Unit 1: 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.
In Unit 1, sixth-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 fourth and fifth grades, 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: Rates & Percent and re-appears in Unit 6: Equations and Inequalities when students analyze and graph relationships between independent and dependent variables. Beyond sixth grade, students extend their understanding of ratios and rates to investigate proportional relationships in seventh grade. This sets the groundwork for the study of functions, linear equations, and systems of equations, which students will study in eighth grade and high school.
Pacing: 21 instructional days (18 lessons, 2 flex days, 1 assessment day).
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The following assessments accompany Unit 1.
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.
Pre-Unit Student Self-Assessment
Have students complete the Mid-Unit Assessment.
Use the resources below to assess student mastery of the unit content and action plan for future units.
Post-Unit Assessment Answer Key
Post-Unit Student Self-Assessment
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.
Suggestions for how to prepare to teach this unit
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.
The central mathematical concepts that students will come to understand in this unit
Terms and notation that students learn or use in the unit
double number line
part to part ratio
part to whole ratio
To see all the vocabulary for Unit 1, view our 6th Grade Vocabulary Glossary.
The materials, representations, and tools teachers and students will need for this unit
To see all the materials needed for this course, view our 6th Grade Course Material Overview.
Topic A: Understanding & Describing Ratios
Define ratio and use ratio language to describe associations between two or more quantities.
Represent ratios using discrete drawings. Understand that the order of numbers in a ratio matters.
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Topic B: Equivalent Ratios
Define and find equivalent ratios.
Reason with equivalent ratios and determine if two ratios are equivalent.
Represent ratios using double number lines and identify equivalent ratios.
Solve ratio problems using strategies including double number lines.
Find equivalent ratios using ratios with “per 1” unit.
Compare situations using equivalent ratios and double number lines.
Use ratio reasoning to solve a three-act task.
Topic C: Representing Ratios in Tables
Represent ratios in tables.
Understand the structure of tables of equivalent ratios. Solve ratio problems using tables.
Solve ratio problems using tables, including those involving total amounts.
Compare ratios using tables.
Solve ratio problems using different strategies.
Topic D: Solving Part:Part:Whole Ratio Problems
Solve part:part ratio problems using tape diagrams.
Solve part:whole ratio problems using tape diagrams.
Solve more complex ratio problems using tape diagrams.
Solve ratio problems using a variety of strategies, including reasoning about diagrams, double number lines, tables, and tape diagrams. Summarize strategies for solving ratio problems.
The content standards covered in this unit
— 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."
— 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.
— 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.
— 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?
Standards covered in previous units or grades that are important background for the current unit
— Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two column table.
For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), …
— Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?
— Interpret multiplication as scaling (resizing), by:
— Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
— Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.
— Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.
— Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane.
For example, given the rule "Add 3" and the starting number 0, and given the rule "Add 6" and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.
Standards in future grades or units that connect to the content in this unit
— Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.
For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.
— Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 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."
— 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.
— Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
— Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.
For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction <sup>1/2</sup>/<sub>1/4</sub> miles per hour, equivalently 2 miles per hour.
— Recognize and represent proportional relationships between quantities.
— Use proportional relationships to solve multistep ratio and percent problems.
Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
— Make sense of problems and persevere in solving them.
— Reason abstractly and quantitatively.
— Construct viable arguments and critique the reasoning of others.
— Model with mathematics.
— Use appropriate tools strategically.
— Attend to precision.
— Look for and make use of structure.
— Look for and express regularity in repeated reasoning.
Unit Rates and Percent