# MA.6.AR.3.2

Given a real-world context, determine a rate for a ratio of quantities with different units. Calculate and interpret the corresponding unit rate.

### Examples

Tamika can read 500 words in 3 minutes. Her reading rate can be described as which is equivalent to the unit rate of words per minute.

### 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.)
Strand: Algebraic Reasoning
Status: State Board Approved

## Benchmark Instructional Guide

• 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?

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

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

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?

### 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?

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

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.2: Given a rate, calculate the unit rate for a ratio with different units.

## Related Resources

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

## Formative Assessments

Unit Rate Length:

Students are asked to write ratios and unit rates from fractional values.

Type: Formative Assessment

Unit Rate Area:

Students are asked to convert a ratio of mixed numbers to a unit rate and explain its contextual meaning.

Type: Formative Assessment

Comparing Unit Rates:

Students are asked to compute unit rates from values that include fractions.

Type: Formative Assessment

Computing Unit Rates:

Students are asked to compute and interpret unit rates in two different ways from values that include fractions and mixed numbers.

Type: Formative Assessment

Writing Unit Rates:

Students are given verbal descriptions of rates and asked to write them as unit rates.

Type: Formative Assessment

Identifying Unit Rates:

Students are asked to decide if given statements express unit rates.

Type: Formative Assessment

Explaining Rates:

Students are asked to explain the meaning of given rates and identify any that are unit rates.

Type: Formative Assessment

Book Rates:

Students write and explain the meaning of a ratio and corresponding unit rate in the context of a word problem.

Type: Formative Assessment

## 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.

Type: Lesson Plan

The students will rank the local produce markets by using qualitative and quantitative data. The students will have to calculate unit rates and compare and order them.

Type: Lesson Plan

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.

Type: Lesson Plan

The Best Domestic Car:

In this MEA students will use problem-solving strategies to determine which car to recommend to Americans living in India.

Type: Lesson Plan

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.

Type: Lesson Plan

Orange Juice Conversion:

In this MEA, the students will be able to convert measurements within systems and between systems. They will be able to use problem solving skills to create a process for ranking orange juices for a Bed and Breakfast.

Type: Lesson Plan

## Original Student Tutorial

Farmers Market: Ratios, Rates and Unit Rates:

Learn how to identify and calculate unit rates by helping Milo find prices per item at a farmer's market in this interactive tutorial.

Type: Original Student Tutorial

## Perspectives Video: Expert

Statistical Sampling Results in setting Legal Catch Rate:

Fish Ecologist, Dean Grubbs, discusses how using statistical sampling can help determine legal catch rates for fish that may be endangered.

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiasts

Unit Rates in Swimming:

In this video, David Fermin demonstrates real-time estimates for monitoring swimming performance and physiology.

Type: Perspectives Video: Professional/Enthusiast

Unit Rate: Spring Water Bottling:

Nestle Waters discusses the importance of unit rate in the manufacturing process of bottling spring water.

Type: Perspectives Video: Professional/Enthusiast

Unit Rate and Florida Cave Formation:

How long does it take to form speleothems in the caves at Florida Caverns State Parks?

Type: Perspectives Video: Professional/Enthusiast

Pizza Pi: Area, Circumference & Unit Rate:

How many times larger is the area of a large pizza compared to a small pizza? Which pizza is the better deal? Michael McKinnon of Gaines Street Pies talks about how the area, circumference and price per square inch is different depending on the size of the pizza.

Type: Perspectives Video: Professional/Enthusiast

Amping Up Violin Tuning with Math:

Kyle Dunn, a Tallahassee-based luthier and owner of Stringfest, discusses how math is related to music.

Type: Perspectives Video: Professional/Enthusiast

Building Scale Models to Solve an Archaeological Mystery:

An archaeologist describes how mathematics can help prove a theory about mysterious prehistoric structures called shell rings.

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

Bicycle Mathematics: Selecting Gear Ratios for Performance:

Don't let math derail you. Learn how bicycle gears use ratios to help you ride comfortably on all kinds of terrain.

Type: Perspectives Video: Professional/Enthusiast

KROS Pacific Ocean Kayak Journey: Calories, Distance, and Rowing Rates:

Food is fuel, especially important when your body is powering a boat across the ocean.

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: Professional/Enthusiast

KROS Pacific Ocean Kayak Journey: Calories, Exercise, and Metabolism Rates:

How much food do you need to cross the Pacific in a kayak? Get a calculator and a bag of almonds before you watch this.

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: Professional/Enthusiast

## Perspectives Video: Teaching Idea

Bicycle Mathematics: Speed and Distance Calculations:

Cycling involves a lot of real-time math when you use an on-board computer. Learn about lesson ideas and how computers help with understanding performance.

Type: Perspectives Video: Teaching Idea

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

Determining Rates with Fractions:

This video demonstrates finding a unit rate from a rate containing fractions.

Type: Tutorial

Rate Problem With Fractions:

Watch as we solve a rate problem finding speed in meters per second using distance (in meters) and time (in seconds).

Type: Tutorial

Solving Unit Rate Problem:

This video demonstrates solving a unit rate problem using equivalent ratios.

Type: Tutorial

Solving Unit Price Problem:

This video demonstrates solving a unit price problem using equivalent ratios.

Type: Tutorial

Converting Speed Units:

In this lesson, students will be viewing a Khan Academy video that will show how to convert ratios using speed units.

Type: Tutorial

## STEM Lessons - Model Eliciting Activity

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.

Orange Juice Conversion:

In this MEA, the students will be able to convert measurements within systems and between systems. They will be able to use problem solving skills to create a process for ranking orange juices for a Bed and Breakfast.

The students will rank the local produce markets by using qualitative and quantitative data. The students will have to calculate unit rates and compare and order them.

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.

The Best Domestic Car:

In this MEA students will use problem-solving strategies to determine which car to recommend to Americans living in India.

## 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 6-8

Farmers Market: Ratios, Rates and Unit Rates:

Learn how to identify and calculate unit rates by helping Milo find prices per item at a farmer's market in this interactive tutorial.

## Student Resources

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

## Original Student Tutorial

Farmers Market: Ratios, Rates and Unit Rates:

Learn how to identify and calculate unit rates by helping Milo find prices per item at a farmer's market in this interactive tutorial.

Type: Original Student Tutorial

## Perspectives Video: Professional/Enthusiast

Building Scale Models to Solve an Archaeological Mystery:

An archaeologist describes how mathematics can help prove a theory about mysterious prehistoric structures called shell rings.

Type: Perspectives Video: Professional/Enthusiast

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

Determining Rates with Fractions:

This video demonstrates finding a unit rate from a rate containing fractions.

Type: Tutorial

Rate Problem With Fractions:

Watch as we solve a rate problem finding speed in meters per second using distance (in meters) and time (in seconds).

Type: Tutorial

Solving Unit Price Problem:

This video demonstrates solving a unit price problem using equivalent ratios.

Type: Tutorial

Converting Speed Units:

In this lesson, students will be viewing a Khan Academy video that will show how to convert ratios using speed units.

Type: Tutorial

## Parent Resources

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

## Perspectives Video: Professional/Enthusiast

Building Scale Models to Solve an Archaeological Mystery:

An archaeologist describes how mathematics can help prove a theory about mysterious prehistoric structures called shell rings.

Type: Perspectives Video: Professional/Enthusiast