Winplot and 3-Space

1

WinPlot and 3-Space

WinPlot and 3-Space

Using WinPlot Graphics to Enhance Teaching and Learning

of Mathematical Concepts and Ideas in 3-Space

Education 6639

Technology and the Teaching and Learning of Mathematics

Paul Gosse

Faculty of Education

Memorial University of Newfoundland

Submitted by: Jamie Loveless

Date: November 6, 2002

The teaching and learning of mathematics can certainly present numerous challenges for teachers and students alike. Despite what many people may think, not all mathematical concepts and ideas can be explained with a marker and whiteboard, or a piece of paper and pencil. The limitations of these ‘teaching tools’ are especially noticeable once we delve into the realm of 3-Space. It can be very difficult to represent three dimensions on a two-dimensional surface. The challenge of helping students visualize planes in 3-space, for instance, can prove very frustrating for both the teacher and the student. How do we successfully combat these difficulties?

Computer software is available to facilitate the teaching and learning of mathematical concepts and ideas relating to three-dimensional space. WinPlot is one such program, with features allowing for the construction of three-dimensional graphs. The options made available through this software serve not only as a tool to improve the presentation of documents, but can be very valuable as an interactive piece of software in facilitating students’ understanding of three-dimensional mathematics.

The first unit in mathematics 2204/05, a second level course in the new senior high mathematics curriculum in the Atlantic Provinces, is entitled Investigating Equations in 3-Space. When I taught this course for the first time, I discovered it to be challenging task to help students visualize points, lines, and planes in three-space. As a class, we piled blocks on a two dimensional grid and I encouraged students to think of the meeting of the wall and the floor in the corner of the classroom as the x,y and z axes. Probably the greatest challenge was in attempting to have students graph and visualize planes and the intersection of planes in three dimensions. In teaching for understanding it is crucial that students be able to see what is happening in order to fully comprehend the mathematics being studied. WinPlot is definitely a piece of software I will use the next time I teach the math 2204/05 course to help students grasp these novel concepts and ideas.

The following interactive activity using WinPlot and subsequent worksheet will hopefully provide a good introduction for students as they expand their mathematical knowledge into the realm of three-dimensional mathematics. Solutions have been provided for the worksheet.

Activity/Tutorial Discovering the Amazing World of

3-Space in Mathematics

How can we plot points in 3-space? How can we visualize the graphs of equations such as ? Or ? What happens when the graphs of planes intersect each other?

Questions such as those given above, among others, will be addressed in the following tutorial. The tutorial requires you to interact with WinPlot, a program of Peanut Software. Remember to follow all instructions carefully to ensure successful completion of this activity.

Part I: Three Axes Instead of Two!

Setting Up 3-Space In WinPlot and Examining Some of its Features

Before we actually begin the investigation of 3-space, we need to choose

WinPlot from our Desktop and choose 3-dim F3 from the Window menu.

We obtain a screen similar to the one given below:

We see that to represent points in three dimensions, we use a coordinate grid with three axes at right angles to each other. Two of these axes, the x and y axes are on the same horizontal plane and are perpendicular to each other. The other axis, the z-axis, is vertical to that plane. Normally, in 3-space, the x-axis and the y-axis represent values of independent variables and the z-axis represents values of a dependent variable.

If your axes are not already labeled, as shown in the previous diagram, you can complete this task by clicking on Misc, Axes, and Show labels.

You may also want to change the background colour and/or the colour of your axes. You can investigate these features under Misc, and Colors.

Before we plot any points in 3-space we will define the scale on our x, y, and z axes by clicking on View, and Lock lengths…:

We see a screen where you can input the desired lengths of your axes. Change your values to match those provided in the screen below and click lock when complete:

Part II: Plotting Points in 3-Space and Rotating/Changing

the View of the Graph

Points can be easily plotted in 3-space using WinPlot. Let’s plot the point (1,2,3). In other words, x = 1, y = 2, and z = 3. To plot this point we click on Equa, and Point…, as shown:

Then, we enter the point (1,2,3). We will not include the anchors at this stage and we will also change the colour of the point:

When we click ok, we see the following:

NOTE: If the “Inventory” screen does not appear, you can access it by clicking on Equa and Inventory (or by using Ctrl-I)

As you proceed through this tutorial, you will need to remain cognizant of several important options available to you in relation to the manner in which you view the graph. You can rotate, or change the view of your graph using options as illustrated below:

Please note the short-cut keys, F8, F9, and F10, and the associated opposite actions Ctrl-F8, Ctrl-F9, and Ctrl-F10. You can briefly and CAREFULLY experiment with these options. You can also zoom in and out using Ctrl-S and Ctrl-E, respectively. If you happen to “lose yourself in 3-space”, you can return to the original, familiar window by clicking View, Observer, and Coordinates, as shown:

The coordinates you would need to enter are:

Activity: Change the positioning of your graph (using the keys given above) so that it is similar to the picture provided below:

Remember: To label your point or graph, use Ctrl-I, and click on equa in the hide/show box.

Save your graph at this point before continuing any further in the tutorial.

Activity: Using the graph that you have already created, plot the following points with different colours and label them.

(i). (-1, 0, 5) (ii). (4, -1, 0) (iii). (0, 6, -2)

(iv). (1, 7, 6) (v). (0, 0, 0)

Your resulting graph should look similar to the following:

Now, rotate your graph (use F9 and Ctrl-F9) to get a better idea of where these points actually exist in 3-space. Before we do this, however, we will hide the labeling of our points. The easiest way to do this is to click Equa, Hide/show all, and Equations.

Part III: Constructing Planes in 3-Space

A plane is a flat two-dimensional surface in three-dimensional space.

In Part III of this tutorial we will construct planes in 3-space in a couple of different ways. Before we begin, however, it is necessary to choose Equa, and Blank slate.

You should double-check to make sure your axes are again locked at the previously specified values using View and Lock lengths.

First of all, we will illustrate the plane y = 3 using a series of points, all having the same y-value. Plot the following points, using anchors this time (Don’t forget that once the first point is plotted you can speed up the process of entering the remaining points using the dupl button in the Inventory (Ctrl-I) menu!):

(-1, 3, 0) (-1, 3, 1) (-1, 3, 2) (-1, 3, 3) (-1, 3, 4)

(1, 3, 0) (2, 3, 1) (1, 3, 2) (2, 3, 3) (1, 3, 4)

(3, 3, 0) (4, 3, 1) (3, 3, 2) (4, 3, 3) (3, 3, 4)

(5, 3, 0) (6, 3, 1) (5, 3, 2) (6, 3, 3) (5, 3, 4)

(7, 3, 0) (7, 3, 1) (7, 3, 2) (7, 3, 3) (7, 3, 4)

(-1, 3, 5) (-1, 3, 6)

(2, 3, 5) (1, 3, 6)

(4, 3, 5) (3, 3, 6)

(6, 3, 5) (5, 3, 6)

(7, 3, 5) (7, 3, 6)

Your graph should now resemble the following:

Another view (from the top):

You are again encouraged to rotate the image to examine the plane from various perspectives in 3-space…remember the special keys!

Before moving on to the second method to graph planes, make sure you have a blank slate, as before.

A second method to graph planes involves the use of the z = f(x,y)… feature under Equa (or simply press F1). The equations entered here must be of the form , where a, b, and c are real numbers.

First, let’s create a graph for the equation . To do this we click Equa,

z = f(x, y)…(or F1), and obtain the following screen, where we enter our equation, change the settings for x and y, and change the colour:

Our resulting graph looks like the following:

From a slightly different view:

Next, we graph the plane represented by the equation

First of all, however, we need to rearrange the equation as follows:

When we graph this equation, with some adjustments to the view, we obtain the following:

And from a different perspective:

Part IV: Intersecting Planes

The final section of this tutorial involves an examination of possible results of two planes intersecting in 3-space and three planes intersecting in 3-space. Any ideas before we begin?

We will change the lengths of our axes before commencing this section:

Intersection of Two Planes:

Using Equa, z = f(x, y)…(or F1), graph the following two equations as indicated in the given screens:

And,

You should obtain a screen similar to the following:

From a different perspective:

From these illustrations, we discover that the two planes intersect in a straight line. The line where one plane intersects another is called a trace.

Intersection of Three Planes:

Using Equa, z = f(x, y)…(or F1), graph the following equation, in addition to the two previous equations, as indicated in the following screen:

You should now have a graph similar to the following (…the background colour has been changed!):

With some minor modifications, we obtain the following:

Thus, we see from the diagram above that the graphs of the three equations intersect at a point. In other words, these three equations have a single solution. Such a system of equations is said to be consistent and independent.

Part V: A Phone Plan

Problem:

Jane’s phone company charges 15¢/min for calls within Canada and 25¢/min for calls to the U.S.. She is also charged a $10/month base rate.

(A). If f is the total monthly fee, c represents the time in minutes for calls within Canada, and u represents the time in minutes for calls to the U.S., then determine an equation giving the total charges per month.

(B). Graph the equation found in (A).

Solution:

(A). Using the given variables, the equation would be:

(B). We will use WinPlot to generate a graph for this equation.

We will change the variables on the axes to correspond with the variables in the problem. To do that we click Misc, Axes, and Labels:

Then, change the labels as follows:

You can now proceed to produce a graph similar to the one provided. You will be required to use various features of WinPlot including Box visible (Ctrl-B) and Box coords visible (Ctrl-C) under View. To insert text, click on Btns and Text (right-click to insert text).

Remember, as well, that when we enter our equation we must use the variables x, y, and z. Thus, we use f = z, u = x, and c = y.

The graph of the equation in 3-Space is:

There are many other features of WinPlot you can use when graphing in three dimensions. Feel free to experiment with these other options in your continued study of 3-Space in mathematics.

Journal Entry

After working through this tutorial, outline the various strengths and weaknesses you may have noticed in using WinPlot to graph in three dimensions.

How has your understanding of 3-Space increased as a result of completing this tutorial? Explain.

Worksheet Investigating Equations and

Solving Problems in 3-Space

1. Plot the following points in 3-Space and label them.

(A). P(1, 2, -4) (B). Q(-1, 3, 2) (C). R(2, -1, -3)

2. Graph the following plane:

3. Joe is privileged to have a couple of jobs. He earns $ 6.50 an hour working at Moe’s garage and $ 7.00 an hour working at a call center.

(A). Write an equation to describe the relationship between Joe’s total earnings (e), and the number of hours working at Moe’s garage (m) and the call center (c).

(B). Sketch the plane of the equation and describe it.

(C). Why is the intercept of the vertical axis zero?

(D). When would Joe earn $150?

Worksheet{Solutions} Investigating Equations and

Solving Problems in 3-Space

1. The points have been plotted using WinPlot:

2. First of all, in order to graph, we need to solve the given equation for z so that we can input the equation in WinPlot:

When we construct the graph, we obtain a picture similar to the following:

(The x, y, and z intercepts are also highlighted in the graph}

3. (A). Using the given variables, an equation to describe the relationship between

Joe’s total earnings and the number of hours worked at each job is as follows:

(B). When we sketch the plane we let e = z, m = x, and c = y. The graph of the equation should resemble the following:

The graph above has only one intercept, at the origin (0,0,0), and rises in both the positive m and positive c directions.