# Unit Rates and Percent

Students investigate rates and percentages by identifying the rates associated with a ratio, defining a percent as a rate per 100, and applying strategies to solve rate and percent problems.

Math

Unit 2

## Unit Summary

In Unit 2, 6th grade students continue and extend their study of ratios to investigate rates and percentages. Using their knowledge of ratios, students will learn to identify two rates associated with a ratio and use them as efficient strategies to solve rate problems. Students will draw on their prior knowledge of the two measurement systems, and see unit conversions as applications of ratio and rate problems. Lastly, students will define a percent as a rate per 100, and understand that the strategies you use to solve a rate problem can also be applied to solve a percent problem. Throughout the entire unit, students will draw on their ability to reason abstractly and quantitatively (MP.2). As they work with different unit rates and conversion factors, students will need to attend to the units at hand, what the quantities mean, and how all of the pieces fit together. Calculators are recommended for several lessons in order to provide students with the option to use the tool in their calculations (MP.5).

In 4th grade and 5th grade, students interpreted fractions as division problems and began to make the connection between fractions and decimals. They multiplied fractions by whole numbers and other fractions in context of real-world problems, and they reasoned about what happens to a quantity when you multiply it by a number greater than one or less than one. Students will draw on these prior skills and understandings as they make connections between unit rates and fractions, and between fractions, decimals, and percentages.

Beyond this unit, students will revisit percentages in Unit 6 when they study equations as another strategy to solve percent problems. In 7th grade, students will solve even more complex ratio, rate, and percent problems involving, for example, tax and percent increase or decrease. Additionally, students will investigate and analyze proportional relationships between quantities, and use more efficient and abstract methods to solve problems. For example, in 7th grade, students will learn to set up and solve proportion equations, using their knowledge of unit rates and equivalent ratios. (Note that using proportions to solve ratio or rate problems is not an expectation in sixth grade).

Pacing: 19 instructional days (14 lessons, 4 flex/review days, 1 assessment day)

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## Assessment

The following assessments accompany Unit 2.

### Pre-Unit

Have students complete the Pre-Unit Assessment and Pre-Unit Student Self-Assessment before starting the unit. Use the Pre-Unit Assessment Analysis Guide to identify gaps in foundational understanding and map out a plan for learning acceleration throughout the unit.

### Mid-Unit

Have students complete the Mid-Unit Assessment after lesson 7.

### Post-Unit

Use the resources below to assess student understanding of the unit content and action plan for future units.

Expanded Assessment Package

Use student data to drive instruction with an expanded suite of assessments. Unlock Pre-Unit and Mid-Unit Assessments, and detailed Assessment Analysis Guides to help assess foundational skills, progress with unit content, and help inform your planning.

## Unit Prep

### Intellectual Prep

Unit Launch

Before you teach this unit, unpack the standards, big ideas, and connections to prior and future content through our guided intellectual preparation process. Each Unit Launch includes a series of short videos, targeted readings, and opportunities for action planning to ensure you're prepared to support every student.

#### Internalization of Standards via the Post-Unit Assessment

• Take the Post-Unit Assessment. Annotate for:
• Standards that each question aligns to
• Strategies and representations used in daily lessons
• Relationship to Essential Understandings of unit
• Lesson(s) that Assessment points to

#### Internalization of Trajectory of Unit

• Read and annotate the Unit Summary.
• Notice the progression of concepts through the unit using the Lesson Map.
• Essential Understandings
• Connection to Post-Unit Assessment questions
• Identify key opportunities to engage students in academic discourse. Read through our Teacher Tool on Academic Discourse and refer back to it throughout the unit.

#### Unit-Specific Intellectual Prep

Model Example
double number line For every 2 cups of milk, there are 3 cups of flour.
table of equivalent ratios A turtle travels 3 feet every 9 seconds.
 Distance (ft) Time(sec) 3 9 6 18 9 27 30 90
tape diagram The ratio of girls to boys in a 6th grade class is 4 to 5.

### Essential Understandings

• A rate, associated with a ratio a:b, is a/b or b/a units of one quantity per 1 unit of the other quantity. For example, if a person walks 6 miles in 2 hours, the person is traveling at a rate of 3 miles per hour, or equivalently a rate of 1/3 hour per mile.
• A percent is a rate per 100; percent problems can be solved using the same strategies that are used for other rate problems.
• There are many applications to rate, including unit price, constant speed, and measurement conversions, that can be solved using several different strategies, such as using double number lines, tables, tape diagrams, and the unit rate.

### Vocabulary

conversion rate

customary system

equivalent ratio

metric system

percent/ percentage

rate

ratio

unit rate

To see all the vocabulary for Unit 2, view our 6th Grade Vocabulary Glossary.

### Materials

• Calculators (1 per student) — It is recommended that students use calculators in this lesson since some of the problems involve decimals. Students have experience with calculating with decimals from fifth grade, but they will study them more in depth and with fluency in Unit 3. Calculators will ensure the focus of the lesson is on the concepts and reasoning, rather than calculations (MP.5).
• Massachusetts Comprehensive Assessment System Grade 6 Mathematics Reference Sheet (1 per student) — We recommend the MCAS reference sheet, however, other reference sheets may suffice as well.

To see all the materials needed for this course, view our 6th Grade Course Material Overview.

## Lesson Map

Topic A: Defining Rate & Solving Rate Problems

Topic B: Measurement Unit Conversions

Topic C: Percent

## Common Core Standards

Key

Major Cluster

Supporting Cluster

### Core Standards

#### Ratios and Proportional Relationships

• 6.RP.A.2 — Understand the concept of a unit rate a/b associated with a ratio a:b with b â‰  0, and use rate language in the context of a ratio relationship. Expectations for unit rates in this grade are limited to non-complex fractions. For example, "This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar." "We paid $75 for 15 hamburgers, which is a rate of$5 per hamburger."
• 6.RP.A.3 — Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
• 6.RP.A.3.B — Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
• 6.RP.A.3.C — Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
• 6.RP.A.3.D — Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

• 4.MD.A.1
• 4.MD.A.2
• 5.MD.A.1

• 5.NBT.B.6

• 4.NF.B.4.B
• 4.NF.B.4.C
• 4.NF.C.6
• 5.NF.B.3
• 5.NF.B.4.A
• 5.NF.B.5
• 5.NF.B.5.A
• 5.NF.B.5.B
• 5.NF.B.6

• 6.RP.A.1

• 6.EE.B.7
• 6.EE.C.9

• 7.RP.A.1
• 7.RP.A.2
• 7.RP.A.3

### Standards for Mathematical Practice

• CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.

• CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.

• CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.

• CCSS.MATH.PRACTICE.MP4 — Model with mathematics.

• CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.

• CCSS.MATH.PRACTICE.MP6 — Attend to precision.

• CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.

• CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.

Unit 1

Understanding and Representing Ratios

Unit 3

Multi-Digit and Fraction Computation

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