104

TRAINING LAB – SHOE PRINTS AS EVIDENCE: SUSPECT HEIGHT

NAME________________________________________ Background: You have just arrived at a crime scene and, as usual, you begin looking for evidence. Your partner reports that there are apparently no witnesses to the crime and you growl under your breath. Without witnesses you have absolutely no description of the suspect. Then you see the clue you have been looking for – a single shoe print left behind on the floor. You now know who you are looking for. This is a man who wears expensive shoes, however, what will make him easy to find is not the fact that he wears the perfect shoe. The man you are after is almost seven feet tall! Continue reading to learn how you made this important discovery of the criminal’s height! 1. You will be trained to estimate the height of a person by simply measuring the length of their shoe. 2. You will be trained to construct a SCATTER PLOT graph and interpret any relationships that are present on this graph. 3. You will be trained to develop an equation for a straight line and use this equation to help you estimate the height of a person from their shoe length. Procedures:

Part 1 – Taking Measurements 1. Measure your shoe length IN INCHES to the nearest O.5 INCH. Record your shoe length in Table 1 below. 2. Measure your height IN INCHES to the nearest 0.5 INCH. Record your height in Table 1 below. 3. Record both your shoe length (IN INCHES) and your height (IN INCHES) on the board OR on the Class Data Form supplied by your supervisor. Part 2 – Graphing Your Measurements And Looking For A Relationship

Between Foot Length And Height

1. Before beginning your graph you must first have access to your class’s shoe and height measurements (see the board or Class Data Form provided by your supervisor). You should also have a list of shoe and height measurements that were collected from other classes (Measurement Data Provided By Your Supervisor). 2. Use a piece of the provided graph paper. This graph paper already has the “X” axis and “Y” axis numbered, however, you need to label them. Label the numbers on the “X” axis as “SHOE LENGTH” (don’t forget to write in the units!!) and the numbers on the “Y” axis as “HEIGHT” (don’t forget to write in the units!!).

Table 1 – Your height and shoe length to the nearest 0.5 inches.

Your Shoe Length

IN INCHES!

Your Height

IN INCHES!

105

3. Now it’s time to start adding the dots. Look at someone’s shoe and height measurements from your class’s data. Find their shoe length on the graph’s “X” axis, then move straight up that line to their height (from the “Y” axis). Stop there and place a dot on the graph. You have just plotted this person’s shoe length versus their height. Move on to the next person in your class and plot their shoe length versus their height. Continue until you have plotted your entire class’s measurements on our graph. If two people end up having the same measurements (their points would be on top of each other) simply draw a circle around the point to show that two points are present. 4. Time to plot even more points! Find the “MEASUREMENT DATA PROVIDED BY YOUR SUPERVISOR” page that includes the measurements of students from other classes. You must also plot the shoe length versus height for each of these people on your graph! It may seem a lot of dots, however, the more data you place on your graph, the more accurate your results! 5. When you are finished you will have dots everywhere on your graph. You have just constructed a SCATTER PLOT. The goal of a scatter plot is not to connect all the dots together in a line – that would be impossible. The goal is to look for some kind of relationship! As you look for a relationship you will have one of three choices: A. Do the scattered dots tend to fall lower and lower as you move from left to right across the graph?? The relationship is that as shoe size INCREASES a person’s height would DECREASE (short people have larger shoes). B. Do the scattered dots tend to rise higher and higher as you move from left to right across the graph?? The relationship is that as shoe size INCREASES a person’s height would also INCREASE (tall people have larger shoes). C. Do the scattered dots tend to be random, not definitely rising or falling as you move from left to right across the graph?? There is no relationship between shoe size and a person’s height (a short person will have large shoes just as often as a tall person). In the space below, write a single sentence that describes the relationship you discovered in your SCATTER PLOT.

106

6. There is one final step that will allow you to use your SCATTER PLOT GRAPH to its full potential. You must draw a single, straight line that best fits through the scattered points and that shows the relationship you described in Step #5. To make a best fit line you should draw your line through the majority of points so that, when finished, about half the remaining points end up above the line and about half the remaining points end up below the line. Show your supervisor where you plan to draw your line BEFORE you actually draw it. Draw your line carefully as it will be an important tool that may help you solve a crime! 7. Once you have your best fit line drawn you can forget about all the points on your graph. Just pretend the points are no longer there and concentrate on your line. This line is very useful!! It represents the average of all the measurements you just plotted on the graph. You can take the length of any shoe and use your line to find the average height of the person who would wear that shoe. For example, I have found an old tennis shoe along the highway and I want to return it to its owner. The shoe is 10 ½ inches long. How tall is the person I should be looking for? Grab your graph and move your finger along the “X” axis to find the 10 ½ inch shoe length. Now move your finger straight up along the 10 ½ inch shoe line. Stop when you come to your best fit line. Now move your finger straight across to the left until you reach the “Y” axis. Carefully read the height that the owner of this shoe should be. Remember, however, this will be the average height for a person with a shoe this size. It may not be exact, but it gives you a good estimate of the person’s height! 8. Get a Question Page from your supervisor and answer questions 1-8 using your graph. Part 3 – Finding And Using A Mathematics Equation That Represents Your

Line

1. In the real world there is one more important step that must be completed when using a scatter plot graph. Moving your finger up from the “X” axis to your line, then over to the “Y” axis is not considered to be the most accurate way to use this graph. The most accurate method for estimating the height of a person from their shoe length is to use MATH! Just like every person has a name, every line on a graph has a mathematics equation as its “name”. It might sound complicated but it’s really no big deal. Let’s first find your line’s “name”. 2. The equation for a straight line is: y = (m)x + b. y = the value on the “Y” axis you are trying to find (the person’s height) m = slope of the line (how steep the line angles up or down) x = the value on the “X” axis you already know (the length of the shoe) b = the y-intercept (this is where your line crosses the “Y” axis when X = 0) Don’t panic! Just follow along step by step with me!

107

3. To find your line’s equation or “name” you must first find its slope (how steep it angles up or down). The equation for slope (m) is: m =

X1, Y1 and X2, Y2 are just two points somewhere/anywhere on your line. Let’s find a couple of points together. You have a shoe that is 10 inches long (X1). How tall is the person that belongs to this shoe (Y1)? Use your graph to find the answer and write your answer in the Y1 space below. X1 = 10 Y1 = __________ Now find a second X and Y value. The shoe is 14 inches long (X2). How tall is the person (Y2)?. Write your answer in the Y2 space below. X2 = 14 Y2 = __________ Use the X1, Y1 and X2, Y2 values you found to calculate your line’s slope. Record your line’s slope in the box below. EXAMPLE: m = m = m = m = 2.5 My slope (m) = 4. To find your line’s equation or “name” you must next find the Y – Intercept (b). To find your line’s Y-Intercept complete this simple calculation. The equation for Y-Intercept (b) is: b = y – (m)x

Just take any X and Y value from your line and put them in the X and Y spaces of the equation. To make it simple just use the X1 and Y1 values from your slope calculations! Place your slope number in the slope space and you’re ready to calculate the Y-intercept! Record your line’s Y-Intercept in the box below. EXAMPLE: b = y – (m)x b = 60 – (2.5)10 b = 60 – 25 b = 35

My Y-Intercept (b) =

Y2 – Y1

X2 – X1

Y2 – Y1 X2 – X1

70 – 60 14 – 10

10 4

108

5. Now you can finally give your line its own equation “name”! The equation for a straight line is, once again, y = (m)x + b . To “name” your line simply replace the “m” in the equation with your line’s slope and the “b” with your line’s Y-intercept. Write out your line’s equation below!

y = ( ___ )x + ____ 6. Also, neatly record your line’s equation on your graph. 7. So what does this all mean??? You can determine the approximate height of anyone by using your line’s equation! Simply find the shoe length of a person, plug this number in your line’s equation where the “X” is, complete the easy math, and without any effort you will end up with the person’s height! See below for an example. EXAMPLE: You are holding a 9 inch shoe in your hands. How tall is the owner of the shoe?

The formula for the line is: y = (2.5)x + 35

Plug in the shoe length for “X”: y = (2.5)9 + 35

Complete the simple math: y = 22.5 + 35 y = 57.5 inches tall

57.5 inches = 4 feet 9.5 inches 8. Answer questions 9-13 on the Question Page using your line’s equation!!

109

7

8

9

10

14

15

11

13

12

55

60

65

70

75

80

16

Est

imate

d H

eig

ht

of

a P

ers

on

Base

d o

n t

he L

en

gth

of

Th

eir

Sh

oe

110

MEASUREMENT DATA PROVIDED BY SUPERVISOR

NumberShoe Length To

Nearest 0.5 Inch

Height to Nearest

0.5 InchNumber

Shoe Length To

Nearest 0.5 Inch

Height to Nearest

0.5 Inch

1 10.5 65.5 26 10.5 72

2 12 72 27 12 72

3 12 69 28 10 67

4 10 68 29 11 71

5 11 68 30 12.5 71.5

6 8.5 62.5 31 10.5 68

7 12.5 70 32 11.5 70

8 10.5 64 33 13 73

9 12 71 34 9.5 65.5

10 10 65 35 11 67

11 11 65 36 11.5 73

12 11.5 67 37 12 72

13 12 71 38 10 64

14 9.5 63.5 39 12 74.5

15 12.5 73.5 40 11 70

16 10.5 66 41 13 72

17 12.5 75 42 10.5 68

18 11 72 43 13 78

19 12.5 75.5 44 11.5 65

20 10 61 45 13 74

21 12 74 46 13.5 76

22 11.5 72 47 13 77.5

23 12 68 48 14 77

24 13 70 49 9 62

25 13 75 50 10.5 69

111

Questions - Shoe Prints As Evidence: Suspect Height NAME_____________________ ANSWER QUESTIONS 1-8 USING YOUR GRAPH. 1. Once again, describe the relationship you discovered in your scatter plot graph. 2. What is the length of your shoe? _________________ 3. Use your graph to determine your height and record your answer here. _______________ 4. What is your actual height?______________ 5. Ask your teacher for their shoe length and record here._______________ 6. Use your graph to determine your teacher’s height and record your answer here. _______ 7. What is your teacher’s actual height?_____________ 8. A shoe print you discover at a crime scene is 9.5 inches long. About how tall of a suspect will you be looking for? Estimated height = ___________ inches OR ___________ feet ___________ inches ANSWER QUESTIONS 9-12 USING YOUR LINE’S EQUATION (DO NOT use your graph)! 9. What is the length of your shoe? _______________________________ 10. Using the length of your shoe and your line’s equation, calculate your height and record your answer here.______________________________ 11. How did your calculated height differ from your graph height in question #3? 12. The shoe print found at the front of the room was left behind at a crime scene. Use your line’s equation to calculate the suspect’s approximate height. Estimated height = ___________ inches OR ___________feet ____________inches 13. You can also use your equation to calculate the shoe length of a person if you know their height. The tallest man to have ever lived was Robert Wadlow (from Alton, IL). He stood 8 feet, 11 inches tall. Use your equation to calculate how long his shoes were. (Place his height in the equation’s Y space and solve for X (shoe length)). Robert Wadlow’s shoes were about ________________ inches long!!