AM_S_C_L_3.indd Page // 3: PM s-user /Volumes//GO/CORE_READING/TENNESSEE/ANCILLARY... Proportionalit and Linear Relationships Teach the Concept Lesson - Direct Variation Interactive Stud Guide See pages for: Getting Started Real-World Link Notes Essential Question How are linear functions used to model proportional relationships? Common Core State Standards Objective identif and use direct variation Building on the Essential Question What You ll Learn Identif direct variation. Use direct variation to solve problems. At the end of the lesson, students should be able to answer What steps would ou take to determine whether or not an equation has a constant of variation? Real-World Link Video Games According to a recent surve, about % of U.S. households own n a current ese sstems video game sstem. Games for these can get epensive, so man storess offe offer er sales on pre-owned video games. The e prices for pre-owned games at a local ocal store are shown in the table below. w. A direct variation equation can be used to represent this situation. Number of Games Total Cost ($). 33... Content Standards.RP.,.RP.a,.RP.b,.EE. Mathematical Practices, 3, Vocab Vocabular direct variation constant of variation Ke Concept Creatas/PunchStock Words Direct Variation A direct variation is a relationship in which the ratio of to is a constant, m. We sa varies directl with. Smbols _ = m or = m, where m Eample = Graph = O When the ratio of two variable quantities is constant, their relationship is a direct variation. Since (, ) is a solution of = m, the graph of a direct variation alwas passes through the origin and represents a proportional linear relationship. In the equation = m, m is called the constant of variation or constant of proportionalit. It is the slope of the graph of = m. Lesson - Direct Variation
Eample Directl Proportional Since k is a constant rate of change in a direct variation, we can sa the following: varies directl with. is directl proportional to. The graph shows the cost of different amounts of trail mi. Determine if the relationship between the cost and the weight of the trail mi is a direct variation. To determine if the relationship is a cost direct variation, find _ for points weight on the graph. _ $ lb = $3./lb _ $ lb = $3./lb _ $ lb = $3./lb The ratio, $3./lb, is a constant rate. Since the graph passes through the origin and the ratios are constant, the cost of the trail mi varies directl with the weight of the trail mi. Cost ($) 3 Trail Mi Weight (lb) Do this problem to find out.. Tler charges his customers $ per week plus $ ever time he walks their dogs. Determine whether the relationship between total weekl cost and the number of times the dog is walked is a constant variation. number of walks No; the ratio pa is different for ever pair of values. Eample The equation = represents the distance in miles an ostrich can travel in hours. Determine whether there is a constant of variation. If so, eplain what it represents in this situation. = m Compare the equation to = m, where m is the constant of variation. = = _ The constant of variation is. This means that an ostrich can travel miles per hour. Do these problems to find out. a. The equation P = s relates the perimeter P of a regular heagon to the length of a side s. Determine if there is a constant of variation. If so, eplain what it represents in this situation. The constant of variation is. The perimeter of a b. There is no constant heagon varies directl with the length of a side. of variation. Kelvin temperature does not b. The equation K = C + 3 relates the Kelvin temperature K to Celsius var directl with temperature C. Determine if there is a constant of variation. If so, eplain what it Celsius temperature. represents in this situation.
Eample 3 Graphs The graph of a direct variation is alwas a line that goes through the origin. The time it takes to burn amounts of information on a CD varies directl with the amount of information. Write and solve an equation to find how long it will take to fill a -megabte CD. Step Use the equation = m. Choose an point on the graph. Then solve for m. = m Direct variation equation = m(.) Replace with and with.. = m Simplif. Step Use m to write an equation. = m Direct variation equation = Replace m with. Step 3 Find how long it will take to fill the CD. = Write the direct variation equation. = () or Replace with and simplif. It will take seconds to fill the CD. Time (s) 3 (, ) (3., ) (, ) (., ) 3 Amount of Information (megabtes) Do these problems to find out. 3. The cost of bulk peanuts varies directl with the weight of the peanuts. At a local grocer store, pounds of peanuts costs $.. Write and solve an equation to find how much pounds of peanuts would cost. =.; $. Compare Direct Variations Eample. Rabbit; Sample answer: The unit rate for the rabbit is 3 beats per second, which is greater than beats per second. The equation =. represents the relationship between the number of heartbeats and time in seconds for a dog. The graph shows the heartbeats for a cat. Which animal has a faster heart rate? Eplain. In the equation =., the slope or unit rate is. beats per second. In the graph, the point (, ) represents the unit rate, which is beats per second. Since >., the cat has a faster heart rate. Do this problem to find out. Cat Heart Rate. The equation = 3 represents the heart rate of a rabbit, where is the time in seconds and is the number of heartbeats. Does the rabbit or cat have a faster heart rate? Eplain. Number of Heartbeats (, ) (, ) (3, ) (, ) 3 Time (s)
Guided Practice Check. Recall the graph from Lesson 3 of the circular design on an Internet advertisement, shown at the right. The design has two circles, one that is decreasing in size and one that is increasing in size. (Eample ) a b. See Answer Appendi. a. Determine if the relationship between the number of seconds and the radius of circle is a direct variation. b. Determine if the relationship between the number of seconds and the radius of circle is a direct variation.. Financial Literac The equation = represents the number of dollars Olivia earns in weeks. Determine if there is a constant of variation. If so, eplain what it represents. (Eample ) The constant of variation is. This means that Olivia earns $ per week. 3. The length that a spring stretches varies directl with the amount of weight attached to it. When an -ounce weight is attached, a spring stretches inches. (Eample 3) a. Write an equation relating the weight and the amount of stretch. =. b. Predict the stretch of the spring when it has a -ounce weight attached. in. Radius of Circle (cm) 3 Circle Designs Circle Circle 3 Number of Seconds. The table below shows the changes in height for a kitesurfer. Assume that the height varies directl with the number of seconds. The height of a second kitesurfer increases 3. feet per second. Does the height of the first or second kitesurfer increase faster? Eplain our reasoning. (Eample ) Time (s) Height (ft) the second kitesurfer; Sample answer: The unit rate for this kitesurfer is 3. feet per second, which is greater than feet per second. Independent Practice Go online for Step-b-Step Solutions ehelp Determine if the relationship between the two quantities is a direct variation. (Eample ) Lifting Weights This is a direct. Getting Tickets This is not a direct variation since the variation since the 3 graph passes graph does not pass 3 through the origin through the origin and the ratio of the and the ratio of number of weights tickets available to to the total weight is 3 tickets sold is not constant. constant. 3 3 Number of Iron Weights Tickets Sold Total Weight (lb) Tickets Available. The constant of variation is.. This means that Conrad runs. miles in each marathon.. The equation =. represents the number of miles Conrad runs in marathons. Determine if there is a constant of variation. If so, eplain what it represents. (Eample )
AM_S_C_L_3.indd Page // 3: AM u-s /Volumes//GO_del_R/Accelerated_Maths_/PRE_ALGEBRA/... Proportionalit and Linear Relationships. The equation = 3. + represents the number of dollars Kristin charges for driving ou miles in her tai. Determine if there is a constant of variation. If so, eplain what it represents. (Eample ) There is no constant of variation. The cost to ride in Kristin s tai does not var directl with the number of miles she takes ou. Water pressure is measured in pounds per square inch (psi), which Depth (ft) Pressure (psi) varies directl with the depth of the water. (Eample 3) a. Write an equation that relates the depth and the water pressure. Round to 33. the nearest tenth. =.. b. The deepest dive ever recorded b an orca whale is feet. What is the approimate water pressure at this depth? 3 psi.. Model with Mathematics The cost of cheese varies directl with the number of pounds bought. Suppose pounds cost $.. Write and solve an equation to find the cost of 3. pounds of cheese. (Eample 3) =.; $. 3.. Erica s earnings varies directl with the number of hours she works. The Erica; Sample answer: The relationship is shown in the table below. Javier s earning can be represented unit rate for her pa is b the equation =.. Who earns more mone per hour? Eplain. (Eample ) $. per hour, which is more than $. per hour. Hours.... a. There is a direct variation. 3a. false; Sample b. There is a linear relationship. answer: the true; Sample answer: the rate of graph does not change is constant. pass through the origin. Height of Plant (in.). The graph shows the cost of pounds of apples. The cost of pounds of pears can be represented b the equation =.. Which fruit has the lower price? Eplain our reasoning. (Eample ) apples; Sample answer: The unit cost for apples is $. per pound, which is less than the unit cost for pears, $. per pound. B 3. Use the graph at the right to determine Growth of Plant whether each statement is true or false. Eplain our reasoning. 3 Apples Cost ($) Mone Earned ($) Create Your Own Homework Online can be used to create worksheets for the suggested assignments on page, or create our own worksheets for differentiated homework or review. MATHEMATICAL PRACTICES Emphasis On Eercise(s) 3 Construct viable arguments and critique the reasoning of others. Model with mathematics. Mathematical Practices, 3, and are aspects of mathematical thinking that are emphasized in ever lesson. Students are given opportunities to be persistent in their problem solving, to epress their reasoning, and to appl mathematics to real-world situations. (,.) (., ) (.,.) Weight (lb) 3 Number of Weeks. Sample answer: Compare the equation to = m, where m is the constant of variation. =, where is the constant of variation. = + does not have a constant of variation. H.O.T. Problems C. Higher Order Thinking Find the Error Ramiro is determining whether a line through the points with coordinates (, ) and (, ) represents a direct variation relationship. Find his mistake and correct it.. Ramiro incorrectl said the relationship was a direct variation. It is not. Graphs that are direct variation must go through the origin.. Building on the Essential Question Describe the steps ou take to determine whether an equation has a constant of variation. Give an eample of an equation that has a constant of variation and one that does not. O Watch Out! Find the Error For Eercise, students need to see that the must also look at the graph because the cannot determine from just two points whether there is a constant of variation. Lesson - Direct Variation
Standardized Test Practice. Which is not a true statement about the graph shown below? B Animal Race.. 3. 3...... 3 Distance (in.) Time (s) A It is a linear relationship. B The constant of variation is. C The slope is - _. D The -intercept is (, ).. The equation = 3 describes the number of miles per gallon that Melanie s car gets. Predict how man miles Melanie can drive on gallons of gas. H F mi H 3 mi G 3 mi J mi. The cost of cookies at a bake sale is shown below. B Cookie Prices.. 3. 3...... 3 Number of Cookies Cost ($) Which of the following is the best prediction for the cost of cookies? A $. C $. B $. D $.. Short Response At an amusement park, the cost of admission varies directl with the number of tickets purchased. One ticket costs $.. Write an equation that could be used to find the cost of an number of admission tickets. =. Common Core Review. The costs of admission to a water park are shown in the table at the right..rp.b,.rp.d,.ee. a. Find the constant rate of change between the quantities in the table. The constant rate is. b. For this situation, what is the meaning of the constant rate of change? It means the relationship is linear and that for each person the cost is $. c. What is the slope of a line that connects the ordered pairs for this relationship? Water Park Costs Number of People Total Cost ($) 3 3 Find the slope of the line that passes through each pair of points..ee.. R(-, ) and S(, ). A(-, 3) and B(-, ) undefined 3. C(, ) and D(, ) - Solve each problem using the percent equation..rp.3. 3. is what percent of? %. 3. is.% of what number? Interactive Stud Guide See page 3 for: Mid-Chapter Check