Interpreting the Unit Rate as Slope

Transcription

1 L E S S N 3.3 Florida Standards The student is expected to: Expressions and Equations.EE.. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Also.F.1.,.F.. MP..1 Modeling Interpreting the Unit Rate as Slope Engage ESSENTIAL QUESTIN How do you interpret the unit rate as slope? Sample answer: The ratio of the change in y to the change in x is the unit rate. It is also the ratio of the rise to the run, or the slope. Motivate the Lesson Ask: Have you ever been skiing, or watched downhill skiers on television? Discuss types of slopes that might be encountered, from a bunny slope to the black diamond expert slope. Consider how you might describe slight slopes and extreme slopes numerically. Explore ADDITINAL EXAMPLE 1 Every seconds an escalator step rises 6 feet. Draw a graph of the situation. Find the unit rate of this proportional relationship. Step height (ft) feet per second Interactive Whiteboard Interactive example available online Animated Math Proportional Relationships Students explore how changing the parameters of a real-world proportional relationship affects tables and graphs. EXPLRE ACTIVITY Connect Multiple Representations Ask students for the method they use to find the coordinates of points on a line. Point out that it is easiest to find the coordinates of points at the intersection of grid lines. Explain EXAMPLE 1 Questioning Strategies What is the constant change in the input values? 3 What is the constant change in the output values? Where do the rise of and run of 6 come from? The vertical and horizontal distance from (0, 0) to (6, ) What if you used (6, ) and (1, ) to find the slope? You would get a rise of 1, a run of, and a slope of 1 which is equal to 3. Avoid Common Errors Students may reverse the order in the ratio and divide the difference of the x-values by the difference of the y-values. Remind students that slope is always rise over run (y-values over x-values). YUR TURN Avoid Common Errors Note that in the Example the time is given first, and then the volume. In Exercise, the distance is given first, and then the time. Caution students to read the information carefully before deciding on the independent variable. Talk About It Check for Understanding Ask: What would be another point on the line that could be shown with a larger graph and different scales? Sample answer: (1, 3) 3 Lesson 3.3

2 D NT EDIT--Changes must be made through File info CorrectionKey=B D NT EDIT--Changes must be made through File info CorrectionKey=B LESSN 3.3? Interpreting the Unit Rate as Slope.EE.. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Also.F.1.,.F.. ESSENTIAL QUESTIN Graphing Proportional Relationships You can use a table and a graph to find the unit rate and slope that describe a real-world proportional relationship. The constant of proportionality for a proportional relationship is the same as the slope. Math n the Spot EXAMPLE 1 How do you interpret the unit rate as slope?.ee.. Every 3 seconds, cubic feet of water pass over a dam. Draw a graph of the situation. Find the unit rate of this proportional relationship..ee..,.f.. STEP 1 Relating the Unit Rate to Slope Animated Math A rate is a comparison of two quantities that have different units, such as miles and hours. A unit rate is a rate in which the second quantity in the comparison is one unit. In a proportional relationship, how are the constant of proportionality, the unit rate, and the slope of the graph of the relationship related? They are the same. B Find the unit rate of snowfall in inches per hour. Explain your method Water ver the Dam Draw a graph. Find the slope. rise _ slope = run = 6 3 Volume (ft3) = _3 6 Time (sec) The unit rate of water passing over the dam and the slope of the graph of the relationship are equal, _3 cubic feet per second. Reflect 1. inches per hour; Sample answer: The point (1, ) is on the line, and represents inches snowfall in 1 hour. What If? Without referring to the graph, how do you know that the point ( 1, _3 ) is on the graph? Sample answer: The point (1, r) is on any graph of a proportional relationship, where r equals the unit rate. C Compare the slope of the graph and the unit rate of change in the snow level. What do you notice? They are the same. YUR TURN D Which unique point on this graph gives you the slope of the graph and the unit rate of change in the snow level? Explain how you found the point. Because the line goes through the point (0, 0), the amount of snow Tomas rides his bike at a steady rate of miles every minutes. Graph the situation. Find the unit rate of this proportional relationship. in 1 h equals the y-coordinate of the point where x is 1. That point His unit rate and the slope of. (1, ); Sample answer: the unit rate is the amount of snow in 1 h. is (1, ) and in./h is the unit rate and the slope. nline Assessment and Intervention a graph of the ride both equal 1 _ mi/min. Tomas s Ride change in y-value - = = _ = _1 change in x-value -1 Math Talk Image Credits: Cavan Images/ Getty Images A Find the slope of the graph using the points (1, ) and (, ). Remember that the slope is the constant rate of change. STEP 3 Misty Mountain Storm Snowfall (in.) A storm is raging on Misty Mountain. The graph shows the constant rate of change of the snow level on the mountain. STEP Make a table. Amount (cu ft) EXPLRE ACTIVITY Time (min) Lesson 3.3 _MFLESE0671_UM03L3.indd 3 3 //1 : AM Unit _MFLESE0671_UM03L3.indd /0/1 11:1 AM PRFESSINAL DEVELPMENT Integrate Mathematical Practices MP..1 This lesson provides an opportunity to address this standard. It calls for students to analyze mathematical relationships using tools such as tables and graphs. Students find and analyze the unit rate in input and output tables. Then students use graphs to find the slope of a line. In this way, students are led to make the connection between unit rate and slope. Math Background The slope of a line that does not go through the origin can be found by its x-intercept and y-intercept. If the x-intercept is a and the y-intercept is b, the line goes through (a, 0) and (0, b). The slope m of the line containing the points is b-0 m = = -_ba. 0-a Interpreting the Unit Rate as Slope

3 ADDITINAL EXAMPLE The equation y = 1.x represents the rate, in beats per second, that Lee s heart beats. The graph represents the rate that Nancy s heart beats. Determine whose heart is beating at a faster rate. Heart beats > 1.; Nancy s heart is beating faster. Interactive Whiteboard Interactive example available online EXAMPLE Questioning Strategies What does the point (, ) represent in the graph? Well B pumps barrels of oil in hours. In the equation y =.7x, what does.7 represent in the graph of the equation? The slope of the graph Focus on Modeling Have students reproduce the graph on their own sheet of graph paper. Ask them to describe how they can make sure that the graph they draw accurately matches the graph in the book. Encourage the use of a ruler and sharpened pencil. Engage with the Whiteboard Have a student plot the points for Well A from the table on the graph and connect them. Ask students to compare the rise and run for each graph. This will help them confirm that the slope is greater for Well A than Well B. YUR TURN Connect Multiple Representations Students find the greater rate from an equation and a table. In Example, students found the greater rate from an equation and a graph. Students should feel confident they can find the greater rate using any two representations. Elaborate Talk About It Summarize the Lesson Ask: How do you find the slope when you are only given an equation or a table? Sample answer: For an equation such as y =.6x, the slope is the unit rate, which is the coefficient of x. For a table, the change in y divided by the change in x is the unit rate, or slope. GUIDED PRACTICE Engage with the Whiteboard For Exercises 1, after students complete each exercise, have them copy the graph for Akiko onto the graph for Jorge. Have them determine whether Jorge or Akiko is the faster hiker and use the graph to explain their answer. In Exercise 3, students should graph the equation for Henry on the graph for Clark. Avoid Common Errors Exercise 3 Help students understand that comparing rates, or slopes, is the same as comparing rational numbers. Students can write each unit rate as a decimal, if necessary. Integrating Language Arts ELL Encourage English learners to use the text and table in Exercises 1 to help them understand the content of the graphs. Lesson 3.3

4 Using Slopes to Compare Unit Rates You can compare proportional relationships presented in different ways. Guided Practice Give the slope of the graph and the unit rate. (Explore Activity and Example 1) Image Credits: Tom McHugh/ Photo Researchers, Inc. EXAMPLE The equation y =.7x represents the rate, in barrels per hour, that oil is pumped from Well A. The graph represents the rate that oil is pumped from Well B. Which well pumped oil at a faster rate? STEP 1 STEP STEP 3 STEP Use the equation y =.7x to make a table for Well A s pumping rate, in barrels per hour. 1 3 Quantity (barrels) Amount (barrels) Use the table to find the slope of the graph of Well A slope = unit rate = =.7-1 =.7 barrels/hour 1 Use the graph to find the slope of the graph of Well B. slope = unit rate = rise run = =. barrels/hour Compare the unit rates..7 >., so Well A s rate,.7 barrels/hour, is faster. Well B Pumping Rate Reflect 3. Describe the relationships among the slope of the graph of Well A s rate, the equation representing Well A s rate, and the constant of proportionality. Sample answer: The slope and the constant of proportionality equal the value.7 in the equation y =.7x. YUR TURN. The equation y = 37x represents the relationship between x, the time that a plane flies in hours, and y, the distance the plane flies in miles for Plane A. The table represents the relationship for Plane B. Find the slope of the graph for each plane and the plane s rate of speed. Determine which plane is flying at a faster rate of speed EE..,.F.1. A: 37, 37 mi/h; B:, mi/h; B is flying faster. Math n the Spot nline Assessment and Intervention 1. Jorge: miles every 6 hours. Akiko Jorge 3. The equation y = 0.x represents the distance Henry hikes, in miles, over time, in hours. The graph represents the rate that Clark hikes. Determine which hiker is faster. Explain. (Example ) Write an equation relating the variables in each table. (Example )? slope = unit rate = _ 6 mi/h. Time (x) 1 6 Distance (y) ESSENTIAL QUESTIN CHECK-IN Akiko slope = unit rate = _ mi/h Clark is faster. From the equation, Henry s rate is equal to 0., or 1_ mile per hour. Clark s rate is the slope of the line, which is 3_, or 1. miles per hour. y = 1x 6. Describe methods you can use to show a proportional relationship between two variables, x and y. For each method, explain how you can find the unit rate and the slope. Table of values: The ratio of y to x gives the unit rate and slope. Clark. Time (x) Distance (y) y = 3_ x Equation: If the equation can be written as y = mx, then m is the unit rate and the slope. Graph: When the line passes through the origin, then the value of r at the point (1, r) is the unit rate and the slope. Lesson Unit DIFFERENTIATE INSTRUCTIN Technology Have students use a graphing calculator to examine slopes and rate of change. For example, have them graph y = 1.x and use the TRACE function to find points on the line. Lead them through the process of finding a friendly graphing window by first using the ZM ZDecimal function. A good first quadrant window is X Min = 0, X Max =., Y Min = 0, Y Max = 6.. You can multiply these numbers by whole numbers to use larger windows. y = 1.x (3,.) Additional Resources Differentiated Instruction includes: Reading Strategies Success for English Learners ELL Reteach Challenge PRE-AP Interpreting the Unit Rate as Slope 6

5 3.3 LESSN QUIZ nline Assessment and Intervention nline homework assignment available Lesson Quiz available online.ee..,.f.1.,.f.. 1. Every seconds, a ski lift chair rises 1 feet. Draw a graph of the situation. Then describe the relationship between the height of the chair and the time.. Under Plan A, a -minute call costs $0. and a -minute call costs $1.0. Under Plan B, the cost for x minutes is given by y = 0.x. Which plan is cheaper? Why? 3. The equation y = 13x represents the rate, in gallons per minute, that Tank A at an aquarium fills with water. The table represents the rate that Tank B fills with water. Determine which tank fills faster. Time (min) 1 1 Amount (gal) Answers Evaluate GUIDED AND INDEPENDENT PRACTICE.EE..,.F.1.,.F.. Concepts & Skills Explore Activity Relating the Unit Rate to Slope Example 1 Graphing Proportional Relationships Example Using Slopes to Compare Unit Rates Additional Resources Differentiated Instruction includes: Leveled Practice worksheets Practice Exercises 1 Exercises 1, 7 Exercises 3,, 1 Exercise Depth of Knowledge (D..K.) 7 Skills/Concepts MP..1 Modeling 1 Recall Skills/Concepts MP..1 Modeling Strategic Thinking MP.3.1 Logic 13 3 Strategic Thinking MP..1 Reasoning Exercise 1 combines concepts from the Florida cluster Understand the connections between proportional relationships, lines, and linear equations. Chair height (ft) The unit rate of the height of the chair is 7 feet per second.. Plan A; $0.7/min < $0./min 3. Tank A; 13 gal/min > 11 gal/min 7 Lesson 3.3

6 Name Class Date 3.3 Independent Practice.EE..,.F.1.,.F.. 7. A Canadian goose migrated at a steady rate of 3 miles every minutes. a. Fill in the table to describe the relationship. Time (min) nline Assessment and Intervention. Cycling The equation y = 1_ x represents the distance y, in kilometers, that Patrick traveled in x minutes while training for the cycling portion of a triathlon. The table shows the distance y Jennifer traveled in x minutes in her training. Who has the faster training rate? Time (min) Distance (km) 1 Patrick s rate is 1_ kilometer per minute. Jennifer s rate is 1_ kilometer per minute. 1_ < 1_, so Jennifer has the faster training rate. b. Graph the relationship. c. Find the slope of the graph and describe what it means in the context of this Migration Flight problem. 3_ ; The unit rate of migration of the goose and the slope of the graph both equal 3_ mi/min. Time (min). Vocabulary A unit rate is a rate in which the first quantity / second quantity in the comparison is one unit.. The table and the graph represent the rate at which two machines are bottling milk in gallons per second. Machine Machine Amount (gal) Time (sec) a. Determine the slope and unit rate of each machine. Machine 1: slope = unit rate = 0.6 = 0.6 gal/s; 1 Machine : slope = unit rate = 3_ = 0.7 gal/s b. Determine which machine is working at a faster rate. Machine is working at a faster rate since 0.7 > 0.6. Amount (gal) FCUS N HIGHER RDER THINKING 11. Analyze Relationships There is a proportional relationship between minutes and dollars per minute, shown on a graph of printing expenses. The graph passes through the point (1,.7). What is the slope of the graph? What is the unit rate? Explain. slope = unit rate =.7. If the graph of a proportional relationship passes through the point (1, r), then r equals the slope and the unit rate, which is $.7/min. 1. Draw Conclusions Two cars start at the same time and travel at different constant rates. A graph for Car A passes through the point (0., 7.), and a graph for Car B passes through (, 0). Both graphs show distance in miles and time in hours. Which car is traveling faster? Explain. Car B; the slope and unit rate of speed of Car A is = 7. = mi/h. The slope and unit rate of 0. speed of Car B is = 0 = 60 mi/h. 60 >, so Car B is traveling faster. 13. Critical Thinking The table shows the rate at which water is being pumped into a swimming pool. Time (min) Amount (gal) Use the unit rate and the amount of water pumped after 1 minutes to find how much water will have been pumped into the pool after 13 1_ minutes. Explain your reasoning. 3 gallons; sample answer: The unit rate is 36 = 1 gal/min. So 1 1_ minutes after 1 minutes, an additional 1 1 1_ = 7 gallons will be pumped in, so the total is Work Area = 3 gal. Lesson Unit EXTEND THE MATH PRE-AP Activity available online Activity Have students select any of the proportional relationships in the lesson, given as an equation, table, or graph. Then have them consider whether there is still a proportional relationship if each of these operations is performed on the y-values. double the y-values yes divide the y-values by yes add to the y-values no subtract from the y-values no Then have them classify which operations kept the proportional relationship and which did not. Multiplication and division preserve the proportional relationship; addition and subtraction do not. Interpreting the Unit Rate as Slope