Name Period Date LINEAR FUNCTIONS STUDENT PACKET 2: MULTIPLE REPRESENTATIONS 2

Transcription

1 Name Period Date LINEAR FUNCTIONS STUDENT PACKET 2: MULTIPLE REPRESENTATIONS 2 LF2.1 LF2.2 LF2.3 Growing Shapes Use variables, parentheses, and exponents in expressions. Use formulas to find perimeter and area of rectangles. Describe geometric patterns numerically, symbolically, graphically, and verbally. Plot ordered pairs that satisfy a specified condition. Informally connect the slope of a line to its context in a graph. Saving for a Purchase Use tables, graphs, equations, and words to solve problems. Informally introduce the slope-intercept form of a line. Informally connect the y-intercept to a context. Going to the Park Solve time-distance problems. Interpret time-distance graphs. Explore rates of change on a graph. Understand the meaning of the points of intersection of two graphs. Informally connect the slope of a line to its context in a graph. LF2 STUDENT PACKET LF2.4 Vocabulary, Skill Builders, and Review LF2 SP

2 WORD BANK Word or Phrase Definition or Explanation Example or Picture area base height length linear function perimeter point of intersection rate slope of a line (informal) slope-intercept form the equation of a line unit rate y-intercept (informal) LF2 SP0

3 2.1 Growing Shapes Ready (Summary) We will extend square and rectangle patterns. Then we will represent geometric measures in the patterns using an inputoutput table, a graph, with symbols, and with words (the fourfold way ). GROWING SHAPES Go (Warmup) Complete each table. Find a rule that will give the output for any input. Set (Goals) Use variables, parentheses, and exponents in expressions. Use formulas to find perimeter and area of rectangles. Describe geometric patterns numerically, symbolically, graphically, and verbally. Plot ordered pairs that satisfy a specified condition. Informally connect the slope of a line to its context in a graph. Table 1 Table 2 Table 3 Input Output Input Output Input Output Rule: If the input is x, then the output will be Rule: If the input is x, then the output will be Rule: If the input is x, then the output will be LF2 SP1

4 2.1 Growing Shapes GROWING SQUARES This is a pattern of growing squares. Continue the pattern for step 4 and step 5. Then complete the tables and the questions, and make graphs on the following page. Step # Table 1 Table 2 Table 3 Step Length of Step Step Perimeter Area number side number number (P) (A) (n) (L) (n) (n) n n n Rule: L = Rule: P = Rule: A = 1. What is the perimeter of the figure obtained in step #10? rule: substitute: perimeter: If the perimeter of the figure is 84, what is the step number? rule: substitute: step number: 3. Use words or diagrams to explain how the length of the side and the perimeter of a square are related. 4. Use words or diagrams to explain how the length of the side and the area of a square are related. LF2 SP2

5 2.1 Growing Shapes Lengths in units L (one color) Perimeter in units P (a second color) GROWING SQUARES GRAPHS Two Graphs on the Left (Use Tables 1 and 2) Step number n 1. Compare the two graphs on the left axes. Which is steeper? Explain. 2. How is the graph on the right different from the two graphs on the left? Area in square units A (a third color) One Graph on the Right (Use Table 3) Step number n LF2 SP3

6 2.1 Growing Shapes GROWING RECTANGLES 1 1. This is a pattern of growing rectangles. We are going to observe growing lengths (base, height, and perimeter) as the step numbers increase. Continue the pattern for steps 4 and 5. Step # Complete the tables. Table 1 Table 2 Table 3 Step # (n) base (b) Step # (n) height (h) Step # (n) Perimeter (P) n n n Rule: b = Rule: h = Rule: P = 3. What is the base of the rectangle for step #12? 5. What is the perimeter of the rectangle for step #10? 4. What is the height of the rectangle for step #14? 6. If the base of the rectangle is 36, what is the step number? LF2 SP4

7 2.1 Growing Shapes GROWING RECTANGLES 1 (Continued) All the tables on the previous page compare the step number to a length. This is because base, height, and perimeter are all linear measurements. 7. Draw a vertical axis on the grid at the right and label it length. Draw a horizontal axis on the grid at the right and label it step number. 8. Draw graphs from the tables on the previous page and note the color used. a. Base vs. step # (color: ) b. Height vs. step # (color: ) c. Perimeter vs. step # (color: ) 9. Draw a trend line for each graph to show each linear pattern, and label each line by name. 10. Use words such as parallel, slant, flat, steep, intersect, and cross to describe the graphs. a. How are they the same? b. How are they different? LF2 SP5

8 2.1 Growing Shapes GROWING RECTANGLES 2 1. This is another pattern of growing rectangles. We are going to observe growing lengths (base, height, and perimeter) as the step numbers increase. Continue the pattern for steps 4 and 5. Step # Complete the tables. Step # (n) Table 1 Table 2 Table 3 base (b) Step # (n) height (h) Step # (n) Perimeter (P) n n n Rule: b = Rule: h = Rule: P = 3. What is the base of the rectangle for step #18? 5. What is the perimeter of the rectangle for step #20? 4. What is the height of the rectangle for step #24? 6. If the base of the rectangle is 40, what is the step number? LF2 SP6

9 2.1 Growing Shapes GROWING RECTANGLES 2 (Continued) All the tables on the previous page compare the step number to a length. This is because base, height, and perimeter are all linear measurements. 7. Draw a vertical axis on the grid at the right and label it length. Draw a horizontal axis on the grid at the right and label it step number. 8. Draw graphs from the tables on the previous page and note the color used. a. Base vs. step # (color: ) b. Height vs. step # (color: ) c. Perimeter vs. step # (color: ) 9. Draw a trend line for each graph to show each linear pattern, and label each line by name. 10. Use words such as parallel, slant, flat, steep, intersect, and cross to describe the graphs. a. How are they the same? b. How are they different? LF2 SP7

10 2.3 Going to the Park Ready (Summary) We will use input-output equations, tables, and graphs to find out how much time is needed to save for a camera and for a printer. We will begin our study of the slope-intercept form of a line through this context. SAVING FOR A PURCHASE Go (Warmup) 1. Use the rule to determine the output numbers. Set (Goals) Use tables, graphs, equations, and words to solve problems. Informally introduce the slope-intercept form of a line. Informally connect the y-intercept to a context. Rule: Multiply each input number by 3 and then add 5 to get each output number. Input Number (x) Output Number (y) 10 (10)(3) + 5 = What is an equation for this rule? Use x for input and y for output. y = LF2 SP8

11 2.3 Going to the Park A digital camera costs $240. CAMERA: INSTRUCTIONS Julie wants to save for the camera. She has $100 in the bank to start, and she is going to save $10 each month. Christina also wants to save for the camera. She has $40 in the bank to start, and she is going to save $25 each month. How many months will it take Julie and Christina to each save up for the digital camera? 1. What is the cost of the digital camera? 2. What is the amount that Julie still needs to save? 3. What is the amount that Christina still needs to save? We will now learn how to write equations that can be used to determine the amount that each girl saved at the end of any month. Let m represent the amount of money that Julie and Christina are going to deposit in their bank accounts each month. Let b represent the amount that Julie and Christina each already have in the bank to start. In the tables on the next page, keep track of the amounts that each girl has when she starts to save, and how much they have each month until they each reach their goal. Let x represent the number of months that Julie and Christina have been saving and y represent the total amount saved. LF2 SP9

12 2.3 Going to the Park CAMERA: TABLES Use the information on the previous page to complete the tables and find equations JULIE m = $10 per month, b = $100 in the bank x y (# of months) (total amount saved) 0 10(0)+ 100 = ( ) = x Write an equation for the amount that Julie saved (y) at the end of any month (x) y = CHRISTINA m = $25 per month, b = $40 in the bank x y (# of months) (total amount saved) x Write an equation for the amount that Christina saved (y) at the end of any month (x) y = The equations you wrote for Julie and Christina are linear functions in the form y = mx + b. We call this slope-intercept form. You will learn more about the slope-intercept form of a linear function in this lesson and future lessons. For Julie s equation: m = b = For Christina s equation: m = b = LF2 SP10

13 2.3 Going to the Park CAMERA: GRAPHS Use the data from the tables on the previous page to make graphs representing the total amount of money that Julie and Christina will save each month. Use one color for Julie s graph and another color for Christina s graph. Total amount saved (y) $40 $ Number of months (x) LF2 SP11

14 2.3 Going to the Park CAMERA: QUESTIONS 1. Who starts out with more money in the bank? How do you know? What is a special name for this start number, which crosses the y-axis? 2. Who is saving at a faster rate? How do you know? What is a special name for this rate? 3. When will both girls have saved the same amount of money? How do you know? What is the special name for this coordinate? 4. How long will it take Julie to save for the camera? 5. How long will it take Christina to save for the camera? 6. Who will be the first to save enough money for the camera? 7. Write an equation that describes the amount of money y that Julie has saved after x months. y = 8. Write an equation that describes the amount of money y that Christina has saved after x months. y = 9. The linear functions you wrote in problems 7 and 8 are in y = mx + b form. We call this form. LF2 SP12

15 2.3 Going to the Park A printer costs $150. PRINTER: INSTRUCTIONS AND TABLES Theresa wants to save for a printer. She has $10 in the bank to start, and she is going to save $20 each month. Cary also wants to save for the printer. She has $25 in the bank to start, and she is going to save $15 each month. How many months will it take Theresa and Cary to each save up for the printer? Theresa m = $20 per month b = $10 in the bank y = x (# of months) To find the total amount saved, use the equation form y = mx + b. y (total amount saved) Cary m = $ per month b = $ in the bank y = x (# of months) y (total amount saved) LF2 SP13

16 2.3 Going to the Park PRINTER: GRAPHS AND QUESTIONS Use the data from the tables on the previous page to make graphs representing the total amount of money that Theresa and Cary will save each month. Use one color for Theresa s graph and another color for Cary s graph. Label your axes and title your graph. Then answer questions about the graphs. 1. Who starts out with more money in the bank? How do you know? 2. Who is saving at a faster rate? How do you know? 3. When will both girls have saved the same amount of money? 4. How long will it take Theresa to save for the printer? 5. How long will it take Cary to save for the printer? 6. Who will be the first to save enough money for the printer? 7. Write an equation in slope-intercept form that describes the amount of money y that Theresa has saved after x months. 8. Write an equation in slope-intercept form that describes the amount of money y that Cary has saved after x months. LF2 SP14

17 2.3 Going to the Park BRIAN S PROBLEM: INSTRUCTIONS AND TABLE Brian wants to save for a camera and then a printer. A digital camera costs $240 and a printer costs $150. He has $100 saved in the bank and is going to save $20 each month. 1. Cost of camera: 2. Cost of printer: 3. Total amount Brian still needs to save to purchase both: To find the total amount saved, use the equation form: y = mx + b m = the amount of money Brian is going to deposit in his bank account each month b = the amount of money Brian already has in the bank x = the number of months he has been saving y = total amount saved 4. Write an equation in slope-intercept form to show the total amount of money Brian has saved at the end of each month. x (# of months) y (amount saved) LF2 SP15

18 2.3 Going to the Park BRIAN S PROBLEM: GRAPH AND QUESTIONS 1. Make a graph to show the total amount of money Brian has saved each month. Total amount saved (y) $400 $300 $200 $ Number of months (x) 2. Brian wants to buy the camera first. How long will it take Brian to save for the camera? How do you know? 3. Then Brian will buy the printer. How long will it take him to save for the printer, after he purchases the camera? How do you know? 4. Suppose Brian decided to buy the printer first. How long would it take Brian to save for the printer? Why is this number of months different than your answer to #3 above? LF2 SP16

19 2.3 Going to the Park Ready (Summary) We will use information about friends going to a park after school to help us understand time, distance, and rate of speed relationships using numbers, graphs, symbols, and words. GOING TO THE PARK Go (Warmup) Set (Goals) Solve time-distance problems. Interpret time-distance graphs. Explore rates of change on a graph. Understand the meaning of the points of intersection of two graphs. Informally connect the slope of a line to its context in a graph. Wing-Ye and Conchita are racing. Use the graph below to answer the following questions. 1. Who starts out faster? How do you know? 2. Who starts out slower? Does she ever catch up? How do you know? 3. Who wins the race? How do you know? Distance goal Conchita Time Wing-Ye LF2 SP17

20 2.3 Going to the Park PART 1 INTRODUCTION A group of friends are going to meet at the park after school. They will all travel 90 meters straight down Euclid Street from the school to the park. Zoë got a new digital camera and wants to use it to take pictures of her friends journey. She will monitor their progress by taking nine pictures at six-second intervals from a building high above Euclid Street. She will lay the pictures down side-by-side, in order from the first picture to the last. She will then graph these images and analyze their movements. Zoë starts to take pictures at exactly 3:00:00 PM. Amy is walking and got a head start. At 3:00:00 PM, she is already 36 meters from school. Brandy is jumping rope. She leaves the school at 3:00:00 PM. 1. Approximately how far is 90 meters? 2. How many pictures is Zoë going to take? How long will Zoë wait between snapping pictures? 3. Beginning at 3:00:00 PM, record the first six times that Zoë will snap pictures. 3:00:00 4. Complete the table showing distances from school. Picture # Amy s distance from school Brandy s distance from school 36m 42m 48m 54m 60m 66m 72m 78m 84m* 0m 12m 24m 36m 48m *not at the park yet 5. Does Amy reach the park by the time Zoë finishes taking her pictures? 6. Does Brandy reach the park by the time Zoe finishes taking her pictures? LF2 SP18

21 2.3 Going to the Park RECORDING SHEET: PART 1 1. Graph the information from the table about Amy and Brandy :00 (picture 1) :06 (picture 2) :12 (picture 3) :18 (picture 4) :24 (picture 5) :30 (picture 6) :36 (picture 7) Time (in seconds) elapsed since 3:00:00 :42 (picture 8) :48 (picture 9) 2. If possible, mark and label the coordinate that shows when the girls are the same distance from school. 3. If possible, mark and label the coordinates that show when the girls reach the park. 4. For both girls graphs, connect the points to show trend lines. LF2 SP19

22 2.3 Going to the Park BRANDY S JOURNEY Suppose that Brandy continues to jump rope through all 9 of Zoë s pictures. 1. Complete the following information about Brandy s journey. Picture # Time (in seconds) Distance (in meters) Write a rule that describes the relationship between Brandy s time and distance. In words: In symbols: Distance = D = 3. Use the information from problem 1 to complete the table below. Pictures Change in Distance (in meters) Elapsed Time (in seconds) 1 to 2 12m 0m = 12m 6 sec 0 sec = 6 sec 1 to 3 2 to 4 Rate of change meters second 12m 2m = 6 sec 1 sec 4. In the last column in the table above, what do you notice about the rates of change? 5. How is the rate of 2 meters per second represented on the graph of Brandy s trend line? We will call the rate of change as you move from one point to another the slope of the line. Brandy s trend line is an example of a line with a positive slope. LF2 SP20

23 2.3 Going to the Park AMY S JOURNEY Suppose that Amy continues to walk through all 9 of Zoë s pictures. 1. Complete the following information about Amy s journey. Picture # Time (in seconds) Distance (in meters) 36m 42m 48m 54m 2. Write a rule that describes the distance traveled (in meters) by Amy at a given time (in seconds) In words: In symbols: Distance = D = 3. Use the information from problem 1 to complete the table below. Pictures Change in Distance (in meters) Elapsed Time (in seconds) 1 to 2 42 m 36 m = 6 m 6 sec 0 sec = 6 sec 1 to 4 3 to 4 Rate of change meters second 6 m 1 m = 6 sec 1 sec 4. In the last column in the table above, what do you notice about the rates of change? 5. How is the rate of 1 meter per second represented on the graph of Amy s trend line? 6. The rate of change as you move from one point to another is called the. LF2 SP21

24 2.3 Going to the Park PART 2 Now, Zoë will take nine more pictures of Charlie and Diego starting at 3:15:00. Again she will wait 6 seconds between snapping pictures. Charlie left earlier and is waiting 60 meters from the school. Diego is already at the park at 3:15:00. He leaves the park on his roller skates to go back to school for soccer practice. 1. Beginning at 3:15:00, record the first six times that Zoë will snap pictures. 2. Complete the table showing distances from school. Name Picture # (Picture 1 starts at 0 seconds. Zoë took a picture every 6 seconds.) Picture # Charlie s distance from school Diego s distance from school 60m 60m 60m 60m 60m 60m 60m 60m 60m 90m 72m 54m 36m 18m 0m 3. Does Charlie get back to school by the time Zoë finishes taking her pictures? If so, when? How do you know? 4. Does Diego get back to school by the time Zoe finishes taking her pictures? If so, when? How do you know? LF2 SP22

25 2.3 Going to the Park RECORDING SHEET: PART 2 1. Graph the information from the table about Charlie and Diego. Meters from school :00 (picture 1) :06 (picture 2) :12 (picture 3) :18 (picture 4) :24 (picture 5) :30 (picture 6) :36 (picture 7) Time (in seconds) elapsed since 3:15:00 :42 (picture 8) :48 (picture 9) 2. If possible, mark and label the coordinate that shows when the boys are the same distance from school. 3. If possible, mark and label the coordinates that show when the boys reach the school. LF2 SP23

26 2.3 Going to the Park DIEGO S JOURNEY Suppose that Diego continues to roller skate through all 9 of Zoë s pictures. 1. Complete the following information about Diego s journey. Picture # Time (in seconds) Distance (in meters) Write a rule that describes the relationship between Diego s time and distance. 3. Use the information from problem1 to complete the table below. Pictures Change in distance (in meters) Elapsed Time (in seconds) 1 to 2 72 m 90 m = -18 m 6 sec 0 sec = 6 sec 1 to 3 2 to 4 Rate meters second -18 m -3 m = 6 sec 1 sec 4. What do you notice about the rates of change between pictures? How is this represented on the graph of Diego s trend line? 5. How long does it take Diego to roller skate to school? Diego is not traveling at a negative rate of speed, which is impossible. Rather, Diego is traveling in the opposite direction as the others, at a rate of 3 meters per second. Diego s trend line is an example of a line with a negative slope. LF2 SP24

27 2.3 Going to the Park CHARLIE S JOURNEY Suppose that Charlie remains 60 meters from school through all 9 of Zoë s pictures. 1. Complete the following information about Charlie s journey. Picture # Time (in seconds) Distance (in meters) Write a rule that describes the relationship between Charlie s time and distance. 3. Use the information from problem 1 to complete the table below. Pictures Change in distance (in meters) Elapsed Time (in seconds) 1 to 2 60 m 60 m = 0 m 6 sec 0 sec = 6 sec 1 to 4 3 to 4 Rate meters second 0 m 0 m = 6 sec 1 sec 4. What do you notice about the rates of change between pictures? How is this represented on the graph of Charlie s trend line? 5. How is this shown on the graph? Charlie s trend line is an example of a line with a zero slope. LF2 SP25

28 2.4 Vocabulary, Skill Builders, and Review VOCABULARY, SKILL BUILDERS, AND REVIEW FOCUS ON VOCABULARY Select the word(s) from the word bank on SP0 that best complete the sentences. 1. is a measurement used to describe distance. The distance around the outside of a two-dimensional shape or figure is called the. 2. A is a ratio in which the numbers have units attached to them. A is the value of one unit of measure. 3. The length of the side perpendicular to the is the height of the rectangle. The of the rectangle is the product of its and its. 4. An equation of the form y = mx + b is a function in form. 5. The is the point where a graph crosses the y-axis. The of a line tells its rate of change. A of two lines is a point where the lines meet. LF2 SP26

29 2.4 Vocabulary, Skill Builders, and Review Compute. SKILL BUILDER (-8) (-6) (-7) + (-7) (-25) (-15) 10. Graph each point on the coordinate plane. A(5, -2) B(3, 5) C(-3, 0) D(-3, 4) E(-3,-3) F(-5, 1) G(2, -1) H(0, 4) 11. What quadrant are points G and A located? 12. Name the point that lies on the x-axis. 13. Name the point that lies on the y-axis. LF2 SP27

30 2.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 2 Compute. Show your work using positive (+) and negative ( ) symbols if needed. 1. (-7) + (9) = 2. (16) + (-12) = 3. (-32) + (-42) = 4. (-50) (-20) = 5. (50) (-20) = 6. (10) (16) = 7. (10) (-16) = 8. (16) (-10) = 9. (0) (73) = 10. (-7) (-9) = = (-7)(-4) = Place parentheses in the equations below so that each becomes true. Write none needed if the equation is already true = = -48 Simplify (2 6) Solve each inequality. Then graph the solution(s). Check a point on the ray x > -30 Solution: Graph: Check a point on the ray. 18. x + (-4) -2 Solution: Graph: Check a point on the ray. LF2 SP28

31 2.4 Vocabulary, Skill Builders, and Review Simplify (2 6) SKILL BUILDER 3 Translate each verbal expression into a variable expression 3. three times the sum of a number v and (2 6) the sum of 9 and three times a number v 5. Translate the verbal inequality into symbols, solve it mentally, and graph the solution(s). a. Words: A number times 3 is at least 18. c. Solution(s): d. Graph: b. Symbols (choose a variable): 6. Build, draw, record, solve, and check the equation. Picture Equation/Steps What did you do? Check your solution using substitution: 3(x 1) = 3x +1 + x 5 LF2 SP29

32 2.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 4 Draw the next step suggested by this pattern. Then complete the table and find a rule for the number of toothpicks at step n. step1 step 2 step 3 step 4 Step # n Arithmetic # of toothpicks 1. Label the horizontal and vertical axes and graph the data points. 2. Recursive Rule: 3. Explicit Rule: 4. How many toothpicks are in step #50? 5. In what step number are there exactly 64 toothpicks? LF2 SP30

33 2.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 5 Find missing values in each input-output table, and write an explicit rule for the data x y x y x y Rule: y = Rule: y = Rule: y = For each explicit rule, complete the input-output table x Rule: y y = x 2 x Rule: y x y = 2x + 4 Rule: 1 y = 2 x y LF2 SP31

34 2.4 Vocabulary, Skill Builders, and Review TEST PREPARATION Show your work on a separate sheet of paper and choose the best answer. 1. Which of the following is a rule for this input-output table? x y A. y = 4x + 6 B. y = 6x + 4 C. y = x + 4 D. y = x Sierra is saving for a printer that costs $150. She has $75 already saved in the bank and is going to save $35 each month. If x represents the number of months and y represents the total amount saved, which equation shows the total amount of money Sierra will have at the end of each month? A. y = 35x + 75 B. y = 35x C. y = 75x + 35 D. y = 75x Jesse starts running at noon at a constant rate. The table shows the distance he travels at 5-second intervals past noon. Time (in seconds past noon) Distance (in meters) What is a rule for the relationship between the distance and time traveled by Jesse? A. d = t + 5 B. d = 2t C. d = t + 10 D. d = 5t LF2 SP32

35 2.4 Vocabulary, Skill Builders, and Review KNOWLEDGE CHECK Show your work on a separate sheet of paper and write your answers on this page. 2.1 Growing Shapes 1. Write a numerical expression for step (1 1) 2 + (2 2) 2 + (3 3) step 1 step 2 step 3 step 4 2. Write a variable expression for the pattern illustrated in #1 for step n. 2.2 Saving for a Purchase 3. Use the following explicit rule: Rule: to find each output number, multiply each input number by 4, and then add 2. Input (x) Output (y) 4. Write an equation that fits the rule in #3. Use x for the input number and y for the output number. y = 2.3 Going to the Park Jamal starts rollerblading at 8:00:00 AM. The table below shows the distance Jamal travels at four-second intervals after 8:00:00 AM. 5. Complete the table using the established pattern. Time (seconds past 8:00:00 AM) Distance (meters traveled) Write a rule about the relationship between Jamal s time and distance. In words: In symbols: Distance = D = LF2 SP33

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39 HOME-SCHOOL CONNECTION Here are some questions to review with your young mathematician. 1. Write an equation for this explicit rule (input-output rule): To get each output number (y), multiply each input (x) by 25, and then add Complete the table and find an explicit rule for the pattern. The first three steps are given. step1 step2 step3 Step # (x) x Perimeter (y) Boris and Sandy are riding bikes. Based on the graph to the right, who is riding faster? How can you tell? Parent (or Guardian) Signature Distance Time Boris Sandy LF2 SP37

40 COMMON CORE STATE STANDARDS MATHEMATICS 6.RP.3b 6.EE.9 8.EE.5 8.EE.8a 8.F.4 A-CED-2 A-REI-10 F-BF-1a MP1 MP2 MP3 MP4 MP5 MP6 MP7 MP8 STANDARDS FOR MATHEMATICAL CONTENT Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed? Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Determine an explicit expression, a recursive process, or steps for calculation from a context. STANDARDS FOR MATHEMATICAL PRACTICE Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. LF2 SP38