6th Grade Mathematics Curriculum Guide
2015 – 2016
Unit 2: Unit 2 Ratios, Rates, & Proportions
Time Frame: Quarter 1 – about 23 days
Connections to Previous Learning:
The study of ratios and proportional relationships extends students’ work in measurement and in multiplication and division from the elementary grades. It is expected that
students will have prior knowledge and experience related to concepts and skills such as multiples, factors, and divisibility rules. This background knowledge about relationships
and rules for multiplication and division of whole numbers connects to the understanding of how to complete tables to help support the development of ratio and rate
reasoning.
In Grade 6, students develop an understanding of ratio and proportion using ratio tables, tape diagrams, and double number lines.
Focus of this Unit:
Students learn that a ratio expresses the comparison between two quantities. Special types of ratios are rates, unit rates, measurement conversions, and percentages are
concepts that are applied to a variety of real world and mathematical situations. Students gain a deeper understanding of proportional reasoning through instruction and
practice. They develop and use multiplicative thinking to develop a sense of proportional reasoning as they describe ratio relationships between two quantities.
Students extend their understanding of ratios and develop understanding of proportionality to solve single and multi-step problems involving such real world contexts as
percent of increase or decrease and scale drawing.
Connections to Subsequent Learning:
Ratios and proportional relationships are foundational for further study in mathematics and science and useful in everyday life. Students use ratios in geometry and in algebra
when they study similar figures and slopes of lines, and later when they study sine, cosine, tangent, and other trigonometric ratios in high school.
Students use ratios when they work with situations involving constant rates of change, and later in calculus when they work with average and instantaneous rates of change of
functions. An understanding of ratio is essential in the sciences to make sense of quantities that involve derived attributes such as speed, acceleration, density, surface tension,
electric or magnetic field strength, and to understand percentages and ratios used in describing chemical solutions. Ratios and percentages are also useful in many situations in
daily life, such as in cooking and in calculating tips, miles per gallon, taxes, and discounts. They also are also involved in a variety of descriptive statistics, including demographic,
economic, medical, meteorological, and agricultural statistics (e.g., birth rate, per capita income, body mass index, rain fall, and crop yield) and underlie a variety of measures,
for example, in finance (exchange rate), medicine (dose for a given body weight), and technology (kilobits per second). Students will use this understanding of proportionality to
find scale factors between geometric figures and develop understandings of congruence and similarity. They will use ratio tables to study statistics and probability.
Mathematical Practices
1. Make Sense of Problems and Persevere in Solving Them.
2. Reason Abstractly and Quantitatively.
3. Construct Viable Arguments and Critique the Reasoning of Others.
4. Model with Mathematics.
Unit 2
5. Use Appropriate Tools Strategically.
6. Attend to Precision.
7. Look for and Make Use of Structure.
8. Look for and Express Regularity in Repeated Reasoning
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6th Grade Mathematics Curriculum Guide
Stage 1 Desired Results
2015 – 2016
Transfer Goals
Students will be able to independently use their learning to…
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Make decisions based upon the proportional relationships that they see in their lives.
How can I use proportional relationships to solve ratio and percent problems?
Meaning Goals
UNDERSTANDINGS
Students will understand that…
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ESSENTIAL QUESTIONS
A ratio expresses the comparison between two quantities. Special types of ratios are
rates, unit rates, measurement conversions, and percents.
A ratio or a rate expresses the relationship between two quantities. Ratio and rate
language is used to describe a relationship between two quantities
(including unit rates.)
A rate is a type of ratio that represents a measure, quantity, or frequency, typically
one measured against a different type of measure, quantity, or frequency.
Ratio and rate reasoning can be applied to many different types of mathematical
and real-life problems (rate and unit rate problems, scaling, unit pricing, statistical
analysis, etc.).
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When is it useful to be able to relate one quantity to another?
How are ratios and rates similar and different?
What is the connection between a ratio and a fraction?
Acquisition Goals
Students will know…
Students will be skilled at…
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A ratio compares two related quantities.
Ratios can be represented in a variety of formats including each, to, per, for each, %,
1/5, etc.
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.
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
Unit 2
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Use ratio language to describe a ratio relationship between two quantities. (6.RP.1)
Represent a ratio relationship between two quantities using manipulatives and/or
pictures, symbols and real-life situations. (a to b, a:b, or a/b) (6.RP.1)
Represent unit rate associated with ratios using visuals, charts, symbols, real-life
situations and rate language. (6.RP.2)
Use ratio and rate reasoning to solve real-world and mathematical problems.
(6.RP.3)
Make and interpret tables of equivalent ratios. (6.RP.3)
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6th Grade Mathematics Curriculum Guide
tables, tape diagrams, double line diagrams, equations,
equivalent fractions, graphs, etc
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Calculators
6.RP.1
6.RP.2
6.RP.3
6.RP.3a
6.RP.3b
6.RP.3c
6.RP.3d
no
yes
2015 – 2016
Plot pairs of values of the quantities being compared on the coordinate plane.
(6.RP.3)
Use multiple representations such as tape diagrams, double number line diagrams,
or equations to solve rate and ratio problems. (6.RP.3)
Solve unit rate problems (including unit pricing and constant speed). (6.RP.3)
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. (6.RP.3)
Materials Needed:
Holt Course 1
Holt Course 2
Holt Course 3
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Engage NY 6 Module 1
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yes
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yes
Stage 1 Established Goals: Common Core State Standards for Mathematics
6.RP.A Understand ratio concepts and use ratio reasoning to solve problems.
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.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. 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.3a 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.3b 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.3c 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.3d Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
Unit 2
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6th Grade Mathematics Curriculum Guide
2015 – 2016
Explanations, Examples, and Comments
Overview of Suggested Assessments & Performance Tasks for Unit
Vocabulary will be listed next to each standard
End-of-Unit Assessment – Engage NY Module 1
Percent, proportion, rate, unit rate, ratio, quantity, tape diagram, double number line,
numerator, denominator, equivalent, part-to-part, part-to-whole
Formative assessments for this unit
Suggested Performance Tasks/Activities
Georgia
• Ratios and Rates 6.RP.1 6.RP.2, 6.RP.3a & b
• How Many Noses are in Your Arms? 6.RP.1, 6.RP.3d
• Fruit Punch 6.RP.3
• Reaching the Goal 6.RP.3c
• Ice Cream or Cake 6.RP.3c
• Free Throws 6.RP.3c
Illuminations
It’s Not Heavy, It’s Your
Grade http://illuminations.nctm.org/Lesson.aspx?id=3173
Summer Daze
http://illuminations.nctm.org/Lesson.aspx?id=2847
Ti Activities
Ratios & Unit Rates
Exploring Percents
Engage NY Module 1
http://www.engageny.org/resource/grade-6-mathematics-module-1
6.RP.A.1
Vocab – ratio, part-to-part, part-to-whole
A ratio is the comparison of two quantities or measures. The comparison can be part-to-whole
(ratio of guppies to all fish in an aquarium) or part-to-part (ratio of guppies to goldfish).
A ratio is a comparison of two quantities which can be written as
Unit 2
Stage 3 MATERIALS BY STANDARD(S):
Teacher should use assessment data to determine which of the materials
below best meet student instructional needs. All materials listed may not be
needed.
6.RP.A.1 and 6.RP.A.2:
Holt Course 1 Lesson 7-1 Ratios and Rates, 6.RP.A.1 (1)
Holt Course 2 Lesson 5-2 Ratios, Rates, and Unit Rates, 6.RP.A.2 (1)
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6th Grade Mathematics Curriculum Guide
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.
A comparison of 8 black circles to 4 white circles can be written as the ratio of 8:4 and can be
regrouped into 4 black circles to 2 white circles (4:2) and 2 black circles to 1 white circle (2:1).
Students should be able to identify all these ratios and describe them using “For every…., there
are …”
2015 – 2016
Holt Course 3 Lesson 5-2 Ratios, Rates, and Unit Rates OPTIONAL
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Engage NY 6 Module 1 Lessons 9 Tables of Equivalent Ratios, 6.RP.2 (1)
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Engage NY 6 Module 1 Lessons 10 The Structure of Ratio Tables: Additive
and Multiplicative, 6.RP.2 (1)
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Engage NY 6 Module 1 Lessons 11 Comparing Ratios Using Ratio
Tables, 6.RP.2 (1)
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Engage NY 6 Module 1 Lessons 14 From Ratio Tables, Equations, & Double
Number Line Diagrams to Plots on the Coordinate Plane, 6.RP.2 (1)
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Engage NY 6 Module 1 Lessons 18 Finding a Rate by Dividing Two
Quantities, 6.RP.2 (1)
Additional Resources, if necessary
Holt Course 2 Lesson 5-1 Ratios (1)
6.RP.A.2
Vocab – rate, unit rate, numerator, denominator
A unit rate compares a quantity in terms of one unit of another quantity. 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 will begin to notice that related unit rates are
reciprocals as in the first example. 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.
Examples:
o On a bicycle you can travel 20 miles in 4 hours. What are the unit rates in this
situation, (the distance you can travel in 1 hour and the amount of time required to
travel 1 mile)?
Unit 2
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6th Grade Mathematics Curriculum Guide
o
2015 – 2016
A simple modeling clay recipe calls for 1 cup corn starch, 2 cups salt, and 2 cups boiling
water. How many cups of corn starch are needed to mix with each cup of salt?
6.RP.A.3
6.RP.A.3:
Vocab – percent, proportion, quantity, tape diagram, double number line, equivalent
Engage NY 6 Module 1 Lesson 15 A Synthesis of Representations of
Equivalent Ratio Collections 6.RP.A.3a (4) OPTIONAL
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Engage NY 6 Module 1 Lessons 19-23, 6.RP.2, 6.RP.3b and d (5)
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Engage NY 6 Module 1 Lessons 24-28, 6.RP.3c (5)
In 6th grade, students used ratio tables and unit rates to solve problems. Students expand their
understanding of proportional reasoning to solve problems that are easier to solve with crossmultiplication. Students understand the mathematical foundation for cross-multiplication.
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Using the information in the table, find the number of yards in 24 feet.
There are several strategies that students could use to determine the solution to this
problem.
o Add quantities from the table to total 24 feet (9 feet and 15 feet); therefore the
number of yards must be 8 yards (3 yards and 5 yards).
o Use multiplication to find 24 feet: 1) 3 feet x 8 = 24 feet; therefore 1 yard x 8 = 8
yards, or 2) 6 feet x 4 = 24 feet; therefore 2 yards x 4 = 8 yards.
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Compare the number of black to white circles. If the ratio remains the same, how many black
circles will you have if you have 60 white circles?
Unit 2
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Additional Resources, if necessary
Holt Course 1 Lesson 7-7 Percents (1)
Holt Course 1 Lesson 7-8 Percents, Decimals, and Fractions (1)
Holt Course 1 Lesson 7-9 Percent Problems (1)
Holt Course 2 Lesson 6-3 Estimate with Percents (1)
Holt Course 2 Lesson 6-4 Lab Explore Percents (1)
Holt Course 2 Lesson 6-4 Percent of a Number
Holt Course 3 Lesson 6-1 Relating Decimals, Fractions, and Percents (1)
Holt Course 3 Lesson 6-2 Estimate with Percents)
Holt Course 3 Lesson 6-4 Finding a Number When the Percent is Known (1)
CMP Bits and Pieces I Investigation 6.3 Changing Forms
CMP Bits and Pieces I Investigation 6.4 It’s Raining Cats and Dogs
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6th Grade Mathematics Curriculum Guide
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If 6 is 30% of a value, what is that value? (Solution: 20)
2015 – 2016
Performance Tasks: Georgia (on P drive)
• Ratios and Rates 6.RP.1 6.RP.2, 6.RP.3a & b
• How Many Noses are in Your Arms? 6.RP.1, 6.RP.3d
• Fruit Punch 6.RP.3
• Reaching the Goal 6.RP.3c
• Ice Cream or Cake 6.RP.3c
• Free Throws 6.RP.3c
Illuminations
It’s Not Heavy, It’s Your
Grade http://illuminations.nctm.org/Lesson.aspx?id=3173
Summer Daze
http://illuminations.nctm.org/Lesson.aspx?id=2847
Stage 2 - Evidence
Evaluative Criteria/Assessment Level Descriptors (ALDs):
OTHER Assessment Evidence
6.RP.A (SBAC Target A)
Formative assessments (available on P: drive)
Spaced Learning Over Time (SLOT - entrance & exit slips)
Level 4 students should be able to solve unfamiliar or multi-step problems by finding
Checking for understanding (CFU’s)
the whole, given a part and the percent; explain ratio relationships between any two
Lesson Quizzes
number quantities; and identify relationships between models or representations.
Classroom Assessments
Level 3 students should be able to use ratio reasoning to solve and understand the
concept of unit rates in unfamiliar or multi-step problems, including instances of unit
pricing and constant speed, and solve percent problems by finding the whole, given a
part and the percent. They should be able to describe a ratio relationship between any
two number quantities (denominators less than or equal to 12).
SBA Examples
Level 2 students should be able to understand the concept of unit rate in
straightforward, well-posed problems and solve straightforward, well-posed, one-step Claim 1 Item Specs
problems requiring ratio reasoning.
Level 1 students should be able to describe a ratio relationship between two whole
number quantities, find missing values in tables that display a proportional
relationship, and plot the pairs of values from a table on the coordinate plane. They
should be able to find a percent as a rate per hundred and convert measurement
units.
Unit 2
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6th Grade Mathematics Curriculum Guide
LEARNING ACTIVITIES:
Stage 3 – Learning Plan Sample
Summary of Key Learning Events and Instruction that serves as a guide to a detailed lesson planning
NOTES:
2015 – 2016
**Days may change depending on any tasks or assessing you choose to do.
6.RP.A.1 and 6.RP.A.2:
Day 1: Ratios and Rates, 6.RP.A.1 (1)
• Holt Course 1 Lesson 7-1
Day 2: Ratios, Rates, and Unit Rates, 6.RP.A.2 (1)
• Holt Course 2 Lesson 5-2
Optional: Ratios, Rates, and Unit Rates, 6.RP.A.2 (1)
• Holt Course 3 Lesson 5-2
Day 3: Tables of Equivalent Ratios
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• Engage NY 6 Module 1 Lesson 9
Day 4: The Structure of Ratio Tables: Additive and Multiplicative
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• Engage NY 6 Module 1 Lessons 10
Day 5: Comparing Ratios Using Ratio Tables
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• Engage NY 6 Module 1 Lessons 11
Day 6: From Ratio Tables, Equations, & Double Number Line Diagrams to Plots on the Coordinate Plane
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• Engage NY 6 Module 1 Lessons 14
Day 7: Finding a Rate by Dividing Two Quantities
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• Engage NY 6 Module 1 Lessons 18
Unit 2
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6th Grade Mathematics Curriculum Guide
Stage 3 – Learning Plan Sample
6.RP.A.3:
2015 – 2016
Optional: A Synthesis of Representations of Equivalent Ratio Collections
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• Engage NY 6 Module 1 Lessons 15
Day 8: Comparison Shopping – Unit Price & Related Measurement Conversions
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• Engage NY 6 Module 1 Lessons 19
Day 9: Comparison Shopping – Unit Price & Related Measurement Conversions
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• Engage NY 6 Module 1 Lessons 20
Day 10: Getting the Job Done – Speed, Work, & Measurement Units
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• Engage NY 6 Module 1 Lessons 21
Day 11: Getting the Job Done – Speed, Work, & Measurement Units
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• Engage NY 6 Module 1 Lessons 22
Day 12: Problem-Solving Using Rates, Unit Rates, & Conversions
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• Engage NY 6 Module 1 Lessons 23
Day 13: Percent and Rates per 100
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• Engage NY 6 Module 1 Lessons 24
Day 14: A Fraction as a Percent
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• Engage NY 6 Module 1 Lessons 25
Day 15: Percent of a Quantity
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• Engage NY 6 Module 1 Lessons 26
Day 16: Solving Percent Problems
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• Engage NY 6 Module 1 Lessons 27
Day 17: Solving Percent Problems
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• Engage NY 6 Module 1 Lessons 28
Unit 2
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6th Grade Mathematics Curriculum Guide
Stage 3 – Learning Plan Sample
Daily Lesson Plan
Learning Target:
Warm-up:
Activities:
• Whole Group:
• Small Group/Guided/Collaborative/Independent:
• Whole Group:
Checking for Understanding (before, during and after):
Assessments:
Unit 2
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2015 – 2016
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