Understanding and Representing Ratios

Students are introduced to the concept of ratios, learning ratio language to describe the association between two or more quantities and different strategies to solve ratio problems.

Math

Unit 1

6th Grade

Unit Summary


In Unit 1, 6th grade students have the opportunity to study a concept that is brand new to them: ratios. They learn how to use ratio language to describe the association between two or more quantities, expanding their abilities to analyze relationships and see multiplicative patterns. Students learn many ways to represent ratios, starting with discrete drawings and working their way to abstract tables. These representations become important tools in their ratios toolkit, enabling students to be strategic about which tools to use for different problems (MP.5). When students work with tables and double number lines, they discover how structure can shine light on a relationship, especially when comparing multiple ratio situations (MP.7).

Throughout the unit, students see similar problems posed to them in different lessons. This is to support students learning new strategies to solve ratio problems and to compare and contrast different approaches. By the end of the unit, students should be able to select a strategy they think is best for a problem and to explain their choice. 

In 4th grade and 5th grade, students learned the difference between multiplicative and additive comparisons and they interpreted multiplication as a way to scale. Students will access these prior concepts in this unit as they investigate patterns and structures in ratio tables and use multiplication to create equivalent ratios. 

The work students do in this unit connects directly to Unit 2 and re-appears in Unit 6 when students analyze and graph relationships between independent and dependent variables. Beyond 6th grade, students extend their understanding of ratios and rates to investigate proportional relationships in 7th grade. This sets the groundwork for the study of functions, linear equations, and systems of equations, which students will study in 8th grade and high school

Pacing: 21 instructional days (18 lessons, 2 flex days, 1 assessment day).

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Assessment


The following assessments accompany Unit 1.

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.

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 your planning with an expanded suite of unit assessments to help gauge students’ facility with foundational skills and concepts, as well as their progress with unit content.

Unit Prep


Intellectual Prep

Unit Launch

Prepare to teach this unit by immersing yourself in the standards, big ideas, and connections to prior and future content. Unit Launches include a series of short videos, targeted readings, and opportunities for action planning.

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.
  • Do all Target Tasks. Annotate the Target Tasks for: 
    • 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
Discrete drawing The ratio of teaspoons of cinnamon to cups of raisins is 4:8.
Double number line For every 2 cups of milk, there are 3 cups of flour.
Ratio table 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 ratio is a set of numbers that associates two or more quantities. The order of the values in a ratio relates directly to the order of the quantities described. 
  • Equivalent ratios are useful in understanding a situation more deeply or in comparing multiple situations. Two ratios are equivalent if there is a nonzero number that can be multiplied by both quantities in one ratio to equal the corresponding quantities in the second ratio. 
  • Ratio problems can be represented and solved using a variety of strategies, tools, and representations, including a discrete diagram, a double number line, a ratio table, and tape diagrams. Each tool has its advantages and disadvantages and can be selected strategically.

Vocabulary

double number line

equivalent ratio

multiplicative relationship

part to part ratio

part to whole ratio

ratio

ratio table

tape diagram

To see all the vocabulary for Unit 1, view our 6th Grade Vocabulary Glossary.

Materials

  • Calculators (1 per student) — This lesson involves calculations with decimals. Students have worked with decimals in 5th grade, however, in order to keep the emphasis on the ratio reasoning, it is recommended to give students calculators to support their computations throughout this lesson (MP.5).
  • Ratio Shapes Handout (1 per student)

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

Lesson Map


Topic A: Understanding & Describing Ratios

Topic B: Equivalent Ratios

Topic C: Representing Ratios in Tables

Topic D: Solving Part:Part:Whole Ratio Problems

Common Core Standards


Key

Major Cluster

Supporting Cluster

Additional Cluster

Core Standards

Ratios and Proportional Relationships

  • 6.RP.A.1 — Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, "The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak." "For every vote candidate A received, candidate C received nearly three votes."
  • 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.A — Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
  • 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?

Foundational Standards

Measurement and Data

  • 4.MD.A.1

Number and Operations—Fractions

  • 5.NF.B.3
  • 5.NF.B.5
  • 5.NF.B.5.A
  • 5.NF.B.5.B

Operations and Algebraic Thinking

  • 4.OA.A.2
  • 5.OA.B.3

Future Standards

Expressions and Equations

  • 6.EE.C.9

Ratios and Proportional Relationships

  • 6.RP.A.2
  • 6.RP.A.3.C
  • 6.RP.A.3.D
  • 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.

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Unit 2

Unit Rates and Percent

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