3-D Graphing, Contour Graphs, Topographical Maps, and Matrices Using Spreadsheets

3-D Graphing, Contour Graphs, Topographical Maps, and Matrices Using SpreadsheetsAuthor(s): Louis FeichtSource: The Mathematics Teacher, Vol. 92, No. 2 (FEBRUARY 1999), pp. 166-174Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27970889 .

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Graphing, Contour

Graphs, Topographical Maps, and Matrices

Using Spreadsheets Louis Feicht

Photograph by Meg Feicht and Louis Feicht; all rights reserved

odern spreadsheets can make powerful mathemati cal concepts accessible to students at a younger age than ever before. Contours and three-dimensional

graphing are topics that were previously reserved until well into the first year of college calculus. Three-dimensional graphing now can be success

fully taught to middle school students with the assistance of a computer spreadsheet. This combi nation of the computer with hands-on activity exposes students to numerical and graphical repre sentations of data on the same spreadsheet "page" and forces them to make connections between the two forms of data.

This lesson will probably take several days. The lesson first emphasizes a conceptual understanding of three-dimensional graphing while the teacher

guides students through a discovery lesson using examples on the computer spreadsheet. Next, work

ing in teams at the computers, students try a hands-on activity with the computer spreadsheet. Finally, using topographical maps, students com

plete a hands-on small-group computer activity that

applies the concepts to a real-world setting. For the final project, students create a surface plot from a

topographical map. I was introduced to using spreadsheets for three-dimensional plots by the

Systemic Initiative for Montana Mathematics and Science project; more ideas for three-dimensional

graphing lessons can be found in the SIMMS Proj ect Integrated Mathematics textbook series (Fong 1996). I have done variations of this lesson with students from middle school through precalculus.

MAKING THE THREE-DIMENSIONAL SURFACE CHART Students are first introduced to a matrix of num bers representing heights above a two-dimensional

plane by looking at teacher-generated examples on the spreadsheet. Students understand the concepts better when the teacher begins the activities by presenting spreadsheet examples to the class on some form of projection device, such as an overhead

projector and LCD projection panel, connected to the computer. I like to start with an easy example, such as a triangular pyramid (see fig. 1). The trian

gular base of the pyramid is intentionally not equi lateral to give students some frame of reference between the matrix and the chart. The obvious tri

angular shape of the numbers corresponds to con tour lines on the chart.

To create a chart example similar to the pyramid for use with the whole class, enter a matrix of data into the spreadsheet; to do so, type the numbers into the spreadsheet cells. Next create a three dimensional-surface plot of the matrix using the built-in charting features of the spreadsheet (see fig. 1). In Microsoft Excel (1995), select the option for data series in columns instead of rows for a more accurate correspondence between the numeri cal data and the graph. If you use the default value of data series in rows, the graph will still look the same but the right side of the graph will be created from numbers on the left side of the matrix; the pic ture will be reversed. The original graph then can be rotated to correspond visually with the data (see fig. 2). My surface plot is at an "angle" because I

prefer it that way (see fig. 2a); some teachers or students may want to rotate it so that the tip of the

triangle points toward the top of the computer screen, as in the matrix (see fig. 2b). Most graph ing software also has other controls available for

setting the viewpoint and rotation of axes.

Louis Feicht, lfeicht%capnet.kl2.or.us, is a technology specialist for Silver Falls School District, Silverton, OR 97381. He is interested in how technology affects commu nication, learning, and teaching in the classroom.

166 THE MATHEMATICS TEACHER



012345678 0000000000 ?000000000 2 3 4 5 6 7

O O O O

0 0 0 0 00000000

O 1 2 3 4 s 6 78

2.5-3

2-2.5

1.5-2

1-1.5

0.5-1

0-0.5

Fig. 1 The chart's appearance when it is first generated using Microsoft Excel. The data are entered into the

spreadsheet, and a three-dimensional-surface chart is then created. When creating the chart in Microsoft Excel, select the option that enters your data series in columns rather than in rows. The pyramid is not at exactly the

same orientation as the triangular-shaped numerical data.

Contours

and three

dimensional

graphing were topics for college calculus

1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8

0 0 0 0

0 0

0 0

0 0 0 0 0 1 0 1 2 1 2 2 1 1 1

0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0

0 0 0 0

To rotate the chart, double-click on it to

get into chart-editing mode. Then click on

any one of the corners of the chart until the black selection handles appear. Drag any one of the black handles until the chart is in the orientation you want.

(a)

1 2 3 4 5 7 8 0 1 2 3 4 5 6 7 8 0

0 0 0 0

0 0 0 0 0 0 0 1 0 0 12

0 0 0 0

0 0 0

0 0

This chart has been rotated so that the

long side of the triangle is horizontal toward the bottom of the screen.

0 0 0 0 0 0 0 0

0 1

2.5-3

2-2.5

1.5-2

1-1.5

0.5-1

0-0.5

(b) Fig. 2

The chart after it has been rotated to orientations that more closely resemble the numerical representation. Figure 2(a) gives a more three-dimensional look, but figure 2(b) shows more of the connection between

the matrix and the chart.

Vol. 92, No. 2 ? February 1999 167



Students can see

connections, as well as

cause and

effect

Since I have only four numbers in the matrix?0, 1, 2, 3?and three intervals, I want only three intervals in my chart legend and only three colors on the chart. I therefore change the intervals between tick marks on the z-axis. See figure 3 for directions. The cells in the matrix are then colored to match the colors of the chart (see fig. 4). This

simple yet powerful technique of using correspond ing colors for the graph and matrix soon makes it obvious to most students how the graph is connect ed to the numbers in the matrix.

The chart and matrix in figure 4 are now ready for the teacher to use for a classroom presentation. The spreadsheet provides an environment in which students can form mental links between the differ ent representations of the same data. A constant flow of students' attention between representations becomes possible. The teacher can change numbers in the spreadsheet, and the spreadsheet program will make the corresponding changes in the graph as soon as the data are entered. See figures 5 and

6. Students can see connections, as well as cause and effect. In figure 6, two 0's were changed to 4's in the upper-right-hand corner. Notice that the col ors do not change automatically in the matrix, whereas they do change in the graph; the color coordination between the matrix and the graph must be performed manually. Making changes in the numbers of the matrix is an excellent way for teachers to help those students who cannot easily see the ties between the two data forms.

If you are not happy with the default colors of the

chart, you can easily change them in Excel by format

ting the individual legend keys. To format individual

legend keys, double-click on the chart to get into

chart-editing mode and then select the individual

legend keys while holding down the command key.

MAKING THE CONTOUR CHART After students are comfortable with how the num bers correspond to heights above the ry-plane, a contour map can easily be created by copying, past

012345678 0 0 0 0 0 0 1 0 0 0 0 0 2 0 0 0 0 1 3 0 0 0 1 2 4 0 0 1 2 3 5 0 1 2 2 2 6 11111 7 0 0 0 0 0 8 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 1 0 1 1 0 0 0 0

To change the tick marks on the z-axis, double-click on the chart to get into chart

editing mode. Then double-click on the z-axis. The Format Axis window should

appear. Select the tab labeled Scale, and set the Major Unit to 1.

Fig. 3 The chart now has only three colors, which correspond to the three height intervals of our matrix.

012345678 0000000000 7 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0

Fig. 4 The colors of the matrix match the colors of the chart.




By changing this cell to a 6, the chart

automatically changes._

Fig. 5

Changes anywhere in the matrix are immediately seen in the chart.

1 2 3 4 5 7 8 000000004 700000000

80000000 0

The changes in the matrix are reflected in

changes in the chart, although the colors in the matrix will not change automatically.

Fig. 6

Another illustration showing how changing the numbers in the matrix results in changes in the chart.

ing, and then rotating the three-dimensional-surface chart (see fig. 7). Even though a separate contour

map can be created by repeating the charting process, the process of copying, pasting, and then

rotating seems to have a larger impact on students. When students can actually see one chart turned into another, the connections between representa tions of the data are retained and most students

immediately understand how to interpret the con

tour lines. Again, numbers can be changed by the

teacher, and the class can watch the corresponding changes in the matrix, the three-dimensional surface chart, and the contour chart.

The spreadsheet can automatically label the axes on the right and bottom edges of the graph, and the matrix was manually labeled on the left and top edges. One obvious method of making more

straightforward the correspondence between label

ing on the different representations would be to

label both sides of the matrix (see fig. 8). It is

important that students understand that the labels can be placed on either set of edges. However, Excel will not recognize labels on the right and bottom

edges of the matrix and will not automatically put them in the chart. The type of labeling of the axes

and the visual or mental conflict that it can produce

Changing the matrix helps students see

the ties between the

two data

forms

Vol. 92, No. 2 ? February 1999 169



Most students

immediately understand

how to

interpret the contour lines

o 1 2 3 4 5 6 7

012345678 000000000 000000000 0 0

0 0 0 0 0 0 0

00000000 8000000000

3. A view of the chart from the top; although the triangle is not perfect, we can see how the data match both the matrix and the three dimensional chart.

Fig. 7 Three representations of the same data on one screen: a matrix, a three-dimensional

surface chart, and a topographical view

y-axis

012345678 :00000000000

?000000000? a 2 0 0 1 0 0 0 0 2

70000000 80000000008

012345678 Fig. 8

The matrix with axes labeled on both sides, making it easier to understand how the graph is labeled. The colored numbers used as axes on the matrix were not part of the graph. They were added and colored,

both on the chart and matrix, after the chart was created.

in students are a direct result of the technology used and an example of how technology can affect communication and learning in unanticipated but

important ways. This approach contrasts with the traditional chalkboard approach to a three dimensional system in which axes are always labeled in the same way.

I like to compare the spreadsheet's representa tion of a three-dimensional surface with other forms of axes so that students are sure to gain a grasp of the complete three-dimensional world in which they are working. Figures 9a-c show a similar graph, but with larger x-, y-, and z-axes, ranges, and differ ent styles of axes. Not all students will recognize




that the graphs with which we previously worked are positioned in just one octant from a complete three-dimensional world. Figure 9d shows the same

graph but from a different viewpoint. Unfortunately, changing the scale on the axes for three-dimensional surface charts cannot easily be accomplished in some spreadsheets; figure 9 was generated using

Maple V, Release 4 (1996).

THE FIRST STUDENT PROJECTS After three-dimensional systems have been intro duced and explained, students are given matrices and asked to give a description of a graph (see fig. 10). Then the problems are reversed?students are

given graphs and asked to give the matrix (see fig. U). This exercise can be done as a whole-class

activity or in groups. The teacher and students can use the computer to check their hypotheses and revise them if necessary. Most students are quick to grasp the concepts and can make accurate pre dictions with very little practice.

To check students' understanding and to provide practice in generating these charts, the students' first assignment will be to generate charts of such

simple surfaces as a rectangular pyramid, some

stairs, and then a bowl (see fig. 12). These begin ning problems will reveal to a teacher whether stu dents grasp the concepts of topographical views and contour lines and their relation to the matrix. I pre fer to have students work in pairs for these begin ning activities. For younger students and maybe

Vol. 92, No. 2 ? February 1999 171



o 10 6 6 6 6 6 6 6 6 10

1 6 10 7 7 7 7 7 7 10 6

3 6 7

10 8 8 10

8 8 9 9

8 10 10 8

10 10 9 10 10 9

8 10 10 8 8 8

7 7 6 6

9 10 8 8 10 7 7 6 6

8 6 10 7 7 7 7 7 7 10 6

9 10 6 6 6 6 6 6 6 6 10

Fig. 10 Students will be able to look at a matrix similar to this one and predict what its graph will look like.

4-6

2-4

0-2

Fig. 11

Graphs such as this one are used to develop students' ability to generate a matrix given a graph.

even some older students, a good preliminary or

accompanying activity would be to use Cuisinaire rods to build representations of such surfaces as

stairs or pyramids before starting in the more abstract environment of the computer.

CREATING A THREE-DIMENSIONAL SURFACE CHART FROM A TOPOGRAPHICAL MAP The final project may take several days; it applies the

concepts presented previously and combines them with a more traditional hands-on approach. The

objective is to make a three-dimensional-surface

plot from a real topographical map. Students can

choose their favorite geographical area, or the teacher can assign an area. Topographical maps are easily obtained from sporting-goods stores, the

World Wide Web, or the Forest Service (see fig. 13).

It is a good idea to laminate the original maps so

they can be reused. After students have a copy of a

topographical map, they receive a sheet of graph paper on clear acetate and a sheet of regular graph paper on which to record data. The acetate with the

grid can be placed over the topographical map, and students can record the height of the closest line for each cell, either on paper or on the acetate with erasable markers (see fig. 14). A student originally had the idea of using the clear acetate, and I have found it very effective. The teacher can supply older or more advanced students with the maps and grids, tell them to make a three-dimensional surface, and see what types of solutions they find and how they divide the tasks. Younger students may need more

direct instruction. I have found that groups of at least three have worked well when doing the final

project, with different students performing differ ent tasks to complete the work.

Once students have recorded the data, they enter it in the spreadsheet and create the three-dimensional surface of their topographical maps (see fig. 15). The three-dimensional-surface chart can be copied, pasted, and rotated to produce a colored contour map, which can be compared with the original (see fig. 16).

When used for this activity in the classroom, the

topographical maps should be enlarged before the data are recorded. The problems associated with

having multiple contour lines in a single grid square are thereby limited. For schools with more

advanced technology, I have found it entertaining to place the surfaces into three-dimensional

rendering programs and apply different surfaces or use software with more advanced graphing and matrix capabilities. Computer-algebra systems or

three-dimensional-rendering programs can create the three-dimensional surfaces from the matrix or

from a color-coded contour map (see fig. 17). This method of producing three-dimensional sur

faces is not intended to produce professional-quality maps, and the software used limits the quality of the surface. As with many computer activities, when we try to use discrete measuring tools to model such continuous data as elevation, we must realize in advance that our accuracy will be limited. Students will begin to intuitively question how to

get smoother, more realistic graphs, which can lead into discussions of modeling continuous phenomena with a discrete machine. Such questioning is not a

drawback to using spreadsheets for these types of charts but rather furnishes an opportunity to pro mote classroom discussion and understanding. The teacher and students must also reach a compromise as to how many points will be recorded on the final

project. More points will result in a smoother graph but will require more time and computer power.

This activity is excellent for a multidisciplinary project. Matrix representations are certainly not

THE MATHEMATICS TEACHER



a m ; -? ; * ' ? $ Is ? ?t: *f ?fe

p?-f*. * : r ; $ ^ # s s ? ? * 11 $

Fig. 14 A topographical map with overlay of a

clear acetate grid

Students can

choose their

favorite

geographical area, or the teacher can

assign an

area

3800 3600 3400 3200 3000 2800 2600 2400 2200 2000 1800 1600 1400 1200 1000 800 600 400 200

0

3600-3800 3400-3600 3200-3400

3000-3200 2800-3000

2600-2800

2400-2600

2200-2400

2000-2200

1 800-2000 1600-1800 1400-1 600 1200-1400

1000-1200

800-1000

O 600-800

400-600

200-400

0-200

Fig. 15 A three-dimensional-surface chart generated from a topographical map. This chart is a rough three-dimensional

rendering of the topographical map in figure 14 using Microsoft Excel. It has been rotated from the original position; the lowest levels are on the top of the contour map in figure 14.

Vol. 92, No. 2 ? February 1999 173



3600-3800 a 3400-3600

3200-3400 3000-3200 2800-3000 2600-2800 2400-2600

?2200-2400

2000-2200

1800-2000 1600-1800 1400-1600 1200-1400

1000-1200

800-1000

600-800

400-600

200-400

0-200

Fig. 16 A three-dimensional color-coded contour chart. This map was generated by copying and pasting the

three-dimensional-surface chart in figure 14. Setting the scale on the legend will control how many different colors are generated.

Fig. 17 The same three-dimensional world as in figure 15

but graphed using Maple V

limited to elevation as in these problems. A three dimensional-surface plot or scatterplot could be used to represent many different forms of data, including temperature, air pollutants, or any fune

tion that depends on two variables. Data collected from science experiments are well suited for this

type of analysis and will further enhance students'

understanding of three-dimensional surfaces where

they must interpret and analyze the surface plot in relation to real-world variables.

Students should be fluent with the spreadsheet before trying to use it for analyzing real data. Oth

erwise, they focus too much attention on how to use the computer and software, and the lesson deterio rates into a computer lesson rather than a mathe matics and science lesson.

BIBLIOGRAPHY Cuisenaire Company of America, P.O. Box 5026, White Plains, NY 10602-5026; (800) 237-3142.

Fong, Peter, ed. Integrated Mathematics: A Modeling Approach Using Technology. Montana Council of Teachers of Mathematics: The Systemic Initiative for Montana Mathematics and Science. Levels 1-6. Needham Heights, Mass.: Simon & Schuster, 1996.

Maple V, Release 4, Student Edition. Waterloo, Ont.: Waterloo Maple, 1996. Software.

Mason, J., and D. Pimm. "Generic Examples: Seeing the General in the Particular." Educational Studies in Mathematics 15 (1984): 277-89.

Microsoft Excel, Ver. 5.0, Redmond, Wash.: Microsoft

Corporation, 1995. Software. National Council of Teachers of Mathematics (NCTM).

Curriculum and Evaluation Standards for School Mathematics. Reston, Va.: NCTM, 1989.

Robertson, Bruce. How to Draw Charts and Diagrams. Cincinnati: North Light, 1988. @)




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3-D Graphing, Contour Graphs, Topographical Maps, and Matrices Using Spreadsheets