. Students must provide a "Best Value" lawn mower to the company owner and explain how they arrived at their solution. 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
Original Student Tutorials
Perspectives Video: Expert
Perspectives Video: Professional/Enthusiasts
Perspectives Video: Teaching Idea
Problem-Solving Tasks
  • sandundertheswingset2024 # The 7th graders at Sunview Middle School were helping to renovate a playground for the kindergartners at a nearby elementary school. City regulations require that the sand underneath the swings be at least 15 inches deep. The sand under both swing sets was only 12 inches deep when they started. The rectangular area under the small swing set measures 9 feet by 12 feet and required 40 bags of sand to increase the depth by 3 inches. How many bags of sand will the students need to cover the rectangular area under the large swing set if it is 1.5 times as long and 1.5 times as wide as the area under the small swing set?
  • Price Per Pound and Pounds Per Dollar # Students are asked to use a given ratio to determine if two different interpretations of the ratio are correct and to determine the maximum quantity that could be purchased within a given context.
  • Running at a Constant Speed # Students are asked apply knowledge of ratios to answer several questions regarding speed, distance and time.
Tutorials
STEM Lessons - Model Eliciting Activity
  • 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
  • Cool Special Effects # In this MEA, students will apply the concepts of heat transfer, especially convection. Students will analyze factors such as temperature that affect the behavior of fluids as they form convection currents. Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.
  • For Students by Students # Students are presented with the task of evaluating several types of fabric based on each of its characteristics. They need to analyze their current uniform needs and decide by choosing which type of fabric will best fit their uniform needs. Then they have to write a report explaining the procedure they used to analyze their choices, reasoning for their ranking and make the requested recommendations. Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.
  • Happy Lawns: Lawn Care Service MEA # This Model Eliciting Activity (MEA) is written at a 6th grade level.
    This MEA asks the students to decide on a lawn mower that will provide the Happy Lawns: Lawn Care Service with the best value for their money. Students are asked to rank order the lawn mowers in term of gas tank capacity, customer rating, speed, amount of time the mower takes to cut an acre of grass, shipping, and cost of the lawn mower. Students must provide a "Best Value" lawn mower to the company owner and explain how they arrived at their solution. 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
  • Lightyear Rockets # Students are asked to evaluate and test several rocket fin designs to determine the most effective design. After launch, the students are asked to test an additional design and also design their own rocket fin. Additionally, students will record and graph their results. Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.
  • Rate Your Local Produce Market # In this Model Eliciting Activity, MEA, the students will rank the local produce markets by using qualitative and quantitative data. The students will have to calculate unit rates of produce prices and then compare and order them. 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
  • Real Estate Rental Agency # In this Model Eliciting Activity, MEA, students will choose the best location for a family relocating and will find the monthly costs per month to make the best decision. Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.
  • The Best Domestic Car # In this MEA students will use problem-solving strategies to determine which car to recommend to Americans living in India. 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
  • Travel Troubles # This activity engages the students into time scheduling, budgeting, and decision making to maximize time efficiency. 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
  • Book Rates # Students write and explain the meaning of a ratio and corresponding unit rate in the context of a word problem.
  • Comparing Unit Rates # Students are asked to compute unit rates from values that include fractions.
  • Computing Unit Rates # Students are asked to compute and interpret unit rates in two different ways from values that include fractions and mixed numbers.
  • Explaining Rates # Students are asked to explain the meaning of given rates and identify any that are unit rates.
  • Identifying Unit Rates # Students are asked to decide if given statements express unit rates.
  • Unit Rate Area # Students are asked to convert a ratio of mixed numbers to a unit rate and explain its contextual meaning.
  • Unit Rate Length # Students are asked to write ratios and unit rates from fractional values.
  • Writing Unit Rates # Students are given verbal descriptions of rates and asked to write them as unit rates.
Original Student Tutorials Mathematics - Grades K-5
  • Una’s Unique Unit Rate Reunion # Join Una, the unicorn, as she learns the power of calculating unit rates while planning her family reunion in this interactive tutorial.
Original Student Tutorials Mathematics - Grades 6-8
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