SYMBOL OF EACH SECTION HEADER TO RETURN HERE.
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TEKS Support |
This unit contains activities that support the following knowledge and skills elements of the TEKS.
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(1) (C) |
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(2) (A) |
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(2) (B) |
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(2) (D) |
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(3) (A) |
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(3) (B) |
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The mathematical prerequisites for this unit are
The mathematical topics included or taught in this unit are
The equipment list for this unit is
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Context Overview |
This unit investigates statistical concepts involved with prediction. Archaeologists, criminologists, and doctors all have an interest in predicting people’s heights from knowledge of another variable, such as the length of bones in the body or the distance between footsteps. In this unit, students collect their own data on height, forearm length, and stride length. Based on their data, they determine models to predict height from forearm or stride length. For the final project, students analyze skeletal data from the Forensic Anthropological Data Bank and determine models to predict height from the length of long bones in the arms and legs.
In addition to predicting height, students investigate an ecological problem. The manatee is an endangered species with a rising death rate. About one-third of all manatee deaths are attributable to human causes, and among these the leading cause is contact with powerboats. Based on data collected by the Florida Department of Environmental Protection, students decide how increases in powerboat registrations are affecting the life of the manatee.
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Mathematical Development |
The two major mathematical goals of the unit are to explore bivariate data analysis and to further understanding of linear relationships.
This unit expands students’ acquaintance with data analysis, emphasizing not simply describing data but also using data to make predictions. The unit begins by examining a proportional relationship between people’s head length and their height. Based on data collected from their classmates, students determine the best multiplier for this relationship. In addition, they interpret the meaning of slope in this context.
Later in the unit, students display data using dot plots (one-variable data) and scatter plots (two-variable data) and then use their displays to make predictions. They assess the precision of their predictions based on the variability in the data. Midway through the unit, students encounter a major problem: Given a scatter plot, how do you select the "best" line to describe the data? After looking at several methods and comparing the resulting models, students are introduced to the least-squares line. They use residual plots to judge the adequacy of the linear regression model to describe the pattern in a scatter plot. In addition, they examine the effect that outliers have on the least-squares line and learn to select the best variable for making a prediction.
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Teacher Notes |
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Preparation Reading—Let the Bones Speak! |
This reading introduces the major contextual theme of the unit and leaves students with a question: What can you predict about a person from a few bones?
Make sure students begin this unit by reading this preparation reading. Information from this reading is used throughout the unit.
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Activity 1—Using Your Head |
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| Materials Needed |
Meter sticks (at least 2) Rulers |
Mathematically, relationships between height and head length, examined from the perspective of artists, focus attention on models of the form y = mx and the meaning of slope. At the end of the activity, students use one such model to predict a person’s height based on the length of his or her skull bone.
Students should work in small groups (3 to 4 students) on this activity. For Item 2(a) each group will have to share its data with another group.
For Item 1 it is not important that students arrive at "correct" answers. What is important is the reasoning that they use to arrive at their answers. For example, some students may decide that the 416-mm tibia belongs to the same person as the 413-mm femur. They may have arrived at this conclusion based on the data provided in Table 1. Other students may argue that the femur is the longest bone in the body, and thus this tibia belongs to the same person as the 508-mm femur. Let students argue this out for themselves. Don’t give them the answer.
If students struggle making a guess in Item 1(e), remind them to use their general knowledge about people’s heights.
At the conclusion of this activity, discuss Item 2(d and e). Students should understand that, when a residual error is positive, the prediction underestimates the actual value; when a residual error is negative, the prediction overestimates the actual value. You would like to choose a model where, in some sense, the positive and negative errors are balanced or tend to cancel each other out.
Discuss Item 4 with your students to continue developing the concepts of slope and rate of change. Here are some suggested approaches: numeric reasoning, algebraic approach, and graphical approach.
Numeric reasoning:
Make a table of values similar to Example 1. Explain to the students that you are using the notation D(Head length) and D(Height) as shorthand for the change in head length and the change in height, respectively. Stress that the direction of the change must be consistent. Positive values indicate increases, negative values indicate decreases.
In this table, start with the preliminary head length, make the indicated change to head length, and note the corresponding change in height. Point out to students that each time the head length increases by 1 cm, the height increases by 7.5 cm. This is true no matter how large the preliminary head lengths are. If the head length increases by 2 cm, then the height increases by 15 or 2 × 7.5 cm.
| Head-length of preliminary sketch | D(Head length) from preliminary to final sketch | Height of preliminary sketch (cm) | Height of final sketch (cm) | D(Height) |
|---|---|---|---|---|
| 8 | 1 | 7.5 | ||
| 9 | 1 | 7.5 | ||
| 10 | 1 | 7.5 | ||
| 10 | 2 | 15 | ||
| 11 | 2 | 15 |
Example 1. Table of height values
Algebraic approach:
Make sure the students understand the distributive-law-based answer to Item 4.
Graphical approach:
Use Transparency T.1 to illustrate the change in height corresponding to a 1 cm change in head length.
Be sure that students see the connections between the three approaches.
When discussing Item 5, note that groups may have arrived at different predictions if they decided that the length of a deceased person’s skull is smaller than the person’s head length when he or she was living. Using the length of the skull to estimate the length of the person’s head introduces a source of uncertainty (or error) into the prediction process. The second source of error may be the artists’ guidelines. They were meant to be rough guides for drawing figures and not precise methods for predicting height.
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Homework 1—Leg Work |
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For this assignment, students work with linear models developed by Dr. Mildred Trotter to predict people’s heights based on femur and tibia lengths. At the conclusion of this assignment, they discover that a person’s femur is generally longer than his or her tibia.
This assignment foreshadows results from the final project. Here students are introduced to several of Dr. Trotter’s equations relating height to lengths of leg bones. At the end of this Homework, they should understand that a person’s femur is longer than his or her tibia.
As background, here is some information on Dr. Trotter. Dr. Mildred Trotter had a long and distinguished physical anthropology career that included working as a special consultant to the U. S. government during World War II. Her task during the war involved the identification of skeletal remains of servicemen. At the time, she realized that bone sizes and proportions vary based on age, sex, race, and ethnic background. Forensic scientists and law enforcement agencies are still using Trotter’s formulas for estimating people’s stature based on the lengths of their bones.
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Supplemental Activity 1—Under Investigation |
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In this activity students investigate graphs of members of the y = mx + b family and learn how changes in m and b affect the graphs. In addition, students discover that the appearance of a graph can be altered by changing the scaling on one or both of the axes.
When discussing Item 2, note that, because the graph’s origin is often not displayed on the calculator screen when plotting data, students should keep in mind the location of the origin in relation to the graph. Give students practice locating the origin with sample calculator windows. Show them the screens and ask them to sketch the origins. Below are some sample windows.
[100,200] × [50,90]
[–10,–5] × [5,15]
[75,100] × [–40,–10]
[–10,–5] × [50,90]
Note: Generally, students will use windows in the first quadrant or select the standard viewing screen.
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Activity 2—Measuring Up |
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| Materials Needed |
Handout 1 (to record class data) Tape measures, rulers, meter sticks Background Reading: Dr. Mildred Trotter’s Study of Military Personnel |
In this activity, students plan how they will measure and collect data on students’ heights and forearm lengths. Then they collect the data from students in their class.
The quotation below sheds light on how the military measured the height of personnel in the 1940s. (Dr. Trotter’s equations were based on military personnel from this time period.) Share this reading with your class; students may be surprised at the level of detail in the regulation. Can they find the one important detail that’s missing? The regulation appears in a paper by Mildred Trotter and Goldine Gleser (1952).
In Mobilization Regulations, War Department, October 15, 1942 (Regulation 10):
"Directions for taking height. Use a board at least 2 inches wide by 80 inches long, placed vertically, and carefully graduated to 1/4 inch between 58 inches from the floor and the top end. Obtain the height by placing vertically in firm contact with the top of the head, against the measuring rod, an accurately square board of about 6 by 6 by 2 inches, best permanently attached to graduated board by a long cord. The individual should stand erect with back to the graduated board, eyes straight to the front."
As detailed as the regulation appears, something was forgotten. In another set of mobilization regulations dated April 19, 1944, the same essential directions were given with the following sentence added:
"The shoes should be removed when the height is taken."
Mobilization Regulations, War Department, April 19, 1944 (Regulation 10). (Trotter and Gleser 1952, 469-470).
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Homework 2—Follow in My Footsteps |
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Determining a standard method for measuring stride length is a bit more complicated than measuring height. For this assignment, students write a set of instructions for measuring a person’s stride length.
At some time prior to Activity 6, you should collect the stride length data from students in the class.
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Supplemental Activity 2—Line Up |
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This activity reviews determining an equation for a line given its graph. Students use both slope-intercept and point-slope forms to determine equations of lines from their graphs.
This is a review activity for students who are rusty in use of the slope-intercept and point-slope forms for determining the equation of a line. This is optional.
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Activity 3—I Predict That |
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Materials Needed |
Class data on heights and forearm lengths |
This activity focuses on the idea of variability in data and its relation to the precision of predictions made from the data. Students analyze one variable, student heights. They assess the precision of using the mean as a predictor of height.
The purpose of this activity is for students to see that precision in prediction is linked to variability in the data. Dot plots are introduced as a graphical tool for analysis of one-variable data, and the mean is suggested as a simple predictor for such data.
Discuss Item 3. Notice that, instead of using the minimum and maximum heights for the prediction interval, we narrow the interval by omitting the three smallest and three largest observations. Although this allows for a more precise prediction (the interval is narrower), omitting data also increases the chance that the prediction will be false. Discuss why the increase in precision is probably worth the increased risk of being wrong.
In Item 4, students consider the mean as a predictor. Point out that asking how far off a prediction might be is another way of asking how large the prediction error might be. Point out that the error depends on the spread (variability) of the data.
In Item 5, note that, whenever you see data that are bimodal (appear roughly as two mounds), you should ask if the data contain two subpopulations. If you can identify the subpopulations, which in this case are the boys and the girls, you should analyze each separately and then compare the results.
In Item 6, the girls’ data are less variable (exhibit less spread) than the entire data set. This reduction in variability allows a more precise prediction.
The purpose for Item 8 is to acquaint students with calculator output from one-variable statistics calculations. Check to see that students understand the mathematical notations for sum and mean.
If you run short of time, students can complete Items 10 and 11 on their own. Note that the term outlier is defined in Item 10. In Item 10, students discover the effect that outliers have on the mean and the importance of adjusting predictions when outliers are present. For Item 11, check to see that students are aware of the link between the precision of predictions and the variability of data. This concept will reappear when students analyze precision of predictions that are based on linear models.
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Homework 3—Exercising Judgment |
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In Item 1, although students will notice that, on average, the mothers who smoked had babies that weighed less than those of mothers who did not smoke, they may not notice that all of the babies who weighed under 6 lb. had mothers who smoked.
In Item 2, each set of data contains an outlier that inflates the mean. Be sure that students recognize this, remove the outliers, and compute the means of the data that remain. For Items 2(f) and (g), check that students understand why a scatter plot is an inappropriate way to display these data. Scatter plots are used when there is an assumption that two quantities obtained as matched pairs are related. There is no such pairing here, and no natural reason to pair particular numbers for the two groups.
This activity reviews determining an equation for a line given its graph. Students use both slope-intercept and point-slope forms to determine equations of lines from their graphs.
This is a review activity for students who are rusty in use of the slope-intercept and point-slope forms for determining the equation of a line. This is optional.
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Activity 4—Forearmed Is Forewarned |
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Materials Needed |
Graph paper Ruler Spaghetti or toothpicks |
This activity emphasizes the use of scatter plots in identifying and describing relationships. Students face the problem of selecting the "best model" to describe the pattern of a scatter plot. In deciding between two contenders for the best model, students analyze both models’ residuals.
Item 2 is designed to connect analyses of one-variable data (discussed in Activity 3) to two-variable settings, bridging dot plots to scatter plots. By drawing a vertical line to specify a single value of the independent variable, students can interpret the data that fall along that line (or close to it) as a vertical dot plot.
For example, to view the variability in heights for students with forearm length 27 cm, draw the vertical line x = 27. Then look at the range of heights for students with 27-cm forearms (or close to 27-cm forearms).
After students have calculated a few predicted values and residual errors in Item 4, you may wish to help them use calculator lists to speed their work. See Handout 4 for TI-83 calculator instructions.
For Item 5, students may find it helpful to use a tangible object such as uncooked spaghetti (or toothpicks if they’re working on calculator screens) to use as lines. That way, they can easily adjust the line until they are satisfied with how it fits the data.
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Homework 4—The Nature of Our Relationship |
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This assignment introduces the ideas of direction (positive or negative), form (linear or nonlinear), and strength (strong or weak) of a relationship.
Note: After students have completed this assignment, review some of the new vocabulary words, such as positively and negatively related, linear and nonlinear form, and weak and strong relationships.
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Activity 5—Dangerous Waters |
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Students fit a least-squares line to describe the relationship between the number of manatees killed per year and the number of powerboat registrations. They use their model for analysis and prediction. In addition, they learn to use residual plots to assess whether their model is adequate to describe the data.
For Item 2, you will need to teach students to calculate the equation for the least-squares line on the graphing calculator or computer. Handout 2 contains TI-83 instructions for computing the least-squares line. Handout 3 provides similar instructions for Excel. For other calculators or spreadsheets, check your manual.
Item 4 states two essential criteria related to good fits and defines "residual plot." Stress student understanding of what this plot really means. The randomness of this plot should be the primary criterion for deciding that a model is reasonable. You may refer students to Handout 4 if they need help calculating the residuals on their calculators.
For Item 6, check that students realize that the number of powerboat registrations is in units of 1,000.
You may want to point out that a "good" residual plot looks like a bunch of dots thrown haphazardly at a piece of paper; the dots should appear randomly scattered around the x-axis. If the dots do not look randomly scattered around the x-axis but instead form a clear pattern, you should look for another model to describe your data.
Some calculators give the value of Pearson’s correlation coefficient, r, as part of the output from a linear regression. If this is the case, you may want to tell students that r is a measure of the strength and direction of a linear relationship. However, stress that judging the goodness of a fit should begin with examining the graphs of the original data and the residual plot.
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Homework 5—Anscombe’s Data |
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Students fit least-squares lines to four data sets and discover that they get the same equation in all four cases. After examining scatter plots of the data, students learn that the least-squares equation is an adequate model for describing the pattern in only one of the data sets.
This is a famous data set. You will find it in numerous statistics texts.
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Activity 6—The Plot Thickens |
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Materials Needed |
Tape measures or meter sticks Partially completed Handout 1 |
Students decide which of two independent variables, forearm or stride length, is a better predictor of student height. In addition, students work through an analysis illustrating how forensic data can help solve crimes.
For Item 1, one method for deciding which of two independent variables yields more precise predictions for the same dependent variable is to select the relationship that has the smaller average of squared errors. Taking the average instead of the sum adjusts for situations where the scatter plot of one relationship has more data than the scatter plot of another relationship. For example, this method can be used when comparing the regression equation based on the class height-forearm data to predict height to the regression equation based on the boys’ height-forearm data.
The average of squared errors is one estimate of the variance about the least-squares line. It is, however, not the one generally used by statisticians. Statisticians generally use the unbiased estimator SSE/(n–2) where n is the number of cases or the sample size, but this is not relevant to student work in this unit.
To speed the completion of Item 1, you may decide to work the item as a whole class activity.
The sample answers to the remainder of the items in this activity are based on the following set of data collected from a set of 9th and 10th graders.
Name |
Gender |
Height |
Stride Length |
Forearm Length |
| Scott | Male | 166.0 | 58.25 | 28.5 |
| John | Male | 178.0 | 68.5 | 29.0 |
| Matt | Male | 171.0 | 58.5 | 27.2 |
| Will | Male | 165.0 | 50.125 | 28.0 |
| Michael | Male | 177.5 | 58.75 | 31.3 |
| Jeffrey | Male | 166.0 | 62.875 | 28.3 |
| Even | Male | 175.5 | 59.125 | 28.6 |
| Brad | Male | 171.0 | 67.75 | 31.5 |
| Lonnie | Male | 184.0 | 68.875 | 30.5 |
| William | Male | 184.5 | 66.25 | 30.8 |
| Robert | Male | 183.5 | 79.5 | 30.5 |
| Karim | Male | 172.0 | 70.5 | 30.3 |
| Meredith | Female | 164.5 | 55.875 | 24.2 |
| Lee | Female | 166.0 | 52.375 | 27.3 |
| Pilar | Female | 168.0 | 55.375 | 28.0 |
| Ansley | Female | 178.5 | 59.75 | 29.1 |
| Julie | Female | 166.0 | 48.375 | 27.9 |
| Becton | Female | 159.0 | 57.125 | 28.0 |
| Elizabeth | Female | 166.0 | 64.0 | 27.4 |
| Shannon | Female | 154.5 | 57.75 | 25.8 |
| Jamie | Female | 161.0 | 63.5 | 27.0 |
| Jeris | Female | 177.0 | 69.75 | 30.1 |
| Kat | Female | 161.0 | 72.5 | 26.5 |
| Blaie | Female | 164.0 | 75.25 | 28.2 |
| Frances | Female | 174.0 | 58.5 | 28.4 |
| Eliz | Female | 164.0 | 59.75 | 26.8 |
| Baily | Female | 168.0 | 55.25 | 26.4 |
For Item 2, if you have not already collected class data on student stride lengths, you should do so. (See Homework 2.) After deciding on a method for collecting the data, each group can be responsible for collecting the data from its members. After groups have collected their data, pool the results. Students should record these results in the last column of Handout 1. If you have already collected the stride-length data, students can read quickly through Items 2 and 3 and begin their work at Item 4.
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Homework 6—You Are What You Eat |
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In this assignment students discover the drastic effect that outliers can have on a regression line by comparing models computed with and without outliers. Students also learn to seek the interpretation of outliers in particular settings.
After students have completed this assignment, discuss how the presence of outliers affects the values of m and b in the least-squares equation. The least-squares equation can be very sensitive to outliers, particularly if they occur at the extremes. In these situations, the least-squares line does a poor job of describing the pattern of the majority of the data.
In those cases where you can determine that the outliers are "unusual points" that are not representative of the relationship, remove these points and recalculate the equation of the least-squares line using the remaining data. For example, in Item 2 (the situation with the swimming data) a good argument could be made that the first two times were not "typical" because the swimmer was still learning the butterfly. In this case, it seems reasonable to remove the outliers and refit the model.
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Unit Project—Who Am I? |
This project can be adapted for a wide range of student abilities and time constraints. Students can complete their analysis using a spreadsheet or a graphing calculator. Ideally, students work in groups, and each group should present its work in a formal written report. You may want to have groups give oral presentations in addition to or in place of the written report. Work may also be done individually if more time is available.
Below is a brief set of guidelines for reports. You may decide to give more detailed guidelines of your own design.
If possible, let students decide for themselves how they will complete this project. Encourage them to plan what equations they will need to determine and then divide the work among group members. If some groups struggle, you may need to provide additional structure.
The following is a direct method (not necessarily the best method) of addressing the questions in this project.
Students may use several different approaches in developing equations to predict the heights of Bones 1 and Bones 2. First, they should realize that there are two bones that can be used to predict height: the femur and the ulna. So, they should begin predicting height using each of these independent variables.
Note that the data that appear in the student pages of this project are also provided as computer files, as listed:
Column headings are not included in these files. However, the calculator file is a program that stores the data to named lists. See student pages for the heading labels and units of measure.
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Handout 1—CLASS DATA RECORDING SHEET |
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Female |
Male |
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Name |
Forearm |
Height |
Strident |
Name |
Forearm |
Height |
Stride |
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Handout 2—TI-83 INSTRUCTIONS FOR FINDING THE LEAST-SQUARES LINE |
Here’s how to use the TI-83 to calculate the least-squres line and its residuals.
Press ENTER again and the residuals will appear. How many are positive and how many negative? What is the absolute value of the largest residual? How can you use the features on your calculator to find the sum of the residuals?
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Handout 3—EXCEL 4.0 GUIDANCE FOR FINDING THE LEAST-SQUARES LINE |
Complete Activity 5 in your text. Use these instructions to assist your work using Excel 4.0.
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Handout 4—TI-83 INSTRUCTIONS: CALCULATING PREDICTED VALUES AND ERRORS |
Example: Find the errors using the linear model y = 2x + 1.
Recall that prediction errors are defined as Yactual - Ypredicted.
Figure 3. Screen showing formula for L1
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Annotated Student Materials |
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Preparation Reading—Let the Bones Speak! |
Legend has it that, about 100 years ago, somewhere in Arizona’s Superstition Mountains, a Dutchman by the name of Jacob Walz murdered a group of gold miners in order to claim their mine for himself. Over the years, he would periodically be seen in Phoenix with saddlebags filled with rich ore. Many attempted to follow Walz when he returned to the mine, but he always managed to lose trackers in the rugged wilderness. Walz died in 1891. For over a century, people have searched without success for the Lost Dutchman Mine. Some have lost not only time and money, but their lives. At least two searchers are known to have been murdered during their quest. Others, unable to meet the physical challenges of the rugged area, never returned from their treks and remain missing. |
From time to time, human bones are found in rugged areas such as the Superstition Mountains. Suppose that a contemporary gold digger searching for the Lost Dutchman Mine finds a skull, eight long bones, and numerous bone fragments. He notifies the local authorities who, in turn, send out a team to investigate. After first documenting the exact location and position of the bones at the site, the team records information about the bones, such as their size and general condition. A partial list of information similar to what might be recorded is contained in Table 1.
Bone type |
Number found |
Length (mm) |
| Femur | 3 | 413, 414, 508 |
| Tibia | 1 | 416 |
| Ulna | 2 | 228, 290 |
| Radius | 1 | 215 |
| Humerus | 1 | 357 |
| Skull | 1 | 230 |
| Fragments | More than 10 | From 30 to 50 mm |
Table 1. Sample record of bones found at site
In such situations, police frequently request help from forensic anthropologists to identify the deceased and determine the cause of death. The bones tell the forensic scientists a story about who the deceased were and frequently how they died. Sometimes the story is an old one, as would be the case if the bones belonged to the gold miners murdered by Walz. Other times, the bones tell a story of more recent crime and provide police with clues that may help them solve a mystery.
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Activity 1—Using Your Head |
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= 1, 2, 3, 4
In this unit you will be asked to think like a forensic scientist. After studying the data in Table 1 of the preparation reading, you will begin to sort out clues about the deceased from their bones. For the final project at the end of this unit, you will write a report detailing their story.
There are three femurs. Two of the femur bones are very close in length (413 mm and 414 mm) and probably belong to the same person. So, these three bones most likely belong to two people. It is possible that the bones actually belong to more than two people.
Sample answer:
Assume that there are only two people.
Bones 1: femurs–413, 414; tibia–416; ulna–28; and radius–215.
Bones 2: femur–508; ulna–290; humerus–357.
The skull bone could belong to either Bones 1 or Bones 2.
Since the two femurs—413 mm and 414 mm—are close in length, they most likely belong to the same person, Bones 1. From the skeleton in Table 1, it appears reasonable to assume that a person’s tibia should be fairly close in length to her femur, so the 416-mm tibia probably belongs to Bones 1. Assuming that there are only two people, the person with shorter legs probably also has shorter arms. That means that the 228-mm ulna belongs to Bones 1. The length of a person’s radius should be close to the length of the ulna, and hence the 215-mm radius belongs to Bones 1.
Bones 2, the taller of the two people, has a 508-mm femur and a 290-mm ulna. A person’s humerus should be just a bit longer than his ulna (based on the diagram of the skeleton in the preparation reading). Assuming that there are only two people, the 357 mm humerus belongs to Bones 2. (It’s possible that this bone belongs to a third person.)
Students probably will be most uncertain to whom the skull and humerus belong. Some students may know that the femur is the longest bone in the body and may conclude that the 416-mm tibia belongs to Bones 2.
Sample answer:
There is very little evidence to suggest whether the decedents are male or female. Because the femur bones of one of the decedents are considerably shorter than the other, it’s possible that Bones 1 (shorter femur) is female and Bones 2 is male. However, it could just as easily be the case that the decedents are two males, one short and one tall.
Sample answer #1: Most likely the decedents were adults. What would young children be doing out in the Superstition Mountains?
Sample answer #2: When I measured my own forearm to get an estimate of how long an adult’s ulna might be, it measured about 260 mm. (The actual length of my ulna would be a bit less than this measurement.) So, Bones 2 is probably an adult. Bones 1 might be a child.
Sample answer:
Assuming that Bones 1 is female and Bones 2 is male, my guesses are that Bones 1 is 5 feet 5 inches and Bones 2 is 5 feet 10 inches. These guesses were based on my estimates of average heights for adult females and adult males, respectively. While these guesses are fairly rough, they should be within a foot of the actual heights.
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Head |
Predicted height |
Actual height |
Residual error: Actual – Predicted |
See sample answer to (d).
See sample answer to (d).
In almost every situation in which predictions are made from data, it is useful to examine the residual errors. Residual errors are defined as the difference between the actual value and the predicted value for each point in your data.
Name |
Head length (cm) |
Predicted height (cm) |
Actual height (cm) |
Residual error |
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Jan |
23.0 |
161.0 |
168.0 |
7.0 |
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Horrace |
25.0 |
175.0 |
178.0 |
3.0 |
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Betty |
24.0 |
168.0 |
166.5 |
–1.5 |
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John |
25.0 |
175.0 |
184.0 |
9.0 |
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Joy |
23.5 |
164.5 |
170.5 |
6.0 |
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Lora |
22.0 |
154.0 |
159.0 |
5.0 |
Positive residual errors indicate that your predictions are too low; negative errors mean predictions are too high; zero residual errors mean the predicted heights match the actual heights.
Sample answer:
Most of the residual errors were positive. So, the rule of thumb seems to frequently underestimate students’ actual heights.
Sample answer based on sample answer to (d):
Use Average ratio of height/head length: (1/6)(168.0/23.0 + 178.0/25.0 + 166.5/24.0 + 184.0/25.0 + 170.0/23.5 + 159.0/22.0)
» 7.2.Using 7.2 as the multiplier produces the following residuals errors.
Name |
Head length (cm) |
Predicted height (cm) |
Actual height (cm) |
Residual error |
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Jan |
23.0 |
165.6 |
168.0 |
2.4 |
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Horrace |
25.0 |
180.0 |
178.0 |
–2.0 |
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Betty |
24.0 |
172.8 |
166.5 |
–6.3 |
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John |
25.0 |
180.0 |
184.0 |
4.0 |
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Joy |
23.5 |
169.2 |
170.5 |
1.3 |
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Lora |
22.0 |
158.4 |
159.0 |
0.6 |
Using this multiplier, two of the six residual errors are negative; the size of errors tends to be smaller. In addition, the errors sum to zero so that the amount overestimated balances the amount underestimated.
H = 7.5L
H = 7.0L
Sample answer: H = 7.2L
Sample answer:
Note that the shaded sections indicate portions of the graphs that represent the real-world relationship between height and head length. (Students may decide to choose a window cropped to capture the shaded sections.)
These three graphs are all lines that pass through the origin. The multiplier controls the amount of inclination: the larger the multiplier the steeper the incline.
Predicted height: H = 7.5(23.0) = 172.5 cm. This estimate is likely to be too high. The graph H = 7.5L lies above the graph H = 7.0L for L > 0. So, for each positive entry for L, the value for H will be larger from the first relationship than from the second.
Each time head length is changed by 1 cm, the height gets changed by 7.5 cm. This is true regardless of the size of the head in the preliminary sketch.
Sample answer:
The decedent was most likely adult. The length of the skull is somewhat smaller than the length of the person’s head.
Sample answer:
Assume that the person was an adult and that head length was 2 cm larger than the skull bone, or 25.0 cm. (The extra 2 cm leaves room for skin and soft tissue and also accounts for shrinkage of the skull due to drying.)
Prediction: 7.5(25.0 cm) = 187.5 cm.
The sample answer of 187.5 cm is about 6 ft 2 in. This is a reasonable height for a tall person.
Sample answer:
First, skull size was used to estimate head length. In addition, the artists’ guidelines are only rough approximations and are not exact for every individual. So the estimate is a very rough one. It may be very far from the person’s actual height.
Sample answer:
It would be helpful to know the relationships between bone lengths and heights. Perhaps data could be collected from the class. Perhaps an artists’ handbook would contain relationships between arm lengths and heights or leg lengths and heights. Perhaps you could get data on bones and peoples’ heights from the Internet and then use these data to determine models that predict height from lengths of bones.
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Homework 1—Leg Work |
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Dr. Mildred Trotter (1899-1991), a physical anthropologist, was well known for her work in the area of height prediction based on the length of the long bones in the arms and legs.
Here is one of the relationships proposed by Dr. Trotter.
H = 2.38 F + 61.41 |
where H is the person’s height (in cm) and F is the length of the femur (in cm). |
Predicted height for a person with a 38-cm femur: 151.85 cm or approximately 152 cm.
Predicted height for a person with a 55-cm femur: 192.31 cm or approximately 192 cm.
Figure 3. Axes for height and femur length
Notice that the horizontal axis is scaled from around 35 cm to 60 cm (a slightly wider range than the minimum and maximum femur lengths) with tick marks every 5 units. A zigzag has been added to indicate that there is a break in this scale between 0 and 35.
Refer to answer in (b).
Note: Students may choose a different scaling for the vertical axis.
Jason’s height: (40)(2.38) + 61.41 = 156.61
Jason’s brother’s height: (41)(2.38) + 61.38 = 158.99
Difference: 2.38 cm
Each time you increase the length of the femur by one cm, the height changes by 2.38 cm. This difference has the same value as the multiplier of F, the slope of the linear equation.
Sample answer: 47.5 cm.
F = (H – 61.41)/2.38; femur is the dependent variable and height the independent variable.
(172.7 – 61.41)/2.38
» 46.8 cmSecond Formula:
H = 2.52T + 78.62,
where H and T are measured in cm.
H = (2.52)(41.6) + 78.62
» 183.5 cm. This person is approximately 6 feet tall. This is a reasonable height for a person.Second formula H = 2.52T + 78.62 for T.
(A doctor might use such an equation to check that the length of a person’s tibia is normal for a person of that height.)
Step 1: subtract 78.62 from both sides of the equation.
H – 78.62 = 2.52T
Step 2: divide both sides by 2.52.
(H – 78.62)/2.52 = T or T = (H – 78.62)/2.52
Approximately 37.3 cm.
H = 1.30(F + T) + 63.29.
(All measurements are in cm.)
H = 1.30(F + T) + 63.29.
The predicted height is 173.8 cm.
The predicted height is (2.38)(42) + 61.41
» 161.4 cm. This is 12.4 cm (or about 5 inches) shorter than the prediction in a).H = 2.52T + 78.62.
The predicted height is (2.52)(43) + 78.62
» 187 cm. This is 13.2 cm larger than the prediction in (a) and 25.6 cm more than the prediction in (b).Solving 175 = 2.38F + 61.41 for F gives F
» 47.7 cm. Solving 175 = 2.52T + 78.62 for T gives T = 38.2 cm. You would expect the femur to be approximately 9.5 cm longer than the tibia.Solving 160 = 2.38F + 61.41 for F gives F
» 41.4 cm. Solving 160 = 2.52 T + 78.62 for T gives T = 32.3 cm. From these equations, it appears that a person 160 cm tall should have a larger femur than tibia.Based on the answers to (e) and (f), it appears that a person’s femur should be longer than his or her tibia. When the students measured the bones, they found that the femur of the skeleton was shorter than the tibia. This is the reverse of what Dr. Trotter’s equations indicate. Perhaps the students need to recheck their measurements.
Bones 1: H = 2.38(413.5) + 61.41
» 159.823 cm or approximately 5 ft 3 in. (The average of the two femurs closest in length was used to make this estimation.)Bones 2: H = 2.38(508) + 61.41
» 182.314 cm or approximately 6 ft.Using the tibia length from Table 1, H = (2.52)(41.6) + 78.62
» 183.5 cm or approximately 6 ft.Sample answer:
These calculations do not appear to refute the assumption that there were only two deceased. However, based on these calculations, it appears that the 416-mm tibia might belong to Bones 2 (the taller person) rather than Bones 1.
In Activity 1 and in this homework, you examined and interpreted equations established by artists and by a scientist. You used some of Dr. Trotter’s models to estimate the heights of Bones 1 and Bones 2 (described in the preparation reading). Dr. Trotter’s formulas may have challenged some of the assumptions that you made in Item 1(b), Activity 1. However, for the equations given in this homework, she assumed that the deceased were adult white males. If this assumption is not valid, your estimates based on Dr. Trotter’s equations may not be accurate.
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Supplemental Activity 1—Under Investigation |
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Unlike the artists’ guidelines for drawing figures, Dr. Trotter’s equation,
H = 2.38F + 61.41
(where height, H, and femur length, F, are in cm),
is not a member of the y = mx family, but instead belongs to the larger y = mx + b family. You indicate members of this family by choosing values for m and b. (What were Dr. Trotter’s choices for m and b?)
Recall that Dr. Trotter’s equation
H = 2.38 F + 61.41
was designed to work well for a particular population, adult white males. She later modified her formula by modifying the values of m and b to adjust for age, ethnic background, and gender. To make such adjustments, you will need to know how changes in m and b affect the graph. Complete the following investigation to find out what happens when you make changes to m and b.
Because there are two quantities to change, m and b, it may help to divide the investigation into two parts, as described below.
PART I: KEEP m THE SAME AND CHANGE b.
PART II: KEEP b THE SAME AND CHANGE m.
Repeat Part I, reversing the roles of m and b.
Changing the value of b moves the line up (if b is increased) or down (if b is decreased). In addition, the line will cross the y-axis at b.
The slope, m, determines how steeply the line tilts and (depending if m is positive or negative) whether the line tilts upward or downward as you look along the graph from left to right.
The value of b determines where the line crosses the y-axis. So, y-intercept is a descriptive name. The value of m determines the steepness of the line or how much it slopes.
By changing your window settings, you can affect the appearance of a line described by a member of the y = mx + b family without changing the values of m or b. At times, you may want to adjust your window settings to display your graph more effectively. However, you should also be aware that some people, driven by an interest in distorting the truth, will tinker with their window settings until they achieve a graph that satisfies their purpose. Your understanding of how scale change affects the appearance of the line will help you interpret graphs correctly and avoid being misled by their distortions. The next investigation will help you learn the effects on a graph of changing the maximum settings for the horizontal or vertical axis.
Sample answers:
Hand-drawn graph (left) and calculator produced graph (right) with the same scale settings.
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In each case the value of Ymin = 140. Then Ymax was changed from 200 to 300 and then to 160. The appearance of the graph of Dr. Trotter’s formula changed as indicated below.
The graph appeared flatter when the value of Ymax was increased and became steeper when the value of Ymax was decreased.
Display 1 |
Display 2 |
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The settings for Display 1 were Xmin = 35, Xmax = 60, Ymin = 140, Ymax = 200. For Display 2, Xmax has been changed to 120.
For each Display, the equation of the line is the same and each time the x value changes by 1 unit the y-value will change by 2.38 units. However, in Display 2, the distance on the horizontal axis representing 1 unit is smaller than in Display 1 since more units must fit on the same display screen. This makes the line appear steeper.
The graph of y = 5.78x + 5 is steeper; 5.78 is larger than 3.48. This means that every time the x-value is increased by one unit, the y-value for the first graph will increase by 5.78 units compared to only 3.48 units for the second graph.
The graph of y = 15x + 30. The value for b is 30.
y = (1/2)x + 15 or y = –2x + 5?
The graph of y = –2x + 5.
In this activity you discovered how modifying a member of the y = mx + b family by changing the value of m or b affects its graph. You also discovered that rescaling can change your perception of how steeply a line rises or falls, even though you are graphing the same equation. This understanding will come in handy when you want to select members of the y = mx + b family to describe patterns in data.
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Activity 2—Measuring Up |
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= 5, 6, 7, 8
Your analysis in Activity 1 and Homework 1 has included parts of a process known as mathematical modeling. The process begins when you identify a problem for which you need an answer or a situation that requires further understanding.
For example, during World War II, the armed services sometimes had problems identifying the remains of dead soldiers. Dr. Mildred Trotter was asked to help. She wondered if there were relationships between the height of a person and the lengths of his long bones.
Having posed this question, Dr. Trotter’s next step was to collect relevant data. She needed measurements of people’s heights and the lengths of their long bones. Her model
H = 2.38 F + 61.41
where femur length is in cm
expresses the relationship she observed between the height measurements and femur-length measurements from her data.
Because it depends on data, the model is only as good as the quality of the data on which it is based. Dr. Trotter took special care to check that her data were collected by people who followed detailed instructions for taking the measurements. In this way, she was able to keep to a minimum the data variability that was due to the measurement process.
Dr. Trotter used lengths of long skeletal bones to predict height. You can’t directly measure the bones in your body. Instead, in this activity, you will design methods for collecting data on students’ heights and the lengths of their forearms. Later you will develop a model to predict classmates’ heights using the lengths of their forearms.
Before you collect your data (height and forearm length from each student in your class), you need to establish a method for taking the measurements. Remember, the worth of your model will depend on the quality of the data that you collect. Everyone who will be doing the measuring must use the same method and then record their data to the same degree of precision (for example, to the nearest eighth of an inch or to the nearest millimeter).
Sample answer:
For height: Stack two meter sticks vertically and tape them to the wall. Have the student to be measured stand in front of the meter sticks. The student should take his or her shoes off and stand up straight. Place a cardboard on top of the student’s head. Hold the cardboard so that it is parallel to the floor. Read the spot where the cardboard touches the meter stick. Record height to the nearest millimeter.
For forearm: Have student hold arm flat on a desk with his or her hand placed palm down. Measure along outside of the arm from the point on the elbow to the top of the knobby bone at the wrist. Measure to the nearest millimeter.
Note: Save your data for use in Activities 3 and 6.
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Homework 2—Follow in My Footsteps |
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Sometimes all that’s left at a crime scene is a few footprints. However, the length of a person’s stride is also related to the person’s height. Now you will develop a method for measuring a person’s stride. Later you will gather these data and then use your measurements in a model to predict height.
To collect reliable data, you need to carefully plan the method you will use to collect the data. Remember, your model will be only as good as the data on which it is based.
Design a method for measuring the length of a person’s stride.
Here are some items to consider.
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How will the person walk? Do you plan to measure from heel to heel or heel to toe? Since step lengths for the same person can vary, does it makes sense to have the person take more than one step and average the results? If so, how many steps should the person take? Determine the measurement instrument (e.g., ruler, tape measure, meter stick) you will use to make the measurement. Specify the precision of the measurement. After you have decided on your method, test your method as you did the methods for measuring height and forearm length. When you are satisfied with your method, describe it with a set of written instructions. Give your instructions to a friend to see if someone else understands what you mean. If necessary, revise your instructions. Save them until your class is ready to collect the stride-length data needed later in this unit. |
Sample answer:
A tape measure with metric reading will be used for measuring. Measurements will be recorded to the nearest tenth of a centimeter.
Setup: Mark a line about 15 feet long with adhesive tape. Mark the starting position with another piece of tape.
Have the person put his or her heels at the edge of the starting position and then tell the the person to take four steps along the marked line. Measure from starting point to back of heel after the person has stopped. Divide by four to get the stride length.
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Supplemental Activity 2—Line Up |
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This activity gives you an opportunity to practice determining an equation of a line from its graph.
Figure 4. Graphs of four lines
y = 1 when x = 0. This is the y-intercept, the value of b.
When Dx = 2, Dy = 1.
D
y/Dx = 1/2; this ratio is the same as the slope.When
Dx = 3, Dy = –6.D
y/Dx = –2; this ratio is the same as the slope.D
y = 1 when Dx = 4; slope = Dy/Dx = 1/4; y = (1/4)x.y = (1/4)x + 3.
Lines A and B have the same steepness and they are parallel; they cross the y-axis at different locations. The slopes for the two lines are equal, m = 1/4; the y-intercepts are different, b = 3 for Line A and b = 0 for Line B.
Figure 5. Understory trees in a forest
Species A: approximately 2.6 m; Species B: approximately 1.5 m.
Species A: 3 m; Species B: 6 m.
The two lines in Display 2 are examples of straight-line relationships between two variables. In this case the variables are tree height and crown width. The official name for such relationships is linear relationships, and the equations that describe these relationships are called linear equations.
In the section "Let the Bones Speak," you studied linear relationships between bone length and height. Dr. Trotter’s equations are examples of linear equations relating bone length and height variables.
The line for species A. The line goes through the origin. The approximate value for m is 2/3. We found this value for m by starting at the origin. Then we got to another point on the line by moving up 2 meters and across 3 meters.
m
» 2/7. We started at the point x = 2.5 and y = 1. We moved 1 unit up and 3.5 units across to get to another point on the line. So the slope is approximately 1/3.5 = 2/7 » .29.When x = 0, y
» 0.25. This gives you the value of b.The equation for Species B is y = (2/7)x + (1/4) or approximately y = 0.29x + 0.25.
