Equations and the Coordinate Plane

Transcription

1 Equations and the Coordinate Plane Unit 3 0 College Board. All rights reserved. Unit Overview In this unit ou will compare and contrast linear and non-linear patterns and write epressions to represent these patterns. You will stud functions, domain, range, slope, and forms of linear equations. You will model and solve problems involving sstems of equations and ou will collect and analze bivariate data. Academic Vocabular Add these words to our vocabular notebook. bivariate data continuous data discrete data domain function linear data range rate of change relation slope solution to a sstem of linear equations sstem of linear equations trend line -intercept -intercept?? Essential Questions Wh is it important to consider slope, domain, and range in problem situations? How can graphs be used to interpret solutions of real world problems? EMBEDDED ASSESSMENTS This unit has three Embedded Assessments after Activities 3., 3. 5, and 3.7. These embedded assessments allow ou to demonstrate our understanding of linear relations and linear equations, sstems of equations, and applications of bivariate data. Embedded Assessment Linear Relationships and Functions p. 35 Embedded Assessment Slopes and Intercepts p. Embedded Assessment 3 Bivariate Data and Sstems p. 75 5

2 UNIT 3 Getting Read Write our answers on notebook paper. Show our work.. On the grid below, draw a figure that illustrates the meaning of linear Name five ordered pairs that would be on a graph made from the following table. input output A line contains the points (, 5) and (, ): a. Where does it cross the -ais? b. Where does it cross the -ais?. Use the graph below to: a. Plot and label the points R(3, 5) and S(, 0). b. Give the coordinates of point T Draw a horizontal line that contains the point (, 3) and a vertical line that contains (, ). 8 T 3. Complete the table below so that the data is linear. input output 3. Name 3 ordered pairs that satisf the equation = Write a ratio that compares the shaded region in the figure below to all the regions. 0 College Board. All rights reserved. SpringBoard TM Mathematics with Meaning TM Level 3

3 Linear and Non-Linear Patterns Fill It Up SUGGESTED LEARNING STRATEGIES: Predict and Confirm, Use Manipulatives, Create Representations, Look for a Pattern, Quickwrite Using the template that our teacher gave ou, create the cube and cone, which ou will use in an eperiment. You will be filling the cube and the cone b adding 0 beans at a time. Before ou begin the eperiment, make the following predictions.. Predict how man groups of 0 beans ou can add to the cube until it is full.. Predict how man groups of 0 beans ou can add to the cone until it is full. 3. Are our predictions different? Eplain how the shape of the figure affected our predictions. Do the following eperiment for the cube. Each stage of the eperiment consists of 3 steps. Each time ou complete the three steps ou complete a stage. Step : Step : Step 3: Add 0 beans to the cube. Shake the figure gentl to allow the beans to settle. Measure the height of the beans in centimeters.. Complete the eperiment for the cube. M Notes ACTIVITY 3. 0 College Board. All rights reserved. a. Enter the data in the table. b. Do ou see a pattern in the data in the table? c. Plot the values from the table on the grid. Stage # Height of Beans (cm) d. Looking at the graph, what do ou notice about the relationship between the stage number and the height of the beans? Height of Beans Stage Number Unit 3 Equations and the Coordinate Plane 7

4 ACTIVITY 3. Linear and Non-Linear Patterns Fill It Up M Notes SUGGESTED LEARNING STRATEGIES: Use Manipulatives, Create Representations, Look for a Pattern, Quickwrite 5. Complete the eperiment for the cone. a. Fill in the table for each stage and plot the points on the grid. Stage # Height of Beans (cm) 5 Height of Beans Stage Number b. Look at the data in the table. What, if an, patterns do ou notice? 0 College Board. All rights reserved. c. Look at the graph. What patterns do ou notice about the relationship between the stage number and the height of the beans? 8 SpringBoard Mathematics with Meaning TM Level 3

5 Linear and Non-Linear Patterns Fill It Up ACTIVITY 3. SUGGESTED LEARNING STRATEGIES: Create Representations, Look for a Pattern, Quickwrite The following data was collected as beans were added to a clinder. M Notes Stage # Height of Beans (cm) Plot the points on the grid for the clinder. 0 Height of Beans 5 0 College Board. All rights reserved Stage Number a. What patterns do ou notice for the data in the table? b. Looking at the graph, what patterns do ou notice about the relationship between the stage number and the height of the beans? ACADEMIC VOCABULARY The rate of change in a relationship represents the ratio of vertical change in the output to the horizontal change in the input. The output is often represented b the variable. The input is often represented b the variable. c. What conjecture can ou make about the rate of change of the height of the beans as the stages increase? Unit 3 Equations and the Coordinate Plane 9

6 ACTIVITY 3. Linear and Non-Linear Patterns Fill It Up M Notes SUGGESTED LEARNING STRATEGIES: Create Representations, Look for a Pattern, Quickwrite The following data was collected as beans were added to an irregular polhedron. Stage # Height of Beans (cm) Plot the points on the grid for the irregular polhedron. 0 Height of Beans Stage Number a. What patterns do ou notice for the data in the table? b. Looking at the graph, what patterns do ou notice about the relationship between the stage number and the height of the beans? 0 College Board. All rights reserved. c. What conjecture can ou make about the rate of change of the height of the beans as the stage numbers increase? SpringBoard Mathematics with Meaning TM Level 3

7 Linear and Non-Linear Patterns Fill It Up ACTIVITY 3. SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Quickwrite, Group Discussion, Group Presentation, Create Representations, Simplif a Problem M Notes 8. How does the rate of change from the clinder eperiment differ from the rate of change for the irregular polhedron eperiment? 9. Eplain how the shape of the object affects the rate of change.. If the height of the clinder and the irregular polhedron were etended indefinitel, eplain how the height of the beans would change as the stage number increased.. The graphs and tables below show what happened when the bean eperiment was performed with each of the vases shown. Match each vase to a graph and a table. Eplain the reasoning behind our choices. a. b. c. 0 College Board. All rights reserved Unit 3 Equations and the Coordinate Plane

8 ACTIVITY 3. Linear and Non-Linear Patterns Fill It Up M Notes ACADEMIC VOCABULARY linear data SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Quickwrite, Create Representations Data is linear if it has a constant rate of change. When ou plot the points of linear data on a coordinate plane, the lie on a straight line.. Compare and contrast the graphs and tables of the three figures in Item. Which of the figures appeared to encourage a linear relationship? Eplain our reasoning. A person is drinking water from a clindrical cup using a straw. The following graph gives the height of the water at different time intervals. Height of Water Time 3. Using the data from the graph fill in the table below. Time (sec) Height (cm) 0 College Board. All rights reserved. SpringBoard Mathematics with Meaning TM Level 3

9 Linear and Non-Linear Patterns Fill It Up ACTIVITY 3. SUGGESTED LEARNING STRATEGIES: Quickwrite, Create Representations, Group Presentation, Identif a Subtask, Discussion Group M Notes. Using the table, describe an patterns ou see in the height of the water over time. 5. Is the relationship between time and the height of the water linear? a. Eplain using the graph. b. Eplain using the table.. What is the rate of change in the water level from 0 seconds to 30 seconds? 7. Connect the data points, and determine what the rate of change in the water level is from 0 seconds to seconds. 0 College Board. All rights reserved. 8. Predict the height of the water at 5 seconds. How did ou make our prediction? If ou wanted to look at man different times, would our method still be effective? 9. Create an epression that gives the height of the water in terms of the time (t). 0. How long will it take for the water to completel empt out of the cup? Eplain using multiple representations. Unit 3 Equations and the Coordinate Plane 3

10 ACTIVITY 3. Linear and Non-Linear Patterns Fill It Up CHECK YOUR UNDERSTANDING Write our answers on notebook paper. Show our work.. Which equation matches the data in our work. the table?. Find the rate of change for a. = + 5 the table. 0 5 b. = c. = d. = Graph the following points and determine if the data is linear Find the rate of change for the table. 3. Determine which of the following tables displas linear data. Eplain our reasoning. a. b {(5, -3), (7, -), (9, 0), (, )}. Determine which of the following epressions displas a linear relationship. Use multiple representations to eplain our reasoning. a. b. - + c. () d MATHEMATICAL REFLECTION In this activit, ou eplored three was to represent linear data: in a table, graphicall, and with an epression. Which representation of linear data do ou understand most easil and wh? 0 College Board. All rights reserved. SpringBoard Mathematics with Meaning TM Level 3

11 Functions Who Am I? SUGGESTED LEARNING STRATEGIES: Create Representations, Quickwrite Relationships can eist between different sets of information. For eample, the pairing of the names of students in our class and their heights is one such relationship.. Collect the following information for 5 members of our class. M Notes ACTIVITY 3. Student Number First Name Height (cm) Length of Inde Finger (cm) College Board. All rights reserved Write the student numbers of 5 students in the class and their height in the following form: (Number, Height). Unit 3 Equations and the Coordinate Plane 5

12 ACTIVITY 3. Functions Who Am I? M Notes MATH TERMS An ordered pair is two numbers written in a certain order. Most often, the term ordered pair will refer to the and coordinates of a point on the coordinate plane, which are alwas written (, ). The term can also refer to an values paired together according to a specific order. SUGGESTED LEARNING STRATEGIES: Create Representations, Think/Pair/Share The number and height that ou wrote for the five students in our class is called an ordered pair. Given the input () of a student s number, ou can get an output () of that student s height. In a cartesian coordinate plane, ordered pairs are represented b (,). 3. Graph and label the coordinates of a point for each of the five ordered pairs ou wrote in Question Using the information from the table, what other relationships can ou create? College Board. All rights reserved. 5. Using one of the relationships ou described in Question that contains numeric values onl, create five ordered pairs of students in our class. SpringBoard Mathematics with Meaning TM Level 3

13 Functions Who Am I? ACTIVITY 3. SUGGESTED LEARNING STRATEGIES: Question the Tet, Marking the Tet, Vocabular Organizer, Note Taking, Think/ Pair/Share, Summarize/Paraphrase/Retell, Quickwrite A relation is a set of ordered pairs. For eample, the pairing of students inde finger length with their height is a relation. The set of all the starting values or inputs is called the domain. In a relation, all domain values must be matched with an output value. The set of all output values is called the range. EXAMPLE Find the domain and range of the following set: {(,), (,), (,5), (8,3)} Step : Look at the first number in each pair to identif the domain. Step : Look at the second number in each pair to identif the range. Solution: The domain is {,,, 8}, and the range is {, 3,, 5} M Notes MATH TERMS A set is a collection of objects, like points, or a tpe of number. The smbols { } indicate a set. ACADEMIC VOCABULARY domain range relation TRY THESE A Determine the domain and range of the following sets. a. {(,), (,5), (,), (,7), (,8)} b. {(3,), (,), (,), (,8)}. From Question, what would be the domain of the relation that associates Length of Inde Finger to Height? 0 College Board. All rights reserved. 7. What would be the range of the relation that associates Length of Inde Finger to Height? A function is a special kind of relation. Like a relation, a function must match an input to an output, but functions have the additional restriction that each element in the input can match onl one element in the output. 8. Is the relation that associated student numbers and their height a function? Eplain our reasoning. Unit 3 Equations and the Coordinate Plane 7

14 ACTIVITY 3. Functions Who Am I? M Notes SUGGESTED LEARNING STRATEGIES: Marking the Tet, Vocabular Organizer, Note Taking, Create Representations, Quickwrite One tpe of representation that helps to determine if a relation is a function is a mapping. The illustration to the left is a mapping. The particular relation that was mapped is a function. Ever input (-value) is mapped to eactl one output (-value). Note that a -value can be associated with more than one -value. Each input () has eactl one output (). ACADEMIC VOCABULARY A function is a special kind of relation in which each element of the domain is paired with eactl one element of the range. EXAMPLE Consider the relation {(,), (,), (,5), (8,3)}. Use mapping to determine if the relation is a function. Eplain. Step : Step : Step 3: Write all domain values in an oval. Write all range values in another oval. Connect the input values with their output values using arrows. CONNECT TO AP Functions and relations, which describe how two varing quantities are related, form the basis of much of the work ou will do in calculus. Solution: From the mapping, we can see that ever element in the input set is mapped to eactl one element in the output set. Therefore, the relation is a function. TRY THESE B Use mapping to determine if the following are functions. Eplain. a. {(,3), (,), (,5)} b. + for = {0,,, 3, } 0 College Board. All rights reserved. c Use mapping to determine if the relation that associates height to student number is a function. Eplain our reasoning. 8 SpringBoard Mathematics with Meaning TM Level 3

15 Functions Who Am I? ACTIVITY 3. SUGGESTED LEARNING STRATEGIES: Create Representations, Look for a Pattern, Group Presentation Another representation that helps to determine if a relation is a function is a table. M Notes EXAMPLE 3 Determine if the following relations are functions. a. {(,), (,), (,), (8,)}. b. {(,), (,3), (8,3)} Step : Look at the number of output values for each input. Solution: The relation A is a function since each input has onl one output. The relation B is not a function because one input,, has two different outputs. 0 College Board. All rights reserved. TRY THESE C Determine if the following relations are functions. Eplain wh the are or are not. a. b. c. d Create a table of values that represents a function and a second table that does not represent a function. How would ou identif an table that does not represent a function? Another representation that helps to determine if a relation is a function is graphing. An eample on how to use graphing this wa is on the net page. Unit 3 Equations and the Coordinate Plane 9

16 ACTIVITY 3. Functions Who Am I? SUGGESTED LEARNING STRATEGIES: Look for a Pattern M Notes EXAMPLE Determine which of the following graphs represents a function. Step : Plot the ordered pairs. Relation A Relation B {(-,-), (-,), (0,0), {(-,-), (-,), (0,0), (,-), (,)} (,), (,-)} Step : Look at the graph to determine if an of the -values have more than one -value. Solution: Relation A is a function since each input has onl one output. Relation B is not a function because at least one input has two different outputs. Relation C Relation D = = Solution: Relation C is a function since each input has onl one output. Relation D is not a function because at least one input has two different outputs. TRY THESE D Which of the following graphs represent functions? Eplain our reasoning. a. b College Board. All rights reserved SpringBoard Mathematics with Meaning TM Level 3

17 Functions Who Am I? ACTIVITY 3. SUGGESTED LEARNING STRATEGIES: Look for a Pattern TRY THESE D () M Notes c. d ACADEMIC VOCABULARY. When looking at a graph of a relation, how can ou determine if it is a function? Discrete data are data that can onl have certain values such as the number of people in our class. On a graph there will be a space between ever two possible values. Continuous data can take on an value within a certain range; for eample, height. On a graph continuous data and continuous functions have no breaks, holes, or gaps. In the following eample, Function A is discrete and Function B is continuous. Data are discrete if there are onl a finite number of values possible or if there is a space on the number line or on a graph between each possible values. Data are continuous if there are no breaks in their domain or range or if the graph has no breaks, holes or gaps. 0 College Board. All rights reserved. Function A Function B Unit 3 Equations and the Coordinate Plane 3

18 ACTIVITY 3. Functions Who Am I? M Notes SUGGESTED LEARNING STRATEGIES: Shared Reading, Think/Pair/Share Functions can be represented b epressions.. If a function that is represented b the epression + 5 has inputs labeled and outputs labeled, then the diagram below represents the mapping from the input to the output. + 5 a. If = 5 is used as an input in the diagram, what it the output? b. If = -3 is used as an input in the diagram, what it the output? c. If = 0.03 is used as an input in the diagram, what it the output? d. If = - is used as an input in the diagram, what it the output? e. Is there an limit to the number of input values that can be used with this epression? Eplain our reasoning. 0 College Board. All rights reserved. f. Is the function discrete or continuous? Eplain. 3 SpringBoard Mathematics with Meaning TM Level 3

19 Functions Who Am I? ACTIVITY 3. SUGGESTED LEARNING STRATEGIES: Activate Prior Knowledge, Debrief Mr. Walker collected the following data about shoe size and height from five members in his class. M Notes Approimate Height Shoe Size (in centimeters) Consider the relation that associates the shoe size of one of Mr. Walker s student to his or her approimate height. a. Use a mapping to determine if the relation is a function. Eplain how ou arrived at our answer. b. Draw a graph and eplain how it confirms our answer to part a. 0 College Board. All rights reserved. c. An epression that can be used to represent the relation is 8 + 9, where represents the students shoe size. Is there an limit to the input values that can be used with this epression? Eplain our reasoning. d. Is the relation discrete or continuous? Eplain our reasoning. Unit 3 Equations and the Coordinate Plane 33

20 ACTIVITY 3. Functions Who Am I? CHECK YOUR UNDERSTANDING Use notebook paper to write our answers. Show our work. Find the domain and range for the data in questions and.. {(-,7), (-,), (-,-3), (,-3)} Use mapping to determine if the information in Questions 3 5 represent functions. 3. {(-,), (-3,), (-,5), (-,), (-3,7)}. + 5 for = 3, 5, 7, 9, For Questions 8 determine if the relations represent a function. Eplain our reasoning MATHEMATICAL REFLECTION How do the domain and range of a relation help to determine if a relation is a function? 0 College Board. All rights reserved. 3 SpringBoard Mathematics with Meaning TM Level 3

21 Linear Relationships and Functions EDUCATION PAYS Embedded Assessment Use after Activit 3.. The following data was taken from an article, Education Pas, b Sand Baum and Jennifer Ma. Age Median Annual Income with a High School Diploma Median Annual Income with a Bachelor s Degree $9,88 $,57 3 9,88,57 9,88,57 5 7,73,593 7,73, ,73, ,73, ,73, ,73,593. Does the data relating age with income for those with a high school diploma represent a linear relationship? Eplain our reasoning.. Is the median annual income for either the High School Diploma or the Bachelor s Degree a function of age? Eplain our reasoning. 3. Eplain wh the data in the table is considered discrete. 0 College Board. All rights reserved.. Three relations follow. For each relation, eplain whether: The information represents a linear relationship. The information is discrete or continuous. The information represents a function. a b. = Unit 3 Equations and the Coordinate Plane 35

22 Embedded Assessment Use after Activit 3.. Linear Relationships and Functions EDUCATION PAYS c Math Knowledge #,, Communication #,, 3 Eemplar Proficient Emerging The student: Correctl identifies the data as linear or nonlinear. () Correctl identifies whether or not income is a function of age for both relations. () Correctl identifies data as linear or nonlinear, discrete or continuous, and determines if the data represents a function. () The student: Correctl eplains wh the data is nonlinear or nonlinear. () Correctl eplains wh income is or is not a function of age for both relations. () Correctl describes wh the data is discrete. (3) The student provides complete and correct identification for two of the items. The student gives eplanations for the three items, but onl two are complete and correct. The student provides at least two identifications, but onl one is complete and correct. The student gives at least two of the required eplanations for questions,, and 3, but the are incomplete and incorrect. 0 College Board. All rights reserved. 3 SpringBoard Mathematics with Meaning TM Level 3

23 Eploring Slope High Ratio Mountain SUGGESTED LEARNING STRATEGIES: Create Representations, Look for a Pattern, Activate Prior Knowledge, Discussion Group Mist Flipp worked odd jobs all summer long and saved her mone to bu passes to the ski lift at the High Ratio Mountain Ski Resort. In August, Mist researched the lift ticket prices and found several options. Since she worked so hard to earn this mone, Mist carefull investigated each of her options. M Notes ACTIVITY 3.3 High Ratio Mountain Ski Resort Student Lift Ticket prices Dail Lift Ticket $30 -Da Package $80 upon purchase and $0 per da (up to das) Unlimited Season Pass $390. Suppose Mist purchased a dail lift ticket each time she goes skiing. Complete the table below for the total cost of the lift tickets. 0 College Board. All rights reserved. Number of Das Total Cost of Lift Tickets Use the table to complete the statement: When the number of das in the row increases b, Mist s cost increases b. 3. Does the data in the table represent a linear relationship? Eplain our reasoning.. Determine the following: a. Does the data represent a function? b. Is the data discrete or continuous in this contet? Unit 3 Equations and the Coordinate Plane 37

24 ACTIVITY 3.3 Eploring Slope High Ratio Mountain M Notes SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Create Representations, Look for a Pattern 5. State the domain and the range of the data in the table.. Plot the data from the table on the grid below Total Cost of Lift Tickets Das 3 7. Label the left most point on the graph point A. Label the net points, from left to right, points B, C, D, E, F, and G. 8. Use the graph to complete the statement: When the number of das increases b, Mist s cost increases b. 9. Describe how ou move along the grid to get from one point to another. From A to B: Go Up $ and Go Right Da(s) 0 College Board. All rights reserved. From B to C: Go Up $ and Go Right Da(s) From C to D: Go Up $ and Go Right Da(s) From D to E: Go Up $ and Go Right Da(s) From E to F: Go Up $ and Go Right Da(s) From F to G: Go Up $ and Go Right Da(s) 38 SpringBoard Mathematics with Meaning TM Level 3

25 Eploring Slope High Ratio Mountain ACTIVITY 3.3 SUGGESTED LEARNING STRATEGIES: Marking the Tet, Vocabular Organizer, Think/Pair/Share, Look for a Pattern, Activating Prior Knowledge. The movements ou traced in Question 9 can be written as a up ratio, right. Write ratios in the form up that describe how to right move from: A to B: B to C: C to D: D to E: M Notes MATH TERMS A ratio is an epression that compares two values or quantities. The rate of change of a relation is a ratio. Another wa to think of the movement Go Up is as the change in. Similarl, the movement Go Right is the change up in. With this in mind, the ratio,, can be rewritten change in right as. The illustration to the right shows the change in change in and the change in between two points on a line. Change in Change in. Find the change in, the change in, and write the ratio: Change in Change in = 3 5 From A to C: From B to E: From A to E:. What do ou notice about these ratios? 0 College Board. All rights reserved. 3. What are the units of the ratios ou created?. Eplain how the ratios relate to Mist s situation. 5. Find the change in, the change in, and write a ratio: WRITING MATH When writing a ratio, ou can also represent the relationship b separating each quantit with a colon. For eample, the ratio : is read one to four. From B to A: From E to B:. How do these ratios compare to those ou found in Question? Unit 3 Equations and the Coordinate Plane 39

26 ACTIVITY 3.3 Eploring Slope High Ratio Mountain M Notes ACADEMIC VOCABULARY Slope is the ratio of vertical change to horizontal change or change in change in -intercept SUGGESTED LEARNING STRATEGIES: Quickwrite, Marking The Tet, Vocabular Organizer, Interactive Word Wall change in The slope of a line is determined b the ratio change in between an two points that lie on the line. The slope is the constant rate of change of a line. All linear relationships have a constant rate of change. The slope of a line is what determines how steep or flat it looks on a graph. The -intercept of a line is the -coordinate when the -coordinate is 0. It is the point at which the line crosses the -ais, (0, ). 7. Let d represent the number of das Mist plans to ski and let C represent Mist s total cost. Write an equation for C in terms of d. READING MATH The slope of a line, TRY THESE A change in, is also epressed change in Find the slope and -intercept for the following. smbolicall as Δ Δ. a. b. Δ is the Greek letter, delta. 3 CONNECT TO SPORTS Longboards are larger than the more trick-oriented skateboards. Longboards are heavier and sturdier than skateboards. Some people even use them instead of biccles c. d e. John is longboarding at a constant rate down the road. If min after he leaves his house he is 00 ft awa and at 5 minutes he is 500 ft from his house what would his average rate of change be? College Board. All rights reserved. 8. Draw a line that contains the points ou plotted in Question. Using the graph, find the slope and -intercept of the line. SpringBoard Mathematics with Meaning TM Level 3

27 Eploring Slope High Ratio Mountain ACTIVITY 3.3 SUGGESTED LEARNING STRATEGIES: Create Representations, Look for a Pattern, Shared Reading, Interactive Word Wall M Notes 9. Suppose Mist purchased the -Da Ticket Package that costs $80 plus $0 per da. a. Complete the table below for the total cost of the lift tickets in the -da package for 0 through das. Be sure to include the initial cost of $80. Number of Das Total Cost of Lift Tickets b. Eplain how ou know the data in the table above is linear. 0 College Board. All rights reserved. 0. Plot the data from the table on the given aes. Total Cost of Lift Tickets Das CONNECT TO AP The concepts of slope and -intercept will continue to be developed in later math courses.. Draw a line that contains the points ou plotted in Item 0. Unit 3 Equations and the Coordinate Plane

28 ACTIVITY 3.3 Eploring Slope High Ratio Mountain M Notes SUGGESTED LEARNING STRATEGIES: Quickwrite, Group Presentation, Create Representations, Summarize/Paraphrase/Retell. Find the slope and the -intercept of the line that contains the points in the graph for Question 0, and eplain how the relate to Mist s situation. 3. Compare and contrast the lines associated with the data for the Dail Lift Tickets in Question, and the data for the -Da Package.. Let d represent the number of das Mist plans to ski and let K represent Mist s cost. Write an equation for K in terms d for Mist s cost. 5. Although it seemed like a lot of mone, Mist thought about the unlimited season pass for $390. a. First, she compared the season pass to the dail lift tickets at $30 each. How man times would Mist have to go skiing before she would save mone with the $390 season pass? Show our work. b. Net, Mist compared the price of an unlimited season pass to two -Da packages that she would use for 0 das of skiing. Which package would be the best bu? Eplain our reasoning. 0 College Board. All rights reserved. SpringBoard Mathematics with Meaning TM Level 3

29 Eploring Slope High Ratio Mountain ACTIVITY 3.3 SUGGESTED LEARNING STRATEGIES: Create Representations, Identif a Subtask, Discussion Group, RAFT M Notes. If Mist skis the following number of das, which of the three packages should she purchase? Eplain wh. a. das b. 8 das c. 3 das d. das 7. Write a persuasive letter to Mist based on our analsis that makes a recommendation of which package she should purchase. Include multiple representations (graphs, tables, and/or equations) to support our reasoning. 0 College Board. All rights reserved. Unit 3 Equations and the Coordinate Plane 3

30 ACTIVITY 3.3 Eploring Slope High Ratio Mountain CHECK YOUR UNDERSTANDING Write our answers on notebook paper. Show our work. Mist determined that she gets miles on gallons of gas from her car as she drives from her house to go skiing.. Create a ratio of Mist s miles per gallon.. Using the ratio ou found in Question, determine how far Mist can go on gallon of gas. 3. How man miles could Mist travel on a full tank of gallons of gas?. What is the slope of the line shown? a. - 5 b c. - 3 d. 3 e. 5. Find the slope and -intercept of the following: a b c. d.. If a line has a slope of 3, and contains the point (3, ), then it must also contain which of the following points? a. (-, -) b. (-, -) c. (0, -3) d. (, ) e. (7, 3) 7. MATHEMATICAL REFLECTION of the line? How does the steepness of a line affect the slope 0 College Board. All rights reserved. SpringBoard Mathematics with Meaning TM Level 3

31 Slope Intercept Form The Leak Bottle SUGGESTED LEARNING STRATEGIES: Use Manipulatives, Create Representations, Discussion Group, Think/Pair/Share, Activating Prior Knowledge Owen s water bottle leaked in his bookbag. He did the following eperiment to find how quickl water drains from a small hole placed in a water bottle.. Follow the steps below and fill in the table. Get a water bottle and a container to catch the water. Poke a small hole in the bottom of the water bottle Ensure the hole is facing down, and open the bottle cap. Draw a line on the bottle ever 5 seconds to mark the water level. After the water is drained from the bottle, measure the heights at each of the times that ou marked. Time in Seconds Height of Water (cm). Make a scatter plot of the data on the grid below. M Notes ACTIVITY 3. 0 College Board. All rights reserved. Height of Water Time 3. Does the relationship between time and the height of the water appear to be linear? Eplain our reasoning. 0. Is the data ou collected continuous or discrete? Eplain our reasoning. 5. Draw a line through the points on the scatterplot ou created. a. Find the slope of the line ou drew. b. Find the -intercept of the line ou drew. MATH TERMS The -intercept of a line is the -value when = 0. It is the place where the line crosses the -ais. Unit 3 Equations and the Coordinate Plane 5

32 ACTIVITY 3. Slope Intercept Form The Leak Bottle M Notes SUGGESTED LEARNING STRATEGIES: Group Presentation, Think/Pair/Share, Create Representations. Write an equation that gives the height of the water H given the time t. 7. How does the coefficient of t in our equation relate to the eperiment? Be certain to include appropriate units in our answer. 8. How does the constant term in the equation relate to the eperiment? Be certain to include appropriate units in our answer. 9. For each linear equation below: Make a table of values. Graph using a different color for each line. Determine the slope a. = b. = c. = = = = 0 College Board. All rights reserved. SpringBoard Mathematics with Meaning TM Level 3

33 Slope Intercept Form The Leak Bottle ACTIVITY 3. SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Create Representations, Look for a Pattern. How does the slope ou found for each linear equation relate to the coefficients of in the equations for Question 9? M Notes. For each linear equation below: Make a table of values. Graph using a different color for each line. Determine the slope College Board. All rights reserved. a. = - b. = - c. = - = = = How does the slope ou found relate to the coefficients of in the equations for Question? 3. Write an equation of a line that is: a. Steeper (increasing) than the ones ou graphed in Question 9. b. Steeper (decreasing) than the ones ou graphed in Question. Unit 3 Equations and the Coordinate Plane 7

34 ACTIVITY 3. Slope Intercept Form The Leak Bottle M Notes SUGGESTED LEARNING STRATEGIES: Create Representations, Look for a Pattern, Group Presentation, Think/Pair/Share, Guess and Check. For each linear equation below: Make a table of values. Graph using a different color for each line. Determine the slope a. = b. = c. = 5 0 College Board. All rights reserved. 5. Compare and contrast the slopes ou found in Questions 9,, and. Refer to the representations ou ve created in our comparisons. What conclusions can ou draw about the slope of lines?. Write the equation of a line that is steeper than but less than one. 8 SpringBoard Mathematics with Meaning TM Level 3

35 Slope Intercept Form The Leak Bottle ACTIVITY 3. SUGGESTED LEARNING STRATEGIES: Create Representations, Look for a Pattern, Think/Pair/Share 7. For each linear equation below: Make a table of values. Graph using a different color for each line. Determine the -intercept. Determine the slope. M Notes a. = b. = 3 + c. = College Board. All rights reserved. 8. How is the -intercept related to the constant term in the equations? 9. Identif the slope and -intercept in each of the following equations. a. = b. = - + c. = - 3 Unit 3 Equations and the Coordinate Plane 9

36 ACTIVITY 3. Slope Intercept Form The Leak Bottle M Notes SUGGESTED LEARNING STRATEGIES: Shared Reading, Interactive Word Wall, Discussion Group, Create Representations 0. Identif and plot the -intercept of the equation = + 3 on the coordinate grid and use the slope to find two more points on the line MATH TERMS The slope-intercept form of a linear equation is = m + b, where m is the slope and b is the -intercept.. Sketch a line through the three points. Equations of the form = m + b are written in slope-intercept form, where m is the slope of the line, and b is the -intercept of the line.. Use the -intercept and the slope to graph the following equations of lines. a. = - b. = - + c. = College Board. All rights reserved SpringBoard Mathematics with Meaning TM Level 3

37 Slope Intercept Form The Leak Bottle ACTIVITY 3. SUGGESTED LEARNING STRATEGIES: Work Backwards, Quickwrite, Group Presentation 3. Owen found that the equation = -3 + represented the water leaking from his bottle. M Notes a. What is the -intercept, and what would it represent in this contet? b. What is the slope, and what would it represent in this contet? c. Eplain to Owen what would have to happen to the bottle for the slope to change to -. 0 College Board. All rights reserved.. Eplain how to graph the equation = - 3 without using a table of values. Unit 3 Equations and the Coordinate Plane 5

38 ACTIVITY 3. Slope Intercept Form The Leak Bottle CHECK YOUR UNDERSTANDING Write our answers on notebook paper. Show our work.. Find the slope Graph the linear equations. 7. = + 8. = = 3-5. Write an equation for the line graphed below. Graph the linear equations.. = 3 3. = -5. = - 5. Write an equation of a line that has a slope that is greater than but less than Write the equation of the line graphed below MATHEMATICAL REFLECTION Eplain two was to graph a linear equation of the form = m + b, where m and b represent an real number. 0 College Board. All rights reserved. 5 SpringBoard Mathematics with Meaning TM Level 3

39 Intercepts, Horizontal and Vertical Lines Drive Time SUGGESTED LEARNING STRATEGIES: Shared Reading, Marking the Tet, Create Representations, Interactiving Word Wall, Vocabular Organizer, Activating Prior Knowledge M Notes ACTIVITY 3.5 Matt is driving from Tucson to Flagstaff, Arizona. After driving 0 miles on two-lane roads, he gets on the interstate highwa where he will drive 5 mph.. Write a linear equation that gives Matt s distance from Tucson given the number of hours since Matt has been driving on the interstate.. What are the slope and -intercept of the line in Question? The -intercept of a line is the point where the line crosses the -ais. Its coordinates will be in the form (c, 0) where c is a real number. ACADEMIC VOCABULARY -intercept EXAMPLE Find the -intercept on a graph. Step : Find the intersection of the line with the -ais. 0 College Board. All rights reserved. Solution: The -intercept is, or the point (, 0). TRY THESE A Find the - and -intercepts of the graphs below. a. b. Unit 3 Equations and the Coordinate Plane 53

40 ACTIVITY 3.5 Intercepts, Horizontal and Vertical Lines Drive Time M Notes SUGGESTED LEARNING STRATEGIES: Discussion Group, Think/Pair/Share, Quickwrite To find the -intercept of a line algebraicall, use the fact that the intercept lies at the point (c, 0). EXAMPLE A. Find the -intercept of the line = - algebraicall. Step : Substitute 0 for. 0 = - Step : Solve for. 0 = = = Solution: The -intercept is. The coordinates are (, 0). To find the -intercept of a line algebraicall, use the fact that the intercept lies at the point (0, d). B. Find the -intercept of the line = - algebraicall. Step : Substitute 0 for. = (0) - Step : Solve for. = - Solution: The -intercept is -. The coordinates are (0, -). TRY THESE B Find the - and -intercepts of the following equations. a. = -5 - b. = + 5 c. = d. = 7 + e. + 3 = 9 0 College Board. All rights reserved. 3. Find the - and -intercepts of the equation ou found in Question algebraicall. 5 SpringBoard Mathematics with Meaning Level 3

41 Intercepts, Horizontal and Vertical Lines Drive Time ACTIVITY 3.5 SUGGESTED LEARNING STRATEGIES: Create Representations, Look for a Pattern. Graph each of the following equations. a. = b. = 5 c. = M Notes What happens to the graph of the equation of a line as the slope gets closer to zero?. Predict what a line with a slope that is equal to zero would look like. 0 College Board. All rights reserved. Unit 3 Equations and the Coordinate Plane 55

42 ACTIVITY 3.5 Intercepts, Horizontal and Vertical Lines Drive Time M Notes SUGGESTED LEARNING STRATEGIES: Create Representations 7. Fill in the table values for the following equations. a. = b. = 0 + c. = Graph the equations from Question Simplif and rewrite the equations in Question 7. What patterns do ou notice about equations of lines that have a slope of zero? 0 College Board. All rights reserved. 5 SpringBoard Mathematics with Meaning Level 3

43 Intercepts, Horizontal and Vertical Lines Drive Time ACTIVITY 3.5 SUGGESTED LEARNING STRATEGIES: Create Representations, Look for a Pattern, Interactive Word Wall, Vocabular Organizer. Graph each of the following equations. M Notes a. = 5 b. = 7 c. = 8 8. What happens to the graph of the line as the slope gets larger?. On the coordinate grid above, draw a vertical line through the point (5, 0). 3. Using the line ou drew in Question : a. Plot and label the coordinates of additional points on the line. 0 College Board. All rights reserved. b. Epress the slope of the line in the form Δ Δ. As the slope of a line increases, the line becomes closer to a vertical line. When the denominator of a slope ratio is zero, the slope is said to be undefined. The slope of a vertical line is undefined.. Look at the line ou drew in drew in Question. a. What do ou notice about the values? b. What do ou notice about the values? c. Wh do ou think the equation of the line is = 5? Unit 3 Equations and the Coordinate Plane 57

44 ACTIVITY 3.5 Intercepts, Horizontal and Vertical Lines Drive Time M Notes SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Create Representations 5. Graph the following horizontal and vertical lines. a. = -3 b. = - c. = d. = Write the equations of the following horizontal and vertical lines. a. b. 0 College Board. All rights reserved. 58 SpringBoard Mathematics with Meaning Level 3

45 Intercepts, Horizontal and Vertical Lines Drive Time ACTIVITY 3.5 SUGGESTED LEARNING STRATEGIES: Create Representations M Notes EXAMPLE 3 Graph = + using - and -intercepts. Step : Find the - and -intercepts algebraicall. Find the -intercept Find the -intercept 0 = + = (0) + - = = -3 = Step : Plot the coordinates of the - and -intercepts. Step 3: Connect the intercepts with a line. Solution: TRY THESE C Graph the equations of the following lines using and -intercepts. 0 College Board. All rights reserved. a. = + 5 b. = - - c. = Unit 3 Equations and the Coordinate Plane 59

46 ACTIVITY 3.5 Intercepts, Horizontal and Vertical Lines Drive Time CHECK YOUR UNDERSTANDING Write our answers on notebook paper. Show our work. Graph the lines that have the following intercepts. Find the - and -intercepts of the following graphs. 7. -intercept: -intercept: intercept: -3 -intercept: 7 Graph the following lines 8 9. =. = -. Write the equation of the lines graphed below. a For 3, find the - and -intercepts of the equations. 3. = 8 +. = 3 + b College Board. All rights reserved. 5. = = 8. MATHEMATICAL REFLECTION When would it be easier to graph a line using its slope and -intercept than to graph it using its - and -intercepts? Eplain our reasoning. 8 SpringBoard Mathematics with Meaning Level 3

47 Slopes and Intercepts LINEAR KINDNESS Embedded Assessment Use after Activit 3.5. Ben s Bells, a communit service organization, started hanging ceramic wind chimes randoml in trees, on bike paths, and in parks around the countr in 003 with a written message to simpl take one home and pass on the kindness. The linear equation = represents the total number of bells,, that have been hung b the project given the ears,, since What is the slope of the line and what does it represent?. What is the -intercept of the line? 3. Graph the equation on the grid below. 0 College Board. All rights reserved Write the equation of the line represented b: a. the data in the table Unit 3 Equations and the Coordinate Plane

48 Embedded Assessment Use after Activit 3.5. Slopes and Intercepts LINEAR KINDNESS. b. the data in the graph Math Knowledge #,, 3, Representation #3 Eemplar Proficient Emerging The student: Correctl determines the slope of the line and what it represents in the problem situation. () Correctl identifies the -intercept of the line. () Correctl graphs a line given its equation. (3) Correctl determines the equation of a line from a table of values. (a) Correctl determines the equation of a line given its graph. (b) The student correctl represents the equation as a graph. The student provides complete or correct answers for three or four of the items. The student provides at least two answers for the five items, but the ma be incorrect or incomplete The student is unable to produce a graph of the equation. 0 College Board. All rights reserved. Communication # The student correctl eplains what the slope of the line represents in the contet of this problem situation. () The student is unable to eplain what the slope represents in this contet. SpringBoard Mathematics with Meaning TM Level 3

49 Analzing Bivariate Data Sue Swandive SUGGESTED LEARNING STRATEGIES: Shared Reading, Role Pla The famous bungee jumper, Sue Swandive, is coming to visit our communit to promote her new doll line. There will be a bungee competition with the new doll. The winning group will get a special prize. Rumor has it that the ma get to go bungee jumping with Sue herself. The competition rules are as follows: a. Attach a rock to the back of a Sue Swandive doll. b. Make a bungee cord b connecting rubber bands and attach it to the doll. c. Drop the doll, with bungee cord attached, from a height specified b our teacher. Height:. d. The winning group s Sue doll will come as close to the ground as possible without hitting her head. M Notes. Pull the black band through,. over the gra, 3. and underneath itself. How to Tie a Slipknot ACTIVITY 3. 0 College Board. All rights reserved. To help our group predict how long to make the bungee cord for the competition, ou will collect data in our classroom first. You will use this data to make a prediction for the number of rubber bands it will take to win the competition. When it is time for our doll to bungee from the height our teacher specified, ou will use the prediction our group made. Begin the classroom part of our eperiments as follows: With one rubber band attached to the Sue doll, have a student hold the end of the rubber band and the doll s feet at the 0 position on the tape measure. Let go of the doll s feet but not the bungee cord. Have our group watch carefull to record the height of the doll s head at its lowest position. (It ma be helpful to tie the doll s hair back.) Be prepared to repeat each jump a few times to get an accurate measurement. Record our findings in the table on the net page. Add rubber bands and continue to take readings until just before Sue s head touches the floor. Unit 3 Equations and the Coordinate Plane 3

50 ACTIVITY 3. Analzing Bivariate Data Sue Swandive M Notes ACADEMIC VOCABULARY Bivariate data can be written as ordered pairs where each numerical quantit represents measurement information recorded about a particular subject. SUGGESTED LEARNING STRATEGIES: Create Representations, Think/Pair/Share. Number of Rubber Bands Attached to the Sue Doll Length of Bungee Jump The data ou have recorded is an eample of bivariate data. Bivariate data is data with two variables.. Create a scatter plot of the data on the grid below. Length of Bungee Jump College Board. All rights reserved Number of Rubber Bands 3. Does the data represent a linear relationship? Eplain our answers using both the scatterplot and the table. SpringBoard Mathematics with Meaning TM Level 3

51 Analzing Bivariate Data Sue Swandive ACTIVITY 3. SUGGESTED LEARNING STRATEGIES: Quickwrite, Interactive Word Wall, Create Representations, Think/Pair/Share. Describe how the length of the bungee jump changes as the number of rubber bands increases. ACADEMIC VOCABULARY trend line 5. What tpe of association does the data represent? A trend line is a line that indicates the general course or tendenc of data.. Use a tool like spaghetti or a ruler, and place it on the scatter plot in a position that has about the same number of points above and below the line. On the coordinate grid, mark two points that the line passes through. The do not have to be data points. MATH TERMS A collection of data points has a positive association if it has the propert that tends to increase as increases. It has a negative association if tends to decrease as increases. If the data have no clear relationship, the have no association. 7. Draw the line that passes through the two points. Positive Association 0 College Board. All rights reserved. 8. Write an equation for our trend line in slope intercept form. 9. Eplain what the variables in the equation of our trend line represent. Negative Association. How does the slope relate to the Sue Doll situation? No Association Unit 3 Equations and the Coordinate Plane 5

52 ACTIVITY 3. Analzing Bivariate Data Sue Swandive M Notes SUGGESTED LEARNING STRATEGIES: Quickwrite, Think/Pair/Share, Work Backwards. Could ou use the equations ou wrote to predict the length of the bungee jump with 3.5 rubber bands?. Use our equation to predict how man rubber bands it will take to give Sue a maimum bungee jump without touching the ground in the contest. The following data was collected on a group of students. There are man possible was to pair the data: TV to homework, homework to TV, TV to test scores, test scores to TV, homework to test scores, test scores to homework. Hours of TV per Week Percent of Homework Completed Test Score Which pairs of data seem to have a positive association? Eplain our reasoning.. Which pairs of data seem to have a negative association? Eplain our reasoning. 0 College Board. All rights reserved. 5. Which pairs of data seem to have no association? Eplain our reasoning. SpringBoard Mathematics with Meaning TM Level 3

53 Analzing Bivariate Data Sue Swandive ACTIVITY 3. SUGGESTED LEARNING STRATEGIES: Create Representations, Activating Prior Knowledge, Discussion Group. For each pair of variables listed below, create a scatter plot with the first variable shown on the -ais and the second variable on the -ais. Find a trend line that represents the data. M Notes a. Hours of TV per week versus the percent of homework completed b. Hours of TV per week versus Test Score c. Percent of homework done versus Test Score a CONNECT TO AP b In AP Statistics, ou will find trend lines for bivariate data using a line called the Least Squares Regression. 0 College Board. All rights reserved. c One student came in late to take the test. He had watched 30 hours of TV during the week, but he scored 0 on the test. How would adding this student s data change the trend line? Unit 3 Equations and the Coordinate Plane 7

54 ACTIVITY 3. Analzing Bivariate Data Sue Swandive M Notes SUGGESTED LEARNING STRATEGIES: Discussion Group, Think/Pair/Share 8. Does the data tell ou that watching TV causes ou to score lower on tests? Eplain our reasoning. CHECK YOUR UNDERSTANDING Write our answers on notebook paper. Show our work. Determine if the following graphs have a positive, negative, or no association Graph Find the equations of the trend lines for an of the questions,, that had a positive or negative association.. MATHEMATICAL REFLECTION What does the association of a set of bivariate data indicate about the slope of the trend line? 5 0 College Board. All rights reserved. 8 SpringBoard Mathematics with Meaning TM Level 3

55 Sstems of Linear Equations Sstems of Trees SUGGESTED LEARNING STRATEGIES: Marking the Tet, Summarize/Paraphrase/Retell, Work Backwards, Create Representations, Group Presentation, Quickwrite, Activating Prior Knowledge M Notes ACTIVITY 3.7 Bob decided to plant some trees in his ard. He bought a -gallon mesquite tree and a 50-gallon desert willow and planted them in his ard. After one ear he was shocked at the growth of both trees, so he measured their heights. The mesquite was 5 ft tall, and the desert willow was 8 ft tall. The net ear he measured again and found the mesquite was ft in. tall, and the desert willow was 8 ft 8 in. tall.. List all the numerical information associated with each tree.. What information in the paragraph is not needed to find an equation that will predict the height of the trees in a given ear? 3. If the trees grew at a constant rate the first two ears, how tall were the when Bob planted them? Mesquite Leaves. Let M be the height of the mesquite tree in inches. Find a linear equation that represents the height of the tree in a given ear, t. 0 College Board. All rights reserved. 5. Find a linear equation that represents the height, W, in inches of the desert willow in a given ear.. Could ou use the equations ou came up with in Questions and 5 to predict the height at.5 ears? 7. Is the domain continuous or discrete? Eplain our reasoning. Desert Willow Leaves 8. What is the domain of the functions M and W? Unit 3 Equations and the Coordinate Plane 9

56 ACTIVITY 3.7 Sstems of Linear Equations Sstems of Trees M Notes SUGGESTED LEARNING STRATEGIES: Create Representations, Quickwrite, Group Presentation, Think/Pair/Share 9. Use the table below to help eplain how the height of the mesquite tree compares to the height of the willow over time. Year M (inches) W (inches). Graph each of the equations on the following grid and use the graph to determine in what ear the mesquite reaches the same height as the desert willow College Board. All rights reserved. 8. When the mesquite tree and the desert willow are the same height, what is true about the values of W and M? 70 SpringBoard Mathematics with Meaning TM Level 3

57 Sstems of Linear Equations Sstems of Trees ACTIVITY 3.7 SUGGESTED LEARNING STRATEGIES: Create Representations, Quickwrite, Shared Reading, Interactive Word Wall, Think/ Pair/Share M Notes. Write and solve an equation to find the value of t when the mesquite tree and the desert willow are the same height. 3. What is the meaning of our solution in Question?. How does the solution ou found in Question relate to the table and the graph? 0 College Board. All rights reserved. One wa to categorize equations M and W is as a sstem of linear equations. The solution to a sstem of linear equations will alwas be the point where the two lines intersect. The value ou determined in Question was the solution to this particular sstem of linear equations. Sstems of linear equations can be solved in man different was. One wa is numericall. 5. Determine which ordered pair in the set {(,), (,3), (,), (3,3)} is the solution to the sstem of linear equations. { = = +. Create a table of values to find the solution to the following sstem of equations. { = - - = ACADEMIC VOCABULARY A sstem of linear equations is a collection of equations which are all considered simultaneousl. The word linear indicates that there will onl be equations of lines in this collection. A point, or set of points, is the solution to a sstem of equations in two variables, when it makes both equations true. WRITING MATH When working with two or more sets of data in a sstem of equations, the output variables can be differentiated b writing them with subscripts. For instance, and are used in problem. 0 Unit 3 Equations and the Coordinate Plane 7

58 ACTIVITY 3.7 Sstems of Linear Equations Sstems of Trees M Notes SUGGESTED LEARNING STRATEGIES: Create Representations, Think/Pair/Share 7. Create a table of values to find the solution to the following sstem of equations (use the M Notes space): { = 5 + = + 8. What problems came up while solving the sstems of equations numericall? Another wa to solve sstems of linear equations is b graphing. 9. Graph the following sstem of equations and write out the solution. { = - = Graph the following sstem of equations and write out the solution. { = 3 - = College Board. All rights reserved SpringBoard Mathematics with Meaning TM Level 3

59 Sstems of Linear Equations Sstems of Trees ACTIVITY 3.7 SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Quickwrite, Interactive Word Wall, Note Taking. Graph the following sstem of equations and write out the solution. { = 3 + = M Notes 8 8. What problems came up while solving the sstems of equations graphicall? You can also solve a sstem of linear equations algebraicall b using the transitive propert of equalit. 0 College Board. All rights reserved. EXAMPLE Solve the following sstem of equations algebraicall. { = - = - + Step : Set the equations equal to each other. - = = Step: Solve for. 5 - = = 5 = Step 3: Substitute into one of the original equations, and solve for. = () - = 3 Step : Check our solution using the other equation = 3 Solution: Write the solution as an ordered pair. The lines intersect at the point (,3). MATH TERMS The transitive propert of equalit states: If a = b and b = c, then a = c. Unit 3 Equations and the Coordinate Plane 73

60 ACTIVITY 3.7 Sstems of Linear Equations Sstems of Trees M Notes TRY THESE A SUGGESTED LEARNING STRATEGIES: Create Representations Solve the following sstems of linear equations algebraicall. a. { = - - = = - c. { = - b. { = + = d. { + = -5 - = CHECK YOUR UNDERSTANDING Write our answers on notebook paper. Show our work. Solve the sstems of equations graphicall. Show our work.. { = + = + 3 Determine what information is needed to solve the following problem. Do not solve the problem.. A boat on a river traveled miles in 0 minutes going downstream. The boat can hold 5 gallons of gas. It takes 30 minutes for the boat to travel back upstream to where it started. Find the speed of the current.. Determine which of the following points {(,-), (-,), (,), (-,-)} are solutions to the sstem of equations. { 3 - = 5 + = Create a table of values to find the solution to the sstem of equations. { = 5-3 = = 0 5. { + 3 = 9 Solve the following sstems algebraicall.. { = = { + = 8 = = { + 3 = 9 9. MATHEMATICAL REFLECTION Which method(s) ou have learned for solving sstems of equations do ou prefer? Eplain wh. 0 College Board. All rights reserved. 7 SpringBoard Mathematics with Meaning TM Level 3

61 Bivariate Data and Sstems IS IT HOT IN HERE OR IS IT ME? Embedded Assessment 3 Use after Activit 3.7. The weather at places around the world changes dail, sometimes hourl. The average temperature over a period of several ears is used to stud weather trends. The average temperatures for two cities, one in the northern hemisphere and one in the southern, are shown below. Guamas, MX Month 3 5 Temp F Johannesburg, SA Month 3 5 Temp F Plot the data from both cities on the grid below. Use dots for Guamas and triangles for Johannesburg Temperature The jagged part of the vertical ais means that the lower values are not included because there are no points with -coordinates that are these lower values. 0 College Board. All rights reserved. 3 5 Month. Describe the associations for each cit. 3. Draw a trend line for each set of data.. Find the equations of the trend lines for both cities. 5. Eplain what the -intercept means for each line in this contet.. Eplain what the slope of each line represents in this contet. 7. Determine the month in which the temperatures of both cities are the same. 8. Solve the following sstem of equations algebraicall. { = 3 - = 5-8 Unit 3 Equations and the Coordinate Plane 75

62 Embedded Assessment 3 Use after Activit 3.7. Bivariate Data and Sstems IS IT HOT IN HERE OR IS IT ME? 9. Solve the following sstem of equations graphicall. = -3 = 5 Math Knowledge #,, 7, 8, 9 Eemplar Proficient Emerging Correctl identifies associations of data on the graph () Correctl determines both equations of trend lines () Correctl identifies the month when temperatures are the same (7) Correctl solves the sstem of equations (8) Correctl solves the sstem of equations graphicall (9) Can onl identif one of the two associations on the graph Determines the correct equation for one of the trend lines Identifies the common temperature but not the month Identifies onl one coordinate of the solution to the sstem Graphs the equations but does not provide the correct solution Is unable to identif the associations present in the graph Is unable to determine the equation of either line Does not identif the common temperature or the month Is unable to provide a solution to the sstem Does not graph the equations Problem Solving #, 8 Correctl interprets data on a graph to describe both associations () Correctl uses an appropriate method to solve the sstem of equations (8) Solves one of the two items correctl and completel Is unable to solve either of the two problems correctl Representation #, 3, 9 Communication #, 5, Creates representation of data () Correctl represents associations with trend lines (3) Correctl graphs the sstem of linear equations and determines the correct solution (9) Correctl describes association of data plotted on the graph () Correctl eplains the meaning of the -intercept for both trend lines (5) Correctl eplains the meaning of the slope for both of the trend lines () Provides appropriate representations for two of the three problems Clearl communicates an eplanation for two of the three items Provides one of the required representations Clearl communicates an eplanation for onl one of the items 0 College Board. All rights reserved. 7 SpringBoard Mathematics with Meaning TM Level 3

63 Practice UNIT 3 ACTIVITY 3.. Looking at the graph, what do ou notice about the relationship between and? 8 8 ACTIVITY 3. Find the domain and range for the data in Questions 8 and 9 8. {(,), (,-), (-5,3), (58,33)} Looking at the graph, what do ou notice about the relationship between and? 8 8 Use mapping to determine if the information in Questions represents a function.. {(-3,), (-,), (,0), (-,5), (-,)}. - 9 for = -, -3, -5, -7, College Board. All rights reserved. Graph the following data sets and identif each as linear or non-linear. 3. {(,-3), (,-), (-,-5), (0,)}. {(3,0), (,), (-,-), (-,)} 5. {(0,5), (,-3), (3,-), (,)}. Determine which of the following epressions displas a linear relationship. Use multiple representations to eplain our reasoning. a. b. c d Eplain how ou can determine if an epression represents a linear pattern. For Questions 3 5 determine if the relations represent functions. Eplain our reasoning Unit 3 Equations and the Coordinate Plane 77

64 UNIT 3 Practice. 9. Find the slope and -intercept of the following: a b c. 3 3 ACTIVITY 3.3 Veronika rides her bike miles in hours.. Create a ratio of Veronika s miles per hour. d. 7. Using the ratio ou found in Question, determine how far Veronika can ride in 5 hours. 8. If Veronika rode her bike for miles at the rate ou found, how long was she riding? If a line with a slope of - contains the point (, 3), then it must also contain which of the following points? a. (-, ) b. (0, 5) c. (, ) d. (, ) e. (8, 0) 0 College Board. All rights reserved. 78 SpringBoard Mathematics with Meaning TM Level 3

65 Practice UNIT 3 ACTIVITY 3.. Find the slope Graph the following linear equations.. = 5 3. = -. = 5 5. A line with a slope of - goes through the point (3, 5). It also goes through the point (-, p). What is the value of p?. Write the equation of the line graphed below. ACTIVITY 3.5 Find the - and -intercepts of the following graphs College Board. All rights reserved. 5 5 Graph the following linear equations. 7. = 5-8. = + 9. = Write an equation for the line graphed below. 5 5 Find - and -intercepts of the following equations. 33. = = = = 7 For Questions 37 and 38, graph the line with the given intercepts intercept: 7 -intercept: 38. -intercept: -5 -intercept: 3 Graph the following equations of lines. 39. = = Unit 3 Equations and the Coordinate Plane 79

66 UNIT 3 Practice For, write the equation of the line in the graph ACTIVITY 3. Determine if the graphs for Questions 3 through have a positive, a negative, or no association College Board. All rights reserved Find the equations of the trend lines for an of the graphs in 3 that had a positive or negative association. 80 SpringBoard Mathematics with Meaning TM Level 3

67 Practice UNIT 3 ACTIVITY 3.7 Determine what information is relevant to solve the following problem. Do not solve the problem. 8. A monke weighs pounds. He eats pounds of bananas in a da. How man pounds of bananas will he eat in week? 9. Determine which of the following points {(0,3), (-,), (3,0), (,-)} are solutions to the sstem of equations. 3 + = 5 + = 7 { 50. Determine which of the following points {(-3,5), (3,-5), (3,5), (-3,-5)} are solutions to the sstem of equations. 3 - = = 9 { Solve the following sstems of equations b graphing. 5. { = 3 + = { + = - + = Solve the following sstems of equations algebraicall. 53. { = - = - 5. { = = { - = + = - 0 College Board. All rights reserved. Unit 3 Equations and the Coordinate Plane 8

68 UNIT 3 Reflection An important aspect of growing as a learner is to take the time to reflect on our learning. It is important to think about where ou started, what ou have accomplished, what helped ou learn, and how ou will appl our new knowledge in the future. Use notebook paper to record our thinking on the following topics and to identif evidence of our learning. Essential Questions. Review the mathematical concepts and our work in this unit before ou write thoughtful responses to the questions below. Support our responses with specific eamples from concepts and activities in the unit. Wh is it important to consider slope, domain, and range in problem situations? How can graphs be used to interpret solutions of real-world problems? Academic Vocabular. Look at the following academic vocabular words: bivariate data range sstem of linear continuous data rate of change equations discrete data relation trend line domain slope -intercept function solution of a sstem of -intercept linear data linear equations Choose three words and eplain our understanding of each word and wh each is important in our stud of math. Self-Evaluation 3. Look through the activities and Embedded Assessments in this unit. Use a table similar to the one below to list three major concepts in this unit and to rate our understanding of each. Unit Concepts Concept Concept Concept 3 Is Your Understanding Strong (S) or Weak (W)? a. What will ou do to address each weakness? b. What strategies or class activities were particularl helpful in learning the concepts ou identified as strengths? Give eamples to eplain.. How do the concepts ou learned in this unit relate to other math concepts and to the use of mathematics in the real world? 0 College Board. All rights reserved. 8 SpringBoard Mathematics with Meaning TM Level 3

69 Read Solve Eplain. Which situation, when graphed, would be non-linear? A. the amount of water in a tub as it drains B. the height of a wedding cake as 5-inch laers are added C. the speed of each car passing through an intersection D. the weight of a sandbag as shovelfuls of dirt are added. What is the slope of the graph of = - +? 3. Jimm joined Rhapsod internet music service at a cost of $.99 per month. He received an MP3 plaer for a gift and wanted to start downloading songs. Rhapsod charges $0.99 per downloaded song. Part A: Complete the table for the cost of downloading,, 3,, or 5 songs in a month Math Standards Review Unit College Board. All rights reserved. # of songs Cost 3 5 Part B: List the domain and range of the function from the table. Write an equation that Jimm can use to determine the cost C of an number of downloads d. Answer and Eplain Domain: Range: Unit 3 Equations and the Coordinate Plane 83