Algebra II Foundations

Transcription

1 AIIF Algebra II Foundations Linear Functions Teacher Manual

2 Table of Contents Lesson Page Lesson 1: Plotting Points...1 Lesson 2: Linear Equations in Slope-Intercept Form...18 Lesson 3 Applications of Linear Functions in Slope-Intercept Form...43 Lesson 4: Linear Function Notation...63 Lesson : Other Forms of Linear Functions...79 Lesson 6: What is a Function? Lesson 7: Linear Regression Lesson 8: Linear Inequalities...19 Assessment CREDITS Author: Contributors: Graphic Design: Dennis Goette and Dann Jones Robert Balfanz, Doroth Barr, Leonard Bequiraj, Stan Bogart, Robert Bosco, Carlos Burke, Lorenzo Haward, Vicki Hill, Winnie Horan, Donald Johnson, Ka Johnson, Karen Kelleher, Kwan Lange, Dennis Leah, Song-Yi Lee, Hsin-Jung Lin, Gu Lucas, Ira Lunsk, Sandra McLean, Hemant Mishra, Glenn Moore, Linda Muskauski, Trac Morrison, Jennifer Prescott, Gerald Porter, Steve Rigefsk, Ken Rucker, Stephanie Sawer, Dawne Spangler, Fred Vincent, Maria Waltemeer, Tedd Wieland Gregg M. Howell Copright 2009, The Johns Hopkins Universit, on behalf of the Center for Social Organization of Schools. All Rights Reserved. CENTER FOR SOCIAL ORGANIZATION OF SCHOOLS Johns Hopkins Universit 3003 N. Charles Street Suite 200 Baltimore, MD fa All rights reserved. Student assessments, Cutout objects, and transparencies ma be duplicated for classroom use onl; the number is not to eceed the number of students in each class. No other part of this document ma be reproduced, in an form or b an means, without permission in writing from the publisher. Transition to Advanced Mathematics contains Internet website IP (Internet Protocol) addresses. At the time this manual was printed, the website addresses were checked for both validit and content as it relates to the manual s corresponding topic. The Johns Hopkins Universit, and its licensors is not responsible for an changes in content, IP addresses changes, pop advertisements, or redirects. It is further recommended that teachers confirm the validit of the listed addresses if the intend to share an address with students.

3 Linear Functions Planning Document AIIF Page i Planning Document: Linear Functions Overview Unit 2 is based on linear functions. The linear function lessons include: Plotting Points Linear equations and the slope-intercept form Applications of linear functions Linear function notation Other forms of linear functions Absolute value functions Defining linear functions and the vertical line test Linear regression Linear inequalities The total suggested das for the unit is 12 to 14.. Adjustments ma be needed based on student performance during the unit and time available until the end of the semester. Vocabular Coordinate plane Cartesian Coordinate Plane -value -value Ordered pair Reflect Scatter plots Connected scatter plots Origin Intersection Vertical Horizontal -ais -ais Quadrant I Quadrant II Quadrant III Quadrant IV Number line Positive Negative Table of ordered pairs Reflection -minimum -minimum -maimum -maimum -scale -scale Original object Function Input value Output value Scenario Rule in words Rule as an equation Graph Δ Δ Slope Slope of a line Rise Run -intercept Slope intercept Parallel lines Constant Constant rate Term number Term value Input Output Domain Range Function Function notation Standard form of a linear equation Point-slope form of a linear equation Vertical line Horizontal line Parallel slope Perpendicular slope Slope-intercept form of a linear equation Point Vertical line test Regression Least sum Correlation coefficient Line of best fit Line of least squares Least-squares line Correlation Positive correlation Negative correlation No correlation Sum Summation Summation notation Linear inequalit Less than Less than or equal to Greater than Greater than or equal to Inequalit Solution Solution region Sstem of inequalities

4 AIIF Page ii Student journal Setting the Stage transparencies Dr-erase boards Markers and erasers Scissors Materials List Chart paper Graphing calculators Calculator view screen Blank transparencies Paper Lesson specific transparencies Overhead projector Construction paper Poster paper

5 Linear Functions Planning Document AIIF Page iii The following table contains lesson name, timeline, summar of concepts covered, and the Essential Question(s) for each lesson. Lesson Timeline Concepts Covered Essential Question(s) Plotting Points 1 Da Coordinate plane and aes Linear Equations in Slope-Intercept Form Applications of Linear Functions in Slope- Intercept Form Linear Function Notation Other Forms of Linear Functions Identif the four quadrants Graph ordered pairs Create a connected scatter plot on a grid and using technolog 2 to 3 Das Linear functions represented five was: o Scenario o Rule in words o Rule as an equation o Table of ordered pairs o Graph Linear equations in slope-intercept form Applications of linear equations in slopeintercept form Graphic organizer related to lesson vocabular 1 to 2 Das Determine linear equations from real-world scenarios Determine linear equations given: o Number pattern o Polgon dot pattern Use linear equations to determine specific term value for various scenarios and patterns 1 Da Write equations using function notation For linear functions, determine: o Domain o Range o Independent variable o Dependent variable 3 Das Standard form of a linear equation Point-slope form of a linear equation Determine linear equations that match real- How can ou represent a table of ordered pairs visuall? How can a linear function represent a realworld scenario? How does using a linear equation help answer questions related to linear patterns or scenarios? How is function notation useful to represent linear function scenarios? What are the advantages of using linear equations to model linear functions and real world applications?

6 AIIF Page iv What is a Function? 1 to 1. Das world linear function scenarios Determine linear equations for horizontal and vertical lines Determine the equation of a line parallel or perpendicular to a second line and passing through a specific point Use the vertical line test to determine if a graph represents a function Graph various non-linear functions using a graphing calculator Create or match graphs that model real-world scenario. For eample, distance-time graphs. Linear Regression 2 Das Estimate the line of best fit for a set of data and then determine the slope-intercept form of the equation Use a formula to determine the correlation coefficient from a set of data Use a formula to determine the equation for the line of best fit Use the graphing calculator to determine the correlation coefficient and the equation for the line of best fit Make predictions based on the line of best fit Determine if a set of data has a linear trend based upon visual observations of the graph of data and the correlation coefficient Linear Inequalities 1 Da Graph linear inequalities b hand and using a graphing calculator Determine the linear equalit from the graphed solution Graph the solution to a sstem of linear inequalities Appl linear inequalities to model and solve realworld problems involving two or more variables What characteristic of a graph confirms that it can represent a function? How can linear regression be used to predict values and in real world applications? How can the properties of inequalities be used to create a linear inequalit in slope-intercept form? How can shading and dashed lines represent the solution to a linear inequalit?

7 Linear Functions Lesson 1: Plotting Points AIIF Page 1 Lesson 1: Plotting Points Objectives Students will be able to graph a table of ordered pairs as a scatter plot and connected scatter plot on a coordinate grid both b hand and with technolog. Essential Questions How can ou represent a table of ordered pairs visuall? Tools Student Journal Dr erase boards, markers, erasers Graph paper Graphing calculator Warm Up Problems of the Da This lesson is to be used as a review for plotting and graphing ordered pairs and matching scatter plots. Depending on our students, ou ma decide to skip this lesson or use multiple das to implement a strong foundation. Number of Das for Lesson 1 Da Vocabular Coordinate plane Cartesian Coordinate Plane -value -value ordered pair reflect scatter plots connected scatter plots origin intersection vertical horizontal -ais -ais Quadrant I Quadrant II Quadrant III Quadrant IV number Line positive negative table of ordered pairs reflection -minimum -minimum -maimum -maimum -scale -scale original object Notes At the end of each lesson there are Practice Eercises, Outcome Sentences, and a short quiz. Teachers can use these tools as needed and as time allows. Prior to teaching, ou will need to prepare transparencies from the master hard copies supplied in this manual.

8 AIIF Page 2 Teacher Reference Setting the Stage Before this lesson, ou ma want to ask the students to bring in maps, or ou ma have maps donated b a local business. You will need a class set of maps with at least one per group. If this is not possible then check with the Social Science Department and obtain some pull down wall maps. You could also print out maps from the Internet. The maps will be used to show the class the coordinate sstem used b maps to locate places. You could also talk about other features of maps such as scale, orientation, smbols, boundaries (count and state), rivers, lakes, and major roadwas. Use a grouping strateg of our choice to divide the class into groups of four. Two possible strategies are as follows. A heterogeneous strateg is to use our assessment data from Unit 1 to divide our class into groups of four b creating groups that have one high level student, two medium level students, and one below medium level student. Another strateg is to randoml pass out one plaing card to each student and then ask each student to locate the three other students with matching cards to form a group of four. For eample, all the number four cards would form a group. It ma be valuable to have students sta in these groups for the remainder of the unit to develop a team relationship. There ma be occasions that ou need to have the students work with others outside these groups to give some variet. Make sure each group has a map with a grid tpe method to locate points. Give the class two to three minutes to discuss and list everthing the know about maps and reading locations on a map. You ma want to list some questions on the board such as: What methods do the map makers use to distinguish the various components of a map such as roads, boundaries, cities, etc.? What characteristics do the maps have in common and what characteristics are different? How could ou use the map to describe a location? What methods do the map makers use to describe the components? Hint: Legend. How are maps used in video games? What advantage does using a map have over using a list of directions to get from one place to another? How is a map similar and different to a GPS (Global Positioning Sstem)? After the time is up, have groups volunteer to share their list with the class. Ask a volunteer to record the responses on the board. Displa the Setting the Stage Transparenc or have students use their group map. Note: You will need to prepare a transparenc supplied as a hard cop version in this teacher manual. Give the students about two minutes in their groups to locate as man places as possible b using coordinates listing the row first and column second, such as Mt. Washington is at H14. Have volunteers give the coordinates of the places on the maps. The coordinates for some places will var depending on what the student ma think is the closest letter or number. For eample, Mt. Washington could have coordinates of either H14 or I14. The locations of the various places on the Setting the Stage Transparenc could be: Berlin: G14 Laconia: N14 Claremont: O9 Keene: R10 Concord: P13 Rochester: O1 Dover: P1 Portsmouth: Q17 Eeter: R16 Manchester: R13 Nashua: S13 Salem: S14 North East Corner of NH: B1 South West Corner of NH: S9

9 Linear Functions Lesson 1: Plotting Points AIIF Page 3 Setting the Stage Transparenc A B B C D E F G H I J K L M N O P Q R S T

10 AIIF Page 4 Teacher Reference Activit 1 Make sure each group has two dr erase boards and markers. Displa the Activit 1 Transparenc 1 and give the class three to five minutes to represent each of the words on the grid side of the dr erase board and to describe the words on the blank side of the dr erase boards. As the students work, walk around the room and note a few of the responses that ou would like some students to later share with the class. Have the students share. You ma want to point out the words that most of the students could locate or define and the words that no one could locate or define. This is a great time to review words used in previous math classes. You ma want to displa Activit 1 Transparenc 2 and ask students to represent the words and definitions that most of the students had trouble locating. Have the students agree on a description of the four terms at the bottom of Activit 1 Transparenc 1. Ask the class what some of the similarities and differences are between the Cartesian Coordinate plane and the map grid. Obtain the following information from the class with a guided discussion: Each point on the coordinate plane represents an -value and a -value, much like a place on a map. Together these values are called an ordered pair. The first number in an ordered pair is the -value (horizontal coordinate) and the second number is the -value (vertical coordinate). An ordered pair can be represented b numbers in parenthesis such as (-value, -value) or in a table of values with the left column representing -values and the right column representing -values. For eample, using Activit 1 Transparenc 2, plot point A at the ordered pair ( 8, 9), then have a volunteer, or volunteers, plot the points B (3, 4), C ( 3, ), and D (0, 3) on the coordinate plane. Ask the students guiding questions such as, In which quadrant are the -values positive and the -values negative? and On which ais are the -values alwas equal to zero? Have a volunteer reflect point A across the -ais and label this point E. Ask the class, What are the coordinates of point E? Have another volunteer reflect point A across the -ais and label this point F. Ask the class, What are the coordinates of point F? Inform the class that the unique characteristic of one point on a coordinate plane is that it can represent two values: namel an -value and a -value. Now, ask the class, Can all combinations of two numbers be represented on the coordinate plane? Make sure the eplain their answer. Theoreticall, all combinations of two real numbers can be represented on a coordinate plane. However, the coordinate plane theoreticall etends infinitel in all directions, so students will probabl claim that reall large numbers would be off the page. For the net part of this activit students will draw connected scatter plots. Have the students work as pairs in their groups of four to complete Eercises 1 through 7. After the class has completed the eercises, bring the class together and have students share their results on the board or overhead. Before students work on Eercises 9 through 12, introduce the method of plotting points with a graphing calculator b guiding students through Eercise 8. For our particular classroom graphing calculator, go through the graphing window parameters. The screen shots displaed here are from a TI-84 Plus graphing calculator. Adjust our instructions based on the particular classroom graphing calculator supplied at our school.

11 Linear Functions Lesson 1: Plotting Points AIIF Page Remember to have the students clear their calculator before the begin. This will clear an equations in Y= editor, all lists, and prevent problems and errors students ma have and not initiall understand. To clear the TI-84 Plus, have the students press these kes: 2ND, +, 7. Use the view screen and guide students through the method. Pair the students appropriatel to facilitate the use of technolog. You will want to become familiar with plotting points from a list on the graphing calculator before doing this with the students. Students should be able to enter ordered pairs into a list, plot from a list, and change the dimensions of the graph. For Eercise 8, the solutions to a, b, and c will var depending upon the brand of graphing calculator. Here are some screens shots from a TI-84 Plus. After students have completed Eercises 9 through 12, have a discussion about plotting with a graphing calculator. You ma want to point out that even though on Eercises 10 through 12 the ordered pairs didn t change, adjusting the window parameters gave a different impression of how the graph looked. The idea is that students begin to realize how important it is to understand the settings of the window parameters and the effects on displaing graphs. Before students continue with the net lesson, make sure the understand how to use the graphing calculator to graph a set of points. You ma want to discuss with the students the method of changing the unit size of the and -ais on a graphing calculator b changing the -scale and -scale. Knowing how to do this will help students read graphs easier. For eample, when plotting points that have a minimum -value of 0 and maimum of 1000 it is helpful to set the -scale to 100.

12 AIIF Page 6 Activit 1 Transparenc 1 Locate and label the following on the coordinate grid on our dr-erase board. Origin Vertical -ais Quadrant I Quadrant II -value Ordered Pair Negative Intersection Horizontal -ais Quadrant II Quadrant IV -value Positive Number Line

13 Linear Functions Lesson 1: Plotting Points AIIF Page 7 Activit 1 Transparenc 2

14 AIIF Page 8 SJ Page 1 Activit 1 In this activit, ou will draw connected scatter plots. The following table represents a set of ordered pairs. The first ordered pair is ( 2,3 ) where 2 is the -value and 3 is the -value. -value -value Legend has it that René Descartes ( ) developed the Cartesian Coordinate Plane as he lie in bed sick. He noticed a fl on the ceiling and determined that the position of the fl crawling on the ceiling could be recorded as the distance from two adjacent walls. Refer to the coordinate plane below for Eercises 1 through The first three ordered pairs from the table are plotted on the graph and connected in order. Complete the graph b plotting the remaining ordered pairs and connecting them in order What object did ou create? Duck

15 Linear Functions Lesson 1: Plotting Points 3. Wh was the first ordered pair in the table the same as the last ordered pair in the table? AIIF Page 9 SJ Page 2 Sample response: To finish the duck ou had to go back to the first point. 4. If the object were moved to the left units, what effect would it have on the table of -values and - values? Sample response: Moving the object to the left five units would not affect the -values. Moving the object to the left five units would reduce all the -values in the original table b.. If the original object were moved up 3 units, what effect would it have on the table of -values and - values? Sample response: Moving the object up three units would not affect the -values. Moving the object up three units would increase all the -values in the original table b Draw a sketch of the object reflected over the -ais on the coordinate plane in Eercise 1. (Hint: Trace the object and then cut out the trace of the object to reflect it across the -ais.) Complete a new table of ordered pairs that represents this new object. What is the relationship between this new table of ordered pairs and the original table? See the object in the Quadrant II of the coordinate plane in Eercise 1. Sample response: All the - values staed the same but all the -values became negative. -value -value Draw a sketch of the object reflected over the -ais on the coordinate plane in Eercise 1. What effect does the reflection have on the table of -values and -values? See the object in Quadrant IV of the coordinate plane in Eercise 1. Sample response: All the -values staed the same but all the -values became negative.

16 AIIF Page 10 SJ Page 3 8. Follow the directions as our teacher eplains how to plot ordered pairs with a calculator. Input ordered pairs into a table of ordered pairs. -values -values Set the dimensions of the graph to: -minimum = 0 -maimum = 1 -minimum = 0 -maimum = 10 Plot the ordered pairs as a connected scatter plot. a. Write the steps needed to enter a table of ordered pairs in a graphing calculator. Make sure students record the steps needed to enter a table of ordered pairs into their graphing calculator. b. Write the steps needed to set the dimensions of the graph. Make sure students record the steps needed to set the dimensions. c. Write the steps needed to plot the ordered pairs as a connected scatter plot. Make sure students record the steps needed to plot the ordered pairs.

17 Linear Functions Lesson 1: Plotting Points AIIF Page 11 SJ Page 4 9. Plot a connected scatter plot of the table below onto a graphing calculator. The table of ordered pairs is given below. Set the dimensions of the graph to an -minimum of 0, an -maimum of 1, a - minimum of 0, and a -maimum of 10. -value -value This is the same object that was created in Eercise Change the -maimum to 30 and graph. Note the change in the connected scatter plot. Change the -maimum to 60 and displa the graph. Note the change in the scatter plot. Eplain what happened to the scatter plot as the -maimum increased. Sample response: The picture shrunk in width but staed the same height. 11. Change the -maimum back to 1 and displa the graph. Change the -maimum to 10 and graph. Note the change in the scatter plot. Change the -maimum to and graph. Note the change in the scatter plot. Eplain what happened to the scatter plot as the -maimum decreased. Sample response: The picture increased in width but staed the same height. With the -maimum set at the picture increased in width beond the screen. 12. Set the -maimum back to 1. What might ou do to make the duck appear shorter? Sample response: If ou change the -maimum to 20 the duck will appear to be shorter. Note: You ma also want to show students about changing the and -scale to change the unit size on the and -aes.

18 AIIF Page 12 SJ Page Practice Eercises 1. Enter the following ordered pairs into the graphing calculator and plot a connected scatter plot of the object. Make sure the dimensions of the graph have an -minimum of 10, an -maimum of 10, a - minimum of 6., and a -maimum of 6.. -value -value We will refer to this as the original object. Determine the following features of the original object. a. What is the shape of the object? The object should be an arrow. b. Which direction is it pointing? The object is an arrow pointing to the right. c. What quadrant is it in? The object is in Quadrant I. d. Multipl all the -values of the original object b 1. Plot the new object. Describe what happened to the object. Sample Response: The object was reflected through the -ais. It is now in Quadrant II. e. Multipl all the -values of the original object b 1. Plot the new object. Describe what happened to the object. Sample Response: The object was reflected through both the -ais and -ais. It is now in Quadrant III.

19 Linear Functions Lesson 1: Plotting Points AIIF Page Input the table of ordered pairs into a graphing calculator. SJ Page 6 -values -values Set the dimensions of the graph to: -minimum = 0 -maimum = 10 -minimum = 0 -maimum = 10 Plot the ordered pairs as a connected scatter plot. a. What is the shape of the object? Draw the shape. Answers ma var. A sample response might be: "The object is a bo." b. Change the maimum to 20. What happens to the shape? Answers ma var. A sample response might be: "The bo is half as wide as it was before." c. Change the maimum back to 10. Change the maimum to 20. What happens to the shape? Answers ma var. A sample response might be: "The bo is half as tall as it was before."

20 AIIF Page 14 SJ Page 7 d. Change the maimum back to 10. Multipl the values b 2. What happens to the shape? Does the shape fit on the screen? Answers ma var. A sample response might be: "The bo is two times wider than it was before. The shape does not fit on the screen." e. Multipl the values b 2. What happens to the shape? Does the shape fit on the screen? Answers ma var. A sample response might be: "The bo is two times taller than it was before. The shape does not fit on the screen." f. Compare and eplain the differences between changing the window parameters compared to multipling the values and values b a constant value greater than one. Answers ma var. A sample response might be: "Increasing the window parameters makes the shape appear smaller while multipling the values or values b a constant value greater than one makes the shape larger." g. How could ou change the window parameters to make the shape appear larger? Answers ma var. A sample response might be: "If the window parameters are made smaller the shape would appear larger." h. How could ou change the values and the values to make the shape smaller? Answers ma var. A sample response might be: "If we multiplied the values and the values b a constant value less than one then the shape could be made smaller."

21 Linear Functions Lesson 1: Plotting Points AIIF Page 1 Outcome Sentences SJ Page 8 To reflect a shape about the ais To reflect a shape about the ais To make a shape larger To make a shape smaller To make a shape appear larger or smaller I need help with I would like to find out more about

22 AIIF Page 16 Teacher Reference Lesson 1 Quiz Answers 1. Answers will var. A sample table might be: -values -values Answers ma var. A sample response might be: "The last coordinates in the table must be the same as the first to complete drawing the square." 3. Sample response: To draw the square in the third quadrant, both the values and the values need to be multiplied b a Sample response: To make the square appear larger ou would make the window parameters for maimum and maimum smaller than the original values.. Sample response: To make the square appear smaller ou would make the window parameters for maimum and maimum larger than the original values.

23 Linear Functions Lesson 1: Plotting Points AIIF Page 17 Lesson 1 Quiz Name: 1. Fill in the table below so that a connected scatter plot of the ordered pairs would make a square in the first quadrant with sides 4 units or more in length. -values -values 2. Wh must the last coordinates in the table be the same as the first? 3. What must ou multipl the values and the values b so that the square appears in the third quadrant? 4. How can ou make the square appear larger without changing the values and the values?. How can ou make the square appear smaller without changing the values and the values?

24 AIIF Page 18 Lesson 2: Linear Equations in Slope-Intercept Form Objectives The students will be able to represent a linear function with five different representations: scenario, rule in words, rule as an equation, table of ordered pairs, and graph. The students will be able to determine a linear equation in slope-intercept form given a scenario, table of ordered pairs, or graph of a linear function. Students will be able to create a graphic organizer that represents the vocabular related to linear functions. Essential Questions How can a linear function represent a real-world scenario? Tools Student Journal Dr erase boards, markers, erasers Graph paper (optional) Graphing calculator Calculator viewscreen Warm Up Problems of the Da Number of Das for Lesson 2 to 3 Das A suggestion is to complete Activit 1 and Practice Eercises 1 through on the first and second da, then complete Activit 2, Practice Eercises 6 through 1, and quiz on the third da. Vocabular function input value output value scenario rule in words rule as an equation table of ordered pairs -value -value ordered pairs graph -minimum -minimum -maimum -maimum -scale -scale slope Δ Δ slope of a line rise run -intercept slope-intercept parallel lines

25 Linear Functions Lesson 2: Linear Equations in Slope-Intercept Form AIIF Page 19 Setting the Stage Have students work in their groups of four to complete this Setting the Stage. Place the Setting the Stage transparenc on the overhead. Note: You will need to prepare a transparenc supplied as a hard cop version in this teacher manual. Discuss with students the following items related to both graphs on the transparenc. Lines are often used to represent trends in real-world data. Statisticians, engineers, weather forecasters, investment brokers, and other professionals represent data with lines when the can so that the determine approimate answers to trends with linear functions and linear equations. During this unit ou will investigate linear functions and linear equations so that b the end of the unit ou too will use lines and linear functions to represent data. Note: The goal of the transparenc is to give the students a little contet of wh the are about to stud linear functions. It is not meant to be used to actuall fit a line to data at this time. Fitting lines to data will be developed in a later lesson. You ma want to find and displa other real-world linear trend scatter plots from news papers, journals, magazines, or on the Internet. Shut the overhead projector off and make sure each group has a dr-erase board. Have each student from each group draw a line with a ruler on the grid side of the dr-erase board so that each student has a different and distinct line. Tell the groups to write or describe everthing the know about how to describe the lines. Have the students write this information under the K portion of their K-W-L chart in their student journal. Then have the groups discuss what the want to know about lines on the dr-erase board or about lines representing data and write this under the W portion of their K-W-L chart. Draw a large K-W Chart on the board and solicit from the groups what the know and want to know about lines. The answers students give for the K and W will var. You ma want to spend some time discussing the K and W list that students generated. Hopefull, students are mentioning things such as -intercept, slope, origin, horizontal lines, vertical lines, intersection, -ais, and -ais for the K part of the chart. You ma have some groups mention the equation for lines such as = m + b. You ma want to record the information from the chart so that ou can use it to make informal assessment decisions. Have the students complete the L portion of the chart after the conclusion of Activit 2 in this lesson as well as at the end of the unit.

26 AIIF Page 20 Setting the Stage Transparenc Lines are often used to represent trends in data. Free Throws Made Dann s Basketball Free Throw Success During Practice Free Throws Attempted Dann s After-School Job Earnings Pa in Dollars Number of Hours Worked

27 Linear Functions Lesson 2: Linear Equations in Slope-Intercept Form AIIF Page 21 K W L What do I/we know? What do I/we want to Know? What did we learn?

28 AIIF Page 22 Teacher Reference Activit 1 To begin to understand how to use lines and linear functions to represent real-world data, have students look at patterns in which the data is eactl linear b having students complete Eercises 1 through 10. The suggestion is to have students work in pairs b having each student in the group of four pair with another student in the group. The goal is to have students work in pairs to construct knowledge related to the five representations of a linear function: scenario, rule in words, rule as equation, table of ordered pairs, and graph. It ma be helpful to complete a simple pre-reading with students b asking them to glance through Eercises 1 through 10 and predict what the think the ma learn and what main vocabular or concepts will be covered. As the student pairs complete Eercises 1 through 10, walk around the classroom and support pairs as needed. Suggest that if the need help to check with the other pair in their group. After students have completed Eercises 1 through 10, discuss the five representations of a function: scenario, rule in words, rule as equation, table of ordered pairs, and graph. Ask guiding questions such as, Which representation gives the most information?, Which representation do ou prefer?, or What information is in the scenario that ou can t find in the table? Before pairs answer Eercise 11, give the students guidance on how to enter an equation into the graphing calculator in order to graph it. The following directions can be used on the TI-83 Plus or TI-84 Plus. Using the calculator viewscreen, show the class how to enter an equation b pressing the o ke which displas the Y= editor. Tell the students that the will need to press the ke for. The pictures below show the equation entered into the Y= editor and the resulting graph after pressing the s ke. After students can successfull enter the equation for Eercise 11, show them how to look at the table of ordered pairs that can be displaed on the graphing calculator that relates to the entered equation b pressing the s kes. You ma want to discuss the advantages and disadvantages of using the graphing calculator to work with the equation, table, and graph. Some advantages are the ease of going back and forth between table, graph, and equation as well as the precision of the calculator. A disadvantage is there is no permanent record of the table and the graph. Also the calculator doesn t show the labels easil on the graph. Have each group of four students complete Eercises 12 through 14. These eercises are to reinforce and confirm students understanding of the five different representations of a function.

29 Linear Functions Lesson 2: Linear Equations in Slope-Intercept Form AIIF Page 23 Before students complete Eercises 1 and 16, ou ma want to complete a discussion related to slope. Ask the class, What does the measure of steepness or slope mean to ou? Give eamples. Use the Pass It Along strateg to have a group share one of their meanings and an eample b asking a group to share and if the do not want to share the can pass it to another group. Allow onl two passes. Have another group share its meaning and eample using the same method. Have the students record the responses in a blank area in their student journal. Some responses might include the incline of a roof, a road, a ramp, or a set of stairs. If the class doesn t mention an eamples involving horizontal or vertical slope, ou ma want to ask them about such things as the horizontal Bonneville Salt Flats in Utah where speed records for automobiles are often recorded or vertical telephone poles. You ma also want to talk with students about the slope of the graph of scenarios in which there is a constant rate such as Kimberl s earnings or other eamples such as: Snow is accumulating at a rate of three inches per hour. In water, the amount of pressure increases 4.3 pounds per square inch ever 10 feet ou descend. Thirt five cents per three minutes of phone usage at a public pa phone. Net, continue the discussion toward a more mathematical definition of slope. Displa the blank coordinate grid transparenc from Lesson 1. Plot the points ( 3, 4) and (2, 6) on the transparenc. Have the class plot the points ( 2, 1) and (1, ) on their dr-erase boards. Connect the points with a line segment. Lead a discussion about the horizontal and vertical distances from the point ( 3, 4) to the point (2, 6) while the students calculate the horizontal and vertical distances from ( 2, 1) to (1, ) on their dr-erase boards. Create a ratio of the vertical distance to the horizontal distance. Ask the class What does the ratio of a positive number over a positive number give us? Net, start from point (2, 6) and go to the point ( 3, 4), while the class goes from (1, ) to ( 2, 1). The class should realize that we have to go down verticall (negative direction) and to the left horizontall (negative direction). Now ask, What does the ratio of a negative number over a negative number give us? Erase the points and line segment on our graph and have the students do the same. Now plot the points ( 3, 4) and (2, 6) on our transparenc while the class plots the points ( 2, 1) and (1, ) on their dr-erase boards. Go through the same discussion with these two points with the first two points. The ke concept we are tring to conve is that going up from left to right (or down from right to left) ields a positive ratio or slope while going down from left to right (or going up from right to left) ields a negative ratio or slope. Conduct a class discussion on the formal definition for slope given before Eercise 1. Have the students, in pairs, complete Eercises 1 and 16. Ask for volunteers to share their results with the class. Lead a discussion with the class about properl calculating the slope of lines. Discuss the terms rise and run. If students need more eperience with slope refer to Practice Eercises 12 and 13. Have the students work in pairs on Eercises 17 through 22. One student works on the first problem while the other coaches and observes. The student coaching/observing then does a check. When the problem is solved and both partners agree on the answer, the coaching/observing students puts his/her initials on the problem, and the roles are reversed for the net problem. Ask for students to share their result with the class.

30 AIIF Page 24 SJ Page 9 Activit 1 In mathematics, a function is a rule that assigns to each input value eactl one output value. A function allows a person to describe one quantit in terms of another. Let s eplore linear functions b beginning with a scenario related to mone earned. Scenario For each hour that Kimberl works she earns $ What mathematical statement, or rule, could ou use to describe the relationship between the hours Kimberl works and the amount of mone she earns? Write the rule in words and as an equation. Rule in Words Answers ma var. Students should write a statement that represents multipling the number of hours worked times $12.00 to determine the total she earns. Sample response: The mone earned equals 12 times the hours worked. Rule as an Equation You ma want to describe to students that sometimes it is easier to use letters that match the words and sometimes it is easier to use the common letters and. Sample responses: t = 12h or = 12 Let s develop an understanding of what we mean b an input value and output value. 2. If we consider the value Kimberl earns as the output value, what might we consider as the input value? Sample response: The input value for this scenario is number of hours worked. 3. We can use a table to record the input and output values related to Kimberl s earnings. Complete the table below. Table of Ordered Pairs Kimberl s Earnings Input Value Output Value Hours Worked Dollars Earned

31 Linear Functions Lesson 2: Linear Equations in Slope-Intercept Form AIIF Page 2 SJ Page According to the definition, a function is a rule that assigns each input value eactl one output value. Is the rule ou made for Kimberl s earnings a function? Other was to think about this question are to ask ourself, Is there an input that would give two different outputs? or Is there ever at time when Kimberl could work 3 hours and earn two different amounts of mone? Sample response: Yes. The rule is a function. Students should see that there is no hour input that could give two different outputs of mone earned. In Lesson 1, ou learned that a table of values could be considered a set of ordered pairs. The first value of an ordered pair is known as the -value and the second value is called the -value. For eample, the first ordered pair in Kimberl s Earnings table is (1, 12).. Describe two other ordered pairs for Kimberl s earnings scenario. Sample response: (3, 36) and (, 60) 6. Write an ordered pair that would represent the hours and mone earned when the hours worked is 10. (10, 120) 7. Write an ordered pair that would represent the hours and mone earned when the mone earned is $ (12, 144) 8. What does the ordered pair (0, 0) mean in terms of Kimberl s scenario? Sample response: It means that for zero hours of work Kimberl has earned zero dollars. Also in Lesson 1, ou learned about plotting a set of ordered pairs. 9. Plot the ordered pairs from the table in Eercise 3 on the coordinate grid below and on a graphing calculator. Graph below also has line drawn for Eercise 11. Kimberl Scenario What dimensions do ou need to consider when 80 graphing the ordered pairs on our graphing calculator? a. -minimum: 0 0 b. -maimum: c. -scale: 1 30 d. -minimum: 0 20 e. -maimum: f. -scale: Hours Worked Dollars Earned

32 AIIF Page 26 SJ Page On the coordinate grid in Eercise 9, draw a line that goes through all the points. Then graph the line on our graphing calculator b entering the equation. Your teacher will give ou the steps to complete this procedure. Show students how to enter the equation = 12 into the graphing calculator and displa it at the same time as the plotted ordered pairs. Note: It is encouraged at this time to show the students the table feature of the graphing calculator and how it relates to the entered equation. If students understand this feature, the will be able to quickl look at man ordered pairs for the equation. If ou choose to do this at this time, ou ma want to ask the students to use the table to determine the value of the dollars earned at 0 hours. We will use a scenario, a written rule, an equation, a set of ordered pairs, and a graph, to represent linear functions in this unit. 12. To practice these five different representations of a linear function, write our own scenario for mone earned over time and then complete the four different methods of representation. Use our graphing calculator as needed but make sure to fill in the information in the table and graph supplied below. Answers will var. Make sure the scenarios are linear and go through the origin. Scenario Rule as an Equation Rule in Words Table of Ordered Pairs Hours Worked Dollars Earned Graph Dollars Earned Hours Worked

33 Linear Functions Lesson 2: Linear Equations in Slope-Intercept Form AIIF Page Kimberl s rate of pa was 12 dollars per hour. What is the rate of pa for our scenario? SJ Page 12 Answers will var. 14. Use our graphing calculator to graph both Kimberl s linear function and our linear function in the same window. Compare the two graphs and describe the similarities and differences. Answers will var. Sample response: The similarities are that the graphs are both lines and that the both go through (0, 0), also called the origin. The difference is that both lines have a different slope. Students ma not know the term slope et so the ma use a different method to describe steepness. Slope is the net topic in this lesson. The slope of a line can be described b one value. We will often refer to this one value for slope as m. The slope formula is a ratio of the change in -values to the change in -values from two ordered pairs on the line. You will sometimes see slope defined with the word change or the smbol for the word change which is Δ, called delta. Definition: Slope of a Line m = change of -values Δ m = = change of -values Δ To calculate the slope for the line representing Kimberl s earnings we could use the two ordered pairs (1, 12) and (6, 72) as shown below. Slope for line representing Kimberl s earnings: m = = = Use the definition of the slope of a line and two ordered pairs to calculate the slope of earnings per hour. Answers will var. How does the value for rate of pa compare to the slope? Sample response: The slope is the same as the rate of pa. Note: Refer to Practice Eercises 12 and 13 for students who need additional practice and development of the concept of the slope of a line.

34 AIIF Page 28 SJ Page 13 When looking at a graph, mathematicians also often refer to the slope of a line with the ratio, rise run, verbalized as rise over run. For eample, the rise from point (1, 12) to the point (6, 72) is 60 and the run is giving a ratio of 60/ or 12. Kimberl s Scenario rise run = = Run = Hours Worked 16. Stud the different methods to calculate slope and categorize the following items into two lists, then describe each categor with one sentence. change in -values run Δ 2 1 rise 2 1 Δ change in -values Students should create a list related to change in and another list related to change in. Dollars Earned Rise = You have just studied five different methods to represent a linear function with a scenario, a rule in words, a rule with an equation, a set of ordered pairs, and a graph. a. Which method do ou think best represents the concept of linear? Wh? It is important for students to see that the graph of the points and matching equation make a line. Functions that create a line when graphed will be called linear functions. b. Describe how the slope of the line is represented in each of the five methods. Answers will var. Sample response: In the scenario, slope is represented b the rate. In the equation slope is represented b the number in front of. In the rule with words slope is the number that ou multipl b. In the graph, slope is how far up and over from one point on the line to another point on the line. With ordered pairs, slope is the difference of the -values divided b the difference of the corresponding -values.

35 Linear Functions Lesson 2: Linear Equations in Slope-Intercept Form AIIF Page 29 SJ Page 14 You ve just investigated two linear function scenarios that had linear graphs that went through the origin. The equations were of the form = m where the value of the slope was the same as the rate in the scenario. Let s change the scenarios slightl to investigate a different tpe of linear function. Before having students continue, discuss with them that so far all of the lines went through (0, 0), and ask, What might change in the scenario to cause a line to go through a different place on the -ais? Scenario Kimberl earns $12.00 per hour but in addition she received a one-time $0.00 bonus for accepting the job. 18. Complete the four other methods to represent this scenario. Sample responses: Rule as an Equation t = 12h + 0 = Rule in Words The mone earned equals 12 times the hours worked plus 0. Table of Ordered Pairs Sample response: Hours Worked Dollars Earned Dollars Earned Graph Kimberl s Scenario Hours Worked 19. Graph the equation from Eercise 1 and Eercise 18 on the same screen on our graphing calculator. Compare both graphs. What are the similarities? What are the differences? Sample response: The graphs have the same slope, but one line is 0 units higher on the scale. The -intercept of a line is the point where the line crosses the -ais on a graph. 20. What are the -intercepts for the equations from Eercises 1 and Eercise 18? This question is here to begin to have students think about the concept of -intercept. It is not here for students to completel understand the -intercept at this time. The net activit will help students gain a better understanding of -intercept and slope. Sample response: The -intercept for Eercise 1 is (0, 0) and for Eercise 18 is (0, 0).

36 AIIF Page 30 SJ Page In general, how might ou define input value and output value as it relates to an equation? Sample response: The input value is the value ou would substitute for. The output value is the value for that occurs for specific values of. 22. In this activit ou were introduced to man terms. Create a graphic organizer in the space provided below that includes all the terms. A list of terms is given below. Include other terms as needed. Terms function input value output value scenario rule in words rule as an equation table of ordered pairs -value -value ordered pairs graph -minimum -minimum -maimum -maimum -scale -scale slope Δ Δ slope of a line rise run -intercept Graphic Organizers will var.

37 Linear Functions Lesson 2: Linear Equations in Slope-Intercept Form AIIF Page 31 Teacher Reference Activit 2 Tell students that the graphing calculator is a powerful tool that allows them to investigate man situations over a short period of time. This gives students the abilit to discover patterns and make conclusions about various mathematic concepts. Before students begin Eercise 1, ou ma need to remind students how to enter an equation, to set the window parameters, and to determine the slope and -intercept on a graphing calculator. Use the following as an eample. 2 Graph the function = + 4 on a graphing calculator. Set the dimensions of the graph to an -minimum of 3, an -maimum of, a -minimum of, and a -maimum of. It ma be helpful to square the window so that the units for the -ais are equivalent in size to the units for the -ais. B using the trace feature, show students how to determine that the -intercept of the line is (0, 4). Turn the grid feature on and show students how to determine that the slope of the line is 2/3 b counting the units for rise and run. To turn the grid feature on for the TI-83 Plus or TI-84 Plus press 2 ND ZOOM and choose GridOn. Have students complete Eercises 1 through 3 with a graphing calculator. You can have students complete Eercises 1 through 3 two different was. One method is to have each pair or group work on each eercise. Another method is to have different groups work on different eercises and then have the groups share their findings. The goal is for students to realize that the coefficient in front of relates to slope and the constant added represents the -intercept (or vertical translation from the origin). Alternative: An additional or optional method for Eercises 1 through 3 is to have students walk Eercises 1 through 3 on a coordinate plane on the floor of the classroom. Prepare the room before the students arrive. Move the desks to the side of the room to make an area for a large Cartesian Coordinate Plane on the floor. Use masking tape or duct tape to create the - ais, the -ais, and the grid marks. If our classroom alread has tile flooring, then ou ma not need to place grid marks. The units of the grid should be uniform and approimatel one foot in length. The range of the units should be at approimatel ( 10, 10) for both the -ais and -ais. You ma want the students to help ou prepare the room. Have students work in pairs. One partner will stand at a point along the -ais. This person should perform the operation for the number the are standing on as the teacher reads each function. For eample, if ou are standing at the -value of 3 and the teacher tells ou to show the relationship = 2, then ou need to move to the -value of 6. This would place ou at the point (2, 6). If ou have a -value that is too large for the grid, then step off the grid to the side.

38 AIIF Page 32 The other partner will record the information discovered in this activit on the following pages. There are four main questions to consider when recording the information. Where does the pattern cross the -ais? Is the pattern a line or a curve? If the pattern is a line, what is the slope of the line? Does the pattern increase or decrease? Now, as a class complete Eercises 1 through 3. Half the class will walk each equation as the other half records what is created. Assign each student walking a different -value to represent. For Eercise 1, students should record that each equation made a line with the same slope but different -intercept. For Eercise 2, students should record that each equation had a -intercept of zero (the origin) but the slope was larger for large values of m and smaller for smaller values of m. Finall, for Eercise 3, students should record the effect of negative m values. After students have completed Eercises 1 through 3 with whatever method ou choose, have pairs complete Eercise 4 and then have a discussion about the affect of m and b on the graph of = m + b. Refer to the definitions offered below Eercise 4 as needed during our discussion. It ma be helpful for students to understand the connection between parallel lines and slope at this time b completing Eercise. Later, in Lesson, students will work more with parallel and perpendicular lines as well as other forms of lines such as point-slope and standard form. Make sure to have students complete the L part of the KWL that the began in the beginning of this lesson.

39 Linear Functions Lesson 2: Linear Equations in Slope-Intercept Form AIIF Page 33 SJ Page 16 Activit 2 In Activit 1, ou learned about five different methods to represent a linear function. In this activit, ou will investigate the relationship between the equation = m + b and the graph of a linear function. 1. Graph each linear equation from the table on the same grid on the graphing calculator. Describe what ou see on the screen and record the results in the table. Linear Equation -intercept slope Description: Answers will var. = +2 (0, 2) 1 = +1 (0, 1) 1 = +0 (0, 0) 1 = 1 (0, 1) 1 = 2 (0, 2) 1 What does b from the linear equation = m + b represent on the graph? Sample response: b represents the -coordinate of the -intercept of the line on the graph. 2. Graph each linear equation from the table on the same grid on a graphing calculator. Describe what ou see on the screen and record the result in the table. Description: Answers will var. Linear Equation -intercept slope = 2 (0, 0) 2 = 1 (0, 0) 1 1 (0, 0) 1/2 = 2 1 (0, 0) 1/4 = 4 What does m from the linear equation = m + b represent on the graph? Sample response: m represents the slope of the line on the graph. 3. Graph each linear equation from the table on the same grid on a graphing calculator. Describe what ou see on the screen and record the result in the table. Description: Answers will var. Linear Equation -intercept slope = 2 (0, 0) 2 = 1 (0, 0) 1 1 = 2 1 = 4 (0, 0) 1/2 (0, 0) 1/4 Eplain the effect of a negative value for m on the graph of the linear equation = m + b. Sample response: A negative m changes the slope from positive to negative.

40 AIIF Page 34 SJ Page On our dr-erase board, draw a line that has a specific -intercept and specific slope. Trade the board with our partner. a. Write an equation that represents our partner s line. Graph the equation on a graphing calculator. With our partner compare the graph on the dr-erase board and the graph on the graphing calculator. Correct an mistakes to make sure both graphs are the same. Record the following information related to our partners line. -intercept: Answers will var. slope: Answers will var. equation: Answers will var. Slope-Intercept Form of a Linear Equation The slope-intercept form of a linear equation is = m + b. It has this name because the equation includes the slope, m, and the -coordinate of the -intercept, b. For eample, 1 the equation = 1 has a slope of and a -intercept of 1. -intercept RISE 1 RUN 2 Parallel lines are lines that have the same slope and never intersect. For eample, both lines on the grid to the right are parallel and both have a slope of Write two different scenarios that represent linear functions that have the same slope but different -intercepts when graphed. Write the equations that match the scenarios. Answers will var. Sample response might be. Ran earns $8 per hour and Monica earns $8 per hour with an initial onetime bonus of $10. = 8 = Describe what part of the equations represents the parallel slope. Sample response: The numbers in front of represent parallel slope because the are the same value. Note: Remember to have students complete the L part of the KWL before the net lesson.

41 Linear Functions Lesson 2: Linear Equations in Slope-Intercept Form AIIF Page 3 Practice Eercises SJ Page To maintain a health diet and lifestle, nutritionists recommend that the food ou eat each da should not contain more than 30% fat. Most food products contain a nutrition label stating the amount of grams of fat, per serving, of the food item. There are nine calories for each gram of fat that ou eat. The amount of fat,, ou should have each da is based on the number of total calories, ou consume. Since we onl want 30%, or 0.30, of our calorie intake to be fat, our equation to determine the amount of fat we have consumed is = which is the same as 1 =. 30 Create a table of ordered pairs comparing calories and fat and then graph the ordered pairs. Table of Ordered Pairs: Calories () Fat () Graph: Table of values will var. Graphs will var. 2. A cab compan charges $2.0 to enter the cab and $2.2 for each additional mile. Create a table of ordered pairs comparing miles and cab fares. Graph the ordered pairs from the table to create a connected scatter plot. Determine an equation that represents the cab fare for each mile. Table of Ordered Pairs: Graph: Table values will var. Equation: = Graphs will var.

42 AIIF Page 36 SJ Page From the table below, pick three activities that ou participate in or find interesting. Create a table of values comparing the number of hours completing the eercise and the number of calories burned. On the same set of aes, graph the ordered pairs and create three different connected scatter plots. Determine equations that represent the calories a person weighing 10 pounds will burn for each of the activities ou chose. Calories Burned During Eercise Eercise (1 hour) Calories Burned b Person Weighing 10 Pounds Aerobics dancing 46 Basketball (recreational) 40 Biccling 612 Canoeing 174 Dancing (ballroom) 210 Football (touch, vigorous) 498 Jogging (6 miles per hour) 64 Horseback Riding 246 Ling down or sleeping 90 Racquetball 88 Roller Skating 384 Scrubbing Floors 440 Swimming (crawl, 4 ards 22 per minute) Swimming (crawl, 20 ards 288 per minute) Volleball (recreational 264 Walking, 2 miles per hour 198 X countr Skiing ( miles per 690 hour) Weight Training (circuit) 76 Hours Activit 1 Activit 2 Activit Table of values will var. Equations: Answers will var. Graphs will var.

43 Linear Functions Lesson 2: Linear Equations in Slope-Intercept Form AIIF Page 37 SJ Page The distance, d in miles, a lightning flash is from our location can be approimated b counting the number of seconds between the flash of lightning and the sound of thunder and then multipling b Create a table of values comparing seconds and distance. Graph the ordered pairs from the table to create a connected scatter plot. Determine an equation that represents the distance from a lightning strike based on the number of seconds between the lighting flash and sound of thunder. Sample response: Table of Ordered Pairs: Seconds Equation: = 0.21 Miles Miles Graph: Seconds. Fill in the table of ordered pairs for the equation = and then graph the ordered pairs to create a connected scatter plot representing the equation. Table of Ordered Pairs: Graph:

44 AIIF Page 38 SJ Page Write an equation in slope-intercept form given the slope and the -intercept. a. Slope of 3/4 and -intercept of (0, 6) 3 = b. Slope of 4 and -intercept of (0, 8) = 4+ 8 c. Slope of 2/ and -intercept of (0, 4) 2 = 4 d. Slope of 3 and -intercept of (0, 2) = State the slope and the -intercept for the given equations. a. = The slope is 3 and the -intercept is (0, 2) b. 1 = 4 The slope is 1/4 and the -intercept is (0, ) c. 4 = The slope is 4/7 and the -intercept is (0, 9) 8. 4 How might ou change the equation = + 9 so that the graph of the equation has a -intercept of 7 10? How might ou change the equation so that the graph of the equation has a -intercept of zero? Sample response: Change the 9 to a 10. Change the 9 to a zero. 9. How might ou change the equation from left to right? 1 = so that the graph of the equation has a slope that rises 4 Sample response: Change the 1/4 to 1/ Describe how ou might change the linear equation = 2 3 to a new equation so that the graph of the new equation is five units higher than the graph of = 2 3. Sample response: Add to the equation so that it becomes = Describe how ou might change the linear equation = 3 to a new equation so that the graph of the new equation has less of a slope. Sample response: Change the to a smaller number such as 2: = 2 3

45 Linear Functions Lesson 2: Linear Equations in Slope-Intercept Form AIIF Page 39 SJ Page Plot each set of ordered pairs and connect each set of ordered pairs to make line segments. Label the line according to the eercise number. Determine the slope of each line segment. a. ( 4, 3) and (2, 9) The slope is 12/6, 12/( 6), or 2. b. ( 2, 6) and ( 6, 2) The slope is 8/4, or 8/( 4), or 2. c. (6, 6) and (6, 6) The slope is 12/0 or 12/0 which is undefined. d. ( 3, 3) and (, 3) The slope is 0/8, 0/( 8), or 0. e. (, ) and (, ) The slope is 10/10, or 10/( 10), or Calculate the slope for the following sets of ordered pairs. a. ( 2, 4) and (4, 8) b. (6, ) and (2, 1) c. ( 2, 4) and (10, 0) Slope is 2. Slope is 1. Slope is 1/ Fill in the table of ordered pairs for the equation = and then graph the ordered pairs as a connected scatter plot How can a linear function represent a real-world scenario? Answers will var.

46 AIIF Page 40 SJ Page 23 Outcome Sentences The rise and run are used to To graph a line in slope intercept form A line with positive slope A line with negative slope Parallel lines I need more help with

47 Linear Functions Lesson 2: Linear Equations in Slope-Intercept Form AIIF Page 41 Teacher Reference Lesson 2 Quiz Answers 1a. 2 = + 1b. = 6 3 2a. The slope is 1/3 and the -intercept is (0, 8) 2b. The slope is and the -intercept is (0, 4) 3a. The slope is. 8 3b. The slope is.

48 AIIF Page 42 Lesson 2 Quiz Name: 1. Write an equation in slope-intercept form given the slope and the -intercept. a. Slope of 2/ and -intercept of (0, ). b. Slope of 6 and -intercept of (0, 3). 2. State the slope and the coordinates of the -intercept from the given equations. a. = b. = 4 3. Calculate the slope for the following sets of ordered pairs. a. (, 3) and ( 3, 8) b. (2, 1) and (4, 9)

49 Linear Functions Lesson 3: Applications of Linear Functions in Slope-Intercept Form AIIF Page 43 Lesson 3: Applications of Linear Functions in Slope-Intercept Form Objectives The students will be able to determine the linear equation given a real-world scenario. The students will be able to determine the linear equation given a pattern. The students will be able to use a linear equation to determine a specific term value for various scenarios and patterns. Essential Questions How does using a linear equation help answer questions related to linear patterns or scenarios? Tools Student Journal Dr erase boards, markers, erasers Graphing calculator Paper Scissors Warm Up Problems of the Da Number of Das for Lesson 1 to 2 Das (Note: If ou take two das to cover this lesson, the authors suggests that ou complete Activit 1 on the first da with Practice Eercises 1 and 2 followed b Activit 2 with Practice Eercises 3 through 16 on da two.) Vocabular constant constant rate -intercept -intercept term number term value

50 AIIF Page 44 Teacher Reference Setting the Stage Because linear functions can model situations in which there is a constant rate, it ma help for students to think about their current knowledge of the term constant. Displa the Setting the Stage transparenc and have student volunteers read each tpe of meaning for the term constant. Then have students list things that are constant in their life. There are a few different was the term constant is used in mathematics. One method is to describe a value 2 (number) that doesn t change in a specific epression, such as 4 in the equation = + 4. Mathematicians 3 also often refer to a constant rate in application problems such as traveling at a constant rate of speed in miles per hour or in problems related to graphs that have a constant rate in comparison to the two variables. The constant rate is visible on the graph of a line b the slope.

51 Linear Functions Lesson 3: Applications of Linear Functions in Slope-Intercept Form AIIF Page 4 Setting the Stage Transparenc Different Uses of the Term Constant Meaning Ever present Eample Darl had a constant suppl of fresh water for his fish tank. Done repeatedl She makes constant visits to the doctor. Not changing The cave has a constant temperature of 4º. What is constant in our life? How might the term constant be used in mathematics? What does constant rate mean?

52 AIIF Page 46 Teacher Reference Activit 1 You ma want to guide students through this activit. Have a volunteer read Eercise 1 and then solicit answers for parts a, b, and c from other students. You can randoml choose students to answer each part b calling on one student, if that student doesn t want to answer he or she can pass to another student, but onl allow two passes. Continue in this fashion for Eercises 2 through 9. The goal of these nine eercises is for students to appl their knowledge of the slope-intercept form of a linear equation and the five representation of a linear function. You ma need to supplement this guided discussion with guiding questions such as: How might ou easil remember how to find the -intercept? -intercept? If we etended the graph of the line of Ramond s flight into Quadrant IV or Quadrant II, what would the information on the line represent? What advantage does the equation give compared to the graph? to the table of ordered pairs? How is rate represented on the graph? How is rate represented in the table of ordered pairs? How is rate represented in the scenario? How is rate represented in the equation? How is the -intercept represented in the scenario? How is the -intercept represented in the graph? How is the -intercept represented in the table of ordered pairs? How is the -intercept represented in the equation?

53 Linear Functions Lesson 3: Applications of Linear Functions in Slope-Intercept Form AIIF Page 47 SJ Page 24 Activit 1 You can appl linear functions to scenarios in which there is a constant rate. 1. Analze the following scenario b completing the four additional representations. Scenario Ramond is traveling toward his Air Force base in an F/A-18 at a constant speed of 70 miles per hour, which is just under the speed of sound. He is tring to travel as fast as he can without breaking the sound barrier, which is 767 mile per hour. He started at a distance of 220 miles from base. a. Complete the table of ordered pairs that represent the distance Ramond is from base each hour. Table of Ordered Pairs Hour Distance from Base in Miles b. Plot the ordered pairs on the graph and draw a line through the points. Then determine the slope and the -intercept. Graph 200 Slope: intercept: (0, 220) Hours c. Use the slope and -intercept to write a rule as an equation and a rule in words to represent this function. Miles from Base Traveling Jet Rule as an Equation = or d = 70h Rule in Words Sample response: The distance from base is equal to the 70 times the hours traveled plus 220.

54 AIIF Page How is the rate of 70 miles per hour represented in the equation and in the graph? SJ Page 2 Sample response: The rate is the slope. Note: You ma want to discuss with students wh for this scenario the slope of the equation and graph is actuall 70 and not Use the equation, table, or graph to determine how far Ramond will be from base at 1. hours. 1,12 miles from base. Note: The goal for this problem is to have students begin to eplore the value of using the equation or graph to answer questions. Describe how to solve for the distance given the number of hours. Sample response: Substitute the number of hours into the equation and then solve for distance. 4. Use the equation, table, or graph to determine how long Ramond will fl before he is onl 1000 miles from base. Approimatel 1.67 hours or 1 hour and 40 minutes. Note: The goal for this problem is to have students begin to eplore the value of using the equation to answer questions. Describe how to solve for the number of hours given the distance from base. Sample response: Substitute the distance into the equation and then solve for hours. The -intercept is where a graph intersects the -ais. The -intercept is where a graph intersects the -ais.. Determine the -intercept of the equation and describe what it means in the scenario of Ramond fling to base. Sample response: The -intercept is (0, 220). It is the distance from base when Ramond started. 6. Determine the -intercept of the equation and describe what it means in the scenario of Ramond fling to base. Sample response: The -intercept is (3, 0). It is the number of hours Ramond will fl until he is 0 miles from the base (at the base). 7. Stud the table of values in Eercise 1a and describe how to find the -intercept and -intercept in the table. Sample response: For the -intercept find the ordered pair that has 0 for the -value. For the -intercept find the ordered pair that has 0 for the -value.

55 Linear Functions Lesson 3: Applications of Linear Functions in Slope-Intercept Form AIIF Page 49 SJ Page What would change in the graph and in the equation of Ramond s flight, if he traveled 40 miles per hour instead of 70 miles per hour? Create a new table of ordered pairs and graph if needed. Sample response: The -intercept of the graph would change to (, 0). The slope of the graph and equation would change to 40. The equation would be = Complete each of the following. a. Describe what would change in the graph and in the equation of Ramond s original flight if he started 3000 miles from base and flew 70 miles per hour toward base. Sample response: The -intercept on the graph and in the equation would change to The slope would remain 70, but the -intercept would change to 4 hours (4, 0). b. What would be the scenario if the equation for Ramond s flight was = ? Sample response: Ramond is traveling toward his Air Force base in an F/A-18 at a constant speed of 600 miles per hour. He started at a distance of 2400 miles from base. c. What would change in the graph, the equation, and the scenario if Ramond had taken 6 hours to fl 220 miles at a constant rate? Sample response: The graph would have a different slope and the -intercept would change to (6, 0). The slope would change to 37, calculated b (220 0)/(0 6) = 37. The scenario would now read: Ramond is traveling toward his Air Force base in an F/A 18 at a constant speed of 37 miles per hour. He started at a distance of 220 miles from base. d. What would change in the equation if Ramond actuall flew at the speed of sound of 767 miles per hour? How long would it take Ramond to reach base at 767 miles per hour if he started 220 miles from base? Round to the nearest hundredth of an hour. Sample response: The slope in the equation would change to 767: = +. It would take Ramond approimatel 2.93 hours to reach the base.

56 AIIF Page 0 Teacher Reference Activit 2 Tell students that laering objects before cutting them is often done to increase the efficienc in making multiple objects. For eample, printing companies have paper cutters that can easil cut a full ream of paper. Imagine doing that with a pair of scissors. Another eample could be folding fabric to cut multiple patterns for clothing. Mathematics can describe how the number of folds and number of cuts relate to the number of pieces created. Talk to the students about how the power of mathematics can help us with modeling patterns that we see around us in order to make predictions and decisions. In this activit, we will have a chance to look at various patterns and discover a matching equation that represents some of the mathematics related to the pattern. In front of class, fold an 8.11 inch sheet of paper twice into thirds, as shown below. Cut the folded paper once down the middle, parallel to the folds. Ask the students, How man pieces of paper did I create? (Answer should be 4.) Fold another sheet of paper as described above and ask, What if I cut it twice down the middle, how man pieces would I get? Have the students guess first, and then proceed to cut it out and count the pieces in front of them (Answer should be 7). Now transition to the eercises b asking, How might we determine the number of pieces that we would get, if we cut it down the middle 100 times? Would it be practical for us to actuall cut this 100 times and keep track of all the little pieces of paper or could we create an equation to model the scenario? Then, tell the students their goal is to determine a linear equation that describes the number of pieces of paper dependent upon the number of cuts down the middle. Have the students work in groups of four to answer the eercises. You ma want to complete this additional optional etension before students answer Eercise 7. Additional Optional Etension: You ma want to show students another wa to see how a linear equation can be determined for this scenario b using a table. Then have them practice on Eercise 7. Help students organize the information into a table as shown below. Have the students record the number of cuts and number of pieces. The students should notice a common difference between each term. Have the students record what is happening mathematicall to the number of pieces after each cut and then tr to have them write the mathematical pattern based on the number of cuts. Eventuall have the students write the mathematical pattern for an unknown number of cuts, such as. Note: Some math teachers would prefer to use n in this case instead of because the number of paper pieces is alwas a positive integer value and not a continuous real value. Number of Cuts Number of Pieces Mathematical Pattern of Number of Pieces Mathematical Pattern Based on Number of Cuts

57 Linear Functions Lesson 3: Applications of Linear Functions in Slope-Intercept Form AIIF Page 1 Activit 2 SJ Page In our group, continue to work to determine an equation that models the number of pieces of paper created based on the number of cuts down the middle. 1 Cut made pieces 2 Cuts made pieces 3 Cuts make pieces 2. Table of Ordered Pairs Number of Cuts Number of Pieces Equation: = Check the equation to make sure it works for 0, 1, and 2 cuts. Students should show that their equation works for 0, 1, and 2 cuts.. Use the equation to determine the number of pieces of paper created with 100 cuts down the middle. If = 100, then = 3( 100) + 1= 301. There will be 301 pieces. 6. Eplain wh entering negative values for would or would not make sense. Answers will var. The goal is for students to realize that mathematicall it is possible to enter negative values for, but that would not make sense in terms of the scenario because a person cannot make a negative cut.

58 AIIF Page 2 SJ Page 28 Another method to determine a linear equation is to etend the table of ordered pairs to show each constant addition. Complete the following problem. 7. Given the following pattern. Assume the difference between each term stas constant. 8, 13, 18, 23, 28, 33, a. Complete the table of values that model the pattern. Term Number Term Value Addition Notation Multiplication Notation 1 st 8 8 ( 0) nd ( 1) rd ( 2) th ( 3) + 8 th ( 4) + 8 th ( 1) + 8 b. Determine a function that models the pattern. Graph the function on a graphing calculator. = or = + 3 Sample response: ( ) c. Determine the term value if the term number is. = ( ) + 3= 278

59 Linear Functions Lesson 3: Applications of Linear Functions in Slope-Intercept Form AIIF Page 3 SJ Page 29 Practice Eercise Use these eercises for students to practice working with real-world applications and scenarios related to linear functions. There are a variet of methods to assign these eercises. You can have different groups or pairs work on different eercises and then share their findings. Or, ou can assign certain eercises to all of the groups or pairs. 1. Kaila is responsible for billing the customers who purchase items from the online compan for which she works. The following pattern shows the bills for ordering different amounts of the T-shirt tie-de kits that her compan sells. For one kit the bill is $7.00, for two kits the bill is $10.00, for three kits the bill is $ If this pattern continues, determine an equation that could model the bill of tie-de kits based on the number sold. Use the table as needed. Number of Kits Bill in Dollars a. Equation: = 3+ 4 b. Use the equation to determine the bill for 20 tie-de kits. For = 20, = 3( 20) + 4= 74. The bill is $ Two parking garages charge different prices. The tables below show the different pricing per hour. Analze the different parking garages and determine which garage has the best pricing. Parking Garage A Parking Garage B Hour Price Hour Price 1 $1.2 1 $ $ $3.0 3 $2.7 3 $ $3.0 4 $4.0 Answers will var. The goal for this eercise is for students to use the knowledge of linear functions learned in the past two lessons to determine an equation for each parking garage and then b graphing the line associated with each equation, the students should be able to determine when each parking garage has the best prices. The equation for each garage is = and = Sample response: Parking Garage A is the better price when parking for less than 8 hours. The are both the same for parking 8 hours. Parking Garage B is better when parking for more than 8 hours.

60 AIIF Page 4 SJ Page What equation models the number of pieces with one fold when cutting it once, twice, three times, etc. down the middle? = 2+ 1 Use the equation to determine the number of pieces that occur when cutting it 7 times? = 2( 7) + 1 = cuts make 11 pieces. a. What equation models the number of pieces with three folds when cutting it once, twice, three times, etc. down the middle? = 4+ 1 Use the equation to determine the number of pieces that occur when cutting it 7 times? = 4( 7) + 1 = cuts make 301 pieces. b. What function models the number of pieces with four folds when cutting it once, twice, three times, etc. down the middle? Assume pattern of folding is the same as in part a. = + 1 Use the equation to determine the number of pieces that occur when cutting it 7 times? = ( 7) + 1 = cuts make 376 pieces. c. What function models the number of pieces with k folds when cutting it once, twice, three times, etc. down the middle? Assume the pattern of folding is the same as in part a. = ( k+ 1) + 1 Use the equation to determine the number of pieces that occur when cutting it 7 times after folding it 10 times. = ( 10+ 1) 7+ 1= 826

61 Linear Functions Lesson 3: Applications of Linear Functions in Slope-Intercept Form AIIF Page SJ Page The number of points that create the equilateral triangle below is 1. The number of points on each side of the equilateral triangle is 6. a. How man points would be needed to make an equilateral triangle with 7 points on each side? 18 b. How man points would be needed to make an equilateral triangle with 8 points on each side? 21 c. Determine a function that models the number of points needed to make an equilateral triangle with points on each side. Be prepared to eplain and show how ou determined the function. The goal for this eercise is for students to use what the learned about linear functions and equations to determine the equation. The equation is = 3 3, where is the number on each side and is the total number of points. d. First check our equation to make sure it works for 6 points, 7 points, and 8 points on each side, then determine the number of points needed to make an equilateral triangle with 40 points on each side. = = points ( ) e. How man points would each side have if the total number of points is 60? The triangle would have 21 points on each side. This can be determined b substituting 60 in for and solving for : 60 = 3 3. Eercises through 7 are designed to etend Eercise 4 to other regular polgons and eventuall to create an equation that could represent all regular polgons made with points. You ma need to remind students that a regular polgon is a polgon that has sides of equal length.. The number of points that create the square below is 20. The number of points on each side of the square is 6. a. How man points would be needed to make a square with points on each side? 16 b. How man points would be needed to make a square with 4 points on each side? 12 c. Determine a function that models the number of points needed to make a square with points on each side. Be prepared to eplain and show how ou determined the function. Sample response: The equation is = 4 4, where is the number on each side and is the total number of points.

62 AIIF Page 6 SJ Page The number of points that create the regular pentagon below is 2. The number of points on each side of the regular pentagon is 6. a. How man points would be needed to make a regular pentagon with points on each side? 20 b. How man points would be needed to make a regular pentagon with 7 points on each side? 30 c. Determine a function that models the number of points needed to make a regular pentagon with points on each side. Be prepared to eplain and show how ou determined the function. Sample response: The equation is =, where is the number on each side and is the total number of points. 7. The number of points that create the regular heagon below is 30. The number of points on each side of the regular heagon is 6. a. How man points would be needed to make a regular heagon with points on each side? 24 b. How man points would be needed to make a regular heagon with 4 points on each side? 18 c. Determine a function that models the number of points needed to make a regular heagon with points on each side. Be prepared to eplain and show how ou determined the function. Sample response: The equation is = 6 6, where is the number on each side and is the total number of points. 8. Use the knowledge that ou gained b completing Eercises 4 through 7 to determine functions that model other regular polgons. a. Determine a function that models the number of points needed to make a regular heptagon with points on each side. A heptagon has 7 sides. The equation is = 7 7, where is the number on each side and is the total number of points. b. Determine a function that models the number of points needed to make a regular octagon with points on each side. An octagon has 8 sides. The equation is = 8 8, where is the number on each side and is the total number of points. c. Determine a function that models the number of points needed to make a regular n-gon with points on each side. An n-gon has n sides. Sample response: The equation is = n n, where is the number on each side and is the total number of points.

63 Linear Functions Lesson 3: Applications of Linear Functions in Slope-Intercept Form AIIF Page 7 9. Determine a function that models the pattern, 1, 7, 13, 19, 2, SJ Page 33 Sample response: = 6( 1) or = 6 11 where is the term number. 10 Determine the net three numbers for the following pattern. Determine a function that describes the pattern. 1, 23, 31, 39, 47,,,, = or = Sample response: 63, 71, 79 ( ) 11. The following pattern has a constant difference of 0. Determine a function that models the pattern. 4, 4, 4, 4, 4, 4, Sample response: = 0( 1) + 4 or = The following pattern has a constant difference of 4. Determine a function that models the pattern. 10, 6, 2, 2, 6, 10, Sample response: = 4( 1) + 10 or = Determine a function that models the following decreasing pattern. 91, 74, 7, 40, 23, Sample response: = 17( 1) + 91 or = Barrels are sometimes stacked as shown below. a. If the pattern of barrels continues and another row of barrels is placed under the stack, how man barrels will be in the new row? 12 b. Write a pattern of numbers that represent the number of barrels in each row. 8, 9, 10, 11, 12,... c. Determine the constant difference of the pattern. 1 d. Determine a function that models the number of barrels in each row. = or = + 7 Sampler response: ( ) e. Determine the number of barrels that would be in the 20 th row. = = 27

64 AIIF Page 8 SJ Page Hope is raising mone for a charit b attending a dance marathon. Her parents decided to donate $30.00 to Hope for entering the fundraiser with an additional $.00 for each hour she dances. a. Determine an equation that would model the mone Hope would earn just from her parents based on the number of hours she dances. Sample response: = + 30 where represents the number of hours dancing and represents the mone earned for charit. b. Use the equation to determine how man hours Hope would need to dance to earn $ Solving 200 = + 30 for gives 34 hours. c. Graph the equation on a graphing calculator and describe what the -intercept means. Sample response: The -intercept represents the $30.00 given just for entering the dance marathon. d. Use the table feature of the graphing calculator to determine what hour Hope would reach $ hours 16. Sound travels at a slower speed than light. That is wh ou see the flash of lightning before ou hear the thunder. The light travels so fast that ou basicall see it instantaneousl. The sound usuall follows a few seconds later. You know a lightning strike is close when ou see the flash and hear the thunder at the same time. The graph shows the distance the sound of thunder has traveled over time. The lightning flashed at time zero. Distance in Feet Time in Seconds a. If it took 1 second for the sound to reach ou, how far from the lightning strike would ou be? Approimatel 1,12 feet. b. If it took 3 seconds for the sound to reach ou, how far from the strike would ou be located? Approimatel 3,37 feet. c. If ou were 1 mile from the strike, how long would it take for the sound to reach ou? (Remember that there are,280 feet in a mile.) Sample response: Between 4 and seconds d. If ou were 1/2 mile from the lightning strike, how long would it take for the sound to reach ou? Approimatel 2. seconds. e. Approimatel, how man feet per second does sound travel? Approimatel 1,12 feet per second f. What linear equation could model the distance a lightning strike is from ou based on the time between seeing the lightning strike and hearing the thunder? Sample response: = 112, where is time between strike and thunder in seconds and is distance in feet.

65 Linear Functions Lesson 3: Applications of Linear Functions in Slope-Intercept Form AIIF Page 9 SJ Page The space shuttle can orbit the earth at a speed of approimatel 17,00 miles per hour. Graph the distance the space shuttle can travel over time b plotting points on the following coordinate plane and connecting them with a line. 90,000 80,000 70,000 Distance in Miles 60,000 0,000 40,000 30,000 20,000 10, Time in Hours a. According to our graph, how man miles does the space shuttle travel in hours? Approimatel 87,00 miles b. What would change on the graph if the space shuttle traveled at a speed of 18,000 miles per hour? Sample response: The line of points would have a larger slope. c. How man hours would the shuttle take to travel approimatel 0,000 miles? Sample response: A little less than 3 hours. d. What equation could model the distance the shuttle has traveled as a function of time? Sample response: = 1700, where is time in hours and is miles. 18. How does using a linear equation help answer questions related to linear patterns or scenarios? Answers will var. One goal of this question is for students to describe how, for specific scenarios, the can determine the term value based on the term number. For eample, with the paper cutting activit students could determine the number of pieces of paper based on the number of cuts or with Ramond s traveling jet scenario the number of miles left to base could be determined b the number of hours flown.

66 AIIF Page 60 SJ Page 36 Outcome Sentences Finding a linear pattern I now understand wh I still don t understand how I would like to understand wh linear functions Determine the linear equation of

67 Linear Functions Lesson 3: Applications of Linear Functions in Slope-Intercept Form AIIF Page 61 Teacher Reference Lesson 3 Quiz Answers 1a. = where is the temperature in Celsius and is the temperature in Kelvin. 1b. = ( 2) = 248 ; the temperature is 248 Kelvin for 2 Celsius. 2a. The net two sets of squares are: 2b. The equation is = 3( 1) + 9 or = c. There are 36 total squares for 10 shaded squares. 2d. There are 20 shaded squares for 66 total squares.

68 AIIF Page 62 Lesson 3 Quiz 1. The Kelvin temperature scale begins at the theoretical absence of all thermal energ which on the Kelvin scale is represented b zero. There is a linear relationship between the Celsius scale and the Kelvin scale. Use the table below to determine an equation that models the temperature in Kelvin based on the temperature in Celsius. Celsius ºC Kelvin ºK a. Equation: b. Use the equation to determine the temperature in Kelvin if the Celsius temperature is The pattern below describes the number of shaded squares,, compared to the total number of squares,. There is a linear relationship between the number of shaded squares and the total number of squares. Use the pattern to determine an equation that models the number of total squares based on the number of shaded squares. a. Draw the net two sets of squares based on the pattern above. b. Equation: c. Use the equation to determine the number of total squares if there were 10 shaded squares. d. How man shaded squares are there if there are a total of 66 squares?

69 Linear Functions Lesson 4: Linear Function Notation AIIF Page 63 Lesson 4: Linear Function Notation Objectives Students will be able to write an equation in function notation. Students will be able to determine the domain, range, independent variable, and dependent variable of a linear function. Essential Questions How is function notation useful to represent linear function scenarios? Tools Setting the Stage Transparenc Activit Transparenc Student Journal Dr erase boards, markers, erasers Graphing calculator Warm Up Problems of the Da Number of Das for Lesson 1 Da Vocabular input output domain range independent variable dependent variable function function notation

70 AIIF Page 64 Teacher Reference Setting the Stage Displa the Setting the Stage transparenc. Have students determine the rule for the function. At this time, the goal is for students to become familiar with a function as an input/output tpe relationship and to build a small foundation for notation that will be developed in the lesson. You ma want to make other functions for students to practice creating a rule. Students could graph the input versus the output to see that this machine is a linear function. Guiding questions that ou ma want to ask: Would ou ever epect a function machine like this to give two different outputs for the same input? How might the input and output of a function relate to the variables of the matching rule? Could an input value be determined from an output value? What techniques do ou use to assist ou in writing a rule (equation) from an input output scenario?

71 Linear Functions Lesson 4: Linear Function Notation AIIF Page 6 Setting the Stage Transparenc If ou input a number into the function machine below, it will output a new number. The function machine below follows the same rule ever time. Determine the rule the function machine performs. 1 Input Function Machine Output 6 If 1 is the input, then 6 is the output. If 2 is the input, then 9 is the output. If 0 is the input, then 3 is the output. If 3 is the input, then 12 is the output. Write the rule as an equation with as the input and as the output. How do ou know this function is a linear function?

72 AIIF Page 66 Teacher Reference Activit Because the first two pages of this activit have a lot of reading ou ma want to conduct the pre-reading and post-reading activit provided. Have the students put check marks in the What I Think column before the read the first two pages. After the have read those pages, have them put check marks in the What Tet Sas column and answer the two questions. Complete a discussion about the reading, and as needed, complete a guided discussion about the topic and terms brought up in the first two pages. You ma need to epand our guided discussion to include some of the following. A place for discussion notes has been provided below the pre- and post-reading activit. In this activit, students will learn a more formal terminolog and notation for functions. Start with the terms input and output and see if the students remember the more mathematical terms of domain and range. One possible wa to lead the discussion is to talk with students about the website MSpace TM or mabe their room. The idea is that students link their own space (on the web or in their house) as their domain and the range could be all their friends on MSpace TM sites or the range of the neighborhood area where the go to hang out. Another eample is the number of miles a car can drive based on the fuel in the gas tank. The domain is the amount of fuel in the tank while the range would be the miles the car could drive on that fuel. You will also need to discuss with the class independent variable and dependent variable. Eamples that ou can use for independent and dependent variables are: Gallons of fuel and miles traveled Demand for a product and its associated cost Rate of pa and salar Month of the ear and temperature (on average) Interest rate and amount of interest earned in an investment Number of items sold and revenue Refer the students to the definition of a function given at the beginning of the activit. In addition, ou ma want to offer this alternative definition after ou have discussed domain and range: A function is a mathematical correspondence that assigns eactl one element of one set, the domain, to each element of another set, the range. Make sure the students know how to read f(): f is a function of or the short form f of. Tell the class that the function letter replaces the dependent variable. Ask the class Wh did mathematicians use special notation to denote when an equation represents a function? The ke concept is that students understand that when mathematicians see an equation with function notation, the know that each input (domain) will have eactl one output (range) and the will not have to possibl search for other output values. You also might want to discuss that historicall, mathematicians used the letters f, g, and h to rewrite equations in function notation, but when it comes to real-world applications other letters could be used. For eample, the cost function, C, is used to represent the cost of producing a product. Other functions related to producing a product include the revenue function R, and the profit function P. See if the class can come up with a list of possible function letters the ma have come across in their dail lives (we have discussed Suppl and Demand before). Have students work in pairs on Eercises 1 through 3. Bring the class together and have volunteers or groups, share their results with the rest of the class.

73 Linear Functions Lesson 4: Linear Function Notation AIIF Page 67 Pre-Reading and Post-Reading for Activit SJ Page 37 Before reading the tet, put a check mark in the What I Think column for each statement that ou believe is correct. What I Think What Tet Sas The range of a function is how man smbols ou use to write the function. An independent variable is an equation that has no operations. For eample, = is independent and = 2 is not independent. A dependent variable depends on the values of the independent variable. The domain of a function is all the values that could be inputed into the function. Function notation alwas has to be of the form f(), g(), or h(). After reading the tet, put a check mark in the What Tet Sas column for each statement that is correct based on the tet. What changed in our thinking after reading the tet? Answers will var. What new information did ou learn b reading the tet? Answers will var. Notes:

74 AIIF Page 68 SJ Page 38 Activit In this activit, ou will investigate function notation and some of the formal terminolog used with functions. Recall, a function is a rule that assigns to each input value eactl one output value. A function allows a person to describe one quantit in terms of another. In the Setting the Stage, ou investigated a machine that behaves as a function. Here is a different function machine called f. input f output When 1 is the input, is the output. When 2 is the input, 7 is the output. When 3 is the input, 9 is the output. The rule that represents this function is = With formal function notation, the can be replaced with f() and written as f( ) = 2+ 3 and read as, The function of is equal to two times plus 3. When the smbols f() are used to represent a function it means f of or f is a function of and not f times. Using this notation is convenient for man reasons. One reason is that ou can see the input and output values for ordered pairs while ou see the function. For eample, compare the following two different was to represent the input and output statements. When 1 is the input, is the output. f (1) = 2( 1) + 3 = When 2 is the input, 7 is the output. OR f (2) = 2( 2 ) + 3 = 7 When 3 is the input, 9 is the output. f (3) = 2( 3) + 3 = 9 It is common for functions to be named with an f, g, or h and the input value to be represented b. 1. On the dr-erase board, create our own linear function and name it with a letter. Use function notation to represent the equation that represents our rule. Record our function below Answers will var: Sample response: g() = 2 Trade dr-erase boards with our partner and represent three different ordered pairs using function notation with our partner s function. Record the information below. You ma need to walk around and help students as needed on this eercise. g(2) = 2 2= 12 Answers will var: Sample response: ( ) g(4) = ( 4) 2= 22 g( 3) = ( 3) 2= 13 Have our partner check to make sure our ordered pairs are correct. Let s use function notation to look at Ramond s scenario with fling an F/A-18 and learn about a second reason that function notation is convenient. The reason is that the letters used to write the function can match the scenario.

75 Linear Functions Lesson 4: Linear Function Notation AIIF Page 69 SJ Page 39 Scenario Ramond is traveling toward his Air Force base in an F/A-18 at a constant speed of 70 miles per hour, which is just under the speed of sound. He is tring to travel as fast as he can without breaking the sound barrier which is 767 mile per hour. He started at a distance of 220 miles from base. Equation The equation that represents this scenario is = Using function notation, the equation could be f ( ) = , where represents the time in hours flown and f() represents the distance in miles from base. While it is common for functions to be named with letters such as f, g, or h and input values to be labeled with, it ma be more convenient to use letters that match the scenario. In this scenario, the input value is time in hours so instead of using it ma make more sense to use the letter t for time. Because the output value for this scenario is distance from base, it will make more sense to use d for distance instead of f. The new method d t = t+. to write the equation would be ( ) With functions, we often refer to the two quantities that can change as the independent and dependent variables. The dependent variable of a function is the variable that depends on the value of the independent variable. In the case of Ramond fling, the dependent variable is the distance from base and the independent variable is the time in hours that Ramond has flown because the distance from base depends on the time flown. The column heading in the table of ordered pairs could be titled to match this new notation. Table of Ordered Pairs Independent Variable Time in Hours t Dependent Variable Distance from Base in Miles d(t) The domain of a function is all the values that could represent the independent variable and that could be inputs in the function. The domain of this function is 0 t 4, because this represents all the input values or independent variable values that make sense for this scenario. Finall, when graphing functions the aes can be labeled with function notation. Graph Distance from Base in Miles d(t) 0 Traveling Jet Time in Hours t The range of a function is all the values that could represent the dependent variable and that could be outputs of the function. The range of this function is 0 d(t) 220, because this represents all the output values or dependent variable values that make sense for this scenario. The independent variable is tpicall graphed on the horizontal ais and the dependent variable is tpicall graphed on the vertical ais.

76 AIIF Page 70 SJ Page 40 Practice appling this new terminolog with the following eercises. 2. This table of ordered pairs represents a linear function. Table of Ordered Pairs Hours Worked Weekl Salar in Dollars a. Write a scenario that could represent this table. Scenario: Answers will var. Sample response: Kenata, the paroll administrator, uses this table of values to estimate the weekl salar for emploees in the manufacturing plant. b. What are the independent and dependent variables of this linear function? Describe the letters ou used to represent the independent and dependent variables. Sample response: The independent variable is time or hours worked. I will use t to represent time. The dependent variable is salar or dollars earned. I will use s(t) to represent the salar as a function of time. Graph: c. Graph a scatter plot of the table of s(t) Weekl Salar ordered pairs on the coordinate grid and label the ais correctl See graph for student responses. Students should include appropriate labels and units d. Determine an equation that 10 represents this linear function and 100 use function notation to write the 0 equation. 0 t Hours Worked Equation: s( t ) = 9.2t + 0 or st ( ) = 9.2tStudents should be able to determine an appropriate equation based on a -intercept of 0 and slope of 9.2. e. According to the information given in the table, what is the domain and range of this linear function? Sample response: The domain is time between 0 and 40 hours, 0 t 40. The range is between 0 and $370.00, 0 s(t) 370. Another method to represent the domain and range is with discrete data such as, domain is {0, 12, 1, 18, 22, 2, 30, 40} and range is {0.00, , 138.7, 166.0, 203.0, 231.2, 277.0, }. Both of these could be viable depending upon whether the table represents onl the eact discrete ordered pairs or a range of ordered pairs. Weekl Salar in Dollars

77 Linear Functions Lesson 4: Linear Function Notation AIIF Page Use the equation = 4 + to complete the following. SJ Page 41 a. Write a scenario that could represent the equation. Sample response: The da started with the temperature at F and increased 4 ever hour. b. Create a table of ordered pairs. Sample response: c. Graph our table of ordered pairs. Sample graph. d. Write the equations in words. 0 Sample response: equals four times plus fift five. e. Rewrite the equation in function notation using the letter h to represent the function. 0 h() = How is function notation useful to represent linear function scenarios? Answers will var. The goal is for students to write about the conveniences of seeing the input and output as well as the letter representing the words for the independent and dependent variables.

78 AIIF Page 72 SJ Page 42 Practice Eercises 1. For the following tables of ordered pairs, state the domain, range, independent variable, and dependent variable. Also, write an equation with function notation that represents the ordered pairs. a. b. t B Sample responses: domain: 0, 10, 20, 30, 40, 0, 100 or 0 t 100 range: 2, 4, 6, 8, 10, 12, 22 or 2 B 22 independent variable: t dependent variable: B function notation: B(t ) = 2t + 2 Sample responses: domain: 4, 1, 2,, 10, 20, 0 or 4 0 range: 14,, 4, 13, 28, 8, 148 or 148 t 14 independent variable: dependent variable: function notation: g ( ) = 3+ 2 c. Militar Time in Hours t Temperature T in Fahrenheit Midnight (0) 48 F 3 7 F 63 F F 13 (1pm) 87 F 18 (6pm) 102 F Sample responses: domain: 0, 3,, 10, 13, 18 or 0 t 18 range: 48, 7, 63, 78, 87, 102 or 48 t 102 independent variable: t dependent variable: T function notation: T( t) = 3t A certain compan makes hammers that cost $1 per item to make. Fied costs, costs that do not depend on making a hammer, for the compan are $1,000. An equation that could represent this function is = , where is the number of hammers made and is the cost of producing the hammers. Change the equation to function notation and use letters that represent the information in the scenario. Describe the independent variable, dependent variable, domain, and range. Sample response: C(h) = 1h , C(h) is a function of h. The independent variable is the number of hammers, h, that are produced. The dependent variable, C(h), is the cost to produce h hammers. The domain is all h 0 where h is an integer. The range is {1000, 101, 1030, 104, }.

79 Linear Functions Lesson 4: Linear Function Notation AIIF Page 73 SJ Page Rachel recentl gave birth to her first bab. During her pregnanc she was required to gain weight to help guarantee a health bab. Now, Rachel would like to begin an eercise program to loose the etra weight she gained. Rachel plans to start out slowl b stretching, walking and running a certain amount of time each week. Rachel bought a special bab carriage that will allow her to take her bab with her during her eercise. Her eercise program consists of a warm up stretch, a walk, a short run, and a cool down stretch. Rachel will start out with 30 minutes each da for the first week. After the first week, Rachel will increase her eercise routine b 8 minutes each week. Write an equation that represents the amount of time that Rachel eercises each da as a function of the number of weeks eercising. Describe the independent variable, dependent variable, domain, and range. Sample response: E(w) = 8w+ 30, E is a function of w, where E(w) is the dependent variable and w is the independent variable. Students ma describe the dependent variable as minutes eercising each da and independent b number of weeks training. The domain is all w 0 where w is an integer; the range is 30, 38, 46, 4, etc. Note: It is difficult to describe the upper limit of the range of this function. Theoreticall, it could be 1440 minutes (24 hours) but that is not reall phsicall possible. It ma make sense to approimate the upper limit of the range to be around 118 minutes which is close to 2 hours. Which week will Rachel eercise for 110 minutes each da? Solving 110 = 8w + 30, for w gives 10. The 10 th week. 4. Consider a game show in which mone is won b answering questions. Contestants start with $00.00 and for each question answered correctl the contestants win $3, Write an equation that represents the amount of winnings in terms of the number of questions answered correctl. Describe the independent variable, dependent variable, domain, and range. Sample response: W( q) = 300q + 00, W(q) is the total winnings and is the dependent variable. q is the number of questions answered correctl and is the independent variable. The domain is all q 0 where q is an integer; the range is 00, 4000, 400, 00, etc. Use the equation to determine how man correct answers a constant would need to win $200,000. Solving q 00 = +, for q gives 7. There would need to be 7 questions answered correctl to win $200,000.

80 AIIF Page 74 SJ Page 44. Write a linear function that represents the information in this table of ordered pairs. Hot Dogs d P(d) = 0.7d 0 Profit P a. Write a scenario that represents this function. Answers will var. Sample response: Renatta owns a hot dog stand. It costs her $0 each da to run the hot dog stand. She makes $0.7 per hot dog that she sells. b. We see that if no hot dogs are sold, then the profit is $0. Describe what this means. Sample response: It costs Renatta $0 each da to run the hot dog stand so she starts with a negative profit. She doesn t start to make a profit until after the 66 th hot dog sold. For Eercises 6 through 8, write an equation in function notation for the given table, fill in an blank table entries, graph the data table, and write a scenario. If there does not appear to be a real-world scenario, write the rule in words. 6. Table of Ordered Pairs: Graph: See students graphs. s P Equation: 1 P(s) = s 2 Scenario: Sample response: The store is offering a sale of 2 sodas for $1. For eample, if a person purchases 8 sodas, the will spend $4.

81 Linear Functions Lesson 4: Linear Function Notation AIIF Page 7 7. Table of Ordered Pairs: Graph: See students graphs. SJ Page 4 Velocit v Momentum p Equation: p(v) = 2v Scenario: Sample response: Momentum is equal to 2 times velocit. 8. Table of Ordered Pairs: Graph: See students graphs. f() Equation: f () = + 12 Scenario: Sample response: The dependent variable is 12 more than the product of and the independent variable.

82 AIIF Page 76 SJ Page 46 Outcome Sentences The independent variable The dependent variable The difference between input and output To write an equation in function notation The domain is The range is The part I don't understand is

83 Linear Functions Lesson 4: Linear Function Notation AIIF Page 77 Lesson 4 Quiz Answers 1. Sample responses: Domain: 4, 2, 0, 2, 4, 6, 8 Range: 10, 2, 6, 14, 22, 30, 38 independent variable: dependent variable: function notation: f () = Sample response: C(h) = 2h , C(h) is a function of h. The independent variable is the number of headphones, h, that are produced. The dependent variable, C(h), is the cost to produce h sets of headphones. The domain is all h 0 where h is an integer. The range is {8000, 802, 800, 807, }. 3. Equation: c(u) = 0.40u Scenario: Sample response: Cell phone cost is equal to 0.40 times the number of minutes used. Table of Ordered Pairs: Graph: Cell Minutes Used u Total Cost c Total Cost c Cell Minutes Used u

84 AIIF Page 78 Lesson 4 Quiz 1. For the following table of ordered pairs, state the domain, range, independent variable, and dependent variable. Also, write an equation with function notation that represents the ordered pairs Domain: Range: independent variable: dependent variable: function notation: 2. A certain compan makes headphones that cost $2 per item to make. Fied costs, costs that do not depend on making the headphones, for the compan are $8,000. An equation that could represent this function is = , where is the number of headphones made and is the cost of producing the headphones. Change the equation to function notation and use letters that represent the information in the scenario. Describe the independent variable, dependent variable, domain, and range. 3. Write an equation in function notation for the given table of ordered pairs, fill in an blank table entries, graph the data table, and write a scenario. Table of Ordered Pairs: Graph: Cell Minutes Used u Total Cost c Equation: Scenario:

85 Linear Functions Lesson : Other Forms of Linear Functions AIIF Page 79 Lesson : Other Forms of Linear Functions Note: Complete this lesson if ou want students to become familiar with the standard form and point-slope form of linear functions and equations, as well as vertical lines, horizontal lines, parallel lines, and perpendicular lines. Objectives The students will be able determine the standard form of a linear equation based on a table of values or graph. The students will be able to determine the point-slope form of a linear equation based on a table of values or graph. The students will be able to determine a linear equation that matches a real-world linear function scenario. The students will be able to determine the linear equation for vertical and horizontal lines. The students will be able to determine the equation of a line given various conditions of parallel or perpendicular. Essential Questions What are the advantages of using linear equations to model linear functions and real-world applications? Tools Setting the Stage Transparenc Activit Transparenc Student Journal Dr erase boards, markers, erasers Graphing calculator Warm Up Problems of the Da Number of Das for Lesson 3 Das A suggestion is to complete Activit 1 on the first da, complete Activit 2 on the second da, and then Practice Eercises and quiz on the third da. Vocabular Standard form of linear equation Vertical line Parallel slope Ordered pair Run Slope-intercept form of linear equation Point-slope form of linear equation Horizontal line Perpendicular slope Rise Slope Point

86 AIIF Page 80 Teacher Reference Setting the Stage In this lesson students will work with two other forms of linear equations: standard form and point-slope form. To begin to introduce students to these other forms, use this Setting the Stage to show students that a graph of a linear equation can be described b three different equations. Displa the Setting the Stage. 1. Have students help ou complete the tables of ordered pairs. 2 2 = = 12 2= ( 3) Have students describe what is similar in the tables. Students should see that the tables are the same. Each input value gives the same output value in the corresponding equations. 3. Because these equations give the same tables of ordered pairs, the represent the same equation. Students ma need to graph each of the tables to see that the are the same linear equation.

87 Linear Functions Lesson : Other Forms of Linear Functions AIIF Page 81 Setting the Stage Transparenc Recall the slope-intercept form to represent linear equations. = m + b In this lesson, ou will continue to work with two additional forms to represent linear equations called standard form and point-slope form. Standard form: A+ B = C Point-Slope form: = m( + ) Complete the following three tables to match the corresponding form of the linear equation. 2 2 = = What is similar about the tables? + = ( ) 3. Do these three equations represent the same linear equation?

88 AIIF Page 82 Teacher Reference Activit 1 In this activit, students will investigate the standard form of a linear equation. This activit is designed for students to eplore the - and -intercepts of an equation in standard form. The standard form is introduced with a real-world application to give contet. The goal is for students to be able to graph a line from a standard form equation b determining the ordered pairs for the - and -intercepts, plotting the points, and then connecting the points with a line. A suggested strateg for this activit is to have the students work in pairs on Eercises 1 through, followed b a discussion of what the determined. You ma want to begin displaing the different forms of a linear equation on the wall. After the lesson, ou could have the students help ou create a poster that represents the various forms of linear equations. Guide students through Eercise 6. Think out loud as ou work the problem and solicit help from the students. As ou guide students, ou ma want ask questions such as: Wh is the -intercept found when substituting zero in for? If ou substitute zero for, what intercept is found? Wh? What is unique about the intercepts for the equation =0 or =? Have students, in groups of four, complete Eercises 7 through 14. Follow with a discussion confirming their understanding of the standard form. While students are working on these eercises, ou ma need to model how to change equations to standard and to slope-intercept form. Model changing the equation + 1 = 4 to standard form, with a positive coefficient, b multipling both sides b negative one and then model changing it to slope-intercept form, while students use the equation = 48. Remember to emphasize to students that the must complete the same operation to both sides of an equation to keep the equations equivalent. You ma want to have the groups create a presentation that describes how to change from one form to another. Now, etend the concept of standard form to include vertical and horizontal lines b having the students complete Eercises 1 through 18. You ma need to walk around and help the groups as needed.

89 Linear Functions Lesson : Other Forms of Linear Functions AIIF Page 83 Activit 1 Let s look at the scenario of buing two items b the pound at the grocer store to understand the standard form of a linear equation. Standard form of a linear equation: A+ B = C SJ Page 47 Scenario: Trenton s big sister sent him to the store to bu all the grapes and all the apples he could with $16 for Sunda s picnic. At the store he found that grapes were $2 per pound and apples were $4 per pound. The linear function that models this scenario can represent the man different combinations of pounds of grapes and apples that Trenton could purchase. Linear Equation in Standard Form: A linear equation in standard form that could represent this scenario is = 16, where represents the number of pounds of grapes and represents the number of pounds of apples. 1. What are the values of A, B, and C for this standard form of a linear equation? A = 2, B = 4, and C = Complete the table of ordered pairs for this scenario. Table of Ordered Pairs Number of Pounds of Grapes Number of Pounds of Apples Graph the table of ordered pairs and draw a line through the points Use the graph of the line to determine the number of pounds of apples Trenton could purchase if he purchased 6 pounds of grapes. Sample response: The ordered pair on the line that represents 6 pounds of grapes also represents 1 pound of apples. This forms the ordered pair (6, 1).. Use the graph of the line to approimate the number of pounds of apples Trenton could purchase if he purchased 3. pounds of grapes. Sample response: The ordered pair on the line that approimates 3. pounds of grapes also represents about 2.3 pounds of apples. This is the ordered pair (3., 2.3).

90 AIIF Page 84 SJ Page 48 One method for graphing the line of a standard linear equation is to determine the -intercept and -intercept and then connect the two points. Generall, a simple table of ordered pairs is used to record the and - intercept b substituting 0 for and then 0 for. 6. Graph the standard form equation = 18 b completing the following. a. Determine the and -intercepts b completing this table of ordered pairs b. Graph the and -intercept and then connect the points. See graph. c. What would be the slope-intercept form of the equation? 3 = Use the method of graphing lines described above to draw the graphs for the following linear equations in standard form. Determine the slope intercept form of the equation for each. a. + 3 = 30 b. 4 3 = 12 c = 12 4 = + 10 = d. = 1 e. =0 1 = 3 2 = + 1 =

91 Linear Functions Lesson : Other Forms of Linear Functions AIIF Page 8 SJ Page 49 Man mathematicians prefer the coefficient A in the standard form linear equation, A+ B = C, to be positive. For eample, instead of = 36, an equivalent equation would be 3 9 = Discuss in our group of four how ou could change the following equation into standard form so that A is positive. Be prepared to share our group s response = 24 a. Check to make sure our new equation is equivalent to the given equation b determining if the and -intercepts are the same for each equation. b. Will the method ou used work for other equations? Tr it with the following equations. + 7 = 70 4 = = 24 Answers will var. Walk around to check with groups. Also, the coefficients A, B, and C, on standard form linear equations need to be integers. The shouldn t be fractions. 9. Discuss in our group of four how ou could change the following equation into standard form so that A, B, and C are integers. Be prepared to share our group s response = 3 4 a. Check to make sure our new equation is equivalent to the given equation b determining if the and -intercepts are the same for each equation. b. Will the method ou used work for other equations? Tr it with the following equations = = Answers will var. Walk around to check with groups. 3 = 10 4 In Unit 1, ou learned how to manipulate equations to solve for an unknown variable b completing the same operation to both sides of the equal sign and simplifing as needed. You can use a similar method to change one tpe of linear equation into another. 10. Change the following equation from standard form to slope-intercept form as our teacher completes an eample. 6 8 = = = = = = + 4 4

92 AIIF Page 86 SJ Page Change the following linear equation from slope-intercept form to standard form as our teacher completes an eample. 4 = 7 4 = 7 4 = 7 = = = Change the following standard form linear equations into slope-intercept form. a. + = b. 4 = 8 = = 4+ 8 c. 9 4 = 20 d = 78 9 = 4 10 = Change the following linear equations into standard form. a. 2 = 8 b. = = = 77 c = 9 d = = = How might ou determine if an equation in standard form such as = has positive or negative slope? Answers will var. Students should sa something about changing it to slope-intercept form to determine the slope. Some students ma be able to figure out that when A and B are both the same sign the slope will be negative and when A and B are different signs the slope will be positive.

93 Linear Functions Lesson : Other Forms of Linear Functions AIIF Page 87 SJ Page 1 Sometimes the graph of a linear equation will be either a vertical or horizontal line. The following four eercises will help ou construct knowledge around the relationship between the equation and graph of vertical and horizontal lines. 1. Fill in the table of ordered pairs for the standard form linear equation = 6and then graph the ordered pairs to create a connected scatter plot representing the equation See students graphs. a. Describe the graph of the equation. Sample response: The graph is a vertical line with a -intercept of 6. b. Solve the equation for. = 6 c. How does the graph of the line correspond to the equation in part b and the table of ordered pairs? Sample response: The graph, the equation, and the table of ordered pairs all show that ever value on the line has a -value of Fill in the table of ordered pairs for the standard form linear equation = and then graph the ordered pairs to create a connected scatter plot representing the equation See students graphs. a. Describe the graph of the equation. Sample response: The graph is a vertical line with an -value of. b. Solve the equation for. = c. How does the graph of the line correspond to the equation in part b and the table of ordered pairs? Sample response: The graph, the equation, and the table of ordered pairs all show that ever value on the line has an -value of.

94 AIIF Page 88 SJ Page Draw the graphs of the following equations. a. = 3 b. = 2 c. = 7 d. = Write the equations for the following graphs. a. b. = 7 = 3 c. d. = 8 =

95 Linear Functions Lesson : Other Forms of Linear Functions AIIF Page 89 Teacher Reference Activit 2 In this activit, students investigate another form of a linear equation called point-slope form. Guide the class through Eercise 1. You ma need to write the point-slope form of a line on the board or overhead, =, and the slope m. m( ). Tell the class that this form is used when given a point ( ) 1 1 Have the students work in groups of four. Assign each group a different eercise from Eercises 2 through. You ma need to assign more than one group to the same eercise depending on the size of our class. Make sure each student in the group has his/her own dr-erase board. Tell them the will create a table for their equations that are written in point-slope form and then graph them on their dr-erase boards. Have each group complete a group eercise. Make sure to tell the class not to erase the graphed lines until ou tell them to do so. The will need the graphed equations for their questions as well as for sharing with the other groups. After the groups have completed their eercise, have three members of each group go to a different group and share what the have discovered from their activit with the other groups. Then have the new groups answer Eercise 6. These new groups ma need help solving Eercise 6d. While in the new groups, each student should write facts that the other groups have discovered and are sharing. Have the original groups get back together. Within their groups, the students should discuss the information the have obtained from the other groups. Although, theoreticall, each group member should have obtained the same information, the information will probabl be written down and conveed differentl from different group members. Discuss the concept of parallel lines and what it means for two lines to be parallel to each other. Have the class agree on a definition for parallel lines and write the definition in their student journal. Lead a class discussion about the three different forms of linear equations that the worked with so far: slopeintercept, standard, and point-slope form. Ask what the have learned so far in the unit about linear equations and linear functions. Have a volunteer write the responses in the L portion of the K W L chart that the started in Lesson 1. Have students appl their understanding of linear equations b completing Eercises 7 through 10. You ma want to have students etend their understanding of linear equations to situations that involve parallel and perpendicular lines. Have groups complete Eercises 11 through 12. Have students share their results with the class and lead a discussion about perpendicular lines. Have the students determine a definition for perpendicular lines involving the two slopes m 1 and m 2. Have the students write this definition in their student journals. You could also have them solve for one of the slopes given the other and the fact that two lines are perpendicular. For eample, if we were given m 1, or could determine it from the equation, and were told that a line is perpendicular to the given line, then the students should be able to use the product of the slopes to determine that the slope of the second line is m 2 = 1/m 1. This will tell the class that the second slope is the negative reciprocal of the first slope. Discuss Eercise 13 with the students and then have them complete Eercises 14 and

96 AIIF Page 90 SJ Page 3 Activit 2 In this activit, ou will investigate the point-slope form of a linear equation. The point-slope equation is useful when ou know at least one point and the slope of the linear equation. Point Slope Form of a Linear Equation: =, is a m( ), where ( ) given point on the line and m is the slope. 1. In 1980, the average price of a house in the United States was about $73,600. The rate of increase has been approimatel $8, per ear. The point-slope form of a linear equation that could represent 73,600 = this information is ( ) a. Determine what part of the scenario represents the slope. How does this compare to the rate of increase each ear? Sample response: The part that represents the slope is m, which has been replaced b 8,840 which is the approimate rate of increase each ear. This rate corresponds to a slope of 8,840. b. Determine what part of the scenario represents the point. Sample response: The part that represents the ordered pair for the point has the part on the right and the part on the left. These values come from the ear that matches the price. From this eample, at ear 1980 the price was $73,600. The ordered pair for the point would be (1980, 73600) where 1 is 1980 and 1 is 73,600. c. Use the equation to determine what the possible average price ma be toda. Note to teacher: You ma want to look up the current actual average price of homes in the United States and = for compare. A response for 2009 would be $329,960. Solving ( ) gives 329,960.

97 Linear Functions Lesson : Other Forms of Linear Functions AIIF Page Create a table of ordered pairs for each set of linear equations. 1= 2( 3) 3= 2( 3) = 2( 3) -value -value -value -value -value -value SJ Page 4 a. Graph each set of ordered pairs from the table and draw the matching lines on our dr-erase board. Place all three lines on the graph at the same time. Don t erase the lines. Describe the similarities and differences between the three lines. Answers will var. Sample response: The lines are parallel and the all have the same slope. Each line has a different -intercept. b. How might these similarities and differences be described b the matching linear equations? Sample response: The slope is 2 on the graph which is m in the equations. The lines are two units apart verticall which is the difference that occurred to 1 in the three different equations. c. Compare the point (3, 1) to the equation in the first table, (3, 3) to the equation in the second table, and (3, ) to the equation in the third table. Describe how these points might relate to the matching equations. Answers ma var. Students should recognize that the values of these ordered pairs for the points are also in the equation as the values subtracted from and. 3. Create a table of ordered pairs for each set of linear equations. 1= 2( 2) 1= 2( 6) 1= 2( 10) -value -value -value -value -value -value a. Graph each set of ordered pairs from the table and draw the matching lines on our dr-erase board. Place all three lines on the graph at the same time. Don t erase the lines. Describe the similarities and differences between the three lines. Answers will var. Sample response: The lines are parallel and the all have the same slope. Each line has a different -intercept. b. How might these similarities and differences be described b the matching linear equations? Sample response: The slope is 2 on the graph which is m in the equations. The lines are four units apart horizontall which is the difference that occurred to 1 in the three different equations. c. Compare the point (2, 1) to the equation in the first table, (6, 1) to the equation in the second table, and (10, 1) to the equation in the third table. Describe how these points might relate to the matching linear equations. Answers ma var. Students should recognize that the values of these ordered pairs for the points are also in the equation as the values subtracted from and.

98 AIIF Page 92 SJ Page 4. Create a table of ordered pairs for each set of linear equations. + 1= 2( + 3) + 3= 2( + 3) + = 2( + 3) -value -value -value -value -value -value Parallel lines: a. Graph each set of ordered pairs from the table and draw the matching lines on our dr-erase board. Place all three lines on the graph at the same time. Don t erase the lines. Describe the similarities and differences between the three lines. Answers will var. Sample response: The lines are parallel and the all have the same slope. Each line has a different -intercept. b. How might these similarities and differences be described b the matching linear equations? Sample response: The slope is 2 on the graph which is m in the equations. The lines are two units apart verticall, which is the difference that occurred to 1 in the three different equations. c. Compare the point ( 3, 1) to the equation in the first table, ( 3, 3) to the equation in the second table, and ( 3, ) to the equation in the third table. Describe how these points might relate to the matching equations. Answers ma var. Students should recognize that the values of these points are also in the equation as the values subtracted from and.. Create a table of ordered pairs for each set of linear equations. + 1= 2( + 2) + 1= 2( + 6) + 1= 2( + 10) -value -value -value -value -value -value a. Graph each set of ordered pairs from the table and draw the matching lines on our dr-erase board. Place all three lines on the graph at the same time. Don t erase the lines. Describe the similarities and differences between the three lines. Answers will var. Sample response: The lines are parallel and the all have the same slope. Each line has a different -intercept. b. How might these similarities and difference be described b the matching linear equations? Sample response: The slope is 2 on the graph which is m on the equations. The lines are four units apart horizontall which is the difference that occurred to 1 in the three different equations. c. Compare the point ( 2, 1) to the equation in the first table, ( 6, 1) to the equation in the second table, and ( 10, 1) to the equation in the third table. Describe how these points might relate to the matching equations. Answers ma var. Students should recognize that the values of these ordered pairs are also in the equation as the values subtracted from and.

99 Linear Functions Lesson : Other Forms of Linear Functions AIIF Page 93 SJ Page 6 6. In our new groups, use our combined knowledge to determine the point-slope form of a linear equation for the following scenario. Charlie just opened his cell phone bill and noticed the charge was $200 for 400 minutes of use. He knows that the rate is $0.40 per minute. a. What information in the scenario represents the slope? The value of 0.40 represents the slope. b. What information in the scenario represents the point? The values $200 and 400 minutes represents the point (400, 200). c. Write the point-slope form of the linear equation. 200 = 0.40( 400) d. Change the equation to the slope-intercept form and graph the equation on a graphing calculator. = e. How much would Charlie spend if he used zero minutes? What would this fee correspond to? Sample response: Charlie would spend $40.00 because = 0.40( 0) + 40= 40. This fee could correspond to a monthl fee for just having the cell phone. f. How is the rate represented on the graph? The rate is the slope of the graph. g. Eplain if ou believe this is a fair phone plan. Answers will var. Students ma mention that a $40 fee for no minutes isn t fair and that $0.40 a minute is too high of a rate.

100 AIIF Page 94 SJ Page 7 You will continue to investigate writing equations of lines from real-world applications with the point-slope form, but this time ou will be given two points. Point-Slope form: = m( ) Let's look at house prices again in a little more detail. In 1980, the average house sold in the United States was $73, B 200, the average house sold in the United States jumped to $294, a. How can ou convert the information given in the scenario above to two sets of ordered pairs that represent two points? Sample response: The ear represents the first coordinate of the ordered paid and the average price represents the second coordinate of the ordered pair. The ordered pairs are (1980, 73600) and (200, ). b. Determine the slope of the line that would go through both points. The slope is c. Determine the point-slope form of the linear equation that would represent the information. Check the equation to make sure the point (200, ) makes it true = 8840( 1980) = 8840( ) = 8840( 2) = d. Assuming this equation will still represent the average price of a house in the United States in the future; determine the approimate price of an average house 10 ears from now. Sample response: If the ear is 2010 then 10 ears from now the price will be approimatel $427, = 8840( ) = Suppose the cost of a business propert is $900,000 and a Depreciation is the reduction in compan wants to depreciate the propert to a value of $0 in the value of an item due to 360 months. If is the value of the propert after months, usage, passage of time, wear and then the compan s depreciation schedule will have a linear tear, technological outdating, equation passing through the points (0, ) and (360, 0). depletion, or other similar Write the linear equation of this depreciation schedule if the factors. propert depreciated at a constant rate. The propert depreciates $2,00 per ear which in the equation would be a slope of 200. The equation = of the line for the depreciation is = ( ) or ( ) a. Use the equation to determine the value of the propert after 80 months. 0 = for gives $700,000. Solving ( ) b. Use the equation to determine the value of the propert after 10 ears. 0 = for gives $600,000. Solving ( )

101 Linear Functions Lesson : Other Forms of Linear Functions AIIF Page 9 9. Antibiotics given to rabbits to fight an infection require a dosage related to the rabbit s weight. The relationship between the dosage and the size of the rabbit is linear. SJ Page 8 a. If a 3 pound rabbit should receive 67. milligrams of antibiotic medicine and a pound rabbit should receive 112. milligrams, write the equation that represents this relationship. Equations will var. One possible equation of the line for this 4 relationship is: 67.= ( 3). 2 b. What is the slope of the line and what does it represent? Simplif our slope to lowest terms. Also find the decimal equivalent for the slope. The slope is 4/2or 22.. It represents that for ever 1 pound increase the dosage should be 22. milligrams more. c. Approimatel how much antibiotic should a rabbit that weighs 9 pounds receive? 4 Solving the equation 67.= ( 9 3) for gives a value of A 9-pound rabbit should 2 receive 202. milligrams of antibiotic. 10. For the following plotted points, write the coordinates of the points, draw a line through the two points, and write a linear equation in point-slope form. Convert the equation into the slope-intercept form. State the coordinates of the -intercept. a. b. a. Coordinates are ( 7, ) and (4, ); point-slope form is form is 10 1 = + ; and -intercept is (0, 1/11) b. Coordinates are ( 8, 8) and (, 8); point-slope form is form is = ; and -intercept is (0, 24/13 ) = ( 4) ; slope- intercept = (+ 8) ; slope-intercept 13

102 AIIF Page 96 SJ Page 9 In mathematics, it is common to analze and create equations that have graphs with parallel lines or perpendicular lines. You learned in Lesson 2 that parallel lines have the same slope. Complete the following eercises to learn about these tpes of equations. 11. Each eercise below represents the equations of two lines that are perpendicular to each other. For each eercise below determine the slope of each line and determine the product of the slopes. 1 a. = 3 and = Perpendicular lines: The slopes of the lines are 3 and 1/3, respectivel. The product of the slopes is 1. b = 8and 6 3 = 12 The slopes of the lines are 1/2 and 2, respectivel. The product of the slopes is 1. = and + = ( + 7) c. 6 ( 3) 1 The slopes of the lines are and 1/, respectivel. The product of the slopes is What do ou notice about the product of the slopes of perpendicular lines? The product of the slopes for perpendicular lines is How can ou determine if two lines are parallel, perpendicular, or neither parallel nor perpendicular to each other? Answers will var. A sample response might be: Two lines are parallel if the have the same slope; two lines are perpendicular if the product of the slopes is 1; if the slopes are not equal or if the product of the slopes is not 1 then the lines are neither parallel nor perpendicular to each other.

103 Linear Functions Lesson : Other Forms of Linear Functions AIIF Page 97 SJ Page For each problem below, find the equation of a line parallel and a line perpendicular to the given line that passes through the given point. Write the linear equations in all three formats: slope intercept form, point slope form, and standard form. a. = 2 + 4and (2, 1) The equation of the line parallel to the given line and passing through the point (2, 1) is 1= 2( 2), = 2 3, and 2 = 3 ; the equation of the line perpendicular to the given 1 line and passing through the point (2, 1) is 1 = ( 2), = + 2, and + 2 = b. 2 + = 3and ( 3, 4) The equation of the line parallel to the given line and passing through the point ( 3, 4) is + 4= 2(+ 3), = 2 10, and 2 + = 10 ; the equation of the line perpendicular to the 1 1 given line and passing through the point ( 3, 4) is + 4 = (+ 3), =, and 2 = c. 2 = 6 12and (, 0) The equation of the line parallel to the given line and passing through the point (, 0) is 0= 3( ), = 3 1, and 3 = 1 ; the equation of the line perpendicular to the given 1 1 line and passing through the point (, 0) is 0 = ( ), = +, and + 3 = The suppl for a certain product is given b the 3 linear function p= q+ 1, where q represents the quantit supplied and p represents the price of the product. Find the linear function for the demand of the product if it is known that the demand function is perpendicular to the suppl function and that the equilibrium point (the location where both the suppl and demand functions intersect) is (7, 60). Graph both equations on the same grid. Write the linear function for the demand in all three formats: slope intercept form, point slope form, and standard form. The demand function in the three formats is p= q+ 18, p 60= ( q 7), and 3 3 q + 3p = p q

104 AIIF Page 98 SJ Page 61 Practice Eercises 1. State whether the linear equation is in standard form, slope-intercept form, point-slope form, or neither. a. 4 = 3( + 2) Point-slope form. b. + 6 = 7 Standard form. c. = 4 6 Slope-intercept form. d = 20 Neither e Neither. This is an epression not an equation. 2. Fill in the table of ordered pairs for the equation = and then graph the ordered pairs to create a connected scatter plot representing the equation Fill in the table of ordered pairs for the equation = 18 and then graph the ordered pairs to create a connected scatter plot representing the equation

105 Linear Functions Lesson : Other Forms of Linear Functions AIIF Page 99 SJ Page Use the graph of a line to write equations in both the slope-intercept form and the point slope form. Also state the slope, -intercept, and a point (other than the -intercept) of the line. a. b. a. The slope is 3/2, the -intercept is (0, ), a point is ( 6, 4) (points ma var), slope-intercept form is = 3/2 + ; point-slope form is + 4 = 3/2( + 6). b. The slope is 3, the -intercept is (0, 7), a point is ( 3, 2) (points ma var), slope-intercept form is = 3 7; point-slope form is 2 = 3( + 3).. For each of the following, state the form of the equation and rewrite the equation in the two other forms. The equation is in the point-slope form. Other forms are = 3 2 and 3 = 2. a. 4= 3( 2) b. 6 = 7 The equation is in standard form. Other forms are c. = = = 4 2 and 4 = 6. = + and ( ) The equation is in slope-intercept form. Other forms are ( ) 6. For each set of points below, determine the point-slope form and slope-intercept form of a linear equation that goes through the two points. a. (, 4) and (7, 8) b. (7, 7) and ( 2, 11) 8= 7 11= 2( + 2) = + 1 = Describe the advantages and disadvantages of the three different forms of linear equations: standard form, slope-intercept form, and point-slope form. Answers will var. 8. Write about a situation where ou would prefer a certain form of linear equation over the others. Answers will var.

106 AIIF Page 100 SJ Page Wh might ou change from one linear equation form to another? Answers will var. 10. Which linear equation form or forms lends itself to determining parallel lines quickl? Answers will var. 11. Which linear equation form or forms lends itself to determining the -intercept quickl? Answers will var. 12. What advantage does the graphing calculator give ou to understand linear equations? Answers will var. 13. The graph to the right represents linear data collected b a coach over ears of training 100 meter sprinters. The coach has noticed a linear trend comparing temperature during the race and average time to complete the race. a. Complete the data table of ordered pairs that represents the graph. -value Temperature -value Average Time Average Time (Seconds) Meter Times Temperature (Celsius) b. Determine the linear equation that represents the graph and table of ordered pairs. Equation: Sample responses could include ( 18) =, = , or + 20 = 234

107 Linear Functions Lesson : Other Forms of Linear Functions AIIF Page 101 SJ Page Complete the table of ordered pairs. Sketch a graph, determine a matching linear equation, and write a scenario. Table of Ordered Pairs: Graph: 200 Minutes Cost ($) Equation: C(m) = 0.2m Or C(m) = 0.2( m 0) + 40 Scenario: Answers will var. Sample response: The phone compan charges $27.0 a month to just have a phone and then $0.2 a minute. 1. Complete the table of ordered pairs. Sketch a graph, determine a matching linear equation, and write a scenario. Table of Ordered Pairs: Graph: Length of Height (cm) Tibia (cm) Equation: H = 2.l+ 73 Scenario: Answers will var. Sample response: The height of a person can be predicted b the length of their tibia. Starting at a tibia length of 10 centimeters and height of 98 centimeters a person will be 2. centimeters taller for each centimeter increase in the length of the tibia.

108 AIIF Page 102 SJ Page In the following eercises, graph the equations, state the values of the rise and run, and then calculate the rise/run ratio, also known as the slope, of the line. a. 4 + = 20 b. + 4 = 8 a. The rise is 4 and the run is. The slope is 4/. b. The rise is 1 and the run is 4. The slope is 1/4. c. = d = 12 c. The rise is 3 and the run is 1. The slope is 3. d. The rise is 2 and the run is 3. The slope is 2/3

109 Linear Functions Lesson : Other Forms of Linear Functions AIIF Page 103 e. = 4 f. = SJ Page 66 e. Sample response: The rise is 0 and the run is continuous. The slope is 0. f. The rise is continuous and the run is 0. The slope is undefined. 17. For each pair of linear equations below, determine if the equations represent lines that are parallel, perpendicular, or neither. Eplain wh the lines are parallel, perpendicular, or neither. a. 6 + = 7 and + 6 = 3 The lines are neither parallel nor perpendicular because the slopes are 6 and 1/6 which are not equal and the product of the slopes is 1, not 1. b. 4 = 9 and + 4 = 2 The lines are perpendicular because the product of the slopes is 1. c. = 8 1 and 24 3 = 12 The lines are parallel because the slopes are both equal to 8.

110 AIIF Page 104 SJ Page For each problem below, find the equation of a line parallel and a line perpendicular to the given line that passes through the given point. Write the linear equations in all three formats: slope intercept form, point slope form, and standard form. a. = 4 2and ( 3, 4) The equation of the line parallel to the given line and passing through the point ( 3, 4) is 4= 4(+ 3),= 4 8, and 4 + = 8 ; the equation of the line perpendicular to the given line and passing through the point ( 3, 4) is 4 = (+ 3), = +, and 4 = b = and (, 2) The equation of the line parallel to the given line and passing through the point (, 2) is = ( ), =, and 2 3 = 16 ; the equation of the line perpendicular to the given line and passing through the point (, 2) is + 2 = ( ), = +, and =

111 Linear Functions Lesson : Other Forms of Linear Functions AIIF Page 10 Outcome Sentences SJ Page 68 The difference between the standard form of a line and the point-slope form of a line is m, I would To obtain the point and slope from the linear equation, = ( ) 1 1 The difference between the slope-intercept form of a line and the point-slope form of a line is I need help in understanding I can remember how to write an equation for a vertical line b The equation for a horizontal line Parallel lines have The slopes for perpendicular lines are

112 AIIF Page 106 Teacher Reference Lesson Quiz Answers 1. The slope of the line is a. Point slope form: 7 = 2( ) or + 7 = 2( + 2) Slope intercept form: = 2 3 Standard form: 2 = b. Point slope form: + 2 = ( 3) or = (+ 8) Slope intercept form: = Standard form: = 1 3a. The product of the slopes is 1 so the lines are perpendicular. 3b. The slopes are equal so the lines are parallel The equation of the parallel line that passes through the point (, ) is + = ( ). The equation 3 3 of the perpendicular line that passes through the point (, ) is + = ( ). 2

113 Linear Functions Lesson : Other Forms of Linear Functions AIIF Page 107 Lesson Quiz 1. Create a table of values for 4 2 = 4, graph the equation using the table of values, and calculate the slope for the equation. 2. For each set of points below, determine the point-slope form, slope-intercept form, and standard form of a linear equation that goes through the two points. a. ( 2, 7) and (, 7) b. (3, 2) and ( 8, ) 3. For each pair of linear equations below, determine if the equations represent lines that are parallel, perpendicular, or neither. State wh the lines are parallel, perpendicular, or neither. a. + 9 = 4 and 9 = 6 b. 9 3 = 17 and 3 + = Find the equations of the lines that are both parallel and perpendicular to = 6 that pass through the point (, ). Use the point slope form for the equations.

114 AIIF Page 108 Lesson 6: What is a Function? Objectives The students will be able to determine if a graph could represent a function b using the vertical line test. The students will be able to graph various non-linear functions using the graphing calculator. The students will be able to create a graph that models a real-world scenario such as distance time graphs. Essential Questions What characteristic of a graph confirms that it can represent a function? Tools Setting the Stage Transparenc Activit Transparencies Student Journal Dr erase boards, markers, erasers Graphing calculator Ruler Warm Up Problems of the Da Number of Das for Lesson 1 or 1. Das One da could be dedicated to the Setting the Stage and Activit 1. You ma want to assign Practice Eercises 1 through 3 after Activit 1. A second half da would probabl suffice to complete Activit 2, the remaining Practice Eercises, outcome sentences, and quiz. Vocabular function output input vertical line test

115 Linear Functions Lesson 6: What is a Function? AIIF Page 109 Teacher Reference Setting the Stage This Setting the Stage is to introduce students to scenarios that don t have linear graphs. The goal for this activit as well as the lesson is to help students realize that not all functions are linear functions. With this simple introduction, ou should be able to show students that population as a function of time ma not behave linearl and that it can fluctuate. Displa the Setting the Stage transparenc and ask the students which graph the believe matches the scenario. Have students describe wh the believe the graph the chose is correct. Have students show and describe on the graph the three different population growths for the scenario: slow increase, rapid increase, and constant increase. After students have determined the correct graph, have them create their own population graph on a dr-erase board. Then have the students trade with a partner and write a scenario for the graph. Ask volunteer pairs to share their graphs and scenario. Some guiding questions ou ma want to ask during this process could be: What feature on the graph would represent no change in population? What feature on the graph would represent a decrease in population? What feature on the graph would represent an increase in population? What does a larger slope on the graph mean as compared to a smaller slope? Wh would it not make sense for the population to go below the -ais?

116 AIIF Page 110 Setting the Stage Transparenc Determine which of the four graphs below best matches the scenario. Scenario The population of Sunnville increased slowl at first then had a rapid increase and finall remained constant population population time time population population time time

117 Linear Functions Lesson 6: What is a Function? AIIF Page 111 Teacher Reference Activit 1 Displa Activit 1 Transparenc 1. Introduce the students to distance-time graphs and relationships b discussing the distance a race car travels over time. Because the race car maintained a constant speed the graph was linear. You ma want to discuss the feasibilit of a race car traveling at a constant speed for more than 3 hours. More than likel it would slow down and speed up for various reasons such as turning and passing. It ma also need to make a pit stop. If the race car were traveling at 200 miles per hour, it would travel 200 miles in one hour and 00 miles in 2 ½ hours. Pre-Reading Talk about the terms constant, rate, constant rate, speed, units of time, and units of distance. You ma also want to talk about the difference between the distance an object has traveled and the distance an object is from a location. These are often not the same values. For eample, ou could travel awa from home for two miles and then return toward home for one mile. The distance ou would travel would be three miles, but the distance from home would onl be one mile. Activit Displa Activit 1 Transparenc 2. Have 8 to 10 students line up in front of class and demonstrate a model of the first graph b having each student pass an object to the net person in the row. The students should pass the object to a new person ever second. Count the seconds out loud. Separate the class into groups of 8 to 10. Have each group complete one group assignment. After ou have given time for the groups to practice, have them demonstrate to the class each of their assignments. Displa the activit transparencies as each group models. The students should gain an understanding of what makes a straight line or a curve on a distance time graph as well as what a horizontal line means. Have students in pairs complete Eercise 9. Walk around to answer questions and to help as needed. Have 8 to 10 volunteers tr to demonstrate Eercises 10 through 13. The goal is for students to realize that these graphs are impossible with one object, because one object cannot be in two places at the same time. This will give the foundation for students to understand the vertical line test for determining if a graph can represent a function for the net eercises. Before students answer Eercise 1, ou ma want to have a volunteer read the paragraph before Eercise 1, then draw a few graphs on the board and ask students to determine if the graphs ou drew can represent functions based on the vertical line test. You ma want to pass out rulers to students to use for the vertical line test.

118 AIIF Page 112 Activit 1 Transparenc 1 It is common on a coordinate plane to have the -ais represent the distance an object has traveled and to have the -ais represent the time the object has traveled. The following graph represents the distance a race car has traveled over time. Distance in Miles Time in Hours 1. How far did the race car travel in one hour? 2. How far did the race car travel in 2 1 hours? 2 3. Did the race car travel at a constant speed or did it change speeds?

119 Linear Functions Lesson 6: What is a Function? AIIF Page 113 Activit 1 Transparenc 2 10 Distance in 1 Unit Intervals Time in Seconds

120 AIIF Page 114 Activit 2 Transparenc 3 Group Distance Distance Time in Seconds Time in Seconds Group Distance Distance Time in Seconds Time in Seconds

121 Linear Functions Lesson 6: What is a Function? AIIF Page 11 Activit 2 Transparenc 4 Group Distance Distance Time in Seconds Time in Seconds Group Distance Distance Time in Seconds Time in Seconds

122 AIIF Page 116 Activit 1 Transparenc Distance Distance Time in Seconds Time in Seconds Distance Distance Time in Seconds Time in Seconds

123 Linear Functions Lesson 6: What is a Function? AIIF Page 117 Activit 1 SJ Page 69 In this activit, ou will model the movement of an object as it travels along a human assembl line. The assembl line will be made of 8 to 10 students. The students should spread evenl shoulder to shoulder all facing the same direction. The object will pass from person to person along the line. Each pass will be considered a movement of 1 unit in distance. Time will be measured in one-second intervals. The teacher will count out loud for each second. The objective of the activit is to model distance-time graphs. Your teacher will demonstrate the first model of a distance-time graph. Watch and participate as our teacher models a distance-time graph based on the linear function =, where represents distance and represents time. Notice that for each second of time, the object should pass from one person to another. 10 Distance Time in Seconds Group 1 Model the following distance-time graphs b passing the object. Be prepared to eplain our group s reasoning Distance Distance Time in Seconds Time in Seconds For graph 1, pass the object to one person ever two seconds. For graph 2, pass the object to one person ever two seconds and then the person holding it at four seconds holds it for the remaining time.

124 AIIF Page 118 SJ Page 70 Group 2 Model the following distance-time graphs b passing the object. Be prepared to eplain our group s reasoning Distance Distance Time in Seconds Time in Seconds Group 3 For graph 3, pass the object to one person ever second, after five passes, pass it back the wa it came at the rate of one person ever second. For graph 4, pass it from the furthest person back at a rate of one person ever second. Model the following distance-time graphs b passing the object. Be prepared to eplain our group s reasoning Distance Distance Time in Seconds Time in Seconds For graph, the person five units from the first person holds it for the entire time. For graph 6, pass the object to one person during the first 3 seconds, then one person during the net two seconds, then one person the net second, then two people the net second, and finall three people the last second. The object in graph 6 increases in speed over time.

125 Linear Functions Lesson 6: What is a Function? AIIF Page 119 Group 4 SJ Page 71 Model the following distance-time graphs b passing the object. Be prepared to eplain our group s reasoning Distance Distance Time in Seconds Time in Seconds For graph 7, one person holds the object for three seconds then passes it to the net person who holds it for three seconds and then passes it to the net person, and so forth. For graph 8, pass the object through three people the first second, two people the net second, one person the net second, one person over the net two seconds, and then one person the net three seconds. It slows down over time. Let s refer back to the definition of a function that was presented in Lesson 2. In mathematics, a function is a rule that assigns to each input value eactl one output value. Think of time in seconds as the input value and the distance the object is from the start as the output value. 9. Pick one of the graphs that ou just completed for passing an object and answer each of the following. a. If the input time is seconds what is the output? Answers will var depending on the graph the students picked. For Eercise 1, an input of seconds gives an output of 2.. b. If the input time is 8 seconds what is the output? Answers will var depending on the graph the students picked. For Eercise 1, an input of 8 seconds gives an output of 4. c. For the graph ou picked, eplain how the graph can be a function based on the definition of a function? Answers will var. Sample response: For the graph in Eercise 1, each input value has onl one output value.

126 AIIF Page 120 SJ Page 72 As a class let s tr to model passing an object that represents the following graphs. Determine if the following situations could eist for passing the object along the human assembl line Distance Distance Time in Seconds Time in Seconds For graph 10, with one object this situation could not occur because it would have to go back in time. It ma be possible to create with two objects one starting at the first person and one starting at the 10 th person. Each object is passed to the middle person at a rate of one person each second. For graph 11, this pattern could also not occur unless ou had two objects Distance Distance Time in Seconds For graph 12, this could not occur with one object. It would need to be in everone s hand at the same time. If ou had 11 objects then everone could hold the object just during the 4 th second. For graph 13, this also could not occur with one object because the object would need to be with two different people at the same time. It ma be possible to complete this pattern with two objects. 14. Pick one of the graphs ou just tried to model with the class and describe wh it cannot represent a function. Answers will var. Sample response: For Eercise 10, this graph cannot represent a function because for an input value of 4 seconds there are two output values of 4 and 6. For the real-world eample of passing an object this would be like having the object in two different places at the same time Time in Seconds

127 Linear Functions Lesson 6: What is a Function? AIIF Page 121 SJ Page 73 If on a graph there are two points that have the same input value (-value) and different output values (value) the graph cannot represent a function. A test that ou can use to determine if a graph can represent a function is the vertical line test. To complete a vertical line test on a graph, determine if an vertical line intersects the graph at more than one point. If the vertical line intersects at two or more points, the graph cannot represent a function. If each vertical line that ou could draw on the graph onl intersects the graph at one point, the graph can represent a function. 1. Determine which graphs below could represent a function. State which graphs represent a function and which ones do not. Eplain wh a graph does or does not represent a function. a. b. c. d. e. f. Sample response: Graphs a, d, e, and f could represent functions because the pass the vertical line test. For each input there is onl one put. Graphs b and c could not represent functions because the do not pass the vertical line tet. For at least one input value there are two output values.

128 AIIF Page 122 Teacher Reference Activit 2 (Optional) The goal of this activit is to show students that there are equations for other functions besides linear functions. Further development of non-linear functions will be covered in Unit 4. You ma want to review with students how to enter an equation in the graphing calculator. First, have students work on Eercise 1 individuall and then have them check with a partner to see if the graphed correctl. Have the students make adjustments as needed. Second, have the students answer Eercises 2 through 4 individuall and then check with their group of four to compare answers. Have the students make adjustments as needed. Walk around to make sure each group is writing an answer that makes sense. You ma want to show students a variet of other functions that are programmed into the graphing calculator such as cosine, tangent, e, absolute value, greatest integer, natural logarithm, logarithm, etc. You ma also want to show students that the can add, subtract, multipl, and divide numbers to functions. Such as, = Some of these other functions will be covered in Non-Linear Functions. After ou show students these other functions ou ma want to lead students to Eercise, where the can eperiment with combining different functions to make a graph of their choice.

129 Linear Functions Lesson 6: What is a Function? AIIF Page 123 Activit 2 SJ Page 74 Other non-linear equations can also represent functions. Knowing a little about other non-linear equations will help ou understand linear equations better. What ou learn in this activit will be epanded on when ou complete the non-linear functions unit. 1. Graph each of the following on our graphing calculator and then sketch each graph on the coordinate grids provided. Set our graphing window to an -minimum of, a -minimum of, an -maimum of, a -maimum of, an -scale of 1, and a -scale of 1. a. = 2 b. = 3 c. = d. 1 = 2 e. = 4 f. = sin( ) 2. Describe how ou know that each of the graphs in Eercise 1 could be a function. Sample response: Each graph can pass the vertical line test. 3. What characteristics do the graphs in Eercise 1 have in common? Sample response: Each graph has onl one -intercept ecept for =1/ which as no -intercept and each graph has curved parts. Student will later learn that this graph has a horizontal asmptote. 4. What differences do the graphs have? Sample response: Each graph has a different tpe of curved graph.

130 AIIF Page 124 SJ Page 7. It s possible, b eperimenting with combining different functions and changing the dimensions of the graphing calculator view screen, to create some interesting graphs. For eample, combining the 2 function of squaring with the sine function such as = sin( ) gives the graph below when the dimensions are set to an -minimum of 2., -maimum of 2., -minimum of 2, and -maimum of 2. Eperiment with combining two or more functions to create an interesting graph. Write the final equation below as well as sketching the graph. Answers will var. Have some students displa their interesting graphs on the view screen in front of class.

131 Linear Functions Lesson 6: What is a Function? AIIF Page 12 Practice Eercises SJ Page Which of the following graphs could represent a function? Eplain. a. b. c. d. Sample response: Graphs a and b could be functions because the pass the vertical line test. Graphs c and d could not be functions because the do not pass the vertical line test. 2. Draw a graph that represents a function and one that does not. Label our graphs. Eplain wh one graph represents a function and wh the other does not. Answers will var. The Vertical Line Test should be used to eplain wh a graph does or does not represent a function.

132 AIIF Page 126 SJ Page In each set of graphs below, one graph represents a function and the other graph does not. Describe how ou determined which graph is the function and which graph is not a function. a. b. Sample response: I determined that the first graph is a function because I used the vertical line test and it onl intersected the graph one point at time. On the second graph when I used the vertical line test it often intersected the graph at two points. Sample response: I determined that the second graph is a function because I used the vertical line test and it onl intersected the graph one point at a time. On the first graph when I used the vertical line test it often intersected the graph at four points.

133 Linear Functions Lesson 6: What is a Function? AIIF Page 127 SJ Page The following graphs represent the distance an object has moved over time or the distance an object is from another location over time. Match the descriptions with the graph. a. b. Distance Distance Time Time c. d. Distance Distance Time Time b i. A person walked from home to work at a constant speed. She stopped at a coffee shop for a few minutes once on the wa to work. d a c ii. iii. iv. A person walked to work at a constant speed but she had to wait at 3 traffic lights. A person walked to a coffee shop twice as fast as he walked home. In both directions he walked a constant speed. He staed at the coffee shop for a few minutes. A person walked to a coffee shop at a slow pace, staed a few minutes at the coffee shop, and then finished walking to work at a faster pace.

134 AIIF Page 128 SJ Page Draw a graph that could represent each of the given scenarios. Scenario 1: Scenario 2: Scenario 3: Scenario 4: In the first ear of the wetlands creation there were 1,000 birds. For si ears there was a stead increase each ear in the population of the birds. Then a drought came and the population of the birds dropped dramaticall in one ear. Then for several ears the population staed the same before dramaticall increasing as the area recovered from the drought with new plant life leaving the wetlands filled to capacit. The size of the cand bar that I get from the vending machine depends upon the amount of mone entered. The more mone I put in, the bigger cand bar I get. For an increasing negative input, the output, although positive, steadil declines until both the input and output are zero. As the input increases in the positive direction, the output increases at the same rate. Zach started the mile race at a ver fast pace for the first mile then slowed to a medium pace for 3 miles then picked the pace back up to finish the last mile. Answers will var. Make sure students determine the value that each ais represents because graphs will depend on the units students use for the - and -ais. For eample, the -ais could be time and the -ais could be distance or the -ais could be time and the -ais could be rate. Scenario 1 Scenario 2 Scenario 3 Scenario 4. What method do ou use when determining a graph for a scenario? Answers will var.

135 Linear Functions Lesson 6: What is a Function? AIIF Page 129 SJ Page The motion of a vehicle can be modeled b a distance time graph. If a vehicle maintains a constant speed, then the distance it travels over time could be modeled b the function d= rt, where d is the distance, r is the rate (speed), and t is the time. Complete the following table of ordered pairs for the distance a vehicle has traveled over time when the rate (speed) is 60 miles per hour and then plot the ordered pairs on the graph. Time in Hours t d= 60t Distance Traveled in Miles d a. Also, graph the function d= 60t on a graphing calculator and set the range of -values and - values to match the graph above. You ma want to walk around and check students calculators. b. Eplain if the graph represents a linear function or a non-linear function? Distance Traveled in Miles Sample response: The graph represents a linear function because all the points create a line. c. Describe what would be different about the graph if the vehicle slowed to 30 miles per hour at hour and staed at that pace for hours Sample response: The graph would be the same from time 0 to time, but then it would change to a line with a smaller slope from time to time 10. d 0 Time in Hours t d. Describe what would happen to the graph if the vehicle suddenl stopped at hours? Sample response: The graph would be the same from time 0 to time, but then it would change to a horizontal line at 300 from time to time 10.

136 AIIF Page 130 SJ Page Gravit will bring an object back to the ground if it is thrown into the air. The function d= 16t + rt will model the distance of an object off the ground that is thrown from ground level straight up, where d is the distance off the ground in feet, t is time in seconds, and r is the rate (speed) in feet per second that the object was thrown. a. Complete the following table of values for the distance an object is off the ground if the object is thrown straight up at an initial rate of 14 feet per second and then plot the ordered pairs on the graph. 2 d= 16t + 14t d Time in Seconds t Distance Off Ground in Feet d Time in Seconds b. Graph the function d= 16t + 14t on a graphing calculator and set the range of -values and -values to match the graph above. You ma want to walk around and check students calculators. c. Eplain if the graph represents a linear function or a non-linear function? Sample response: The graph represents a non-linear function because the points create a curve. d. At what approimate time was the object at the highest point? Sample response: From the graph it appears the highest point is approimatel 370 feet at five seconds. e. At what two times was the object approimatel 100 feet off the ground? Hint: Use the trace feature or the table feature on the graphing calculator. Sample response: About 0.7 seconds and 8.9 seconds. f. When did the object hit the ground? Sample response: Approimatel 9.6 seconds. g. What does a negative -value mean? Sample response: Because the ground is represented b the -ais, it would represent the distance below ground the object is if it could keep falling. h. Describe what would change on the graph if the object was thrown straight up at a rate (speed) of 200 feet per second? Sample response: The graph would still be an arch but go higher and would take longer to hit the ground. Distance Off Ground in Feet t

137 Linear Functions Lesson 6: What is a Function? AIIF Page 131 Outcome Sentences SJ Page 82 Distance versus time graphs are confusing because I finall understand I have learned I would conclude that a distance versus time graph Not all functions are linear because To determine if a graph can represent a function The vertical line test I am having trouble with

138 AIIF Page 132 Lesson 6 Quiz Answers 1. Answers will var. A sample graph is displaed to the right. A scenario might be: Appling the brake to slow down to a stop. distance time 2. Answers will var. A sample graph is displaed to the right. A scenario might be: The person walked at a constant rate toward school and then staed at school for the rest of the da. Distance from home time 3. Sample response might be: The graph represents a function because it passes the vertical line test. 4. A sample response might be: The graph does not represent a function because it does not pass the vertical line test.

139 Linear Functions Lesson 6: What is a Function? AIIF Page 133 Lesson 6 Quiz 1. Using the grid to the right, model a distance time scenario where the distance decreases over time. Describe our distance time scenario. 2. Using the grid to the right, model a distance time scenario where the distance increases for awhile and then remains stead. Describe our distance time scenario. 3. Using the grid to the right, draw a graph which represents a function. Eplain wh the graph represents a function. 4. Using the grid below, draw a graph which does not represents a function. Eplain wh the graph does not represent a function.

140 AIIF Page 134 Lesson 7: Linear Regression Objectives The students will be able to estimate a line of best fit for a set of data then determine the slope-intercept form of the equation. The students will be able to use a formula to determine the correlation coefficient for a set of data. The students will be able to use a formula to determine the equation of the line of best fit. The students will be able to calculate the correlation coefficient and equation of the line of best fit using the graphing calculator. The students will be able to make predictions based on the line of best fit. The students will be able to determine if a set data has a linear trend based upon visual observations of the graph of the data and the correlation coefficient. Essential Questions How can linear regression be used to predict values and be used in real world applications? Tools Student Journal Dr erase boards, markers, erasers Setting the Stage transparenc Rulers Graphing calculator Warm Up Problems of the Da Number of Das for Lesson 2 Das A suggestion is to complete Activit 1 and Activit 2 on the first da, then complete Activit 3, Practice Eercises, and quiz on the second da. Vocabular regression least sum correlation coefficient line of best fit line of least squares least-squares line correlation positive correlation negative correlation no correlation sum summation summation notation

141 Linear Functions Lesson 7: Linear Regression AIIF Page 13 Teacher Reference Setting the Stage Discuss with the class that in Lesson 2, the briefl looked at lines that could represent real-world data. Remind the students that the then began to eplore linear equations so that later the too could determine the lines that best fit data. Displa the Setting the Stage transparenc. Have the students work in groups of four. Tell the students to discuss, in their groups, which of the graphs on the transparenc could represent linear data and which ones could not. Tell the students the need to be able to eplain and justif their reasons. Have volunteers draw lines on the graphs the think would best represent linear data. Also, have the students discuss how the might find the equation for the lines the drew to represent the data. Give the students 3 to minutes to discuss in their groups. Have each group share at least one of its results with the class along with the drawn lines and how to go about finding the equation of the line drawn or the equation of a line that might better represent the data. For the graphs that do not seem to be linear, have the students determine if an tpe of non linear pattern might be present in the data and eplain wh or wh not. For graph B there appears to be no pattern. For graph C the pattern ma be cubic. Students will not know this et, but ou ma want to talk with them that later in this course and Algebra II the will look at other non-linear equations that could model this pattern. Ask guiding questions such as: How could ou describe to someone what tpe of scatter plot would have a linear trend as compared to one that does not have a linear trend? How might ou mathematicall determine if a scatter plot has a linear relationship? What does it mean to have a linear relationship? Answers to these questions will var. At this time, ou do not need to epect that students will have the correct answers.

142 AIIF Page 136 Setting the Stage Transparenc A B C D

143 Linear Functions Lesson 7: Linear Regression AIIF Page 137 Teacher Reference Activit 1 In this activit, the teacher will guide students in linear regression. Lead a class discussion on the word regression and what it might generall mean in the English language and what students might think it means mathematicall. Have a volunteer record the students' responses on the board or on a blank transparenc on the overhead. Determine how close the students' meaning of regression, mathematicall, matches the real meaning of regression which can be defined as the process of finding a function of a particular form that best represents a set of paired data. In statistics or mathematics, regression generall relates to determining an equation that best models the relationship between dependent variable values and independent variable values. In pscholog, regression can refer to a person s defensive reaction to an unaccepted impulse where a person ma revert back to ounger behavior. In medicine, regression can represent the case where a person with a disease has lighter smptoms without the disease completel disappearing. Refer back to graphs from the Setting the Stage that were either linear or had a noticeable linear pattern. Ask the students to discuss in their groups how the might find the equations which best model the linear graphs from the lines the drew. Ask students to share the results of their discussions. Have a volunteer record the students' responses on the board or on a blank transparenc on the overhead. A couple of guiding questions ou might ask after the class has shared their responses are: How might we determine if the line we drew to represent the data was the best line possible? Is there a wa to measure differences between drawn lines to determine if one line is a better representation than another line? Have the class share their thoughts on these questions with the class and possibl make a list of what the think is the best wa to determine the best possible line drawn. Tell the class that the are going to pla a game that is based on what the just discussed. The game is called Best Fit Challenge. The object of the game is to draw the line that is the best possible fit for a set of ordered pairs. The class will compete as teams of two in their groups of four. Each team will plot the set of ordered pairs and draw the line it thinks best fits the given data. Make sure each team has a dr erase board and two different colored markers. The plotted points and lines drawn will be done on their dr erase board. To determine which line best models the given data, the teams need to measure the vertical distance (all positive) between each -coordinate of the set of ordered pairs and the -coordinate of the line the other team drew. In other words, the swap dr erase boards and measure their opponent's line. Have the students use the metric portion of the rulers and measure the distance accurate to 0.1 centimeter or 1 millimeter. After measuring the distances, each team then sums the distances. The team with the least sum gets 10 points. Both teams are then to determine the equation for the line which had the best fit. Each team will answer the questions in the student journal for each set of ordered pairs. Each team of two in their groups of four with the largest number of points after the three sets of ordered pairs have been used is the winner. You ma also want to determine which team from the whole class had the most total points. You can decide what award each team will get. You ma consider, a small prize of cand, free homework pass, or other simple awards.

144 AIIF Page 138 Have one group of four, as two teams of two, volunteer to model the game on the overhead projector, with our assistance, while the class completes it at their desks. Have the data on the board alread and a blank grid on the overhead projector. Have the group volunteers use the following set of ordered pairs and scenario to model the game. Suppose ou are a scientist studing the migration of a sand dune over time as it approaches a nearb condominium development. You have looked up data from previous ears since the condominiums were built and have found the following information and recorded it into a table. Year from Building Condominiums Distance from Condominiums in Meters Distance between -coordinates Using our dr erase board agree on the unit scale for the and -ais, plot the ordered pairs from the table, and draw our line of best fit. Swap dr erase boards and measure the vertical distance between each -coordinate of the set of ordered pairs and the -coordinate of the line on the dr erase board. Write the distance in the table in the space provided. Sum the distances. The team with the least sum distance would be awarded 10 points. What is the equation of the line with the least sum? The graph below is the possible representation of the migration of the sand dune. Distance from Condominiums Year from Building Condominiums Have the students work on Eercises 1 through 3 in their groups as teams of two. Ask for students to share their results and determine which team or teams had the most points. Award groups as ou have planned. Talk with students about Eercise 4.

145 Linear Functions Lesson 7: Linear Regression Activit 1 AIIF Page 139 SJ Page 83 In this activit, ou will pla the game called Best Fit Challenge. The object of the game is to draw a line that is the best possible fit for a set of ordered pairs. Your group of four will consist of two teams of two. Each team in our group will plot the set of ordered pairs and draw the line the think best fits the given data on a dr erase board. To determine which line best models the given data, our team will measure the vertical distance (all positive values because we are not calculating slope) between each coordinate of the set of ordered pairs and the line the other team in our group drew. Use the metric portion of the rulers and measure the vertical distance accurate to 0.1 centimeter or 1 millimeter. After determining all the distances, sum the distances. The team with the least sum gets 10 points. Both teams in our group are then to determine the equation for the line which had the best fit. The team with the highest total points after the three eercises is the winner. Complete the following eample in our group as a volunteer group demonstrates in front of the class. Eample Suppose ou are a scientist studing the migration of a sand dune over time as it approaches a condominium development. You have looked up data from previous ears since the condominiums were built and have found the following information and recorded it into a table. Year from Building Condominiums Distance from Condominiums in Meters Distance between -coordinates Using our dr erase board agree on the unit scale for the and -ais, plot the ordered pairs from the table, and draw our line of best fit. Swap dr erase boards and measure the vertical distance in millimeters between each -coordinate of the set of ordered pairs and the -coordinate of the line on the dr erase board. Write the distance in the table in the space provided. Sum the distances. The team with the least sum distance will be awarded 10 points. What is the equation of the line with the least sum? Notes: For Eercises 1 through 3 complete the following. As a group of four, agree on the unit scale for the and -ais on the dr erase boards, then plot the ordered pairs from the table and draw our line of best fit. Swap coordinate grids with the other pair in our group of four and measure the vertical distance in millimeters between each -coordinate of the set of ordered pairs and the line on the dr erase board. Write the distance in the table in the space provided Sum the distances. The team with the least sum distance would be awarded 10 points. What is the equation of the line with the least sum?

146 AIIF Page 140 SJ Page Suppose ou are a meteorologist studing data that ou have recorded from a station on an island in the path of a hurricane. The data is seven readings of barometric pressure and wind speed as the storm approaches. Let be the barometric pressure (in millibars) and let be the wind speed (in miles per hour). Barometric pressure in millibars Wind speed in MPH Distance a. What is the sum of the distances for our team? Answers will var. b. What is the sum of the distances of our opponent? Answers will var. c. What is the slope of the line with the least sum? Interpret what the slope represents. Answers will var. The slope represents the change in wind speed per change in the barometric pressure. The negative value of slope means as the barometric pressure increases b one millibar, the wind speed decreases b the value of the slope. d. What is the equation of the line with the least sum? Answers will var. 2. Suppose ou re a statistician who works for an automotive consulting compan and ou need to determine the linear relationship between the weight of a car and its potential miles per gallon. Suppose ou chose several cars at random from a list of cars. Let be the weight of the car (in hundreds of pounds), and let be the car's miles per gallon (mpg) rating. Weight of Car in hundreds of pounds MPG Distance a. What is the sum of the distances for our team? Answers will var. b. What is the sum of the distances of our opponent? Answers will var. c. What is the slope of the line with the least sum? Interpret what the slope represents. Answers will var. The slope represents the change in miles per gallon per change in the weight of the car. The negative value of slope means as the car's weight increases b one hundred pounds, the car's miles per gallon decreases b the value of the slope. d. What is the equation of the line with the least sum? Answers will var.

147 Linear Functions Lesson 7: Linear Regression AIIF Page 141 SJ Page 8 3. Suppose ou work for a state transportation department as a statistician in charge of determining information related to traffic accidents. You have been looking at two different tpes of fatal car accidents for the past ear; accidents caused b speeding and accidents caused b failure to ield the right of wa. For both sets of data ou let represent the age in ears of a licensed automobile driver and be the number of all fatal accidents due to speeding (first table) or failure to ield the right of wa (second table). Fatal Accidents Caused b Speeding Age in Years Number of Fatal Accidents Distance Fatal Accidents Caused b Failure to Yield the Right of Wa Age in Years Number of Fatal Accidents Distance a. What is the sum of the distances for our team for each table? Answers will var. b. What is the sum of the distances of our opponent for each table? Answers will var. c. What is the slope of each line with the least sum? Interpret what the slope represents. Answers will var. The slope, for speeding, represents the change in the number of fatal accidents per change in the age of the licensed driver. The negative value of slope means as the driver's age increases b one ear, the number of fatal accidents due to speeding decreases b the value of the slope. The slope, for failure to ield the right of wa, represents the change in the number of fatal accidents per change in the age of the licensed driver. The positive value of slope means as the driver's age increases b one ear, the number of fatal accidents due to failure to ield the right of wa increases b the value of the slope. d. What are the equations of the line with the least sum? Answers will var. e. At what approimate age does the number of fatal accidents caused b speeding equal the number of fatal accidents caused b failure to ield the right of wa? What is the approimate number of fatal accidents for this age for both speeding and failure to ield the right of wa? The age is approimatel 46 ears where the number of fatal accidents is the same for both speeding and failure to ield the right of wa. The number of fatal accidents for both speeding and failure to ield the right of wa is approimatel Wh do ou think we measure the vertical distance between the line and ordered pairs and not the horizontal distance? Answers will var. A sample response might be: "We're tring to predict the output, or dependent value, from the input, or independent value, and not predict the input from the output."

148 AIIF Page 142 Grid Transparenc

149 Linear Functions Lesson 7: Linear Regression AIIF Page 143 Teacher Reference Activit 2 In this activit, students will learn mathematical representations and terms for linear regression. Prepare students to read the first paragraph b talking with them briefl about fitting lines to data and relating this to the correlation coefficient which will be a value that describes how linear a set of data is. Read the first paragraph together. Think out loud as ou read the paragraph. It is a fairl dense paragraph and students will need help understanding it. Use the three graphs below the paragraph to talk about high and low values of the correlation coefficient as well as positive and negative correlation coefficients. Guide the students through Eercises 1 through 8. Make sure to introduce the students to the Greek letter Sigma, Σ. Tell the class that the letter represents summation or sum. For Eercise 1, tell the students that before we can do an summations, we need to fill in the bottom three rows of the table which represents the product of the -values and -values, the values of 2, and the values of 2. Make sure the students are ver careful in their calculations. Let students know that a mistake in a single calculation will cause an error in the final answer. Model for students how to calculate and then for Eercise 2, have the students calculate,, 2, and 2. Model how to calculate the mean of the -values and then have them answer Eercise 3 to determine the mean of the -values. Guide the students, in Eercise 4, ver carefull on using the correlation coefficient formula to determine its value for the drifting rate of sand. n ( )( ) r = n n ( ) ( ) Let students know that n is the sample size or number of ordered pairs. Have the class calculate the correlation coefficient to three decimal places. Let them know that the formula is quite complicated because of man calculations that have to be performed. But, the r value is ver important because it is a numerical measurement that assesses the linear relationship between two variables and. Remind the class that during the last activit, the drew what the thought was the line that best fit the data. The students measured the vertical distance from their line to the ordered pairs and we called the smallest sum of the distances the least sum. Let the class know that we used this term as a wa to get them read for the mathematical wa to find the equation of the line that best fits the data. For our distances, we alwas used positive values. But if we considered whether the point was below the line ("negative" distance) or above the line ("positive" distance), then if we added these distances, the would normall sum to zero. So, statisticians used a criterion called least squares which states, "The sum of the squares of the vertical distances from the data point (, ) to the line is made as small as possible." Because squaring an number will make it positive, we will have a wa to find the best line possible to make the sum of the squares of the vertical distances as small as possible. Now that we have all the necessar calculations, tell the class that we will now calculate the least squares line, which is the line of best fit. Guide students through Eercises through 8. Because of the number of calculations required, we'll limit the eercises to the first two the did in Activit 1. Have the class work in their groups of four on Eercises 9 through 10 then share their results.

150 AIIF Page 144 SJ Page 86 Activit 2 In this activit, we'll use mathematical formulas to determine how linear a set of data is and the equation of the line that represents the best fit. One of the guiding questions from the Setting the Stage asked if there was something that could be calculated to determine how linear or non linear the data is in the graphs. Another question dealt with a linear relationship. When we talk about a linear relationship, we are talking about a linear relationship between and (or between the independent and dependent variables). Another term for this linear relationship is correlation. In statistics, correlation means that the items, here and, show a tendenc to var together. In a linear equation, how does var in respect to? The slope told us how the line was either increasing or decreasing. Even though the data ma seem to var in a linear fashion, we still need to determine how linearl the value changes with respect to a change in the value. There is a mathematical measurement that describes the strength of the linear relationship between two variables. The measure is called the correlation coefficient and it is denoted b the letter r. The range of r values is 1 r 1. The closer r is to 1 or +1, the more linear the data. A wa to understand the value of r is b looking at the scatter plots of various data. The graphs below show a positive correlation, a negative correlation, and no correlation. In positive correlation, low -values will have corresponding low -values and high -values will have corresponding high -values. A strong positive correlation will have an r-value close to 1. In negative correlation, low -values will have corresponding high -values and high -values will have corresponding low -values. A strong negative correlation will have an r-value close to 1. In no correlation, low -values will correspond to both high and low -values and high -values will also correspond to both low and high - values. In graphs that have no or little correlation, the r-value will be close to 0. Positive Correlation Negative Correlation No Correlation High and high values Low and high values Low and low values High and low values

151 Linear Functions Lesson 7: Linear Regression The eample problem in Activit 1 dealt with a migrating sand dune caused b blowing wind. AIIF Page 14 SJ Page For each set of ordered pairs, calculate the product of and pairs, the square of each, and the square of each. Place them in the table below. Year from Building Condominiums Distance from Condominiums in Meters ,316 19,600 12, The Greek letter Sigma, Σ, is used to represent summation or sum. For eample, the sum of all the -values is: = = Use the data in the table above to calculate the sum of the -values, the sum of the products of and - values, the sum of the squares of the -values, and the sum of the squares of the -values. a. = 717 c. = 416 d. 2 = 816 e. 2 = 79,21 To determine the strength of the linear relationship between two variables and ou can calculate the n ( )( ) correlation coefficient with the formula r = where r is the correlation n n 2 ( ) ( ) coefficient and n is the number of ordered pairs of and -values. 3. Use the information ou calculated in Eercise 2 to determine the correlation coefficient for the migrating sand dune data. Round the value to three decimal places. ( ) ( )( ) ( ) ( ) ( ) ( ) r = = Talk with students about how this is a fairl strong correlation coefficient close to 1. The data is fairl linear in a negative direction.

152 AIIF Page 146 SJ Page 88 A bar over a letter is used to represent the mean of all the values. For eample, the mean of all the -values is: = Use the data in the table above to calculate the mean of the -values. Round to two decimal places The line of least squares or best fit line is the mathematical method to find the equation of the line that gives the best fit for a set of linear data. It can be calculated b determining a special slope and -intercept. The slope of line of least squares is ( )( ) 2 n ( ) 2 n m =, where n is the number of ordered pairs. The -intercept of line of least squares is b = m.. Determine the slope and -intercept of line of least squares for the migrating sand dune data. Round the values to two decimal places. a. m = ( ) ( )( ) 7 ( 816) ( 64) b. b = ( 4.94)( 9.14) Substitute this special slope and -intercept into the slope-intercept form of a linear equation, = m + b, to determine the line of least squares equation. = How does this equation compare to the equation we estimated at the beginning of Activit 1? Answers will var. 7. Plot the and -values from the table of ordered pairs for the migrating sand dune data on a graphing calculator, then graph the line on the graphing calculator on the same screen. Describe how well the line fits the data visuall. Year from Building Condominiums Distance from Condominiums in Meters Answers will var. You ma want to walk around the room and look at students calculators as needed.

153 Linear Functions Lesson 7: Linear Regression AIIF Page 147 SJ Page What might be some uses of the line of least squares for the migrating sand dune data? Answers will var. Students should include in the answer something about predicting the distance the sand dune is from the condominium development over time. 9. Eercise 1 from Activit 1 was about a meteorologist analzing the barometer readings and wind speed of an approaching hurricane. Fill in the table below and determine the sum of each row. Barometric pressure in millibars Wind speed in MPH Σ Use the information ou just calculated to determine the correlation coefficient for the wind speed of the approaching hurricane related to the barometer reading. Round the value to three decimal places. r = n ( )( ) 2 2 ( ) ( ) 2 2 n n 7 ( ) ( 7036)( 16) ( ) ( ) ( ) ( ) = Talk with students about how this is a fairl strong correlation coefficient close to 1. The data is fairl linear in a negative direction. b. Determine the equation for the line of least squares. Remember to calculate the mean of the - values, mean of the -values, slope, and -intercept first. Round the slope and -intercept to two decimal places. Slope of line of least squares is ( )( ) 2 n ( ) 2 n m = ( ) ( )( ) 7 ( ) ( 7036) = 1.42 The mean of the -values: The mean of the -values: = = = ( )( ) -intercept: b m = The equation for the line of least squares: = c. How does the equation for line of least squares compare with our equation in Eercise 1 of Activit 1? Answers will var.

154 AIIF Page 148 SJ Page Eercise 2 from Activit 1 was about a statistician for an automotive consulting compan determining the relationship between the weight of a car and its potential miles per gallon. Fill in the table below and find the correlation coefficient and the equation for the line of least squares. Round the correlation coefficient to three decimal places and the slope and -intercept to two decimal places. Weight of Car in hundreds of pounds MPG Correlation Coefficient: n ( )( ) r = 2 2 n n ( ) 2 ( ) 2 n ( )( ) m = 2 n ( ) 2 Slope of line of least squares is 8( 814) ( 299)( 167 ) ( ) ( ) ( ) ( ) = ( ) ( )( ) 8( ) ( 299) = = 23 The mean of the -values: = The mean of the -values: = intercept: b m = = ( )( ) The equation for the line of least squares: = a. How does the equation of the line of least squares compare with our equation in Eercise 2 Activit 1? Answers will var. Σ b. Use the equation to predict the miles per gallon for a car that weighs 17 hundred pounds. 33 mpg Does this seem reasonable? Eplain wh or wh not. Yes. Man small cars toda get above 30 miles per gallon. c. Use the equation to predict the miles per gallon for a car that weighs 8 hundred pounds Does this seem reasonable? Eplain wh or wh not. No. A car couldn t have negative miles per gallon.

155 Linear Functions Lesson 7: Linear Regression AIIF Page 149 Teacher Reference Activit 3 In this activit, students will learn to use the graphing calculator to find the correlation coefficient and the line of best fit. Use the calculator view screen to review with the students how to enter a set of ordered pairs. The instructions provided here are for the TI 83 Plus / TI 84 Plus graphing calculator. Make sure students take notes on how to complete the steps or make a handout that matches the calculator the will be using. Tell the class that we can save time on all the calculations we had to do in Activit 2 b using the graphing calculator. Also tell the class that we can find both the correlation coefficient and the equation for the line of best fit b just pressing a few kes on the graphing calculator. Before we can obtain a value for the correlation coefficient we need to turn diagnostics on. To do this, press the following ke sequence: N (in Catalog mode the keboard is locked in ALPHA mode. This ke represents the letter D) (down arrow to DiagnosticOn) Í (Diagnostic appears on home screen) Í (Diagnostic command eecutes) The first Í will displa DiagnosticOn on the home screen. The second Í will then eecute the command and the "Done" message will displa on the home screen. Model finding the correlation coefficient and line of best fit b using the migrating sand dune data. Have the students clear their calculators b pressing the following ke sequence: Ã À Á. This will reset RAM and the message RAM cleared will be displaed on the home screen. Have the class enter the data b pressing the ke followed b the Í ke (for the EDIT menu item). Enter the -values of the ordered pairs in L1 and the -values of the ordered pairs in L2. After the data has been entered press the following ke sequence to obtain correlation coefficient and equation for the line of best fit: ~ Í. The following screen shots depict the ke sequence and results. The last screen shot, on the right, is an error message that could occur if the lists, L1 and L2, do not contain the same number of values. Guide the students through Eercises 1 and 2. Then have the students work in pairs to complete Eercises 3 through 8. Walk around and help groups as needed. If a pair of students is struggling, have the pair first ask the other pair in their group for suggestions, then the can ask ou questions. Make sure ou pause to discuss Eercises 3 through 8 before ou have the students work on Eercises 9 and 10.

156 AIIF Page 10 SJ Page 91 Activit 3 So far in this lesson, ou have two different methods to determine the equation of a line that matches data. You learned in Activit 1 how to estimate drawing a line on the scatter plot of the data. In Activit 2, ou learned how to use formulas to determine the line of best fit (line of least squares) and the correlation coefficient. In this activit, ou will investigate using the graphing calculator to find line of best fit and the correlation coefficient. 1. The table below contains the data used in the first two activities. Use this data as our teacher guides ou in the process of setting up our graphing calculator, entering the data, and determining the equation of the line of best fit. Year from Building Condominiums Distance from Condominiums in Meters a. Record steps for entering data here: Answers will var. Students need enough information here to successfull determine the line of best fit and correlation coefficient with the calculator. b. Record steps for determining the equation of the line of best fit and correlation coefficient here: Answers will var. c. What is the equation for the line of best fit? = d. What is the correlation coefficient? Approimatel e. How do these two answers compare to our answers from Activit 2? Sample response: The answers are ver close to the same value. The slope is the same and the -intercept is onl different b about 3 hundredths. The correlation coefficient is the same. 2. Eplain which method ou would rather use to determine the equation for the line of best fit: estimate the line on a scatter plot then determining the equation, using the formulas to determine the equation, or using the graphing calculator to determine the equation. Answers will var.

157 Linear Functions Lesson 7: Linear Regression AIIF Page 11 SJ Page 92 Complete eercises 3 through 8 to gain a better understanding of the relationship between the correlation coefficient, the graphed line, and the set of data related to a line of best fit. 3. Use our graphing calculator to determine the equation for the line of best fit and correlation coefficient for the following set of data Equation: = Correlation Coefficient: r = 1 a. Graph the line and a scatter plot of the data on the same graph on our graphing calculator. Walk around to help students as needed and to look at their graphs. b. Describe how well the line fits the scatter plot visuall and how this corresponds to the correlation coefficient. Answers will var. Sample response: The line fits the scatter plot perfectl. This makes since because the correlation coefficient was Use our graphing calculator to determine the equation for the line of best fit and correlation coefficient for the following set of data Equation: = Correlation Coefficient: r = 1 a. Graph the line and a scatter plot of the data on the same graph on our graphing calculator. Walk around to help students as needed and to look at their graphs. b. Describe how well the line fits the scatter plot visuall and how this corresponds to the correlation coefficient. Answers will var. Sample response: The line fit the scatter plot perfectl. This makes sense; because, the correlation coefficient was 1.

158 AIIF Page 12 SJ Page 93. Use our graphing calculator to determine the equation for the line of best fit and correlation coefficient for the following set of data Equation: = Correlation Coefficient: r a. Graph the line and a scatter plot of the data on the same graph on our graphing calculator. Walk around to help students as needed and to look at their graphs. b. Describe how well the line fits the scatter plot visuall and how this corresponds to the correlation coefficient. Sample response: The line sort of fits the scatter plot because the scatter plot is linear but because the data is not in a perfect line the graphed line doesn t go through all the points. This makes since because the correlation coefficient was Use our graphing calculator to determine the equation for the line of best fit and correlation coefficient for the following set of data Equation: = Correlation Coefficient: r 0.6 a. Graph the line and a scatter plot of the data on the same graph on our graphing calculator. Walk around to help students as needed and to look at their graphs. b. Describe how well the line fits the scatter plot visuall and how this corresponds to the correlation coefficient. Sample response: The line doesn t reall fit the scatter plot because the scatter plot curves down and then back up. The line doesn t curve. This makes sense because the correlation coefficient was Use our graphing calculator to determine the equation for the line of best fit and correlation coefficient for the following set of data Equation: = Correlation Coefficient: r a. Graph the line and a scatter plot of the data on the same graph on our graphing calculator. Walk around to help students as needed and to look at their graphs. b. Describe how well the line fits the scatter plot visuall and how this corresponds to the correlation coefficient. Sample response: The line doesn t reall fit the scatter plot because the scatter plot is random. This makes sense because the correlation coefficient was 0.096, which is close to zero.

159 Linear Functions Lesson 7: Linear Regression AIIF Page 13 SJ Page Describe how the correlation coefficient relates to how well a line fits a scatter plot visuall. Answers will var. Sample response: Lines that fit the scatter plot almost perfectl have a correlation coefficient reall close to 1 or 1. Scatter plots that have a linear pattern will have a correlation coefficient that is probabl between 0.6 and 1 or between 0.6 and 1. Scatter plots that are fairl random or that are not reall linear in pattern will have correlation coefficients between -0.6 and You are the foreman on the Double G llama ranch. A friend of ours has some cria (pronounced cree-ah), bab llamas, for sale. You want to predict how much a health cria weighs based on the age. You have some data that ou took from last ear in the table below. Age in weeks Weight in pounds a. Determine the correlation coefficient and equation of the line of best fit for the data in the table. The value of r is approimatel and = b. Use the equation to predict how much ou would epect a 1 week old cria to weigh? A 1-week-old cria would weigh approimatel 60.7 pounds. c. Approimatel how old, in das, is a cria that weighs 100 pounds? A 100-pound cria would be approimatel 31.8 weeks which is approimatel 223 das old. 10. You are an archaeologist for a universit working at a site in the Wind Mountain region of southwest New Meico. You have discovered some ancient and prehistoric potter vessel sherds (broken pieces) and are carefull reconstructing the potter if enough sherds can be found. Let represent the bod diameter of the reconstructed potter vessel (in centimeters), and let be the height (in centimeters) of the vessel. You have consulted the leading book on ancient potter for New Meico. The following data was taken from this book. Diameter in centimeters Height in centimeters a. Create a scatter plot of the data on our graphing calculator. Check graphs as needed. b. From the scatter plot, would ou conclude the correlation is low, moderate, or high? Is it a positive or negative correlation? Sample response: The correlation is high and is positive. c. What is the correlation coefficient and the equation for the line of best fit? The value of r is approimatel and = d. Predict the height of the potter vessel if the diameter was 13 centimeters. The potter vessel is approimatel 9.9 centimeters.

160 AIIF Page 14 SJ Page 9 Practice Eercises 1. Use the scatter plot to answer the questions below. a. Looking at the scatter plot, would ou sa the correlation is low, moderate, or high? Is the correlation positive or negative? Sample response: The correlation is high and positive because the points make a relative straight line and the slope of the points is positive. b. What is the value of the correlation coefficient? Round to three decimal places. The value of the correlation coefficient is r c. What is the equation for the line of best fit? Round the slope and -intercept to two decimal places. The least squares line is = A breakfast cereal maker puts coupons for a free cand bar in each bo of cereal it makes. You work in the main office and need to weigh the incoming mail to determine how man emploees ou need to assign to collect the coupons and mail out the cand bars on an given da. Your compan answers the mail on the da the receive it (this is wh our compan is a favorite cereal maker all over the countr). You have collected sample data for 8 das to help ou determine how man emploees will be needed on an given da. Let be the weight of the incoming mail in pounds, and let be the number of emploees needed to process the mail and send out the cand bars. Weight of mail in pounds Number of emploees a. Find the correlation of coefficient and the equation for the line of best fit. The value of the correlation coefficient is r The equation for the line of best fit is = b. Using the equation, determine approimatel how emploees ou ma need for 2 pounds of mail. 0.64(2) people

161 Linear Functions Lesson 7: Linear Regression AIIF Page 1 SJ Page Your famil takes a fishing trip to a lake in Canada to fish for Northern Pike. You have checked with people living in the area about the size and weight of Northern Pike the have caught. You compiled a table of the information ou gathered. Let be the length of the Northern Pike in inches that were caught, and let be the weight of the Northern Pike in pounds that were caught. Length in inches Weight in pounds a. Draw a scatter plot of the data. See students graphs. b. Find the correlation coefficient to three decimal places. The value of the correlation coefficient is r = c. Find the equation of the line of best fit. Round to two decimal places. The equation of the line of best fit is = d. How much could ou epect a Northern Pike to weigh that ou caught if it was 30 inches in length? A fish that measured 30 inches in length would weigh approimatel 8. pounds. e. What is the average length and weight of the fish that the local people caught, to the nearest tenth? The average length of a fish caught was 34.1 inches and the average weight was 11.3 pounds. f. What does the slope in the equation for the line of best fit represent? Sample response: The slope represents the increase in weight of a fish caught, in pounds, for each increase in length of one inch. Basicall this is the rate of increase in weight per inch. 4. How can linear regression be used to predict values and be used in real world applications? Answers will var. This is the essential question given at the beginning of the lesson. You ma want to spend some time discussing responses.

162 AIIF Page 16 SJ Page 97 Outcome Sentences The line of best fit The correlation of coefficient tells me Linear regression The best was to communicate mathematical results in a meaningful manner is I can use more help with understanding

163 Linear Functions Lesson 7: Linear Regression AIIF Page 17 Lesson 7 Quiz Answers 1. Equation: = Correlation Coefficient: r Walk around to check students graphs. 3. Answers will var. Sample response: The line fit the scatter plot fairl close. This makes sense; because, the correlation coefficient was = 0.( 60) = 1.2

164 AIIF Page 18 Lesson 7 Quiz 1. Use our graphing calculator to determine the equation for the line of best fit and correlation coefficient for the following set of data Equation: Correlation Coefficient: 2. Graph the line and a scatter plot of the data on the same graph on our graphing calculator. 3. Describe how well the line fits the scatter plot visuall and how this corresponds to the correlation coefficient. 4. Use the equation to predict the value of when the value of is 60.

165 Linear Functions Lesson 8: Linear Inequalities AIIF Page 19 Lesson 8: Linear Inequalities Objectives The students will be able to graph the solution to linear inequalities b hand and with a graphing calculator. The students will be able to determine the linear inequalit from the graphed solution of a linear inequalit. The students will be able to graph multiple linear inequalities and determine the solution region. Students will appl linear inequalities to model and solve real-world problems involving two or more variables. Essential Questions How can the properties of inequalities be used to create a linear inequalit in slope-intercept form? How can shading and dashed lines represent the solution to a linear inequalit? Tools Student Journal Dr erase boards, markers, erasers Construction paper Graphing calculator Warm Up Problems of the Da Number of Das for Lesson 1 Da Vocabular linear inequalit less than less than or equal to greater than greater than or equal to inequalit solution solution region sstem of inequalities

166 AIIF Page 160 Teacher Reference Setting the Stage Discuss the following scenario with the students: To rent a car a person generall has to be at least 24 ears old and have a major credit card with a certain minimum credit limit. Ask students, What do these requirements reall mean? Have a volunteer list the class response about what the requirements reall mean on the board or on a blank transparenc on the overhead projector. Also discuss the following scenario in a similar manner as the first scenario: In general, to graduate from high school a person must be at least 16 ears old, pass a minimum of 23 credits, pass the required core courses, and pass an eit eamination. Again, ask the students what do these requirements reall mean. Have a student volunteer list the responses about what the requirements reall mean on the board or on a blank transparenc on the overhead projector. Have the class get into groups of 4 and discuss other tpes of situations that describe when a person has more than one minimum or maimum requirement. Sample responses could be the age requirements for social securit benefits and maimum allowable income ou can make while receiving the benefits. Bring the class together and have each group share one situation that it discussed along with the required minimum and/or maimum requirements.

167 Linear Functions Lesson 8: Linear Inequalities AIIF Page 161 Teacher Reference Activit 1 In this activit, the teacher will guide students in graphing the solution to linear inequalities. The activit will start with a discussion about one-variable inequalities. Ask the class, When we graphed, on a number line, inequalities with less than (<) or greater than (>), compared to less than or equal to ( ) or greater than or equal to ( ), what was the difference in the graphs? The goal is for the students to remember that for less than (<) and for greater than (>), the end point was hollow as compared to a solid end point for less than or equal to ( ) and greater than or equal to ( ). Now ask the class, What about when we graph inequalities in two-variables, known as linear inequalities. What characteristics will the line have for linear inequalities involving the smbols < or >, compared to linear inequalities involving the smbols or? Can we put circles all up and down the line? We are gauging the students prior knowledge from graphing linear inequalities in algebra for how the line is displaed as either a dashed line or a solid line. We can model whether an inequalit has a dashed or solid line. Talk aloud to model graphing the linear inequalit < + 1. The class could follow along with ou, on their dr-erase boards, or guide ou in the decision process of graphing the inequalit. Have the class create a table of inequalit values for and b filling in the following table: < Tell the class to plot the "ordered inequalit pairs" as if each pair was to be graphed on a number line. The graph above depicts the individual graphed inequalities from the table. Net, have the class create the boundar between the solution region, which is the shaded region, and the non solution region b creating a connected scatter plot of the circled endpoints graphed inequalities. Tell the class that if the now erase the circles the would have a dashed line. The solution region would be all points below the line. Have the class shade the solution region. Ask the class to find the equation of the dashed line. The class should realize that the equation of the line is the inequalit but with an equal sign instead of the inequalit smbol, = + 1.

168 AIIF Page 162 Ask the students, "How would the solution change if the inequalit was less than or equal to instead of just less than?" The ke concept is that the students realize that the line is included in the solution and the hollow circles would become solid circles and the line would now be a solid line instead of a dashed line. Have a student volunteer model a second eample, 2 1, while the class parallels with 3 2. Make sure the class goes through the same process as the first modeled inequalit. Have the volunteer state the steps that he/she is performing out loud to solve the inequalit while the class follows along with their inequalit. Have the students hold up their solutions and visuall inspect the results. Have students share their results with the rest of the class. Now, ask a series of questions about the process the class just went through: 1. What is the first thing that we did to solve the inequalit? Have a volunteer record the agreed upon class responses to the questions ou pose. Hopefull, the first item that the class agreed upon was to create the table of inequalit values. 2. Ok, we now have our first step done. What did we do net? Again, have the student volunteer record the net agreed upon step on the board. 3. Now that we have done the first two steps, we now need to find our net step. What could this net step be? The class should agree that the third step was to determine the tpe of line, solid or dash. 4. "What is the last step we need to do?" The last step should be to shade the solution region. Ask the class if there is an easier process from the steps the just listed that would be just as effective in solving linear inequalities. A possible list of steps might be: 1. Determine if the graphed line will be a solid or dashed line 2. Graph the dashed or solid line 3. Pick a point on either side of the line and substitute the ordered pair values for that point into the inequalit 4. Determine if the inequalit statement is true or false for the selected point. Shade in the solution region that satisfies the inequalit Model a third inequalit of a different tpe while the students do a different one on their dr-erase boards. See if the students can identif the form that the line portion of the inequalit is in. Here s a list of inequalities to model. Eamples: and ( ) and + 4 3( + 2) 6 < and 4 < 2 6 Have the students work in pairs on Eercises 1 through 8. Bring the class together and have volunteers share their results, along with their eplanations of wh their graphs represent the solution to the inequalities.

169 Linear Functions Lesson 8: Linear Inequalities Activit 1 AIIF Page 163 SJ Page 98 In this activit, ou will graph the solutions to linear inequalities. For the following inequalities: a. Determine if the linear inequalit has a solid or dashed line. b. Graph the dashed or solid line of the inequalit. c. Pick a point on either side of the line to substitute the ordered pairs into the inequalit. d. Substitute the ordered pair for the point into the inequalit and determine the solution region. e. Shade in the solution region. 1. < > 3( 2)

170 AIIF Page 164 SJ Page 99. Denise has $0.00 to spend at the local chocolate shop. She can either bu milk chocolate for $2.0 per pound or sweet chocolate for $4.00 per pound or she could bu a combination of milk and sweet chocolate. Write the inequalit that can represent this scenario, then, draw an inequalit graph that represents the number of different was she can bu chocolate and remain at $0.00 or under. Make sure to place labels and units on the ais Let represent the number of pounds of milk chocolate and the number of pounds of sweet chocolate that Denise bus. The inequalit is You are working part-time at a car dealership. The manager of the dealership has come to ou to help him with a math problem he is having trouble solving. He has $1,00,000 available to purchase a fleet of the latest new hbrid cars and the latest new electric cars. Each hbrid car can cost the dealer $2,000, while each electric car costs the dealer $30,000. Determine the number of different hbrid cars and electric cars the manager can purchase and still be under $1,00,000, b writing an inequalit that represents this situation and drawing a graph. Sample response: Let represent the number of hbrid cars the dealer bus. Let represent the number of electric cars the dealer bus. The inequalit is or if we divide both sides b 000 the inequalit is Determine the inequalit that matches the solution graph. The inequalit for the graph is <

171 Linear Functions Lesson 8: Linear Inequalities 8. Determine the inequalities that match the solution graphs. a. b. AIIF Page 16 SJ Page 100 The inequalit for the graph is 3 6. The inequalit for the graph is 2 4. c. d. The inequalities for the graph are: 12 > 4 < 2+ 3 The inequalities for the graph are:

172 AIIF Page 166 Teacher Reference Activit 2 In this activit, students will learn to use their graphing calculators to graph linear inequalities. Use our calculator view screen to show the students how to shade inequalities on their graphing calculator. You ma want to show the students how to activate and inactivate an equation if the would like to do multiple problems at one time. Remind the class that the Y= editor epects equations in the = m + b form. Model using the graphing calculator with the inequalit 3. Ask the class if the graphing calculator can distinguish between when the line should be solid or dashed when graphing inequalities. The graphing calculator ma alwas graph a solid line when graphing an inequalit. Remind the class that the will need to determine which region to shade b picking a point not on the line and determining which region will give a true statement for the inequalit. The screen shots below are for a TI-84 Plus graphing calculator. Please adjust our instructions based on our classroom graphing calculator. For 3 : Shades above the line For 3 : Shades below the line Inactive equation Have the students work in pairs for Eercises 1 through. Have students share their results with the class for Eercises 1 through. Note: Remind the students to complete the L portion of the KWL that was introduced in Lesson 2.