derive 6 was released. there were a lot of exciting new ... · wiris shows “true” sliders. they...

Josef Böhm/Sliders 1

It was in 2003 when DERIVE 6 was released. There were a lot of exciting new features after DERIVE 5 offering programming within the CAS program. Albert Rich himself demonstrated stepwise simplification, the back ground pictures and the SLIDER BARS. All these new fea-tures opened new possibilities for teaching mathematics. One could find sliders in Excel but not in MATHEMATICA and not on the TIs at this time; Nspire did not exist. First versions of GeoGebra offered sliders and I believe that Autograph had the Constant Control imple-mented.

was the headline of a short contribution in the DERIVE Newsletter #52 (2003). In this issue the slider bars appeared the first time in our bulletin (Bézier Curves and some other demon-strations).

In this lecture (paper) I´d like to demonstrate the power of sliders in teaching mathematics and to compare the sliders implemented in various mathematics tools (which are of impor-tance in school mathematics).

Take the following presentation as a picture book divided in some chapters.


I intend to consider a bundle of tools:

Let me take one of the most common examples for using sliders to compare their implementa-tion in various programs which are used in our schools: The parameters of a linear function.

Excel_0.xls

In MS-Excel introducing sliders is not so comfortable and intuitive. Usually the teacher has to provide the file – and there is no CAS attending.

The next picture shows the respective GeoGebra screen. I can imagine providing the line and the sliders and ask the students to find out the meaning of m and b and then add the visualisa-tion of the two parameters in connection with the straight line (slope triangle and intercept segment). Finally come back linking the graph and the equation.

Implementing the sliders is easy and there are a lot of properties which can be given to them.


geogebra_0.ggb

Douglas Butler´s Autograph offers a “Constant Controler”. It is easy to set up – but not really a “Slider” – one has to press buttons (like on a CD-player).

autograph_0.ggb

wiris_0.wiris


WIRIS shows “true” sliders. They have to be introduced by commands. And … WIRIS is a CAS. One can combine the power of symbolic calculation and graphic visualization as well.

Now as we have turned to Computer Algebra Systems I will show DERIVE 6:

derive_0.dfw

The sliders are set up via the Menu in the Plot Window. Unfortunately there are some defi-ciencies: a minor one is that the Properties-Window asks for the number of intervals between Minimum and Maximum (instead of the step width like in other programs); the major one is that the sliders cannot be saved in the file. The sliders must be set up again. Greek characters are not accepted as variable names. This should have been improved with DERIVE 7, but ….

I believe that the software smithies underestimated the potential of sliders (and of the back-ground pictures, as well) for math education.

nspire_0.tns


The screen above shows the TI-NspireCAS. It is no problem at all introducing sliders and setting the appropriate properties. There are two forms of sliders available.

Finally let´s have a look to MATHEMATICA:

http://demonstrations.wolfram.com/LineThroughTwoPoints/

(You can download Mathematica Player 7 and then you have access to a rich resource of demonstrations from all fields of pure and applied mathematics.)

As we saw now, there are two programs which use a command for setting up sliders (WIRIS & MATHEMATICA). The code for a “dynamic box” is very bulky – but I am not an expert for MATHEMATICA at all …. In all other tools you need options from a drop-down-window.

The sliders (Constant Control and Animation Tool in Autograph) cannot be animated in DE-RIVE, WIRIS and MS-Excel. (Maybe that animation of the sliders was also on the To-Do list for a planned DERIVE 7?).

DERIVE, WIRIS, MATHEMATICA and Autograph provide sliders for the 2D and 3D Plot Windows. I will show respective examples later.


We start browsing the "Slider Picture Book":

Rectangular and Polar coordinates:

Using sliders the students will have the opportunity to experience the mutual rela-tion between a pair of numbers and its graphic representation:

From the numbers (sliders) to the point (screen) and from the point (cross) to the numbers (status bar).

coordinates.dfw

The polar coordinates need little preparation done by the teacher.


You can proceed with the representations of x = constant, y = constant, r = constant and ϕ = constant.

The next screen shows a possible tool for visualizing scalar multiplication and sum (differ-ence) of vectors. It is possible to extend to other vector operations and then switch to working with complex numbers.

vectors.dfw

(The arrow tool was provided by Peter Schofield, UK, and Lorenz Kopp, GER.)

Possible questions for students? line_parameter.dfw


Observe the point moving on the line:

Which direction? Which position for t_ = 0, 1, -1, 0.5, …?

What happens if you replace (B – A) by (A – B) in expression #5?

Take a space curve like the helix. We have three parameters in expression #7 below. Explain their roles and make them visible.

Unfortunately DERIVE does not accept Greek characters as variable names in sliders.

WIRIS offers more possibilities for presenting the curve and a moving point (color, size, …).

helix.wiris


The parameter form of a plane:

The next surface is a little bit more complicated (do you know what it is?):

The Bottle of (Wine) Klein


You can see the parameter curves (above) and then A point tracing one of them (below). (Question: Which is the fixed parameter and which is the variable one?

line_parameter.dfw

Calculus is one of the main chapters in secondary school mathematics:


Many extremal value problems (which can be treated with dynamic geometry programs as well) offer chances to introduce sliders and offer also the chance to solve the problem analyti-cally:

It should be no problem for the students to create the dynamic model of the “Magic Box”.

We cannot find the maximum value of the volume. Let the students accomplish the figure making the volume visible! (by adding the volume function in a reasonable scaling)

Zooming in gives a chance to find the respective side of the cut out square approximatively.

superbox.dfw

Finally we can find the exact solution by means of calculus:


Investigation of a family of curves.

Find and plot the loci of the turning points and the points of inflection!

Plot the curves varying the parameter a!

family.dfw and family.wiris


It makes a difference to plot the whole family (use VECTOR in DERIVE or create a list of functions in other programs) or to plot stepwise one curve after the other.

Other programs offer – I talked about this – additionally animating the parameters. This is quite nice but I prefer stepwise activating the slider bar. I am quite sure that the relationship between value and appearance will become much clearer in this way. Otherwise the students might “lean back and enjoy the video”.

One fix point in teaching calculus is the Newton-Raphson algorithm. In fact it has lost its importance for solving equations numerically in times of computers. But it does not disappear from our text books. The sliders can enhance imagination of the method by stepwise linearisa-tion of the function of the equation. I would like to demonstrate the convergence problem in connection with the choice of the initial value. Let’s take the function f(x) = sin 4x and we would like determining a special zero of f.

The function newt(u) returns a point (x0, y(x0)) with y being the 5th iteration of the algo-rithm starting with x0.

My first attempt runs for -3 ≤ x0 ≤ 3 with an increment of 0.05.

The initial value x0 = 1.20 leads to the zero 1.25π and the next initial value x0 = 1.25 leads to the zero 0.75π. The next screen shows all zeros in this interval and we can trace the “scatter diagram” of zeros.

We observe a “nice” pattern. Things are becoming exciting by zooming in.

Take 1.1 ≤ x0 ≤ 1.3 with a step size of 0.001 for the slider. See the following results:


The complete pattern of the zeros in this small interval is given by:

newton.dfw

You can again zoom in and find a similar pattern. “Self Similarity” is a property of “chaotic behaviour”. Produce your own CHAOS using the slider bar. (Zoom in, take other functions, ….)

I am closing the “Differentiating Section” with an animation of Taylor Series:


taylor-dfw

It works in DERIVE but it needs remarkable calculation times when activating the sliders.

GeoGebra does much better – and shows even without an implemented CAS – the approxi-mating polynomial (Yes, I know, it is not in the correct symbolic form.)

This is the exact (DERIVE-) result:


Now let´s turn from differentiation to integration:

arc_length.ggb (Andreas Linder)

This is a GeoGebra screen. We can drag the boundaries of the interval a and b and we use the slider for changing the number of intervals.

I am sure that seeing this you are thinking on visualizing integration. This is also no problem with GeoGebra because of built in functions for lower sum and upper sum:

LowerUppersum.ggb

You can do this also with DERIVE, TI-Nspire and WIRIS but it needs some – interesting and useful – programming. The area works in Autograph but you cannot follow dynamically the values (area, arc length, volume in 3D, …) and you cannot plot two or more methods simulta-neously.


By the way, GeoGebra offers the feature to create a html-file which can be transmitted to the students together with tasks to be treated by them:

What about differential equations?

Can we use sliders in a meaningful way?

The next page shows the direction field and one integral curve with the TI-NspireCAS and with Autograph.


ode.tns

Autograph: ode.agg

y x a y′ = + ⋅


The slider introduced in Nspire varies the parameter(s) within the differential equation. The initial point can be grabbed and dragged. In Autograph and WIRIS (next plots) we need slid-ers for moving the initial point, too.

The left screen shows a family of integral curves (fixed) and a changing single integral curve determined by the changing initial point (x_, y_). The right screen shows the fixed direction field with a movable initial point.

The last screen shows the direction field and a family of integral curves together with a slider for the parameter a of DE y’ = x + a ⋅y. I believe that there is much until now undiscovered potential in the use of sliders for teaching DEs and systems of DEs as well. A well known system describes the Prey-Predator-Model.

y x a y′ = + ⋅


This is the standard Prey-Predator-Model presented with GeoGebra. The initial point describ-ing the initial populations can be moved by dragging in the plane (Dynamic Geometry Tool). The parameters of the DEs can be varied by the sliders. The points of the phase diagram and the diagrams of the populations are calculated in the GeoGebra spreadsheet using the Runge-Kutta-Method (which is not originally implemented but done by me, “programmed” in the GeoGebra-spreadsheet ).

I was not able to produce a similar presentation with DERIVE – even by applying a RUNGE-KUTTA-program. Problems with the variables management and their internal storing cause too long calculation times and memory overflow.

I tried with TI-NspireCAS and … I was successful as you can see watching the next screen shots.


I used two sliders for entering the initial populations (x_start and y_start). The spread sheet below gives an impression how I performed the Runge-Kutta-Algorithm with TI-Nspire. The cells C1 and C2 are linked with the variables x-start and y-start.


Problem: Given are data from counting vehicles on a street. The scatter dia-gram shows the traffic density, i.e. number of vehicles per hour passing a certain point. The diagram shows the number of vehicles/h vs time of the day.

It would be useful for further investigations and calculations modelling the discrete data by a continuous function.

After trying some (unsatisfying) polynomial fittings students find out that the diagram re-minds for the two half days on the Gauß bell curve (which they do know from probability theory) and instead of trying parameters they insert sliders for mean and standard deviation.

The vertical stretch factors (1000 for the morning model and 900 for the noon model) can be read off from the given data.

The TABLE-function produces a “thick” line.


This is the (very) satisfying result:

Another example together with an application of the background pictures: I found a diagram in the Scientific American showing the premature blooming of plants. The diagram reminds – like many processes in nature – on the normal distribution.

Looks pretty good, doesn’t it?


Introducing linear regression using MS Excel and GeoGebra:

Regressionen.xls (slider for the slope, passes ( , )x y )

regressionen1.ggb

You may extend to other regression lines without any problems.

What makes the difference between mean and median?

boxplot.tns

Take a list of values and make one value variable (= z), then you can experience the differ-ence!


Investigate the properties of various distributions:

binomial.dfw

Simulations demand very often answers to “What if?” – questions. It is more comfortable to move a slider than to enter function names with a bundle of parameters. The next example is a simulation without “What if?”, but it is a simulation of a random walk. (simulation.dfw)

m2d shows the position after t steps and pathsstep2d gives the last step.

We could extend to show how the walked way develops.


Loci problems are a fix part of mathematics and geometry. In school they are usually treated (if technology supported) by dynamic geometry (leading to conjectures without analytic cal-culations) or by algebraic methods (supported by CAS with finally plotting the results). Ap-plying sliders we have the possibility to combine the concept of dynamic geometry with the algebraic representation of each single step.

This is one of my favourite applications of sliders within a CAS-program. I have a rich collec-tion of respective examples. One of them is inspired by articles published by Wolfgang Moldenhauer und Wilfried Zappe in the TI-News. Wolfgang is giving a lecture on this objec-tive in the conference, too. W & W investigate loci of special intersection points of “triangle lines”. In their publications mentioned above they didn’t treat the following case:

Given is an arbitrary triangle with one fixed side AB and point C running on straight line. What is the locus of the intersection point of an altitude with a perpendicular bisector?

Let’s look what the dynamic geometry programs are delivering? Starting with CABRI we find out that we have to differ between two cases (altitude through the moving point C and altitude through point B; what can be said about the locus created with altitude through point A?)


GeoGebra does little more; it identifies one of the loci as a hyperbola??

Geometry Expressions is a great dynamic geometry program which gives not only the im-plicit form or the parameter form of the loci but offers the feature to export the equations to DERIVE and to TI-Nspire!!

What can we do with a CAS in connection with sliders? What is the benefit of this?

First step: Create the “mobile” triangle. B is variable on the x-axis and C moves on the vari-able line.


Next steps (always simultaneously: calculate in the Algebra Window – Plot for confirming the calculation): Calculate and plot the altitude and one of the meaningful perpendicular bi-sectors.

Expression #13 gives the intersection point (which can be traced by the t_-slider). But it represents also the parameter form of the locus. Substitute t_ by t – giving #14 – and plot it!

Finally we are successful finding the explicit form of the locus (#16).


As a very special “add in” we can import the red equation (the shorter one) from Geometry Expression (page 27). This equation reminds me on a parabola, but …

… we I have to take into account that d is the y-value of point C which is in fact k t + d and t can be replaced by x. Hence the identity is given.

A possible next step would be analyzing the 2nd order curve (conic) …

By the way, the other locus is an algebraic 3rd order curve. Do the two loci kiss each other? In my opinion is this one of the most appealing applications of the sliders within a CAS-program. We combine the most important and most contrary representation forms of mathe-matics – the algebraic/numerical one and the graphic one. We know that students, and gener-ally spoken, not only students, prefer one of these representation forms for comprehending, learning and transmitting concepts and problems. I believe that this combined approach would deserve some research. In DNL#52 (Wonderful World of DERIVE 6) we had an article contributed by Franz Schlögl-hofer about Bézier Curves in School. I could not resist imitating dynamic geometry “slider supported”:

The sliders move the control points B and C of the 3rd order Bézier curve and influence its shape .


Sliders are useful finding solutions of linear and non linear programming up to three vari-ables. The next two pictures show the same problem, solved first with a pre-slider-time “self made slider” in DERIVE, then solved with the original DERIVE slider moving the goal func-tion line to a corner point of the feasible region.

You can see visualizing quadratic programming (problem given by Bjoern Felsager, Den-mark, in a DERIVE newsletter). I supported the graphic interpretation of the solution intro-ducing sliders in the 2D-model and in the 3D-model as well.

Do you know “Numeric Sliders”? See next page!


A short article in the last TI-News (1/10, Andreas Pallack) iNSPIREd me to use the TI-Nspire sliders together with the “Math Boxes” working with “Numeric Sliders”. It is comfortable to investigate “What If?”- problems changing parameter values in numeric problems. I take an easy example from financial mathematics, calculating the annuity of a loan:

annuity.tns

You can vary the loan, the number of payments (years) and the interest rate using the sliders and you will find immediately the amount of the annuity.

Now let´s take the reverse problem: What is the interest rate with given loan, annuity and number of annually payments?

In this case I don´t want to solve the respective equation. Instead of this I perform a decimal search. Until now I did this using the table and stepwise refining the proce-dure.


Now I change the settings of the interest rate slider:

I refine the search

I am quite sure that you can find a lot of meaningful and informative applications of the nu-merical sliders in school mathematics. I´d like to close my slider picture book presenting some mouth watering specials.

interest_rate.tns


Heinz Rainer was one of the first who used the potential of sliders – together with his students he developed the model of a roboter applying rotation matrices and others.

ROBOT.dfw

I found a great website: http://jmora7.com/ and further http://jmora7.com/Arte/arte.htm:

Francisco Goya, El Quitasol (The Parasol)

I want the students to analyse the composition of this famous picture and added tasks and questions. This is the status of the investigation after two steps; task three is waiting to be performed:

mygoya.ggb

Heinz Raier Geyer´s Roboter


In this way we can provide some additional quality for this application of using background pictures and sliders in a dynamic geometry program. If there is also CAS available you can ask for the equations of the geometric objects. (This can be done in the Algebra Window of Geogebra, too.)

David Sjöstrand from Sweden sent an Excel file demonstrating morphing. This is a meta-morphosis from a circle to a cow produced with MS Excel:

cow.xls

It is a nice application of working with vectors. David did not stop here. He explained how to perform a metamorphosis between three or four appearances. This needs little (CAS-) calcula-tion and it runs perfect.

In Derive it looks like this:

See the initial circle, the process after half of the transmutation to a cow and finally the result-ing cow (0 ≤ t ≤ 1):

cow_circle.dfw


Andreas Lindner shows us his “Spirograph” for experimenting with trochoids:

hypocycloid.ggb

The last screen shot shows a fractal created by the following dynamic system of dif-ference equations. Five sliders support us changing the attractor in a very short time.

This is a MATHEMATICA product:

http://demonstrations.wolfram.com/PeterdeJongAttractors/

As we are in Spain, let me close with a Spanish product. Ramon Eixarch provided a simultaneous 2D- and 3D-plot connected by one slider for variable a:


As we are in Spain, let me close with a Spanish product. Ramon Eixarch provided a simultaneous 2D- and 3D-plot connected by one slider for variable a:

simultaneous.wiris

How can sliders support teaching mathematics? A Summary. Generally spoken, animation is always nice to look at, but only looking at it is not enough. There is a big danger, that students lean back watching a wonderful animation prepared by their teacher – and that was it. I prefer sliders: they can be introduced by the students them-selves and they must be activated – by pressing one or the other key – so the students are ac-tive and each pressed key returns another value for the chosen parameter in connection with a change in the graphic representation.

I can imagine giving a task like the following:

The picture below is a tool for investigating the equilibrium point of Supply and Demand (MATHEMATICA).

(1) Reproduce this slider supported representation with your tool. (2) Explain the parameters. Which parameters make sense? (3) Create a similar tool showing the Break-Even-Point (linear cost

and revenue function).


The students can easily change the range of the parameters and investigate their do-main.

Sliders in connection with the algebraic (symbolic) representation are an excellent means for giving more insight in many concepts and methods. They do not only form a link between dynamic geometry and algebra - as demonstrated in the locus example - but they form also a link between numerical calculation and algebraic representa-tion. Sliders allow simultaneous working with many numeric realisations of algebraic variables considering their structure of connection (discussion with Josef Lechner, Austria).

Many of the mathematics tools used in school provide sliders. I tried to demonstrate that in most cases you can transfer the ideas to the tool of your choice.

Unfortunately sliders are a “nice toy” in the opinion of most users. They all have not recognized and realized the didactical power of sliders (and animation, of course). I believe that the influence of using sliders in math education has not been investi-gated very much until now but it might deserve a deeper research in the future.

(All files are available in request. Write to [email protected])

References

DERIVE Newsletters (can be downloaded from www.austromath.at/dug

TI-Nachrichten 1/07, 2/08, 1/10

Help files and manuals to all software tools mentioned in the paper

http://demonstrations.wolfram.com/

http://jmora7.com/

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