6. Geogebra Handout

University of Western Sydney

Mathematics education Unit

GEOGEBRA A Freeware Program To Share With Your Students And BOOST Your Mathematics Lessons Allan White University of Western Sydney You will be given a CD containing the program which you can install on any computer, and some files courtesy of Tobias Cooper. You may distribute this program to your colleagues, to your school computers and to your students and BOOST your mathematics lessons. By BOOST, I mean By Out Of School Time. The more time your students spend on mathematics outside the classroom the easier your task becomes, so give them the opportunity to experience how technology can enhance their understanding. If the file will not open it is probably because you do not have JAVA installed. This can be downloaded free from the Internet as can GeoGebra.

Activity 1: 1. Run your GEOGEBRA program 2. In the input field type: x^2 and hit enter 3. Pull down view and select grid.

GEOMETRY WINDOW (Drawing Pad)

Instructions Toolbar

Undo/Redo

Algebra Window

Input Field

Input Options

ďƒ“ Allan White "Geogebra - Geometry, Functions and Calculus Adapted with permission and thanks to Tobias Cooper from his MANSW presentation notes

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University of Western Sydney

Mathematics education Unit

Input field: This is where you can enter the equation for a graph, coordinates of a point or one of Geogebra's commands. A list of commands is viewable by clicking on the 'Command' drop down menu near the input field.

Geometry Window: This is the window where your input is displayed graphically.

Free objects can be manipulated using the arrow button Algebraic Window: This is on the left hand side. Every geometrical object has an algebraic representation - eg our example f(x) = x^2. This window can be opened and closed using VIEW. Toolbar: This is the row of buttons. The button selected will turn blue. Each button has a drop down arrow at the bottom right corner. This drop down arrow will reveal more buttons and will give some brief instructions on how it works.

Activity 2: Let Us Improve The Setup 1. To change the font size to 20 point. Go to OPTIONS select Font Size and choose 20 pt.

2. To select snap to grid which is handy for students who are developing mouse skills. Go to OPTIONS select Point Capturing and then select On Grid 3. Finally to select labelling of points. Go to OPTIONS select Labelling then select New Points Only.

Pedagogical activity If I wish to develop within my students an understanding of the effect of the variables A, B, and C on the quadratic curve y = A(x-B)^2 +C then I can use the Algebra window to good effect. For example: If I wish to begin by first helping the students understand the effect of A on the curve y = Ax^2 then I would type in an equation 0.7x^2 in the Input slot. I would then change the colour to red by clicking on the curve, selecting Properties then clicking on Color, then red and then Close (See Activity 4 Editing Objects And Their Properties for more instructions). I would then close the Algebra Window.

ďƒ“ Allan White "Geogebra - Geometry, Functions and Calculus Adapted with permission and thanks to Tobias Cooper from his MANSW presentation notes

2

University of Western Sydney

Mathematics education Unit

Now I am ready to show students my graph and ask them to guess at the equation. I will enter their guesses and get them to discuss the effect of A being greater than one, less than one and negative. I would repeat the process for the other variables. The drill and practice of the algebra can come after the students have developed an understanding.

Activity 3: Points and lines 1. Go to FILE and select NEW. Do not save Activity 1 2. Click on the Point Button (next button after arrow button) and place a point at (2,1) using the left click of your mouse.

3. Click on the Arrow Button and then drag the point around. Notice the coordinates change and in the Algebraic window.

4. Click on the Input field, and type (-3,4) and then hit ENTER. Notice what has happened.

5. To draw a line through the two points, click on the Line Button and then click on the two points on the screen.

6. Now click on the Arrow Button and then drag: (i) the line and watch the equation change, (ii) any point and watch the equation change.

ďƒ“ Allan White "Geogebra - Geometry, Functions and Calculus Adapted with permission and thanks to Tobias Cooper from his MANSW presentation notes

3

University of Western Sydney

Mathematics education Unit

7. To change the equation to the gradient-intercept form, Right Click the equation in the Algebra Window 8. Save your file to My Documents Pedagogical activity One teaching strategy for developing a greater understanding of the gradient-intercept form of linear equations could be: (i) Close the Algebra Window then move the line so that the slope remains the same but the intercept changes. (ii) Ask the class: What is the new equation? (iii) Check by going to View and opening the Algebra Window (iv) You could do similar things by changing the gradient but keeping the intercept constant. (v) You could now change both the gradient and intercept. (vi) Or if the students have GeoGebra on their laptops you could ask them to move the line so that it has the equation y = 3x - 5.

Activity 4: Editing Objects And Their Properties Right clicking and double clicking on any object in the Algebra or Geometry window 1. Right Click on the point A to see the available options (see below)

ďƒ“ Allan White "Geogebra - Geometry, Functions and Calculus Adapted with permission and thanks to Tobias Cooper from his MANSW presentation notes

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University of Western Sydney

Mathematics education Unit

2. Choose Properties and change the coordinates of point A to (2,5) and CLOSE

3. Another way is to Double Left Click on the point in the Algebra Window.

Change the point A to (3,6)

4. Or Double Left Click on the point in the Geometry Window. Change the point A to (3,6) and the colour to red.

ďƒ“ Allan White "Geogebra - Geometry, Functions and Calculus Adapted with permission and thanks to Tobias Cooper from his MANSW presentation notes

5

University of Western Sydney

Mathematics education Unit

So we have three ways of editing the properties of all objects.

Activity 5: Moving The Screen By Using The Moving Drawing Pad Button And The Undo Button 1. Click on the Moving Drawing Pad Button (the last button - four arrows) and use the Left Mouse to drag the screen around 2. Click back on the Arrow Button when you have finished.

3. Click the Undo Button to move back to last position. If you keep clicking it will eventually return to the original position.

Activity 6: Examining Functions And The Zoom Tool 1. Click the File Menu and select New. 2. Type f(x) = e^x into the Input Field using the Drop Down Menu with the degrees symbol

ďƒ“ Allan White "Geogebra - Geometry, Functions and Calculus Adapted with permission and thanks to Tobias Cooper from his MANSW presentation notes

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University of Western Sydney

Mathematics education Unit

3. Use the Moving Drawing Pad Button to move the graph to the right.

4. Click on the Drop Down Menu of the Moving Drawing Pad Button and choose the Zoom Button. Click where the function approaches the x axis to zoom in.

Keep clicking.

5. Another way is to Right Click and drag a selection box around an area you want to zoom in to. When you release it will zoom in. To return to the original view then Right Click and choose Standard View or use the yellow arrow.

ďƒ“ Allan White "Geogebra - Geometry, Functions and Calculus Adapted with permission and thanks to Tobias Cooper from his MANSW presentation notes

7

University of Western Sydney

Mathematics education Unit

Activity 7: Translating Functions 1. Click the File Menu and select New or New Window.

2. Type f(x) = x^2 into the Input Field 3. Drag the parabola around to see the equation dynamically change. This function is unique in the mathematics software world according to Tobias.

Pedagogical activity A teaching strategy for developing a greater understanding of the parabola could be: (i) Develop an understanding for how the equation is affected by changes of the vertex in the y axis only. (ii) Develop an understanding for how the equation is affected by changes of the vertex in the x axis only. (iii) Using the Algebra Window and changing the equation develop an understanding for how the equation is affected by changes of the vertex for both the x and y axes. (iv) Closing the Algebra Window and changing the position of the graph then asking the students for the new equation and reopening the Algebra Window to check their answers.

ďƒ“ Allan White "Geogebra - Geometry, Functions and Calculus Adapted with permission and thanks to Tobias Cooper from his MANSW presentation notes

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University of Western Sydney

Mathematics education Unit

Activity 8: Angles And Triangles 1. Click the File Menu and select New or New Window.

2. Click on the Drop Down Arrow Button Menu and select the Segment Between Two Points Button

3. Construct a triangle using this button by right clicking. You will notice in the Algebra View under Dependent objects, the three side lengths are shown. To check which side is represented by which letter, click on the Arrow Tool and click on one side and watch the Algebra View light up. DONT ASSUME 4. To measure the size of an angle, click on the Angle Button (seventh button) then click on the vertices for the angle ABC. A clockwise clicking will give the internal angle and an anticlockwise clicking will give the external angle.

5. Click on the Arrow Tool and drag the letters and angle values til you have a clear diagram. Now drag a vertex and examine the effect. If you drag enough you will notice reflex angles appearing. To stop this Right click the Algebra View or Geometry Window and choose Properties. Then click Basic and in the Objects column un-tick the box at the bottom that says allow reflex angle.

ďƒ“ Allan White "Geogebra - Geometry, Functions and Calculus Adapted with permission and thanks to Tobias Cooper from his MANSW presentation notes

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University of Western Sydney

Mathematics education Unit

Activity 9: Text And Calculations Using Input Field 1. Using your triangle from Activity 8 2. Click on the Drop Down Slider Button Menu (9th button) and select the Insert Text Button 3. Click once on the screen and type Angle sum of a triangle.

4. Click on the Arrow Button and drag the text to the bottom of the screen 5. Right Click on the text and choose Properties. Then change the colour and font size of the text. Click Close when you are finished.

6. To edit the text, Right Click and choose Edit then you may change the heading to Angle Sum of a Triangle Using Geogebra.

7. Click on the Input Field and click on the Drop Down Menu with the Greek letters. Use the menu to form α + β + γ. Then press Enter on the keyboard to calculate the sum. You will notice in the Algebra Window. Hover the mouse over δ = 180 to see that this pronumeral represents α + β + γ.

8. Use the Arrow Tool to drag the vertices and observe the results.

 Allan White "Geogebra - Geometry, Functions and Calculus Adapted with permission and thanks to Tobias Cooper from his MANSW presentation notes

10

University of Western Sydney

Mathematics education Unit

Activity 10: Dynamic Text The aim of this activity is to create a text box displaying the α + β + γ = δ as dynamic text.

1. Using the sheet from Activity 9 2. Click on the Insert Text Button and click once on the screen to open text dialogue box.

3. Using the Drop Down Menu create the expression α + β + γ = δ

4. In this dialogue box select the letter α using your mouse, go to the Algebra Window and click once on α (this is making the dynamic link). Carry out the same procedure for β, γ and δ.

5. Click the Apply Button.

6. Click on the Arrow Button and drag a vertex of your triangle and watch the dynamic text change.

 Allan White "Geogebra - Geometry, Functions and Calculus Adapted with permission and thanks to Tobias Cooper from his MANSW presentation notes

11

University of Western Sydney

Mathematics education Unit

Pedagogical activity I like to give my students the following instructions. Drag the vertices until you form triangles with sides (i) 3,4,5 (ii) 5,12,13 and (iii) 6,8,10 . I ask them to tell me something interesting about these triangles.

WE CAN NOW DO THE SAME FOR THE SIDES OF THE TRIANGLE. 7. Right click on the side 'a' and choose Properties. Click on the Basic heading in the top Options menu if not shown. From the Show label drop down menu choose Value. Then push Close.

8. Notice the values for the sides are now displayed. You can use the Arrow Button to move the labels. If the snap to grid is not what you want then use the menu sequence Options/ Point Capturing/ On.

ďƒ“ Allan White "Geogebra - Geometry, Functions and Calculus Adapted with permission and thanks to Tobias Cooper from his MANSW presentation notes

12

University of Western Sydney

Mathematics education Unit

Pedagogical activity repeated The previous activity can be repeated but with the following condition. Go to Point Capturing and turn it On (as above) and not On the grid. Drag the vertices until you form triangle with sides 5,12,13. The students will have to use the Zoom Out and to get exact sides by entering the values by clicking twice on the sides in the Algebra Window.

Activity 11: Inflexions, stationary points and derivatives The aim of this activity is to use the available commands (bottom right corner) for the Input Field. We will use the commands Extremum and Inflection Point which can be used only on polynomials.

1. Click on the File Menu and choose New Window.

2. Enter the polynomial f(x) = x^3 - 6x^2 + 9x - 3 in the Input Field using the commands 3. Type in Extremum[f] using the commands

ďƒ“ Allan White "Geogebra - Geometry, Functions and Calculus Adapted with permission and thanks to Tobias Cooper from his MANSW presentation notes

13

University of Western Sydney

Mathematics education Unit

4. Type InflectionPoint[f] using the commands

5. Type f ' (x) to plot the derivative function. Notice the equation of the derivative function appears in the Algebra Window.

6. Colour code f(x) blue and f ' (x) red by Right Click on the curve and then select Properties / Colour and click on the desired colour.

7. Observe what happens as you drag f(x) up and down and left and right.

8. Save the file as Derivative Function.

ďƒ“ Allan White "Geogebra - Geometry, Functions and Calculus Adapted with permission and thanks to Tobias Cooper from his MANSW presentation notes

14

University of Western Sydney

Mathematics education Unit

Activity 12: Tangents 1. Using your Derivative Function file from Activity 11.

2. Use the menu sequence Options/ Point Capturing/ On. 3. Use the Point Button to place a point on the curve f(x).

4. Click on the Drop Down Menu of the Perpendicular Line Button (4th button) and choose the Tangent Button 5. Click once on the point and once on the function 6. Click on the Arrow Button, and use Ctrl and the Left Mouse to stretch the x axis. (If you don't grab the x axis it will merely move the whole graph to the left)

ďƒ“ Allan White "Geogebra - Geometry, Functions and Calculus Adapted with permission and thanks to Tobias Cooper from his MANSW presentation notes

15

University of Western Sydney

Mathematics education Unit

7. Explore by dragging the tangent along f(x). Notice the gradient of the tangent changes in the Algebraic Window 8. To save your work use the menu sequence File/ Save As/ Derivative plus tangent.

Activity 13: Integration 1. Click on the File Menu and choose New Window 2. Click once in the Input Field and type in x^2 and press Enter 3. In the Input Field also type Integral [f,0,3] to display the area under the parabola from 0 to 3. The area is displayed in the Algebraic Window.

4. Click on the Move Drawing Pad Button (10th Button) and drag the graph down to the bottom right of the screen so all the area under the curve is visible.

5. Click on the Slider Button (9th Button) and click once on the screen

6. To set up the number of rectangles that will change with the moving of the slider.

Change the name to n and the min to 0 and the max to 20 and Increment to 1 and press Apply.

7. Click in the Input Field and type UpperSum [f,0,3,n] and press Enter. Use the Arrow Button to separate the 'a' and the 'b' labels if needed.

8. Use the Arrow Button to drag the slider n to see the area of the rectangles approach the exact area. How many are needed? ďƒ“ Allan White "Geogebra - Geometry, Functions and Calculus Adapted with permission and thanks to Tobias Cooper from his MANSW presentation notes

16

University of Western Sydney

Mathematics education Unit

9. If you wish to increase the number of rectangles above 20 then Right Click the slider to get Properties. Then choose the Tab Slider and change the value to 200 and press Close.

10. Again use the Arrow Button to drag the slider n to see the area of the rectangles approach the exact area.

11. To save your work use the menu sequence File/ Save As/ Integration

ďƒ“ Allan White "Geogebra - Geometry, Functions and Calculus Adapted with permission and thanks to Tobias Cooper from his MANSW presentation notes

17

University of Western Sydney

Mathematics education Unit

Activity 14: Sliders and the Quadratic Function (Note: This could follow on from the earlier pedagogical comments regarding quadratics) 1. Click on the File Menu and choose New Window 2. Click once on the Input Field 3. Type a = 1 and press Enter 4. Type b = 1 and press Enter 5. Type c = 1 and press Enter 6. Type in f(x) = a*x^2 + b*x + c and press Enter. The equation should appear in the Algebraic Window.

7. In the Algebraic Window, Right Click on 'a' and choose Show Object to see slider 'a' on the graph. Repeat this set of instructions for 'b' and 'c'.

8. Click on the Arrow Button and use it to drag the sliders and examine the effect on the graph.

9. Right click on each slider and colour code each.

10. To save your work use the menu sequence File/ Save As/Parabola with sliders Note: To change the max, min or increment of the slider, right click and choose properties. ďƒ“ Allan White "Geogebra - Geometry, Functions and Calculus Adapted with permission and thanks to Tobias Cooper from his MANSW presentation notes

18

University of Western Sydney

Mathematics education Unit

Activity 15: Creating Classroom Learning Objects by Fixing Objects It is often a good idea to fix the objects you do not want the students to change. This can be done by right clicking on the object and selecting Properties then choosing Fix Object.

For example: Create a new window and select Point and put a point at (2,3). Select the Arrow tool and right click on the point A and fix it. Now select Centre with circle through point and then click on A and move away from A and click again and you will have a circle centre A which is fixed and B which you can drag to change the equation of the circle. Select Segment between two points and click on A and B then select the Arrow tool. Now click on the segment and go to Properties then change the label to Value. Thus the students are able to see the value of the radius.

Use you knowledge of inserting text to give this sheet a heading such as Changing the radius in the circle equation or something similar and then add a set of instructions that will lead the students to investigating the link between centre, changing radius and the equation.

Other Pedagogical Notes Using GeoGebra there are many possible ways to deliver classroom learning objects and material such as: 1. The provision of written notes outlining a lesson where students log into the school and either install web GeoGebra or use installed version. Students recreate the lesson in Geogebra following written instructions.

2. If students have laptops then a class distribution list allows you to email a GeoGebra learning object which can contain instructions, demonstrations and questions.

3. If students have internet access (home or laptops), GeoGebra learning objects can be stored in lesson profiles on Noodle or the school's elearning system.

KIDS DONT HAVE COMPUTERS OR GEOGEBRA BUT HAVE ACCESS TO THE INTERNET THEN THERE IS STILL A WAY TO ACCESS 4. Export GeoGebra as Web pages that are interactive. If you join the GeoGebra wiki at http://www.geogebra.org/en/wiki/index.php/Main_Page it will explain how to do it.

Students can then explore and interact with the objects without having to have GeoGebra on their computer as they are html files.

ďƒ“ Allan White "Geogebra - Geometry, Functions and Calculus Adapted with permission and thanks to Tobias Cooper from his MANSW presentation notes

19

University of Western Sydney

Mathematics education Unit

FILES PROVIDED WITH THE DISC TO DOWNLOAD File Name Description Teaching Ideas 1st_2nd_derivative.ggb Cubic function Using the arrow button to with the first move the turning points in derivative and order to examine the effect second derivative upon the gradient and functions. second derivative function.

Absolute_Value_Interactive.ggb Absolute function Using the arrow button to f(x) = abs(x) move the vertex around the grid and examining the effect upon the equation alt_coint_angles.ggb Examining the Using the arrow button to alternate and comove the lines to achieve interior angles of parallel lines two lines cut by a transversal angle_alt_segment.ggb Examining the Using the arrow button to tangent to a circle move the points on the and the angle in the circle to examine the effect alternate segment on the angles. Also allows the confirmation of the angle in a semicircle is a right angle. angle_subtended_chord.ggb Examining equal Using the arrow button to chords subtend move the points on the equal angles on the circle to examine the effect circumference on the angles.

Angles of any magnitude.ggb The unit circle for Using the arrow button to Angles of any magnitude(2).ggb the trigonometric move the points on the (Two files) ratios circle to examine the effect on the trig ratios in any quadrant Area_Trapezoid_worksheet.ggb Has a slider and a Using the arrow button to trapezium. move the slider to examine the effect of rotating the trapezium to get the resulting shape of a parallelogram hence the 1/2 in the trapezium formula. bart.ggb A cubic function, a Using the arrow button to tangent at a point move the fixed shape graph and Bart Simpson to examine the effect on the surfing the tangent tangent at the fixed point and of course Bart Basic_Hyperbola.ggb A hyperbolic Using the arrow button to function y = a/bx move the sliders to and two sliders examine the effect on the allowing you to graph change a and b ďƒ“ Allan White "Geogebra - Geometry, Functions and Calculus Adapted with permission and thanks to Tobias Cooper from his MANSW presentation notes

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University of Western Sydney

circle2.ggb

Circle Equation. ggb

Mathematics education Unit

A circle and a choice of examining 5 circle properties, 3 Cyclic quadrilaterals or 4 tangent properties A circle with a centre A and point on the radius B.

Conic_foci.ggb

An ellipse with focii and points on the ellipse

Dynamic limit with cont.ggb

A cubic and two points on the continuous curve A cubic and two points on the discontinuous curve the absolute graph of y=a|bx+c|+k with sliders for a,b,c,and k A hyperbola with sliders The curve y=a√bx+c+k and sliders for a,b,c,and k Using sliders to change the coordinates of the centre and radius Two tangents drawn from an external point to a circle The hyperbola y=a/x with a slider for a Has a map and bearings from Butterworth to Jakata A cubic, a gradient function and a tangent to the curve

Dynamic limit with discont.ggb Dynamic_Absolute_value.ggb

Dynamic_Hyperbola.ggb Dynamic_SquareRoot.ggb

eqn_circle.ggb

Ext_point_Tangents.ggb

Family_of_Hyperbolas.ggb GGB5c_Bearings_worksheet.ggb

Gradient function trace.ggb

Tick a box then use the arrow button to examine the theorem dynamically.

By using the arrow button both the centre and the radius can be changed to examine the effect upon the equation of the circle.

Dynamically change the shape of the ellipse and examine the effect on the focii Watching the effect on the limit as the x value tends towards the limit Watching the effect on the limit as the x value tends towards the limit Watching the effect on the graph as you use the sliders Watching the effect on the graph as you use the sliders Watching the effect on the graph as you use the sliders Watching the effect of the sliders on the circle and its equation.

The external point can be dragged.

Watching the effect of the slider on the graph of the hyperbola Best to go to Navigation Bar for construction steps As curve is changed observe changes to tangent and gradient function

ďƒ“ Allan White "Geogebra - Geometry, Functions and Calculus Adapted with permission and thanks to Tobias Cooper from his MANSW presentation notes

21

University of Western Sydney

Mathematics education Unit

Gradient_function animation.ggb

A cubic and gradient function and the critical points Hyperbola_Drag_Asymptotes.ggb Curve y = 2 + 1/(x+3) madeline median prob.ggb Median of a triangle

Sliders allow you to watch the changes to all features Curve and asymptotes are dynamic Dynamic example allows triangle to be varied to obtain generalisation for medians

mice_problem_2.ggb Parabola_Vertex_Form.ggb Parallelogram.ggb Quadratic_roots_formula.ggb Secant_Tangent_1.ggb

sinx derivative trace.ggb small angles.ggb

Parabola with vertex marked Parallelogram with the diagonal angles and sides Has formula, the graph and the slider for a, b, c Curve and two points A and B with a tangent at A and a secant AB Sine curve and tangent The unit circle

Drag the vertex and notice change to equation Dynamic investigation of properties of the parallelogram Dynamic use of sliders to investigate the quadratic formula Investigate how the tangent approaches the secant as A approaches B - first principles derivative Using the tangent to plot exact values of the gradient function Explore the trig ratios from the unit circle.

ďƒ“ Allan White "Geogebra - Geometry, Functions and Calculus Adapted with permission and thanks to Tobias Cooper from his MANSW presentation notes

22

University of Western Sydney

Mathematics education Unit

I DON'T LIKE TECHNOLOGY BECAUSE MY STUDENTS WILL KNOW MORE THAN I. THEY HAVE MORE TIME TO PLAY AROUND, ESPECIALLY THE BAD STUDENTS (BOYS)!!!!

If this sounds familiar or if you have had similar thoughts then let me quieten these fears by telling you a story. It is one I have used many times, but it still speaks to me when I think of the use of technology in schools. It is called the Golden Eagle and comes from 'The Song of the Bird', by Anthony De Mello.

A man found an eagle's egg and put it in the nest of a backyard hen. The eaglet hatched with the brood of chicks and grew up with them.

All his life the eagle did what the backyard chickens did, thinking he was a backyard chicken. He scratched the earth for worms and insects. He clucked and cackled. And he would thrash his wings and fly a few feet into the air like the chickens. After all, that is how a chicken is supposed to fly, isn't it?

Years passed and the eagle grew very old. One day he saw a magnificent bird far above him in the cloudless sky. It floated in graceful majesty among the powerful wind currents, with scarcely a beat of its strong golden wings.

The old eagle looked up in awe. "Who's that?" he said to his neighbour.

'That's the eagle, the king of the birds," said his neighbour. "But don't give it another thought. You and I are different from him."

So the eagle never gave it another thought. He died thinking he was a backyard chicken.

How many of your students will die thinking they were no good at mathematics?

Are your students going to be eagles or chooks? It is up to you! We want to encourage their mathematical thinking to soar.

Ok serious now, let us consider the disruptive students. Many will be boys who are disengaged with school and use their peer group to experience feelings of worth and recognition. Many boys have decided school is for girls or not for them and have taken up computer games as a way of gaining respect from their peers.

So how to handle one of these students who comes to class with something he has discovered about GeoGebra that he knows his teacher will not know. Perhaps he wants to embarrass the teacher as revenge for times when he has been embarrassed.

How should the teacher act? The teacher could prevent the student from contributing his knowledge but this would only deepen the disengagement and run the risk of confrontation (and I can hear the chickens clucking!).

Whereas, if the teacher encourages the student to share his knowledge and allows the 'spotlight' to be upon the student, allows the student to feel valued and acknowledged, then the process of re-engaging this boy has started. He may soar to heights in the digital world and remember his old teacher when he is counting his millions!!!! ďƒ“ Allan White "Geogebra - Geometry, Functions and Calculus Adapted with permission and thanks to Tobias Cooper from his MANSW presentation notes

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