Margarida Oliveira Escola EB 2,3 Piscinas Lisboa CMAT- Minho University Learn mathematics with Excel

 Although we currently have on disposal animated modules which ease the understanding of mathematics, we defend that the advantages will be greater if computational modules are built by the user himself as a result of the mastering he will get of the mathematical concepts. This suggests in particular the possibility of adopting teaching methods which are support by basic programming techniques.  The goal is to build with students computational simulations and show that the way the students choose to build those simulations, can give them the possibility to deepen their mathematical knowledge. The central idea of the study

- The project was developed during the last year with two classes of seventh grade. - Students worked every week in the classroom with laptop using a spreadsheet or a geometry dynamic software - If the subject of the maths class was about numbers, functions, similarity they used excel. If the subject was plane geometry they used geogebra. - In each class students developed an computational application guided by teacher. At the end they send the work for the mail of the teacher and he give some suggestions to improve their work. - It was developed a web page. The Project “Matemática Dinâmica”

In the webpage we put not only the students work but also the worksheets and computational applications developed by teacher. The opinions of the students about this type of work. And the suggestions to continue this project.

 In the current school year, we developed a project that links mathematics with physics.  The project was developed with the same classes that attended last year’s project. These classes are now formed by students of the eight grade.  In this project the students used the computational simulations developed by the teacher using Excel and Scratch. The Project “Tópicos de Física em Experimentação Virtual”

this example shows a computational application developed to study the behavior of a spring when a force is applied at its end this example shows a computational application developed to study the apparent motion of the Sun, at different latitudes here we can see some reports written by the students regarding the study of light reflection

I will show two examples aimed at high school and university students, in a more developed away 1 – Approximate value to Pi using the Archimedes method 2 – Study of pendulum oscillations using spreadsheets and numerical methods for solving non-linear differential equations

 “Method of exhaustion", as used by Archimedes to obtain an approximate value of Pi.  We begin with two triangles inscribes and circumscribed to a circle with radius 1.  Next, acting on a scroll bar, we increase the number of sides of the inscribed and circumscribed polygons.  We can then observe the graphic representation of the sequences of the correspondent polygons areas converging to the same limit, Pi. Computational module developed in Excel

 Each point on the circumference is associated with a pair of coordinates. The radius r is fixed and the angle varies from 0 0 to

 To draw, in Excel, a polygon inscribed or circumscribed in a circumference just, enter a table with the values of the coordinates of its vertices. Triangle and a square inscribed in the circle, with the vertices given in polar coordinates (a similar method can be used to draw the circumscribed polygons) Students should generalize to the other polygons

Students should introduce a scrollbar to vary the number of sides of the polygons Private Sub ScrollBar2_scroll() ScrollBar2.Min = 3 ScrollBar2.Max = 100 Cells(2, 4) = ScrollBar2.Value End Sub The code in Excel:

For each polygon of n sides, inscribed in circumference of radius r, the expression of the area in relation to the number of sides n is The expression that gives the area of each circumscribed polygon of n sides is

To obtain the differential equation describing the oscillatory motion it is enough to consider that, for each instant, the total amount of all the forces acting on the mass must be zero. A second order non linear differential equation, owing to the presence of the term involving sen, and so with no trivial solution.

 We shall begin with the organization of the spreadsheet for a simultaneous visualization of the parameters m, c, g and L and the initial conditions. Then the value of h is introduced in a cell to implement the numerical methods. How to use EXCEL in order to obtain graphical representations of the solution

 To plot the desired graphics, we select the values in column and choose Chart Wizard button. Inside we should pick a type XY scatter plot graphic. Then it is essential to format the graphic paying special attention to the scale. Graphs time-position, time- velocity and Phase Plan

Analyze the influence of the different parameters on the action of the pendulum system

Assuming both air friction and an initial velocity

The application

 Traditionally the dominant means of communicating mathematics is printed (static) text and ocasionally some (not too many) static pictures.  Software use implies that we use a richer set of communicating vehicles especially dynamic elements like animations.  Until about ten years ago the tools for creating documents with these elements for mathematicians were expensive and complicated to use.  Only recent advances in software technology have given us more easily usable and affordable tools for creating projects of this kind.

 Provide a faster, more dynamic and engaging way to demonstrate mathematical concepts than using transparencies or drawing on the board.  Can help us quickly explore variables, relationships, and the mathematics of change with our students. When we construct objects in these software, we can drag points and lines with the mouse. As shapes and positions change, all mathematical relationships are preserved, allowing us and our students to examine an entire set of similar cases in a matter of seconds.  We can use these programs across the mathematics curriculum, so we don’t need different software for each class, concept, or grade level that we teach.  The friendly user interface of those softwares allows to get a quickly acquaintance with its procedures so we can spend our time teaching mathematics, not software.

 These interactive dynamic models can be a strong motivation for students who want to get a better understanding of the different possible dynamic behaviors of these models.  The use of selectors allows a high degree of interactivity.

 By using software tools which are available on most PCs we have the important effect of showing that mathematical examples can be implemented without relying on specialized software tools.  From a computation point of view, all operations we need are very simple. Therefore, the “conceptual mechanism” of these examples is accessible to those who want to learn mathematics deeply.

 The user sees that interactive mathematical models can be created and used with tools accessible to everybody. This hopefully demonstrates that mathematical modelling is not first and foremost depending on very specialized tools, but upon insights, and can be done with standard tools.

Thank you