Exploring Linear Relations Math B Essentials
Module #2 Student Edition 2017 - 2018 Created in collaboration with Utah Middle School Math Project A University of Utah Partnership Project
San Dieguito Union High School District
Table of Contents MODULE #2: EXPLORING LINEAR RELATIONS * STANDARDS FOR MATHEMATICAL PRACTICE: A GUIDE FOR STUDENTS AND PARENTS ....................................... 4 SECTION 2.1: ANALYZE PROPORTIONAL RELATIONSHIPS AND CONTEXTS* ....................................................... 5 2.1A LESSON: SIMPLIFYING FRACTIONS .......................................................................................................... 6 2.1B LESSON: MULTIPLYING AND DIVIDING FRACTIONS .................................................................................... 7 2.1B EXTENSION: FRACTION CROSSWORD ...................................................................................................... 9 2.1C LESSON: GRAPHING ON A COORDINATE PLANE ..................................................................................... 10 2.1D LESSON: PROPORTIONAL RELATIONSHIPS ............................................................................................. 11 2.1E LESSON: W RITING EQUATIONS FOR PROPORTIONAL RELATIONSHIPS ...................................................... 13 2.1F LESSON: INDEPENDENT AND DEPENDENT VARIABLES ............................................................................. 15 2.1G LESSON: COMPARING PROPORTIONAL RELATIONSHIPS ......................................................................... 17 SECTION 2.1 REVIEW LESSON ...................................................................................................................... 19 SECTION 2.2: NON-PROPORTIONAL RELATIONSHIPS AND LINEAR GROWTH .................................................... 22 2.2A LESSON: RATE OF CHANGE* ................................................................................................................. 23 2.2B LESSON: NON-PROPORTIONAL RELATIONSHIPS ..................................................................................... 26 2.2C LESSON: W RITING EQUATIONS FOR NON-PROPORTIONAL RELATIONSHIPS.............................................. 29 2.2D LESSON: TABLE OF VALUES FOR LINEAR RELATIONSHIPS ....................................................................... 32 2.2E LESSON: LINEAR PATTERNS* ................................................................................................................ 33 2.2E EXTENSION: CHART OF PATTERNS ........................................................................................................ 35 2.2F LESSON: EXAMINING LINEAR GROWTH* ................................................................................................. 36 SECTION 2.2 REVIEW LESSON ...................................................................................................................... 39 TASK: FILLING THE POOL .............................................................................................................................. 42
* Denotes a lesson that was adapted from Utah Middle School Math Project. © Utah Middle School Math Project and University of Utah Partnership
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SDUHSD Math B Essentials Module #2 –STUDENT EDITION 2017-2018
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Module #2: Exploring Linear Relations * Online support for this module can be found at http://goo.gl/s2vDtM (case sensitive) or using the QR code below. This website includes copies of student classwork, homework, and instructional videos for common core standards.
Common Core Standard(s): Curriculum Support Website 8.EE.5 - Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.
8.F.4 -
Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
Academic Vocabulary: proportional relationship, constant of proportionality, unit rate, non-proportional relationship, rate of change, linear relationship, slope (m), y-intercept (b), origin, rise, run, context, geometric model, initial value
Module Overview: Students begin this module by reviewing proportional relationships from 6th and 7th grades, recognizing, representing, and comparing proportional relationships. In eighth grade, a shift takes place as students move from proportional linear relationships, a special case of linear relationships, to the study of nonproportional and linear relationships in general. Students explore the growth rate of a linear relationship using patterns and contexts that exhibit linear growth. During this work with linear patterns and contexts, students begin to explore ideas about the two parameters of a linear relationship, constant rate of change (slope) and initial value (y-intercept).
Connections to Content: Prior Knowledge: This module relies heavily on students’ knowledge about ratios and proportional relationships from 6th and 7th grades. Students should have an understanding of unit rate and how to compute it. In addition, they should to be able to recognize and represent proportional relationships from a story, graph, table, or equation. In addition they must identify the constant of proportionality or unit rate given different representations. Future Knowledge: After this module, students continue to work with linear relationships, applying slopeintercept form as they write and graph equations of lines. This will set the stage for students to be able to graph and write the equation of a line given any set of conditions. Students use their knowledge of slope and proportionality to represent and construct linear functions in a variety of ways. They will expand their knowledge of linear functions and constant rate of change as they investigate other functions in future grades.
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Standards for Mathematical Practice: A Guide for Students and Parents The Standards for Mathematical Practices are central to the Common Core. These practices build fluency and help students become better decision-makers and problem solvers. The practices reflect the most advanced and innovative thinking on how students should interact with math content. Students and parents will develop skill with these standards by asking some of these questions: Make Sense of Problems and Persevere in Solving Them. What is the problem that you are solving for? Can you think of a problem that you recently solved that is similar to this one? How will you go about solving the problem? (i.e. What’s your plan?) Are you progressing towards a solution? How do you know? Should you try a different solution plan? How can you check your solution using a different method? Construct Viable Arguments and Critique the Reasoning of Others. Can you write or recall an expression or equation to match the problem situation? What do the numbers or variables in the equation refer to? What’s the connection among the numbers and variables in the equation? Reason Abstractly and Quantitatively. Tell me what your answers(s) mean(s) How do you know that your answer is correct? If I told you I think the answer should be (a wrong answer), how would you explain to me why I’m wrong? Model with Mathematics. Do you know a formula or relationship that fits this problem situation? What’s the connection among the numbers in the problem? Is your answer reasonable? How do you know? What do(es) the number(s) in your solution refer to? Use Appropriate Tools Strategically. What tools could you use to solve this problem? How can each one help you? Which tool is most useful for this problem? Explain your choice. Why is this tool (the one selected) better to use than (another tool mentioned)? Before you solve the problem, can you estimate the solution? Attend to Precision. What do the symbols that you used mean? What units of measure are you using (for measurement problems) Explain to me what (term from the lesson) means. Look For and Make Use of Structure. What do you notice about the answers to the exercises you’ve just completed? What do different parts of the expression or equation you are using tell you about possible correct answers? Look for and Express Regularity in Repeated Reasoning. What shortcut can you think of that will always work for these kinds of problems? What pattern(s) do you see? Can you make a generalization?
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Section 2.1: Analyze Proportional Relationships and Contexts* Section Overview: This section begins by reviewing simplifying, multiplying, and dividing fractions. Students then review graphing ordered pairs on a coordinate plane. Students analyze proportional relationships by investigating different contexts. They define the constant of proportionality or unit rate in tables, graphs, and equations. They recognize that a proportional relationship can be represented with a line that goes through the origin and the equation for any proportional relationship relates x-values to y-values through multiplication only. Concepts and Skills to Master: By the end of this section, students should be able to: Multiply, divide, and simplify fractions. Plot points on a coordinate plane. Graph and write equations for a proportional relationship and identify the constant of proportionality or unit rate given a table, graph, equation, or context. Compare proportional relationships represented in different ways.
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2.1A Lesson: Simplifying Fractions Name:
Period:
Directions: Simplify the fractions.
1.
2.
3.
4.
5.
6.
7. Sunny converted the mixed number why not.
to an improper fraction,
. Is Sunny correct? Explain why or
Directions: Complete the following table as you convert from improper fractions to mixed fractions and mixed fractions to improper fractions. Improper Fraction
Mixed Fractions
8.
9.
10.
11.
12. What is the value of
?
13. What is the value of
?
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2.1B Lesson: Multiplying and Dividing Fractions Name:
Period:
Directions: Multiply the fractions. Simplify if possible. 1.
2.
3.
4.
5.
6.
7.
8.
9.
10. Lucy made a mistake when dividing the fractions fractions correctly.
. Explain her mistake in words and divide the Lucy’s work
Explanation
Simplified correctly
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Directions: Divide the fractions. Simplify if possible. 11.
12.
13.
14.
15.
16.
17.
18.
19.
20. Simplify the expression:
(
)
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2.1B Extension: Fraction Crossword Name:
Period:
Directions: Complete the crossword by simplifying the expressions below. Write the answers in words by the corresponding number in the crossword.
Across
Down
4.
1.
7.
2.
9.
3.
10.
4.
11.
5.
13.
6.
15.
8. 12. 14.
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2.1C Lesson: Graphing on a Coordinate Plane Name:
Period:
Directions: Fill in the blanks using the provided word bank. quadrants
origin
y-axis
x-coordinate
ordered pair
coordinate plane
x-axis
y-coordinate
(x, y)
(0, 0)
1. The __________________ __________________ is a grid formed by the intersection of two number lines. 2. The __________________ is the point at where the two number lines intersect. This is written as ______________. 3. The horizontal number line is called the __________________. 4. The vertical number line is called the __________________. 5. The axes divide the plane into four __________________. 6. An __________________ __________________ gives the coordinates of a point’s location. An ordered pair is written as __________________. 7. The __________________ shows the position right or left of the y-axis. 8. The __________________ shows the position above or below the x-axis.
9. Label the terms in the word bank onto the coordinate plane. Graph the points below on the coordinate plane. a. (4, 5)
b. (-8, -1)
c. (0, 6)
d. (7, 7)
e. (-3, 0)
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2.1D Lesson: Proportional Relationships Name:
Period:
A unit rate is a comparison of two different quantities, the second of which is one. When a unit rate is written as a fraction, the denominator is 1. 1. Carmen is making homemade root beer for an upcoming charity fundraiser. The number of pounds of dry ice to the number of ounces of root beer extract (flavoring) is proportionally related. If Carmen uses 12 pounds of dry ice, she will need to use 6 ounces of root beer extract. a. Write a ratio that relates pounds of dry ice to ounces of root beer extract.
x
b. State the unit rate for this situation.
y
0 c. What does the unit represents in the context? 2 2. In the table, of dry ice.
= ounces of root beer extract and
= pounds
2
a. Complete the table.
3
6
b. In the table, when x = 0 and y = 0, what does the ratio equal? Explain.
?
3. Graph the relationship. a. What observations can you make about the graph?
b. Show the unit rate on the graph.
This situation represents a proportional relationship.
pounds of dry ice
c. In the table, what do you notice about the ratio
8
15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
1
3
4
5
6
7
8
9
10
ounces of root beer extract
In the table, every x-value is multiplied by to determine the corresponding y-value. The ratio
2
is
constant and is called the For this situation, 2 is the
This situation represents a
proportional relationship because the graph is a straight line that passes through the SDUHSD Math B Essentials Module #2 –STUDENT EDITION 2017-2018
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4. Does the equation
represent this situation? Explain.
5. Bubba’s Body Shop charges $50 per hour to fix a car. a. Complete the table and graph to represent this situation x
y
(time in hours)
(cost in dollars)
140
0 Cost in dollars
120
1 2 3
100 80 60 40 20
b. State the unit rate.
0
1
2
3
4
5
6
7
8
9
10
Time (in hours)
c. Show the unit rate on the graph.
d. State the constant of proportionality.
e. Are cost and time proportionally related in this situation? Justify your answer using the table of values and graph. Table of Values Graph
f. Does the equation
represent this situation? Explain.
The equation for any proportional relationship relates x-values to y-values through The equation is written in the form
, where
is the
In the situation above, cost and time are proportionally related because the equation relates time to cost by multiplying the number of hours by 25, the constant of proportionality.
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2.1E Lesson: Writing Equations for Proportional Relationships Name:
Period:
Directions: Use the context to complete the following. 1. The cat ran away from home at a rate of 4 feet for every 2 seconds. a. Complete the table of values and graph for this situation.
x (seconds) 0
y (feet)
Feet
1 2 3
b. State the unit rate. Time (seconds)
c. State the constant of proportionality.
d. Write an equation that represents this situation.
2. Bill’s Burger Barn has a special deal of 2 hamburgers for $9.00 a. Complete the table of values and graph for this situation. y (cost)
2
Cost
x (# of hamburgers) 1
3 4
b. State the unit rate. # of hamburgers
c. State the constant of proportionality.
d. Write an equation that represents this situation.
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Directions: Examine the situation to complete the following. 3. The mouse scurries away at 4 inches per second. a. State the constant of proportionality.
b. If x represents seconds and y represents inches, write an equation, in the form represents this situation.
, that
4. John can eat 2 slices of pizza in one minute. a. State the constant of proportionality.
b. If x represents minutes and y represents number of pizza slices, write an equation, in the form , that represents this situation.
5. Lucy eats four Hershey bars every 5 minutes. a. State the constant of proportionality.
b. If x represents minutes and y represents number of Hershey bars, write an equation, in the form , that represents this situation.
6. Chocolate gummi bears cost $3.00 for 2 pounds. a. State the constant of proportionality.
b. If x represents pounds and y represents cost, write an equation, in the form this situation.
, that represents
7. Does the equation represent the context below? Explain why or why not. If the equation is wrong, write the correct equation for the situation. Context: DJ took 20 pounds of aluminum cans to the recycling center and received $10.00.
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2.1F Lesson: Independent and Dependent Variables Name:
Period:
In a relationship, the x-value is the independent variable and the y-value is the dependent variable. The dependent variable is determined by or depends on the independent variable. 1. Shari is filling up her gas tank and wants to know how much it will cost. At Grizzly’s Gas-n-Go, gas costs $3.50 per gallon. a. State the independent variable in this situation.
b. State the dependent variable in this situation.
c. Complete the graph and table for this situation. Label the columns in the table and each axis in the graph. x:
y: 0 1 2
12 10 8 6
3 4
4 2
d. State the unit rate.
0
1
2
3
4
5
e. State the constant or proportionality.
f. Write an equation that represents this situation.
Directions: For each situation, draw a circle around and put an I above the independent variable. Underline and put a D above the dependent variable. 2. The drama club is selling tickets to the Fall Ball. The more tickets that they sell, the more money they can spend on decorations.
3. The air pressure inside a tire increases with a rise in temperature.
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4. As the amount of rain decreases, so does the water level of the river.
5. The number of calories you burn increases as the number of minutes that you walk increases.
6. The value of your car decreases with age.
7. The distance you can drive vs. the amount of gas in the tank.
8. The total number of laps run depends on the length of each workout.
9. A tree grows 15 feet in 10 years.
10. During the summer, Ruth eats two vanilla ice cream cones each week. Was the situation graphed correctly? Explain why or why not.
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2.1G Lesson: Comparing Proportional Relationships Name:
Period:
Directions: For each activity below, compare the dependent variable, cost, to the independent variable, hours. Complete the table of values. 1. Story: It costs $10 per hour to talk on the phone long distance. a. State the constant of proportionality.
Hours x 0
Cost ($) y 10
b. Write an equation that represents this activity.
20 c. What is the cost for half an hour?
3
2. Story: Two hours of laser tag costs $5.
Hours x
a. State the constant of proportionality.
Cost ($) y 0 2.5
b. Equation: 5 c. $20 relates to how many hours?
3. Story: It costs $75 to play 3 hours of paintball. a. State the constant of proportionality.
4
Hours x
Cost ($) y 0 75
b. Equation:
150 c. What is the cost for
hours? 9
4. Story: Renting a camping spot for 24 hours costs $6. a. State the constant of proportionality.
b. Equation:
Hours x 0
Cost ($) y
6 12
c. $12.50 relates to how many hours?
SDUHSD Math B Essentials Module #2 –STUDENT EDITION 2017-2018
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17
5. Create your own proportional story, table, and equation that compares cost per hour, and has a greater unit rate than problem #1, but a lesser unit rate than problem #3. Story:
Table:
Constant of Proportionality: Hours x 0
Cost ($) y
Equation:
1 2 3
6. Graph all situations (#1-5) on the graph below. Label each line with the equation that represents the situation.
b. How do you compare the cost per hour by looking at the equations?
Cost
a. How do you compare the cost per hour by looking at the graph?
c. What happens to the lines on the graph as the unit rate increases?
d. What happens to the lines as the unit rate decreases? Hours
e. If a certain activity costs $50 per hour, describe what its graph would look like.
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Section 2.1 Review Lesson Name:
Period:
Directions: Complete the following table as you convert from improper fractions to mixed fractions and mixed fractions to improper fractions. Improper Fraction
Mixed Fractions
1.
2.
3.
Directions: Multiply or divide the fractions. Simplify if possible. 4.
6.
5.
Directions: Use the coordinate plane to complete the following. 7. Write the letter that represents the ordered pair. (2, 4)
(-2, 5)
(-4, -4)
(5, 1)
(1, -5)
(-2, -4)
8. Write the ordered pairs for each point. Q(
,
)
E(
,
)
H(
,
)
R(
,
)
W(
,
)
A(
,
)
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Directions: State the constant of proportionality for each representation. 9. Vanessa is making lemonade. She is supposed to add two scoops of lemonade mix for every one cup of water.
10.
x
y
1
3
2
6
3
9
11.
Distance
45
12.
30 15
0
1
2
3
4
5
6
Time
Directions: For each situation, draw a circle around and put an I above the independent variable. Underline and put a D above the dependent variable. 13. The ball drops 2 feet every second.
14. Olivia bought 5 t-shirts for $50.
15. The tires on Jordyn’s car lose 1 millimeter of tread after traveling 100 miles.
16. Emma is putting together an order for sugar, flour, and salt for her restaurant pantry. The graph shows the cost to buy pounds of sugar and flour. One line shows the cost of buying pounds of flour and the other line shows the cost of buying pounds of sugar. a. From the graph, which ingredient costs more to buy per pound? Justify your answer.
b. The cost to buy salt by the pound is less than sugar and flour. Draw a possible line that could represent the cost to buy pounds of salt.
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17. Captain Cable charges $50.00 per hour to install internet service at your friend’s house.
x:
y: 2
a. State the independent variable.
3 4
b. State the dependent variable.
5 c. Complete the table of values. Label each column. d. State the constant of proportionality. 250
e. What does the constant of proportionality represent in the context? 200
f. Write an equation that represents the situation.
g. Make a graph for this situation. Label each axis. Label your line Captain Cable.
150
100
h. Explain why this situation is proportional using the equation and graph. 50
Equation
Graph
0
2
4
6
18. Super Shop charges $30 per hour to install internet service at your house. a. Complete the table of values. Label each column. b. State the constant of proportionality.
c. Write an equation that represents the situation.
x:
y: 2 3 4
d. Make a graph for this situation on the same coordinate plane you graphed Captain Cable. Label your line Super Shop.
5
19. Examine your graph. Which line is steeper? Explain why this occurs.
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Section 2.2: Non-Proportional Relationships and Linear Growth Section Overview: Students learn that rate of change is a constant ratio comparing change in y-values to change in x-values. Students apply their knowledge of proportional relationships and rate of change to non-proportional relationships, understanding that non-proportional relationships are represented by lines that do not pass through the origin. Students analyze the equations and table of values for non-proportional relationships. Students examine linear patterns and use these patterns to identify rate of change and initial value (yintercept) in different representations (table of values, graph, equation, and geometric model.) Students begin to understand how linear growth changes. Concepts and Skills to Master: By the end of this section, students should be able to: Understand that all linear relationships have a rate of change. Understand how rate of change affects linear relationships. Understand how non-proportional relationships differ from proportional relationships. Examine linear patterns and connect equations to the pattern. Identify the rate of change and y-intercept of a linear relationship in a table of values, graph, equation, context, and geometric model.
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2.2A Lesson: Rate of Change* Name:
Period:
Unit rate is a constant ratio that compares measurements in which one of the terms has a value of 1. Rate of change is a constant ratio comparing change in y-values, the dependent variable, with respect to change in x-values, the independent variable. All linear relationships show a constant rate of change. We can further explore the concept of rate of change by examining three different staircases. On properly built staircases, all of the stairs have the same measurements. The important measurements on a stair are what we call the “rise,” the vertical measurement, and the “run,” the horizontal measurement. When building a staircase, these measurements are chosen carefully to prevent the stairs from being too steep, and to get you where you need to go. One step from three different staircases is given below. Staircase #1
Staircase #2
1. State the rise and run for each staircase. Staircase #1 rise =
run =
Staircase #2
rise =
run =
Staircase #3
Staircase #3 rise =
run =
2. Using the rise and run for each step, graph the first 5 steps for each staircase.
Height after 5 steps
Staircase #2:
Height (rise)
Height (rise)
Staircase #1:
Staircase #3
Horizontal Distance (run)
3. For each staircase on the graph, draw a connecting line from the origin (0,0) to the point representing the final height after 5 steps.
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When a linear relationship is graphed, the constant rate of change is represented by the ratio which is the
. Slope describes how steep a line is. It represents the
4. Find the slope of each line representing a staircase. Simplify your ratio. a. Staircase #1:
b.
Staircase #2:
c. Staircase #3:
5. If you didn’t have the graph to look at, and could only examine the ratios just calculated, how would you know which staircase is the steepest?
6. Calculate the slope for climbing one, two, and three steps on each of the staircases.
Staircase #1 Total Rise
Total Run
Slope (rise/run)
Staircase #2 Total Rise
Total Run
Slope (rise/run)
Staircase #3 Total Rise
Total Run
Slope (rise/run)
one step two steps three steps
7. Does the slope of the staircase change as you climb each step?
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. State the rate of change for the line.
Directions: Examine the graph to write three different ratios for y
y
8.
9. 6
9
4
6
2
3
0
1
2
3
rise
4
5
6
x
0
1
run
2
3
rise
5
6
x
run
1st
1st
2nd
2nd
3rd
3rd
Rate of Change:
Rate of Change:
10.
4
y
11. 3
2
1
0
rise
1
2
3
rise
run
1st
1st
2nd
2nd
3rd
3
Rate of Change:
Rate of Change:
4
5
6
x
run
rd
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2.2B Lesson: Non-Proportional Relationships Name:
Period:
1. Last Saturday at Sunnyside Park, there was a hot dog eating contest that lasted 5 minutes. Landon participated in the contest and ate 5 hot dogs every 2 minutes. Nate also participated, and before the competition even began, Nate ate 4 hot dogs (he was really hungry!). Once the competition started, Nate ate 3 hot dogs every 2 minutes. The graph below represents the contest. a. State the rate of change for Landon and show his rate on the graph.
# of hot dogs
b. At 0 minutes, how many hotdogs has Landon eaten?
c. State the rate of change for Nate and show his rate on the graph.
Nate
Landon
d. At 0 minutes, how many hotdogs has Nate eaten?
# of minutes
Number of minutes e. At what time have both boys eaten the same number of hot dogs? How is this shown in the graph?
f. Who won the contest? How do you know?
2. In the tables, = time in minutes and
= number of total hot dogs consumed. Complete the tables.
Landon t
Nate
h
t
0 1
h
0 1
2.5
5.5
2 3
8.5
7.5
a. What do you notice about the ratio
for Landon?
b. What do you notice about the ratio
for Nate?
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c. Which situation is proportional? Justify using the table of values and graph. Table of Values
Graph
.
Nate’s situation represents a non-proportional relationship. In the table, every x-value is not multiplied by the same value to determine the corresponding y-value. The ratio
is
Nate’s graph is a line that Nate has eaten 4 hotdogs at 0 minutes. The graph passes through the point and does not pass through the origin.
3. Oliver makes $26 for selling 13 bags of popcorn at the Starling County Fair. Sara makes $8 a day plus $2 for every bag of popcorn she sells at the Fair. a. In the tables, = number of popcorn bags and = amount of dollars. Complete the tables to show the amount of money that Oliver and Sara make for selling up to three bags of popcorn in one day at the Fair.
Oliver
Sara
0
0
1
1
2
2
3
3
b. State the rate of change for Oliver and explain what it represents in the context.
c. State the rate of change for Sara and explain what it represents in the context.
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d. Graph each situation on the coordinate plane. Label your graphs and axes. y 14 12 10 8 6 4 2
0
2
4
6
x
e. When Oliver sells zero bags of popcorn, how much money does he earn? On the graph, circle the ordered pair that represents this situation.
f. When Sara sells zero bags of popcorn, how much money does she earn? On the graph, circle the ordered pair that represents this situation.
g. Which person earns more money for every bag of popcorn sold? Use the tables of values and graph to justify your answer.
4. Which situation represents a non-proportional relationship? Justify using the table of values and graph. Table of Values
SDUHSD Math B Essentials Module #2 –STUDENT EDITION 2017-2018
Graph
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2.2C Lesson: Writing Equations for Non-Proportional Relationships Name:
Period:
Directions: Use the context to complete the following. 1. Burt the Bumble Bee is 2 inches away from a flower, and then suddenly flies away from it at a rate of 3 y inches per second.. a. Complete the table of values and graph for this situation. y (feet)
0 1 2 3
b. Use the graph to state the rate of change.
8
Feet away from flower
x (seconds)
10
6
4
2
0
2
4
6
x
Time (seconds)
c. At 0 seconds, how far is Burt from the flower? Circle on the graph and state the ordered pair that represents this.
d. This situation is represented by the equation equation?
. How is the rate of change shown in the
e. How is Burt’s initial distance shown in the equation?
The equation for any non-proportional relationship relates x-values to y-values through The equation is written in the form and
, where
is the
is the value of y when x equals 0. In the situation above, feet and time are not
proportionally related because the equation
relates time to feet by multiplying the
number of seconds by 3, and adding the initial distance, 2. The initial value is the on the graph. The y-intercept is the point where the
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2. Jumpers-r-us charges $10 per person per hour to jump and a $3 shoe rental fee. a. State the rate of change.
b. If Jane jumps for 0 hours, what is her cost?
c. If x represents hours and y represents cost, write an equation, in the form represents this situation.
, that
3. DJ has $100 in the bank and saves $5 per week. a. State the rate of change.
b. Initially, how much money does DJ have in the bank?
c. If x represents number of weeks and y represents the amount of money in the bank, write an equation, in the form , that represents this situation.
Directions: For each non-proportional graph, state the rate of change, circle and state the y-intercept, and write an equation in the form . Graph 4.
Rate of change
y-intercept
Equation (
)
6
4
2
0
5.
1
2
3
4
5
6
1
2
3
4
5
6
x
15
10
5
0
x
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Graph 6.
Rate of change
y-intercept
Equation (
)
9
6
3
0
7.
1
2
3
4
5
6
1
2
3
4
5
6
x
30
20
10
0
x
8. Complete the table below comparing graphs, tables of values, and equations for proportional and non-proportional linear relationships. Graph
Table of Values
Equation Example
Proportional relationships
non-proportional relationships
SDUHSD Math B Essentials Module #2 –STUDENT EDITION 2017-2018
Constant of Proportionality (yes or no) Rate of Change (yes or no)
Constant of Proportionality (yes or no) Rate of Change (yes or no)
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2.2D Lesson: Table of Values for Linear Relationships Name:
Period:
Directions: Complete the tables of values for the given equations. 1.
2.
x
y
3.
x
y
x
-2
-6
-3
-1
-3
-1
0
2
2
1
4
4
2
7
5
4.
5.
x
6.
y
x
19
-6
7 3
y
x
y
-6 -19
-5
0
3 4
y
-10 16
5
-1 4
7. Circle the equation that represents the table of values. A.
B.
C. SDUHSD Math B Essentials Module #2 –STUDENT EDITION 2017-2018
32
2.2E Lesson: Linear Patterns* Name:
Period:
1. Use the pattern to complete the following. a. Draw the figure at stage 4.
b. Complete the table. Stage (s) c. How many blocks are being added from one stage to the next?
# of Blocks (b)
1 2
d. How many blocks are in stage 5? 3 4 e. How many blocks are at stage 0?
2. This pattern can be represented by the equation and s represents the stage number.
where b represents the number of blocks
a. Show how the equation relates to the pattern.
b. How is the constant in the equation, 1, shown in the pattern?
c. How is the coefficient of
in the equation, 2, shown in the pattern?
d. Use the equation to determine the number of blocks in stage 100.
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3. Use the pattern to complete the following. a. Draw the figure at stage 4.
b. Complete the table. Stage (s) c. How many blocks are being added from one stage to the next?
# of Blocks (b)
1 2
d. How many blocks are in stage 5?
3 4
e. How many blocks are at stage 0?
4. This pattern can be represented by the equation and s represents the stage number.
where b represents the number of blocks
a. Show how the equation relates to the pattern.
b. How is the constant in the equation, 2, shown in the pattern?
c. How is the coefficient of
in the equation, 4, shown in the pattern?
d. Use the equation to determine the number of blocks in stage 100.
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2.2E Extension: Chart of Patterns Name:
Period:
Directions: Complete the chart below for each linear pattern.
Stage 1
Stage 2
Stage 3
# of blocks at stage 0
Rate of Change
Equation s = stage b = # of blocks
1.
2. 4
3. 3
2
4. b = 3s + 1
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2.2F Lesson: Examining Linear Growth* Name:
Period:
In lesson 2.2E, you examined patterns and found how an equation can represent their growth. In this lesson, you will examine the same patterns and justify that their growth is linear. 1. Use the pattern to complete the following.
Stage 2
Stage 1
Stage 3
y
a. Complete the table of values and graph. 14
Stage (s)
# of Blocks (b) # of blocks
0
12
1 2
10 8 6 4
3
2 0
2
4
6
8 10 Stage
12
14
x
b. State the rate of change and show the value in the graph and table of values.
c. State the y-intercept. Circle this point on the graph.
The equation
represents the growth of this pattern. The constant, 1, represents the
number of blocks at stage 0. The constant,1, is the
on the graph. The coefficient of s,
2, represents the number of blocks added from one stage to the next. The coefficient of s, 2, represents the
in the table of values and on the graph. The graph is a line that does
not pass through the origin. This pattern is a
and has
growth.
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2. Use the pattern to complete the following.
a. Complete the table of values and graph.
Stage (s)
y
# of Blocks (b)
14 12
1
10
# of blocks
0
2 3
8 6 4 2 0
2
4
6
8
10
x
Stage
b. State the rate of change and show the value in the graph and table of values.
c. State the y-intercept. Circle this point on the graph.
d. The equation represents the growth of this pattern. Explain how each part of the equation is shown in the table of values.
e. Explain how each part of the equation,
, is shown on the graph.
f. Does the pattern represent linear growth? Use the table of values and graph to justify your reasoning. Table of Values
Graph
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37
Directions: Determine if the statement is true or false. If false, explain why. 3. All linear relationships are proportional.
4. Some linear relationships have a constant rate of change.
5. For some linear relationships, the constant of proportionality is the same value as the rate of change.
6. All linear relationships have a constant of proportionality.
7. The equation for a proportional linear relationship is y = mx + b.
8. Some linear relationships create a line that passes through the origin.
9. All linear relationships have a constant rate of change.
10. In a non-proportional linear relationship, the y-intercept is (0,0)
11. All proportional and non-proportional linear relationships are linear if they show a constant rate of change.
12. The equation for a non-proportional linear relationship is y = mx.
SDUHSD Math B Essentials Module #2 –STUDENT EDITION 2017-2018
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Section 2.2 Review Lesson Name:
Period:
1. Complete the table below to describe the qualities of proportional and non-proportional linear relationships. Table of Values
Graph
Equation
Proportional linear relationship
Non-proportional linear relationship
Directions: Complete the tables of values for the given equations. 2.
3. x
y
x
-2
y
-8 4
1
7
10
4. Circle the equations that represent proportional linear relationships. A.
B.
C.
D.
E.
F.
(
)
5. Circle the equations that represent non-proportional linear relationships.
A.
B.
D.
E.
C.
(
)
F.
SDUHSD Math B Essentials Module #2 –STUDENT EDITION 2017-2018
39
. State the rate of change and y-intercept
Directions: Examine the graph to write two different ratios for for the line. Write the equation for each line. y
6.
y
7.
18
6
12
4
6
2
0
1
2
3
rise
4
5
6
x
0
1
2
3
rise
run
1st
1st
2nd
2nd
Rate of Change:
Rate of Change:
y-intercept:
y-intercept:
Equation of line:
4
5
6
x
run
Equation of line:
Directions: Use the context to complete the following. 8. Splashy Splash charges $20 admission and then $4 per hour to swim. a. State the rate of change.
b. If you go to Splashy Splash with your friends, but choose not to swim, how much will you pay?
c. If x represents hours and y represents cost, write an equation that represents this situation.
d. Does this situation represent a proportional or non-proportional linear relationship? Explain.
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9. Jan drives 70 mph on the freeway. a. State the rate of change.
b. How many miles does Jan drive in 0 hours?
c. If x represents hours and y represents miles, write an equation that represents this situation.
d. Does this situation represent a proportional or non-proportional linear relationship? Explain
10. Use the pattern to complete the following. a. Complete the table of values. b. How many blocks are being added from one stage to the next?
Stage (s)
# of Blocks (b)
1
c. How many blocks are in stage 4?
2 d. How many blocks are at stage 0? 3 e. Graph this situation. y
f. State the rate of change and show the value in the graph and table of values.
10
g. State the y-intercept. Circle this point on the graph.
# of blocks
8 6 4 2
h. Does the pattern represent linear growth? Use the table of values and graph to justify your reasoning. Table of Values
0
2
4
6
8
10
x
Stage
Graph
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Task: Filling the Pool Name:
Period:
Summer is about to begin and you just got a job at Sparkle Pool Service. You are in charge of pool maintenance, and next week you are responsible for cleaning and filling the local YMCA pool. The pool holds 8,000 gallons of water and fills at a rate of 1,400 gallons per hour. 1. Complete the table below to show how much time it will take to fill the entire pool. x (hours)
0
1
2
y (gallons)
3
4
4200
5
6 8400
2. Graph time in hours versus gallons of water on the coordinate plane. On your graph, label each axis and label your line “YMCA.”
3. State the independent and dependent variables.
4. Write the equation of the line that relates , hours, to , gallons.
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5. Are hours and gallons of water proportionally related in this situation? Use the table of values and graph to explain your reasoning. Table of Values
Graph
6. Next week, you will be cleaning and filling the pool at the Boys and Girls Club. This pool is smaller, and holds only 4,000 gallons of water. This pool fills at a rate of 800 gallons per hour. x (hours)
0
1
2
3
4
5
6
y (gallons)
7. Write the equation of the line for this situation that relates , hours, to , gallons. 8. Graph the Boys and Girls Club information on the same coordinate plane as your “YMCA” graph. Label the line “Boys and Girls Club.” 9. At the end of the month, you will be cleaning and filling the pool at Newport Villa. This pool is the largest pool you have ever cleaned, and holds 16,000 gallons of water. The pool already has 2,000 gallons of water before you begin filling it, and fills at a rate of 4000 gallons per hour. Complete the table below to show how much time it will take to fill the entire pool. x (hours)
0
1
2
3
4
5
6
y (gallons)
10. Write the equation of the line for this situation that relates , hours, to , gallons. 11. Graph the Newport Villas information as a line on the same coordinate plane. Label the line “Newport Villas.” 12. Are hours and gallons of water proportionally related in this situation? Use the table of values and graph to explain your reasoning. Table of Values
Graph
13. Comparing the graphed lines for YMCA, Boys and Girls Club, and Newport Villas, which place fills their pool at the fastest rate? Justify your answer using your equations and graphs.
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