Grade 6 - Module 9 - Ratio and Proportion
NATIONAL TRAINING NETWORK
2013-2014
Big Idea
Ratios and unit rates can be represented mathematically, in tables, and in real-world problems like percentage problems and converting measurements.
Vocabulary
ratio, equivalent ratios, part-to-part, part-to-whole, quantity, compare, relationship, unit rate, coordinate plane, x-axis, y-axis, scale, ordered pair, variable, tape
part
percent
diagram, constant rate, percent, identity property of multiplication, conversion factor, convert,
=
whole
100
Prior Learning
Students in Grade 5 work with fraction relationships, equivalency and simplifying fractions
Grade 6 - Module 9 - Ratio and Proportion | 2013-2014
Essential Questions
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How are ratios and percents related?
How do you determine a unit rate given a table of values?
Given the quantity and price of two objects, how can you determine which one is the better buy?
Why is it important to know how to solve for the unit rate?
How are ratios used in the real world?
What does a percent represent?
How can knowledge about percents help solve real-world problems?
How are ratios like fractions? How are they different?
How are rates and unit rates used in the real world?
How can you find equivalent ratios?
How can a table or graph of equivalent ratios be used to solve problems and analyze situations?
Competencies
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Students will use ratios to describe proportional situations.
Students will represent ratios and percents with concrete models, fractions, and decimals.
Students will apply knowledge of rations and proportions to percent problems.
Students will solve problems involving percents and percent problems using ratios and ratio tables
Students will make conversions between different measurements and unit ratios.
Students will express a ratio in fractional form, using the word “to” and with a colon
Students will compute equivalent ratios using a multiplicative strategy and/or unit rates
Students will simplify ratios and determine unit rates
Students will solve mathematical and real-world problems involving unit rates
Students will create and graph ratio tables
Students will convert measurements and solve problems using ratios and ratio tables
Misconceptions
•
Students may confuse the comparisons for fractions and ratios. Fractions always express a part-to-whole comparison, but ratios can express a part-to-whole
comparison or a part-to-part comparison.
Students may incorrectly substitute values when determining percent of a number, find a number using percents or determine a whole value when the part
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and percent is given.
Students may have difficulty plotting ratios on the coordinate plane because they confuse the x and y axes.
Students may use the wrong conversion values when changing units of measurements.
Resources from The Key Elements to Mathematics Success - KEMS Level F
(Unless noted) for Building the Conceptual Understanding of this Module
NEW LESSON - RATIOS AND UNIT RATES
To access the online teacher lesson notes, a video clip, and the student homework go to:
http://ntnmath.kemsmath.com/CCSSFpages/lesson 4.html?utm_source=google&utm_medium=web&utm_campaign=Level+F++Lesson+4+Ratios+and+Unit+Rates
NEW LESSON- RATIOS AND TABLES
To access the online teacher lesson notes, a video clip, and the student homework go to:
http://ntnmath.kemsmath.com/CCSSFpages/lesson 5.html?utm_source=google&utm_medium=web&utm_campaign=Level+F+-+Lesson+5+Ratios+and+Tables
Grade 6 - Module 9 - Ratio and Proportion | 2013-2014
NEW LESSON- UNIT RATE PROBLEMS
To access the online teacher lesson notes, a video clip, and the student homework go to:
http://ntnmath.kemsmath.com/CCSSFpages/lesson 6.html?utm_source=google&utm_medium=web&utm_campaign=Level+F+-+Lesson+6+Unit+Rate+Problems
2
NEW LESSON- PERCENTS AS A RATE PER 100
To access the online teacher lesson notes, a video clip, and the student homework go to:
http://ntnmath.kemsmath.com/CCSSFpages/lesson 7.html?utm_source=google&utm_medium=web&utm_campaign=Level+F++Lesson+7+Percent+as+a+Rate+per+100
NEW LESSON- CONVERTING MEASUREMENTS WITH RATIOS
To access the online teacher lesson notes, a video clip, and the student homework go to:
http://ntnmath.kemsmath.com/CCSSFpages/Converting Measurements with Ratios.html?utm_source=google&utm_medium=web&utm_campaign=Level+F++Converting+Measurements+with+Ratios
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Additional Manipulatives/Tools/Resources to Enhance the Module
Understanding Rational Numbers and Proportions: In this lesson, students use real world models to develop an understanding of fractions, decimals, unit rates,
proportions, and problem solving. http://illuminations.nctm.org/LessonDetail.aspx?id=L284
Bean Counting and Ratios: By using sampling from a large collection of beans, students get a sense of equivalent fractions, which leads to a better understanding
of proportions. Equivalent fractions are used to develop an understanding of proportions. http://illuminations.nctm.org/LessonDetail.aspx?id=L722
Feeding Frenzy: In this activity, students will multiply and divide a recipe to feed groups of various sizes. Students will use unit rates or proportions and think
critically about real world applications of a baking problem. http://illuminations.nctm.org/LessonDetail.aspx?id=L781
Capture-Recapture: In this lesson, students experience an application of proportion that scientists actually use to solve real-life problems. Students learn how to
estimate the size of a total population by taking samples and using proportions. The ratio of “tagged” items to the number of items in a sample is the same as the
ratio of tagged items to the total population. http://illuminations.nctm.org/LessonDetail.aspx?id=L721
Grade 6 - Module 9 - Ratio and Proportion | 2013-2014
Grid It and Percent It: In this 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://illuminations.nctm.org/LessonDetail.aspx?id=L249
3
Fractional Clothesline: In this lesson, a string will be stretched across the classroom and various points will be marked for 0, 1, 2, 3, and 4. This classroom number
line will be used to show that all proper fractions are grouped between 0 and 1, and that improper fractions or mixed numbers are all grouped above 1. Students clip
index cards with various proper fractions, improper fractions, and mixed numbers on the clothesline to visually see groupings. Students then play an estimation
game with groups using the same principle. Encouraging students to look at fractions in various ways will help foster their conceptual fraction sense.
http://illuminations.nctm.org/LessonDetail.aspx?id=L784
Now and Then: In this activity, students work in groups as contestants on a fictitious game show, Now and Then. They use their knowledge of percent
computations and percent change to answer each game show question. As a culminating activity, the students create their own game show and host it in front of the
rest of the class. http://illuminations.nctm.org/LessonDetail.aspx?id=L837
Real Estate Tycoon: This article describes a year-long project. Students design, "build," and "sell" a house; after which they simulate investment of the profits in
the stock market. Along the way, students make scale drawings, compute with fractions and decimals in various contexts, and even solve simple equations. This
lesson plan was adapted from an article by David B. Smith, which appeared in the September 2000 edition of Mathematics Teaching in the Middle School.
http://illuminations.nctm.org/LessonDetail.aspx?id=L279
Fraction Models: Use this computer applet to explore different representations for fractions including improper fractions, mixed numbers, decimals, and
percentages. Additionally, there are length, area, region, and set models. Adjust numerators and denominators to see how they alter the representations and models.
Use the table to keep track of interesting fractions. http://illuminations.nctm.org/activitydetail.aspx?ID=11
Different Looks, Same Values (TI-73): In this activity, students explore the meaning and purpose of equivalent fractions. They also practice writing fractions that
meet a given criteria. http://education.ti.com/calculators/timiddlegrades/US/Activities/Detail?s=Math&id=11731
©Copyright National Training Network 2013
What’s the Unit of Choice (TI-73): In this activity, students will select appropriate units for given situations. Then they will use the Convert menu to convert
measures from metric to customary and from a small unit to a larger unit of the same system.
http://education.ti.com/calculators/timiddlegrades/US/Activities/Detail?s=Math&id=12290
Grade 6 - Module 9 - Ratio and Proportion | 2013-2014
Percents – Small and Large (TI-73): In this activity, students will explore percents greater than 100% and less than 1%. Students will use a 100 grid to represent
percents and convert between fractions, decimals, and percents. http://education.ti.com/calculators/timiddlegrades/US/Activities/Detail?s=Math&id=11268
Performance Tasks
4
Fruit Salad
http://www.illustrativemathematics.org/illustrations/1032
Converting Square Units
http://www.illustrativemathematics.org/illustrations/79
Currency Exchange
http://www.illustrativemathematics.org/illustrations/134
Dana's House
http://www.illustrativemathematics.org/illustrations/118
Data Transfer
http://www.illustrativemathematics.org/illustrations/815
Friends Meeting on Bicycles
http://www.illustrativemathematics.org/illustrations/137
Games at Recess
http://www.illustrativemathematics.org/illustrations/76
Jim and Jesse's Money
http://www.illustrativemathematics.org/illustrations/498
Kendall's Vase - Tax
http://www.illustrativemathematics.org/illustrations/131
Mangos for Sale
http://www.illustrativemathematics.org/illustrations/77
Mixing Concrete
http://www.illustrativemathematics.org/illustrations/53
Overlapping Squares
http://www.illustrativemathematics.org/illustrations/899
Price per pound and pounds per dollar
http://www.illustrativemathematics.org/illustrations/549
Running at a Constant Speed
http://www.illustrativemathematics.org/illustrations/193
Running at a Constant Speed, Assessment Variation
http://www.illustrativemathematics.org/illustrations/1175
Security Camera
http://www.illustrativemathematics.org/illustrations/115
Shirt Sale
http://www.illustrativemathematics.org/illustrations/54
The Escalator, Assessment Variation
http://www.illustrativemathematics.org/illustrations/1181
Voting for Three, Variation 1
http://www.illustrativemathematics.org/illustrations/66
Voting for Three, Variation 2
http://www.illustrativemathematics.org/illustrations/67
Voting for Three, Variation 3
http://www.illustrativemathematics.org/illustrations/68
Walk-a-thon
http://www.illustrativemathematics.org/illustrations/711
©Copyright National Training Network 2013
CCSS-Mathematics Content Standards
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.”
Examples
A ratio is the comparison of two quantities. 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).
Example 1: A comparison of 6 guppies and 9 goldfish could be expressed in any of the following forms:
6
, 6 to 9 or
9
6:9. If the number of guppies is represented by black circles and the number of goldfish is represented by white
circles, this ratio could be modeled as
These values can be regrouped into 2 black circles (guppies) to 3 white circles (goldfish), which would reduce the
Grade 6 - Module 9 - Ratio and Proportion | 2013-2014
ratio to:
5
2
, 2 to 3 or 2:3.
3
Students should be able to identify and describe any ratio using “For every _____, there are _____” In the example
above, the ratio could be expressed saying, “For every 2 guppies, there are 3 goldfish”.
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 ¾ cup of flour for each cup of
sugar.” “We paid $75 for 15
hamburgers, which is a rate of $5 per
hamburger.”1
A unit rate expresses a ratio as part-to-one. Common unit rates are cost per item or distance per time.
Students are able to name the amount of either quantity in terms of the other quantity. Students will begin to notice
that related unit rates (i.e. miles / hour and hours / mile) are reciprocals as in the second example below. At this level,
students should use reasoning to find these unit rates instead of an algorithm or rule.
In 6th grade, both the numerator and denominator of the original ratio will be whole numbers.
Example 1: There are 2 cookies for 3 students. What is the amount of cookie each student would receive? (i.e. the
unit rate)
Solution: This can be modeled as shown below to show that there is
1
Expectations for unit rates in this
grade are limited to non-complex
fractions.
2
2
of a cookie for 1 student, so the unit rate is :
3
3
1.
1 1
2 3
2
3
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Example 2: On a bicycle Jack can travel 20 miles in 4 hours. What are the unit rates in this situation, (the distance
Jack can travel in 1 hour and the amount of time required to travel 1 mile)?
Solution: Jack can travel 5 miles in 1 hour written as
5 mi
1 hr
and it takes
1
5
of a hour to travel each mile written as
1
hr
5
1 mi
.
Students can represent the relationship between 20 miles and 4 hours.
1 mile
Grade 6 - Module 9 - Ratio and Proportion | 2013-2014
1 hour
6
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.
a. Make tables of equivalent ratios
relating quantities with wholenumber measurements, find missing
values in the tables, and plot the
pairs of values on the coordinate
plane. Use tables to compare ratios.
Ratios and rates can be used in ratio tables and graphs to solve problems. Students need to begin using multiplicative
reasoning. Scaling up or down with multiplication maintains the equivalence. When working with ratio tables and
graphs, whole number measurements are the expectation for this standard.
Example 1: At Books Unlimited, 3 paperback books cost $18. What would 7 books cost? How many books could be
purchased with $54.
Solution: To find the price of 1 book, divide $18 by 3. One book costs $6. To find the price of 7 books, multiply $6
(the cost of one book times 7 to get $42. To find the number of books that can be purchased with $54, multiply $6
times 9 to get $54 and then multiply 1 book times 9 to get 9 books. Students use ratios, unit rates and multiplicative
reasoning to solve problems in various contexts, including measurement, prices, and geometry. Notice in the table
below, a multiplicative relationship exists between the numbers both horizontally (times 6) and vertically (ie. 1 • 7 = 7;
6 • 7 = 42). Red numbers indicate solutions.
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Cost
1
6
3
18
Number
of
Books
(n)
(C)
7
42
9
54
Cost
(C)
4
20
8
40
Another bookstore offers paperback books at the prices to the right above. Which bookstore has the best buy? Explain
your answer. Students can write equations to express the cost of any number of books. For example, the equation for
the first table would be C = 6n, while the equation for the second bookstore is C = 5n.
The numbers in the table can be expressed as ordered pairs (number of books, cost) and plotted on a coordinate plane.
Books Unlimited
Cost
Grade 6 - Module 9 - Ratio and Proportion | 2013-2014
7
Number of
Books (n)
35
30
25
20
15
10
5
0
0
1
2
3
4
5
6
Number of Books
©Copyright National Training Network 2013
Example 2: The ratio of cups of orange juice concentrate to cups of water in punch is 1: 3. If James made 32 cups of
punch, how many cups of orange did he need?
Solution: Students recognize that the total ratio would produce 4 cups of punch. To get 32 cups, the ratio would need
to be duplicated 8 times, resulting in 8 cups of orange juice concentrate.
Example 3: Using the information in the table, find the number of yards in 24 feet.
Feet
3
6
9
15
24
Yards
1
2
3
5
?
Solution: There are several strategies that students could use to determine the solution to this problem:
o
Grade 6 - Module 9 - Ratio and Proportion | 2013-2014
o
8
Add quantities from the table to total 24 feet (9 feet and 15 feet); therefore the number of yards in 24 feet
must be 8 yards (3 yards and 5 yards).
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.
Example 4: Compare the number of black circles to white circles. If the ratio remains the same, how many black
circles will there be if there are 60 white circles?
Black
4
40
20
60
?
White
3
30
15
45
60
Solution: There are several strategies that students could use to
determine the solution to this problem
Add quantities from the table to total 60 white circles (15 + 45). Use the corresponding numbers to determine the
number of black circles (20 + 60) to get 80 black circles.
o Use multiplication to find 60 white circles (one possibility 30 × 2). Use the corresponding numbers and
operations to determine the number of black circles (40 × 2) to get 80 black circles.
Students recognize the use of ratios, unit rate and multiplication in solving problems, which could allow for the use of
fractions and decimals.
Peanuts
Chocolate
Example 1: In trail mix, the ratio of cups of peanuts to cups of
chocolate candies is 3 to 2. How many cups of chocolate candies
would be needed for 9 of peanuts?
3
2
o
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, how many lawns could be
mowed in 35 hours? At what rate
were lawns being mowed?
©Copyright National Training Network 2013
Solution: One possible solution is for students to find the number of cups of chocolate candies for 1 cup of peanuts by
2
cup of chocolate for each cup of peanuts. To find the amount of
3
2
chocolate needed for 9 cups of peanuts, students multiply the unit rate by nine (9 • ), giving 6 cups of chocolate.
3
dividing both sides of the table by 3, giving
Example 2: If steak costs $2.25 per pound, how much does 0.8 pounds of steak cost? Explain how you determined
your answer.
Solution: The unit rate is $2.25 per pound so multiply $2.25 x 0.8 to get $1.80 per 0.8 lb of steak.
c. Find a percent of a quantity as a rate
Grade 6 - Module 9 - Ratio and Proportion | 2013-2014
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.
This is the students’ first introduction to percents. Percentages are a rate per
Models, such as percent bars or 100s grids would be beneficial to model
percents.
Part
Whole
Example 1: What percent is 12 out of 25?
12
25
Solution: One possible solution method is to set up a ratio table:
?
100
100.
Multiply 25 by 4 to get 100. Multiplying 12 by 4 will give 48, meaning
that 12 out of 25 is equivalent to 48 out of 100 or 48%.
Example 2: What is 40% of 30?
Solution: There are several methods to solve this problem. One possible solution using rates is to use a 10 x 10 grid
to represent the whole amount (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.
See the weblink for more information.
http://illuminations.nctm.org/LessonDetail.aspx?id=L249
Students also determine the whole amount, given a part and the percent.
Example 3: If 30% 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?
(Solution: 20)
0%
30%
?
100%
6
9
©Copyright National Training Network 2013
Example 4: A credit card company charges 17% interest fee on any charges not paid at the end of the month. Make a
ratio table to show how much the interest would be for several amounts. If the bill totals $450 for this month, how
much interest would you have to be paid on the balance?
Solution:
Charges
$1
$50
$100
$200
$450
Interest
$0.17
$8.50
$17
$34
?
One possible solution is to multiply 1 by 450 to get
450 and then multiply 0.17 by 450 to get $76.50.
d. Use ratio reasoning to convert
measurement units; manipulate and
transform units appropriately when
multiplying or dividing quantities.
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 numerator and
denominator describe the same quantity.
Students use ratios as conversion factors and the identity property for multiplication to convert ratio units.
Grade 6 - Module 9 - Ratio and Proportion | 2013-2014
Example 1: How many centimeters are in 7 feet, given that 1inch ≈ 2.54 cm?
10
Solution:
7 feet × 12 inches
× 2.54 cm =
1 foot
1 inch
7 feet × 12 inches
× 2.54 cm
1 foot
= 7 ×12 × 2.54 cm = 213.36 cm
1 inch
Note: Conversion factors will be given. Conversions can occur both between and across the metric and English
systems. Estimates are not expected.
Works Referenced in the Development of the Module
Common Core State Standards Initiative http://www.corestandards.org/
North Carolina Mathematics Wiki
http://maccss.ncdpi.wikispaces.net/
Illustrative Mathematics Project http://illustrativemathematics.org
PARCC http://parcconline.org/
Mathematics Assessment Project http://map.mathshell.org.uk/materials/index.php
Smarter Balanced Assessment Consortium
http://www.smarterbalanced.org/
Ohio Department of Education
http://education.ohio.gov/GD/Templates/Pages/ODE/ODEPrimary.aspx?page=2&TopicRelationID=1704
Utah Education Network
http://www.uen.org/commoncore/
A detailed progression of the Ratios and Proportional Relationships domain with examples can be found at http://commoncoretools.wordpress.com/
The direct link is http://commoncoretools.me/wp-content/uploads/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf
©Copyright National Training Network 2013