Title
Goals
Standard
Addressed
by the Unit
Proportional Reasoning
Proportions and Proportion Problems
By the end of this session, the teachers will be able to:
 Distinguish between equivalent ratios vs. ratios that are not
equivalent.
 Distinguish equivalent ratios from equivalent fractions.
 Distinguish proportional situations from nonproportional
situations
 How do I know proportional situation when I see it?

Define proportions.

Solving proportion problems.
California Standards:
Common Core Standards: Understand ratio concepts and use ratio
reasoning to solve problems.
6.RP.1 Understand the concept of ratio and use ratio language to
describe a ratio relationship between two quantities.
6.RP.2 Understand the concept of unit rate a/b associated with a:b with
b ¹ 0 , and use rate language in the context of ratio relationships.
6.RP.3Use ratios and rate reasoning to solve real-word and mathematical
problems, e.g., be 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 the missing values in the tables, and
plot the pairs of values on the coordinate plane. Use tables to
compare ratios. 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 100; 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.
Handouts (Smart board), Color Tiles, Grid paper, Ruler
Materials
for Teacher
Color pencils, Color Tiles, Grid paper, Ruler
Materials
for Students
Proportions and Proportion Problems
1
Comparing Ratios
Compare the ratios in the situations given below:
a. The ratio of gray to white circles in both Figure 1 and Figure 2.
Figure 1
Figure 2
b. Use the array of circles given below to find the ratios defined below:
i. Gray to white in the first row.
ii. Gray to white in the first row four rows.
iii. Gray to white in the entire square.
Proportions and Proportion Problems
2
c. Suppose Abby’s orange paint is made by mixing 1 cup of red paint for every 3 cups of
yellow paint. Zack’s orange is made by mixing 3 cups of red paint for every 5 cups of yellow
paint. Whose orange is more yellow? Whose orange is redder?
d. Given two rectangles, one with length 38 inches and width 34 inches and one with length 28
inches and width 25 inches, which rectangle is more square? Describe what this problem has
to do with proportionality.
e. Mr. Ndilinge drove from Los Angeles to San Francisco (a distance of 390 miles) in 6 hours.
On the same day, Ms. Fernandez drove from Los Angeles to Phoenix (a distance of 325
miles) in five hours to attend a Lakers vs. Suns game. How do the distance/time ratios
compare for these two travelers compare?
Proportions and Proportion Problems
3
Definition: Proportional relationships involve collections of pairs of measurements in equivalent
ratios. Thus saying that two measurements x and y are proportional means that for any ordered
y
pair
always gives the same number!
x
More Examples (Distinguishing between Proportional and Non-Proportional
Relationships)
1. Kris and Rich like to skate laps together around an ice rink since they both skate at the same
constant rate. Today, Kris started skating first. By the time that Kris had completed 9 laps,
Rich had completed 3 laps. How many laps will Kris complete by the time that Rich
completes 15 laps? Explain. (Adapted from Lamon, 1999).
2. Three identical bags of mulch weigh 21 pounds. How many pounds do 8 bags of mulch
weigh?
3. Key Questions:
a. Does knowing that a problem involves a constant rate guarantee that it can be represented
by a proportional relationship? Why or why not?
b. How do I know it when I see it?
How do I know a proportional relationship when I see it …
i.
In the problem statement?
Proportions and Proportion Problems
4
ii.
In table representation?
Which of the following tables represents a proportional relationship?
Explain how you know.
x
y
x
y
1
13
2
13
2
17
3
16.5
3
20
4
26
4
23
5
32.5
Relationship A
iii.
Relationship B
In graphical representation?
Which of the following graphs represents a proportional relationship?
Explain how you know
Relationship A
iv.
Relationship A
In equation representation?
How does an equation representing a proportional relationship between x
and y look like? Explain how you know.
Proportions and Proportion Problems
5
Equivalent Ratios vs. Equivalent Fractions
Consider the following equivalent ratios and equivalent fractions.
Use tape diagrams to explain the difference(s) between equivalent
ratios and equivalent fractions
Blue Cups 2 4 6
Total Cups 3 6 9
Proportions and Proportion Problems
Fraction Number 1 2 3
2 4 6
Total Cups
3
6
9
6
Strategies of Solving Proportion Problems
Worth noting: Recommendations for Developing Students’ Conceptual Understanding of Strategies
for solving ratio, rate and proportion problems:
a. Develop students’ understanding of proportional relations before teaching computational
procedures that are conceptually difficulty to understand (e.g., cross multiplication). Build on
students’ developing strategies for solving ratio, rate, and proportion problems.
b. Encourage students to use visual representations to solve ratio, rate, and proportion problems.
c. Provide opportunities for students to use and discuss alternative strategies for solving ratio,
rate, and proportion problems.
d. Note: Although it is traditional to move students quickly to solving proportions by setting up
an equation, the Standards do not require this method in Grade 6!
Solution Strategies (Adapted from Cramer, Post, 1993)
Strategy
Description
Unit-rate strategy As the name implies, this strategy involves using a unit rate. In the above problem, we
are given 3 teaspoons per 4 ounces of water or 3⁄4 of a teaspoon per 1 ounce of water.
Students should also be flexible with using the reciprocal of this rate as well
depending upon what is being asked for. That is, 4 ounces of water for every 3
teaspoons or 4/3 ounces per 1 teaspoon. In this case, since one is looking for the
number of teaspoons for 2 ounces of water, one would take 3⁄4 teaspoon/ounce and
multiply it by 2 ounces to obtain 1.5 teaspoons. If the question asked to find ounces of
water for a certain number of teaspoons, the other unit rate would be used.
Factor-of-change A student using a factor-of-change strategy is using a “times as many strategy.”‖ In
strategy
this case, a student would use the following reasoning: Since 4 ounces of water calls
for 3 teaspoons of flour and 2 ounces is half of 4 ounces, I need to use half as much
flour or 1.5 teaspoons of flour.
Fraction strategy
CrossMultiplication
The fraction strategy is similar to the unit-rate strategy but with the labels dropped on
the rates and the idea of equivalence used. In the above story problem, due to the noninteger solution, a fraction strategy may not often be used. If the story problem asked
for how many teaspoons of flour are needed for 16 ounces of water, a student using
the fraction strategy would reason that 3⁄4 is equivalent to 12/16 by multiplying the
numerator and denominator by 4 and conclude that 12 teaspoons are needed. This
strategy amounts to using a common denominator approach. A student who has a solid
understanding of fractions might reason that one and a half halves is equivalent to
three- fourths. Notice that the word problem in Table 7.1 lends itself well to this type
of reasoning.
To solve by this strategy a student would set up a proportion (the equivalence of two
ratios), find the cross-products, and solve by using division.
Proportions and Proportion Problems
1
Putting the Strategies to Work!
I. Reasoning with Tape Diagrams
1. Mr. Cooper’s class has a female and male student ratio of 3:2. Mr. Cooper’s class has 18
girls. How many boys does he have?
2. Ms. Green’s class has the same number of students as Mr. Cooper’s class. Her female-tomale ratio is 2:1. Which class has the greater number of females? How do you know?
3. Slimy Gloopy mixture is made by mixing glue and liquid laundry starch in a ratio of 3 to
2. How much glue and how much starch is needed to make 85 cups of Slimy Gloopy
mixture?
4. Mr. Reins put students into groups of 5. Each group had 3 girls. If he has 25 students, how
many girls and how many boys does he have in his class?
5. Yellow and blue paint were mixed in a ratio of 5 to 3 to make green paint. After 14 liters
of blue paint were added, the amount of yellow and blue paint in the mixture was equal.
How much green paint was in the mixture first?
II. Reasoning with Double Number Lines
1. Andy Worm travels 6 cm every 4 minutes. Betty Worm travels 15cm every 10 minutes.
Are Andy and Betty travelling at the same rate? If not, who is travelling “faster”? Explain
how you know.
2.
If 75% of the budget is $1200, what is the full budget?
3.
A photocopier can print 12 copies in 48 seconds. At this rate, how many copies can it print
in 1 minute?
Proportions and Proportion Problems
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4.
There are 24 students in a class. A hexagonal table can seat 6 students. How many
hexagonal tables do we need to seat all students?
III. Reasoning with a Table
1. If two pounds of beans cost $5.00, how much will 15 pounds of beans cost?
Mixing Paint
Mark was mixing blue paint and yellow paint in the ratio of 2:3 to make green paint. He wants
to make 45 liters of green paint. He began to make a table to help him think about the
2. Mark was mixing blue paint and yellow paint in the ratio of 2:3 to make green paint. He
problem,wants
but istounsure
of what
topaint.
do next.
make 47.5
liters of
He began to make a table to help him think about the
problem, but is unsure of what to do next.
Liters of Liters of Liters of
Blue
Yellow
Green
Paint
Paint
Paint
2
3
5
4
6
10
a. Explain how to continue to add values to the table.
a.)
b. How many liters of yellow paint will he need to make 47.5 liters of green paint?
Explain how to continue to add values to the table.
b.)
Write an explanation to Mark about how he can use his table to find how many liters of
c. paint
Markand
decides
buy 15
litersof
of yellow
paint. He
still will
wants
mix blue
paint and
yellowof green
blue
howtomany
liters
paint
hetoneed
to make
45 liters
paint in the ratio of 2:3 to make green paint. How many liters of yellow paint should
paint. he buy and how many gallons of green paint can he make?
Mark decides to buy 15 liters of blue paint. He still wants to mix blue paint and yellow
Proportions and Proportion Problems
3
paint in the ratio of 2:3 to make green paint. How many liters of yellow paint should he
c.)
buy, and how many gallons of green paint he can make? Use mathematical reasoning to
justify your answer.
3.
On a United States map, 24 centimeters represents 18 miles.
a. How many centimeters represent one mile?
b. How long is the line segment between A and B in centimeters?
c. If A and B represent two cities what is the actual distance between the two cities?
Proportions and Proportion Problems
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IV. Reasoning with a Graph
Describe the ratio for the proportional relationship given on the graph below
How many pages will the reader have read in 15hours, 24 hours?
Proportions and Proportion Problems
5