ࡱ > bjbj Ě " " e e 8 & I W ( ;$ G% m o o o o o o o - Ӏ U " w " Ӏ Ӏ o e e m m m Ӏ e 8 m m Ӏ m m m 1VJ L H Y 0 G 4 4 & 4 5 $ % ! F m y^ ]q v % % % o o % % % Ӏ Ӏ Ӏ Ӏ 4 % % % % % % % % % " + : Unit 5: Investigation 3: Forensic Anthropology: Technology
and Linear Regression 4 days
Course Level Expectations1.1.9 Illustrate and compare functions using a variety of technologies (i.e., graphing calculators, spreadsheets and online resources).
1.1.10 Make and justify predictions based on patterns.
1.2.2 Create graphs of functions representing real-world situations with appropriate axes and scales.
1.2.4 Recognize and explain the meaning and practical significance of the slope and the x- and y-intercepts as they relate to a context, graph, table or equation.
1.3.1 Simplify and solve equations and inequalities.
4.1.1 Collect real data and create meaningful graphical representations (e.g., scatterplots, line graphs) of the data with and without technology.
4.2.1 Analyze the relationship between two variables using trend lines and regression analysis.
4.2.2 Estimate an unknown value between data points on a graph or list (interpolation) and make predictions by extending the graph or list (extrapolation).OverviewIn this investigation, students will use technology to fit a trend line to data. They will use the correlation coefficient to assess the strength and direction of the linear correlation and judge the reasonableness of predictions.Assessment ActivitiesEvidence of success: What students will be able to do
Students will be able to answer a question about the world that can be analyzed with bivariate data.
For given bivariate data, student will use a guess and check strategy to manipulate the slope and y intercept of a trend line on a calculator to find their best estimate for the trend line.
For given or student-generated bivariate data, students will be able to use technology to graph a scatter plot, calculate the regression equation and correlation coefficient, tell the strength and direction of a correlation, solve the equation for y given x, interpolate and extrapolate, explain the meaning of slope and intercepts in context, identify a reasonable domain, and distinguish between data that is correlated compared to causal.
Assessment strategies: How they will show what they know
Students will make a reasonable prediction from the forensic anthropology data
Students will estimate the value of the correlation coefficient for various scatter plots. (Short Quiz)
Using computer applets, or other grapher, students will create a scatter plot with a given correlation coefficient. (in-class activity)
Using a computer applet or other grapher, students will fit a line to data by manipulating the parameters for slope and y intercept. (In-class activity)
After being given bivariate data or collecting data, students will use technology to graph scatter plots, calculate regression line and correlation coefficient, describe the real world meaning of slope, x intercept and y intercept in context, interpolate and extrapolate from the given data, and solve for the dependent variable given the independent variable. Students will write about the reasonableness of their analyses and the confidence with which they make predictions (classroom activities may have students record on paper a sketch of what the graph and regression equation they see on their calculator screens. Students can show the teacher their calculator screens for scatter plots, regression equations and correlation coefficient as the teacher walks around the room checking student progress during activities and group work.)
Exit slips: Use an exit ticket asking students to name at least two advantages of using technology to analyze data and make predictions. Use an exit ticket asking students to sketch three different scatter plots using three points to approximate relations that have r = 1, r = 0 and r = 0.7.
In writing, students will pose a question to be answered using regression analysis of bivariate data. They will then formulate a plan for collecting data to answer the question they pose.Launch Notes:
This investigation involves two stages: 1) students collect their own data, and 2) students see how data that is messy to work with by hand may be better graphed and analyzed using technologies such as the graphing calculator, spread sheet software or free graphing applets.
Begin with the PowerPoint on forensic anthropology to develop with the students the idea that lengths of various long bones such as the tibia, femur and ulna length are related to height of the person the taller the individual, the longer the bones. Height is not directly proportional to bone length, as you will see when you calculate the regression equation for the ulna, because it has a nonzero intercept. The linear regression equations developed by Professor Trotter are found at HYPERLINK "http://science.exeter.edu/jekstrom/A_P/Puzzle/Files/LivStat.pdf"http://science.exeter.edu/ jekstrom/A_P/Puzzle/Files/LivStat.pdf.
One example of the Trotter equations for determining stature is Stature of white female = 4.27 " U l n a + 5 7 . 7 6 ( + / - 4 . 3 0 ) w h e r e m = 4 . 2 7 c e n t i m e t e r c h a n g e i n h e i g h t f o r e v e r y c e n t i m e t e r c h a n g e i n s t a t u r e a n d b = 5 7 . 7 6 .
T h e P o w e r P o i n t c o n c l u d e s b y p o s i n g a p r o b l e m t o t h e s t u d e n t s : H o w d o y o u f i n d t h e h e i g h t o f a p e r s o n w h o s e s k e l e t a l r e m a i n s i n c l u d e an ulna and no other long bones? Instead of or in addition to the PowerPoint, you could bring in a variety of washed and boiled bones from the butcher. Another prop might be dolls and action figures. After you discern a regression equation for height as a function of ulna length, it might be interesting to test whether the dolls and action figures measurements satisfy the regression equation.
Ask the students how they can find the height of a person of ulna length 28.5 centimeter. Have them brainstorm about how they might find the data needed to write an equation relating height to ulna length. You may guide a discussion about how to design an experiment using measurements from the students in the class.
Additional resources for you are the teacher notes for the activity and background information on the correlation coefficient.
See the Teacher Notes Investigation 3a and Teacher Notes Investigation 3b: Background Information on Correlation Coefficient.Closure Notes:
Once the investigation is complete, and the students have predicted the height of the missing person whose ulna was found, and they have analyzed other real-world data, be sure to review the mathematical ideas:
1) Technology is useful for graphing, especially if the data is messy.
2) Using the least squares regression, rather than fitting a line to data by visual estimation, takes into account all the data points by minimizing the sums of squares of the error.
3) If you fit a linear regression to data, you can make a prediction. The confidence with which one makes a prediction depends on the strength of the correlation and keeping within a reasonable domain for the regression.
4) Correlation does not guarantee causation.
As interesting as the applications may be, continue to emphasize and review the math content and skills that were learned and applied.Important to Note: vocabulary, connections, common mistakes, typical misconceptions
Vocabulary: Linear Regression or Regression means Least Squares Regression in this course, and is the linear model calculated using technology. However, there are other methods for calculating lines of best fit, such as the median-median line. The word trend line implies that the equation was found by estimating the placement of the trend line by eye and calculated by hand.
Using technology, rather than working by hand, is the preferred medium for professionals.
Typical misconception: the more data points that a regression line passes through, the better the fit.
If the correlation coefficient is near zero, then one cannot have much, if any, confidence in ones predictions based on the linear regression, (except if the data is horizontal and nearly collinear). If the correlation is close to zero, the average of the y data is a better predictor of a y value at a given x value than is evaluating a regression line for a given x. (If the data is horizontal and nearly collinear, then the regression equation itself is the average of the y values, because the coefficient of x, the slope, will be near zero.)
Correlation does not imply that variation in the independent variable caused the variation in the dependent variable. Just because two variables occur together, one cannot infer that one causes the other. Correlation is a necessary, but not sufficient, condition for causation. For more information on this: do a Web search on causation versus correlation. How to determine causation is a much-debated problem in the philosophy of science.Learning StrategiesLearning Activities
You may begin the Investigation by presenting the PowerPoint on forensic anthropology to the whole class (Activity 3.1 PowerPoint Presentation). Ask the class to speculate about some of the questions raised in the PowerPoint. You may consider using props, such as washed bones from a butcher or dolls and figurines from a childs toy chest. Ask students to estimate and show with their outstretched arms how large the animal was from which the bone came. Are doll and figurine heights related to their ulna lengths? Perhaps measure the dolls ulna and height. Lead them in a discussion about how they might determine the height of the person with an ulna 28.5 centimeters long. Guide a discussion about how to design an experiment using measurements from the students in the class. How will they measure? What tools do they need? What should they record and graph? If students need more support, you may consider selecting parts of Activity Sheet 3.1a Student Handout Forensic Anthropology to provide more structure. Collect and tabulate the data for ulna length and height for each student. Conduct a discussion about how to graph the data which is the independent and which is the dependent variable in this situation? What should be the scale? How should we label the axes? Is the relationship causal? Have students graph the data, sketch a trend line by hand and find an equation of the trend line by hand. Be sure to let students struggle with the messy data long enough to be motivated to use technology, but not so long as to be frustrated. Several students may show their graphs and trend lines to the class so that everyone sees that there are several different models for the same data. Which line of fit appears to be the BEST model of the data?
Present the calculator as an alternative to graphing data and modeling trend lines by hand. Distribute the calculator directions to students. Guide the whole class in using technology to create a scatter plot. Observe that the same decisions you make in graphing by hand also need to be made when using the calculator: What are the two variables? Which variable depends on which? Which axis is which? What is a good scale i.e., window? Calculate the regression equation and the correlation coefficient. Keep your explanations of each very short. Explain that the calculator can display a trend line based on the data. For the correlation coefficient, point out that the +/- sign of r indicates the direction of the correlation, and the closer r is to 1 or -1, the stronger the correlation. You will expand on these ideas later. Mention that r assigns a numerical value to the concept of the direction and strength of a correlation. It answers the questions: On a scale of -1 to 1, how strong is the correlation between x and y? How close are the data points to the line? Ask students how they might use the regression equation to calculate an answer to the question How tall is the person with an ulna 28.5 centimeters long? Then show students how to use the calculator to find the height of the person with an ulna length of 28.5 centimeters. (See the Unit 5 Graphing Calculator Directions handout for ways to find y given x.) With or without using the worksheet as a recording device, have the students write the regression equation, compare the regression equation they found on the calculator with the trend line they found by hand, discuss the advantages and disadvantages of doing the work by hand versus technology, and discuss how confident they are in their estimate of the missing persons height based on the strength of the relationship between the variables. (Note: The linear regression from technology takes every data point into account when calculating the line of best fit. Experimenting with different graphing windows and editing is tedious by hand but easy by calculator. The graphs drawn by the technology are more accurate than those drawn by hand are.) You might conclude the activity with a review of the main processes:
A question was posed about stature given ulna length.
A linear function was needed, so we collected appropriate data and found the linear regression to model the data.
We used technology because technology removed the tedium of working with messy data, created more accurate scatter plots than what could be draw by hand, calculated a correlation coefficient, and calculated a line of best fit also called a regression equation more accurately than the lines the students had previously created by hand.
If you fit a linear regression to data, you can make a prediction. Once we had an equation that modeled the relationship between height and ulna length we could find height (y) given the ulna length (x).
The confidence with which one makes a prediction depends on the strength of the correlation and keeping within a reasonable domain for the regression.
During the rest of this investigation, there are many different ways to have students explore data and make predictions using trend lines and correlation coefficients. Below are four examples of activities that may be done with students working in small groups. Two of them (centers A and C) may need more teacher direction at the beginning, and the other two (centers B and D) are more open to immediate discovery. One way to proceed might be to do the centers A and B on one day and the other two during a second day so that you have the opportunity to support students who need more attention, as well as be able to launch the sessions that need a short introduction to the software. On the first day, you might assign half the class to Center A and the other students to Center B. Center A allows students to experiment with the appearance of a scatter plot and its correlation coefficient using a computer applet. In discussion with half the class, project the computer screen showing the NCTM Regression Line applet found at HYPERLINK "http://illuminations.nctm.org/LessonDetail.aspx?ID=U135"http://illuminations.nctm.org/Lesson Detail.aspx?ID=U135. Show students how to create a scatter plot by clicking on the coordinate plane. Ask them to estimate the correlation coefficient for the scatter plot and write their estimate on a piece of paper. Then show that clicking on show line will automatically give the regression equation and the correlation coefficient. You may challenge the students to create a scatter plot with a positive correlation coefficient. Then ask another student to add more data to lower the r-value. Ask a third student to create a scatter plot that is moderately correlated and positive. Then challenge another student to add more data to raise the r-value. Ask students what will be the correlation coefficient if there are exactly two points in the scatter plot. Test the student hypothesis by plotting two points on the applet and finding its r-value. Have students sketch points on a paper at their seats to show a scatter plot with a correlation of -0.7. Then students may test the scatter plot on the applet. Now you might put the students in teams of three or four. Have students create scatter plots on the computer applet and have each team member write an estimate of the correlation coefficient. Students should write reasons for their estimates such as the scatter plot show a decreasing relationship that is somewhat strong, so I estimate r = -0.8.
At Center B, students have the opportunity to collect and analyze data. See resources below for some places to go for data and tools. Some ideas for contexts and data sources are listed in the paragraph below. This center provides practice with the calculator key strokes used to graph data and to calculate the regression line and correlation coefficient. Students need practice interpreting what they see on the calculator screen, so be sure to have students make a prediction or answer a question from the data. Students also need practice seeing that data generates questions. You might have students in a think-pair-share come up with their own questions based on the data and then share them.
Data ideas are in the daily news, available at Web sites pertaining to your student interests, and at Web sites for social justice. The United Nations Web site contains a section called cyber school bus HYPERLINK "http://www.cyberschoolbus.un.org/" http://www.cyber schoolbus.un.org/, which includes free downloadable videos for important world issues such as child labor, world hunger, discrimination, environment and more. Using the InfoNation interactive portion of the Web site, generate sets of data on countries of your choice. Students can decide which variables to compare for which countries: HYPERLINK "http://www.cyberschoolbus.un.org/infonation3/basic.asp" http://www.cyber schoolbus.un.org/infonation3/basic.asp. For example, the student may choose five nations, find their gross domestic product, and then view their carbon dioxide emissions.
Lessons are available on the NCTM Illuminations Web site HYPERLINK "http://illuminations.nctm.org/" http:// illuminations.nctm.org. Conduct an Internet search on linear regression or explore the Data and Story Library (DASL) HYPERLINK "http://lib.stat.cmu.edu/DASL/" http://lib.stat.cmu.edu/ DASL/.
Ideas for a physical activity that generates data include the hand squeeze activity or the sport stadium wave activity whereby the students time how long it takes to pass a hand squeeze or a wave along a row of 5, 10, 12, 15, 18 and 30 students. Predict how long a hand squeeze or wave will last if the entire school participated. Estimate how many students are needed to create a wave or hand squeeze long enough to fill a 30-second television commercial. Or you may have students make paper airplanes and measure distance flown as a function of length of airplane or width of wing or number of paper clips added for weight. (Average the measurements from several trials at a given weight or wingspan to reduce the wide variation due to how a person tosses the plane.)
On the next day, divide the class again. At Center C, explain to half the class that the least squares regression equation is the trend line that minimizes the sum of the squares of the error (SSE). Have them look at an applet that shows the sum of the areas of the square of the error as the trend line is moved dynamically by the user. One such applet is available at HYPERLINK "http://www.dynamicgeometry.com/JavaSketchpad/Gallery/Other_Explorations_and_Amusements/Least_Squares.html"http://www.dynamicgeometry.com/JavaSketchpad/Gallery/Other _Explorations_and_Amusements/Least_Squares.html. In a demonstration, display two or three scatter plots using the dynamic graphing capabilities of an applet such as the one on the NCTM Illuminations Web site: HYPERLINK "http:// illuminations.nctm.org/ActivityDetail.aspx?ID=146" http:// illuminations.nctm.org/ActivityDetail.aspx?ID=146 or the applet by Dr. Robert Decker called Function and Data found under Applets, Calculus/Pre-Calculus at his Web site HYPERLINK "http://uhaweb.hartford.edu/rdecker/mathlets/mathlets.html"http://uhaweb.hartford.edu/rdecker/ mathlets/mathlets.html. Ask the class to help you estimate the slope and y intercept for a trend line. This is a good time to review lessons from the past such as increasing lines have positive slope, or steeper lines have a slope of greater magnitude than less steep lines. Enter an initial guess, then use the slider or click and drag possibilities to manipulate the line to fit the data. Observe the resulting equation.
The parameters are adjusted to fit the data better. Have students experiment on their own, trying to find a trend line by guessing and checking various values for m and b with the graphing calculator. Have the students enter the data in their own lists and make a scatter plot. Input a student guess for a trend line in the y = calculator screen. Graph the equation with the data plot to test the student estimate for slope and y intercept. Have students amend their guess to improve the trend line. Use the linear regression feature to calculate the least squares regression and store it in Y2. Graph both the student estimate and the regression equation to check student work. As a possible exit slip, you may have the students enter simple data such as 0, 1, 2 in List 1 and 2, 5, 8 in List 2. Skip the scatter plot step and have them write down the linear regression and the correlation coefficient they calculate with technology. Have them explain how they could have figured out the slope, y-intercept and correlation coefficient if their calculator were broken.
At Center D, the students will prepare to choose a topic or question for the Unit 5 project: Is linearity in the air? Continue having students analyze data sets by graphing them on the calculator, finding the regression equation and the correlation coefficient. Be sure to include data that is not well correlated, data that is negatively correlated, and data that provides a discussion about causation versus correlation. Have students formulate the questions that the data might answer. Different groups may work with different data sets, using a jigsaw puzzle style of cooperative learning, or all students could work on the same data. Have students share contextual questions and discuss whether the variables will work in the context of answering questions using linear regression. Have students brainstorm about how they might collect the data they would need to know to answer the question. Encourage them to begin to think about rudimentary experimental design. If a student-designed question does not lend itself to analysis by linear regression, provide more time for students to find another question or another topic, or both.
Examples of data sets with a low correlation coefficient:
Home runs Ted Williams hit each year during his career is very scattered and nearly horizontal, so r is close to zero. Use the regression to estimate how many home runs he would have had if he did not serve in the Korean War and World War II. If Ted Williams had not taken time out of his career during the 1943, 1944, 1945, 1952 and 1953 seasons to serve his country, would he have broken Hank Aarons record? Discuss how this data is not highly correlated, and how any prediction is not made with much confidence. Ask some students to graph Williams accumulated or career home runs to date for each year since he started playing, which will give a high correlation coefficient.
Brain size and IQ are not correlated. Do people with greater brain mass score higher on IQ tests? Answer is no. HYPERLINK "http://lib.stat.cmu.edu/DASL/Datafiles/Brainsize.html" http://lib.stat.cmu.edu/DASL/Datafiles/Brainsize.html
An example of negatively correlated data that may spark a discussion about correlation versus causation is HYPERLINK "http://lib.stat.cmu.edu/DASL/Stories/WhendoBabiesStarttoCrawl.html"http://lib.stat.cmu.edu/DASL/ Stories/WhendoBabiesStarttoCrawl.html. The data shows that warmer temperatures correlate with crawling at a younger age. Is it the warmer temperature that causes babies to crawl, or the less restrictive clothing, or the fact that parents are more likely to put the baby on the floor in warm weather, or something else? How would one design a study to test your hypothesis?Differentiated Instruction
Transfer data to student calculator lists by linking or to student computer with a flash drive
If students are distracted from the class discussion because of attention focused on note taking, you might provide students with scaffolded activities. For example, give the student a page of notes on the activities and data analysis where the student must fill in the blank or do a sentence completion rather than have to take notes on the entire activity or lesson.
Allow students to use their Algebra 1 hands-on toolkit or Formula Reference Section of their notebook, which could include a procedure card on how to graph data. The procedure card might prompt the student to 1) decide which variable is on the horizontal and which is on the vertical axis; 2) decide on a scale for labeling the axes; 3) plot the data points; 4) adjust the scale if necessary and replot the data; 5) sketch a line of best fit; and 6) choose two points on the line of best fit to find the equation of the line.
There might be another procedure card in the math toolkit or the Formula Reference Section of the students notebook that describes how to find the equation of a line give two points.
One homework idea is to have students make a procedure card that lists the key strokes for plotting data and calculating the regression. Allow students to use the card that is placed in the math toolkit.
Have students create a mnemonic device or rap for the steps in plotting data and calculating the regression.
Extension: Which is better correlated with height? Length of foot, shoe size or length of the ulna? Students may measure foot length and record shoe size, then plot height as a function of each. Should female data be grouped with male data for shoe size? Research the history of detective work and how footprints or shoe prints are used to help identify criminals and solve crimes.Resources:
Props such as bones, action figures, dolls
Classroom set of graphing calculators
A whole-class display for the calculator: either an overhead projector with view screen or computer emulator software, such as SmartView, that can be projected to the whole class
Rulers and tape measures with centimeter scales
PowerPoint presentation
Applet that gives meaning to least squares
Geometers Sketchpad has an online resource center that contains a gallery of downloadable applets, including one that shows geometrically how the area of the squares changes as the slope and y intercept of a line of fit is changed by the viewer. See the Java Sketchpad Download center to obtain zip files to download these applets on your server.
HYPERLINK "http://www.dynamicgeometry.com/JavaSketchpad/Gallery/Other_Explorations_and_Amusements/Least_Squares.html" http://www.dynamicgeometry.com/JavaSketchpad/Gallery/Other_Explorations_and_Amusements/Least_Squares.html
Prepared lessons for linear regression, correlation and outliers:
Applet on Regression Line available at HYPERLINK "http://illuminations.nctm.org/LessonDetail.aspx?ID=U135" http://illuminations.nctm.org/LessonDetail.aspx?ID=U135
Impact of a Superstar: investigate effect of outlier NCTMs illuminations grade 9-12 data analysis section. HYPERLINK "http://www.dynamicgeometry.com/JavaSketchpad/Gallery/Other_Explorations_and_Amusements/Least_Squares.html" http://www.dynamicgeometry.com/JavaSketchpad/Gallery/Other_Explorations_and_Amusements/Least_Squares.html
Regression line and correlation: four lesson series including interactive applet where user plots arbitrary number of points, applet fits regression line and tells correlation coefficient. NCTMs illuminations grade 9-12 data analysis section. HYPERLINK "http://illuminations.nctm.org/LessonDetail.aspx?ID=U135" http://illuminations.nctm.org/LessonDetail.aspx?ID=U135
Least Squares Regression 9 lessons HYPERLINK "http://illuminations.nctm.org/LessonDetail.aspx?ID=U117" http://illuminations.nctm.org/LessonDetail.aspx?ID=U117
Applet where the student plots data, makes a guess about the line of best fit, and tests his guess against the line calculated by the technology HYPERLINK "http://illuminations.nctm.org/ActivityDetail.aspx?ID=146" http://illuminations.nctm.org/ActivityDetail.aspx?ID=146
Similar applet is available at Professor Robert Deckers Web site
HYPERLINK "http://uhaweb.hartford.edu/rdecker/mathlets/mathlets.html" http://uhaweb.hartford.edu/rdecker/mathlets/mathlets.html
Sources of data for creating your own lessons or having students research a topic that interests them:
United Nation Cyberschool bus HYPERLINK "http://www.cyberschoolbus.un.org/" http://www.cyberschoolbus.un.org/ (search data on infonation) or the United Nations general site HYPERLINK "http://www.un.org/"http://www.un.org.
Your town budget for the last few years.
Data and Story Library HYPERLINK "http://lib.stat.cmu.edu/DASL/" http://lib.stat.cmu.edu/DASL/, compiled by Cornell University, is intended for use by students and teachers who are creating statistics lessons. As a teacher, I click on list all methods and go to regression, correlation, causation or lurking variable.
Students may wish to search for data by topic that interests them.
National Oceanic and Atmospheric Administration is the federal government Web site for all things involving climate, weather, oceans, fish, satellites and more. You will find an educator page as well HYPERLINK "http://www.noaa.gov/" http://www.noaa.gov/.
Articles on Correlation Coeffiecient:
Barrett, Gloria B. The Coeffiecient of Determination: Understanding R and R-squared Mathematics Teacher Vol. 93, Number 3, March 2000
Kader, Gary D. and Christine A Franklin. The Evolution of Pearsons Correlation Coefficient. Mathematics Teacher, vol. 102, number 4, November 2008
Attached are
PowerPoint on forensic anthropology
Handout 3.2 on forensic anthropology
Teacher notes on forensic anthropology activity,
Teacher notes on correlation coefficient
TI 84 calculator key strokes for plotting data, calculating regression line, and calculating correlation coefficient. Homework:
Create a worksheet with a screen shot from the calculator home screen that shows the parameters a and b after some linear regression was calculated. Ask students to write the equation in slope intercept form that corresponds to the screen shot. Have them use the equation to find y given an x value.
Ask students to give a rough sketch of three scatter plots having correlation coefficients of -0.9, 0.6 and -0.2. Alternatively, you could create matching problems by sketching four scatter plots and giving four numbers between n -1 and 1 as the r-values to match to the scatter plots.
From the calculator directions included in this investigation, have students create a quick-key guide for the calculator that will remind them how to plot data, and calculate the linear regression and correlation coefficient. Tell them that they will be allowed to use these notes on a test. These Quick Calculator Key strokes could be put on a process card in the Hands On Algebra 1 Toolkit.
Create a worksheet about a scenario of interest to the students, one that may spark discussion or raise consciousness about an important issue, or one that is allied with content in another class they are taking. Ask a question, explain what data was collected, and include screen shots from calculator showing the scatter plot and the regression line, and the home screen where the calculator identifies the values of the parameters a and b and tells r. Based on the calculator screen shots, the students should be able to answer the question that asks them to extrapolate or solve for x. The goal is for students to be able to interpret and use the information they see on the calculator screen to make a prediction.
If students have access to technology at home (either calculator or computer) have them graph the scatter plot of data you give them, calculate the regression line and the correlation coefficient. If students do not have technology at home, give them a copy of the data, a graph, the regression equation and the correlation coefficient. Have all students answer questions about the data, such as meaning of slope and intercepts in context, interpolation, extrapolation, solve for y given x, what is a reasonable domain, and what is the strength and direction of the correlation? How confident are you in your predictions?
If you havent already, now is the time to begin asking students what they would like to investigate. This will prepare them for the Unit Investigation. The topics could pertain to a class fundraiser, environmental issues, social injustices or political oppression, to name a few ideas. Tell students that you want them to formulate a question and find the data necessary to answer the question. For homework, they are to complete the following sentence for something that interests them:
I would like to know ___________________? I think I can answer this question by finding the linear regression for the data: ____ and ________.
Examples: I would like to know How long will it take to collect 500 food items for the food drive? I can answer this question if I find the linear regression for the data: number of days and number of total food items to date.
I would like to know How tall was the person whose ulna bone was found? I can answer this question if I find the linear regression for the data length of the ulna and height of the person.
I would like to know Do richer countries pollute more than poorer countries? I can answer this question if I find the linear regression for gross domestic product for a variety of countries and the corresponding carbon dioxide emission.
I would like to know Are richer states more or less likely to sentence criminals to death? I can answer this if I find the linear regression for the median income for several states and the number of people on death row for each of those states.
Tell students to write three sentences about what data they will collect, and how they will collect data that would answer the student-posed question from the previous homework. If they are going to do an Internet search, the three sentences should include the key word or phrase that was searched, two Web sites the student viewed as a result of the search, and whether the search was productive or unproductive. The purpose of this homework is to have students lay the groundwork for their Unit Performance Task.Post-lesson reflections:
Did the students have enough practice analyzing data with technology?
Did the data sets analyzed include information from various disciplines?
Were some data sets generated by student activity as opposed to simply collected from an Internet or printed source?
Did students have the opportunity to analyze data with positive and negative, strong and weak correlation?
Did students have an opportunity to analyze data with correlation, but not causation, or with data that is causal?
Were students able to identify data sets of their own that might be linearly correlated?
Unit 5, Investigation 3
Teacher Notes 3a, p. 1 of 2 Teacher Notes
Forensic Anthropology
This lesson involves two activities: the students will collect their own data, then technology is used to graph the data and calculate the linear regression.
Begin the lesson with the PowerPoint Forensic Anthropology. You can also bring in some bones from the local butcher boiled and washed. Another prop might be dolls and action figures. Go through the slides with the students. Encourage their participation. Try to elicit from them the idea that bigger animals have bigger bones. Then you can extend that generalization to the idea that height is related to the long bones such as the ulna, tibia and femur.
If you want more background information, search Mildred Trotter, Wyman forensic anthropology, and Bill Bass forensic anthropology.
A famous use of forensic anthropology was by Mildred Trotter (1899-1991) who identified the remains of soldiers from WWII at the Central Identification Laboratory in Hawaii. A history she wrote about her 14-month experience at the CIL is at HYPERLINK "http://beckerexhibits.wustl.edu/mowihsp/words/TrotterReport.htm" http://beckerexhibits.wustl.edu/mowihsp/words/TrotterReport.htm.
Jeffries Wyman and the birth of forensic anthropology are described in HYPERLINK "http://knol.google.com/k/michael-kelleher/the-birth-of-forensic-anthropology/2x8tp9c7k0wac/3#" http://knol.google.com/k/michael-kelleher/the-birth-of-forensic-anthropology/2x8tp9c7k0wac/3#.
A Web site about the work of Bill Bass is HYPERLINK "http://www.jeffersonbass.com/" http://www.jeffersonbass.com/. He was famous for his books on bones and the body farm. Though he is not included in the PowerPoint, people may have heard of his work from radio and television broadcasts.
Once the slide show is completed, you may distribute the student activity sheet if students need more structure working through the steps in the experiment, or need specific places to respond to questions.
To gather the data, you might want to place four or five tape measures around the room, creating stations for students to go to for measuring their height. Have students pair up to measure each others ulnas and height. Have at least two people measure a persons height and ulna length, three measures are preferred. Average the two or three measurements. First, this will reduce measurement error in the data, and secondly, the students can always use practice measuring. Have each student write the average of his or her height and ulna data at a central collection place such as the blackboard, interactive whiteboard, on a computer spreadsheet or on an overhead transparency. Tell the students to record all their classmates data on their activity sheets.
As a whole class discussion, use the data as means of reviewing vocabulary such as independent and dependent variables. Ask the students to decide on a scale and labels for the x and y axes. The students can work in small groups to graph the data by hand and find a trend line. Let them struggle for a short while with scale, the tight clustering of the data points on the scatter plot, inaccuracy, and other issues associated with graphing and calculating by hand. Ideally, the students will be motivated to use technology. Ask them to visually sketch a trend line and compare the wide variety of trend lines that the students have even though they all started with the same data. The students lines cannot all be lines of BEST fit. As a whole class, lead the students step by step as they plot the data and calculate the linear regression and correlation coefficient using a graphing calculator, Excel, or one of the many free graphers available on the Internet. Be sure that the students round the numbers in the regression equation to the same number of decimal places in the data.
A list of steps for plotting data and calculating a regression line with the TI 83-84 is provided.
Unit 5, Investigation 3
Teacher Notes 3a, p. 2 of 2
When you are using technology, you will want to find the height estimate for the given ulna length using the value menu on the calculator, or using an equation in Excel.
Compare the results from the calculations by hand with those from the technology. Discuss the advantages and disadvantages of using technology. Note that you would want to use technology if you had messy data, if you were making a presentation, and to be most accurate. Technology will find the line of BEST fit. The Least Squares Regression calculation takes every data point into account. The equation of the trend line students calculate by hand uses two data points.
You can ask the students to test the accuracy of the regression equation by substituting in their ulna lengths and comparing the regression height value with the students actual height. A possible extension is to measure dolls and/or action figures to determine if the toys fit the same equation as the students.
Now you can introduce the Pearson correlation coefficient. See the separate teacher notes for background information. Explain that the r is a numerical indication of the direction and strength of the correlation, and that the closer r is to 1 or -1, the stronger the linear relationship. If r is close to zero, then x and y are not linearly related.
Finally, you are able to revisit all the ideas developed in the previous lessons interpreting slope, for example but now with the aid of technology. Once the students have answered the original question to estimate the height of the person whose ulna bone was found, be sure to reinforce the mathematical concepts that they are to develop: plot data, find regression equations, use regression equations to predict y given x or vice versa, tell the meaning of slope in context, state a reasonable domain (note that the intercepts fall outside a reasonable domain), and determine the strength and direction of a linear relationship between two variables (r).
Students will need to practice the technology key strokes necessary to analyze data, so provide or have students generate data sets to practice using technology to create a scatter plot, find the linear regression and make predictions.
Unit 5, Investigation 3
Teacher Notes 3b, p. 1 of 1 Teacher Notes
Background Information on Correlation Coefficient
What is the correlation coefficient?
The information contained on this page is for the teachers background knowledge.
The r value is Pearsons Sample Correlation Coefficient or just the correlation coefficient. The word correlation refers to the co-relation between the two variables being analyzed. The correlation coefficient is one indication of how well the linear regression fits the data. The value of r is always a number between -1 and 1. It provides two pieces of information: the direction and strength of the linear relationship between the two variables. If r is positive, then the relationship is increasing: as x values increase, so do y. If r is negative, then the data is negatively correlated: as x values increase, y values decrease. The closer r is to 1 or -1, the stronger the linear relationship between the two variables. If all the data points are collinear, nonhorizontal, then r = 1 or -1.
If there is no linear correlation between the two variables, then r = 0. If r = 0, that does not mean that there is no relationship between the variables, only that the relationship is not linear. For example, data that is in the shape of a circle, a v, or a parabola will have r = 0. Data that is horizontal will also have r = 0, even collinear horizontal data. The magnitude of the correlation coefficient indicates whether the linear regression is a better estimator of y than is the simple arithmetic mean of the y data. So, for example, if the slope of the linear regression is zero, then the linear regression can do no better predicting the y variable than the average of the y data, because, for horizontal data, the linear regression is y = average of the y data.
Some students may ask about the formula for r. If n is the number of ordered pairs, QUOTE is the mean of the x values, QUOTE is the mean of the y data, sx is the standard deviation of the x values, sy is the standard deviation of the y values in the ordered pairs that are your data, then QUOTE . Do not ask students to calculate r by hand except in a statistics course.
Information on the TI calculators stat list editor, plotting stat data and regression equations and coefficient correlations, see the TI guidebook chapter on Statistics. Read through the activity titled Getting Started: pendulum lengths and data. If you lost your guidebook, go to HYPERLINK "http://www.education.ti.com/"http://www.education.ti.com and click Guidebooks under the Downloads menu. A free PDF copy of all guidebooks for all calculators is available.
Unit 5, Investigation 3
Activity 3.1a Page 1 of 3
Forensic Anthropology
Name: _________________________________________________ Date _______________________
PURPOSE:
The purpose of this activity is to have students collect their own data, and to learn to use technology to graph data, find the equation of a line of best fit, and find and interpret the correlation coefficient.
BACKGROUND:
While excavating for the new school building, construction workers found partial skeletal human remains. Who was this person? How tall was he or she? Was the victim a male or a female? How long ago had the person died? Forensic anthropologists were called in on the case to read the bones. Police will want to know the victims height to begin to match the bones with the missing persons on file. Can you estimate the persons height from his or her bones?
The long bones such as the femur (thigh), tibia (shin) and ulna (forearm) predict height better than the shorter bones. The only intact long bone is the ulna, which is 28.5 centimeters in length.
Your job as the forensic anthropologists assistant is to estimate the height of the victim whose bones were found.
PROCEDURE:
Gather height and ulna length data.
Measure and record the ulna lengths and heights of each of your classmates. A good way to have someone measure your ulna is to place your elbow on the table with the thumb pointed toward your body. Then have the classmate measure from the round bone in your wrist, just below your pinky finger to the bottom of your elbow, which is resting on the desk. Because people may measure differently, it is a good idea to have at least two or three people measure your ulna, and then take the average of the two or three measurements.
To measure heights, you may want to tape four or five rulers or tape measures at stations around the room. People can line up with the ruler on the wall and read each others height. Again, have two or three people measure your height and take the average of the two or three measurements.
It is important that you pair each persons ulna length with that persons height.
Unit 5, Investigation 3
Activity 3.1a, Page 2 of 3
Data Sheet
Subject #Ulna Length (cm)Height (cm)Subject #Ulna Length (cm)height (cm)116217318419520621722823924102511261227132814291530
Plot the data and find trend line.
Which variable is the dependent variable?
Which is the independent variable?
Graph the data on the coordinate axis below. Be sure to label the axes. Create a reasonable scale and title.
Unit 5, Investigation 3
Activity 3.1a, p. 3 of 3
Does the data appear linear or not? Explain.
Sketch a trend line on your scatter plot above.
Write the equation found by hand here:
Use technology to graph data, and calculate regression equation and correlation coefficient.
Use a calculator or computer software to graph a scatter and calculate a linear regression for height as a function of ulna lengths.
Write the equation from the technology here:
Write the correlation coefficient: r =
Comment on the direction and the strength of the linear relationship between ulna lengths and height.
Compare the hand-calculated trend line from part 2d with the linear equation you found using technology in part 3a. How close are the two? What are the comparative advantages of doing the work by hand or using technology?
Make a prediction.
Use the linear regression that you found using technology to estimate the height of the missing person. Remember that the ulna bone is 28.5 centimeters long.
How accurate or reliable is your prediction?
Explain your answer by referring to the graph and the correlation coefficient r.
Congratulations. You are now ready to report back to the investigators. You have completed a piece of the puzzle!
Connecticut Algebra 1 Model Curriculum Page PAGE 7 of NUMPAGES 15
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