Olive & Oppong: Transforming Mathematics with GSP 4, page 103
Chapter 9: Data Representations and Mathematical Modeling
The Box Problem: Using GSP to simulate a situation and plot data
You have been given the following problem to solve for a box company: Form a tray (open topped box) from a given square sheet of cardboard that will hold the maximum volume of cat litter. Assume that the tray is formed by cutting squares out of each corner of the square sheet of cardboard and folding up the sides as in Figure 9.1.
Figure 9.1: Making a Tray from a Square Sheet of Cardboard.
Activity 9.1: Making trays from construction paper.
Starting with a square sheet of construction paper (approximately 20 centimeters on each side), cut and fold the edges to make what you think will be the tray with the largest volume (tape the corners to maintain the tray shape). Compare your tray with the other trays in the class. Measure the dimensions of your tray and calculate the volume. Compare your results with others in the class. Which trays appear to have the largest volume?
Activity 9.2: Construct the diagram in Figure 9.1 in GSP in such a way that you can vary the amount cut from each corner by moving a free control point on one side of your square. As this point should not go past the midpoint of a side of your square (why?), you might want to actually place the control point on a segment joining a corner to the midpoint of one side. Calculate the volume of the resulting tray using the GSP calculator and measurements of appropriate segments in Figure 9.1. Also measure the side-length of the removed squares. Plot this side-length against the calculated volume using the Plot as (x, y) option under the Graph menu. Trace the plotted point and move your control point to vary the size of the cut-out squares. Use this plot to find the tray with maximum volume. If you have constructed your simulation appropriately, you should be able to construct the locus of your plotted point as your control point varies (select your plotted point and select your control point then choose Locus from the Construct menu). Figure 9.2 illustrates a possible solution.
Figure 9.2: Plotting Volume Against Length of Side of Cut-out Square.
Vary the size of your square. How does this change the solution to your problem? Could you simulate the problem given a non-square rectangular piece of cardboard to make your tray? Is there an optimal dimension for your rectangle that would give you the greatest volume for the least amount of cardboard (area of original rectangle)?
With GSP 4 it is possible to plot functions directly on the given coordinate system. Using the measures from your diagram, create a New Function f(x) (under the Graph menu) that gives the volume of your box where x is the length of the side of your cut-out square (length CD in Figure 9.2). Plot your function f(x). Does it coincide with the locus of your plotted point? Why is your locus only part of the function graph and not all of it? What kind of function should give you the volume based on side-length? One possible solution to the problem using GSP 4 is shown in Figure 9.3
Figure 9.3: Biggest Box Problem Showing Function Plot and Locus in GSP 4
The View Tube Experiment
The purpose of this experiment is to find mathematical relations among the variables involved when viewing a portion of a vertical wall through cardboard tubes of various lengths and diameters, at various distances from the wall.
Activity 9.3: You will need to work in pairs or small groups for this activity. Gather a variety of cardboard tubes such as the tubes found in toilet paper rolls, paper towels, and rolls of wrapping paper. You will also need masking tape, measuring tape and meter sticks. You will conduct three experiments as follows:
Experiment A: Vary the distance from the wall
1. Find a vertical wall that you can see clearly from 1 to 20 meters away (no obstructions between you and the wall). Tape a vertical strip of paper or masking tape to the wall, starting at the floor and going as high as possible.
2. Select a cardboard tube and measure (and record) its length and internal diameter .
3. Stand with your toes at a mark one meter from the base of the wall (measured on a line perpendicular to the base of the wall).
4. Place one end of the tube up to your eye and look straight ahead at the tape on the wall through the tube.
5. Your partner will move a pencil slowly up the tape from the base of the wall until you just see the pencil though the tube. Mark this point as the lower extremity of your viewing area.
6. Your partner continues moving the pencil up the wall until it disappears from your view through the tube. Mark this point as the upper limit of your viewing area. It is very important NOT to move the tube at all during this measurement process!
7. Measure and record the distance between your two marks.
8. Move one more meter away from the wall and repeat steps 4 through 8. Stop when you have reached 20 meters from the wall or the lower limit is at the base of the wall.
9. Decide which is your independent variable and which is the dependent variable in this experiment and graph the results.
10. What kind of relation exists between the distance from the wall and the viewable portion of the wall?
Experiment B: Vary the length of the tube
1. Find a number of tubes with the same diameter but different lengths or make a telescoping tube out of two tubes, one sliding inside the other. Record the diameter of your tubes.
2. Stand two meters away from the tape on the wall (see experiment A).
3. Measure and record the length of your tube.
4. Measure and record the viewable portion of the wall through this tube as in Experiment A, steps 4-7.
5. Stay at the two-meter mark but change the length of your tube and repeat steps 3 and 4 for 5 different lengths of the same diameter tube.
6. Decide which is your independent variable and which is the dependent variable in this experiment and graph the results.
7. What kind of relation exists between the length of the tube and the viewable portion of the wall?
Experiment C: Vary the Diameter of the tube
- Find a number of tubes of the same length but different diameters. Record the fixed length of your tubes.
- Stand two meters away from the tape on the wall (as in Experiment B).
- Measure and record the diameter of your first tube.
- Measure and record the viewable portion of the wall through this tube as in Experiment A, steps 4-7.
- Stay at the two-meter mark but use a tube with a different diameter but same length and repeat steps 3 and 4 for 5 different diameters of the same length tube.
- Decide which is your independent variable and which is the dependent variable in this experiment and graph the results.
- What kind of relation exists between the diameter of the tube and the viewable portion of the wall?
Assignment 9.4: Using GSP to simulate the three experiments
Construct a working GSP sketch that represents the variables in the View-Tube experiments: Length of tube, diameter of tube, distance of tube from the wall, height of viewable portion of the wall. Use height of viewable portion of the wall as your DEPENDANT variable and plot this against each of the other variables. For example, using the construction and measurements in figure 9.4 (taken from the View Tube.gsp sketch provided on the CD), in order to plot height of viewable portion of wall (HI) against length of tube, you would do the following:
Figure 9.4: Simulation of the View Tube experiment using GSP.
1. Select the measurements Length and HI IN THAT ORDER and then select Plot as (x, y) from the GRAPH menu. A new point should appear in your sketch, along with a coordinate system.
2. With this new point still selected, select point D that controls the length of the tube and then select Locus from the CONSTRUCT menu. The locus of the path of your plotted point should appear (see figure 9.5 below).
Derive functions for each of these relations. Copy your construction onto three pages in your GSP document (or use the three pages in the document provided in View Tube.gsp) and plot one function on each page, using the data generated by your sketch. Check that your functions match your data plots. For example, using the sketch in figure 9.5 you would double click on the f(x)=0 function and edit it to create a function plot that matched the locus of your plotted data point. You would need to use the measurements in the sketch (by clicking on them) as parameters in your function definition. Do NOT type in any numerical values. Your function plot should remain coincident with your locus when you change the dimensions of the tube or its distance from the wall.
Figure 9.5: Plotted point and locus for Length of tube versus HI.
Reflection: Reflect back on all the different modeling activities that you explored in this section of the chapter (both with physical materials and with GSP). Write a brief summary of the ways in which modeling simulations can be used to build connections between geometry and algebra.