Introduction to Geogebra

Transcription

1 Aims Introduction to Geogebra Using Geogebra in the A-Level/Higher GCSE Classroom To provide examples of the effective use of Geogebra in the teaching and learning of mathematics at A-Level/Higher GCSE. To develop your skills in using Geogebra. To consider how to integrate ICT into your teaching. Useful Links Geogebra Website: Geogebra Forum: Geogebra Resource Repository: MEI Use of ICT Tasks for Geogebra: Patrick Cobb, Further Maths Support Programme

2 Views Getting started with GeoGebra There are 3 views (or panels) that can be displayed: Algebra, Graphics and Spreadsheet. The default on opening is the Algebra and Graphics view. To see the Spreadsheet select: View > Spreadsheet view Entering objects There are 3 main ways to enter objects: Geometrically: The square buttons across the top of the screen are drop-down menus. The current tool will be highlighted. e.g. select New Point from the second menu and click on screen to add a point, if you click in one of quadrants it will be a free point, if you click on one of the axes it will be fixed to the axis. The menus are grouped into similar objects: e.g. all the objects related to lines are in the third menu. The text when hovering over a tool indicates which objects should be selected. Algebraically: Enter an equation, constant or pair of coordinates in the Input bar at the bottom of the screen (and press enter): e.g enter y=x^2+2, k=2 or (3,1). Spreadsheets: Enter a number or expression into the cells of the spreadsheet. Scaling the axes To move the axes select Move Graphics View: from the last menu. With this selected you can scale in or out on one of the axes by hovering over it until the cursor changes then clicking and dragging. You can zoom in and out easily if you have a scroll button on your mouse. Alternatively zoom in or zoom out can be found in the last menu.

3 Useful features in GeoGebra Graphics style bar The graphics style bar can be displayed by clicking the triangle in the Graphics bar. This allows the axes and grid lines to switched on/off easily and for the style of objects to be edited. Sliders Sliders can be added by selecting Slider from the 11 th menu: and then clicking on the screen: e.g. add 2 sliders and call them m and c then enter the equation y=m*x+c. Text Text can be inserted by selecting Insert Text from the 10 th menu: Static text can entered directly. Dynamic text can be built using the Objects drop-down in the Edit Text box. Generating constants based on the coordinates of a point To convert the coordinates of the point A to variables enter p=x(a) and q=y(a) into the input bar. Copying the screen (to paste into a document) To copy the screen select Edit > Graphics View to Clipboard. A useful tip is to reduce the window so the picture is the desired size for the document before copying. Alternatively print screen can be used. Images Images can be added by selecting Insert Image from the 10 th menu: The transparency can be altered using Object properties > Style: Filling. The corners can be locked to points using Object properties > Position.

4 Generating a dynamic line in the form y y 1 = m(x x 1 ) Adding a point (x 1, y 1 ), a slider, m, for the gradient and the line Add a New Point, A, (2 nd menu). In the input bar type x_1=x(a) and press enter. In the input bar type y_1=y(a) and press enter. In the input bar type y-y_1=m*(x-x_1) and press enter. Adding a general point and creating the third point in a gradient triangle Add a New Point, B, (2 nd menu) on the line (the cursor should change as you hover over the line). Use Perpendicular Line (4 th menu) to construct perpendiculars to the y-axis through A and the x-axis through B. Use Intersect Two Objects (2 nd menu) to find the intersection of the perpendicular lines, C. Creating the gradient triangle Hide the perpendicular lines: click on the blue circles next to the lines in the Algebra pane. Use Segment between Two Points to add segments AC and BC. Right-click on each of the segments and select Object Properties then enable Label > Value. Right-click on the segments and rename them as diff_x and diff_y. Adding the dynamic text Use Insert Text (10 th menu) to add a text-box. Enter diff_y=m diff_x. diff_x, diff_y and m should be selected from Objects. can be found in Symbols. Use Insert Text (10 th menu) to add a text-box. Enter y-y_1=m(x-x_1). x_1, y_1 and m should be selected from Objects. Enable the LaTeX formula.

5 Generating a dynamic graph transformation Creating and translating the graph of In the input bar type f(x)=x^2-2*x and press enter. Select the curve and then use the Graphics Style bar to change the colour. In the input bar type g(x)=f(x+a). Geogebra will automatically create a slider called a. Move the slider to move the graph. Go to Options Algebra Descriptions and change to Definitions. Changing the type of transformation or the graph Click on g(x) in the Algebra view and change it to g(x)=f(x)+a. Then change to f(a*x). f(x) can also be changed by clicking in the Algebra view. The transformation can be animated by right clicking on the slider a and turning animation on. Adding a point and a tangent to a curve Adding the function and a point and the tangent In the input bar type f(x)=x^2 and press enter. Add a New Point, A, (2 nd menu) on the curve (the cursor should change as you hover over the line). Add a Tangent (4 th menu) to the curve at the point A. Measuring the slope and adding the dynamic text Measure the Slope (8 th menu) of the tangent. In the input bar type x_1=x(a) and press enter. Use Insert Text (10 th menu) to add a text-box. Enter The gradient at x=x_1 is m x_1 and m should be selected from Objects.

6 Investigating Graphs Straight lines 1. Generate a dynamic y = mx + c. Questions: a. How does m affect the line? b. How does c affect the line? c. For a given x-value what is the y-value (or vice-versa)? d. Can the line ever be horizontal/vertical? e. What does m tell you about any two points on the line? f. Give me an example of a line i. that passes through (2,5). ii. that passes through the origin. iii. that passes through the x-axis at x = Generate a line from a point and a gradient using y y 1 = m(x x 1 ) Questions: a. Is this always equivalent to some y = mx + c? b. What does the equation tell you about other points on the line? Give an example of another point on the line? c. Can the line ever be horizontal/vertical? d. Give me an example of a line i. that passes through (2,5) (where x 1 2) ii. that passes through the origin. iii. that passes through the x-axis at x = 3. Quadratics 3. Generate a dynamic version of f(x) = ax² + bx + c Questions: a. What is the effect of varying c? b. What is the effect of varying a? c. What is the effect of varying b? hint consider the turning point using the command turningpoint(f) and turning the trace on. d. Give me an example of a quadratic i. that passes through the origin. ii. that passes through (2,5) iii. that cuts both the positive and negative x-axis. iv. that passes through (1,1) and (2,5). v. that doesn t cut the x-axis.

7 4. Generate a dynamic version of y = ax² + bx + c Questions: Circles a. With a = 1 can you describe the conditions on b and c needed so that: i. The curve crosses the x-axis at two different points. ii. The curve touches the x-axis at a single point. iii. The curve doesn t cross or touch the x-axis? b. With a 1 can you describe the conditions on a, b and c needed so that: i. The curve crosses the x-axis at two different points. ii. The curve touches the x-axis at a single point. iii. The curve doesn t cross or touch the x-axis? 5. Create a dynamic circle: (x a)² + (y b)² = r² Questions: a. Give an example of values of a, b, and r such that the circle: i. is completely in the top-right quadrant ii. cuts the positive x-axis twice and the positive y-axis twice iii. cuts the positive and negative x-axis and the positive and negative y- axis. iv. cuts the positive x-axis twice and touches the y-axis. v. goes through the origin. b. If the scales are ignored how many different cases are there? N.B. A distinct case is defined by the quadrant/axis that contains the centre and the number of distinct intersections with the positive and negative x and y axes. c. Can you give values of a, b, and r that are an examples of each case. 6. Soddy circles (an investigation) a. Construct a triangle with one of the vertices at the origin and one on the x-axis. b. Construct three circles so that each pair of circles meet each other at a single point wherever the vertices are (these are known as Soddy circles).

8 More advanced commands Trigonometric Functions For trig functions sin, cos, tan, cosec, sec, cot, arcsin, arccos, arctan can all be input directly e.g. f(x)=sinx + arctanx. Degrees/Radians Geogebra defaults to using radians. To change the values on the x-axis to multiples of π you need to go to the Graphics settings - ensure none of the objects in the Graphics panel are selected and right click on a blank section of the Graphics panel. Select the bottom option Graphics, click on the x-axis tab then select Distance and your chosen units. To use degrees input sin(xᵒ) - the degree symbol ᵒ can be found by clicking on the α symbol on the right of the Input bar (or press Alt and o together), note that the brackets are important! You will need to go into Graphics settings and in the Basic tab change the max and & min x-values to something sensible (-90 and 670 for example), you can then go to the x-axis tab and change the Distance to 30. Using Trig Functions Trig Example 1 Input f(x)=sinx^2+cosx^2 (conveniently Geogebra interprets sinx^2 to be sin 2 x). Trig Example 2 Input f(x)=3sinx+4cosx Make this dynamic by with two sliders, a & b, and then defining f(x)=a*sinx+b*cosx (the * to denote multiplication is required here otherwise Geogebra will read this as arcsinx)

9 Parametric Curves To define a curve parametrically use the command Curve in the input bar. The syntax for Curve is Curve[ <Expression for x>, <Expression for y>, <Parameter Variable>, <Start Value>, <End Value> ] Parametric Example 1 To draw the ellipse defined parametrically by, with type the following into the Input bar Curve[2sin(t), 3cos(t),t, 0, 2pi]. Slow Plots The Curve command can be used to create an animated or slow plot. Define a slider a with min=-5, max=5 and increment To animate y=x 2 type the following into the Input bar Curve[t,t^2,t,-5,a]. This will plot a curve x=t and y=t 2, ie y=x 2, for. As a is increased the graph will be drawn for more values of x. Parametric Example 2 In a new window, define a slider a with min=0, max=2pi and increment Then define the following Curve[4cos(7t),4sin(4t),t,0,a] (this is a Lissajou curve). Move, or animate, the slider from 0 to 2π to see the curve being plotted.

10 Other ideas and useful commands Geogebra has an enormous range of functions and commands as well as a spreadsheet view, scripting and CAS view (Computer Algebra System). Below is just a selection of some useful commands. To convert the coordinates of a point A to variables p and q, use p=x(a) and q=y(a). Right click on a point or object and select Trace On to see the previous positions of the point or object as it moves. To integrate a function f(x) use Integral[f]. To obtain a polynomial through given points use Polynomial[<list of points>] e.g. Polynomial[(1,2),(3,4),(5,8)] creates a quadratic through those points. To define a complex numbers type z=3+4i or z=e^(3i). Alternatively right click on an existing point and change it to a complex number in the algebra tab. When converting between radians and degrees Geogebra actually uses ᵒ as the constant, therefore if you input pi/ᵒ this gives 180. This can be useful if trig graphs aren t being drawn to the scale you expect! To get Greek letters easily use alt and the Latin equivalent e.g. Alt and t gives θ. For subscripts. z_1 will display z 1. Matrices go to the spreadsheet view and type in the values for your matrix. Select the relevant cells, right click and select Create Matrix.