Area and Perimeter: a Study in Limits

Bees and Fathom Area and Perimeter: A Study in Limits

How are areas and perimeters of rectangles related? We all know the formula- area = length * width, perimeter = 2(length + width) – but if you look at them from a data perspective, what do you get? In this activity, we’ll use Fathom to generate the measurements for rectangles randomly and explore the relationships. You can also generate rectangles haphazardly using any other technique you like (asking people to make them, using The Geometer’s Sketchpad, etc.) and use those data. 1. Open a new Fathom document and create a new case table by selecting Case Table from the Insert menu. 2. Make two attributes – length and width by clicking on new and type length and press enter. After you press enter a new column will be formed; click on the word new again and type width. 3. With the case table selected, chose New Cases… from the Data menu. Tell Fathom make 200 cases. They appear, though their attributes have no values.

Making Random Rectangles We won’t actually make the rectangles here; we’ll just make the measurements. 4. Select length, and choose Edit Formula from the Edit menu to bring up the formula editor. 5. In that box, enter the formula random(10). Close the box by clicking OK. Now length is filled with random numbers ranging from 0 to 10. 6. Use that same formula to define values for width. Now you have random numbers for both attributes. 7. Make a scatter plot of these two attributes by selecting Graph in the Insert menu and drag the length on the vertical axis and the width on the horizontal axis; it ought to look random.

Calculating Area and Perimeter Collection 1 Scatter Plot 10 If you got your rectangle honestly (for example, by cutting them out of 8 h paper), join us here. Enter your data in the table and proceed. t 6 d i

8. Make two new attributes, area and perimeter (remember click on w 4 new in the case table and type area). Make formulas (in the Edit 2 menu under Edit Formula) to determine these; the formula 0 should depend only on length and width. Copy the first two or 0 1 2 3 4 5 6 7 8 9 10 three cases from your table here: length

9. Make a scatter plot with perimeter on the horizontal axes and area on the vertical axis. Sketch the graph at right; label and scale the axes.

10. If everything went well, there should be no points in the upper-left part of the graph- a region bounded by a curve. Explain why there are no points in that region.

Data in Depth revised 2001 Key Curriculum Press The Boundary

Now your goal is to figure out the equation for that curved upper boundary of the cloud of points. There are two strategies: The theoretical one and the empirical one. In the theoretical strategy, you figure if out from the equations and just plot it. These directions will concentrate on the empirical.

The boundary curves upward. It might be exponential, or it might be some polynomial (such as a quadratic). Since finding areas can involve squaring, we’ll try a quadratic first, area = perimeter2.

11. Select the graph and choose Plot Function from the Graph menu. The formula box

appears. Collection 1 Scatter Plot 100 12. Enter perimeter^2 into the box and close it. The curve will 90 appear, as in the illustration. (Remember that you don’t 80 70 60 enter the left-hand side of a function in Fathom.) a e

r 50 Unfortunately, the curve is too steep. a 40 30 You can alter the formula on your own , but the following is fun… 20 10 13. We’ll make a slider. Choose Slider from the Insert menu. 0 A slider appears. Change the name V1 to denom. 0 5 10 15 20 25 30 35 40 perimeter 14. Edit the plotted function (double-click the equation at the bottomarea of = perimeter the graph). Change it to perimeter2/denom. Now you can control the value of denom by dragging the slider’s thumb. As you do so, the function changes. You will have to change the scale of the slider. Do it the way you change the axis-by dragging in the numbers with the hand, or CTRL-SHIFT-clicking to zoom out. 15. Change the value of denom unit the curve accurately forms the boundary. What value of denom did you get? (It should be greater than 10.)

16. Study the points near the boundary. What is the shape of the rectangles near the boundary curve? That is, what do they have in common?

17. Explain why your answer to the preceding question makes sense. That is, why do all those rectangles have that property?

Going Further

1. Show, algebraically, what the curve’s equation must be, and show that it’s the same as the one you got by looking at the data. 2. Depending on how you made the rectangles, there may be another empty zone in the lower-right corner of the scatter plot. Why are there no rectangles there? What is the equation of that border? Why?

Data in Depth revised 2001 Key Curriculum Press