## How Tall?

• Lesson
6-8,9-12
1

We often hear that there are measurements in the body that can be used to predict a person’s height. By graphing different body measurements versus height and comparing their correlation coefficient, students decide which body measurement is the best predictor.

Students will probably have heard about some ratios that are said to exist in the body. For examples, it is often said that if you open your arms wide and measure the length from the fingertips of one hand to the fingertips of the other and compare this to your height, the two measurements are equal. Another example is the claim that if you measure your height and compare it to the distance from the floor to your navel, then the resulting ratio is approximately 1.618:1, which is also known as the golden ratio.

If students measure these ratios on their own bodies, they may notice that the ratios are not exactly what the students predict, but they are probably pretty close. One question that will naturally arise from this is, Which body ratio will be closest to being the same for every person? This activity allows students to find the answer. All of the ratios they will explore compare body measurements to height. This will allow students to discover which body measurement is the best predictor of height.

### Data Collection

First, you need to decide how many body measurements each student will record. The Class Reporting Activity Sheet identifies 10 body measurements, and allows your class to choose up to 4 additional body measurements to work with. Decide as a class what other measurements might be good predictors.

Students should work in pairs to take each other's measurements. Before students begin working, assign each student a row number on the activity sheet in which to record their personal data. If you have more than 20 students, copy the activity sheet and adjust the numbering on the second page.

Create an overhead using the Class Reporting Activity Sheet for students to report their individual data. Once the overhead is filled in, each student or pair should copy all the data onto their own activity sheet.

### Analyzing the Data

Allow students to continue working in the same pairs. Assign each pair one body measurement to investigate. Distribute the How Tall? Activity Sheet to all students. Assign each pair a body measurement to compare to height. If you have fewer body measurements than student pairs, assign a particular measurement to two or more pairs, and have those pairs compare their results before reporting to the class. If you have more body measurements than students, assign more than one measurement to each student pair.

To calculate the correlation coefficient, r, and linear equation, y = mx + b, students can use the Line of Best Fit applet or a graphing calculator, depending on the technology available in your classroom.

As students begin calculating r and y = mx + b, ask them which measurement should be the independent (x) variable, and which should be the dependent (y) variable. Biologically speaking, your height does not determine the length of your foot, and the length of your foot does not determine your height—they are both controlled by your genetic makeup. There may be a correlation between these values, but one does not determine the other. However, in this activity, students are predicting height. The goal is to determine height when you already know another measurement, making height dependent on this other measurement. The other measurement (e.g., length of your foot) is therefore the independent (x) variable.

Discuss the meaning of r, the correlation coefficient, with students. It measures how close data are to forming a perfect line. The closer the value of r is to 1 or –1, the closer the data are to forming a line. If r is 0, the data have no correlation, meaning that the variables are not related to each other and there is no single line that can be drawn through all the data points. The graph that produces the r-value closest to 1 represents the best predictor of height for your class.

Student groups should report the r-values and linear equations for their measurement(s) on the How Tall? Overhead. Student can then copy the data onto their How Tall? Activity Sheets. If you wish to collect these activity sheets later, make sure students mark which measurement(s) they were responsible for.

Give students some time to discuss Question 1 on the How Tall? Activity Sheet, and then discuss the question as a class. The measurement that produced the r value closest to 1 is the answer because that measurement and height have the highest correlation. You may also ask these questions during the discussion:

• What happens if there are 2 r-values that are similar?
[If 2 r-values are exactly equal, your class could conclude that they are equally good at predicting height. An interesting follow-up question in this situation is, What is the correlation coefficient between these 2 body measurements?]
• If there are 2 r-values that are approximately equal, how much of a difference would be needed in order to determine which of the 2 measurements is actually a better predictor? A thousandth? A hundredth? A tenth?
[This is a very difficult question to answer in an absolute sense. You may wish to encourage a discussion about how accurate students believe their measurements were. Could they have been off by a centimeter? a millimeter? It would be reasonable for students to believe that their measurements are accurate to the nearest centimeter. Looking at their r-values, then, they could conclude that values similar to the nearest hundredth could be considered as being equal.]

To test your class's results, invite students (of similar age) in from another class and have your students measure them on the specific body measurement that was the best predictor of height. Have your students then use the linear equation to predict the heights of the visitors. Your students can then measure the heights of the visitors to see how accurate their height predictor is. Have students record their values and answer Questions 2 and 3 on the How Tall? Activity Sheet. If it is difficult to bring in other students, have students complete this part of the assignment with a friend or family member at home.

If the results of measuring another student are not what was expected, take time to discuss why this might be. Some possibilities are:

• Humans grow in a non-uniform way—different body parts grow at different rates. This is called allometric growth.
• The height predictor is making a prediction based on a population of a classroom, not the characteristics of a single individual.
• Gender differences were not taken into account in this activity. There may be a different body measurement that is a better predictor for one gender than the other.

Assessment Options

1. Use the Line of Best Fit Rubric to evaluate students’ thinking and progress through this activity.
Line of Best Fit Rubric
2. Have students write a journal entry that includes their personal reactions to the activity and the questions discussed.

Extensions

1. Have students explore the correlations between different body measurements, not just height. For example, students could explore whether a relationship exists between the length of their pinkie fingers and their shoulder width. Is there a relationship between any 2 measurements that is a better correlation than the one they found in the activity?
2. Students could combine their height predictor calculations with those of another class who are in a different age group. Does this affect the results? Why would this be? This works well if you use this activity in more than one class or collaborate with another teacher.
3. The data could be grouped by gender, plotting data for males and females on different graphs. Does this affect what body measurement is the best predictor for height? Do the equations of the lines change?

Questions for Students

1. What is the meaning of r?

[The r-value is called the correlation coefficient. It measures how close data are to forming a perfect line.]

2. How do you know whether 2 measurements are related to each other?

[The closer r is to 1 or –1, the closer the data are to forming a line, and the more one variable is related to another. If r is 0, the data have no correlation, meaning that the values plotted on the graph are not related to each other.]

3. If you were to graph hair length vs. height and found that r = 0.3, what would this tell you?

[There is very little relationship between a person's hair length and their height.]

4. Do ratios exist in the body that can be used to accurately predict a body measurement?

[The answer is the relationship that gives the r-value closest to 1 for your group of students. If there were no relationships that gave an r-value close to 1, then students could conclude that ratios in body measurements cannot be used to predict specific body measurement lengths.]

Teacher Reflection
• Were students engaged by the idea that ratios exist in the body?
• How did your lesson address auditory, tactile and visual learning styles?
• This lesson has many transitions embedded in its structure. What worked with classroom behavior management? What didn't work? How would you change what didn’t work?

### Learning Objectives

Students will:
• Gather and graph data.
• Determine correlation coefficients.
• Use correlation coefficients to determine which data set is closest to being linear, and therefore is the best predictor.

### NCTM Standards and Expectations

• Select and apply techniques and tools to accurately find length, area, volume, and angle measures to appropriate levels of precision.
• Make conjectures about possible relationships between two characteristics of a sample on the basis of scatterplots of the data and approximate lines of fit.
• For bivariate measurement data, be able to display a scatterplot, describe its shape, and determine regression coefficients, regression equations, and correlation coefficients using technological tools.

### Common Core State Standards – Mathematics

• CCSS.Math.Content.8.SP.A.1
Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

• CCSS.Math.Content.8.SP.A.2
Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

• CCSS.Math.Content.8.SP.A.3
Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.

• CCSS.Math.Content.6.SP.A.1
Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, ''How old am I?'' is not a statistical question, but ''How old are the students in my school?'' is a statistical question because one anticipates variability in students' ages.

• CCSS.Math.Content.6.SP.A.2
Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

### Common Core State Standards – Practice

• CCSS.Math.Practice.MP1
Make sense of problems and persevere in solving them.
• CCSS.Math.Practice.MP4
Model with mathematics.
• CCSS.Math.Practice.MP5
Use appropriate tools strategically.
• CCSS.Math.Practice.MP7
Look for and make use of structure.