Title
Goals
Standard
Addressed
by the Unit
Proportional Reasoning
More Proportion Problems, Percents, and Fractions
By the end of this session, the teachers will be able to:
 Represent percents as ratios.
 Solve word problems involving percents using proportional
reasoning.
 Use proportional reasoning to explain, “flip and multiply” in the
context of division of fractions.
 Solve word problems involving fractions using proportional
reasoning.
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
More Proportion Problems, Percents, and Fractions
1
Percents
Definition: A percent is a number expressed as units per 100.
Examples: Use this definition of percent and proportional reasoning to respond to the following tasks.
1. What number is 60% of 25?
2. What percent of 20 is 5?
3. 25 is 40% of what number?
4. What percent of the rectangle given below is shaded?
More Proportion Problems, Percents, and Fractions
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5. Percent word problems and proportional reasoning: Solve the following problems using
proportional reasoning.
a. If 80% of the students in a certain class are female and there are 40 females students, what is
the total number of students in the class?
b. Joe bought new clothes for his bank job. Before the tax was added on, his total was $180. How
much tax did he pay if the sales tax is 5%? What was the total amount he paid?
c. Sylvia paid $8400 in federal and state income taxes, which amounted to 28% of her annual
income. What was her income last year?
d. On a 120-question test a student answered 84 correctly. What percent of the problems did the
student work correctly?
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3
e. How much acid is in a 5-liter container of acid and water that is marked 80% acid? How much
is water?
f. Of 420 students enrolled in a basic math class, only 25% are first year students. How many are
first-year students? How many are not?
g. If 45 students enrolled in the Success Academy, but only 30 complete their course, what
percent of the students completed their courses?
h. In a class of 32 people, there are 10 people under 25 years of age. What percent of the
population is under 25 years of age?
i. In a shipment of airplane parts, 2% are known to be defective. If 8 parts are found to be
defective, how many parts are in the whole shipment?
More Proportion Problems, Percents, and Fractions
4
Using proportional Reasoning in Multiplication and Division of Fractions
Review: Multiplication of Fractions.
1. Use a number line to illustrate the process of the following operations and find the product.
3
a. 2 ´
4
1 5
b. 3 ´
2 6
c.
1 3
´
2 4
d.
6 3
´
5 4
More Proportion Problems, Percents, and Fractions
5
2. Using the Number Line provided below indicate, by using ✕, where 4 ´
3
is on the number line.
4
Review: Division of Fractions.
1. Use a number line to illustrate the process of the following operations and find the product.
1
2
a.
2¸
b.
1 5
3 ¸
2 6
1 3
¸
2 4
c.
d.
6 3
¸
5 4
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2. Using the Number Line provided below indicate, by using ✕, where 2 ¸
3
is on the number line.
4
3. Using proportional reasoning to explain, “flip and multiply”: Consider the following problems.
a. How many half dollars are there in:
i. $5.00
ii. $3.25
iii. $0.80
iv. $300.00
v. $x
One way of answering this question is by using proportional reasoning and determining the
unit rate (i.e., the number of half dollars in one dollar) and multiplying the rate by the dollar
amount we have. So the ratio of dollars to half dollars is 1:2 and we can create a table to help
us create the table below:
Amount (Dollars) Number of Half Dollars
1
2
0.80
3.25
5.00
300.00
x
How would you express the number of quarters in terms of the amount?
How would a graph of this table look like?
How would a graph of this relationship look like?
More Proportion Problems, Percents, and Fractions
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b. Use the model in (a) above to respond to the question below:
3
How many are there in:
4
i. 1 unit? (Use this to write a ratio of one unit to the number of
3
in every unit)
4
ii. 3 units?
5
iii.
units?
6
3
iv. 4 units?
4
5 2
c. Given the operation ¸ :
4 3
Use words rewrite the operation: How
many_____________________________________________________
How many
2
are there in 1 unit?
3
Sketch a picture to illustrate your answer
Use this to write a ratio of one unit to the number of
2
in every unit.
3
Complete the table below:
Expression
2
1¸
3
2
2¸
3
5 2
¸
4 3
2
3¸
3
3 2
4 ¸
4 3
2
x¸
3
Solution
More Proportion Problems, Percents, and Fractions
8
4. Solving fractions word problems using proportional reasoning:
a. Carlton had $72. Then he spent 1⁄4 of his money on books and 3/8 of his money on sports
equipment. How much money did Carlton have left?
b. If 3⁄4 pound of nails fills a container 2/3 full, then how many pounds of nails will fill the
container?
c. Owen owns 5/6 of an acre of farmland. He grows beets on 2/5 of the land. On how many acres
of land does Owen grow beets?
d. At Devak's Arcade, 1/2 of the games are racing games. Among the racing games, 9/10 are
motorcycle-racing games. What fraction of the games at Devak's Arcade are motorcycle-racing
games?
e. With 1 pint of paint, we can paint 6/5 m2. How much area can you paint with 2/3 pint of paint?
More Proportion Problems, Percents, and Fractions
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f. If 3⁄4 pints of paint can paint 2/3 m2, how much area can you paint with 1 pint?
g. A girl spends 1/3 of her money, loses 2/3 of the remainder, and goes home with 12 cents. How
much money did she have originally?
h. A 160-pound man is involved in a serious auto accident. While in the hospital he loses 1/4 of
his weight. How much does he weigh when discharged from the hospital?
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