Part 1: Investigating Golden Ratios

1) Look at the different rectangles (Appendix 1). Pick out the most appealing or best-looking rectangle.

2) Why did you pick that particular rectangle? Ask other students why they picked their rectangle. 

In a text titled Discovering Geometry, Michael Serra states that most people prefer a rectangle shaped like rectangle A. Find the ratio of the sides of rectangle A (8/5 = 1.6). This ratio is close to the Golden Ratio of 1.618… The Greeks in their architecture repeatedly used this ratio (e.g., the Parthenon). Other places where the Golden Ratio occurs include spiral seashells, Fibonacci sequences, and ratios of certain bones in the human body. You may want to do an Internet search to find examples of the Golden Ratio or complete the Geometer’s Sketchpad activity entitled Exploring the Golden Rectangle located at:

http://curry.edschool.virginia.edu/teacherlink/math/activities/gs/GoldenRec/home.html 

3) Turn to the Mona Lisa picture (Appendix 2). With a ruler, find the dimensions (in centimeters) of the rectangle surrounding the face of Mona Lisa and then calculate the ratio of the sides. Do human faces inscribe Golden Rectangles?

4) Now take two measurements:

5) Collect finger measurement data from the entire class, and construct, on a sheet of paper, a scatter plot representing the data. The x-axis should represent distances from the tip of the finger to the middle of the second knuckle, while the y-axis should represent distances from the tip of the finger to the base of the finger (see example below).

6) What is the general trend of the data points?  Where are most of the points located?  Explain how a data point describing a tall person and a data point describing a short person would differ in location on the scatter plot.

Part 2: Comparing lines of best fit interactively

7) Point your web browser to http://www.exploremath.com.  Select the Lines/Linear equations category, and then select the Least Squares Fit Line Activity. When the activity loads, it should appear similar to the screen shot below.

8)  Select data points from six different students in the class.  Make sure to pick students throughout the range of heights.  Then drag each of the blue points to a location on the graph corresponding to a selected data point.  The coordinates of the point appear when you grab it.

9) Select the ‘manipulate a line’ box. Drag the green line to the location that visually ‘fits’ the data most appropriately.

10) Using the coordinates on the graph, calculate the equation of the median-median line that best fits the data. How does this equation compare to the equation of the green line in section 9?

11) Click on the ‘show error squares’ box. What is represented by the red squares tangent to the green line?  Interpret the ‘total error’ box. Manipulate the green line in such a way as to minimize the total amount of squared error.

12) Select the ‘compute least squares fit line’ box. Are the green and red lines close to being concurrent?

13) Select the ‘green to red’ box. What happens to the total area of squares when the green line coincides with the red line? Can the total area of squares ever be less than this value?

14) Predict how the total area of squares is affected as a point moves away from the green line; toward the green line. Grab a point and move it to test your predictions. When would the total area of squares be 0? Replace point to original location.

Part 3: Using the line of best fit to analyze the data

15) What are the characteristics of the line of best fit? What is its slope? Is the slope close to the Golden Ratio?

16) How could the equation of the line of best fit be used to determine the length of someone’s index finger if the distance from the tip to his or her second knuckle is 7cm?

17) Are there a wide variety of proportions depicted in the graph, or are the proportions fairly uniform? (i.e., Do most people have a ratio close to 1.6?)

18) How can lines of best fit be used in other applications?