Unit 1: Ratios and Proportional Relationships (6.RP)
Big Ideas:
Connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate
to solve problems:
Students use reasoning about multiplication and division to solve ratio and rate problems about
quantities. 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. Thus
students expand the scope of problems for which they can use multiplication and division to solve
problems, and they connect ratios and fractions. Students solve a wide variety of problems involving
ratios and rates. (CCSC Grade 6 p. 39)
Overview (Big Ideas), Enduring Understandings, Essential Questions, Common Misconceptions:
HCPSS Curriculum Framework Grade 6 Unit 1.docx
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Common Core Content Standards:
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.”
HCPSS UDL Lessons:
Which cup has a lighter color?
6.RP.1 Which cup has a lighter color, conceptual reasoning.doc
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6.RP.1 UDL Checklist Which Cup.docx
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Ratio Relationships
6.RP.1 Lesson Ratio Relationships.doc
6.RP.1 UDL Checklist Ratio Relationships.docx
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PARCC Assessment Limit/Clarification:
This standard is part of the major content cluster assessed on PARCC. Students' prior understanding of
an skill with multiplication, division, and fractions contribute to their study of ratios, proportional
relationships and unit rates.
PARCC/UTA Dana Center Prototype Assessment Item:
"Inches and Centimeters" (6.RP.1, 6.EE.9, MP 2)
http://ccsstoolbox.agilemind.com/parcc/about_middle_3789.html
Web Resources:
http://illuminations.nctm.org/LessonDetail.aspx?ID=L510
NCTM Illuminations Lesson: Students learn about ratios, including the "Golden Ratio", a ratio of length to
width that can be found in art, architecture, and nature. Students examine different ratios to determine
whether the Golden Ratio can be found in the human body.
http://illuminations.nctm.org/LessonDetail.aspx?ID=L722
NCTM Illuminations Lesson: By using sampling from a large collection of beans, students get a sense of
equivalent fractions which leads to a better understanding of ratios and proportions.
http://www.uen.org/Lessonplan/preview?LPid=21559
Lesson: This site provides lesson activities involving a number line to help students understand
equivalencies among rational numbers.
http://illustrativemathematics.org/illustrations/76
Task: Games at Recess. Students write sentences to compare the number of boys and girls playing
different games at recess.
http://illustrativemathematics.org/illustrations/496
Task: Ratio of Boys to Girls. Students analyze a part-whole relationship and build connections between
fractions and ratios. Tape diagrams are illustrated as a solution strategy.
http://illustrativemathematics.org/illustrations/61
Task: Voting for Two, Variation 1. Students analyze a part-part relationship to determine how many votes
each candidate received for class president. Ratio tables are illustrated as a solution strategy.
http://illustrativemathematics.org/illustrations/62
Task: Voting for Two, Variation 2. Students extend their thinking from Voting for Two, Variation 1.
http://illustrativemathematics.org/illustrations/63
Task: Voting for Two, Variation 3. Students extend their thinking from Voting for Two, Variation 1 and 2 by
exploring part-whole relationships as well.
Text Resources For Lessons/Tasks:
Lobato, J.E., Ellis, A.B, Charles, R.I., & Zbiek, R.M. (2010). Developing Essential Understanding of
Ratios, Proportions, & Proportional Reasoning for Teaching Mathematics in Grades 6-8. Reston, VA:
National Council of Teachers of Mathematics. p. 20.
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"Reflect 1.3" Task p. 20: In this task students determine
various ways to mix blue and yellow paint to make the
same shade of green paint.
"Figure 1.14" Task p. 22: Students analyze the ratio of
orange concentrate to water in orange juice.
"Reflect 1.4" Task p. 29: Students interpret the rate at
which water is being pumped into a swimming pool.
"Clown and Frog: Task p. 79: Students compare the
various speeds of Clown and Frog.
Van de Walle, J.A., & Lovin, L.H. (2006). "Chapter 18: Proportional Reasoning." Teaching StudentCentered Mathematics. Boston: Pearson. p. 248-268.
NCTM Journal Article Lesson Ideas/Resources:
Billings, E.M.H. (2002). Cocoa: Teaching Notes. Making Sense of Fractions, Ratios, and Proportions:
2002 Yearbook of the NCTM. p. 38-40. [Potential math task: Cocoa Reasoning Problem]
Langrall, Cynthia. (December 2000). "Three Balloons for Two Dollars: Developing Proportional
Reasoning." Mathematics Teaching in the Middle School. Vol. 6. No. 4. p. 254-261.
6.RP.A.2. Understand the concept of a unit rate a/b associated with a ratio a:b with b not equal
to 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 a 3/4 cup of flour for each cup of sugar.”
“We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”
HCPSS UDL Lesson:
Gas Mileage
6.RP2&3b Gas Mileage.doc
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6.RP.2&3b UDL Checklist Gas Mileage.docx
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PARCC Assessment Limit/Clarification:
This standard is part of the major content cluster assessed on PARCC. Students' prior understanding of
an skill with multiplication, division, and fractions contribute to their study of ratios, proportional
relationships and unit rates.
Web Resources:
http://illuminations.nctm.org/LessonDetail.aspx?ID=L511
NCTM Illuminations Lesson: Students experiment to find out how many jumping jacks, how many times
you can sing ABCs, or hop on one foot in one minute and then use that rate to figure out how many of
each of those you can do in 60 minutes (or some other time amount).
http://illuminations.nctm.org/LessonDetail.aspx?ID=L284
NCTM Illuminations Lesson: Students use real-world models to develop an understanding of fractions,
decimals, unit rates, proportions, and problem-solving.
http://illustrativemathematics.org/illustrations/77
Task: Mangos for Sale. Students explore whether a given situation may be represented by more than one
ratio.
http://illustrativemathematics.org/illustrations/549
Task: Price per Pound and Pounds per Dollar. Students develop the concept of unit rate and explore it in
a grocery store setting. Ratio tables are illustrated as a solution strategy.
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.
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.
HCPSS Math Tasks:
World Series of Pop Culture
6.RP.3a Task World Series of Pop Culture.doc
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Grocery Store Gas Points
6.RP.3 Task Grocery Store Gas Points.doc
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HCPSS Disciplinary Literacy Lesson Seed:
Pencils
6.RP.3a Disc Lit Lesson Seed Pencils.docx
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PARCC Assessment Limit/Clarification:
This standard is part of the major content cluster assessed on PARCC. Students' prior understanding of
an skill with multiplication, division, and fractions contribute to their study of ratios, proportional
relationships and unit rates. When students work toward meeting standard 6.RP.A.3, they use a range of
reasoning and representations to analyze proportional relationships.
Web Resources:
http://science.kennesaw.edu/~twatanab/DeKalb%20Title%20I%20Summit%202012.pdf
PowerPoint Presentation on Tape Diagrams & Double Number Line Graphs, designed by Dr. Tad
Wattanabe (former Towson University professor).
http://illuminations.nctm.org/LessonDetail.aspx?ID=L515
NCTM Illuminations Lesson: Students measure the heights and shadows of familiar objects and use
indirect measurement to find the heights of things that are much bigger in size, such as a flagpole, a
school building or a tree.
http://www.uen.org/Lessonplan/preview?LPid=15425
Lesson: In this lesson, students will graph growing patterns using ordered pairs on a coordinate grid.
http://illustrativemathematics.org/illustrations/53
Task: Mixing Concrete. Students determine how to mix concrete by examining part-part and part-whole
relationships. Ratio tables are illustrated as a solution strategy.
Text Resources For Lessons/Tasks:
Van de Walle, J.A., & Lovin, L.H. (2006). "Chapter 18: Proportional Reasoning." Teaching StudentCentered Mathematics. Boston: Pearson. p. 248-268.
Lobato, J.E., Ellis, A.B, Charles, R.I., & Zbiek, R.M. (2010). Developing Essential Understanding of
Ratios, Proportions, & Proportional Reasoning for Teaching Mathematics in Grades 6-8. Reston, VA:
National Council of Teachers of Mathematics. p. 36. "Figure 1.26" Task p. 36: Students use the base and
height of a ramp to determine the dimensions of other ramps that would have the same steepness.
Lannin, J.K., Ellis, A.B., & Elliott, R. (2011). Developing Essential Understanding of Mathematical
Reasoning for Teaching Mathematics in Prekindergarten-Grade 8. Reston, VA: National Council of
Teachers of Mathematics. p. 42. "Same Speed Problem" Task p. 42: Students use the distance and time
of a clown to determine different ways a frog can walk the same speed.
NCTM Journal Article Lesson Ideas/Resources:
Ercole, L.K., Frantz, M., & Ashline, G. (April 2011). "Multiple Ways to Solve Proportions." Mathematics
Teaching in the Middle School. Vol. 16. No. 8. p. 482-490.
This article presents multiple strategies for solving proportions, including the use of ratio tables.
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.
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?
HCPSS Math Task:
Neighborhood Snow Removal
6.RP.3b Task Neighborhood Snow Removal.doc
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6.RP.3b Task Student Sheet.doc
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HCPSS UDL Lesson:
Geysers
6.RP.3b Lesson Geysers.doc
6.RP.3b UDL Checklist Geysers.docx
PARCC Assessment Limit/Clarification:
This standard is part of the major content cluster assessed on PARCC. Students' prior understanding of
an skill with multiplication, division, and fractions contribute to their study of ratios, proportional
relationships and unit rates. When students work toward meeting standard 6.RP.A.3, they use a range of
reasoning and representations to analyze proportional relationships.
PARCC/UTA Dana Center Prototype Assessment Items:
"Gasoline Consumptions" (6.RP.3, 6.EE.9, MP 4)
http://ccsstoolbox.agilemind.com/parcc/about_middle_3788.html
Web Resources:
http://illuminations.nctm.org/LessonDetail.aspx?id=L783
NCTM Illuminations Lesson: Students use dimensions of round and square hay bales to calculate and
compare volumes. They also calculate unit prices to determine which hay bale is the better value.
http://www.uen.org/Lessonplan/preview?LPid=6396
Lesson: In this lesson, students will use a trundle wheel to measure the distance traveled outside after
one minute of walking. After traveling a total of 10 minutes, students will return to the classroom to graph
and analyze the data collected.
http://illustrativemathematics.org/illustrations/137
Task: Friends Meeting on Bicycles. Students determine the speed one girl travels on her bike to meet up
with her friend, knowing how fast the friend traveled and when they met up. Students solve constant
speed problems related to travel between two friends.
http://illustrativemathematics.org/illustrations/193
Task: Running at a Constant Speed. Students use a runner's distance and time to solve problems related
to her speed. Ratio tables are illustrated as a solution strategy.
Text Resources For Lessons/Tasks:
Lobato, J.E., Ellis, A.B, Charles, R.I., & Zbiek, R.M. (2010). Developing Essential Understanding of
Ratios, Proportions, & Proportional Reasoning for Teaching Mathematics in Grades 6-8. Reston, VA:
National Council of Teachers of Mathematics. p. 52, 84.
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"Reflect 2.1" Task p. 52: Students use constant speed to
determine a rabbit's distance from home. Figure 2.2
provides a number line diagram solution.
"Figure 3.9" Task p. 84: Students use a tape diagram to
explore the connection between partitioning and unit
ratios.
NCTM Journal Article Lesson Ideas/Resources:
Horak, V.M. (April 2006). "A Science Application of Area and Ratio Concepts." Mathematics Teaching in
the Middle School. Vol. 18, No. 8. p. 360-364.
Students investigate the ratio of an animal's weight to the area of its paw print (its sinking value) as an
indication of the depth of its imprint into a soft surface.
Sinn, R., Spence, D.J., & Poitevint, M. (December 2010/January 2011). "A Geometric Approach to
Solving Rate Problems." Mathematics Teaching in the Middle School. Vol. 16, No. 5. p. 302-310.
Disciplinary Literacy:
Slate, M. & Hammond, S.E. (October 2010). "Media Clips: Brown-Bagging It." Mathematics Teacher. Vol.
104. No. 3. p. 172-175.
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.
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. Note:
Include finding percent of any number.
HCPSS Math Tasks:
Designing a Kitchen Backsplash
6.RP.3c Backsplash Task.doc
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6.RP.3c Backsplash Task Handout.docx
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HCPSS UDL Lesson:
What's My Grade?
6.RP.3c Lesson What's My Grade.doc
6.RP.3c Student Resource Sheet.docx
6.RP.3c UDL Checklist What's My Grade.docx
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PARCC Assessment Limit/Clarification:
This standard is part of the major content cluster assessed on PARCC. Students' prior understanding of
an skill with multiplication, division, and fractions contribute to their study of ratios, proportional
relationships and unit rates. When students work toward meeting standard 6.RP.A.3, they use a range of
reasoning and representations to analyze proportional relationships.
Web Resources:
http://illuminations.nctm.org/LessonDetail.aspx?ID=L265
NCTM Illuminations Lesson: Students participate in an activity in which they develop number sense in
and around the shopping mall. They develop skills in determining percents and estimating area.
http://illuminations.nctm.org/LessonDetail.aspx?ID=L266
NCTM Illuminations Lesson: Students participate in an activity in which they develop number sense in
and around the shopping mall. They solve problems involving percent and scale drawings.
http://illuminations.nctm.org/LessonDetail.aspx?ID=L849
NCTM Illuminations Lesson: This lesson is designed to enlighten students about how to calculate percent
of calories from fat, carbohydrates, and protein. The calculations are made to determine if a person can
follow the Zone Diet with only McDonald's food items.
http://illuminations.nctm.org/LessonDetail.aspx?ID=L249
NCTM Illuminations Lesson: Students use a 10×10 grid as a model for solving various types of percent
problems. This model offers a means of representing the given information as well as suggesting different
approaches for finding a solution.
http://illustrativemathematics.org/illustrations/132
Task: Finding a 10% Increase. Students are encouraged to use mental math to calculate a percent
increase.
http://illustrativemathematics.org/illustrations/118
Task: Dana's House. Students apply knowledge of area to determine what percent of a lot will not be
covered by the house Dana builds on it.
http://illustrativemathematics.org/illustrations/131
Task: Kendall's Vase - Tax. Students calculate tax and determine the total cost of a vase. Percent tables
are illustrated as a solution strategy.
http://illustrativemathematics.org/illustrations/115
Task: Security Camera. Students analyze a picture of a shop floor plan to determine visibility by a security
camera.
http://illustrativemathematics.org/illustrations/54
Task: Shirt Sale. Students use the percent and amount of discount to determine the original price of a
shirt. Tape diagrams are illustrated as a solution strategy.
NCTM Journal Article Lesson Ideas/Resources:
Mooney, E.S. & Meier, S. (February 2011). "Solve It! Percent Tile." Mathematics Teaching in the Middle
School. Vol. 16. No. 6. p. 320.
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.
3d. Use ratio reasoning to convert measurement units; manipulate and transform units
appropriately when multiplying or dividing quantities.
HCPSS Math Tasks:
Huckleberry Trail Mix
6.RP.3d Task Huckleberry Trail Mix.doc
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6.RP.3d Huckleberry Trail Mix Recipe.docx
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Where Should You Go?
6.RP.3d Task Where Should You Go.doc
HCPSS UDL Lessons:
Maps
6.RP.3d Lesson Maps.doc
6.RP.3d Map 1.docx
6.RP.3d Map 2.docx
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6.RP.3d Maps UDL Checklist.doc
Conversions with Inches to Centimeters
6.RP.3d Lesson Conversions with Inches and Centimeters.doc
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6.RP.3d UDL Checklist Conversions with Inches and Centimeters.docx
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PARCC Assessment Limit/Clarification:
This standard is part of the major content cluster assessed on PARCC. Students' prior understanding of
an skill with multiplication, division, and fractions contribute to their study of ratios, proportional
relationships and unit rates. When students work toward meeting standard 6.RP.A.3, they use a range of
reasoning and representations to analyze proportional relationships.
PARCC Prototype Assessment Item:
"Slide Ruler" (6.RP.3d)
http://www.parcconline.org/samples/mathematics/grade-6-slider-ruler
Web Resources:
http://illuminations.nctm.org/LessonDetail.aspx?ID=L516
NCTM Illuminations Lesson: Students use real-world examples to solve problems involving scale as they
examine maps of their home states and calculate distances between cities.
http://illuminations.nctm.org/LessonDetail.aspx?ID=L743
NCTM Illuminations Lesson: This lesson focuses on the concept of 1,000,000. Students see first hand the
sheer size of 1 million while at the same time providing time with an introduction to sampling and its use
in mathematics. Students will use grains of rice and a balance to figure out the approximate volume and
weight of 1,000,000 grains of rice.
http://www.rda.aps.edu/mathtaskbank/pdfs/instruct/6-8/i69heigh.pdf
Math Task: Students are given a photograph and need to determine how tall a person in it is in real life.
http://www.uen.org/Lessonplan/preview?LPid=15435
Lesson: In this lesson, students complete a variety of measuring activities.
http://illustrativemathematics.org/illustrations/79
Task: Converting Square Units. Students convert the dimensions of a board from inches to feet and
determine the area in square feet.
http://illustrativemathematics.org/illustrations/134
Task: Currency Exchange. Students convert U.S. dollars to Canadian dollars and then to Mexican pesos.
Text Resources For Lessons/Tasks:
Lobato, J.E., Ellis, A.B, Charles, R.I., & Zbiek, R.M. (2010). Developing Essential Understanding of
Ratios, Proportions, & Proportional Reasoning for Teaching Mathematics in Grades 6-8. Reston, VA:
National Council of Teachers of Mathematics. p. 65. "Figure 3.3" Task p. 65: Students analyze the scale
on a city map in centimeters and miles.
Khoury, H.A. (2002). Classroom Challenge: Making Sense of Fractions, Ratios, and Proportions. 2002
Yearbook of the NCTM. p. 100-102. [Potential Task: Exploring Proportional Reasoning: Mr. Tall/Mr. Short]