Investigating Linear Functions - Weeblycarengarza.weebly.com/uploads/3/7/7/4/37742877/unit3lesson1.pdf · Algebra 1 HS Mathematics Unit: 03 Lesson: 01 Suggested Duration: 10 days

Algebra 1 HS Mathematics

Unit: 03 Lesson: 01

Suggested Duration: 10 days

Investigating Linear Functions

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Lesson Synopsis: In this lesson, students will identify the linear parent function and describe the effects of parameter changes on the graph of the linear parent function. Characteristics of linear functions, including slope, intercepts, and forms of equations will be investigated. Linear functions will be written and evaluated in function notation. Linear inequalities will be represented graphically. TEKS:

A.1 The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways. The student is expected to:

A.1C Describe functional relationships for given problem situations and write equations or inequalities to answer questions arising from the situations.

A.1D Represent relationships among quantities using concrete models, tables, graphs, diagrams, verbal descriptions, equations and inequalities.

A.2 The student uses the properties and attributes of functions. The student is expected to:

A.2A Identify and sketch the general forms of linear (y=x) and quadratic (y=x2) parent functions.

A.2C Interpret situations in terms of given graphs or creates situations that fit given graphs.

A.2D Collect and organize data, make and interpret scatterplots (including recognizing positive, negative, or no correlation for data approximating linear situations), and model, predict, and make decisions and critical judgments in problem situations.

A.5 The student understands that linear functions can be represented in different ways and translates among their various representations. The student is expected to:

A.5A Determine whether or not given situations can be represented by linear functions.

A.5B Determine the domain and range values for linear functions in given situations.

A.5C Use, translate, and make connections among algebraic, tabular, graphical, or verbal descriptions of linear functions.

A.6 The student understands the meaning of the slope and intercepts of the graphs of linear functions and zeros of linear functions and interprets and describes the effects of changes in parameters of linear functions in real-world and mathematical situations. The student is expected to:

A.6A Develop the concept of slope as rate of change and determine slope from graphs, tables, and algebraic representations.

A.6B Interpret the meaning of slope and intercepts in situations using data, symbolic representations, or graphs.

A.6C Investigate, describe, and predict the effects of changes in m and b on the graph of y = mx + b.

A.6E Determine the intercepts of the graphs of linear functions and zeros of linear functions from graphs, tables, and algebraic representations.

A.6G Relate direct variation to linear functions and solve problems involving proportional change.

Related TEKS: A.1 The student understands that a function represents a dependence of one quantity on another and can be

described in a variety of ways. The student is expected to:

A.1A Describe independent and dependent quantities in functional relationships.

A.1E Interpret and make decisions, predictions, and critical judgments from functional relationships.

A.2 The student uses the properties and attributes of functions. The student is expected to:

A.2B Identify the mathematical domains and ranges and determine reasonable domain and range values for given situations, both continuous and discrete.

A.3 The student understands how algebra can be used to express generalizations and recognizes and uses the power of symbols to represent situations. The student is expected to:

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A.3A Use symbols to represent unknowns and variables.

A.3B Look for patterns and represent generalizations algebraically.

A.4 The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations. The student is expected to:

A.4A Find specific function values, simplify polynomial expressions, transform and solve equations, and factor as necessary in problem situations.

A.4C Connect equation notation with function notation, such as y = x + 1 and f(x) = x + 1.

Process TEKS: 8.14 Underlying processes and mathematical tools. The student applies Grade 8 mathematics to solve problems

connected to everyday experiences, investigations in other disciplines, and activities in and outside of school. The student is expected to:

8.14A Identify and apply mathematics to everyday experiences, to activities in and outside of school, with other disciplines, and with other mathematical topics.

8.15 Underlying processes and mathematical tools. The student communicates about Grade 8 mathematics through informal and mathematical language, representations, and models. The student is expected to:

8.15A Communicate mathematical ideas using language, efficient tools, appropriate units, and graphical, numerical, physical, or algebraic mathematical models.

8.16 Underlying processes and mathematical tools. The student uses logical reasoning to make conjectures and verify conclusions. The student is expected to:

8.16B Validate his/her conclusions using mathematical properties and relationships.

GETTING READY FOR INSTRUCTION Performance Indicator(s):

Identify and sketch the linear parent function. Describe and predict the effects of changing “m” and “b” on the graph of the parent function. (A.2A; A.5A, A.5B; A.6C)

ELPS: 1E, 2I, 3F, 3G, 5G Determine how to identify linear functions by making connections between the various representations, including

tables, graphs, algebraic generalizations, and verbal descriptions. (A.5C) ELPS: 1E, 2I, 3F, 3G, 5G

Define slope and intercepts of linear functions and determine slope (rate of change) and intercepts from various representations modeling real world problem situations. (A.1C, A.1D; A.2C, A.2D; A.5A, A.5B, A.5C; A.6A, A.6B, A.6E)

ELPS: 1E, 2I, 3F, 3G, 5G Represent direct variation data using tables, graphs, verbal descriptions, and algebraic generalizations. Describe

how direct variations are related to linear functions. (A.6G) ELPS: 1E, 2I, 3F, 3G, 5G

Represent linear inequalities graphically on a coordinate plane. (A.1C, A.1D) ELPS: 1E, 2I, 3F, 3G, 5G

Key Understandings and Guiding Questions:

Parent functions identify various relations in data, and changes in coefficients and constants cause transformations of the parent functions. — What are the parameters of the linear parent function? — How do these parameters affect the graph of the linear parent function?

Linear functions can be identified by analyzing characteristics of their tables, graphs, and algebraic representations. — How can you determine from a table that the data can be represented by a linear function? — Why is the graph of a linear function not curved? — How can you determine from an algebraic rule that the function is linear?

Slope describes the rate of change and can be found in a graph, table, or algebraic equation. — Why can slope be called “rate of change”? — How do the different ways of finding slope compare?

Slopes, x-intercepts, and y-intercepts have specific meanings and effects in real world situations. — What are the effects of changes in slope and y-intercept in mathematical situations?


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— Why is the interpretation of slope and intercepts important in real world problems? x-intercepts and y-intercepts are the points where lines cross the axes and can be found in a graph, table and

algebraic equation. — What are the differences in the x- and y-intercepts? — How do the different ways of determining intercepts compare?

Direct variations are linear functions that pass through the origin, have a constant of proportionality, and can be written in the form y = kx. — How can it be determined that a linear function is a direct variation by comparing the x- and y-values in a data

table of the function? — How can it be determined that a linear function is a direct variation by looking at a graph of the function? — How can it be determined that a linear function is a direct variation by looking at the algebraic generalization

of the function? Solutions to linear inequalities are represented graphically on a coordinate plane.

— How does a linear inequality differ from a linear function when graphed? — How do you determine how to graph a linear inequality?

Vocabulary of Instruction:

parent function direct variation constant of proportionality parameter change

linear function rate of change slope y-intercept

x-intercept zero of a function function notation linear inequality

Materials:

chart paper chart markers sticky dots meter stick

colored pencils card stock colored toothpicks

Starburst or similar wrapped candy (20 per group)

Resources:

State Resources

— Mathematics TEKS Toolkit: Clarifying Activity/Lesson/Assessments http://www.utdanacenter.org/mathtoolkit/index.php

— TMT3 Algebra 1: Elaborate – Volumes of Vessels http://www.tea.state.tx.us/math/index.html

— TMT3 Algebra 1: Student Lesson 1 – Modeling Data Using Linear Functions http://www.tea.state.tx.us/math/index.html

— TEXTEAMS: Algebra 1: 2000 and Beyond: I – Foundations of Functions; 3. Interpreting Graphs, 3.1 Interpreting Distance versus Time Graphs, Act. 1 (Walking Graphs), Act. 2 (Walking More Graphs, Student Act. (Walk This Way), 3.2 Interpreting Velocity verses Time Graphs, Act. 1 (Matching Velocity Graphs), Act. 2 (Connecting Distance and Velocity Graphs); II – Linear Functions; 1. Linear Functions, 1.1 The Linear Parent Function, Act. 1 (ACT Scores), Act. 2 (Temperatures), Act. 3 (Symbolic), Student Act. 1 (Age Estimates), Student Act. 2 (Sales Goals), 1.2 The Y-Intercept, Act. 1 (The Birthday Gift), Act. 2 (Spending Money), Act. 3 (Money, Money, Money), Student Act. (Show Me the Money), 1.3 Exploring Rates of Change, Act. 1 (Wandering Around), Act. 2 (Describe the Walk), Student Act. (What’s My Trend), 1.4 Finite Differences, Act. 1 (Rent Me), Act. 2 (Guess My Function), Act. 3 (Finite Differences), Student Act. (Graphs and Tables)

— Algebra 1 End of Course Success: Vocabulary: Objective 3 Lesson 1 – Graphing Lines, Let’s Hit the Slope, On Your Own: Let’s Hit the Slope, Getting in Shape, Graphing Polygons; Online: Objective 2 Lesson 2 – For What It’s Worth, Not 1 But 2, Looking at the Process, Multiple Representation Cards; Online: Objective 1 Lesson 2 – Round Robin: What’s My Line?, In the Shade, On Your Own: Graphing Inequalities, Tell Me More


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Advance Preparation: 1. Handout: Meet Your Parent (1 per student) 2. Transparency: Meet Your Parent (1 transparency of Birth Dates of Actors per teacher) 3. Transparency: What’s Up with “m”? (1 per teacher) 4. Handout: Parameter Changes on the Linear Parent Function (1 per student) 5. Transparency: What’s Up with “b”? (1 per teacher) 6. Handout: Investigating Slope (1 per student) 7. Handout: Slalom on the Slopes (1 per student) 8. Handout (optional): Headache Card Game (1 per student) 9. Handout (optional): Headache Card Game Cards (1 set per student) 10. Handout: Investigating Intercepts (1 per student) 11. Handout: The Great Hawaiian Race (1 per student) 12. Handout (optional): Battery Up! (1 per student) 13. Handout: Battery Up! Student Activity Sheet (1 per student) 14. Handout: Direct Variation (1 per student) 15. Handout: Patterns Can Be Sweet (1 per student) 16. Handout: Linear Inequalities (1 per student) 17. Cards: Images of Inequalities Cards (1 per pair, run off on cardstock, laminated and cut) 18. Handout: Images of Inequalities Recording Sheet (1 per student) 19. Handout: Functioning on a High Rise (1 per student)

Background Information: Previous Algebra 1 units have developed the basic foundations of functions, including relations, functions, independent and dependent variables, domain and range, and intercepts. This unit will concentrate on the linear function and characteristics that distinguish it as a linear function.

GETTING READY FOR INSTRUCTION SUPPLEMENTAL PLANNING DOCUMENT Instructors are encouraged to supplement, differentiate and substitute resources, materials, and activities to address the needs of learners. The Exemplar Lessons are one approach to teaching and reaching the Performance Indicators and Specificity in the Instructional Focus Document for this unit. A Microsoft Word template for this Planning document is located at www.cscope.us/sup_plan_temp.doc. If a supplement is created electronically, users are encouraged to upload the document to their Lesson Plans as a Lesson Plan Resource for future reference.

INSTRUCTIONAL PROCEDURES

Instructional Procedures Notes for Teacher

ENGAGE NOTE: 1 Day = 50 minutes Suggested Day 1 (1/2 day)

1. Put students into small groups. 2. Distribute the handout: Meet Your Parent to each student. Give students a

couple of minutes to estimate the ages of the actors and fill in the last column of the table.

3. Display the transparency: Meet Your Parent birth dates of the actors. Have students calculate and fill in the actual age column of the table on their worksheet. Have students graph their estimates as a function of the actual ages.

4. Have students complete #1-5. 5. Debrief with scaffold questions.

What do you notice about the actual value and the estimated value of the “perfect” estimator? The values are equal.

What do you notice about the x and y coordinates of the scatterplot for the “perfect” estimator? The values are equal.

How did the shape of the scatterplot of your estimated values versus actual values compare with the “perfect” estimator scatterplot? Individual scatterplot is more spread out even though there may be a linear tendency. The “perfect” estimator scatterplot

MATERIALS Handout: Meet Your Parent (1 per

student) Transparency: Meet Your Parent (1

transparency of Birth Dates of Actors per teacher)

chart paper chart markers sticky dots meter stick

TEACHER NOTE Students will make a scatterplot of data points that define the linear parent function and analyze the results. TEACHER NOTE It might be helpful to ask students,


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Instructional Procedures Notes for Teacher forms a perfect line going through (0, 0).

How did you differentiate between over-estimators and under-estimators? Over – data points above perfect estimator, under – data points under perfect estimator.

6. Have groups finish Meet Your Parent by completing #6-10 and making poster charts of linear parent functions to be displayed around the room. Groups will need a sheet of chart paper, markers, sticky dots, and a meter stick.

7. Use Scaffold questions to debrief. 8. Scaffold questions:

What do you notice about the shape of the points when they are plotted? They form a line.

Why are arrowheads put on the ends of the line? The line is infinite in both directions.

Is the function increasing or decreasing? Explain. Increasing; as x values increase, y values increase.

What would be the domain and range of the linear parent function? Domain is all real numbers. Range is all real numbers.

What are the independent and dependent variables? Independent is x; dependent is y.

“When is being close to “perfect” estimation important?” An emergency case comes into the hospital and must be given a dose of medication, but the doctor does not have time to weigh the patient. The doctor must estimate the patient’s weight to prescribe a dosage of medication. TEACHER NOTE Sample answers are given for Scaffold questions. Student answers may vary.

STATE RESOURCES Mathematics TEKS Toolkit: Clarifying Activity/Lesson/Assessments may be used to reinforce these concepts or used as alternate activities. TEXTEAMS: Algebra 1: 2000 and Beyond: I –Foundations of Functions; 3. Interpreting Graphs, 3.1 Interpreting Distance versus Time Graphs, Act. 1 (Walking Graphs), Act. 2 (Walking More Graphs, Student Act. (Walk This Way), 3.2 Interpreting Velocity verses Time Graphs, Act. 1 (Matching Velocity Graphs), Act. 2 (Connecting Distance and Velocity Graphs), Student Act. 1 (Age Estimates), Student Act. 2 (Sales Goals) may be used as alternate activities.

EXPLORE/EXPLAIN 1 Suggested Day 1 (1/2 day)1. Display the transparency: What’s Up with “m”? Give students time to

record predictions. 2. Distribute the handout: Parameter Changes on the Linear Parent

Function to each student. Tell students they will now verify their predictions using a graphing calculator.

3. Have students group into pairs. Each student will need a graphing calculator. Have students complete Chart A. Students will use the calculators to explore and complete the chart.

4. Display the transparency: What’s Up with “b”? Give students time to record predictions. Tell students they will now verify their predictions using a graphing calculator.

5. Have students complete Chart B. Students will use the calculators to explore and complete the chart.

6. Because students will need to use the calculators to complete both charts during class time, they should answer the questions for Parts A and B after completing both charts. The questions can be completed as homework if necessary.

MATERIALS Transparency: What’s Up with

“m”? (1 per teacher) Handout: Parameter Changes on

the Linear Parent Function (1 per student)

Transparency: What’s Up with “b”? (1 per teacher)

graphing calculator

TEACHER NOTE Students will use the graphing calculator and describe changes to the linear parent function caused by changing the parameters of m and b. Students will then generalize the effects of those parameter changes on the linear parent function. INFORMAL OBSERVATION Monitor students as they complete the


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Instructional Procedures Notes for Teacher charts. Make sure students are entering the function correctly in the calculator. Did you put the linear parent

function in y1? Did you put the new function in

y2? Are you on the standard window?

STATE RESOURCES TEXTEAMS: Algebra 1: 2000 and Beyond: II – Linear Functions; 1. Linear Functions, 1.1 The Linear Parent Function, Act. 1 (ACT Scores), Act. 2 (Temperatures), Act. 3 (Symbolic) may be used to reinforce concepts.

EXPLORE/EXPLAIN 2 Suggested Days 2-3Day 1 1. Debrief Parameter Changes on the Linear Parent Function by going

over in whole-group and discussing the questions and conclusions on Charts A and B. Check for student understanding and have students correct work as necessary.

2. Distribute the handout: Investigating Slope to each student. 3. Have students complete the first two pages in pairs. Discuss results in

whole-group. 4. Refer back to Chart A from Parameter Changes on the Linear Parent

Function to relate rate of vertical and horizontal change to the “m” value in y=mx. Have students look at selected graphs, select two points, count the vertical change and horizontal change, and calculate the rate of change. Then have students compare this number to the “m” value. How does the ratio of the vertical change to the ratio of the

horizontal change compare with the “m” value found in the table? Answers will vary according to points selected.

Where does this occur in the equation? Answers will vary according to points selected.

5. Have students answer questions 1-3 on the second page. Share out answers to check for understanding.

6. Go over Finding Slope on a Graph in whole-group. Work Examples 1-2 as students fill in the worksheet.

7. Distribute the handout: Slalom on the Slopes to each student. Have students complete the activity for homework if necessary.

Day 2 8. Go over Finding Slope on a Table in whole-group. Work Examples 1-2 as

students fill in the worksheet. 9. Go over Special Lines in whole-group. 10. Have students work Guided Practice, then share answers with a partner. 11. Assign Practice Problems to be worked individually. If necessary, these

may be taken for homework. The supplementary material Headache Card Game is extra practice for finding slope from two points and comparing parallel slopes.

MATERIALS Handout: Investigating Slope (1

per student) Handout: Slalom on the Slopes (1

per student) graphing calculator

TEACHER NOTE Students will define slope as a rate of change and determine slope from graphs, tables, and algebraic representations.

MISCONCEPTION Students may have the misconception that all linear functions with fractional slopes are represented with lines whose steepness will be less than the parent function. Students should recognize that a fractional slope whose numerator is greater than the denominator is greater than 1 and will have a slope greater than the parent function. SUPPLEMENTARY MATERIALS Handout: Headache Card Game (1

per student) Handout (optional): Headache Card

Game Cards (1 set per student) The supplementary materials may be used if students need additional practice on concepts of slope. The Headache Card Game distinguishes between parallel and perpendicular slopes. TEACHER NOTE When looking at two or more points the


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Instructional Procedures Notes for Teacher slope will be calculated by using a table. This can be done by looking at the change on a number line, increasing (+) and decreasing (-).

x y 2 De-

crease 8

-3 In-crease

6 -6 3

Change in y is +6. Change in x is -8

Slope equals x

y

= 8

6

= 4

3

MISCONCEPTION Students may have the misconception that the ratio for rate of change (slope) in a linear function is x

y

, since the x

variable (horizontal) always comes before the y variable (vertical), instead of the correct representation that rate of change (slope) in a linear function is

y

x

.

STATE RESOURCES TEXTEAMS: Algebra 1: 2000 and Beyond: I –Foundations of Functions; 3. Interpreting Graphs, 3.1 Interpreting Distance versus Time Graphs, Act. 1 (Walking Graphs), Act. 2 (Walking More Graphs, Student Act. (Walk This Way); II – Linear Functions; 1. Linear Functions, 1.3 Exploring Rates of Change, Act. 1 (Wandering Around), Act. 2 (Describe the Walk) may be used to reinforce concepts.

EXPLORE/EXPLAIN 3 Suggested Day 4 1. Distribute the handout: Investigating Intercepts to each student. 2. Go over the first page in whole-group. Stress that a change in b only

changes the location of the lines, but the slopes remain the same therefore the lines are parallel.

3. Go over the various methods for finding intercepts and work sample problems in whole-group to check for understanding.

4. Scaffold questions What distinguishes the y-intercept on the graph? It is where the

line crosses the y-axis. What distinguishes the x-intercept on the graph? It is where the

line crosses the x-axis. What distinguishes the y-intercept in point form? It has a y-

coordinate value and the x-coordinate value is 0, (0, 5).

MATERIALS Handout: Investigating Intercepts

(1 copy per student)

TEACHER NOTE Students will determine y-intercepts and x-intercepts from various representations. TEACHER NOTE Sample answers are given for Scaffold questions. Student answers may vary.


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Instructional Procedures Notes for Teacher What distinguishes the x-intercept in point form? It has a x-

coordinate value and the y-coordinate value is 0, (5, 0). How can the intercepts be found in an equation? To find the x-

intercept, substitute 0 for y. To find the y-intercept, substitute 0 for x. 5. Assign Practice Problems to be worked individually. If necessary, these

may be taken as homework.

MISCONCEPTION Students may have the misconception that the intercept coordinate is the zero term instead of the non-zero term, since intercepts are associated with zeros. In other words, students may think (0, 4) would be the x-intercept because the 0 is in the x coordinate.

STATE RESOURCES TEXTEAMS: Algebra 1: 2000 and Beyond: I –Foundations of Functions; 3. Interpreting Graphs, 3.1 Interpreting Distance versus Time Graphs, Act. 1 (Walking Graphs), Act. 2 (Walking More Graphs, Student Act. (Walk This Way); II –Linear Functions; 1. Linear Functions, 1.2 The Y-Intercept, Act. 1 (The Birthday Gift), Act. 2 (Spending Money), Act. 3 (Money, Money, Money), Student Act. (Show Me the Money) may be used to reinforce concepts. Algebra 1 End of Course Success Online: Objective 2 Lesson 2 – For What It’s Worth, Not 1 But 2, Looking at the Process, Multiple Representation Cards These activities can be used as scaffolding with students who demonstrate gaps in their understanding of transforming equations to slope-intercept form. Algebra 1 End of Course Success Vocabulary: Objective 3 Lesson 1 – Graphing Lines, Let’s Hit the Slope, On Your Own: Let’s Hit the Slope, Getting in Shape, Graphing Polygons These activities allow students to demonstrate their understanding of graphing y mx b devoid of any contextual situations.

ELABORATE 1 Suggested Day 5 1. Distribute the handout: The Great Hawaiian Race to each student. 2. Allow students to work in pairs or small groups on Part I. 3. If students need extra help, you may use Part I as whole-group discussion

to check for student understanding. Have students work Parts II and III in groups.

4. If necessary, students can complete Part III as homework. 5. Supplementary Materials require students to use technology to collect and

analyze real world data on linear functions.

MATERIALS Handout: The Great Hawaiian

Race (1 per student) TEACHER NOTE Students will apply linear function to a problem situation and observe the


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meaning and effects of the slope and intercepts in the problem. SUPPLEMENTARY MATERIALS Handout: Battery Up! (1 per

student) Handout: Battery Up! Student

Activity Sheet (1 per student) graphing calculator CBL with voltage sensor link cords 6 AA or AAA batteries 2 meter sticks per group masking tape chart paper markers graphing calculator with Data Mate The supplementary materials may be used to have students collect and analyze real world data on concepts of slope and intercept. Since data will vary, an answer key is not provided for this activity.

STATE RESOURCES TMT3 Algebra 1: Elaborate – Volumes of Vessels TEXTEAMS: Algebra 1: 2000 and Beyond: II –Linear Functions; 1. Linear Functions, Student Act. (What’s My Trend), 1.4 Finite Differences, Act. 1 (Rent Me), Act. 2 (Guess My Function), Act. 3 (Finite Differences), Student Act. (Graphs and Tables) may be used as alternative activities.

EXPLORE/EXPLAIN 4 Suggested Day 6 1. Distribute the handout: Direct Variation to each student. 2. Read and discuss the sentence at the top of first page in whole-group. 3. Have students work in pairs to complete questions 1-3. 4. Discuss in whole-group with the following scaffold questions. Have

students correct questions as needed. How are all the graphs similar? They are all linear and go through

the origin. How are the graphs different? The slope of all the lines is different. How do the y-intercepts of all the graphs compare? All graphs

have a y-intercept of (0, 0). Which of the functions graphed have an increasing slope? y = 3x,

y = 0.5x, y = x, y = 1.75x Which of the functions graphed have a decreasing slope? y = -.3x,

y = -3.25x What are the characteristics of a proportion? Proportions are

linear, pass through the origin, can be written as y = kx, and have a

MATERIALS Handout: Direct Variation (1 per

student)

TEACHER NOTE Students will investigate direct variation as a type of linear function and define direct variation as a proportional relationship. TEACHER NOTE Sample answers are given for Scaffold questions. Student answers may vary.


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constant of proportionality (y

x).

Do all the functions graphed meet the characteristics of proportions? Yes

5. Have students work with their partner to complete the remainder of the activity. If necessary, the activity can be completed as homework.

ELABORATE 2 Suggested Day 7 1. Debrief the remaining questions on Direct Variation in whole-group as

needed. 2. Distribute the handout: Patterns Can Be Sweet to each student. 3. Distribute candies and toothpicks to each group. 4. Allow students to work in pairs or small groups on Parts A-C. 5. If students need extra help, you may use Part A as whole-group discussion

to check for student understanding. Have students work Parts B, C, and D in groups. (See Teacher Note.)

6. If necessary, students can complete Part D as homework.

MATERIALS Handout: Patterns Can Be Sweet

(1 per student) toothpicks Starburst or similar wrapped candy

(20 pieces per group) colored pencils

TEACHER NOTE Students will collect data and compare proportional (direct variation) relationships with non-proportional relationships by analyzing various representations. TEACHER NOTE If time permits, groups may make chart displays of the Part D results.

EXPLORE/EXPLAIN 5 Suggested Day 8 1. Distribute the handout: Linear Inequalities to each student. 2. Go over notes and examples as whole-group discussion while students fill

in worksheets. 3. Put students in pairs to work on Guided Practice. Debrief results to check

for student understanding. 4. Assign Practice Problems to be completed independently as homework.

MATERIALS Handout: Linear Inequalities (1 per

student)

TEACHER NOTE Students will represent linear inequalities in a coordinate plane by graphing the boundary line and shading the appropriate region.

STATE RESOURCES Algebra 1 End of Course Success Online: Objective 1 Lesson 2 – Round Robin: What’s My Line?, In the Shade, On Your Own: Graphing Inequalities, Tell Me More These activities allow students to test multiple points on the graph of a linear inequality to determine how to interpret and shade inequality representations.

ELABORATE 3 Suggested Day 9 1. Debrief Practice Problems from Linear Inequalities in whole-group. 2. Distribute the handout: Images of Inequalities Recording Sheet to each

student.

MATERIALS Cards: Images of Inequalities

Cards (1 per pair)


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Instructional Procedures Notes for Teacher 3. Put student in pairs. Distribute a set of Images of Inequalities Cards to

each group. Have students match the cards and record results on the recording sheet.

Handout: Images of Inequalities Recording Sheet (1 per student)

TEACHER NOTE Students will continue to investigate linear inequalities with a matching activity.

EVALUATE Suggested Day 10 1. Distribute the handout: Functioning on a High Rise to each student. 2. Students should complete the worksheet independently.

MATERIALS Handout: Functioning on a High

Rise (1 per student)

TEACHER NOTE Students should work this activity independently to assess student understanding.

TAKS CONNECTION Grade 9 TAKS 2003 #2,7,18,39,48 Grade 10 TAKS 2003 #24,26,46,49 Grade 11 TAKS 2003 #11,16,29,37, 43,50,53,57 Grade 9 TAKS 2004 #19,24,34,42,47 Grade 10 TAKS 2004 #6,8,20,26,52 Grade 11 TAKS 2004 #1,8,13,17,20, 45,47 Grade 11 July TAKS 2004 #1,3,26,40, 43,54,56 Grade 9 TAKS 2006 #5,11,12,17,19,22, 30,51 Grade 10 TAKS 2006 #5,12,26,40,55 Grade 11 TAKS 2006 #26,28,49,52 Grade 11 July TAKS 2006 #14,15,20, 33,36,42,58


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Meet Your Parent (pp. 1 of 3) KEY You have already investigated several relationships. Some were linear and some were not. Each one is a member of a family of functions. In this activity, you will “meet the parent” of one member of the family of functions.

You are meeting your future in-laws. One of them asks, “Just how old do you think I am?” Wow, what a loaded question! How are your estimation skills in determining his/her age? The table below lists 12 famous actors. In the last column write your age estimate of the given person. Birth dates are given for each actor, but Actual Age and Estimated Age will vary.

Actors Actual Age Estimated Age

Antonio Banderas (8/10/60) Answers vary by year Answers vary by student Hillary Duff (9/28/87) Answers vary by year Answers vary by student Dakota Fanning (2/23/94) Answers vary by year Answers vary by student Elijah Wood (1/28/81) Answers vary by year Answers vary by student Denzel Washington (12/28/54) Answers vary by year Answers vary by student Harrison Ford (7/13/42) Answers vary by year Answers vary by student Brad Pitt (12/18/63) Answers vary by year Answers vary by student Angelina Jolie (6/4/75) Answers vary by year Answers vary by student Eva Longoria (3/15/75) Answers vary by year Answers vary by student Jack Nicholson (4/22/37) Answers vary by year Answers vary by student Queen Latifah (3/18/70) Answers vary by year Answers vary by student Goldie Hawn (11/21/45) Answers vary by year Answers vary by student Will Smith (9/25/68) Answers vary by year Answers vary by student Lindsay Lohan (7/2/86) Answers vary by year Answers vary by student Leonardo DiCaprio (11/11/74) Answers vary by year Answers vary by student

1. Plot a scatterplot of the estimated ages as a function of the actual ages on the coordinate

plane below.


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Meet Your Parent (pp. 2 of 3) KEY

2. If a person were a “perfect” estimator, how would the table be altered? Complete a table for a “perfect” estimator. The actual values and the estimated values will be equal, i.e. y = x.


Antonio Banderas Hillary Duff Dakota Fanning Elijah Wood Denzel Washington Harrison Ford Brad Pitt Angelina Jolie Eva Longoria Jack Nicholson Queen Latifah Goldie Hawn Will Smith Lindsay Lohan Leonardo DiCaprio

3. Plot the points of the “perfect” estimator in another color on the coordinate plane. How do the

scatterplots compare? Individual scatterplot is more spread out even though there may be a linear tendency. The “perfect” estimator scatterplot forms a perfect line going through (0, 0).

4. How could you tell if you are an over-estimator of age? Most of the individual data points are above the “perfect” estimator.

5. How could you tell if you are an under-estimator of age? Most of the individual data points are under the “perfect” estimator.


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Meet Your Parent (pp. 3 of 3) KEY

The “perfect” estimator can be represented by one member of the family of functions. Let’s look further at this parent function. Materials: Chart paper, markers, sticky dots, meter stick

6. Given the following set of data, complete the table. In the last two rows of the table, create your own two points to be graphed.

Independent

value (x)

Dependent value

(y) Point (x, y)

0 0 (0, 0)

2 2 (2, 2)

- 5 -5 (5, 5)

- 3 ½ - 3 ½ (-3 ½, -3 ½)

7 7 (7, 7)

- 9 - 9 (-9, -9)

1 1 (1, 1)

- 1 -1 (1, 1)

4.5 4.5 (4.5, 4.5)

10 10 (10, 10)

7. Draw a coordinate plane on the chart paper. Scale and label the axes. Plot each point using a

sticky dot. 8. Use the meter stick to draw a smooth line through the points. Put arrows on both ends of the

line. 9. This is the graph of the linear parent function. The equation for the parent is y = x. Title the

chart Linear Parent Function. Label the graph of the line with the equation y = x. 10. Put the answers to the following on the chart.

a. The arrows indicate that the line goes on forever in both directions. What are the domain and the range of the function? All Real Numbers.

b. What do you notice about the x-values and y-values in the table and scatterplot? How is this reflected in the equation y = x? The coordinates are always equal. The equation states that the y values will equal the x values.


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Birth Dates of Actors

Calculate the age of each actor using their dates of birth. Enter their actual age into your table.

Actor Date of Birth

Antonio Banderas 8/10/60

Hillary Duff 9/28/87

Dakota Fanning 2/23/94

Elijah Wood 1/28/81

Denzel Washington 12/28/54

Harrison Ford 7/13/42

Brad Pitt 12/18/63

Angelina Jolie 6/4/75

Eva Longoria 3/15/75

Jack Nicholson 4/22/37

Queen Latifah 3/18/70

Goldie Hawn 11/21/45

Will Smith 9/25/68

Lindsay Lohan 7/2/86

Leonardo DiCaprio 11/11/74


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Meet Your Parent (pp. 1 of 3)

You have already investigated several relationships. Some were linear and some were not. Each one is a member of a family of functions. In this activity, you will “meet the parent” of one member of the family of functions.

You are meeting your future in-laws. One of them asks, “Just how old do you think I am?” Wow, what a loaded question! How are your estimation skills in determining his/her age? The table below lists 12 famous actors. In the last column write your age estimate of the given person.

Actors Actual Age Estimated Age Antonio Banderas Hillary Duff Dakota Fanning Elijah Wood Denzel Washington Harrison Ford Brad Pitt Angelina Jolie Eva Longoria Jack Nicholson Queen Latifah Goldie Hawn Will Smith Lindsay Lohan Leonardo DiCaprio

1. Plot a scatterplot of the estimated ages as a function of the actual ages on the coordinate

plane below.


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2. If a person were a “perfect” estimator, how would the table be altered? Complete a table for a “perfect” estimator.


Antonio Banderas Hillary Duff Dakota Fanning Elijah Wood Denzel Washington Harrison Ford Brad Pitt Angelina Jolie Eva Longoria Jack Nicholson Queen Latifah Goldie Hawn Will Smith Lindsay Lohan Leonardo DiCaprio

3. Plot the points of the “perfect” estimator in another color on the coordinate plane. How do the

scatterplots compare?

4. How could you tell if you are an over-estimator of age?

5. How could you tell if you are an under-estimator of age?


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The “perfect” estimator can be represented by one member of the family of functions. Let’s look further at this parent function. Materials: Chart paper, markers, sticky dots, meter stick

6. Given the following set of data, complete the table. In the last two rows of the table, create your own two points to be graphed.

Independent

value (x)

Dependent value

(y) Point (x, y)

0 0

2 2 (2, 2)

- 5 (5, 5)

- 3 ½ - 3 ½

7 7

- 9 - 9

1 (1, 1)

- 1 (1, 1)

4.5 4.5

10

7. Draw a coordinate plane on the chart paper. Scale and label the axes. Plot each point using a sticky dot.

8. Use the meter stick to draw a smooth line through the points. Put arrows on both ends of the line.

9. This is the graph of the linear parent function. The equation for the parent is y = x. Title the chart Linear Parent Function. Label the graph of the line with the equation y = x.

10. Put the answers to the following on the chart. a. The arrows indicate that the line goes on forever in both directions. What are the domain

and the range of the function?

b. What do you notice about the x-values and y-values in the table and scatterplot? How is this reflected in the equation y = x?


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What’s Up with “m”? (Transparency)

What if a coefficient is inserted in front of x?

mxy =

What effects do you predict the “m” will have on the graph of the linear parent function, y = x?

Function Prediction

xy 2=

xy2

1=

2y x

1

2y x


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Parameter Changes on the Linear Parent Function (pp. 1 of 7) KEY

Chart A: The parent function of a linear relationship is y = x. If a coefficient is placed in front of the x, it causes the steepness or slope of the graph to be changed. Use the graphing calculator to investigate and describe the changes in steepness “m” has on the graph of the linear parent function y = x.

Function

Sketch (both parent

function and given function on each coordinate plane)

Value of “m” in y = mx

Up or down

from left to right?

Is the steepness more, less,

or the same as y = x?

When x increases

by 1, y (increases, decreases)

by ____

Coordinates of the

y-intercept

y = x 1 Up Same Increases

by 1 (0, 0)

y = -x -1 Down Same Decreases

by 1 (0, 0)

y = 2x 2 Up More Increases

by 2 (0, 0)

y = 0.5x 0.5 Up Less Increases

by 0.5 (0, 0)

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10 y = x


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Function Sketch Value

of “m” in y = mx

Up or down

from left to right?



When x increases


by ____

Coordinates of the

y-intercept

y = -0.5x -0.5 Down Less Decreases

by 0.5 (0, 0)

y = 3.5x 3.5 Up More Increases

by 3.5 (0, 0)

y = -3.5x -3.5 Down More Decreases

by 3.5 (0, 0)

xy3

4=

3

4 Up More

Increases

by 3

4

(0, 0)


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Up or down

from left to right?



When x increases


by ____

Coordinates of the

y-intercept

1

3y x

1

3 Down Less

Decreases

by 1

3 (0, 0)

1

4y x

1

4 Up Less

Increases

by 1

4 (0, 0)

1.5y x

-1.5 Down More Decreases

by 1.5 (0, 0)


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Parameter Changes on the Linear Parent Function (pp. 4 of 7) KEY Use the information from Chart A to help you answer the following:

1. When “m” is positive, does the line go up or down from left to right? Up

2. When “m” is negative, the line goes down from left to right.

3. When the line is less steep than the graph of y = x what is true about the value of “m”? The value of m is between -1 to +1.

4. Does changing the value of “m” cause the line to curve or become a figure other than a line?

No, it does not change the linearity.

5. Does changing the value of “m” change where the line intersects the y-axis? What is the point of intersection?

No, the point of intersection with the y-axis is always (0, 0), the origin.

6. Predict the graph of y = -5x. Sketch your prediction. Remember to label the x-axis and y-axis.

Prediction: steeper, goes down from left to right

7. How do you know if the measure of steepness is positive or negative? If it goes up from left to right, it is positive. If it goes down from left to right, it is negative.


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Parameter Changes on the Linear Parent Function (pp. 5 of 7) KEY Chart B: The parent function of a linear relationship is y = x. If a number is added to or subtracted from x, it changes the position of the line with respect to the x and y axes. Use the graphing calculator to investigate and describe the effect the addition or subtraction of a “b” value has on the location of the graph of the linear parent function y = x.

Function Sketch Value of “m” in y=mx

Point of intersection

on y-axis

Up or down from

left to right?

Is the line more, less,

or the same

steepness?

y = x 1 (0, 0) Up

y = x + 2 1 (0, 2) Up Same

y = x – 2 1 (0, -2) Up Same

y = x – 5

1 (0, -5) Up Same


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on y-axis

Up or down from

left to right?


or the same

steepness?

1

2y x

1 (0, 1

2) Up Same

y = x + 5 1 (0, 5) Up Same

1

2y x 1 (0,

1

2 ) Up Same

Use the information from Chart B to help you answer the following:

8. Does addition or subtraction of a value of “b” change where the line intercepts the y-axis? Explain these changes. Yes. If the b value is positive, it moves the graph of the parent function up that many units. If the b value is negative, it moves the graph of the parent function down that many units.

9. Does addition or subtraction of a value of “b” change whether the line goes up or down from

left to right? No. As long as the x-term is positive, it goes up from left to right.

10. Does addition or subtraction of a value of “b” change the steepness of the line?

No. As long as the x-term has a coefficient of 1, it has the same steepness as the parent function.


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11. Does addition or subtraction of a value of “b” cause the line to curve or become a figure other than a line? No. The addition of a b value does not affect linearity.

12. Predict the graph of y = x + 6. Sketch your prediction. Remember to label the x-axis and y-axis. Prediction: same steepness of 1, up from left to right, up six units, y-intercept of (0, 6)


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Parameter Changes on the Linear Parent Function (pp. 1 of 7)

Chart A: The parent function of a linear relationship is y = x. If a coefficient is placed in front of the x, it causes the steepness or slope of the graph to be changed. Use the graphing calculator to investigate and describe the changes in steepness “m” has on the graph of the linear parent function y = x.

Function

Sketch (both parent

function and given function on each coordinate plane)

Value of “m” in y = mx

Up or down

from left to right?



When x

increases by 1, y

(increases, decreases)

by ____

Coordinates of the

y-intercept

xy = -10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

1 Up Same Increases

by 1 (0, 0)

y x -10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

-1

xy 2= -10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

Increases

by 2

xy 5.0= -10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y


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Up or down

from left to right?



When x

increases by 1, y


by ____

Coordinates of the

y-intercept

0.5y x

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

Down

xy 5.3= -10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

(0, 0)

3.5y x

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

xy3

4= -10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

Increases

by 3

4


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Up or down

from left to right?



When x

increases by 1, y


by ____

Coordinates of the

y-intercept

1

3y x -10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

xy4

1= -10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

1.5y x

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y


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Parameter Changes on the Linear Parent Function (pp. 4 of 7) Use the information from Chart A to help you answer the following:

1. When “m” is positive, does the line go up or down from left to right? __________

2. When “m” is negative, the line goes ______________ from left to right.

3. When the line is less steep than the graph of y = x what is true about the value of “m”?

4. Does changing the value of “m” cause the line to curve or become a figure other than a line?

5. Does changing the value of “m” change where the line intersects the y-axis? What is the point of intersection?

6. Predict the graph of y = -5x. Sketch your prediction. Remember to label the x-axis and y-axis.

7. How do you know from the graph if the measure of steepness is positive or negative?


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Parameter Changes on the Linear Parent Function (pp. 5 of 7) Chart B: The parent function of a linear relationship is y = x. If a number is added to or subtracted from x, it changes the position of the line with respect to the x and y axes. Use the graphing calculator to investigate and describe the effect the addition or subtraction of a “b” value has on the location of the graph of the linear parent function y = x.



on y-axis

Up or down from

left to right?


or the same

steepness?

xy = -10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

2+= xy -10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

y = x – 2 -10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

y = x – 5 -10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y


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2

1+= xy -10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

5+= xy -10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

1

2y x

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

Use the information from Chart B to help you answer the following:

8. Does addition or subtraction of a value of “b” change where the line intercepts the y-axis? Explain these changes.

9. Does addition or subtraction of a value of “b” change whether the line goes up or down from left to right?

10. Does addition or subtraction of a value of “b” change the steepness of the line?


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11. Does addition or subtraction of a value of “b” cause the line to curve or become a figure other than a line?

12. Predict the graph of y = x + 6. Sketch your prediction. Remember to label the x-axis and y-axis.


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What’s Up with “b”? (Transparency)

What if a numerical constant is added to x?

bxy +=

What effects do you predict the “b” will have on the graph of the linear parent function?

Function Prediction

2+= xy

2

3y x

7y x

5

24+= xy


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Investigating Slope (pp. 1 of 8) KEY Defining Slope Study the pattern below and complete the table.

Row Diagram Verbal description Process Total number

of tiles

1

1 row of 3 3

1(3) 3

2

2 rows of 3 3 + 3 2(3)

6

3

3 rows of 3 3 + 3 + 3

3(3) 9

4

4 rows of 3 3 + 3 + 3 + 3

4(3) 12

x x rows of 3 x(3) 3x

Graph the total number of square tiles as a function of row number on the provided coordinate plane.

How is rate of change demonstrated in the diagram of the situation? A row of three tiles is added on to the bottom of the preceding diagram.

How is rate of change demonstrated in the verbal description of the situation? It increases by one row of three from the preceding row.

How is rate of change demonstrated in the process column of the situation? Another three tiles are added for each new row. This repeated addition is simplified to multiplication.

How is rate of change demonstrated in the final generalized algebraic representation of the situation? The three representing the tiles added in each subsequent row becomes the coefficient of the x variable.

How is rate of change demonstrated in the graph? The ratio of vertical change to horizontal change between two points equals 3.


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Investigating Slope (pp. 2 of 8) KEY Defining Slope In the previous activity, you investigated the steepness of lines and connected that measure to the equation, graph, and change in y-values. The measure of steepness of a line, m, is called slope. In a linear function, the slope represents a comparison of the rate of change in the variables. It is the ratio of the change in vertical distance to the change in horizontal distance.

Ratio of vertical change to horizontal change

vertical change 41

horizontal change 4

If another two points were selected and the ratio of vertical change to horizontal change calculated, would it also equal 1? The ratio of vertical change to horizontal change between any two points would equal 1. Verify your answer. Answers will vary.

If the y-value of a linear equation increases by positive 4.5 when the x-value increases by 1, what is the measure of steepness? 4.5

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

Vertical change

Horizontal change


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Investigating Slope (pp. 3 of 8) KEY Slope is represented by the variable m in the equation y = mx.

1. How could you find the slope by looking at an equation of a line such as y = 3.2x? The slope would be the coefficient of the x-term. Slope is equal to 3.2.

2. The slope is one of the parameter changes that affect the linear parent function. Summarize the effects of changing the slope, m, on the graph:

a. When the slope, m, is positive, the graph goes up from left to right.

b. When slope, m, is negative, the graph goes down from left to right.

c. When the slope, m, is greater than 1, the graph steeper than the parent function.

d. When the slope, m, is between -1 and 1, the graph less steep than the parent function.

e. When the slope, m, is changed, the y-intercept does not change and remains at (0,0), origin.

3. The slope of the graph of the parent function y = x is 1. That means when x increases by 1, y increases by 1.

Finding slope from a graph

Locate two points on the graph (preferably integral values) and compare the vertical change to the horizontal change.

Slope = change horizontal

change vertical

Vertical change is also called Rise: up (+) or down (-)

Horizontal change is also called Run: right (+) or left (-)


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Investigating Slope (pp. 4 of 8) KEY

Finding slope from a table

Compare the change in y-values (dependent) with the change in x-values (independent) by writing

a ratio. The Greek letter delta, , is used to represent “change in”.

Change in or

Change in

y y

x x

Example 1: Complete the table to calculate the slope:

Number of

Hours, x

Distance, y

Change in x, x

Change in y, y Slope =

yx

1 4

2 8 2 – 1 = 1 8 – 4 = 4 1

4 = 4

4 16 4 – 2 = 2 16 – 8 = 8 8/2 = 4

6 24 6 – 4 = 2 24 – 16 = 8 2

8 = 4

10 40 10 – 6 = 4 40 – 24 = 16 16/4 = 4

Slope = 4, which means that when x increases by 1, y increases by 4. Example 2: Complete the table to calculate the slope:

x y Change in x,

x Change in

y, y Slope = yx

-2 27

5 13 5 –(-2) = 7 13 – 27 = -14 -14/7 = -2

6 11 6 – 5 = 1 11 – 13 = -2 -2/1 = -2

8 7 8 – 6 = 2 7 – 11 = -4 -4/2 = -2

13 -3 13 – 8 = 5 -3 – 7 = -10 -10/5 = -2

Slope = -2, which means that when x increases by 1, y decreases by 2.


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Special Lines

What is the slope of a horizontal or vertical line? Plot the points on the graphs provided and draw a smooth line through the points. Complete the tables to find the slopes of these special cases.

y = 2

x = -4

x y yx

x y yx

-2 2 -4 -5

0 2 0/2 = 0 -4 -3 2/0 und.

4 2 0/4 = 0 -4 1 4/0 und.

7 2 0/3 = 0 -4 10 9/0 und.

a. The slope of a horizontal line is 0.

b. The slope of a vertical line is undefined.

Guided Practice 1. Which line is steepest and why? Answers will vary.

a. y = 2x

b. y = -x

c. y = 0.5x

2. Write one example of an equation that makes the graph of y = x less steep. Test your hypothesis

with the graphing calculator. Sketch both the linear parent function and your example function the grid below.

y = 3

5x


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Practice Problems Find the slope (rate of change) of the following linear functions.

1. 2.

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

Slope = undefined Slope = -2

3. 4.

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

Slope = 3

4 Slope = 0


Unit: 03 Lesson: 01

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Find the slope (rate of change) from given data in the tables. 5. 6.

x y yx

x y yx

0 0 3 -5 7 -2 4 12 3 1 -5 -2 6 18 3 4 -11 -2 9 27 3 7 -17 -2 10 30 3 9 -21 -2

Slope = 3 Slope = -2

7. 8.

x y yx

x y yx

-3 -3 0 -4 -13 1 1 -3 0 -1 -10 1 6 -3 0 5 -4 1 7 -3 0 8 -1 1 10 -3 0 10 1 1

Slope = 0 Slope = 1 Find the slope (rate of change) for given points and equations.

9. Points (-2, 3) (6, -13) 10. Points (7, 5) (-3, 1) m = -2 m = 0.4

11. y = 5

2 x 12. y = 3x

m = 2

5 m = 3

14. y = -12 15. x = 9 m = 0 undefined


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16. Represent each rate of change a) graphically, b) with two points in a table, and c) as an algebraic function. Answers will vary depending on the line chosen. Sample answers are given.

Rate of change = 2 Rate of change = - 1

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

x y -1 -2 4 8

2y x

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

x y -1 1 4 -4

y x

Rate of change = 0 Rate of change is undefined.

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

x y -3 3 6 3

3y

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

x y -5 1 -5 -4

5x


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Investigating Slope (pp. 1 of 8) Defining Slope Study the pattern below and complete the table.

Row Diagram Verbal description Process Total number

of tiles

1 1 row of 3 3

1(3) 3

2 2 rows of 3

3

4

x

Graph the total number of square tiles as a function of row number on the provided coordinate plane.

How is rate of change demonstrated in the diagram of the situation? How is rate of change demonstrated in the verbal description of the situation?

How is rate of change demonstrated in the process column of the situation?

How is rate of change demonstrated in the final generalized algebraic representation of the situation?

How is rate of change demonstrated in the graph?


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Investigating Slope (pp. 2 of 8) Defining Slope In the previous activity, you investigated the steepness of lines and connected that measure to the equation, graph, and change in y-values. The measure of steepness of a line, m, is called ____________. In a linear function the slope represents a comparison of the ____________________________ in the variables. It is the ratio of the change in vertical distance to the change in horizontal distance.

Ratio of vertical change to horizontal change

vertical change 41

horizontal change 4

If another two points were selected and the ratio of vertical change to horizontal change was calculated, would it also equal 1? Verify your answer. If the y-value of a linear equation increases by positive 4.5 when the x-value increases by 1, what is the measure of steepness?

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y


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Investigating Slope (pp. 3 of 8) Slope is represented by the variable m in the equation y = mx.

1. How could you find the slope by looking at an equation of a line such as y = 3.2x?

2. The slope is one of the parameter changes that affect the linear parent function. Summarize the effects of changing the slope, m, on the graph:

a. When the slope, m, is positive the graph _______________________________.

b. When slope, m, is negative the graph __________________________________.

c. When the slope, m, is greater than 1 the graph __________________________.

d. When the slope, m, is between -1 and 1 the graph _______________________.

e. When the slope, m, is changed, the y-intercept ___________________________.

3. The slope of the graph of the parent function y = x is __________. That means when x increases by 1, y __________________ by _________.

Finding slope from a graph

Locate two points on the graph (preferably integral values) and compare the vertical change to the horizontal change.

Slope = change horizontal

change vertical

Vertical change is also called Rise: up (+) or down (-)

Horizontal change is also called Run: right (+) or left (-)


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Investigating Slope (pp. 4 of 8)

Finding slope from a table

Compare the change in y-values (dependent) with the change in x-values (independent) by writing

a ratio. The Greek letter delta, , is used to represent “change in”.

Change in or

Change in

y y

x x

Example 1: Complete the table to calculate the slope:

Number of

Hours, x

Distance, y

Change in x, x

Change in y, y Slope =

yx

1 4

2 8 2 – 1 = 1 8 – 4 = 4 1

4 = 4

4 16

6 24 6 – 4 = 2 24 – 16 = 8 2

8 = 4

10 40

Slope = __________, which means that when x increases by 1, y ________________ by __________. Example 2: Complete the table to calculate the slope:

x y Change in x,

x Change in

y, y Slope = yx

-2 27

5 13 5 –(-2) =

6 11 11 – 13 = -2

8 7

13 -3 -3 – 7 =

Slope = __________, which means that when x increases by 1, y ________________ by __________.


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Special Lines

What is the slope of a horizontal or vertical line? Plot the points on the graphs provided and draw a smooth line through the points. Complete the tables to find the slopes of these special cases.

y = 2

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

x = -4

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

x y yx

x y yx

-2 2 -4 -5

0 2 -4 -3

4 2 -4 1

7 2 -4 10

a. The slope of a horizontal line is ___________________.

b. The slope of a vertical line is _____________________.

Guided Practice 1. Which line is steepest and why?

a. y = 2x

b. y = -x

c. y = 0.5x

2. Write one example of an equation that makes the graph of y=x less steep. Test your hypothesis

with the graphing calculator. Sketch both the linear parent function and your example function.


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Practice Problems Find the slope (rate of change) of the following linear functions.

1. 2.

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

Slope = Slope =

3. 4.

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

Slope = Slope =


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Find the slope (rate of change) from given data in the tables. 5. 6.

x y yx

x y yx

0 0 -5 7 4 12 1 -5 6 18 4 -11 9 27 7 -17 10 30 9 -21

Slope = Slope =

7. 8.

x y yx

x y yx

-3 -3 -4 -13 1 -3 -1 -10 6 -3 5 -4 7 -3 8 -1 10 -3 10 1

Slope = Slope = Find the slope (rate of change) for given points and equations.

9. Points (-2, 3) (6, -13) 10. Points (7, 5) (-3, 1)

11. y = 5

2 x 12. y = 3x

14. y = -12 15. x = 9


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16. Represent each rate of change a) graphically, b) with two points in a table, and c) as an algebraic function.

Rate of change = 2 Rate of change = - 1

Rate of change = 0 Rate of change is undefined.


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Slalom on the Slopes KEY Find the slope of the line joining each pair of points. Write the answer in the box next to the problem. Connect the answers in order in the puzzle below.

1. (3, 4) (4, 7) 3 2. (4, 0) (6, -2) -1

3. (0, 9) (1, 11) 2 4. (7, -8) (-3, 12) -2

5. (-1, -1) (1, 11) 6 6. (2, -3) (1, 5) -8

7. (3, 4) (5, 5) 1/2 8. (0, 0) (3, 5) 5/3

9. (-3, -1) (5, 6) 7/8 10. (4, 0) (-4, 2) -1/4

11. (3, -9) (5, -1) 4 12. (7, -8) (8, -7) 1

13. ( 1 1,

3 5) ( 2 4

,3 5

) 9/5 14. ( 1 4,

4 9 ) ( 1 7

,4 9

) 2/3

15. ( 3,9

7 ) ( 4

,47

) -5 16. (2, 3) (3, 8) 5

17. (1, -4) (2, 3) 7 18. (9, 3) (4, 3) 0

19. (-3, -11) (-1, 5) 8 20. (6, 11) (5, 0) 11

21. (2, 4) (3, -3) -7 22. (-2, -3) (-1, 6) 9

23. (0, 10) (4, -2) -3 24. (49, -50) (50, 50) 100

25. (8, -12) (4, 12) -6

Dot-to-dot forms a skier.


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Slalom on the Slopes Find the slope of the line joining each pair of points. Write the answer in the box next to the problem. Connect the answers in order in the puzzle below.

1. (3, 4) (4, 7) 2. (4, 0) (6, -2)

3. (0, 9) (1, 11) 4. (7, -8) (-3, 12)

5. (-1, -1) (1, 11) 6. (2, -3) (1, 5)

7. (3, 4) (5, 5) 8. (0, 0) (3, 5)

9. (-3, -1) (5, 6) 10. (4, 0) (-4, 2)

11. (3, -9) (5, -1) 12. (7, -8) (8, -7)

13. ( 1 1,

3 5) ( 2 4

,3 5

) 14. ( 1 4,

4 9 ) ( 1 7

,4 9

)

15. ( 3,9

7 ) ( 4

,47

) 16. (2, 3) (3, 8)

17. (1, -4) (2, 3) 18. (9, 3) (4, 3)

19. (-3, -11) (-1, 5) 20. (6, 11) (5, 0)

21. (2, 4) (3, -3) 22. (-2, -3) (-1, 6)

23. (0, 10) (4, -2) 24. (49, -50) (50, 50)

25. (8, -12) (4, 12)


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Headache Card Game KEY

Find the slope for each card. If lines are parallel, the slopes are equal. Match the cards that have parallel slopes. Fill in the letter with the matching numbers in the boxes below to answer Mrs. Healy’s question.


Unit: 03 Lesson: 01

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Headache Card Game Cards KEY

E (10, 10) (-1, -1) m = 1

1 (-4, 2) (-2, 3) m = 1/2

H (8, 3) (10, 4) m = 1/2

2 (11, 11) (8, 5) m = 2

I (-6, 4) (-2, 0) m = -1

3 (-2, 2) (2, 3) m = 1/4

L (0, -9) (2, -5) m = 2

4 (32, -6) (33, -5) m = 1

M (-6, -1) (-2, 0) m = 1/4

5 (-10, 8) (-5, 3) m = -1

N (-1, -1) (0, 4) m = 5

6 (-1, 3) (3, 2) m = -1/4

P (-1, 3) (2, -3) m = -2

7 (-5, -6) (5, -2) m = 2/5

R (4, 3) (8, 2) m = -1/4

8 (0, 2) (8, 8) m = 3/4

T (4, -1) (7, -3) m = -2/3

9 (3, -5) (4, 0) m = 5

S (-4, 2) (0, 5) m = 3/4

10 (-2, 2) (4, -2) m = -2/3

U (-4, 0) (1, 2) m = 2/5

11 (4, 3) (7, -3) m = -2


Unit: 03 Lesson: 01

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Headache Card Game

Find the slope for each card. If lines are parallel, the slopes are equal. Match the cards that have parallel slopes. Fill in the letter with the matching numbers in the boxes below to answer Mrs. Healy’s question.


Unit: 03 Lesson: 01

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Headache Card Game Cards

E (10, 10) (-1, -1)

1 (-4, 2) (-2, 3)

H (8, 3) (10, 4)

2 (11, 11) (8, 5)

I (-6, 4) (-2, 0)

3 (-2, 2) (2, 3)

L (0, -9) (2, -5)

4 (32, -6) (33, -5)

M (-6, -1) (-2, 0)

5 (-10, 8) (-5, 3)

N (-1, -1) (0, 4)

6 (-1, 3) (3, 2)

P (-1, 3) (2, -3)

7 (-5, -6) (5, -2)

R (4, 3) (8, 2)

8 (0, 2) (8, 8)

T (4, -1) (7, -3)

9 (3, -5) (4, 0)

S (-4, 2) (0, 5)

10 (-2, 2) (4, -2)

U (-4, 0) (1, 2)

11 (4, 3) (7, -3)


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Investigating Intercepts (pp. 1 of 7) KEY

1. Can more than one line have the same slope? If more than one line has the same slope, what makes the lines different? Yes; location on the y-axis

a. Graph the following set of equations on the same set of axes. Label each line.

i. y = x

ii. y = x - 6

iii. y = x + 1

2

iv. y = x + 3

v. y = x - 1

2

b. What observations can you make about the lines? All have the same slope and are parallel.

c. What is the slope of all the lines? The slope is equal to 1.

d. How does addition or subtraction of a “b” value change the line? It changes the line with respect to location of the y-intercept.

e. What is the name of “b?” y-intercept

f. Predict the graph of y = x + 7. Sketch your prediction. Use the graphing calculator to verify your prediction. Prediction: Up, y-intercept of (0, 7), slope of 1

g. Complete the sentence: Lines with the same slope are parallel.

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y


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Finding Intercepts

In a function an intercept is the point at which a line crosses an axis. If it crosses the y-axis, it is called the y-intercept and the point is (0, y). If it crosses the x-axis, it is called the x-intercept and the point is (x, 0). The x-intercepts are also known as the zeros of a function, because the x-intercepts are where the value of the function is zero.

Finding intercepts from a graph

2. Study the graphs of the lines below.

Examples:

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

x-intercept: (2, 0) x-intercept: (8, 0)

y-intercept: (0, 3) y-intercept: (0, -2)


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Investigating Intercepts (pp. 3 of 7) KEY Finding intercepts from a table

y-intercept (0, y)

x-intercept

zero of function (x, 0)

0 y x 0

3. Use patterns to complete the tables and find intercepts.

x y x y

-1 3 -3 -14

0 2 -1 -10

1 1 0 -8

2 0 1 -6

3 -1 3 -2

4 0

5 2

a. Determine the slope m = -1

b. Circle the x-intercept (zero of function)

c. Write the coordinates of the x-intercept. (2, 0)

d. Circle the y-intercept

e. Write the coordinates of the y-intercept (0, 2)

a. Determine the slope m = 2

b. Complete the pattern to find where y = 0.

c. Circle the x-intercept (zero of function)

d. Write the coordinates of the x-intercept (4, 0)

e. Complete the pattern to find where x = 0.

f. Circle the y-intercept

g. Write the coordinates of the y-intercept (0, -8)


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Finding intercepts from an equation

One form of linear equations is called the slope-intercept form. Any linear function can be written in this form in order to determine the slope and y-intercept.

y mx b or ( )f x mx b

m represents the slope of the line. b represents the y-intercept of the line. Use algebraic manipulation to transform the following equation to the slope-intercept form. Determine the slope and y-intercept from of the function.

6x – 3y = 9 -6x -6x -3y = -6x + 9

3 6 9

3 3 3

y x

y = 2x – 3 m = 2, y-intercept = -3, (0, -3)

Solve for y. Subtract the x term from both sides. Divide both sides by the coefficient of the y

term. (Divide by -3.) Simplify. Identify the slope and y-intercept form.

4. Find the slope and y-intercept for each function.

y = -4

5x + 7 f(x) = 12x – 35 y = 60 – 6x

m = -5/4, b = 7 m = 12, b = -35 m = -6, b = 60

3x + 2y = 5 4y – x = 16 y = -3/2x + 5/2, m = -3/2, b = 5/2 y = 1/4x + 4, m = 1/4, b = 4, (0, 4)

Special Cases: Find the slope and y-intercept.

5. y = -10 6. x = 6

m = 0, b = -10 m is undefined, b = none


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Practice Problems 1. Find the slope and the x- intercept and y-intercept of the following graphs of lines.

a. b.

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

slope 3/2 slope undefined x-intercept (zero of function) (4/3, 0) x-intercept (zero of function) (-7, 0)

y-intercept (0, -2) y-intercept none c. d.

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

slope -2 slope 0 x-intercept (zero of function) (1, 0) x-intercept (zero of function) none y-intercept (0, 2) y-intercept (0, 4)


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Find the slope and intercepts from the data in the tables.

2. 3. x y x y -2 6 -1 -20 0 4 0 -16 2 2 1 -12 4 0 3 -4 6 -2 4 0 5 4 7 12

slope = -1 slope = 4 x-intercept (zero) = (4, 0) x-intercept (zero) = (4, 0) y-intercept = (0, 4) y-intercept = (0, -16) Find the slope and y-intercept of each equation.

4. y = 2.5x 5. 3

427

y x

m = 2.5, b = 0 m = -3/7, b = -42

6. 4

( ) 23

f x x 7. 4x + 3y = 12

m = 4/3, b = 2 y = -4/3x + 4, m = -4/3, b = 4, (0, 4) 8. y = -1 9. x = 4

m = 0, b = -1 m is undefined, b = none

10. 2x – 5y = 15 11. 6y = 2x – 18 y = 2/5x - 3, m = 2/5, b = -3, (0, -3) y = 1/3x - 3, m = 1/3, b = -3, (0, -3)

12. 5y + 2x = -8

y = -2/5x – 8/5, m = -2/5, b = -8/5, (0, -8/5)


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13. A line contains the points (-6, -5) and (3, 1). a. Sketch a graph of the line.

b. What is the slope of the original line? m = 2/3 c. If the slope is multiplied by 3 and the y-intercept stays the same, sketch a transformed

graph on the same coordinate plane of the resulting line. d. What is the slope of the transformed line? m = 2

14. A line with a slope of one-half, contains the point (-4, -5). a. Sketch a graph of the line.

b. What is the y-intercept of the original line? (0, -3) c. If the slope remains the same and the y-intercept increased by 3 units, sketch a

transformed graph on the same coordinate plane of the resulting line. d. What is the y-intercept of the transformed line? (0, 0) e. How would you describe the relationship between the two lines? Explain. The lines are

parallel. They will never cross each other. The slopes are parallel.


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Investigating Intercepts (pp. 1 of 7)

1. Can more than one line have the same slope? If more than one line has the same slope, what makes the lines different? a. Graph the following set of equations on the same set of axes. Label each graph.

i. y = x

ii. y = x - 6

iii. y = x + 2

1

iv. y = x + 3

v. y = x - 2

1

b. What observations can you make about the lines?

c. What is the slope of all the lines?

d. How does addition or subtraction of a “b” value change the line?

e. What is the name of “b?”

f. Predict the graph of 7y x . Sketch your prediction on the above graph. Use the graphing calculator to verify your prediction.

g. Complete the sentence: Lines with the same slope are ________________.

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y


Unit: 03 Lesson: 01

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Finding Intercepts

In a function an intercept is the point at which a line crosses an axis. If it crosses the y-axis, it is called the ___________ and the point is (0, y). If it crosses the x-axis, it is called the ___________ and the point is (x, 0). The x-intercepts are also known as the ___________________, because the x-intercepts are where the value of the function is zero.

Finding intercepts from a graph

2. Study the graphs of the lines below.

Examples:

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

x-intercept: __________ x-intercept: __________

y-intercept: __________ y-intercept: __________


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Investigating Intercepts (pp. 3 of 7) Finding intercepts from a table

y-intercept (0, y)

x-intercept

zero of function (x, 0)

0 y x 0

3. Use patterns to complete the tables and find intercepts.

x y x y

-1 3 -3 -14

0 2 -1 -10

1 1

2 0 1 -6

3 -1 3 -2

5 2

a. Determine the slope

b. Circle the x-intercept (zero of function)

c. Write the coordinates of the x-intercept.

d. Circle the y-intercept

e. Write the coordinates of the y-intercept

a. Determine the slope

b. Complete the pattern to find where y = 0.

c. Circle the x-intercept (zero of function)

d. Write the coordinates of the x-intercept

e. Complete the pattern to find where x = 0.

f. Circle the y-intercept

g. Write the coordinates of the y-intercept


Unit: 03 Lesson: 01

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Finding intercepts from an equation

One form of linear equations is called the __________________________________ form. Any linear function can be written in this form in order to determine the slope and y-intercept.

y mx b or ( )f x mx b

m represents __________________________________. b represents __________________________________. Use algebraic manipulation to transform the following equation to the slope-intercept form. Determine the slope and y-intercept from of the function.

6x – 3y = 9

Solve for y.

4. Find the slope and y-intercept for each function.

y = -4

5x + 7 f(x) = 12x – 35 y = 60 – 6x

3x + 2y = 5 4y – x = 16

Special Cases: Find the slope and y-intercept.

5. y = -10 6. x = 6


Unit: 03 Lesson: 01

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Practice Problems 2. Find the slope and the x- intercept and y-intercept of the following graphs of lines.

a. b.

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

slope ______ slope ______ x-intercept (zero of function) ______ x-intercept (zero of function) ______

y-intercept ______ y-intercept ______ c. d.

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y


y-intercept ______ y-intercept ______


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Find the slope and intercepts from the data in the tables.

3. 3. x y x y -2 6 -1 -20 0 4 2 2 1 -12 4 0 3 -4 6 -2 5 4 7 12


y-intercept ______ y-intercept ______ Find the slope and y-intercept of each equation.

4. y = 2.5x 5. 3

427

y x

6. 4

( ) 23

f x x 7. 4x + 3y = 12

8. y = -1 9. x = 4

10. 2x – 5y = 15 11. 6y = 2x – 18

12. 5y + 2x = -8


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13. A line contains the points (-6, -5) and (3, 1). a. Sketch a graph of the line.

b. What is the slope of the original line? c. If the slope is multiplied by 3 and the y-intercept stays the same, sketch a

transformed graph on the same coordinate plane of the resulting line. d. What is the slope of the transformed line?

14. A line with a slope of one-half, contains the point (-4, -5). a. Sketch a graph of the line.

b. What is the y-intercept of the original line? c. If the slope remains the same and the y-intercept increased by 3 units, sketch

a transformed graph on the same coordinate plane of the resulting line. d. What is the y-intercept of the transformed line? e. How would you describe the relationship between the two lines? Explain.


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The Great Hawaiian Race (pp. 1 of 5) KEY

Part I Lani and George each claim to have the fastest sailboats in Hawaii. They decide to settle the dispute with a 60-mile sailboat race starting from the island of Maui and ending on the Big Island (Hawaii). During the race, Lani is able to sail her boat at a speed of 12 miles per hour. George on the other hand had difficulty with a sail and was only able to average 10 miles per hour.

1. Make a table showing the relationship between time (hours) and distance (miles) for each boat from start to finish of the race.

Lani George Time

(hours) Distance (miles)

Time


0 0 0 0 1 12 1 10 2 24 2 20 3 36 3 30 4 48 4 40 5 60 5 50 6 60 x 12x x 10x

2. Graph the data using different colors to represent each. Use appropriate domain

and range values to scale the graph.


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3. Put the data points into the graphing calculator. (STAT-EDIT-xL1 and yLaniL2 and yGeorgeL3 ) Graph the data in the graphing calculator after setting an appropriate window. (2nd STAT PLOT-PLOT1)

4. Which parent function best represents these data points? Use parameter changes on this parent function to determine the function rule for Lani and George. How does this function rule compare with the ones found using the table? Both are best represented by a linear parent function. Function rules are the same by table or parameter changes.

5. What is the slope of the linear function for each?

Lani has a slope of 12 and George has a slope of 10. 6. What are the real world units for the slope values?

The units are miles per hour. 7. What does the slope represent in the original problem?

The slope represents the speed each is traveling in miles per hour.

8. What are the y-intercepts of the linear function of each? The y-intercepts for both are at (0, 0).

9. What do the y-intercepts represent in the problem?

The y-intercepts represent that at time zero, before the race begins, neither have sailed any distance toward the Big Island.

10. What are the x-intercepts of the linear function of each?

The x-intercepts for both are at (0, 0).

11. What do the x-intercepts represent in the problem? The x-intercepts represent when they are on Maui. They are both on Maui at time zero when the race begins.

12. Predict the effect on the problem situation if Lani’s slope was to change to 8 miles

per hour. The slope would be less steep because Lani would be going slower.


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The Great Hawaiian Race (pp. 3 of 5) KEY Part II An outrigger leaves the Big Island and paddles toward the two sailboats, traveling at an average speed of 3 miles per hour. Make a table showing the relationship between time (hours) and distance (miles) for the outrigger. (HINT: Remember it begins from the Big Island which is the finish of the race, therefore its beginning distance is 60 miles.)

Outrigger Time


0 60 1 57 2 54 3 51 4 48 5 45 6 42 x 60 – 3x

13. Put the data for the outrigger into the graphing calculator. (STAT-EDIT youtriggerL4) Graph all three sets of data on the calculator. Using a different color, put the graph of the outrigger on the graph in #2.

14. What is the function rule for the outrigger? y = 60 – 3x

15. Compare the slopes for Lani, George, and the outrigger. How do these relate to the

real-world situation? Lani and George both have a positive slope because their distance from Maui is increasing as time increases. The outrigger has a negative slope because its distance from Maui is decreasing as time increases. Lani is going the fastest because her slope is steepest with a greater rate of change. The outrigger is going slowest with the least rate of change.

16. Who does the outrigger meet first? At what time and distance?

Meets Lani first at four hours and 48 miles from Maui.

17. Put the equation for the outrigger into y =. Use the table function of the graphing calculator to find the x-intercept and y-intercept of the outrigger. How do these relate to the real-world situation? The y-intercept is at (0, 60). This is when it started from the Big Island at time zero and was 60 miles from Maui. The x-intercept is (20, 0). This is twenty hours after the race began when the outrigger arrives at Maui (distance is zero).


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The Great Hawaiian Race (pp. 4 of 5) KEY Part III Because of George’s damaged sail, Lani decides to give George another chance. She agrees to give George an eight-mile head start and repeat the race from the island of Maui to the Big Island (Hawaii). Strong currents cause Lani’s average speed to drop to 8 miles per hour. George who begins with an eight-mile head start averages 6 miles per hour.


Lani George

Time (hours)

Distance (miles)

Time


0 0 0 8 1 8 1 14 2 16 2 20 3 24 3 26 4 32 4 32 5 40 5 38 6 48 6 44 7 56 7 50 8 64 8 56

9 62 x 8x x 8 + 6x




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21. Find the function rules for Lani and George, representing them in function notation.

l(x) = 8x g(x) = 6x + 8

22. Compare the slopes of the function rules and relate them to the real-world situation. Lani’s slope is 8, which means her speed is 8 miles per hour. George’s slope is 6, which means his speed is 6 miles per hour. He is still going slower than Lani.

23. Determine the y-intercepts and x-intercepts for each person and relate them to the

real-world situation. Both of Lani’s intercepts occur at (0, 0). When the race begins at time 0 hours, she is on Maui at distance 0 from Maui. George’s y-intercept is (0, 8). When the race begins at time 0, he is already eight miles from Maui. George’s x-intercept would be (-4/3, 0) which means at his speed he had to start 1 hour and 20 minutes before the race to get to his 8-mile head start position.

24. Who wins the race if it goes all the way to the Big Island? Explain.

Lani wins the race. Because she is sailing faster, she overtakes George after 4 hours.

25. If a thunderstorm threatens to halt the race, up to what time would this be of benefit

to George? Explain your reasoning. If the race is halted before four hours, George would be in the lead.

26. If Lani was more generous and gave George a head start of 15 miles, how would

the final results of the race be affected? They would have arrived on the Big Island at the same time (7.5 hours) and the race would be a tie.


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The Great Hawaiian Race (pp. 1 of 5)

Part I Lani and George each claim to have the fastest sailboats in Hawaii. They decide to settle the dispute with a 60-mile sailboat race starting from the island of Maui and ending on the Big Island (Hawaii). During the race, Lani is able to sail her boat at a speed of 12 miles per hour. George on the other hand had difficulty with a sail and was only able to average 10 miles per hour.


Lani George Time


Time


0 0 0 0 1 1 x x




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4. Which parent function best represents these data points? Use parameter changes on this parent function to determine the function rule for Lani and George. How does this function rule compare with the ones found using the table?

5. What is the slope of the linear function for each? 6. What are the real world units for the slope values? 7. What does the slope represent in the original problem? 8. What are the y-intercepts of the linear function of each? 9. What do the y-intercepts represent in the problem?

10. What are the x-intercepts of the linear function of each?

11. What do the x-intercepts represent in the problem?

12. Predict the effect on the problem situation if Lani’s slope was to change to 8 miles per hour.


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The Great Hawaiian Race (pp. 3 of 5) Part II An outrigger leaves the Big Island and paddles toward the two sailboats, traveling at an average speed of 3 miles per hour. Make a table showing the relationship between time (hours) and distance (miles) for the outrigger. (HINT: Remember it begins from the Big Island which is the finish of the race, therefore its beginning distance is 60 miles.)

Outrigger Time


0 60 1 60 – 3(1) = 57 2 60 – 3(2) = 3 60 – 3( ) = 4 5 6 x

13. Put the data for the outrigger into the graphing calculator. (STAT-EDIT youtriggerL4) Graph all three sets of data on the calculator. Using a different color, put the graph of the outrigger on the graph in #2.

14. What is the function rule for the outrigger?

15. Compare the slopes for Lani, George, and the outrigger. How do these relate to the real-world situation?

16. Who does the outrigger meet first? At what time and distance? 17. Put the equation for the outrigger into y =. Use the table function of the graphing

calculator to find the x-intercept and y-intercept of the outrigger. How do these relate to the real-world situation?


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The Great Hawaiian Race (pp. 4 of 5) Part III Because of George’s damaged sail, Lani decides to give George another chance. She agrees to give George an eight mile head start and repeat the race from the island of Maui to the Big Island (Hawaii). Strong currents cause Lani’s average speed to drop to 8 miles per hour. George who begins with an eight-mile head start averages 6 miles per hour.


Lani George Time


Time


x x




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21. Find the function rules for Lani and George, representing them in function notation.

22. Compare the slopes of the function rules and relate them to the real-world situation. 23. Determine the y-intercepts and x-intercepts for each person and relate them to the

real-world situation. 24. Who wins the race if it goes all the way to the Big Island? Explain.

25. If a thunderstorm threatens to halt the race, up to what time would this be of benefit

to George? Explain your reasoning.

26. If Lani was more generous and gave George a head start of 15 miles, how would the final results of the race be affected?


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BATTERY UP! (pp. 1 of 3)

Background How many electronic devices do you use that run on batteries? Some battery operated devices are shown below. Can you think of others?

If you look at the position of the batteries in many flashlights, you will notice that they are lined up in a column or a series. The batteries in the flashlight are lined up so that the positive terminal (+) touches the negative terminal (−). Study the position of the batteries in the CBL. Notice that the batteries do not stack like they do in a flashlight. Instead there is a piece of metal connecting the positive and negative terminals. These batteries are also connected in series.

Batteries in a series

Batteries supply electrical energy when a circuit is created. A circuit is a path linking the positive terminal to a load (like the light bulb in the flashlight) back to the negative terminal. Objectives

Collect voltage data for individual batteries using a CBL and voltage sensor. Collect voltage for batteries in a series. Analyze data collected to identify characteristics of batteries in a series and analyze

representations of the data.


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Extend representations of the data to make decisions, predictions and critical

judgments. Materials

Graphing calculator, CBL, link cord, voltage sensor, 6 batteries (AA or AAA), chart paper, markers, two meter sticks, masking tape

Procedure

Part A 1. Collect the materials needed to complete the activity. 2. Connect the CBL to the calculator with a link cord. Connect the voltage sensor to

the CBL in the Channel 1 port. 3. On calculator, go to APPS, Easy Data. 4. Lay the meter sticks on the table side by side to create a trough in between large

enough to hold the batteries lengthwise. Secure the meter sticks with tape. 5. Place one battery in the trough. Touch and hold the voltage leads to the appropriate

terminal, red to (+) and black to (-). 6. Read the voltage at the top of the calculator screen. Record it on the Student

Activity Sheet in the table under Part A. 7. Remove that battery and place a second battery in the trough. Find the voltage of

the second battery. Record it in the table. 8. Continue until the voltage of all batteries has been recorded. 9. Find the average voltage of all six batteries. Record it on the Student Activity Sheet

under Part A.


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Part B

1. On the main screen of the Easy Data APP, press WINDOW to select Setup. 2. Press 3 for EVENTS WITH ENTRY. 3. Press ZOOM to select Start. 4. Place one battery in the trough. Touch and hold the voltage leads to the appropriate

terminal, red to (+) and black to (−). 5. Press ZOOM to select Keep. 6. Enter 1 for the number of batteries in the box and then GRAPH to select OK. 7. Add another battery in the trough so that it links to the first with positive to negative

terminals. 8. Touch and hold the voltage leads to the end terminals, red to (+) and black to (-). 9. Enter 2 for the number of batteries in the box and then GRAPH to select OK. 10. Continue until a series of 6 batteries is reached.

11. When all data has been collected, press WINDOW to select Stop to see the graph. 12. To begin data analysis, exit the Easy Data APP using the following steps:

a. Press TRACE to select Main. b. Press GRAPH to select Quit. c. Press GRAPH to select OK.

13. Number of batteries is listed in L1. 14. Voltage is listed in L2. 15. Adjust WINDOW and STAT PLOT settings. 16. Use calculator to complete the analysis of the data.

+ + + − + − − −


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BATTERY UP! Student Activity Sheet (pp. 1 of 3)

Part A

1. Record the voltage for each battery in the table below.

Battery 1st 2nd 3rd 4th 5th 6th

Voltage

2. Find the average of the voltage for the six batteries.

Part B

1. Use the graphing calculator to obtain a scatterplot of the data collected from measuring the batteries in a series. Sketch the scatterplot below. Scale and label axes appropriately.

2. What is the general shape of the graph? What parent function represents this shape?

3. Change parameters on the parent function to find a trend line to represent this function. What is the representative function?


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4. Using the trace feature of the graphing calculator, trace along the points on the scatterplot to collect data points to put in the table below.

# of batteries

x Voltage

y

1

2

3

4

5

6

5. What patterns do you observe in the voltage measurements as additional batteries are placed

in the circuit?

6. Use the graphing calculator to find the line of regression for the data points. Compare the trend line from #3 with the line of regression.

7. Use the derived representations to predict the voltage of the following number of batteries in series?

a. 7 batteries __________ b. 10 batteries __________

c. 20 batteries __________

d. 50 batteries __________

8. For the equation of the line y = ax + b, the “a” is called the ____________ of the line. The “b” is called the ____________ of the line. What does each represent in the experiment?


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9. How does the average of the batteries from Part A compare with the “a” value from the representative function?

10. Summarize the investigation. Include the following a. A description of the total voltage if batteries are in a series b. How to obtain a representative function for batteries in a series c. How the numeric and symbolic parts of the function are related to the batteries in the

series d. A sketch of batteries in a series connected to a load.


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Direct Variation (pp. 1 of 4) KEY

One type of linear function is a direct variation. Direct variation represents a proportional relationship. 1. The following are all equations of direct proportions. Graph the equations on the graphing

calculator. Sketch and label the graphs below. Describe the characteristics of the graphs. Answers will vary. Sample: They are all lines. They all go through the origin.

y = 3x

y = 0.5x

y = x

y = 0.3x

y = 1.75x

y = 3.25x

2. What are the characteristics of a proportional relationship?

Linear

Extend through the origin

Written as y = kx

Have the same constant of proportionality (k = y

x)

3. Give three examples of non-proportional relationships.

Answers will vary. Sample answers given.

y = x + 1

y = 3x – 2

y = x2

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y


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4. Compare and contrast representations of proportional situations by completing the chart.

Verbal Description Table Graph Function

The number of miles represented on a map is equal to 50 times the number of inches on the map.

x y 1 50 2 100 3 150 4 200 5 250 6 300

y = 50x

y varies directly as x, and when x is 8, y is 46.

x y 1 5.75 2 11.5 3 17.25 4 23 5 28.75 6 34.5

y = 5.75x

The number of calories burned is equal to 3.5 times the number of minutes exercising.

x y 1 3.5 2 7 3 10.5 4 14 5 17.5 6 21

y = 3. 5x

The total cost of an item at a store is the cost of the item plus the sales tax of 8.25%.

x y 1 1.08 2 2.17 3 3.25 4 4.33 5 5.41 6 6.50

y = 1.0825x

a. State at least three reasons the relationships all show a direct variation relationship. Use

examples from the different representations and the idea of proportionality to defend your reason. Answers will vary. Sample: The lines contain the point (0, 0). The ratio of y to x is a constant. Any two ordered pairs in the table will create a proportion. The equations are of the form y = kx. All are linear.


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5. Complete the table below by calculating the circumference and areas of the circles with the given radius. Round your answers to the nearest 0.01.

Radius (cm)

Circumference Ratio of

Circumference

to Radius Area

Ratio of Area to Radius

3 18.85 6.28 28.27 9.42

4 25.13 6.28 50.27 12.57

5 31.42 6.28 78.54 15.71

7 43.98 6.28 153.94 21.99

10 62.83 6.28 314.16 31.42

a. Graph the function of circumference versus radius and the function of area versus radius on the same coordinate plane. Label and scale the axes appropriately. Draw a straight line or smooth curve through each set of points. Sample graph


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b. Which relation(s) is a function? How do you know? Both relationships are functions. Circumference depends on radius and there is only one circumference for each given radius. Area depends on radius and there is only one area for each given radius.

c. Which relation(s) shows a direct variation? Explain your response. Circumference and radius. The graph is a line that goes through the origin. The ratio of y to x is a constant (2).

d. Which relation(s) does not show a direct variation? Explain why not. Area and radius. The graph is curved. The ratio of y to x is not constant.

e. Write the equation for the direct variation relationship. y = 6.28x

f. How is the equation related to how you calculated the dependent value? Answers will vary. Sample: Circumference was found by multiplying the radius times 2 or 6.28 is approximately equal to 2.

g. What is the constant of proportionality? What does the constant of proportionality represent? The constant of proportionality is 6.28, which is approximately 2.

h. If y varies directly with x, and y is 42 when x is 8, what direct variation could be used to

represent the situation? 21

4y x

i. If y is directly proportional to x and y = 8 when x = 10, what is the value of y when x = 15? y = 12

j. If y is directly proportional to x and the graph of the function contains the point (-16, 28), what is the value of y when x = 32? y = -56


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Direct Variation (pp. 1 of 4)

One type of linear function is a ________________________. Direct variation represents a ___________________ relationship.

1. The following are all equations of direct proportions. Graph the equations on the graphing

calculator. Sketch and label the graphs below. Describe the characteristics of the graphs.

y = 3x

y = 0.5x

y = x

y = 0.3x

y = 1.75x

y = 3.25x

2. What are the characteristics of a proportional relationship?

3. Give three examples of non-proportional relationships.

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

x

y


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4. Compare and contrast representations of proportional situations by completing the chart.

Verbal Description Table Graph Function

The number of miles represented on a map is equal to 50 times the number of inches on the map.

x y 1 50 2 3 4 200 5 250 6 300

y varies directly as x, and when x is 8, y is 46.

x y 1 5.75 2 3 17.25 4 5 28.75 6 34.5

The number of calories burned is equal to 3.5 times the number of minutes exercising.

x y 1 3.5 2 7 3 4 5 6 21

The total cost of an item at a store is the cost of the item plus the sales tax of 8.25%.

x y 1 1.08 2 3 4 4.33 5 5.41 6 6.50

State at least three reasons the relationships all show a direct variation relationship. Use examples from the different representations and the idea of proportionality to defend your reason.


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5. Complete the table below by calculating the circumference and areas of the circles with the given radius. Round your answers to the nearest 0.01.

Radius (cm)

Circumference

(cm)

Ratio of Circumference

to Radius

Area

(cm2)

Ratio of Area to Radius

3

4

5

7

10

a. Graph the function of circumference versus radius and the function of area versus radius on the same coordinate plane. Label and scale the axes appropriately. Draw a straight line or smooth curve through each set of points.


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b. Which relation(s) is a function? How do you know?

c. Which relation(s) shows a direct variation? Explain your response.

d. Which relation(s) does not show a direct variation? Explain why not.

e. Write the equation for the direct variation relationship.

f. How is the equation related to how you calculated the dependent value?

g. What is the constant of proportionality? What does the constant of proportionality represent?

h. If y varies directly with x, and y is 42 when x is 8, what direct variation could be used to

represent the situation?

i. If y is directly proportional to x and y = 8 when x = 10, what is the value of y when x = 15?

j. If y is directly proportional to x and the graph of the function contains the point (-16, 28),

what is the value of y when x = 32?


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Patterns Can Be Sweet (pp. 1 of 3) KEY Materials for the activity: Starburst or similar candies, colored toothpicks, chart paper, markers, colored pencils

Part A

1. Begin with four pieces of candy connected in a row with toothpicks. This is the first row. How many total candies are in this structure?

2. Add toothpicks and another four candies to make a second row. How many total candies are in this structure?

3. Continue the process through row five. 4. Complete the diagrams below through row five.

Diagrams will vary, but should show toothpicks and candies. Samples through three rows are shown below.


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Patterns Can Be Sweet (pp. 2 of 3) KEY

Part B Complete the table for the rectangles.

Row Number

Verbal Description Process Total Number of Candies

1 One row of 4 1(4) 4 2 Two rows of 4 2(4) 8 3 Three rows of 4 3(4) 12 4 Four rows of 4 4(4) 16 5 Five rows of 4 5(4) 20 9 Nine rows of 4 9(4) 36 x x rows of 4 x(4) 4x

Part C

1. How many total candies would be in row 9? 36 2. What is the algebraic generalization for row “x”? y = 4x

3. Sketch a graph on the coordinate plane below.

4. Is the relationship proportional? Explain your reasoning. Answers will vary. Sample: Yes, it passes through (0, 0), it is linear, it has a constant of proportion = 4, and it can be written in the form y = kx.


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Patterns Can Be Sweet (pp. 3 of 3) KEY

5. Enter the data into a graphing calculator, draw a scatter plot, and verify that the algebraic generalization fits the data. Compare the graph in the graphing calculator to the one on the coordinate plane. Justify your conclusions. See the graph. The algebraic function does fit the scatterplot.

6. How do the domain of the function and the domain for the problem situation compare? The domain and range of the function is all positive integers. The domain and range of the problem situation can only be whole numbers, {0, 1, 2, 3, 4, 5, …}.

7. Use the algebraic generalization to find the number of candies needed to complete the structure through row 50? Row 100? 200, 400

Part D Repeat Parts A, B and C comparing the relationship between number of toothpicks versus row number. Determine if the relationship is proportional. Display the results including a table, graph, equation, verbal description, and a prediction for number of toothpicks for row 50 and row 100. The relationship is not proportional. f(50) = 346, f(100) = 696


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Patterns Can Be Sweet (pp. 1 of 3) Materials for the activity: Starburst or similar candies, colored toothpicks, chart paper, markers, colored pencils

Part A

1. Begin with four pieces of candy connected in a row with toothpicks. This is the first row. How many total candies are in this structure?

2. Add toothpicks and another four candies to make a second row. How many total candies are in this structure?

3. Continue the process through row five. 4. Complete the diagrams below through row five.


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Patterns Can Be Sweet (pp. 2 of 3)

Part B Complete the table for the rectangles.

Row Number

Verbal Description Process Total Number

of Candies 1 2 3 4 5 9 x

Part C

1. How many total candies would be in row 9? 2. What is the algebraic generalization for row “x”?

3. Sketch a graph on the coordinate plane below.

4. Is the relationship proportional? Explain your reasoning.


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Patterns Can Be Sweet (pp. 3 of 3)

5. Enter the data into a graphing calculator, draw a scatter plot, and verify that the algebraic generalization fits the data. Compare the graph in the graphing calculator to the one on the coordinate plane. Justify your conclusions.

6. How do the domain of the function and the domain for the problem situation compare?

7. Use the algebraic generalization to find the number of candies needed to complete the structure through row 50? Row 100?

Part D Repeat Parts A, B and C comparing the relationship between number of toothpicks versus row number. Determine if the relationship is proportional. Display the results including a table, graph, equation, verbal description, and a prediction for number of toothpicks for row 50 and row 100.


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Linear Inequalities (pp. 1 of 4) KEY

The solution to a linear inequality in two variables involves a line (forms the boundary of the region) and a shaded region (describes the solution area of the inequality).

1. Be sure the inequality is in slope-intercept form. 2. Graph the line using the y-intercept and slope.

a. Solid line for or b. Dotted line for or

3. Determine which side of the line to shade by selecting a point to test on one side of the line.

Plug it into the original equation. Hint: (0, 0) is an easy point to test if it is not on the line. a. If the final statement is true, shade the region containing the point b. If the final statement is false, shade the region not containing the point.

4. Use one of the following two methods with the graphing calculator to verify solutions. a. Solve the inequality for y, put the function in y1. Scroll to the left and change to greater

than (shaded above) or less than (shaded below). Graph the function. b. Place the left side of the inequality in y1 and the right side in y2. Analyze the table to

see where the inequality statement is true. Examples:

1. 2 3y x 2. 2

23

xy

-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10

x

y

-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10

3. Which of the following ordered pairs are in the solution set of 2 3y x ?

a. (0, 0) yes b. (-1, 0) yes c. (2, 8) no


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Linear Inequalities (pp. 2 of 4) KEY Guided Practice Plot the graph of the linear inequality. Verify with the calculator. Give three points in the solution set of each. Answers will vary on the points. They must either be on the line if it is solid or from the shaded region of the graph.

1. 2 4y x 2. 3

35

y x

-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10y

x -10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10

3. 5y 4. 6x

-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10

x

y

-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10


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Practice Problems Show the solution to the linear inequalities by plotting the graphs. Label all lines with the y = mx + b form of the equation. Verify all solutions with the graphing calculator. Give three points in the solution set of each. Answers will vary on the points. They must either be on the line if it is solid or from the shaded region of the graph.

1. 3

24

y x

2. 3

45

y x

3. 3

45

y x

4.

3

14

xy

5. 2 3y x

6. 5y

7. 5x

8. 3y


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Unit: 03 Lesson: 01

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Linear Inequalities (pp. 1 of 4)

The solution to a linear inequality in two variables involves a line (forms the boundary of the region) and a shaded region (describes the solution area of the inequality).

1. Be sure the inequality is in slope-intercept form. 2. Graph the line using the y-intercept and slope.

a. Solid line for or b. Dotted line for or

3. Determine which side of the line to shade by selecting a point to test on one side of the line.

Plug it into the original equation. Hint: (0, 0) is an easy point to test if it is not on the line. a. If the final statement is true, shade the region containing the point b. If the final statement is false, shade the region not containing the point.

4. Use one of the following two methods with the graphing calculator to verify solutions. a. Solve the inequality for y, put the function in y1. Scroll to the left and change to greater

than (shaded above) or less than (shaded below). Graph the function. b. Place the left side of the inequality in y1 and the right side in y2. Analyze the table to

see where the inequality statement is true. Examples:

1. 2 3y x 2.

2

23

xy

-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10

x

y

-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10

x

y

3. Which of the following ordered pairs are in the solution set of 2 3y x ?

a. (0, 0) b. (-1, 0) c. (2, 8)


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Linear Inequalities (pp. 2 of 4) Guided Practice Plot the graph of the linear inequality. Verify with the calculator. Give three points in the solution set of each.

1. 2 4y x 2. 3

35

y x

-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10y

x -10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10

3. 5y 4. 6x

-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10

x

y

-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10


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Linear Inequalities (pp. 3 of 4) Practice Problems Show the solution to the linear inequalities by plotting the graphs. Label all lines with the y = mx + b form of the equation. Verify all solutions with the graphing calculator. Give three points in the solution set of each.

1. 3

24

y x

2. 3

45

y x

3. 3

45

y x

4.

3

14

xy

5. 2 3y x

6. 5y

7. 5x

8. 3y


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Linear Inequalities (pp. 4 of 4)


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Images of Inequalities Cards (pp. 1 of 2)

A

B

C

D

E

F

G

H

I


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Images of Inequalities Cards (pp. 2 of 2)

J

K

L

M

N

O

P


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Images of Inequalities Recording Sheet (pp. 1 of 2) KEY

Jathan had just finished his inequality sketch graphs for algebra class, when he realized he had forgotten to number the graphs. Use your knowledge of linear inequalities to help Jathan match his sketch graphs with the corresponding equations below.

Problem Letter of

Corresponding Graph Sketch of Graph

1. 1

12

y x B See graphs.

2. 4 2y x J See graphs.

3. y x H See graphs.

4. 2x N See graphs.

5. 3 3x y C See graphs.

6. 2 0x y P See graphs.

7. 3y D See graphs.


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Images of Inequalities Recording Sheet (pp. 2 of 2) KEY

8. 2 4x y A See graphs.

9. 2x y I See graphs.

10. 3y K See graphs.

11. 3 0x y E See graphs.

12. 2 2x y O See graphs.

13. 2 2x y F See graphs.

14. 4 2x y L See graphs.

15. 2x G See graphs.

16. 2 3 6x y M See graphs.


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Images of Inequalities Recording Sheet (pp. 1 of 2)

Jathan had just finished his inequality sketch graphs for algebra class, when he realized he had forgotten to number the graphs. Use your knowledge of linear inequalities to help Jathan match his sketch graphs with the corresponding equations below.

Problem Letter of

Corresponding Graph

Sketch of Graph

1. 1

12

y x

2. 4 2y x

3. y x

4. 2x

5. 3 3x y

6. 2 0x y

7. 3y


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Images of Inequalities Recording Sheet (pp. 2 of 2)

8. 2 4x y

9. 2x y

10. 3y

11. 3 0x y

12. 2 2x y

13. 2 2x y

14. 4 2x y

15. 2x

16. 2 3 6x y


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Functioning on a High Rise (pp. 1 of 4) KEY

1. Builders Inc. has been contracted to build a high rise in downtown El Paso from ground level.

The following function represents the number of stories (y) Builders, Inc. can build on the new high rise according to the number of weeks (x) they have been working.

y = 2x

a. What parent function best represents this function?

Linear parent function

b. How will the 2 affect the parent function?

Steeper graph, as x increases by 1, y increases by 2 c. Build a table of four values to represent Builders, Inc.

Tables will vary. Sample table is given. x (weeks) y (stories)

1 2 4 8 8 16 12 24

d. Sketch the graph of the function in the coordinate plane. Be sure to label and scale the

axes. Represent the relationship in function notation. ( ) 2b x x

e. How do the domain of the function and the domain for the problem situation compare? The domain of the function would be all real numbers. The domain in the problem situation would be weeks of construction are greater than or equal to zero. 0x

f. Does this function represent a direct proportionality? Explain your reasoning? Yes, y varies directly as x. It is a linear function. It goes through the origin. It has a constant of variation k = y/x, k = 2. It is written in the form y = kx.


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g. Explain how to determine the slope from the given function. In slope-intercept form, the slope is the coefficient of the x-term. m = 2

h. Explain how to determine the slope from the table of values.

From a table of values the slope is the change in y over the change in x. 4/2 = 2

i. Explain how to determine the slope from the graph. From a graph the slope is the vertical change over the horizontal change. 2/1 = 2

j. What are the units on the slope and what is the meaning of the slope in the problem

situation? Stories/number of weeks In the problem slope is the number of stories that can be completed in one week. Builders, Inc. can complete two stories per week.

k. What is the y-intercept? What does it represent?

The y-intercept is (0, 0). When construction begins, there are no stories in the building.

l. What is the x-intercept? What does it represent? The x-intercept is (0, 0). There are no stories at time zero when construction begins.

2. Construction Corp. has been contracted to increase the number of stories on a six-story high rise in downtown Amarillo. The following function represents the number of stories (y) Construction Corp. can build on the high rise according to the number of weeks (x) they have been working.

y – 0.75x = 6

a. Transform the function to y = mx + b form. How could this be expressed in function notation? y = 0.75x + 6 c(x) = 0.75x + 6

b. How will the 0.75 affect the parent function?

Less steep graph, as x increases by 1, y increases by 0.75 or ¾.

c. Build a table of four values to represent Construction Corp. Tables will vary. Sample table is given. x (weeks) y (stories)

1 6.75 4 9 8 12 12 15


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d. Sketch the graph of the function in the coordinate plane. Be sure to label and scale the axes.

e. What is the slope for this function? m = ¾ or 0.75

f. What are the units on the slope and what is the meaning of the slope in the problem situation? Stories/number of weeks In the problem, slope is the number of stories that can be completed in one week. Construction Corp. can complete 0.75 stories in one week or three stories every four weeks.

g. What is the y-intercept? What does it represent?

The y-intercept is (0, 6). This represents the fact that when building begins at time zero, there are already 6 stories in the building.

3. After 16 weeks, how many floors would each company have completed?

b(16) = 2(16) c(16) = 0.75(16) + 6 b(16) = 32 stories c(16) = 18 stories

4. If both buildings were to be built to 24 stories, how long would it take for each company to complete the job?

b(12) = 24 stories c(24) = 24 stories Builders, Inc. would take 12 weeks to build the building to 24 stories. Construction Corp. would take 24 weeks to build the building to 24 stories.


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5. If Builders, Inc. was working on a high rise that already had four stories, how would it affect the function rule and the graph? The function rule would be y = 2x + 4. The graph would have the same slope but shift up by four. The y-intercept would be (0, 4).

6. If Construction Corp. was able to hire more workers and increased their rate from 0.75 stories per week to 1.5 stories per week, how would it affect the function rule and the graph? The function rule would be y = 1.5x + 6. The graph would have the same y-intercept of (0, 6), but the slope would be steeper. As x increased by 1, y will increase by 1.5.

7. Builders, Inc. has the fastest construction crews in El Paso. Sketch an inequality on the coordinate plane below that shows the region all other builders would fall into if they could build a number of stories less than Builders, Inc. per week. (Hint: 2y x )

8. Construction Corp. does the best construction work in Amarillo, but they are the slowest

construction crew. Sketch an inequality on the coordinate plane below that shows the region all other builders would fall into if they could build a number of stories greater than or equal to Construction Corp. per week. (Hint: 0.75 6y x )


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Functioning on a High Rise (pp. 1 of 4)

1. Builders Inc. has been contracted to build a high rise in downtown El Paso from ground level. The following function represents the number of stories (y) Builders, Inc. can build on the new high rise according to the number of weeks (x) they have been working.

y = 2x

a. What parent function best represents this function? b. How will the 2 affect the parent function? c. Build a table of four values to represent Builders, Inc.

d. Sketch the graph of the function in the coordinate plane. Be sure to label and scale the axes. Represent the relationship in function notation.

e. How do the domain of the function and the domain for the problem situation compare?

f. Does this function represent a direct proportionality? Explain your reasoning?


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g. Explain how to determine the slope from the given function.

h. Explain how to determine the slope from the table of values.

i. Explain how to determine the slope from the graph.

j. What are the units on the slope and what is the meaning of the slope in the problem situation?

k. What is the y-intercept? What does it represent?

l. What is the x-intercept? What does it represent?

2. Construction Corp. has been contracted to increase the number of stories on a six-story high rise in downtown Amarillo. The following function represents the number of stories (y) Construction Corp. can build on the high rise according to the number of weeks (x) they have been working.

y – 0.75x = 6

a. Transform the function to y = mx + b form. How could this be expressed in function

notation?

b. How will the 0.75 affect the parent function?

c. Build a table of four values to represent Construction Corp.


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d. Sketch the graph of the function in the coordinate plane. Be sure to label and scale the axes.

e. What is the slope for this function?

f. What are the units on the slope and what is the meaning of the slope in the problem situation?

g. What is the y-intercept? What does it represent?

3. After 16 weeks, how many floors would each company have completed?

4. If both buildings were to be built to 24 stories, how long would it take for each company to complete the job?


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5. If Builders Inc. was working on a high rise that already had four stories, how would it affect the function rule and the graph?

6. If Construction Corp. was able to hire more workers and increased their rate from 0.75 stories per week to 1.5 stories per week, how would it affect the function rule and the graph?

7. Builders, Inc. has the fastest construction crews in El Paso. Sketch an inequality on the coordinate plane below that shows the region all other builders would fall into if they could build a number of stories less than Builders, Inc. per week. (Hint: 2y x )

8. Construction Corp. does the best construction work in Amarillo, but they are the slowest

construction crew. Sketch an inequality on the coordinate plane below that shows the region all other builders would fall into if they could build a number of stories greater than or equal to Construction Corp. per week. (Hint: 0.75 6y x )

Graph #7 Graph #8

Documents

Investigating Linear Functions - Weeblycarengarza.weebly.com/uploads/3/7/7/4/37742877/unit3lesson1.pdf · Algebra 1 HS Mathematics Unit: 03 Lesson: 01 Suggested Duration: 10 days