Proportional and Nonproportional Situations

Transcription

1 L E S S N. Florida Standards The student is epected to: Functions.F.1. Compare properties of two functions each represented in a different wa (algebraicall, graphicall, numericall in tables, or b verbal descriptions). Also.F.1.3,.F.. MP..1 Precision Proportional and Nonproportional Situations Engage ESSENTIAL QUESTIN How can ou distinguish between proportional and situations? Sample answer: Determine whether the relationship is linear and the -intercept is 0. Motivate the Lesson Ask: What are some things where the unit price changes as ou bu more of them? Is the price proportional to the number of items ou bu? Eplore ADDITINAL EXAMPLE 1 The graph shows the water level as a bathtub fills. Does the graph show a linear relationship? Is the relationship proportional or? Water level (cm) 1 es; Interactive Whiteboard Interactive eample available online ADDITINAL EXAMPLE The change in a test score for each incorrect answer is represented b the equation = -, where is the number of incorrect answers. Is the relationship between the number of incorrect answers and the change in score proportional or? proportional Interactive Whiteboard Interactive eample available online 113 Lesson. 10 Time (min) EXPLRE ACTIVITY Two batteries cost $3.00. A twelve-pack of the same batteries costs $15. Is the relationship between the number of batteries and the price proportional? Eplain EXAMPLE 1 Questioning Strategies Is a -intercept of 0 enough to conclude that the relationship is proportional? Eplain. No; the relationship must also be linear. Engage with the Whiteboard Have students add a line to the graph that would show a linear relationship. YUR TURN Connect Vocabular ELL Saing that an equation is or nonlinear is the same as saing that an equation is not proportional or not linear. The prefi non- means not. EXAMPLE Questioning Strategies How old was Keith when he graduated from middle school? 1 ears old Focus on Modeling In the linear equation = m + b, just as m and b both represent an constant, and represent an two variables. Therefore, = 1-1 is equivalent to = a - 1. YUR TURN Avoid Common Errors Tell students that two different variables are needed for a proportional relationship. In Eercise, the ma see that there is onl one term on each side of the equation and assume that it is proportional.

2 LESSN.? D NT EDIT--Changes must be made through "File info" CorrectionKe=B Proportional and Nonproportional Situations ESSENTIAL QUESTIN.F.1. Compare properties of two functions each represented in a different wa (algebraicall, graphicall, numericall in tables, or b verbal descriptions). Also.F.1.3,.F.. If a relationship is nonlinear, it is. If it is linear, it ma be either proportional or. When the graph of the linear relationship contains the origin, the relationship is proportional. EXAMPL 1 EXAMPLE.F.1.3 The graph shows the sales ta charged based on the amount spent at a video game store in a particular cit. Does the graph show a linear relationship? Is the relationship proportional or? Sales ta ($).0 Houghton Mifflin Harcourt Publishing Compan The graph shows a linear proportional relationship because it is a line that contains the origin Amount spent ($) Math n the Spot = a - 1 The slope is $0.0. It is the change in the amount of sales ta paid for each dollar spent. The -intercept is 0, meaning ou pa no sales ta if ou don t bu anthing. Animated Math 3. Communicate Mathematical Ideas In a proportional relationship, the ratio _ is constant. Show that this ratio is not constant for the equation = a - 1. Sample answer: (1, ) and (1, 7) are solutions, 7 but _ = = _1 and _ = = _ What If? Suppose another equation represents Keith s age in months given his age in ears a. Is this relationship proportional? Eplain. Yes; the ratio of age in months to age in ears is constant.. YUR TURN Determine if each of the following equations represents a proportional or relationship. 10 The equation is in the form = m + b, with a being used as the variable instead of. The value of m is 1, and the value of b is -1. Since b is not 0, the relationship between the number of ears since Keith graduated and his age is. Reflect 10.F.. The number of ears since Keith graduated from middle school can be represented b the equation = a - 1, where is the number of ears and a is his age. Is the relationship between the number of ears since Keith graduated and his age proportional or? What do the slope and the -intercept of the graph represent in this situation? Determine if each of the following graphs represents a proportional or relationship. If an equation is not a linear equation, it represents a relationship. A linear equation of the form = m + b ma represent either a proportional (b = 0) or (b 0) relationship. EXAMPLE Math Talk YUR TURN How can ou distinguish between proportional and situations? Distinguish Between Proportional and Nonproportional Situations Using a Graph 1. Math n the Spot Distinguish Between Proportional and Nonproportional Situations Using an Equation proportional nline Assessment Lesson. _MFLESE0571_UM0L.indd /0/1 11:5 AM 5. d = 5t. p = 0.1s n = 50-3p. 3 = 1d proportional nline Assessment Houghton Mifflin Harcourt Publishing Compan D NT EDIT--Changes must be made through "File info" CorrectionKe=B 11 Unit _MFLESE0571_UM0L.indd 11 10/0/1 11:5 AM PRFESSINAL DEVELPMENT Integrate Mathematical Practices MP..1 This lesson provides an opportunit to address this standard. It calls for students to communicate mathematical ideas and arguments using precise mathematical language. Students analze relationships represented b words, tables, equations, and graphs and must describe the relationships with terms such as proportional,, linear, and nonlinear. Math Background A function with a restricted domain ma also be a linear, proportional relationship. For eample, the line segment from (, ) to (, 1) is a linear, proportional relationship with a domain of { }. Similarl, a portion of a piecewise function ma be a linear, proportional relationship. For, the piecewise function has a linear, proportional relationship. < f() = { } Proportional and Nonproportional Situations 11

3 ADDITINAL EXAMPLE 3 The table shows the distance of a train from a station and the time it will take to arrive. The relationship is linear. Is it proportional or? Time (min) Distance (mi) Interactive Whiteboard Interactive eample available online ADDITINAL EXAMPLE A John has a choice of hiring two plumbers. In both cases, is the number of hours and is the total charge in dollars. Compare and contrast these two situations. Plumber A: = 75 Plumber B: Total cost ($) Time (h) Plumber A s charges are a proportional relationship; Plumber B s charges are not. Plumber B charges an initial fee of $100 and a lower hourl rate. B The bowling club has a choice between two bowling alles. In both cases, is the number of games and is the total charge in dollars. Compare and contrast these two situations. Nite wl Lanes: = Luck Five Lanes: EXAMPLE 3 Questioning Strategies What number could ou multipl the number in the first column b to get the number in the second column? 13 What would a graph of this data look like? The points (130, 190), (55, 3315), and (505, 55) would lie along a straight line. Focus on Critical Thinking Suppose a fourth tourist traded 50 U.S. dollars and received 30 Meican pesos in return, and this data were added to the table. Would this change our answer? Eplain. Students should realize this would make the relationship nonlinear as the ratio is no longer 13. Integrating Language Arts ELL Encourage a broad class discussion on the Math Talk. English learners will benefit from hearing and participating in classroom discussions. YUR TURN Focus on Math Connections Point out to students that in Eercise 10, is less than so the will find ratios that are less than 1. Assure them that the can compare fractions or decimal equivalents. EXAMPLE Questioning Strategies How could ou write the total charge for Arena B in = m + b form? = How could ou write the total charge for Painter B in = m + b form? = Engage with the Whiteboard Have a student add the line representing the total charge for Arena A to the graph for Arena B. Ask students for a range of hours when each arena appears to be less costl Luck Five s charges are a proportional relationship; Nite wl s charges are not. Nite wl charges for shoes but its per-game rate is lower. Interactive Whiteboard Interactive eample available online 115 Lesson.

4 D NT EDIT--Changes must be made through "File info" CorrectionKe=B D NT EDIT--Changes must be made through "File info" CorrectionKe=B Comparing Proportional and Nonproportional Situations If there is not a constant rate of change in the data displaed in a table, then the table represents a nonlinear relationship. Math n the Spot Math n the Spot EXAMPLE A linear relationship represented b a table is a proportional relationship when the quotient of each pair of numbers is constant. therwise, the linear relationship is. A A laser tag league has the choice of two arenas for a tournament. In both cases, is the number of hours and is the total charge. Compare and contrast these two situations. Arena A Arena B How might graphing the equation for Arena A = 5 Math Talk.F.. The values in the table represent the numbers of U.S. dollars three tourists traded for Meican pesos. The relationship is linear. Is the relationship proportional or? U.S. Dollars Traded Meican Pesos Received 130 1, , ,55 Math Talk Houghton Mifflin Harcourt Publishing Compan Image Credits: Jupiter Images/ Hemera Technologies/Gett Images How could ou confirm that the values in the table have a linear relationship? 3,315 = 1 = 13 Simplif the ratios to compare the pesos received, = = 13 to the dollars traded The ratio of pesos received to dollars traded is constant at 13 Meican pesos per U.S. dollar. This is a proportional relationship. YUR TURN Determine if the linear relationship represented b each table is a proportional or relationship. 9. help ou to compare the situations? Animated Math 1,90 = 19 = proportional.f.1. Sample answer: B graphing the two situations together, ou can more easil see when one option is better than another. Compare the change in Meican pesos to the change in U.S. dollars from each pair of numbers to the net. The results should all be the same. 500 Total cost ($) EXAMPL 3 EXAMPLE You can use what ou have learned about proportional and relationships to compare similar real-world situations that are given using different representations Arena A s equation has the form = m + b, where b = 0. So, Arena A s charges are a proportional relationship. The hourl rate, $5, is greater than Arena B s, but there is no additional fee Hours Arena B s graph is a line that does not include the origin. So, Arena B s charges are a relationship. Arena B has a $50 initial fee but its hourl rate, $00, is lower. B Jessika is remodeling and has the choice of two painters. In both cases, is the number of hours and is the total charge. Compare and contrast these two situations. Painter A = $5 Painter A s equation has the form = m + b, where b = 0. So, Painter A s charges are proportional. The hourl rate, $5, is greater than Painter B s, but there is no additional fee. Painter B Painter B s table is a relationship because the ratio of to is not constant. Because the table contains the ordered pair (0, 0), Painter B charges an initial fee of $0, but the hourl rate, $35, is less than Painter A s. Houghton Mifflin Harcourt Publishing Compan Distinguish Between Proportional and Nonproportional Situations Using a Table nline Assessment Lesson. _MFLESE0571_UM0L.indd /0/1 11:5 AM 11 Unit _MFLESE0571_UM0L.indd 11 10/0/1 11:5 AM DIFFERENTIATE INSTRUCTIN Auditor Cues Critical Thinking Additional Resources Proportional relationships generall have one condition, while ones have an additional condition. Emphasize connecting words like and and but for situations that indicate a second condition, and teach students to do the same. Rounding or taking the integer part of an otherwise proportional relationship can make the relationship. Students might consider a situation where the price is rounded up to the net cent. An item that sells for 3 for $1 will cost $0.3 for one item, for eample. Differentiated Instruction includes Reading Strategies Success for English Learners ELL Reteach Challenge PRE-AP Eamples might be boats rent for $1 an hour and a dail fee of $10 or the cost is $5 an hour but she has a coupon for $50 off. Have students determine whether these situations might still be considered proportional for specific purposes, even though the technicall are not. Proportional and Nonproportional Situations 11

5 YUR TURN Connect to Dail Life Have students consider the coupon for Test-Prep Center B and create a table or graph for this situation. Students should see that for less than hours, the cost is negative. Encourage students to provide some likel restrictions on the use of the coupon. Elaborate Talk About It Summarize the Lesson Ask: How do ou know that a linear relationship given b a graph, a table, or an equation represents a relationship? Sample answer: The -intercept, the value of the dependent variable when = 0, is not 0. GUIDED PRACTICE Engage with the Whiteboard Have students write the ratio of to for each row of the table in Eercises 5 and. Avoid Common Errors Eercise Remind students that in = m + b form, the value of m does not need to be an integer in order for a relationship to be proportional. Eercise 7 Students should take care in dividing, due to the size of the numbers. Even using a calculator, the ma find that two quotients differ b a power of ten, although the should all be the same. 117 Lesson.

6 YUR TURN 11. Compare and contrast the following two situations. Test-Prep Center A The cost for Test-Prep Center A is given b c = 0h, where c is the cost in dollars and h is the number of hours ou attend. Test-Prep Center B Test-Prep Center B charges $5 per hour to attend, but ou have a $100 coupon that ou can use to reduce the cost. Test-Prep Center A s charges are a proportional relationship, but B s charges are not. Center B offers a coupon that gives ou an initial credit, but its hourl rate, $5, is higher than Center A s hourl rate of $0. Center B will cost more if ou attend more than 0 h. Guided Practice nline Assessment The tables represent linear relationships. Determine if each relationship is a proportional or situation. (Eample 3, Eample ) Find the quotient of and. Proportional; the quotient of and is constant,, for ever number pair The values in the table represent the numbers of households that watched three TV shows and the ratings of the shows. The relationship is linear. Describe the relationship in other was. (Eample ) Number of Households that Watched TV Show TV Show Rating No; the quotient of and is not constant for ever number pair. Houghton Mifflin Harcourt Publishing Compan Determine if each relationship is a proportional or situation. Eplain our reasoning. (Eample 1, Eample, Eample ) Look at the origin. Proportional; the line includes the origin. 3. q = p + 1_ Compare the equation with = m + b. Nonproportional; when the equation is written in the form = m + b, the value of b is not v = 1 10 u Nonproportional; the line does not include the origin. Proportional; when the equation is written in the form = m + b, the value of b is 0.? 15,000, ,000, ,000,000 0 Sample answer: The TV show rating is proportional to the number of households that watched, because the quotient when ou divide the rating b the number of households is alwas ESSENTIAL QUESTIN CHECK-IN. How are using graphs, equations, and tables similar when distinguishing between proportional and linear relationships? Sample answer: Proportional relationships eist if the -intercepts for a graph and an equation are 0, and if the table has, or would have, a value of 0 for when is 0. Houghton Mifflin Harcourt Publishing Compan Lesson Unit Proportional and Nonproportional Situations 11

7 . LESSN QUIZ nline Assessment nline homework assignment available Eplain whether each shows a proportional relationship. 1..F.1.,.F.1.3,.F.. Evaluate GUIDED AND INDEPENDENT PRACTICE.F.1.,.F.1.3,.F.. Concepts & Skills Eample 1 Distinguish Between Proportional and Nonproportional Situations Using a Graph Eample Distinguish Between Proportional and Nonproportional Situations Using an Equation Eample 3 Distinguish Between Proportional and Nonproportional Situations Using a Table Eample Comparing Proportional and Nonproportional Situations Practice Eercises 1, 9 Eercises 3, 11 Eercises 5 7, 15 Eercise Eercise Depth of Knowledge (D..K.) 3. 1 = 5. Compare and contrast. Without Discount Card The cost for cat food is Cases 3 Cost ($) Lesson Quiz available online With Discount Card The cost is given b c = 9.5n + 5 where n is the number of cases. 9 Skills/Concepts MP..1 Reasoning Skills/Concepts MP..1 Modeling 1 13 Skills/Concepts MP..1 Precision 1 3 Strategic Thinking MP..1 Reasoning 15 3 Strategic Thinking MP.3.1 Logic Additional Resources Differentiated Instruction includes: Leveled Practice worksheets Eercises combine concept from the Florida cluster Define, evaluate, and compare functions. Answers 1. No; the graph does not go through the origin.. Yes; the ratio of to is constant. 3. Yes; the equation can be written in the form = m + b, and b = 0.. The cost without a discount card is proportional, with a discount card it is not. The cost is more epensive with the discount card initiall, but the per-case rate without it is more epensive. 119 Lesson.

8 Name Class Date. Independent Practice.F.1.,.F.1.3,.F.. nline Assessment and Intervention 13. Represent Real-World Problems Describe a real-world situation where the relationship is linear and. Sample answer: Amanda bus a flute for $500 and then pas $35 per week for lessons. 9. The graph shows the weight of a cross-countr team s beverage cooler based on how much sports drink it contains. 0 FCUS N HIGHER RDER THINKING Work Area Houghton Mifflin Harcourt Publishing Compan a. Is the relationship proportional or? Eplain. Nonproportional; the graph does not pass through the origin, so b 0. b. Identif and interpret the slope and the -intercept. m = 0.5, b = 10; each cup sports drink weighs a half pound. The empt cooler weighs 10 pounds. In 10 11, tell if the relationship between a rider s height above the first floor and the time since the rider stepped on the elevator or escalator is proportional or. Eplain our reasoning. 10. The elevator paused for 10 seconds after ou stepped on before beginning to rise at a constant rate of feet per second. Nonproportional; sample answer: The graph of this situation will etend from (0, 0) to (10, 0) on the -ais before rising Weight (lb) height above floor Sports drink (cups) 11. Your height, h, in feet above the first floor on the escalator is given b h = 0.75t, where t is the time in seconds. Proportional; this equation has the form = m + b where b = Analze Relationships Compare and contrast the two graphs. Graph A Graph B = 1_ 3 = _ Both include the origin, but onl A is a line, making it linear and proportional. B is nonlinear and. height above floor Lesson Mathematical Reasoning Suppose ou know the slope of a linear relationship and one of the points that its graph passes through. How can ou determine if the relationship is proportional or? You can plot the point, use the slope to find another point, and draw a line through the points to see if it passes through the origin. 15. Multiple Representations An entrant at a science fair has included information about temperature conversion in various forms, as shown. The variables F, C, and K represent temperatures in degrees Fahrenheit, degrees Celsius, and kelvin, respectivel. Equation A F = 9_ 5 C + 3 Equation B K = C Degrees Celsius Table C kelvin a. Is the relationship between kelvins and degrees Celsius proportional? Justif our answer in two different was. No; using Equation B ou see that the -intercept is 73.15, not 0, so the graph does not include the origin. Using Table C ou see that the quotient of K and C is not constant: about 35.1, 19.1, and.575. b. Is the relationship between degrees Celsius and degrees Fahrenheit proportional? Wh or wh not? No; Equation A is in the form = m + b, with F being used instead of and C being used instead of. The value of b is 3. Since b is not 0, the relationship is not proportional. 10 Unit Houghton Mifflin Harcourt Publishing Compan EXTEND THE MATH PRE-AP Activit available online Activit A geometric progression is a sequence or list of numbers with a common ratio r between the terms. ne geometric progression is a sequence with a common ratio of : 1,,,, 1, 3... Have students make a geometric progression using the following rule: Choose a number between 1 and 5 for the first value of and choose another number between and 5 for r. Multipl b r to get the first value of. Use this value of as the net value of. Multipl b r again to get the net value of. Have them write the geometric epression and determine if it is a linear relationship. Sample answer:,, 1, 5, 1,.; es, it is linear. Proportional and Nonproportional Situations 10