These courses are realized in the ESF project JPD /NP1 045 MOVYP, English

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1 RESOURCES TO ENGLISH VERSION OF COURSES FOR ESF PROJECT JPD /NP1 045 MOVYP These courses are realized in the ESF project JPD /NP1 045 MOVYP, English version. The project is founded by European Social Funds. There are three courses available: Derive (work with CAS system Derive in teaching of mathematics at secondary schools and universities) Cabri (using this excellent tool for teaching of geometry in the secondary schools mathematics) Excel (interactive excel spreadsheets in teaching of mathematics at secondary schools and universities) Courses were realized in October November 2007, for mathematics pre service teacher students. Now we introduce briefly content of abovementioned courses: DERIVE Derive is a mathematical computer program, called a computer algebra system (CAS). It processes algebraic variables, expressions, equations, functions, vectors, and matrices. Derive can perform numeric and symbolic computations, algebra, trigonometry, calculus, and plot graphs in 2 and 3 dimensions. It is a powerful tool for doing and applying mathematics, for documenting mathematical work, and for teaching and learning mathematics. Through solving typical mathematical school level problems, students will learn to handle Derive and experience how to use it as a pedagogical tool. In the course we would present some functions of this software those are useful for the teaching of mathematics in these areas: General Mathematics Operations

2 Entering, Simplifying and Approximating Expressions Specifying Numeral Precision Using Built In Functions Symbolic Math Expanding Polynomial and Rational Expressions Factoring Polynomial Expressions Solving Equations Differentiating Expressions Integrating Expressions Graphing Expressions Worksheets Plotting Functions in a 2D Plot Window Framing 2D Plots Tracing, Annotating and Deleting Plots Creating Mathematical Documents Saving and Sharing Your Work For teaching of this stuff we will use program Derive 6 bought by the Department of Algebra, Geometry and Didactics of Mathematics, Faculty of Mathematics, Physics and Informatics, Bratislava and videos from the page: COPYRIGHT 2001, TEXAS INSTRUMENTS INCORPORATED. The mathematical problems on those we will display the stuff would by created by us or on the basis of available materials. For better illustration of Derive and possibilities of its use we will now present information from the page: Derive is easy to use but very powerful and accurate. Floating toolbars give quick access to Greek and other mathematical symbols. It verifies results and explores

3 different approaches to problems. Derive 6 now shows also intermediate steps in calculation (see fig. 1). Fig. 1 Example of Derive screen Another useful Derive Tools are: Maths Worksheets Create, edit, print and save interactive mathematical worksheets Format mathematical expressions, text, graphic images, and OLE objects Open multiple worksheets, 2D, and 3D plot windows Cut, copy and paste objects between worksheets Customise print headers and footers and print preview Write worksheets in Rich Text Format Load and analyze numeric data files Generate C, Fortran, Pascal, and Basic program files Expression Entry Easy linear entry and editing of expressions Enter Greek variables and mathematical symbols using dockable toolbars 2D matrix entry and editing Templates for common Algebra and Calculus functions Expressions displayed using standard mathematical notation View the elapsed time and memory used during long computations Subexpression highlighting, extraction and substitution Move and automatically renumber expressions

4 Communicate with TI CAS calculators: import data from and export data to TI 89, TI 92+, Voyage 200 calculators Customise menus, toolbars and shortcut keys Optional multi line editing Fully scaleable Derive Unicode font State variables saved in DfW files Parentheses matching Plot Windows Plot space curves and complex valued functions Trace plots with moveable cross Plot data points and Boolean combinations of surfaces Zoom in or out, or use automatic scaling Specify plot colours and aspect ratio Annotate plots, number and label axes Embed plots into math worksheet or export as a TIFF, JPEG or BMP Print plots in colour or black & white 2D Plots Explicit, implicit, and parametric plots Plot using rectangular or polar coordinates Specify plot range using a "zoom" box Plot real and imaginary parts Make plots dynamic: animate expression plots with slider bars Let plots be labelled with defining equations 3D Plots Plot surfaces in rectangular, cylindrical and spherical coordinates Rotate one or more plots in real time with the mouse or direction controls Make plots dynamic: animate expression plots with slider bars Let plots be labelled with defining equations Controllable display of 3D mesh lines

5 Controllable point size of 3D data point plots Arithmetic Exact rational arithmetic with no roundoff error Adjustable precision approximate arithmetic Rational, decimal, and scientific notation Decimal, hexadecimal, octal, or binary arithmetic Real, imaginary, complex, and infinite numbers Prime number recognizers and generators Efficiently factors large integers Functions for gcds, lcms, and mods Pseudo random number generator Algebra Expand and factor polynomials Partial fraction expansion of rational functions Simplify expressions using powerful identities Solve equations and inequalities exactly or numerically Solve systems of polynomial equations, linear and nonlinear Exponential, trigonometric, and hyperbolic functions Specify angles in degrees or radians Piecewise continuous functions Riemann zeta, gamma, and error functions Probability, statistics, and financial functions Substitute values for variables or subexpressions Integer, real, and complex variable declarations Define functions and assign variable values Number theory utility library Calculus Display the steps in the simplification of an expression with optional display of transformation rules

6 Finite and infinite symbolic limits First and nth order partial derivatives Antiderivatives and definite integrals Approximate numerical integration Finite and infinite sums and products Curvatures, tangents, and osculating circles Implicit and parametric differentiation Taylor and Fourier series approximations Arclengths, areas, and volumes Laplace transforms Solve first and second order ODEs exactly Runge Kutta approximations for systems of ODEs Utility libraries for special functions Vector, Matrices, and Sets Symbolic and numeric elements Uses standard subscript notation Dot, cross, and outer products Transpose, determinant, inverse, and trace Matrix reduction to row echelon form Eigenvalue and eigenvectors Noncommutative vector and matrix algebra Differential and integral vector calculus Efficient functions for vector and matrix operations Least squares curve fitting Set union, intersection, and power sets Programming Both functional and multi line, procedural programming Block, loop, and if then else control constructs Boolean and relational operators for predicates Define default initial values for local variables Expression type recognizer functions

7 Expression component extraction functions Map functions over the elements of vectors and sets Sort and search the elements of sets and vectors Extract and replace elements of vectors using subscripts Make variable substitutions serially or in parallel String processing functions Documentation and Support Manual with step by step visual examples Reference guide and improved and extended on line help Free support groups Training videos on CD ROM

8 The following screenshots are courtesy of Terence Etchells, who has used Derive to teach both at Bolton Sixth Form College and at Liverpool John Moores University (taken from the web page:

9 The following screenshots are courtesy of David Bowers, who has used Derive to teach at Suffolk College.

10 Below: Change the value of variables with on-screen sliders to dynamically graph new values. Below: Rotate plots with mouse or direction arrows.

11 Below: Derive worksheet containing annotated 3D graphic and Derive maths session

12 Below: Derive worksheets let you present questions, workings (now with intermediate steps) graphs and results in one transportable document.

13 Below: Derive shows intermediate steps in calculations to help students learn the correct procedure and to verify their results and methods.

14 CABRI Cabri II Plus is a geometry construction tool allowing users to produce figures in the same way as one would using a sheet of paper, pencil, rule, compass and eraser, bringing a whole new dimension to their constructions: Constructions elements can be freely manipulated (dragged by the user), Constructions are updated in real time, Images can also be integrated into word processing documents or distributed via the Web (CabriJava). In our course students will learn how to use basic features of Cabri for the preparation of their mathematics lessons and also for the process of geometry teaching. In order to this we will use materials available on the Internet or by other sources. The manuals for the use of Cabri are from the official Cabri web site: The worksheets for our course are based on the worksheets from the site: ry.html The worksheets are dealing with these themes: 1. Cabri Geometry Tour 2. Properties of Parallel Lines Cut by a Transversal 3. Exterior and Interior Angle Theorem

15 4. Isosceles Triangle 5. Bisectors 9. Properties of Parallelograms 10. Special Parallelograms 12. Transformation 13. Similar Figures 14. Sin, Cos, and Tan of Right Triangles 15. Triangle Proportionality 16. Radius, Diameter, and Circumference of a Circle 17. Perimeter and Area of a Square 18. Relationship of Angles to the Circle For better illustration we now present some screenshots from the worksheets: Worksheet: Cabri Geometry Tour, download from the web site: ry.html TEXAS INSTRUMENTS INCORPORATED Name Date Cabri Geometry Tour

16 In this activity, you will practice some of the basic functions of the Cabri Geometry software, including use of the Pointer, Point, Line, Measure, and Construct toolbars. 1. Create and label points. 1a. From the Points Toolbar, select Point. 2b. Move the pencil to any location in the plane and click. Immediately type the name of the point (for example, A, B, C). Repeat this process to make 5 points. 3c. If you forget to label a point, select Label from the Display Toolbar. Move the crosshair near a point. When the message This point appears, click once. A box should appear. Type the label you wish to give this point (for example, A, B, C). Move the crosshair away from the point and click to remove the label box. Repeat this process for each of the points that are still unlabeled. 2. Correct an error. 1a. From the Pointer Toolbar, select Pointer. 2b. Move the crosshair to the point or line you wish to remove. When the appropriate message appears (for example, This point), click to select that figure. (The selected item will begin flashing). Press the DELETE key. The selected figure will disappear. 3. Clear the screen. Click on Edit and then click on Select All. Press the DELETE key. or Press and hold the CRTL key and type A. Press the DELETE key. 4. Create two intersecting lines. a. From the Lines Toolbar, select Line.

17 b. Move the pencil to the plane and click once. Type a point name (for example, A, B, C). c. Move the pencil to a second spot and click once. This will create a line. d. Repeat the first three steps to create a second line so that it intersects the first. e. From the Points Toolbar, select Intersection Point. f. Move the pencil to the intersection point until the message Point at this intersection appears. Click once. Type the point name. g. From the Points Toolbar, select Point On Object. h. Move the pencil until the message On this line appears. Click once. Type the point name. 5. Clear the screen. 6. Create an angle. a. From the Lines Toolbar, select Ray. b. Click on the screen where you want the vertex of the angle. Type B. c. Move the pencil and click once to establish one side of the angle. d. Move the pencil to the vertex until the message This point appears. Click once. e. Move the pencil to create an acute angle and click to establish second side. f. From the Points Toolbar, select Point On Object. g. Move the pencil to one ray until the message On this ray appears. Click once. Type A. h. Move the pencil to the second ray until the message On this ray appears. Click once. Type C. 7. Measure an angle. a. From the Measure Toolbar, select Angle. b. Move the cursor to point A. The message This point appears. Click once. c. Move the cursor to point B. The message This point appears. Click once.

18 d. Move the cursor to point C. The message This point appears. Click once. e. The measure of the angle appears with a flashing bar. This allows you to type the name of the angle (for example, ABC = 45.5). 8. Change the angle. a. From the Pointer Toolbar, select Pointer. b. Move the cross bar until you see the message This ray. c. Click on one of the rays on the angle and hold until it becomes a hand. d. You can now drag the angle and watch the angle change. 9. Bisect an angle. a. From the Construct Toolbar, select Angle Bisector. b. Move the cursor to point A. The message This point appears. Click once. c. Move the cursor to point B. The message This point appears. Click once. d. Move the cursor to point C. The message This point appears. Click once. e. Place a point on the bisector in the interior of the angle. f. Using the new point, measure one of the angles created by the bisector. g. This measure should be half of the original.

19 1.10. Clear the screen Create parallel lines. a. From the Lines Toolbar, select Line. b. Move the pencil to the plane and click once. c. Move the pencil to a second spot and click once. This will create a line. d. From the Construct Toolbar, select Parallel Lines. e. Move the pencil to the line until the message Parallel to this line, appears. Click once. f. Move the pencil off the line and click once. This creates the parallel line.

20 12. Clear the screen. 13. Create perpendicular lines. a. From the Lines Toolbar, select Line. b. Move the pencil to the plane and click once. c. Move the pencil to a second spot and click once. This creates a line. d. From the Construct Toolbar, select Perpendicular Lines. e. Move the pencil to the line until the message Perpendicular to this line appears. Click once. f. Move the pencil off the line and click once. This creates the perpendicular line Clear the screen Create and label a triangle. a. From the Lines Toolbar, select Triangle. b. Click once, type A, and drag on the screen. c. Click again to end the segment and create one side of the triangle. Type B, drag, and click again to finish the triangle. Type C.

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22 EXCEL Microsoft Excel is a spreadsheet program that allows users to insert and manipulate data. In our excel courses we will teach student how to use Interactive Excel Spreadsheets in teaching of mathematics on secondary schools or universities. Doing this we will use these materials and internet resources: Interactive Excel Spreadsheets: A Computational and Conceptual Learning Tool for Mathematics and Science at Computation and Science for Teachers (CAST) Workshop held at the Pittsburgh Supercomputing Center, in Pittsburgh, PA in June Interactive Excel Spreadsheets: A Visualization Tool in Mathematics and Sciences at MICCA: An Association for Educators Using Technology 2006 Conference in Baltimore in May Excelets: A Learning Tool to Captivate the Google Generation at MICCA: An Association for Educators Using Technology 2007 Conference in Baltimore in April Click and Think over Chalk and Talk: The New Pedagogy via Interactive Excel Spreadsheets in Mathematics and the Sciences, at AFACCT 17th Annual Conference at Harford Community College in January Interactive Excel Spreadsheets: Constructing Visualization Tools to Enhance Your Learner centered Math and Science Classroom at Powering Up with Technology at Northwestern High School in November Additional resources are included in electronic documentations.