6th Grade Mathematics
UNIT 2: RATIOS AND UNIT RATES
Ratios and Unit Rates (8 weeks)
Students develop an understanding of ratio concepts and using ratio reasoning to solve problems.
By viewing equivalent ratios and rates as deriving from, and extending, pairs of rows (or columns)
in the multiplication table, and by analyzing simple drawings that indicate the relative size of
quantities, students connect their understanding of multiplication and division with ratios and
rates. Students learn to use ratio and rate language to describe relationships; expand the scope of
problems for which they can use multiplication and division; connect ratios and fractions; and solve
a wide variety of problems involving ratios and rates.
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6th Grade Mathematics
Key Ideas:
 A ratio is a multiplicative comparison of
two quantities.
 A ratio compares two related quantities.
 If one quantity in a ratio is multiplied or
divided by a particular factor, then the
other quantity must be multiplied by the
same factor to maintain the proportional
relationship.
 Ratios can be represented in a variety of
formats including each, to, per, for each,
%, 1/5, etc.
 Knowledge of equivalent fractions and
equivalent ratios can be used to model
ratio, rate and proportional reasoning.
 This knowledge can be generalized into
algorithms for solving ratio, rate and
proportion problems.
 A percent is a type of ratio that compares
a quantity to 100.
 A unit rate is the ratio of two
measurements in which the second term
is 1.
 Determine when it is appropriate to use
ratios/rates to solve mathematical or real
life problems.
 Mathematical strategies for solving
problems involving ratios and rates,
including tables, tape diagrams, double
line diagrams, equations, equivalent
fractions, graphs, etc.
Guiding/Focus Questions:
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How can reasoning with ratios and rates help
solve real-world and mathematical problems?
How are ratios and percents alike? How are
they different?
How do you determine a unit rate given a table
of values?
What are the differences between converting
measurements in the metric and standard
measurement systems?
Given the quantity and price of two objects,
how can you determine which one is the better
buy?
How are fractions,decimals andpercents
related?
How are common factors and multiples applied
to solve real-world scenarios?
How can I use multiplication and division to
solve ratio and rate problems?
What is the difference between a ratio and a
rate?
How can I use tables of equivalent ratios, tape
diagrams, double number line diagrams, or
equations to compare rates of change?
How and where are ratios and rates used in the
real world?
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6th Grade Mathematics
NYS Common Core Standards for Mathematics
https://www.engageny.org/resource/grade-6-mathematics]
Focus Standards:
Understand ratio concepts and use ratio reasoning to solve problems.
6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship
between two quantities
6.RP.2 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.
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.
b. Solve unit rate problems including those involving unit pricing and constant speed.
c. Find a percent of a quantity as a rate per; 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.
Foundational Standards(Prerequisites):
Use the four operations with whole numbers to solve problems.
4.OA.2 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.3
Apply and extend previous understandings of multiplication and division to multiply and divide
fractions.
5.NF.3 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.
Convert like measurement units within a given measurement system.
5.MD.1 Convert among different-sized standard measurement units within a given measurement
system and use these conversions in solving multi-step, real world problems.
Graph points on the coordinate plane to solve real-world and mathematical problems.
5.G.1 Define a coordinate system, with the intersection of the lines (the origin) arranged to
coincide with the 0 on each line and a given point in the plane located by using coordinates.
Understand that coordinates indicate how far to travel from the origin in the direction onboth axis
5.G.2 Represent real world and mathematical problems by graphing points in the first quadrant of
the coordinate plane, and interpret coordinate values of points in the context of the situation.
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6th Grade Mathematics
Focus Standards for Mathematical Practice:
MP.1 Make sense of problems and persevere in solving them. Students make sense of and solve
real world and mathematical ratio, rate, and percent problems using representations, such as tape
diagrams, ratio tables, the coordinate plane, and/or double number line diagrams. Problems
include ratio problems involving the comparison of three quantities, multistep changing ratio
problems, using a given ratio to find associated ratios, and constant rate problems including two or
more people or machines working together.
MP.2 Reason abstractly and quantitatively. Students solve problems by analyzing and comparing
ratios and unit rates given in tables, equations, and graphs. Students decontextualize a given
constant speed situation, representing symbolically the quantities involved with the formula.
MP.5 Use appropriate tools strategically. Students become proficient in selcecting and using a
variety of representations that are useful in reasoning with rate and ratio problems such as tape
diagrams, double line diagrams, ratio tables, a coordinate plane and equations.
MP.6 Attend to precision. Students define and distinguish between ratio, the value of a ratio, a
unit rate, a rate unit, and a rate. Students use precise language and symbols to describe ratios and
rates. Students learn and apply the precise definition of percent.
MP.7 Look for and make use of structure. Students recognize the structure of equivalent ratios in
solving word problems using tape diagrams. Students identifying the structure of a ratio table and
use it to find missing values in the table.
Content/Skills
6.RP1
Explain the concept of a ratio as a comparison of two quantities
Describe in written form (words and symbols) the relationship between two quantities.
Express ratios in simplest form and understand how the reduced value relates to the original
ratio.
Create and label a visual model of a given ratio.
6.RP2
Convert a rate to a unit rate.
Explain the concept of a unit rate as part-to-one.
Unit rate is related to a ratio.
Apply unit rates in real world situations.
6.RP3
Apply ratio and rate relationships and reasoning to solve real-world situations using tables, tape
diagrams, number lines and equations.
Make and interpret tables of equivalent ratios.
Plot pairs of values of the quantities being compared on the coordinate plane.
Use multiple representations such as tape diagrams, double number line diagrams or equations
to solve rate and ratio problems.
Solve unit rate problems (including unit pricing and constant speed).
Solve percent problems, including finding a percent of a quantity as a rate per 100 and finding
the whole, given the part and the percent.
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6th Grade Mathematics
Vocabulary/ Key Terms
 Ratio
 Equivalent ratio
 Rate
 Unit Rate
 Constant speed
 Equivalent Ratios
 Percent
 Proportion
 Double Number Line
 Ratio Table
 Convert
 Tape Diagram
 Coordinate Plane
 Ordered pair
 Axis
 Origin
 Equation
For ELLs, have them create a mini-glossary; instruct them to list each term, its meaning, and an
illustration for each. When students have finished their glossaries, have them share them with the
class. Allow them to use the glossaries as they work the exercises in the lesson and remind them to
use them when they do their homework.
ASSESSMENT EVIDENCE
Initial Assessment: Unit Readiness Test
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Formative Assessments:
Quizzes
Exit Slips
Checks for Understanding
Short and Extended-Response questions used throughout the unit.
Mathematical Reflections
Formative Assessments Tasks
 Performance Task 1 (Ratios and Equivalency)
 Performance Task 2 (Ratio and Unit Rates)
 Performance Task 3 (Percent)
Summative Assessment/s: Unit Test
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6th Grade Mathematics
TEACHING PLAN
Teaching and Learning Activities:
I.
Administer the Unit Readiness Test.
II.
Use the guiding questions to focus each lesson.
Lesson 1: Introduction to ratios
Lesson 2: Writing ratios
Lesson 3: Equivalent ratios
Lesson 4: Equivalent ratios
Lesson 5: Comparing ratios using ratio tables
Lesson 6: Ratio tables to double number line diagrams
Lesson 7: Equations using the value of ratios
Lesson 8: Plotting ratios on the coordinate plane
Lesson 9: Ratio problem solving
Lesson 10: Ratios: Review and Assess
Lesson 11: Understanding rates
Lesson 12: Unit rates
Lesson 13: Rate problem solving
Lesson 14: Consumer math using rates
Lesson 15: What is a percent?
Lesson 16: Percents and tape diagrams
Lesson 17: Percent problems – missing part
Lesson 18: Percent problems – missing percent
Lesson 19: Percent problems – missing whole
III.
Assess students on the unit.
 Final Performance Assessment
 Unit Test
Resources Needed:
 Connected Math Project 3 (CMP3): Comparing Bits and Pieces
 NYS Common Core Math Module 1: Ratio and Unit Rates
 https://www.engageny.org/resource/grade-6-mathematics-module-1
 Impact Mathematics Course 1
 Math Handbook
 MathXL (Pearson’s online homework, tutorial, and assessment system)
 Glencoe New York State Review Series
 Ready New York CCLS (6th Grade)
 njctl.org
 http://www.algebrahelp.com/
 https://grade6commoncoremath.wikispaces.hcpss.org
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6th Grade Mathematics
CALENDAR
Time
Spent
3 weeks
Standards
6.RP.1
Topics To Cover
 Ratio and
Equivalency
Resources
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3 weeks
6.RP.2
6.RP.3
 Rates and Unit
Rates
2 weeks
6.RP.3
 Percents
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Connected Math Project 3 (CMP3):Comparing
Bits and Pieces
NYS Common Core Math Module 1: Ratio and
Unit Rates
https://www.engageny.org/resource/grade-6mathematics-module-1
Impact Mathematics Course 1
Math Handbook
MathXL (Pearson’s online homework, tutorial,
and assessment system)
Glencoe New York State Review Series
Ready New York CCLS (6th Grade)
njctl.org
http://www.algebrahelp.com/
https://grade6commoncoremath.wikispaces.hcp
ss.org
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