Ratios and Proportional
Relationships
Unit Guide Math 6
Big Idea (Cluster):
Understand ratio concepts and use
ratio reasoning to solve problems.
(6.RP.1-6.RP.3)
Edited 5/18/14
Renton School District
Domain: Ratios and Proportional Relationships (6.RP)
Big Idea (Cluster): Understand ratio concepts and use ratio reasoning to solve problems
Standard 6.RP.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.”
Standard 6.RP.2
Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0 (b not equal to zero), and use rate language in the context of a ratio
relationship. 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."
Standard 6.RP.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.
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. Plotting of pairs of values on the coordinate plane will be addressed in
Variables and Patterns.
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.
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.
d.
Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
2
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (6.RP.1-3)
Relevant Math Practices and Student Actions
MP1 Makes sense of problems and persevere in solving them.
 Understands what the problem asks and the relationship among
the problem’s parts.
 Uses multiple strategies and representations.
 Checks process, answers and ask “Does this make sense?”
 Explains why a solution is reasonable.
MP 2
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Reason abstractly and quantitatively.
Interprets problems in context.
Uses representations to make meaning of problems.
Translates a problem from situation to equation.
Understands the meaning of quantities and units.
MP 3 Construct viable arguments and critique reasoning of others.
 Uses definitions and draws on prior mathematical knowledge
when constructing arguments.
 Justifies conclusions with mathematical evidence and responds to
arguments of others.
 Asks clarifying and probing questions to improve argument.
MP 4 Model with mathematics.
 Applies prior mathematical knowledge to describe, analyze, and
solve problems arising in everyday life, society and workplace.
 Checks to see if an answer makes sense within the context of a
situation and improves model when necessary.
MP 5 Use appropriate tools strategically.
 Selects tools strategically for visualizing, exploring, comparing,
predicting, and solving problems.
MP 6 Attend to precision.
 Communicates mathematical thinking accurately both orally and
in writing.
 Understands the meaning of mathematical symbols and
vocabulary and uses them appropriately.
 Labels consistently and accurately graphs and diagrams.
 Specifies the units of measure when labeling.
 Calculates accurately and efficiently.
MP 7 Look for and make use of structure.
 Looks for, identifies, develops and generalizes patterns and
relationships.
 Makes connections to prior mathematical knowledge to solve
new problems.
MP 8 Look for and express regularity in repeated reasoning.
 Notices repeated calculations and looks for general methods and
shortcuts to solve a problem.
 Evaluates reasonableness of intermediate and final results.
3
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (6.RP.1-3)
SBAC Required Evidence (Claim 1)
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Student uses ratio and rate language to describe a ratio relationship (SBAC Grade 6 Claim 1 Target A, required evidence #1).
Student determines the unit rate associated with a real-world ratio (SBAC Grade 6 Claim 1 Target A, required evidence #2).
Student finds missing values in tables of equivalent rations (SBAC Grade 6 Claim 1 Target A, required evidence #3)
Student makes tables of equivalent ratios relating quantities with whole-number measurements (SBAC Grade 6 Claim 1 Target A, required
evidence #5)
Student solves real-world and mathematical problems involving unit rate (SBAC Grade 6 Claim 1 Target A, required evidence #6).
Student solves mathematical problems involving finding the whole, given a part and the percent (SBAC Grade 6 Claim 1 Target A, required
evidence #7).
Student solves real-world and mathematical problems involving finding a percent of a quantity as a rate per 100 (SBAC Grade 6 Claim 1 Target
A, required evidence #8).
Student uses ratio reasoning to convert measurement units (SBAC Grade 6 Claim 1 Target A, required evidence #9).
Student uses ratio reasoning to manipulate and transform units appropriately when multiplying and dividing quantities (SBAC Grade 6 Claim 1
Target A, required evidence #10).
For more information on the assessment of this set of standards, read the Claim 1 SBAC Target A Item Specifications. This cluster of standards will
also be assessed through Claim 2 (Problem Solving), Claim 3 (Communicating Reasoning), and Claim 4 (Modeling and Data Analysis) assessment
items.
Vocabulary
Mathematically proficient students communicate precisely by engaging in discussions about their reasoning using appropriate mathematical
language. Students should learn the following terms with increasing precision within the cluster. The bolded terms will be used on Smarter
Balanced assessment items.
Additive
Compare
Coordinate Plane
Denominator
Double number line
Equation
Equivalent Ratio
Measurement
Model
Multiplicative
Numerator
Ordered Pair
Part-to-part
Part-to-whole
Per
Percent
Proportional relationship
Quantities
Rate
Ratio
Ratio Table
Relationship
Tape Diagram
Unit Price
Unit Rate
Units
Whole
4
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (6.RP.1-3)
Standard 6.RP.1
Understand ratio concepts and use ratio reasoning to solve problems. 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.”
See Grade 6 Math Flip Book pages 7-9 for explanations and examples of this standard.
Learning Objectives
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Describe a ratio relationship between any two number quantities with denominators less than or equal to 12 (SBAC Grade 6 Claim 1 Target A
ALD Level 3*).
Make comparisons (part-to-part and part-to-whole) of two quantities using ratio language verbally, symbolically, or in written form.
Create and analyze comparison statements from given data.
Express ratios in different forms (fraction, decimal, and percent).
Apply multiplicative reasoning to explain the concept of ratio.
Apply ratio reasoning to solve problems.
Use multiple representations and strategies to represent and compare ratios, including tape diagrams.
Find equivalent ratios and use to solve problems.
SBAC Required Evidence (Claim 1)

Student uses ratio and rate language to describe a ratio relationship
(SBAC Grade 6 Claim 1 Target A, required evidence #1).
For more information on the assessment of this standard, read the
Claim 1 Target A SBAC Item Specifications. This cluster of standards
will also be assessed through Claim 2 (Problem Solving), Claim 3
(Communicating Reasoning), and Claim 4 (Modeling and Data
Analysis) assessment items.
Questions to Develop Mathematical Thinking
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Describe the connection between a ratio and a fraction. How are they
similar? How are they different?
How is the ratio being used to compare two quantities?
How might you model the ratio differently to verify your solution is
accurate?
How do you determine whether the comparison is part-to-part or partto-whole?
What other ratio could you use that is equivalent and easier to use?
How is finding an equivalent ratio like finding an equivalent fraction?
How might a tape diagram or table help you explain the ratio
relationship?
5
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (6.RP.1-3)
Requisite Prior Knowledge
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Connections to Prior Learning
Ratio is a first time concept for students in Grade 6. Beginning in Grade 3,
students will have an understanding of a fraction as a part-to-whole
relationship.
Fluently multiply and divide whole numbers.
Recognize and extend patterns involving whole numbers.
Generate primes, factors, and multiples.
Multiply and divide with fractions and decimals.
Understand fraction equivalence.
Understand part-to-whole relationship.
In Grade 5, students will have interpreted a fraction as division.
Connections to Curriculum Resources
Additional Resources/Technology Resources
CMP2 Comparing and Scaling Investigations 1 and 2 fairly aligns to
this standard. Teachers will need to supplement with introductory
learning on what is a ratio and ratio language. See additional resources
for suggested supplementary lessons. The pacing guide also has
suggested supplemental resources.
Understanding ratios with tape diagrams video
“Sharing Costs: Travelling to School” task (MAP)
Pictorial and Fraction models for ratios PowerPoint
Strip Diagram (tape diagram) article (MTMS) filter by Teacher Resources
Ratio and Proportion module lessons 1-4 (engage NY)
Ratio and Proportional Relationships PowerPoint
Ratio and Proportion tasks (Trinity University-click download)
Ratios and Proportions module (NWPS)
Instructional Strategies
Grade 6 is the first year in which students develop the multiplicative thinking rooted in proportional reasoning. Examples with ratios must involve
measurements, prices and geometric contexts that are relevant to sixth graders. Experience with proportional and non-proportional relationships and
comparing and predicting ratios to previously learned unit fractions will facilitate the development of proportional reasoning. Although algorithms
provide efficient means for finding solutions, the cross-product algorithm commonly used for solving proportions will not aid in the
development of proportional reasoning. Delaying the introduction of rules and algorithms will encourage thinking about multiplicative situations.
Students develop the understanding that ratio is a comparison of two numbers or quantities. Ratios that are written as part-to-whole are comparing a
specific part to the whole. Fractions and percents are examples of part-to-whole ratios. Provide students with multiple examples of ratios, fractions
and percents of this type. Part-to-part ratios are used to compare two parts.
Students should use multiple representations, such as tape diagrams, tables, and number lines, to make sense of ratio relationships before they work
with ratios numerically. Students should be able to compare ratios verbally and in writing, and be able to describe a ratio in part-to-part and part-towhole relationships.
6
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (6.RP.1-3)
Explanations and Examples
A ratio is the comparison of two quantities or measures. The comparison can be part-to whole or part-to-part. Students need to understand each of
these ratios when expressed in the following forms: 6 to 15 or 6:15. These values can be simplified to, 2 to 5 or 2:5; however, students would need to
understand how the simplified values relate to the original ratio numbers.
For example, a comparison of 8 black circles to 4 white circles can be written as the ratio of 8:4.
The ratio then can be regrouped into 4 black circles to 2 white circles (4:2) or 2 black circles to 1 white circle
(2:1).
Students should be able to identify all these ratios and describe them using phrases such as, “for every,” “out of every,” and “parts to parts.”
7
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (6.RP.1-3)
Domain: Ratios and Proportional Relationships (6.RP)
Big Idea (Cluster): Understand ratio concepts and use ratio reasoning to solve problems
Standard 6.RP.2
Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0 (b not equal to zero), and use rate language in the context of a ratio
relationship. 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."
See Grade 6 Math Flip Book page 10 for explanations and examples of this standard.
Learning Objectives
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Understand the concept of unit rates in problems (SBAC Grade 6 Claim 1 Target A ALD Level 3*).
Use rate language (“per,” “for each, ‘for every”) in context of a proportional relationship.
Express a unit rate as a part-to-whole ratio using ratio language in verbal, written, and symbolic form.
Understand and apply the concept of unit rate a/b associated with a ratio a:b with b≠0.
Distinguish difference between ratio and rate.
SBAC Required Evidence (Claim 1)
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Student determines the unit rate associated with a real-world ratio
(SBAC Grade 6 Claim 1 Target A, required evidence #2).
For more information on the assessment of this standard, read the
Claim 1 SBAC Item Specifications. This cluster of standards will also
be assessed through Claim 2 (Problem Solving), Claim 3
(Communicating Reasoning) , and Claim 4 (Modeling and Data
Analysis) assessment items.
Requisite Prior Knowledge
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Fluently multiply and divide whole numbers.
Recognize and extend patterns involving whole numbers.
Multiply and divide with fractions and decimals.
Understand fraction equivalence.
Questions to Develop Mathematical Thinking
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How might solving for a unit rate be helpful at home?
How is a rate used to compare two quantities with different measures?
How is a rate similar and/or different from a ratio?
How might a rate table or graph help identify or verify the unit rate?
After finding the unit rate, is there a different unit rate using the same
ratio?
Connections to Prior Learning
Rate is a first time concept for students in Grade 6. In Grade 5, students
will have interpreted a fraction as division. Students will have also used
tables in Grade 5 to extend numerical patterns.
8
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (6.RP.1-3)
Connections to Curriculum Resources
Additional Resources/Technology Resources
CMP2 Comparing and Scaling Investigation 3 fairly aligns to this
standard. Teachers will need to supplement with introductory learning
on what is a rate compared to a ratio and solving for basic unit rates.
See additional resources for suggested supplementary lessons. The
pacing guide also has suggested supplemental resources.
Calculating Speed (Cyberchase)
Ratio and Proportion module 1 lessons 16-18 (engage NY)
Ratio and Proportion tasks (Trinity University – click download)
Ratios and Proportions module (NWPS)
Instructional Strategies
Grade 6 is the first year in which students develop the multiplicative thinking rooted in proportional reasoning. Examples of rates should include,
miles per hour or portions per person within contexts that are relevant to sixth graders. Experience with proportional and non-proportional
relationships and relating unit rates to previously learned unit fractions will facilitate the development of proportional reasoning. Although
algorithms provide efficient means for finding solutions, the cross-product algorithm commonly used for solving proportions will not aid in the
development of proportional reasoning. Delaying the introduction of rules and algorithms will encourage thinking about multiplicative situations.
A rate is a ratio where two measurements are related to each other. When discussing measurement of different units, the word rate is used rather than
ratio. Understanding rate, however, is complicated and there is no universally accepted definition. When using the term rate, contextual
understanding is critical. Students need many opportunities to use models to demonstrate the relationships between quantities before they are
expected to work with rates numerically.
Explanations and Examples
It is important for students to focus on the meaning of the terms “for every,” “for each,” “for each one” and “per” because each of these
understandings are at the center of the structure and multiplicative reasoning with rates. A unit rate expresses a ratio as part-to-one or one unit of
another quantity. The unit rate is the numerical part of the rate; the “unit” in “unit rate” is often used to highlight the 1 in “for each 1” or “for every
1.” Students understand the unit rate from various contextual situations. Students will often use unit rates to solve missing value problems. Cost per
item or distance per time unit are common unit rates, however, students should be able to flexibly use unit rates to name the amount of either quantity
in terms of the other quantity.
Students should identify both unit rates for a given ratio. For example, I can buy 10 oranges for $2. A student could write a unit rate of 5 oranges
per dollar (5:1) or $0.20 per orange (0.2:1). Students will begin to notice that related unit rates are reciprocals. It is not intended that this be taught
as an algorithm or rule because at this level, students should primarily use reasoning to find these unit rates. In Grade 6, students are not expected to
work with unit rates expressed as complex fractions. Both the numerator and denominator of the original ratio will be whole numbers.
9
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (6.RP.1-3)
Domain: Ratios and Proportional Relationships (6.RP)
Big Idea (Cluster): Understand ratio concepts and use ratio reasoning to solve problems
Standard 6.RP.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.
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. Plotting of pairs of values on the coordinate plane will be addressed in
Variables and Patterns..
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.
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.
d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
See Grade 6 Math Flip Book pages 11-13 for explanations and examples of this standard.
Learning Objectives
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Solve unit rate problems (SBAC Grade 6 Claim 1 Target A ALD Level 3*).
Use ratio reasoning to solve and understand the concept of unit rates in multi-step problems including instances of unit pricing and constant speed
(SBAC Grade 6 Claim 1 Target A ALD Level 3*).
Solve percent problems by finding the whole, given a part and the percent (SBAC Grade 6 Claim 1 Target A ALD Level 3)
Use tables, tape diagrams, double number lines, and equations to compare, solve problems with ratios and rates and find missing values.
Apply concepts of rate and ratio to solve real-world problems.
Find a percent of a number as a rate per 100 in real-world and mathematical problems.
Find the whole given a part and a percent in real-world and mathematical problems.
Use ratios to covert measurement units in real-world and mathematical problems.
Compare ratios by graphing points of a coordinate plane that represent equivalent ratios (for example there are two cats for every six dogs, (2,6),
(3,9), and (4,12)).
10
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (6.RP.1-3)
SBAC Required Evidence (Claim 1)
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Student finds missing values in tables of equivalent rations (SBAC
Grade 6 Claim 1 Target A, required evidence #3)
Student makes tables of equivalent ratios relating quantities with
whole-number measurements (SBAC Grade 6 Claim 1 Target A,
required evidence #5)
Student solves real-world and mathematical problems involving
unit rate (SBAC Grade 6 Claim 1 Target A, required evidence #6).
Student solves mathematical problems involving finding the
whole, given a part and the percent (SBAC Grade 6 Claim 1 Target
A, required evidence #7).
Student solves real-world and mathematical problems involving
finding a percent of a quantity as a rate per 100 (SBAC Grade 6
Claim 1 Target A, required evidence #8).
Student uses ratio reasoning to convert measurement units (SBAC
Grade 6 Claim 1 Target A, required evidence #9).
Student uses ratio reasoning to manipulate and transform units
appropriately when multiplying and dividing quantities (SBAC
Grade 6 Claim 1 Target A, required evidence #10).
Questions to Develop Mathematical Thinking
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How might solving for a unit rate be helpful at home?
How is a rate used to compare two quantities with different measures?
How might a rate table or graph help identify or verify the unit rate?
After finding the unit rate, is there a different unit rate using the same
ratio?
For more information on the assessment of this standard, read the
Claim 1 SBAC Item Specifications. This cluster of standards will also
be assessed through Claim 2 (Problem Solving), Claim 3
(Communicating Reasoning) , and Claim 4 (Modeling and Data
Analysis) assessment items.
Requisite Prior Knowledge






Fluently multiply and divide whole numbers.
Recognize and extend patterns involving whole numbers.
Multiply and divide with fractions and decimals.
Convert between measurement units.
Graph points in first quadrant of coordinate plane.
Understand fraction equivalence.
Connections to Prior Learning
In Grade 5, students will have graphed ordered pairs in the first quadrant.
In Grade 5, students will have interpreted a fraction as division. Also in
Grade 5 (standard 5.MD.1), students will have converted different-sized
measurement units within a given measurement system (e.g., convert 5 cm
to 0.05 m).
11
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (6.RP.1-3)
Connections to Curriculum Resources
Additional Resources/Technology Resources
Comparing and Scaling Investigations 1-3 fairly align to this
standard. Teachers will need to supplement with resources on tape
diagrams, missing values in rate tables and unit rates with constant
speed. Teachers will also need to supplement the percent and
measurement conversion portion of this standard. CMP2 does not have
a measurement conversion investigation and the Bits and Pieces III
Investigations 4 and 5 are above the Grade 6 percent standards. See
additional resources for suggested supplementary lessons. The pacing
guide also has suggested supplemental resources.
Introduction to rate tables video
Ratio table applications video
Solving with tape diagrams video
Ratio and Proportional Relationships PowerPoint
Ratio and Proportion module 1 lessons 5-15, 19-23 (engage NY)
Ratio and Proportion tasks (Trinity University – click download)
Ratios and Proportions module (NWPS)
Percent lessons Engage NY Grade 6 Module 1 lessons 24-29
Measurement Conversion lessons Engage NY Grade 6 Module 1 Lesson
21
Instructional Strategies
Using ratio tables develops the concept of proportion. By comparing equivalent ratios, the concept of proportional thinking is developed and many
problems can be easily solved. As students are making sense of solving with ratios and rates in contextual problems, ratio tables will support their
multiplicative reasoning.
Together with tables, students can also use tape diagrams and double number line diagrams to represent collections of equivalent ratios. Tape
diagrams are best used when the two quantities have the same units. Double number line diagrams are best used when the quantities have different
units. A collection of equivalent ratios can be graphed in the coordinate plane. The unit rate appears in an equation and on a graph as the slope of
the line, and in the coordinate pair when the x-value is 1.
Multiplicative reasoning is also used when finding the missing element in a proportional relationship. For example, use 2 cups of concentrate to 5
cups of water to make fruit punch. If 6 cups of concentrate are used to make punch, how many cups of water are needed? Recognize that the
relationship between 2 and 6 is 3 times; 2 · 3 = 6 To find the amount of cups of water, the relationship between 5 and the missing amount must also
be 3 times. 3 · 5 = cups of water, therefore, you would need 15 cups of water.
This is the students’ first introduction to percents. Percentages are a rate per 100. Models, such as percent bars or 10 x 10 grids should be used to
model percents. Percents are often taught in relationship to learning fractions and decimals. This cluster indicates that percents are to be taught as a
special type of rate. Provide students with opportunities to find percents in the same ways they would solve rates and proportional relationships.
Students should also solve real-life problems involving measurement units that need to be converted. Representing these measurement conversions
with models such as ratio tables, t-charts or double number line diagrams will help students internalize the size relationships between same system
measurements and relate the process of converting to the solution of a ratio.
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This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (6.RP.1-3)
Explanations and Examples
Ratios and rates can be used in ratio tables and graphs to solve problems. Previously, students have used additive reasoning in tables to solve
problems. To begin the shift to proportional reasoning, students need to begin using multiplicative reasoning. When working with ratio tables and
graphs, whole number measurements are the expectation for this standard.
To help understand the multiplicative relationship between a ratio or rate, students write equations by identifying the independent and dependent
variables (parts). This portion of the standard will be addressed in Variables and Patterns.
Students use percentages to find the part when given the percent, by recognizing that the whole is being divided into 100 parts and then taking a part
of them (the percent). For example, to find 40% of 30, students could use a 10 x 10 grid to represent the whole (or 30). If the 30 is divided into 100
parts, the rate for one block is 0.3. Forty percent would be 40 of the blocks, or 40 x 0.3, which equals 12. Student should also find the whole, given
a part and the percent. For example, if 25% of the students in Mrs. Rutherford’s class like chocolate ice cream, then how many students are in Mrs.
Rutherford’s class if 6 like chocolate ice cream? Students can reason that if 25% is 6 and 100% is 4 times the 25%, then, 6 times 4 would give 24
students in Mrs. Rutherford’s class.
A ratio can be used to compare measures of two different types, such as inches per foot, milliliters per liter and centimeters per inch. Students
recognize that a conversion factor is a fraction equal to 1 since the quantity described in the numerator and denominator is the same. For example, 12
inches to 1 foot (12:1). Students use ratios to convert between measurement systems using ratios. For example, if 2.54 cm = 1 inch, then how many
cm in 1 foot? Students could set up equivalent fractions or use a ratio table to
find the number of centimeters in 1 foot.
Items 632 and 651 are from the SBAC Grade 6 Practice Test.
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This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (6.RP.1-3)
Examples from Engage NY Module 1: Ratios and Unit Rates
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This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (6.RP.1-3)
The Engage NY Grade 6 Module 1 is being used to supplement this cluster of standards where CMP2 curricular materials do not provide resources
aligned to Common Core Standards 6.RP.1-3 beyond what is provided in Comparing and Scaling Investigations 1-3. Please be selective in the
resources within the Engage NY modules you use for your students to engage with in order to meet proficiency or higher on this standard. The
Engage NY is fairly absent of the Standards for Mathematical Practice which should be equally weighted with content on a daily basis in classroom
instruction. For the time-being, teachers will need to use the Engage NY resources as a foundation to access these standards. But, teachers are
highly encouraged to improve the recommended tasks to be less direct instruction based, have students engage in rich mathematical talk and provide
opportunities for students to work collaboratively. The Engage NY modules are a temporary solution as we transition to the Common Core State
Standards for Mathematics.
Ratio and Proportional Reasoning Unit Pacing Guide using Comparing and Scaling
Comparing and Scaling
Investigations and CCSS-M
Aligned Lessons
Visual representations of ratios and
using ratio language (MISSING
LESSONS)
Spend 2-3 lessons developing ratio
language before you move into
Comparing and Scaling.
Investigation 1.1 (this lesson may
be skipped)
Problem
Set
Alignment Notes
CCSS-M
Standards
Students will need an introduction to ratio and using ratio language before they 6.RP.1
begin the Comparing and Scaling unit. Use the provided resources or teacherdeveloped resources to have students use ratio language, connect visual and
pictorial models to ratio notation. Have students identify part-to-part and partto-whole relationships. Begin to introduce tape diagrams and ratio tables to
work with equivalent ratios with whole number values, initially. Move beyond
pictorial models of ratios and use a fraction diagram model to identify part-topart and part-to-whole relationships. Extend to real-world contextual situations.
Two PowerPoint presentations and lessons 1-4 of the Engage NY Ratio and
Proportion module are hyperlinked within this document under 6.RP.1
and the PowerPoint presentations may be found on the Secondary Math
SWIFT page. The Engage NY module may also be found in your Math 6
Notebook (white binder).
A-D, E
Lesson begins to develop language of ratios by using comparison statements
6.RP.1
extension that are part-to-part, part-to-whole, difference, equivalent ratio, and percent.
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This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (6.RP.1-3)
Investigation 1.2
Instead of Investigation 1.3,
modify and use page 11 ACE 4-7
Investigation 2.1
Investigation 2.2
Investigation 2.3
A-B, C-D Letter B allows for students to use tape diagrams to justify reasoning. Continue
extension to review part-to-part and part-to-whole statements. Part B reinforces the
question “is a percent part-to-part or part-to-whole,” and “is the fraction being
used as part-to-whole relationship where the whole has changed or remained as
a total.” Continue to ask, “what is the part” and “what is the whole.” Have
students justify letter B reasoning in two ways in order to reinforce multiple
representations. Equivalent ratios can be highlighted between B2, B3, and the
narrative. “Where did the 2 and 1 come from?” Reinforce ratio notations.
Have students use tape diagrams to make sense of the ratio relationships when
needed. Reinforce using multiple representations to justify accurate statements.
These ACE questions provide an opportunity to review decimals and nonbenchmark percent. Push students away from using difference as their strategy
of comparison between part-to-part and part-to-whole. Students need to move
towards multiplicative reasoning. Reinforce “what is the part” and “what is the
whole.” Have students identify if statement is part-to-part or part-to-whole.
A-C, D
Have students identify what makes a juice most orangey? Is orangey-ness a
and E
part-to-part or part-to-whole relationship? Students could use equivalent ratios,
extension ratio tables, models, percent and/or common denominators to make sense of the
ratio comparison. Students need to understand what they are solving for; what
determines orangey-ness.
A, B, D
On letter D, students complete a multi-step contextual problem which requires
students to convert tables to number of seats in order to solve. Students may
use a table or tape diagram to solve once tables are changed to number of seats.
A, B, C
Lesson reinforces ratio tables to find missing values. Lesson also reinforces
ratios written part-to-part form is re-written part-to-whole when fraction form is
used. To extend to 6.RP.3a coordinate graphing, graph the two ratios on a
coordinate grid using different colors so students see the ratio comparisons.
6.RP.1
6.RP.1
6.RP.1
6.RP.3
6.RP.3a, 6.RP.1
6.RP.1, 6.RP.3a
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This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (6.RP.1-3)
CMP2 Supplemental lessons
CC Investigation 1: Ratios and
Rates (possible lessons to help
extend rate content)
Investigation 3.1
Investigation 3.4
Investigation 3.3
Investigation 3.2
Engage NY Module 1 Lesson 21
“Speed, work and measurement
units”
Inv 1.11.3
These lessons could be used to deepen the Comparing and Scaling unit if
students need more practice with rates and unit rates following Investigation 3.
As you look through investigations 3.1-3.4 in the text, determine where these
lessons may or may not be used to support depth of student learning around
ratios and rates. You could also use lessons 16-18 from Engage NY module to
deepen Investigation 3 learning with rates and unit rates in connection with
tables and graphs. You could also use lessons from Newark Public Schools on
Ratios and Proportional Relationships to strengthen rate and unit rate concepts
in Investigation 3 of Comparing and Scaling.
A-F (add Students should be introduced to rate language. Take time on the launch of this
on unit
lesson in order for students to make meaning of rate as a comparison of
rate for
quantities with two different units of measure. The lesson does not call out
each
specifically to identify unit rate. But, this is the first opportunity to make this
calculator connection. Have students write the unit rate for each calculator. Have
on A)
students continually identify the differing units of measure found in the rate.
A, C (B, This lesson reinforces that every rate has two different unit rates. Having
D&E
students stay focused on the meaning of each rate and their usefulness will be
are
important, especially as they solve B, D and E. Reinforce the differing phrases
extension used to describe a unit rate, “for every,” “for each,” “for each one” and “per.”
)
Encourage students to use rate tables, graphs, and models in order to solve.
A,B, C, E Encourage students to use a rate table or double number line in order to
determine the unit rate. Reinforce the differing phrases used to describe a unit
rate, “for every,” “for each,” “for each one” and “per.”
A-D
This lesson provides an opportunity to talk about non-proportional relationships
as you look at the race in its entirety. These relationships surface again in
Variables and Patterns. Have students continually identify the differing units of
measure found in the rate. In order to reinforce coordinate graphing, students
could graph the rates for the different stops to visually connect the differing
rates to compare fastest and slowest times.
Comparing and Scaling is missing application of rates with measurement
conversions. Engage NY Module 1 lesson 21 will begin to model for students
how to use rates to covert between measurements. This lesson will likely take 2
days. CMP2 supplemental investigation 1.4 is another lesson to support this
standard.
6.RP.2
6.RP.2, 6.RP.3a,
6.RP.3b
6.RP.2, 6.RP.3a,
6.RP.3b
6.RP.2, 6.RP.3b,
6.RP.3c
6.RP.2, 6.RP.3b
6.RP.3d
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This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (6.RP.1-3)
Engage NY Module 1 Lessons 2429 Percent Lessons
GIVE THE DISTRICTDEVELOPED COMMON
ASSESSMENT
CMP2 Bits and Pieces III investigations 4 and 5 will no longer be taught in
Grade 6 since the lessons address 7.RP, not 6.RP. In the meantime, we will use
Engage NY Module 1 lessons 24-29 to develop understanding of percent as rate
and solve for whole, given part and percent or solve for part, given whole and
percent.
Common assessment should be given at the end of this unit. The
assessment will include level 3 and level 4 questions (based on the rubrics
at end of unit guide) covering ratio, unit rate, percent, representations of
rate and measurement conversion.
6.RP.3c
6.RP.1, 6.RP.2,
6.RP.3
Hard copies of these lessons can be found at http://staff.rentonschools.us/renton/secondary-math/math-6-ccss-m-resources.
The following resources were used to create this curriculum guide: Grade 6 Mathematics SBAC Item Specifications Claim 1 for Target A, Grade 6
level Common Core State Standards Flip Book compiled by Melisa Hancock, 6-7 Ratios and Proportional Relationships progression of learning
document.
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This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (6.RP.1-3)
Standard 6.RP.1
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.
6.RP.1
4
In addition to being proficient
on the standard, student can
demonstrate one or more of the
following:
 Applies understanding of
standard to unfamiliar
situations and/or to solve
complex problems.
 Use precise and relevant
communication to justify
mathematical thinking.
 Connects knowledge to
other learning targets and/or
advance problem sets.


3
Use, write and identify
statements that use ratio
language to describe a ratio
relationship.
Describe a ratio relationship
between any two number
quantities with denominators
less than or equal to 12.


2
Identify statements that use
ratio language to describe a
ratio relationship.
Describe simple,
straightforward ratio
relationship between two
quantities.


1
With help, minimal
success identifying and
describing simple ratio
relationships.
Describe a ratio
relationship between two
quantities with errors
and/or misconceptions.
For example, explain ratio
relationships between any two
number quantities.
Rubric constructed from the SBAC Claim 1 item specifications and SBAC Achievement Level Descriptors. A PLC may decide to further develop
these rubrics.
For more information on the assessment of this standard, read the Claim 1 SBAC Item Specifications-Target A.
19
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (6.RP.1-3)
Standard 6.RP.2
Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0 (b not equal to zero), and use rate language in the context of a ratio
relationship.
4
In addition to being proficient
on the standard, student can
demonstrate one or more of the
following:
 Applies understanding of
standard to unfamiliar
situations and/or to solve
complex problems.
 Use precise and relevant
communication to justify
mathematical thinking.
 Connects knowledge to
other learning targets and/or
advance problem sets.


6.RP.2
3
Identify and determine the

unit rate associated with a
ratio.

Use ratio reasoning and
language to understand the
concept of unit rate.
2
Understand concept of
simple unit rates.
Determine unit rates given
two whole number
quantities where one evenly
divides the other.


1
With help, minimal
success determining the
unit rate given numerical
values that evenly divide
into each other.
Determine unit rates given
two whole number
quantities where one
evenly divides the other
with errors and/or
misconceptions.
For example, explain the ratio
relationship between two
quantities with different
measurement units.
Rubric constructed from the SBAC Claim 1 item specifications and SBAC Achievement Level Descriptors. A PLC may decide to further develop
these rubrics.
For more information on the assessment of this standard, read the Claim 1 SBAC Item Specifications-Target A.
.
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This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (6.RP.1-3)
Standard 6.RP.3 a - d
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.
4
In addition to being proficient on
the standard, student can
demonstrate one or more of the
following:
 Applies understanding of
standard to unfamiliar
situations and/or to solve
complex problems.
 Use precise and relevant
communication to justify
mathematical thinking.
 Connects knowledge to other
learning targets and/or advance
problem sets.
For example, solve unfamiliar or
multi-step problems by finding the
whole, given a part and the
percent.






6.RP.3
3
Create tables of equivalent ratios 
to find missing values and
compare ratios.
Solve real-world and

mathematical problems
involving ratios.
Solve real-world and

mathematical problems
involving rates.
Use ratio reasoning to solve and
understand the concept of unit

rates in multi-step problems
including instances of unit
pricing and constant speed.
Solve real-world and

mathematical problems
involving percent (find part
given percent and whole and
find whole given part and
percent).
Convert measurement units
using ratio reasoning between
same units of measure and
differing units of measure.
2
Creates tables of equivalent
ratios to compare simple
ratios.
Use ratio reasoning to solve
and understand the concept of
unit rates in one-step
problems.
Solve real-world or
mathematical problems
involving rates or ratios.
Solve mathematical problems
involving percent by finding
the part given percent and
whole.
Convert measurements
between same unit of measure.


1
With help, minimal success
using ratio reasoning to
solve and understand the
concept of unit rates in onestep problems.
Solve real-world or
mathematical problems
involving rates or ratios with
errors and/or
misconceptions.
Rubric constructed from the SBAC Claim 1 item specifications and SBAC Achievement Level Descriptors. A PLC may decide to further develop
these rubrics. For more information on the assessment of this standard, read the Claim 1 SBAC Item Specifications-Target A.
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This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (6.RP.1-3)