CONCEPTS SEC MATH
CONCEPTS SEC MATH EMAT 3500
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Section 4 Analyzing Bivariate Data With Fathom Section 4 Analyzing Bivariate Data with Fathom Summary Building from ideas introduced in Section 3 teachers continue to analyze automobile data using Fathom to look for relationships between two quantitative attributes They use the concept of variation and deviations from the mean in each univariate attribute to help conceptualize correlation and least squares regression as ways of describing the relationship in the bivariate data and developing a linear model for making predictions The teachers will consider pedagogical issues concerning the difficulties students may have in analyzing bivariate data and the benefits and drawbacks for using conceptual underpinnings from univariate analysis to develop bivariate analysis techniques Objectives Mathematical Teachers will be able to use concepts and techniques used in univariate data analysis to understand how two attributes covary and the techniques used to analyze two quantitative variables in bivariate data analysis describe how to create a linear model using the method of least squares describe the strength of the relationship between two quantitative attributes using correlation coefficient analyze the appropriateness of a linear function as a model for a bivariate data using a residual plot use the coefficient of determination to explain the amount of variation for a predicted value that can be attributed to a domain value by a least squares line determine whether a linear function is an appropriate model for a set of data Technological Teachers will be able to use Fathom to create box plots and scatterplom plot functions and values on graphs create a movable line and show squares create a leastsquares lineofbestfit create a residual plot enter formulas use the summary table to compute statistical measures Pedagogical Teachers will discuss the benefim and drawbacks related to different representations of data using dynamic files to conceptualize correlation and linear mo e s become familiar with some difficulties studenm have with creating and using a least squares linear model consider benefits and drawbacks of using tasks to assist students in reasoning about how to use univariate data analysis techniques to develop a better understanding of bivariate data analysis techniques Prerequisites univariate analysis techniques understanding mean and standard deviation as measures of center and spread dot plots and box p om Vocabulary covariation bivariate data correlation coefficient residuals Sum of squares Least Squares line coefficient of determination interpolation extrapolation influential poinm Learning to Teach Mathematics With Technology An Integrated Approach Page 1 DRAFT MATERIALS DO NOT DISTRIBUTE Modified 1132006 Section 4 Analyzing Bivariate Data with Fathom Technology Files 2006Vehiclesftm Correlationftm Outlierftm Emergency Technology Files 2006VehiclesPa113ftm 2006VehiclesPa1t4ftm 2006VehiclesPa115ftm 2006VehiclesPa1t6ftm Required Materials Fathom v 2 Leaming to Teach Mathematics with Technology An Integrated Approach Page 2 DRAFT MATERIALS DO NOT DISTRIBUTE Modified 1132006 Section 4 Analyzing Bivariate DataWith Farhom Section 4 Analyzing Bivariate Data While measures of center and spread provide us with important information about the distribution of a single variable often more than one variable is collected and relationships between two or more variables are examined In Section 3 we used two attributes in the 2006 Vehicle data to answer a question about vehicles with which engine type seemed to have the best City mpg performance To answer that question we were using bivariate data with one quantitative attribute City mpg and one qualitative attribute Engine type Bivariate data is the term used to describe data that have two variables for each observation In this lesson we will focus on ways to examine relationships between two quantitative attributes When examining two quantitative measures in a data set our attention is on how the measures co vary To help studenm conceptualize co variation we are going to build on what they already know about partitioning data sem and using measures of center and spread to describe univariate distributions Learning to Teach Mathematicswith Technology An Integrated Approach Page 3 DRAFT MATERIALS DO NOT DISTRIBUTE Modified 1132006 Col leetion menu Tecth ee graph click on any a h ases in a w te Space in the aph Section 4 Analyzing Bivan39ate Data with Fathom Part 1 Examining Relationships Between Two Quantitative Variables Open the 2006Vehicleltm le and clear all previous graphs There are several quantitative attributes in this data set When considering a purchase of a new vehicle a buyer is likely interested in the gas mileage for both city and highway When looking at the 2006 vehicle data we can ask the question Is there a relationship between City mpg and Hwy mpg for this set of vehicles We can begin by examining the distributions of the City mpg and the Hwy mpg n separate dot plots Drag down two empty Graph objects and create dot plots for each attribute ZUUE VENUES EVEWUES 3 3 470 o o o 000 o co oo o canoe a o co w 0 0000009 00 1 000000 94 e Awowwu wooaomo ea c o a U 1U 2U an AU an EU 70 U 1U 2U 3U 4U 511 ED 711 Figure4 1 We can utilize Fathom r linked representations to informally investigate the nature of the relationship between the two distributions by selecting a portion of the cases in the City distribution and noticing the corresponding position of those cases in the Hwy distribution To select a portion qfcases in a my in the graph window click and drag to draw a dashed rectangle around a subset of points When e mouse is released the selected points will appear red in the graph However because the representations are all linked the same cases will be highlighted in all other open representations of this data N Figure 4 2 Learning to Teach Mathematics with Technology An Integrated Approach Page 4 RAFT MATERIALS D0 NOTDISTRIBUTE Modi ed 1132005 Seetinn 4 Analyzing Brvarrate Datawrth Fathom mus Vehmes 10mm 2n Evemmes umwm 288 Sf o8 3330 88 u m 2m an an an en 7n n m 2n n 4 an en 7n Figure4 3 Focus 0N MATHEMATIES Mte nan w 39 39 city mpg and Hwy mpg My r a r a values for mend 39 39 represented by these points M1 or ity and Hwy 39 nisne 39 39 39 L hn nlm 39 r in different parts of a box p1ot What do you notice M424 npa e prorarion in M r you made in MQz Focus 0N PEDAGOGY Pte r prain 39 39 39 39 A prut ur 39 39 39 39 L quantitative attnbutes Learning tn Teaeh Mathematies wrth Teehnnlngy An integrated Apprnaeh Page 5 DRAFT MATERIALS DO NOT DISTRIBUTE Mndr ed 11mm Section 4 Analyzing Bivariate Data with Fathom dot plots and box plots that are only in one dimension The twodimensional space ed to represent bivariate data is a scatterplot Scatterplots are an amine whether there is an association between two quantitative 39 there is an association between City mpg and Hwy mpg City and Hwy mpg are both outcome measures of a vehicle s erformance where one is not dependent on the other We are interested if there is arelationship between these attributes however unlike some situations we are 39 there is a causeeffect independentdependent relationship ities Thus it does not matter which attribute we use as commonly us not ass between these two quant In creating a scatterplot the attribute assigned as the predictor variable is represented on the xaxis while the response variable is on the yaxis For our example we will assign City mpg as the predictor variable and Hwy mpg as the response variable FOCUS ON PEDAGOGY P7Q2 Create several pairs of variables that can help students understand the difference between an association between two variables that are independent dependent or where both are outcome measures and would only have a predictor response relationship Explain why each pair is either independentdependent or a predictorresponse example To help students tmnsition from examining the two attributes as distributions in one dimension to inscribing the data in two dimensions we are going to reorient the univariate distributions such that the distribution of the predictor variable City is horizontal and the response variable Hwy is vertical To move an attribute nm the xwa39s tn y aja39s l in the graph window drag the attribute label Hwy from the xaxis and drop it on the yaxis 2 The graph will be redmwn with the distribution displayed along the yaxis 2n Memes E was Vehtctes tW 7n en 5n gt4EI a 00 i an 2m to u Figure 4 4 Learning to Teach Mathematics with Technology An Integrated Approach Page 5 TERIALS DO NOT DISTRIBUTE Modi ed 1 132005 Section 4 Analyzing Bivariate Data With Fathom Before continuing change the window sizes and orient the two box plots as shown in Figure 45 so as to leave room to add another graph object to display the scatterplot Figure 4 5 The boxplots for each attribute show how each distribution is partitioned by the quartiles with the second quartile represented by the line inside the box representing the median To analyze how the two attributes covary we are going to inscribe the data as a scatterplot where each case icon in the graph will represent the ordered pair City Hwy for that particular vehicle To create a scatterplot QUEEVEW 25 l JSEW E39P Vquot 7D 1 drag down an empty Graph 6D object to the workspace 5D Q 2 Click and drag the attribute 0 representing the predictor or E at independent variable to the x in am x axis 2n 8 Z 3 Click and drag the attribute m representing the response or dependent variable to the y axis my Figure 4 6 Learning to Teach Mathematics With Technology An Integrated Approach E Page 7 DRAFT MATERIALS DO NOT DISTRIBUT Modi ed 1132006 Tech Tip To display the mean on a graph select the graph window then choose Plot Value or Plot Function for the attribute on the yaxis under the Graph menu rrnula meanaltrlbuleiname eg mean ity Tech Tip Remember thatyou can click on data points or select a cluster of data points in a graph and see highlighted in all graphs and in the le Section 4 Analyzing Bivariate Data with Fathom Adjust the scales of the three graphs until they are aligned see Figure 47 It will also be helpful to display the location ofthe mean for the City and Hwy on each graph zonex aaxwm 2on6 Vem les lsca evle my I mean Hwy mean CW 28 2195 W mean hwy 2on6 Vehicles I gure 4 7 The display of the horizontal and vertical lines representing the mean values can help students examine how each data point varies from the means The lines can also serve as a reference to notice the placement of data points in comparison to the general trend of data points When describing relationships between two variables we typically describe the form linear exponential etc direction positive or negative and strength weak moderate or strong of the general trend and relationship FocUs ON MATHEMATICS MQS Explain why you use the command Plot Value to display the mean City MQ7 Describe the location of the data points in relation to the mean City and mean Hwy mpg What does this tell you about the general trend of the data mpg and use the command Plot Function to display the mean Hwy mpg 1n a scatterplot MQ6 Use form direction and strength to describe the relationship between City and Hwy mpg Learning to Teach Mathematics with Technology An Integrated Approach Page 8 DRAFT MATERIALS DO NOT DISTRIBUTE Modified 1132006 Section 4 Analyzing Bivariate DataWith Farhom M Q8 Describe a typical City and Hwy mpg for this set of vehicles Explain how you determined what you would consider as typical FOCUS ON PEDAGOGY PQ3 How can displaying the means in a scatterplot help or hinder studenm ability to think about variation in bivariate data In the next part we will more closely examine how to quantify the strength of a linear relationship between two quantitative variables Learning to Teach Mathematicswith Technology An Integrated Approach Page 9 DRAFT MATERIALS DO NOT DISTRIBUTE Modified 1132006 Tech 2 Fathom sottware ineludes e eral prer ereated documents that eusefulfor doeum ents Seeuon 4 Analysing Rivariate Data with FaLham Part2 Conceptualizing Correlation L r 39 my a u Hwy mpg L Tr L r r r r 1 rr 14L 39 39 linearfunction 39Lr quot 39 r Thus L L correlation 39 39 39L 39 39 39 L 39 ren an an association between the variation in each attribute A correlation coelrrcieng usually represented by the letter r is ameasure used to describe the strength and direction ofalinear relationship between two variables The most 39 39 39 39 39 39 39 product Moment correlation coeificient There are several equivalent fonns ofthe formula used to compute this measure all ofwhich are based on h the data points vary from the mean ofthe predictor x and response y variables For example consider the fonnula below as one expression ofthe Pearson s correlation coef cient r z 2 x rm J 420 r9220 17 Y39 u u r 39 ll quot are unsedon L A 39 A a themean 39 39 394 0pm the le Correlation I Lm To help visualize how the correlation coef cient is ameasure ofcovariation and the spread ofthe bivan39ate data we are going to se this interactive diagram as shown in Figure 48 In this diagram we have a slider to control the value of t or u s c u scatterplotA39 quot 39 39 asthis can L is ameasure ofthe he een the variables Figure 4 a Learning to T aeh Mathematics with Technology An IntegratedApproaeh Page 10 DRAFT MATERIALS DO NOT DISTRIBUTE Modi ed 1132006 Padzgnu TIE Students ean test out estrrnate a eonelauon values for agrven seatterplotusrng the java applet at Seeuon 4 Analyzing Blvarlate Data wrth FaLham i FOCUS ON MATHEMATICS M429 Drag the slider for r and observe how the scatterplot changes Based on your exploration ll in the blank in each ofthe following statements a A 39 39 39 39 gt7 linear association between two Variables e linear association between two Variables c 39 39 association between the two Variables Meow Use the slider to create a scatterplot that can help you estimate a value for L 39 39 39 39 39 39 nn ann Hwy educoursesstatlUUJ avaGCAppletGCA ppleLFrame html FOCUS ON PEDAGOGY PVQ AL x relationship between two Van39 ables Generally a correlation coef cient with absolute value greater than 08 is an indicator ofa strong linear relationship while a correlation coef cient with glue nnwevpvr 39 39 39 ur rue ucu lu m tmrmuuu 39 39 iatiuu for strung mnderale weak linlenr weak mnderale strung enrrelal39 enrrelalinn enrrelalinn nnenrrelalinn enrrelalinn enrrelalinn enrrelalinn u rvalues Figur249 Leamlng to Teaeh Mathernaues wrth Teehnology An Integrated Approaeh DRAFT MATERIALS DO NOT DISTRIBUTE Page ll Modified 1132006 Section 4 Analyzing Bivariate Data with Fa mm Part 3 Using a Line to Describe a Relationship Between Two Quantitative Variables1 Return to the 2006Vehicleltm le The scatterplot oway vs City mpg suggests that there may be a linear relationship between the two variables That is ifwe know a vehicle s City mpg en we can use a linear mction rule to predict an approximate value for that vehicle s Hwy mpg We can use Fathom to compute the correlation coef cient r To compute r 1 drag an empty Summary object to the no data W P 39 Click and drag a quantitative attribute ii label onto the Summary table Once the i cursor is over the table a down arrow and a right arrow will appear Drop the quantitative attribute below the down arrow Click and drag a second quantitative attribute label onto the Summary table to the right of the right arrow The default measure that will be displayed is the correlation between the two 3 53915434 attributes S1 torrelatmn N Figure410 w k I 2006 VEl HEiBS va Figure4 11 FOCUS ON MATHEMATICS Mtel Compare the calculated correlation coefficient with the one you stimated using the Correlation ftm file M7Q12 What does the value of the correlation coef cient imply about the relationship between City and Hwy mpg Since we have a high correlation value it makes sense to try to use a linear mction to model the vehicle data The model represents an estimate of the response variable o en denoted as y given a value for the predictor variable uuu eiiieieimu 39 unable to complete Part 1 with the technology Learning to Teach Mathematics with Technology An Integrated Approach TE Page 12 MATERIALS DO NOT DISTRIBU Modi ed 1132005 Section 4 Analyzing Bivariate DataWith Farhom often denoted as x This model could be thought of as a measure of center for this bivariate distribution To create a movable line to t bivariate data 1 click on the graph and under the Graph menu select Add Movable Line The graph of a line appears in the scatterplot with is equation at the bottom of the graph window Dragging the line by its middle changes the intercept translates the line while dragging by either end changes the slope rotates the line N M Q13 Insert a movable line and adjust it so that you feel it best models the data 5 I Describe the method you used for determining where to place the line to model 0 the data I M Q14 Interpret the slope and y intercept in the equation of your linear model M Q15 We can use the equation of a line to estimate a value for the response variable for an input for the predictor variable For example a Jeep Liberty gets 22 mpg in the City If we think of a linear model for this data with an equation Hwy 101 City5 then we can use this equation to estimate a predicted Hwy mpg when the City mpg is 22 Based on where you placed the moveable line use your equation to predict the Hwy mpg for a vehicle with a City mpg of 31 FOCUS ON PEDAGOGY PQS How can the ability to overlay a moveable line on a scatterplot help or hinder studenm understanding of the use of a linear equation to model a relationship between two variables PQ6 One of the dif culties in using learning activities such as this is that studenm do not have confidence in their solution when their resulm differ from fellow students or from a teacher Think of two strategies that you could employ to help studenm understand that differences in solutions are acceptable and expected in the context of to estimate a linear model How could you capitalize on this difficulty should it arise When we try to create a linear model by visually inspecting a graph it is unlikely that two different people will generate the exact same line If we have two or more different lines how do we determine which is really best There are Learning to Teach Mathematicswith Technology An Integrated Approach Page 13 DRAFT MATERIALS DO NOT DISTRIBUTE Modified 1132006 Section 4 Analyztng Biv ariate Data with Fa lam 39 39 39 39 quot model and analyzmg how good ofa t it is With each method the goal is to minimize the distance each predicted value is from the actual data value similar to how we examined deviations from the mean with Pr f Hal univariate data we can examine 393 quot3 3 7 residual the deviations ofthe actual data points from the predicted values Visually these are the vertical Actual Value points and the line Figure 412 We are trying to minimize the vertical distance between an actual data value andthe predicted value output that would fall on the line for the same input value s coordinate ofthe actual data point Thus we are comparing the ycoordinate for an actual data point in the quot 39 39 r 439 A 39 39 linear function The ditference ycoordinate ofactual data point ycoordinate ofpredicted value is called aresidual Figure 4 12 Recall that for univariate data we described deviation from the mean Pre iranzd Value using variance and standard deviation Pan 4 of Section 3 which are both based on summing the squared u For bivariate data this sum is called Anna Value ofvariation A linear modelt at Figure 4 13 minimizes the sum ofsquares is called the Least Squares regression line Tu visualize mestm ofsquares 1 t t t Tu 1 from L L Nnticethe sum M q 39 r A A displayed below the equation ofthe line quot 39 39 quot linewill 39 39 quot alue of the Sum of squares Learning to Teach Mathematics with Technology An lntegated Approach Page 14 DRAFT MATERIALS DO NOT DISTRIBUTE Modified 1132006 Section 4 Analyzing Bivariate D ata with Fathom You should notice that there are tan and maroon squares shown in the scatterplot Since the meanHWy Was used With the Plot Function command Fathom is also displaying the squared deviations from the horizontal line Before continuing it is important to remove the meanHwy and meanCity lines from the scatterplot To remove aplntted value urfunetiunfmm a graph point to the plotted value or function you Wish to remove and rightclick With the mouse In the popup menu choose Cut or Clear Formula The graph of the r fiea39FD39mla value or function should be removed from the graph mean to 23395 k Edit Formula n at Square 2 7527 Cut Fermuia Dov Favmula N Figure 4 14 Figure 415 displays the moveable line that is being used to estimate a linear model for estimating Hwy mpg given City mpg The squares represent the squared residuals for how much the line over or under predicts for each data point zoos ehicl Scattev PM to 2h an on an an m va115cnw25 Swims Mavessl mt l Figure 4 15 FOCUS ON MATHEMATI s M7Q16 Manipulate the movable line to explore Whether it is possible to cre te a 1 line that is far from several points but still has a small sum of squares Explain Learning to Teach Mathematics with Technology An Integrated Approach MATERIALS DO NOT DISTRIBUTE Page 15 Modi ed 1132005 Section 4 Analyzing Bivariate DataWith Farhom M Q17 Compare and contrast the method of squaring residuals for bivariate data with the calculation of the standard deviation in univariate data gt M Q18 Adjust the movable line so as to minimize the sum of squares Record your new equation for the linear model Use your new linear model to compare the predicted Hwy mpg and the actual Hwy mpg for the following two vehicles a Honda Accord Standard engine vehicle and b Toyota Prius Hybrid engine Does your line underpredict or overpredict for the each of these vehicles By 7 how much FOCUS ON PEDAGOGY PQ7 Describe the benefits and drawbacks of building on what studenm already lcnow about deviations from a mean with univariate data and standard deviation to find a linear model by minimizing the sum of squared residuals Learning to Teach Mathematicswith Technology An Integrated Approach Page 16 DRAFT MATERIALS DO NOT DISTRIBUTE Modified 1132006 Section 4 Analyzing Bivariate Data with Fathom Part 4 Visualizing the Residuals In addition to nding the smallest Sum of squares a plot ofthe residuals is help il in deciding Whether your line is a good model for the data A residual plot displays the order pairs xvalue yvalue predicted yvalue To view aplut of the residuals click on a scatterplot that has a linear model displayed 2 Under the Graph menu choose Make Residual Plot 3 The residual plot Will be displayed at the bottom of the graph Window t Scattev PM 2on6 Vehicles u Residual Hwy u smooth6 a sum of squares 564 a Figure 4 16 In general We Want the residuals to be near zero and the plotted points should be mndomly dispersed above and below the horizontal line y and not reveal any trends or patterns I FoeUSONMATHEMATI s i Mte 9 Consider the residual plot for your linear model If you continue to adjust the moveable line you should notice the residual plot update accordingly What does the residual plot reveal about the usefulness of your linear model for predicting Hwy mpg for various vehicles Learning to Teach Mathematics with Technology An Integrated Approach Page 17 DRAFT MATERIALS DO NOT DISTRIBUTE Modi ed 1 132005 Section 4 Analyzing Biv ariate Data VIth Fathom MiQZO A student placed their movable line in the scatteiplot fothVhe 2006 1 Vehicle data that resulted in the following residual plot u 02 be a m sxa 8 372D o 0 gram 0 lit4 e K 5D 0 c an 2n an 40 an an CW Sketch the location of the predicted linear model based on the residual plot above in the following graph 1 Scattev mm FOCUS ON PEDAGOGY P7Q8 Describe some ofthe conceptual di iculties students may have in interpreting and using the residual plot How will you help them understand the residual plot and its usefulness in analyzing a linear model L mm Ma nnr h PagelS DRAFT MATERIALS DO NOT DISTRIBUTE Modified 1132006 Tech Tip displayed When the Least Squares line is added to a Squares line Section 4 Analyzing Bivariate DataWith Farhom Part 5 The Least Squares Regression Line2 While we can use the techniques of minimizing the sum of squares and viewing the residual plot to help find a really good linear model technologies like Fathom can easily compute a linear model using the Least Squares method To nd the least squares line 1 click on a graph displaying a scatterplot Under the Graph menu select LeastSquares Line The LeastSquares regression line computed by Fathom will appear on the graph along with the equation for this line and the value of r N The square of the correlation coefficientr2 is called the coefficient of determination and can be interpreted as the proportion of variation in the response variable that can be attributed to the variation in the predictor variable by the least squares line As the value gem closer to l the variation is better defined by the predictor variable and increasingly accurate predictions for the response variable can be assumed The difference between r and l l r2 indicates the proportion of the variation in the response variable that is attributed to other variables besides the predictor variable L rows ON39MATrrEMAnCs MQ23 Interpret the coefficient of determination for the least squares line MQ21 Compare the function rule for the least squares linear model with the function rule for your estimated linear model your moveable line MQ22 What is the predicted Hwy mpg for the Honda Accord Standard and the Toyota Prius Hybrid using the least squares linear model How do these compare to the predicted Hwy mpg for both vehicles using your estimated linear model found with the moveable line refer to your solution of MQl 8 Now that we have the least squares line we can remove the Movable line and the Residual Plot from the graph To remove a movable line from a graph 1 with the gra h selected under the Graph menu choose Remove Moveable L1n The moveable line and its associated residual plot should be removed 2 The technology file 2006Vehicles7Part5 m is available for students to use for Part5 if they were unable to complete through Part 4 With the technology Learning to Teach Mathematicswith Technology An Integrated Approach Page 19 DRAFT MATERIALS DO NOT DISTRIBUTE Modified 1132006 Section 4 Analyzing Bivariate Data VIth Fathom The least squares line is computed using means and standard deviations for each variable and the correlation coef cient 7 To help visualize the location ofthe least squares line in relation to the mean mpg for both City and Hwy use Plot Value and Plot Function to display the meanCity and meanHwy me emcles tier Plnt 7n en 5a a e a 3n 39 a 2 a a m Figure 4 17 Algebraically the slope andyintercept of the least squares line are r slope 71 5x y intercept islope Z 39 L L 39 Jul line can quot represented as A 5y 5 yr 4moew a 5x 5x Or an alternative form of A r yrioeny 5x When using a least squares line to make predictions interpolation is the process of predicting a response based on a value within the domain of the predictor variable In answering previous questions we used interpolation Extrapolation quot 39 on a quot the original data Learning to Teach Mathematics VIth Technology An Integated Approach Page 20 RAFT MATERIALS DO NOT DISTRIBUTE Modified 1132006 Section 4 Analyzing Bivariate Datawith Farhom for which the least squares line was computed For example we would be extrapolating if we used the least squares line to predict the Hwy mpg for a vehicle with a City mpg of 5 or 80 When finding a linear model we should take into account at least four different factors in considering whether the line is a good model for the data 1 the value of the correlation coefficient and coefficient of determination 2 how the data is positioned in the scatterplot in relation to the linear model 3 the residual plot and 4 the situation and whether the line makes sense for all data points In many cases the domain of the linear model needs to be specified in order for it to fit the situation and to avoid the potential dangers of extrapolation FOCUS ON MATHEMATICS M Q24 The least squares regression line passes through the intersection of the mean City mpg and the mean Hwy mpg Will this always happen Justify your answer algebraically M Q25 Do you believe the least squares line is a good model for the 2006 Vehicle City and Hwy data Explain FOCUS ON PEDAGOGY PQ9 How could you assist studenm in thinking about the dangers of extrapolating using the 2006 Vehicle data PQ10 If technologies like Fathom as well as others such as Excel and graphing calculators will compute and display the least squares line would you choose to show students the algebraic form for computing the least squares line Defend your position Learning to Teach Mathematicswith Technology An Integrated Approach Page 21 DRAFT MATERIALS DO NOT DISTRIBUTE Modified 1132006 Section 4 Analyzing Bivariate DataWith Fathom Part 6 Exploring Additional Attributes on a Scatterplot3 The analysis of the scatterplot and least squares line for the City and Hwy mpg raises several issues First there are several vehicles all of which are Hybrids for which the linear model grossly overestimates their Hwy mpg based on their City mpg Second the I value suggesm that there may be other variables besides City mpg that are contributing to the variation in the Hwy mpg Fathom can facilitate students analyzing data in a scatterplot in such a way as to add a third variable or dimension to the analysis This can help studenm visualize the relationship between three variables rather than only considering two Deselect the Least Squares line before continuing Tech Tip To overlay a legend attribute on a scutterplot To remove a legend 1 drag the attribute of interest to the center of the scatterplot attribute from a 2 If the attribute is quantitative then each data point will be displayed along graph Under the a color gradient continuum If the attribute is qualitative categorical then graph men Choose each data point will be displayed using different shapes and colors A key Remove Legend Amibme w1ll appear at the bottom of the graph Cl FOCUS ON MATHEMATICS M Q26 Which quantitative and qualitative attributes in the 2006 vehicle data set could be related to the City and Hwy mpg for a vehicle M Q27 Explore overlaying the attribute Weight on the Hwy vs City scatterplot i Explain whether a vehicle s weight seems to be related to the City and Hwy mpg FOCUS ON PEDAGOGY PQll Consider how the differences between the use of color to highlight different attributes in Fathom and in TinkerPlots could affect studenm reasoning about relationships between attributes Earlier we noticed that many of data poinm that did not seem to fit the general trend between City and Hwy mpg were Hybrid vehicles It may make sense then to compute a least squares line for the data in each of the subcategories of engine 3 The technology file 2006Vehicle7Part6 is available for students to use for Part 6 ifthey were unable to complete through Part5 with the technology Learning to Teach Mathematicswith Technology An Integrated Approach Page 22 DRAFT MATERIALS DO NOT DISTRIBUTE Modified 1132006 Section 4 Analyzing Bivariate DataWith Fafhom type Hybrid Diesel and Standard If we overlay a qualitative attribute on a scatterplot and display a LeastSquares Line on it Fathom will compute a least squares line using the qualitative attribute as a filter and will compute a different linear model for subsem of data according to the qualitative attribute FOCUS ON MATHEMATICS M Q28 Overlay the Engine type attribute onto the Hwy vs City scatterplot 7 Display the least squares line Interpret the resulting least squares equations FOCUS ON PEDAGOGY PQ12 What are the benefits and drawbacks of having the ability in Fathom to overlay a qualitative attribute as a filter for computing Least Squares linear models for subsem of data As we have seen overlaying an attribute in a scatterplot can allow students to simultaneously consider three attributes and relationships among them While at first this may seem confusing to studenm this feature can help them consider relationships among more than one variable and realize that linear models that only consider a relationship between two variables are often not sufficient in explaining the phenomenon In the case of the 2006 Vehicle data we have learned that the type of engine in a car appears to be a significant factor that affects the relationship between a vehicle s fuel economy when driving in a city and on the highway Learning to Teach Mathematicswith Technology An Integrated Approach Page 23 DRAFT MATERIALS DO NOT DISTRIBUTE Modified 1132006 Secuon 4 Analyzmg Rivariate Data with Fathom Part7 Exploring the Effects of Outliers orl Correlation and the Least Squares Line I L r39 39 ofthe the location ofamoveable line in the scatterplot Now we are going to reverse our locus ofcontrol and capitalize on the ability to move data points in a scatterplot and observe the effect on the measures ofthat data L39I 1 LI 1 II comml 0pm the le OutliersJ tm This file has 5 data points displayed in the table and a scatterplot The correlation between the two variables is also computed and displayed ecusrtuu t u FigureA Is E FOCUS ON MATHEMATICS II M Q2 The correlation coefficient is currently about 102506 Explain why ltlt this value makes sense for these 5 data points v Meni AAm I t r L Tumt u u Ir cu leasI pk squares lin r 39 t l a l R McQ31 Drag the center point located at 61 75 to the upper right and bottom 5 left comer ofthe graph Describe the effects on the conelation coefficient and ltlt I left ltlt e e comers p Merz Describe two translations ofthe center point that have little effect on p g the slope ofthe quot Learning to Teach Mathematics with Technology An Integrated Approach Page 24 DRAFT MATERIALS Do NOT DISTRIBUTE Modified 1132006 Section 4 Analyzing Bivariate DataWith Farhom H M Q33 By moving the five poinm nd at least three very different arrangemenm of poinm that result in a corresponding least squares line that has a positive slope and a correlation coefficient greater than 08 How are the three arrangemenm different and what does this difference imply about the relationship between the location of the points on the scatterplot the value of the correlation coefficient and the slope of least squares line FOCUS ON PEDAGOGY PQ13 Often middle or high school studenm only consider the correlation coefficient when modeling data Explain why this information alone is not sufficient when determining an appropriate mathematical model PQ14 What are the benefim and drawbacks of changing the data poinm in a graphical representation and observing the effects on the measures of correlation and the least squares line The exploration in the Outliers ftm le is yet another example of how a technology tool can be used to create an interactive diagram Such diagrams allow studenm to engage in dynamic manipulations observe effecm of their activities and reflect on those effecm to develop a more meaningful conception of a mathematical idea These type of diagrams can be used in a variety of settings such as l for individuals to complete working alone at a computer 2 small groups of studenm working together with one computer 3 small groups of studenm working on individual computers but allowed to discuss their results as a group and 4 whole group discussion with the interactive diagram displayed using a projector and students and teacher discussing the activities and the effecm of the activities together When considering how you use such technology files in your own classroom you will need to balance what your goals are for students learning with the time allotted and computers available Learning to Teach Mathematicswith Technology An Integrated Approach Page 25 DRAFT MATERIALS DO NOT DISTRIBUTE Modified 1132006 Section 4 Analyzing Bivariate Datawith Farhom SUGGESTED ASSIGNMENTS H Ql Mathematical In this problem you will be investigating relationships between Weight and City mpg a First determine if there is a strong linear relationship between Weight and City mpg Consider which attribute should be the predictor variable and which should be the response variable Explain your results b Investigate the relationship between Weight and City mpg using a third attribute engine type and describe what additional information about the data this attribute provides c Remove the Engine type attribute from the scatterplot Construct a plot of Residuals versus Weight Describe this plot and what it tells you about the relationship between City mpg and Weight H Q2 Pedagogical In many classrooms studenm and teachers use graphing calculators to enter data create scatterplots and compute linear regression Discuss the advantages and disadvantages between using graphing calculators and Fathom for helping studenm understand these concepts and perform these procedures H Q3 lVIathematical Question about computing residual on graphing calculator and displaying a residual plot Give directions H Q4 Pedagogical Create a task where a linear relationship exism between two variables but a linear model would be inappropriate such as shoe size and scores on an achievement test Create a series of questions that would help students to see that though the scatterplot reveals a linear trend and the correlation coefficient is strong it does not make sense to predict one from the other H QS lVIathematical The medianmedian line is another linear model that is more resistant to outliers than the least squares line Below are directions for creating the medianmedian line without technology To fit a medianmedian line to the poinm divide the poinm into three groups Do this by taking the set of onethird of the poinm consisting of those with the smallest Xvalues a middle group a set of onethird of the poinm with the larges Xvalues For the SAT data set of n51 each one third will include 17 data poinm If the number of points is not divisible by three extra poinm need to be assigned symmetrically Thus if there is Learning to Teach Mathematicswith Technology An Integrated Approach Page 26 DRAFT MATERIALS DO NOT DISTRIBUTE Modified 1132006