Exploring Linear Relations Math B Honors
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 i SDUHSD Math B Honors Module #2 – TEACHER EDITION
Table of Contents MODULE #2: EXPLORING LINEAR RELATIONS * ....................................................................... 3 STANDARDS FOR MATHEMATICAL PRACTICE: A GUIDE FOR STUDENTS AND PARENTS......................................... 5 SECTION 2.1: ANALYZE LINEAR RELATIONSHIPS, PATTERNS AND CONTEXTS* ................................................... 6 2.1A LESSON: COMPARING PROPORTIONAL RELATIONSHIPS* ........................................................................... 7 2.1A EXTENSION: A PROPORTIONAL STORY* ................................................................................................. 10 2.1B LESSON: RATE OF CHANGE* .................................................................................................................. 11 2.1C LESSON: NON-PROPORTIONAL RELATIONSHIPS* .................................................................................... 14 2.1C EXTENSION: PICKING APPLES................................................................................................................ 17 2.1D LESSON: LINEAR PATTERNS*................................................................................................................. 18 2.1E LESSON: EXAMINING LINEAR GROWTH*.................................................................................................. 23 SECTION 2.2: INVESTIGATE THE SLOPE OF A LINE* ......................................................................................... 26 2.2A LESSON: DILATIONS AND PROPORTIONALITY * ........................................................................................ 27 2.2B LESSON: SIMILAR TRIANGLES AND SLOPE* ............................................................................................. 31 2.2B EXTENSION: TREASURE HUNT ............................................................................................................... 34 2.2C LESSON: FINDING SLOPE FROM TWO POINTS* ........................................................................................ 35 2.2C EXTENSION: SLOPE IN A PARALLELOGRAM ............................................................................................. 40 2.2D LESSON: DERIVING THE EQUATIONS Y = MX AND Y = MX + B * .................................................................. 41 2.2D EXTENSION: SIMILAR TRIANGLES ........................................................................................................... 47
* Denotes a lesson that was adapted from Utah Middle School Math Project © Utah Middle School Math Project & University of Utah http://utahmiddleschoolmath.org/ This work is licensed under a Creative Commons Attribution-NonCommercial 2.5 Generic License http://creativecommons.org/licenses/by-nc/2.5/ This work is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported License http://creativecommons.org/licenses/by-nc/3.0/legalcode
SDUHSD Math B Honors 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 lessons, homework, and instructional support videos.
Common Core Standard(s): 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 (for example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has a greater speed).
Curriculum Support Website
8.EE.6 -
Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane. Derive the equation y mx for a line through the origin and the equation y mx b for a line intercepting the vertical axis at b.
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, discrete, continuous, constant of proportionality, unit rate, rate of change, constant first difference, linear relationship, slope (m), translation, dilation, y-intercept (b), linear, right triangle, origin, rise, run, context, geometric model, difference table, initial value, slope-intercept form.
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 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), and gain a conceptual understanding of the slope-intercept form of a linear equation. This work requires students to move fluently between the representations of a linear relationship and make connections between the representations. After exploring the rate of change of a linear relationship, students are introduced to the concept of slope and use the properties of dilations to show that the slope is the same between any two distinct points on a non-vertical line. Finally, students synthesize concepts learned and derive the equation of a line.
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. SDUHSD Math B Honors Module #2 – STUDENT EDITION 2017-2018
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Future Knowledge: After this module, students continue to work with linear relationships and begin work with functions. They will work more formally with slope-intercept form as they write and graph equations for 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 Linear Relationships, Patterns and Contexts* Section Overview: This section begins by reviewing proportional relationships that were studied in 6th and 7th grades. By investigating several contexts, students study 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 compare proportional relationships represented in different ways. Students apply their knowledge of proportional to non-proportional relationships, understanding that all linear relationships have a rate of change. Students examine linear patterns and write rules to represent them. Students use linear patterns to identify the rate of change and initial value (y-intercept) in different representations (table of values, graph, equation, and geometric model.) Students begin to understand how linear functions change.
Concepts and Skills to Master: By the end of this section, students should be able to: 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. Understand how non-proportional relationships differ from proportional relationships. Understand that all linear relationships have a rate of change. Write rules for linear patterns and connect the rules to the pattern (geometric model.) Understand how rate of change affects linear relationships. 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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2.1A Lesson: Comparing Proportional Relationships* Name:
Period:
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 8 ounces of root beer extract. a. Write a ratio that relates pounds of dry ice to ounces of root beer extract.
b. A unit rate is a comparison of two different quantities, in which one of the quantities is 1. State the unit rate for this situation and what it represents in the context.
c. In the table, = ounces of root beer extract and = pounds of dry ice. Complete table. d. In the table, what do you notice about the ratio
x
y
0
?
2 e. Graph this relationship. Label each axis. 3
f. In a discrete graph, the ordered pairs are not connected. In a continuous graph, the ordered pairs are connected. Should your points on the graph be connected? Explain.
g. Use the table of values and graph to justify that this situation is proportional.
h. Define constant of proportionality. State the constant of proportionality for this situation.
6
15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
1
2
3
4
5
6
7
8
9
10
i. Write an equation that represents the relationship between the number of ounces of root beer extract, , and the number of pounds of dry ice, , needed to make homemade root beer.
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In the previous situation, the number of ounces of root beer represents the independent variable and the number of pounds of dry ice represents the dependent variable. 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. 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.
4. As the amount of rain decreases, so does the water level of the river.
5. During the summer, Ruth eats two vanilla ice cream cones each week. Was the situation graphed correctly? Explain why or why not.
Directions: For each activity below, compare the dependent variable, cost, to the independent variable, hour. Fill in the missing values. 6. Making Phone Calls Story: It costs $10 per hour to talk on the phone long distance.
7. Playing Laser Tag Hours 0
Cost ($)
Story:
Hours
Cost ($)
2
5
Hours
Cost ($)
24
6
10
Equation:
20
Equation:
3 What is the cost for of an hour?
8. Playing Paintball
$12 relates to how many hours?
4
Hours
Cost ($)
9. Going Camping
Story: It costs $75 to play 3 hours of paintball.
Story:
Equation:
Equation:
What is the cost for 4 hours and 20 minutes?
$12.50 relates to how many hours?
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10. Create your own proportional story, table, and equation that compares cost per hour, and has a greater unit rate than problem #6, but a lesser unit rate than problem #8. Story:
Table:
Unit rate: Hours
Cost ($) Equation:
11. Graph all situations (#6-10) on the graph below. Label each axis and label each line with the equation that represents the situation. a. How do you compare the cost per hour by looking at the graph?
b. How do you compare the cost per hour by looking at the equation?
c. What happens to the lines on the graph as the unit rate increases? What happens to the lines as the unit rate decreases?
d. If a certain activity costs $50 per hour, describe what its graph would look like.
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2.1A Extension: A Proportional Story* Name:
Period:
1. Create your own story that represents a proportional relationship.
2. Exchange your story with another student, and have that student complete a table of values and graph to represent your story.
3. Exchange your paper with a third student, and have that student write an equation to represent the proportional relationship.
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2.1B 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 #3
Staircase #2
rise =
Staircase #3
run =
rise =
run =
2. Using the rise and run for each step, graph the first 5 steps for each staircase.
45 40 Height for Staircase #1:
35 Height for Staircase #2:
Height (rise)
30 25
Height for Staircase #3
20 15 10 5 0
5
10
15
20
25
30
35
40
45
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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4. When a linear relationship is graphed, how is the constant rate of change represented?
5. Find the slope of each line representing a staircase. Simplify your ratio. a. Staircase #1:
b.
Staircase #2:
c. Staircase #3:
6. 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?
7. Calculate the slope for climbing one, two, and three steps on each of the staircases.
Total Rise
Staircase #1 Total Slope Run (rise/run)
Total Rise
Staircase #2 Total Slope Run (rise/run)
Total Rise
Staircase #3 Total Slope Run (rise/run)
one step two steps three steps 8. Does the slope of the staircase change as you climb each step?
9. Suppose you want to make a skateboard ramp that is not as steep as the one shown. Write down two different slopes you could use.
3 ft 5 ft
𝑥
10. If the slope of the line is , what is the value of ? Show all of your work.
6
11. Find the slope of the waterslide. Calculate it in two different ways.
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Directions: Each staircase below has been drawn to the scale of actual rise height, run depth, and slope for each staircase. (adapted from Illuminations) 12.
13.
14.
Actual rise = Actual run = Slope =
Actual rise = Actual run = Slope =
. Use a ruler to find the
Actual rise = Actual run = Slope =
15. Which staircase is the steepest? Explain why.
16. Which staircase is built closest to building code? Justify your answer using research from the internet.
Directions: Determine if the table of values represents a linear relationship by verifying whether or not there is a constant rate of change. If linear, state the rate of change. 17.
Week
2
4
18
26
Cost
6
12
54
78
18.
Time
0
0.5
1.5
2
Height
0
18
31
26
If a fraction has a numerator or denominator that contains a fraction or decimal, the fraction is called a complex fraction. Directions: Simplify each complex fraction. Show all of your work.
19.
20.
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2.1C 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. Label each axis and the graph to illustrate which line represents Landon and which line represents Nate. b. State the rate of change for each boy. Show each rate of change on the graphs.
c. At what time have both boys eaten the same number of hot dogs? How is this shown in the graph?
d. Who won the contest? How do you know?
Number of minutes
e. In the tables, = time in minutes and = number of total hot dogs consumed. Complete the tables.
Landon t
h
0 f. What do you notice about the ratio What do you notice about the ratio
for Landon?
1
2.5
3
7.5
for Nate?
g. Which situation is not proportional? Justify your answer using the table of values and graph.
Nate t
h
0 1
5.5
2 8.5
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2. 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 Sara and explain what it represents in the context.
c. Which table of values represents a proportional situation? State the constant of proportionality. .
3. The constant first difference in a table of values is the difference in y-values for consecutive x-values. All linear relationships have a constant first difference. a. State the constant first difference for each table.
b. How does the constant first difference relate to the rate of change?
c. How does the constant first difference relate to the constant of proportionality?
4. Graph each situation on the coordinate plane. Label your graphs and axes. a. Should the graph be discrete or continuous? Explain.
b. Which person earns more money for every bag of popcorn sold? Use the graphs and tables of values to justify your answer.
Bags of Popcorn
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c. Write an equation that represents Oliver’s situation.
d. Write an equation that represents the Sara’s situation.
e. Examining your equations for each situation, what conclusion can you make about non-proportional relationships?
5. 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
Constant of Proportionality (yes or no) Rate of Change (yes or no) Constant First Difference (yes or no)
non-proportional relationships
Constant of Proportionality (yes or no) Rate of Change (yes or no) Constant First Difference (yes or no)
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2.1C Extension: Picking Apples Name:
Period:
Anna thought it would be fun to go to Julian to pick apples this fall. When she arrived, she saw two orchards next to each other. One orchard was named Bill’s Apples and the other was named Jorge’s Apples. The following signs were posted at the entrance to both orchards:
Bill’s Apples First 10 pounds - $2 per pound Each additional pound - $1 per pound
Jorge’s Apples $10 entry fee First 10 pounds - $1.50 per pound Each additional pound - $0.75 per pound
1. How much does 40 pounds of apples cost at each orchard?
2. How many pounds of apples can Anna get at each orchard for $30?
3. When is Jorge’s orchard cheaper than Bill’s? For how many pounds of apples would both orchards charge the same amount?
4. Do either of these orchard’s costs represent proportional relationships? Explain.
5. Anna wants in on the apple picking sales in Julian. Describe a way she can be competitive with Bill and Jorge, but at the same time have a proportional relationship.
6. How do Anna’s prices compare and contrast to Bill’s and Jorge’s. Critique the pricing structure at all three orchards.
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2.1D Lesson: Linear Patterns* Name:
Period:
1. Use the pattern to complete the following. a. Draw the figure at stage 4. Explain and show on the picture how you see the pattern growing from one step to the next.
b. How many blocks are in stage 4? Stage 10?
c. Write an equation that gives the number of blocks, b, for any stage, s. Show how your equation relates to the geometric model (pattern).
d. Write a different equation that gives the number of blocks, b, for any stage, s. Show how your equation relates to geometric model. Stage 1
Stage 2
Stage 3
e. Simplify the equation you found in part d. How does it relate to the equation you found in part c?
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f. Use your equation to determine the number of blocks in stage 100.
g. Explain how each part of the equation is shown in the geometric model.
h. Complete the table of values for the pattern.
Stage (s)
# of Blocks (b)
1 i. State the rate of change and show the value in the table.
2 3 4
j. State the constant first difference.
k. Explain how each part of the equation,
, is shown in the table of values.
y
l. Use the table of values to create a graph. 14
m. Show the rate of change on the graph.
12
o. Explain how each part of the equation, is shown on the graph.
,
# of blocks
n. Is the relationship proportional? Explain.
10 8 6 4 2 0
2
4
6
8
10
12
x
14
Stage
p. Does the pattern represent linear growth? Use the table of values and graph to justify your reasoning.
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2. Use the pattern to complete the following.
a. Draw the figure at stage 5 in the space above. Explain and show on the picture how you see the pattern growing from one step to the next.
b. How many blocks are in stage 5? Stage 10?
c. Write an equation that gives the number of blocks, b, for any stage, s. Show how your equation relates to the geometric model. Simplify your equation.
d. Write a different equation that gives the number of blocks, b, for any stage, s. Show how your equation relates to the geometric model.
e. Simplify your equation from part d. How does it relate to the equation you found in part c?
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f. Use your equation to determine the number of blocks in Stage 100.
g. Use your equation to determine which stage has 58 blocks.
h. Explain how each part of the equation is shown in the geometric model.
i. Complete the table of values for the pattern.
Stage (s)
# of Blocks (b)
1 j. State the rate of change and show the value in the table.
2 3 4
k. State the constant first difference.
5
l. Explain how each part of the equation,
, is shown in the table of values.
m. Use the table of values to create a graph. n. Show the rate of change on the graph.
o. Is the relationship proportional? Explain.
p. Explain how each part of the equation, is shown on the graph.
,
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q. Does the pattern represent linear growth? Use the table of values and graph to justify your reasoning.
3. Use the pattern to complete the following.
a. Write an equation that gives the number of blocks, b, for any stage, s.
b. Explain how each part of the equation is shown in the geometric model.
c. If graphed, what would the constant from your equation represent on the graph?
d. If graphed, what would the coefficient of
from your equation represent on the graph?
e. Does the pattern represent linear growth? Justify your reasoning.
4. Create 4 stages of a pattern that does not have linear growth. Explain why your pattern is not linear. Pattern
SDUHSD Math B Honors Module #2 – STUDENT EDITION 2017-2018
Explanation
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2.1E Lesson: Examining Linear Growth* Name: 1. Use the context to complete the following. a. State the independent variable. b. State the dependent variable.
Period: Context: You and your friends go to the state fair. It costs $6 to get into the fair, and $2 each time you go on a ride. Consider the relationship between number of rides and total cost.
c. Complete the table.
x:
d. State the rate of change. What does it represent in the context?
y: 4 5
e. Is this relationship proportional? Use the context to justify.
6 f. Make a graph of the data. Label each axis. g. Should the graph be discrete or continuous? Explain. 16 12
h. State the coordinates of the y-intercept. 8
i. What does the y-intercept represent in the context?
4 0
2
4
6
8
10
j. Define your variables and write an equation that represents the situation.
k. Where do you see the y-intercept in the equation?
l. Explain how you know the situation is linear by using the context.
m. How many rides can you go on if you have $25? Show all of your work.
n. How would you change the context so that the relationship between total cost and number of rides is modeled by the equation ?
o. How would you change the context so that the relationship between total cost and number of rides is modeled by the equation ?
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2. Use the context to complete the following. a. State the independent variable. b. State the dependent variable. c. Complete the table.
Context: You are taking a road trip. You start the day with a full tank of gas and your tank holds 16 gallons of gas. On your trip, you use 3 gallons per hour. Consider the relationship between time in hours and amount of gas remaining in the tank.
d. State the rate of change. What does it represent in the context? x:
y:
e. Is this relationship proportional? Use the context to justify. 0 1 f. Make a graph of the data.
2
g. Should the graph be discrete or continuous? Explain
3 4
h. State the coordinates of the y-intercept. y
i. What does the y-intercept represent in the context? 20
j. Define your variables and write an equation that represents the situation.
16 12
k. Where do you see the y-intercept in the equation?
8 4
l. Explain how you know the situation is linear by using the context. 0
2
4
6
x
m. How many hours can you drive before you have an empty tank of gas? How is this shown on the graph?
n. How would your equation change if your gas tank held 18 gallons of gas and used 4.5 gallons per hour of driving? What would these changes do to your graph?
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3. Use the table of values to complete the following. a. State the rate of change.
b. State the coordinates of the y-intercept.
Hours
Cost
2
50
3
75
4
100
5
125
c. Define your variables and write an equation that represents the table of values.
d. Describe in words what the graph for this table of values would look like.
e. Create a context that represents the table of values.
f. Change your context to represent the equation
.
g. Describe in words how the graph would change to represent the equation
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Section 2.2: Investigate the Slope of a Line* Section Overview: This section uses proportionality to launch an investigation of slope. Transformations are integrated into the study of slope by looking at the proportionality exhibited by dilations. Students dilate similar triangles on lines to show that a dilation produces triangles that have proportional parts and thus the slope is the same between any two distinct points on a non-vertical line. Students apply their investigation of slope to derive the slope formula and the equations and . Students use proportionality produced by a dilation to do this derivation.
Concepts and Skills to Master: By the end of this section, students should be able to:
Show that the slope of a line can be calculated as rise/run for any two points on a line. Explain why the slope is the same between any two distinct points on a non-vertical line. Find the slope of a line from a graph, set of points, or table. Given a context, find slope from various starting points (2 points, table, line, equation). Recognize that m in y = mx and y = mx + b represents the rate of change or slope of a line. Understand that b is where the line crosses the y-axis or is the y-intercept. Derive the equations and using dilations and proportionality.
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2.2A Lesson: Dilations and Proportionality * Name:
Period:
When an object, such as a line, is moved in space, it is called a transformation. A special type of transformation is called a dilation. A dilation is a shape that is the same, but has a different size. A dilation transforms an object in space from the center of dilation, usually the origin, by a scale factor called . The scale factor can be found as the quotient of . The dilation moves every point on the object so that the point is times away from the center of dilation as it was originally. This means that the object is enlarged or shrunk. For example, if you dilate the set of points (0,0), (3, 0), and (3,4) with a scale factor of 2, and the center of dilation is at the origin (0,0), then the distance of each point from the center will be 2 times as far as it was originally.
1. Graph and connect the ordered pairs (0,0), (3,0) and (3, 4) on the coordinate plane. This original object is called the pre-image, and can be labeled A. a. Find the length of each segment and dilate it by a scale factor of two. Draw the new lengths from the center of dilation (the origin) in a different color on the coordinate plane. This new object is called the image, and can be labeled as A’, read “A prime.” b. Compare the length of a side of your image, A’, with your pre-image A. What do you notice?
c. Write the ordered pairs of the vertices for your image A’. Compare them to the ordered pairs of the vertices for your pre-image A. What do you notice? .
d. Find the area of pre-image A and image A’. pre-image A
image A’
e. How does the area of image A’ compare to the area of pre-image A? Justify your reasoning using the formula for area of a triangle.
f. How do the line segments in the pre-image and image compare?
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g. Dilate the pre-image A by a scale factor of . Draw the new lengths from the center of dilation (the origin) in a different color. Label this image A’’. h. Compare the length of a side of the image, A’’, to the pre-image A. What do you notice?
i. Find the area of image A’’.
j. How does the area of image A’’ compare to the area of pre-image A? Justify your answer using the formula for area of a triangle.
2. Apply the given instructions to the image below. a. Connect point B to point C. b. Label triangle ABC as the pre-image. c. Dilate triangle ABC by a scale factor of 3. Label the new endpoints B’ and C’.
D
d. Label triangle AB’C’ as the image. e. What do you notice about B ' C ' in relationship to BC ?
C
f. Compare the areas of the pre-image and image. Justify your reasoning using the formula for area of a triangle.
A
B
E
g. Complete the ratios below that compare the corresponding parts of the image to the pre-image. What do you notice?
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3. What type of figure does a dilation produce? .
4. Draw two different figures that are similar to the figure shown.
5. Are the two figures similar? Explain why or why not.
6. The sides of one right triangle have lengths of 6, 8 and 10. The sides of another right triangle have lengths of 10, 24 and 26. Are these two triangles similar? Explain why or why not.
Directions: Each pair of figures is similar. Find the missing value. Write a proportion and show all of your work. 7.
8.
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9.
10.
Directions: Use similar triangles to solve the word problem. Show all of your work. 11. If a 6 inch yardstick casts a shadow that is 21 feet, how tall is a building whose shadow is 168 feet? Draw two similar figures.
12. At the same time a 15-meter building casts a shadow, a man 1.8 meters tall casts a 2.4 meter shadow. How long is the building’s shadow? Draw two similar figures.
13. A rectangle has a length of 4 feet and a perimeter of 14 feet. What is the perimeter of a similar rectangle with a width of 9 feet?
14.
and are similar triangles. Find the value of Write a proportion and show all of your work.
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2.2B Lesson: Similar Triangles and Slope* Name:
Period:
In Section 2.1, rate of change was introduced as the ratio , which is the slope of a line, denoted by the variable . Slope describes how steep a line is. It represents the change in y-values divided by the change in x-values. Similar triangles can be used to find the slope of a line. 1. Choose any two points that fall on the line below. To make your examination easier, choose two points that fall on an intersection of the gridlines. From the two points you chose, create a right triangle. The line will become the hypotenuse and the legs will extend from the two points and meet at a right angle.
2. Compare the points that you chose and your triangle with those of another student in class. Discuss and answer the following questions: a. Did you both choose the same points?
b. How are your triangles the same?
c. How are your triangles different?
d. What relationship exists between your triangles?
3. The graph has different triangles formed from two points that lie on the given line. These triangles are dilations of one another and are similar. a. How many triangles do you see? b. State the ratio
c. Does the ratio Explain why.
for each triangle.
always simplify to the same fraction?
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4. Compare the graphs and complete the following. Graph #1 a. State the ratio
for each triangle you see. Simplify your ratios.
Graph #1
Graph #2
b. What do you notice about your ratios?
Graph #2 c. Do the lines in both graphs have the same steepness? Explain.
d. How does “rise” related to “run” of a negative slope affect the steepness of a line?
5. Compare the graphs and complete the following. Graph #1 a. State the ratio
for each triangle you see. Simplify your ratios.
Graph #1
Graph #2
b. Are your ratios positive or negative? Explain.
Graph #2 c. Why are the slopes for the above graphs the same, even though the lines are different?
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6. What is the slope of each line below? Justify your reasoning using a right triangle and the ratio y
a.
y
b.
5
-5
.
5
5
-5
x
-5
5
x
5
x
-5
y y 5 5
c.
d.
-5 -5
5
x
-5 -5
7. For any linear relationship, how does the ratio
describe the steepness of a line?
Directions: Determine if the given statement is true or false. If false, explain why. 8. A line with slope is less steep than a line with slope
.
9. A line with slope 4 has the same steepness as a line with slope -4.
10. A line with slope -2 is steeper than a line with slope -3 because -2 is a greater value than -3.
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2.2B Extension: Treasure Hunt Name:
Period:
Directions: Working in numerical order, draw lines with the slopes listed below. When done, you will discover the correct route to the treasure box.
1. 3
2.
3.
4. 0
5. 1
6. -1
7. undefined slope
8.
9.
10.
11.
12. 3
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2.2C Lesson: Finding Slope from Two Points* Name:
Period:
1. Do the lines below have positive, negative, zero or undefined slopes? Explain your reasoning. a.
b.
c.
d.
2. Explain how you know whether a line has positive or negative slope by looking at a graph.
3. Explain how you know whether a line has zero or undefined slope by looking at a graph.
Directions: Graph the following pairs of points on the coordinate plane and connect them to create a line. y
4. Points: (4, 3) and (0, 1)
10
a. What is the rise between points (0,1) and (4,3)?
5
b. How can the y-values from both ordered pairs be used to find the rise? -10
-5
5
10
x
c. What is the run between the points (0,1) and (4,3)? -5
d. How can the x-values from both ordered pairs be used to find the run?
e. Using the ratio
-10
, find the slope of the line. Confirm that your slope is correct by drawing a right
triangle on the line with the same
ratio.
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5. Points: (1, 4) and (-2, 6)
y
a. What is the rise between points (1,4) and (-2,6)?
10
b. How can the y-values from both ordered pairs be used to find the rise?
5
-10
-5
c. What is the run between the points (1,4) and (-2,6)?
10
x
-5
d. How can the x-values from both ordered pairs be used to find the run?
e. Using the ratio
5
-10
, find the slope of the line. Confirm that your slope is correct by drawing a right
triangle on the line with the same
ratio.
6. Develop and explain a procedure for calculating slope without graphing each point. Show that your procedure works for points (4,3) and (0,1).
7. What is the slope formula?
8. Using the line that passes through the points (5,3) and (-4,1), answer the following questions. y
a. Use the slope formula to find the slope. Show all of your work.
10
5
b. Graph the ordered points and connect them to create a line.
-10
c. Draw a right triangle on your graph to calculate slope. d. What do you notice about your answers from part a and part c?
SDUHSD Math B Honors Module #2 – STUDENT EDITION 2017-2018
-5
5
10
x
-5
-10
36
Directions: Fill in the missing information in the problems below. Show your work to calculate slope using the slope formula. 9.
10.
11.
Points:
Points: (-4,4) and (3,-2)
Points:
m=
m=
m = undefined
12. For the line below, Jason and Steph both use the points (4,6) and (-4,0) to find slope.
Jason’s work
Steph’s work
a. How does Jason’s work differ from Steph’s?
b. Even though their methods are different, why do they both get the same value for slope? Explain your reasoning.
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Directions: Calculate the slope of the line that passes though the given points using the slope formula. Show all of your work. 13. (3, 2) and (-2, -1)
14. (-1, -7) and (1, 0)
15.
16.
x 0
y 3
4
-2
12
8
24
23
17. (-10, ) and (10, -1)
18.
(-3, -1) and (-3, 8)
x -6
y 3.75
2
3.75
11
3.75
24
3.75
Directions: Calculate the slope of the line that passes though the given points using the slope formula. After finding the slope, find another point that lies on the same line. 19. (10, -6) and (-5, 4)
20.
(-5, 1) and (-5,-2)
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Directions: Graph the line with the given description. 21. Passes through the point (-2,1) and has a slope of -3.
Directions: Find the value of of your work.
or
22. Passes through the point (2,-5) and has zero slope.
so that the line passing through the points has the given slope. Show all
23. (2, y) and (-7, 0) with slope
24. (1, 4) and (-8, y) with slope zero
25. (5, 2y) and (1, 3y) with slope
26. (3x, 5) and (-9x, -5) with slope
27. (8, 5y) and (-10, 4 + y) with slope
28. (x - 1, 15) and (3x + 5, 7) with slope
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2.2C Extension: Slope in a Parallelogram Name:
Period:
Directions: Use the coordinate plan to complete the following. 1. The vertices of a parallelogram are ( on the coordinate plane.
),
(
), (
) and (
). Draw the parallelogram
2. Determine the slope of each side of the parallelogram. Show all of your work.
3. Find the slope of the two diagonals that connect the opposite vertices. Show all of your work.
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2.2D Lesson: Deriving the Equations y = mx and y = mx + b * Name:
Period:
1. Use the coordinate plane to the right to complete the given instructions. a. Graph a line on the coordinate plane that goes through the origin and has a slope of . Label the rise and run on your graph with a right triangle. Redraw and label your triangle in the space provided below the graph.
b. Does this line describe a proportional relationship? Explain why or why not.
c. Choose any point (x,y) on your line and draw a right triangle that describes the rise and run. Redraw and label your triangle below the graph.
d. Write a proportional equation with your ratios. Explain why the ratios can be set equal to each other.
e. Solve the equation that you wrote above for y.
f. Where do you see the slope in the equation?
g. Does your equation represent a proportional relationship? Justify your reasoning.
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2. Use the coordinate plane to the right to complete the given instructions. a. Graph a line on the coordinate plane that goes through the origin and has a slope of m (slope is the same as a unit rate which compares a y-value to an x-value of 1). Label the rise and run on your graph with a right triangle. Redraw and label this triangle below the graph.
b. Does this line describe a proportional relationship? Explain why or why not.
c. Choose any point (x,y) on your graph and draw a right triangle that describes the rise and run. Redraw and label this triangle below the graph.
d. Write a proportional equation with your ratios.
e. Solve the equation that you wrote above for y.
f. What is the general equation that represents any proportional linear relationship?
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What about linear relationships that are not proportional? If a line does not go through the origin, it does not have a constant of proportionality, however it still has a unit rate or slope. A translation is another type of transformation that relates a proportional relationship to a non-proportional relationship. Shifting a line or moving all the points on the line the same distance and direction is a translation. For example, if you transform the line upwards by 3 units, every ordered pair that lies on that line moves up 3 units. Algebraically, that means you add 3 to every y-value since this is a vertical shift. 3. Investigate the equation a. Make a table of values for the equation . x 4
. b. Graph the equation.
c. On the same coordinate plane, transform every point 3 units up and draw the new graph or image in a different color.
y 10
y d. Using your table of values, add 3 to every y-value.
5
2 0
-10
-5
5
x
10 x
-2
y
y+3
4 -5
2 0
-10
-2 e. Compare the ordered pairs on your new line to those in the table from part d. What do you notice?
f. What is the equation of your new line? How does it compare to
?
The general form of any linear equation is where m represents the slope or rate of change and b is the initial value or y-intercept. This form is called the slope-intercept form of a linear equation. In the previous example, the equation represents a linear relationship with a slope of and a y-intercept of 3.
4. Compare and contrast the two equations using the Venn Diagram below.
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5. Use the coordinate plane and instructions below to investigate the slope-intercept form, y = mx + b. a. Graph a line on the coordinate plane that goes through the point (0,4) and has a slope of . Label the rise and run on your graph with a right triangle. Redraw and label this triangle below the graph.
b. Does this line describe a proportional relationship? Explain why or why not.
c. Choose any point (x,y) on your graph and draw a right triangle that describes the rise and run. Redraw and label this triangle below the graph.
d. Write a proportional equation with your ratios. Explain why the ratios can be set equal to each other.
e. Solve the equation that you wrote above for y.
f. Where do you see the slope in the equation? Where do you see the y-intercept in the equation?
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6. Use the coordinate plane and instructions below to investigate the slope-intercept form, y = mx + b. a. Graph a line on the coordinate that goes through any point on the y-axis (represented by the yintercept b) and has a slope of m. Label the rise and run on your graph with a right triangle. Redraw and label this triangle below the graph.
b. Does this line describe a proportional relationship? Explain why or why not.
c. Choose any point ( ) on your graph and draw a right triangle that describes the rise and run. Redraw and label this triangle below the graph. d. Write a proportional equation with your slope ratios.
e. Solve the equation for y.
f. Where do you see slope
in the equation? Where do you see y-intercept
SDUHSD Math B Honors Module #2 – STUDENT EDITION 2017-2018
in the equation?
45
7. How do you know if a relation is proportional based on an equation? How do you know if a relation is non-proportional based on an equation?
8. How are the equations
9. Graph
and
related to each other?
on the coordinate plane.
a. Write the equation of a line shifted 5 units up from
b. Graph your new equation on the same coordinate plane as
c. How does the graph of your new equation relate to the graph of ?
10. Graph
on the coordinate plane.
a. Write the equation of a line shifted 7 units down from
b. Graph your new equation on the same coordinate plane as
c. How does the graph of your new equation relate to the graph of
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?
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2.2D Extension: Similar Triangles Name:
Period:
Directions: In the coordinate plane, of y. Show all of your work.
is similar to
Show two different methods to find the value
Method #1
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Method #2
47
