Graphical Spreadsheets

Graphical SpreadsheetsAuthor(s): Richard BridgesSource: Mathematics in School, Vol. 20, No. 5 (Nov., 1991), pp. 2-5Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30216544 .

Accessed: 09/04/2014 15:39

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

The Mathematical Association is collaborating with JSTOR to digitize, preserve and extend access toMathematics in School.

http://www.jstor.org

This content downloaded from 173.73.163.236 on Wed, 9 Apr 2014 15:39:46 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=mathas

http://www.jstor.org/stable/30216544?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


GRAPHICAL File Edit Gallery Chart Format Macro Window

-kDecimal Expansion A I B CJOE IF

G H I K L N O Q S T V X Z

~3Nu-mertor Decimal Expansion Chart 1I- 4 Deno mi nato r 17,

6 470 5:883529 782370 5 8 8:2 35 29 941:

Decimal EHpansion Chart

10

5

0 S 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76

Fig. 1 Decimal Expansion Spreadsheet and Chart (on Macintosh)

SPREADSHEETS by Richard Bridges,

King Edward's School, Birmingham

Although spreadsheets are still regarded as relatively new on the mathematical scene, they are beginning to appear more frequently in books and articles on the teaching of mathematics',2. Many of these sources restrict themselves to purely numerical sheets to be constructed by pupils themselves; in this article I wish to investigate some of the potential of spreadsheets that draw graphs, and of complex sheets used purely as demonstrations.

The spreadsheet I used in constructing all the examples mentioned was Microsoft's Excel, which is available for the RM Nimbus (and all PC compatibles) and also the Apple Macintosh. It is a powerful package, which has virtually become the industry standard. It offers large sheets, a wide range of easy to use formatting facilities, very easy formula copying, and a range of graphing ("chart- ing") functions. A valuable feature is that once a graph has been attached to a sheet it is updated whenever the sheet is changed, giving instant feedback. Colour is available to distinguish between graphs plotted on the same axes. Its only weaknesses in a school context seem to me to be a) the graphs are very business oriented, useful types for mathematics such as Cartesian or scatter graphs being difficult to construct and functions such as counting frequencies and histograms being absent altogether, and b) it is a large program, and tends to run fairly slowly on a network, though not impossibly so.

Investigations Using Graphs

A topic which could generate some interesting investigations is the decimal expansion of a fraction. BASIC programs to produce this expansion already exist', but a spreadsheet with attached graph is easy to construct also (see Figure 1). The instant response of the graph to vari- ations of the numerator with the same denominator (eg 13) is striking, and the periodicity is easier to see than by looking at the numbers. The relationship between the period of the recurrence and the denominator is by no means obvious; hence the potential for investigation!

An obvious topic for spreadsheets is Number Sequences2. Graphs can be used to enhance this work considerably. A possible sequence for a module of work might be: Arithmetic Sequences, using graphs to relate to y=mx+c. Geometric Sequences, using a log scale to produce a straight line, which can be used as the recognition point for such a sequence (log-linear graphs are available in Excel). Sequences of squares, cubes, fourth powers and their sequences of first, second etc differences, the graphs showing the relation between the differences and the sequences with power one less (Excel's log-log graph shows this clearly). The Fibonacci sequence, a log graph showing it is nearly Geometric, a graph of ratios of successive terms showing approach to the Golden Ratio, a log graph of

2 Mathematics in School, November 1991



differences from the final value of the GR showing that this approach itself is Geometric (Figure 2). All of these sequence sheets are simple enough to be constructed by the pupils themselves.

Here the built-in functionality of the spreadsheet can be used as a qualitative introduction to a range of sophisticated ideas in a fairly painless way; log graphs, for instance, are extremely common in practice because of the wide occur- rence of geometric (exponential) or power laws in practical situations. The fact that the mathematical theory of logs only appears at A level shouldn't stop us using their simplifying power when appropriate much lower down the school. Again, the idea of using differences leads naturally into calculus, the qualitative idea that differencing ("differentiating") a quartic gives a cubic, then a quadratic, then a linear function being probably more important than knowing the precise rules of differentiation involved. The fact that differencing a Geometric (exponential) gives a Geometric with the same log gradient (common ratio) is one of the main reasons for its usefulness throughout calculus.

Demonstrations Using Graphs A topic which is not so investigational, but where instant graphical feedback may be used to give greater insight, is the representation of the solution of two linear simultaneous equations as the crossing point of two straight line graphs. The graphs can be presented below the algebraic solution using a 2 x 2 inverse matrix if desired, as Excel will perform matrix multiplication and inversion. Matrix operations can also be used to show the effect of a 2 x 2 matrix representing a geometrical transformation on a 2 x n matrix representing a shape; with a little encouragement Excel can be made to plot the original shape and its image as what it thinks is a

scatter graph, but which is effectively an x-y plot (Figure 3). Variation of the object co-ordinates or the elements of the transformation matrix produces instant visual feedback.

As a final example, progressing into A level territory it is possible to use the number crunching power of a spreadsheet to compute the projection on screen of a 3 dimensional object such as a line or plane (though much of the "workspace" needed would not normally be displayed on screen). Figure 4 shows a spreadsheet and associ- ated graph being used to investigate the intersection of three planes in 3D (the case shown is intersection in three parallel lines). As well as the plane equations, the direction of view may be freely altered, so the pupil may "walk around" the planes in space.

Excel as a Language for Program Writing The most complex sheets are beyond being constructible by pupils themselves, and indeed there would be no great benefit in their doing so; it is what the sheet does once complete that is of interest in these cases, rather than the method of doing it. They really serve as alternatives to class demonstration software written in BASIC. Indeed, I was struck while constructing some of these sheets by the idea that I was really using Excel as a super-high-level language for program writing. They took me far less time than writing in BASIC, because the input and graphical output was already built into the functionality of Excel. Even when the number crunching gets heavy, as with constructing the 3D views, duplicating spreadsheet formu- lae tends to be easier than writing loops to work through arrays. Because writing is easier than via BASIC I remained more motivated to create a number of demonstrations. Thus one could contemplate building up a bank of CAL

File Edit Gallery Chart Format Macro Window Help S1P9 =SERIES(SE DATA.XLS!$D 1,rSE DATA.XLS! 0 2: D020A)

SEQ0ATA.XLS B C D

S.Sequenc.Ratio ....iRatio- GR 10000

2 1 1 ....................4. sIe . .. . 1000 - - - - - - - - - - - - -

3 1 1 ..0:618 03 4 30 1.1: 0.618034 1

4 2 2 .- 0 38 966100 0 11 111AIRII IN 11 415 11 11111 411 11 11111111

s

....................... ... .....

.......... ........ .... ... ...o. dao o - - - - - - - - - -1- --0-

6 5 1.666667

T0.048633 . 1.1................... ............7 8 ....................... ... . .

1 . . ............. a................ 1

7 8: -6 0.018034 .

s............. ... 1 , 4............... ... .......... 0 0 ..,60

1 - R a t io -

G R i 8 13 1-625 0.006966 0.1 -----------------

9 21 1.615385.- 0.002649 00 ..5.....- ~ Sqec 9_ . ................. .1.....1,....

1. . 85.0.0..02... 49.'

0.01 - -- --Sequence 10 .....

34 1..519048.

0.001014 I"T

000000

.................. ,5,,.,., .... ,..:.5..1. .,5,4...

i .,0:.,0,.0.,.0..0...8.,7.,000

0.001 11 55 ,,

1 .61..,.. 0.0003871

1_ illegal

#Got..80844................... ,I..,..0 ...... ., , ,I.., . ...,.,,,. .. .

........ . ....0 .5

.. .............. .. ........ . ..... ....

....1....... 1

.....

...... .. ...

......B 0 0 0 0 0 1m00-ID N 19 2584.-'1.618034: 2.42E'07] .0001""" 20 4181 1 .618034 0

Ready

Fig. 2 Fibonacci Sequence and approach to the Golden Ratio (on Nimbus)

Mathematics in School, November 1991 3



File Edit Gallery Chart Format Macro Window Help

GTRAN.XLS Geometrical Transformation of 4 Cornered Shape

Object (Red) Image (Blue) -1 0Ox 4 4 5 4 = -4 -4 -5 -4 0 -1 2 6 5 4 -2 -6 -5 -4

Ready

Fig. 3 A Spreadsheet and Chart for Geometrical Transformations

1 A

File Edit Gallery Chart Format Macro Window Help

SMEQ3X3.XLS intersecting planes View DirI Black@ 1 xS+ *-

-11'yi+ Z 8

Red 2x:+ 3 : lz-

3 Blue 3 x + 2 2 Z = 5I10

.,,4s..s,. SO... ...:... ,,,x.!+ i...,z ,y ,,+.......... a.II

Aslet

sz,60A06-,94 1......... ,,1,91..........

6841

so"ii

Matrix eluation : :: : 101lx x 0 V31X 2

3-,2-,-

Intersection pp nt

z 9

Lines of intersection are arallel to: 4.4 -4 1 8

-0.S 1 Blue .-

-545 -10

Ready

Fig. 4 Solving Simultaneous Equations in 3 Variables

4 Mathematics in School, November 1991



material that could be disseminated on disc quite cheaply because the investment of effort was not too high, in contrast, I suspect, with material based on a conventional high level language.

Not all demonstrations lend themselves to being set up on a spreadsheet. One problem is that control over the graph format, though far easier to exert, is not as complete as via BASIC. Axis labelling and scaling aren't always ideal, for example. Another problem is that all the data for any graph have to be present somewhere on the spreadsheet. For isolated points and straight lines this creates little difficulty, but curved lines such as the parabola corresponding to a quadratic need quite a large number of points to achieve satisfactory resolution, making the sheet more bulky than it need be; this data need not be displayed, however. Some deficiencies in the functionality of the spreadsheet program can be overcome if it has an extensive "macro language" as Excel does. In Excel's case this is a language similar to BASIC, and allows the same sort of control over numbers within the macro, albeit in a slightly cumbersome way. For example, it is possible to write a macro that counts the frequencies of the values appearing in a highlighted column on the spreadsheet, a function not built into Excel as it stands. It is even possible to write macro functions that return the eigenvalues of a matrix, though these are so complex that a more conventional programming language is probably more efficient. One defect of the macro language is that it tends to execute fairly slowly in comparison with, say, BBC BASIC on the same machine. This is a particular problem for simulating random processes, as it takes longer to reach decent sample sizes such as 1,000 rolls of a die than via a more conventional language. Another problem is that macros are

designed only to communicate the results of their pro- cessing when they finish and return control to the spreadsheet; thus statistical demonstrations where an on-going random process such as rolling a die or tossing a coin gradually builds up a graph on screen are not possible.

Summary I have tried to show how the graphing abilities of a powerful spreadsheet such as Excel can be used to advan- tage in the mathematics classroom. I have also tried to assess the scope of such software as an alternative to conventional programming languages for writing demonstration material.

Much of this work has been stimulated by my member- ship of the Computers in Mathematics Teaching subcommit- tee of the Mathematical Association's Teaching Committee, which is chaired by David Green of Loughborough Univer- sity Maths Dept. At present we have members interested in graphical calculators, "Numerator" and computer algebra; I co-ordinate interest in spreadsheets. Anyone who feels they have a contribution to make is most welcome to contact us. It is eventually hoped to produce some material, either written or on disc, for use with spreadsheets, but anyone sending me a (720K MSDOS formatted) 3.5" disc is welcome to a copy of the example sheets so far constructed.

References 1. Miller, D. (1990) Micromathematics Levels 5 and 6, Causeway Press

Ltd. 2. Keeling, R. and Whiteman, S. Simply Spreadsheets, KW Publications.

WHAT I/*' 1. Stacker numbers by Jim Smith, Mathematics Education Centre, Sheffield Polytechnic

This investigation can be introduced by the teacher using counters on an OHP, or by sketching on the board, patterns of the following type;

16 12

18 35

and then discussing the construction of a Stacker Number. Pupils should be able to come up with a reasonable definition for themselves. Perhaps something on the lines of the following will emerge;

"Start with a row of counters, then add a row which is two shorter than the previous row. Keep going until you feel like stopping, or it's impossible to carry on."

After this brief introduction, pupils could be allowed to explore the situation, finding out which numbers are Stacker Numbers and which are not. You might call the non-stacker numbers Surprise Numbers, but perhaps pupils will also have their own name for them.

Those who find it difficult to cope with too much freedom to explore might be directed to consider what is special about the Stacker Numbers made with exactly two rows; having got a result they could move on to three rows, and so on. I would be reluctant to do this too early however, as I like to allow pupils the chance to impress me with their own system and organisation {rather than depressing them with mine}.

As usual we can generate more ideas by thinking about what we have been given as a framework, and then asking what would happen if we were to change this or that. Some of the things that we could change include the "rules" for generating the numbers, and perhaps we could go into three dimensions (or more). What if we asked pupils for more ideas ...?

Mathematics in School, November 1991 5



Documents

Graphical Spreadsheets