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11/2/2015 National Go Math Middle School, Grade8
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ProportionalRelationships
ESSENTIAL QUESTION
How can you useproportional relationshipsto solve real-worldproblems?
MODULE
LESSON 3.1
8.EE.6, 8.F.4
8.F.4
8.EE.5, 8.F.2, 8.F.4
RepresentingProportionalRelationships
LESSON 3.2
Rate of Changeand Slope
LESSON 3.3
Interpreting the UnitRate as Slope
Real-World Video
Speedboats can travel at fast rates while sailboatstravel more slowly. If you graphed distance versustime for both types of boats, you could tell by thesteepness of the graph which boat was faster.
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Complete these exercises to review skills you will need forthis module.
Write Fractions as Decimals
EXAMPLE Multiply the numerator and thedenominator by a power of 10 so that thedenominator is a whole number.
Write the fraction as a division problem.Write a decimal point and zeros in thedividend.Place a decimal point in the quotient.Divide as with whole numbers.
Write each fraction as a decimal.
1.
4.
7.
2.
5.
8.
3.
6.
9.
Solve ProportionsEXAMPLE
7 × 2 =14, so multiply the numerator anddenominator by 2.
5 × 2 =10
Solve each proportion for x.
10.
13.
16.
11.
14.
17.
12.
15.
18.
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Visualize VocabularyUse the words to complete the diagram.
VocabularyReview Words
constant (constante)
equivalent ratios (razonesequivalentes)
proportion (proporción)rate (tasa)
ratios (razón)
unit rates (tasas unitarias)
Preview Wordsconstant of proportionality(constante deproporcionalidad)proportional relationship(relación proporcional)
rate of change (tasa decambio)
slope (pendiente)
Understand VocabularyMatch the term on the left to the definition on the right.
1. unit rate A. A constant ratio of two variables related proportionally.
B. A rate in which the second quantityin the comparison is one unit.
C. A relationship between twoquantities in which the ratio ofone quantity to the otherquantity is constant.
2. constant ofproportionality
3. proportionalrelationship
Active ReadingKey-Term Fold Before beginning the module,create a key-term fold to help you learn thevocabulary in this module. Write the highlightedvocabulary words on one side of the flap. Writethe definition for each word on the other side of the flap. Use the key-term fold to quiz yourselfon the definitions used in this module.
Module 3 69
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MODULE 3
Understanding the standards and the vocabulary termsin the standards will help you know exactly what youare expected to learn in this module.
8.EE.5
Graph proportionalrelationships, interpreting theunit rate as the slope of thegraph. Compare two differentproportional relationshipsrepresented in different ways.
What It Means to YouYou will use data from a table and agraph to apply your understandingof rates to analyzing real-worldsituations.
UNPACKING EXAMPLE8.EE.5Key Vocabulary
proportional relationship(relación proporcional)A relationship between twoquantities in which the ratio ofone quantity to the otherquantity is constant.
slope (pendiente)A measure of the steepnessof a line on a graph; the risedivided by the run.
unit rate (tasa unitaria)A rate in which the secondquantity in the comparison isone unit.
The table shows the volume of waterreleased by Hoover Dam over acertain period of time. Use the datato make a graph. Find the slope ofthe line and explain what it shows.
Water Releasedfrom Hoover Dam
Volume ofwater (ft3 )Time (s)
5
10
15
20
75,000
150,000
225,000
300,000
The slope of the line is 15,000. This means that for every secondthat passed, 15,000 ft3 of water was released from Hoover Dam.
Suppose another dam releases water over the same period of timeat a rate of 180,000 ft3 per minute. How do the two rates compare?
180,000 ft3 per minute is equal to 3,000 ft3 per second. This rate isone fifth the rate released by the Hoover Dam over the same timeperiod.
70 Unit 2
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L E S S O N
3.1RepresentingProportionalRelationships
8.EE.6
…derive the equationy = mx for a line throughthe origin… Also 8.F.4
ESSENTIAL QUESTIONHow can you use tables, graphs, and equations to representproportional situations?
Prep for 8.EE.6
Representing ProportionalRelationships with TablesIn 1870, the French writer Jules Verne published 20,000 Leagues Under the Sea,one of the most popular science fiction novels ever written. One definition ofa league is a unit of measure equaling 3 miles.
Complete the table.
20,000
What relationships do you see among the numbers in the table?
For each column of the table, find the ratio of the distance in miles tothe distance in leagues. Write each ratio in simplest form.
What do you notice about the ratios?
Reflect1. If you know the distance between two points in leagues, how can you
find the distance in miles?
2. If you know the distance between two points in miles, how can you find
the distance in leagues?
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Representing ProportionalRelationships with EquationsThe ratio of the distance in miles to the distance in leagues is constant. Thisrelationship is said to be proportional. A proportional relationship is arelationship between two quantities in which the ratio of one quantity to theother quantity is constant.
A proportional relationship can be described by an equation of the formy = kx, where k is a number called the constant of proportionality.
Sometimes it is useful to use another form of the equation,
EXAMPLE 1 8.EE.6
For every hour Meghanworks, she earns $12. So,for 8 hours of work, sheearns 8 × $12 = $96.
Meghan earns $12 an hour at her part-time job. Show that the relationshipbetween the amount she earned and the number of hours she worked is aproportional relationship. Then write an equation for the relationship.
STEP 1 Make a table relating amountearned to number of hours.
Number of hours
Amount earned ($)
1
12
2
24
4
48
8
96
STEP 2 For each number of hours, write the relationship of the amountearned and the number of hours as a ratio in simplest form.
Since the ratios for the two quantities are all equal to therelationship is proportional.
STEP 3 Write an equation.
Let x represent the number of hours.Let y represent the amount earned.
Describe two real-worldquantities with a proportional
relationship that can bedescribed by theequation y = 25x.
First tell what thevariables represent.
Use the ratio as the constant of proportionality in the equation y = kx.
The equation is or y = 12x
3. Fifteen bicycles are produced each hour at theSpeedy Bike Works. Show that the relationshipbetween the number of bikes produced and thenumber of hours is a proportional relationship.Then write an equation for the relationship.
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72 Unit 2
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Representing ProportionalRelationships with GraphsYou can represent a proportional relationshipwith a graph. The graph will be a line that passesthrough the origin (0, 0). The graph shows therelationship between distance measured in milesto distance measured in leagues.
EXAMPLE 2 8.EE.6
The graph shows the relationship betweenthe weight of an object on the Moon andits weight on Earth. Write an equation forthis relationship.
STEP 1 Use the points on the graph to makea table.
Earth weight (lb)
Moon weight (lb)
6
1
12
2
18
3
30
5
STEP 2 Find the constant of proportionality.
The constant of proportionality is
STEP 3 Write an equation.
Let x represent weight on Earth.
Let y represent weight on the Moon.
The equation is Replace k with
The graph shows the relationship betweenthe amount of time that a backpacker hikesand the distance traveled.
4. What does the point (5, 6) represent?
5. What is the equation of the relationship?
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Lesson 3.1 73
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Guided Practice
1. Vocabulary A proportional relationship is a relationship between twoquantities in which the ratio of one quantity to the other quantity
is / is not constant.
2. Vocabulary When writing an equation of a proportional relationship in the
form y = kx, k represents the
3. Write an equation that describes the proportional relationship betweenthe number of days and the number of weeks in a given length of time.(Explore Activity and Example 1)
a. Complete the table.
Time (weeks)
Time (days)
1
7
2 4
56
10
b. Let x represent
Let y represent
The equation that describes the relationship is
Each table or graph represents a proportional relationship. Write anequation that describes the relationship. (Example 1 and Example 2)
4. Physical Science The relationshipbetween the numbers of oxygen atoms andhydrogen atoms in water
Oxygenatoms
Hydrogenatoms
2
4
5
34
120
5.
ESSENTIAL QUESTION CHECK-IN
6. If you know the equation of a proportional relationship, how can youdraw the graph of the equation?
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Name Class Date
3.1 Independent Practice8.EE.6, 8.F.4
The table shows the relationship between temperatures measured on theCelsius and Fahrenheit scales.
Celsius temperature
Fahrenheit temperature
0
32
10
50
20
68
30
86
40
104
50
122
7. Is the relationship between the temperature scales proportional? Whyor why not?
8. Describe the graph of the Celsius-Fahrenheit relationship.
9. Analyze Relationships Ralph opened a savings account with a depositof $100. Every month after that, he deposited $20 more.
a. Why is the relationship described not proportional?
b. How could the situation be changed to make the situation
proportional?
10. Represent Real-World Problems Describe a real-world situation thatcan be modeled by the equation Be sure to describe what eachvariable represents.
Look for a Pattern The variables x and y are related proportionally.
11. When x = 8, y = 20. Find y when x = 42.
12. When x = 12, y = 8. Find x when y = 12.
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13. The graph shows the relationship between the distance that a snailcrawls and the time that it crawls.
a. Use the points on the graph to make a table.
Distance (in.)
Time (min)
b. Write the equation for the relationship and tell what eachvariable represents.
c. How long does it take the snail to crawl 85 inches?
FOCUS ON HIGHER ORDER THINKING Work Area
14. Communicate Mathematical Ideas Explain why all of the graphs in thislesson show the first quadrant but omit the other three quadrants.
15. Analyze Relationships Complete the table.
Length of side of square 1 2 3 4 5
Perimeter of square
Area of square
a. Are the length of a side of a square and the perimeter of the squarerelated proportionally? Why or why not?
b. Are the length of a side of a square and the area of the square relatedproportionally? Why or why not?
16. Make a Conjecture A table shows a proportional relationship where k isthe constant of proportionality. The rows are then switched. How doesthe new constant of proportionality relate to the original one?
76 Unit 2
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