with “sketchpad”hermis.di.uoa.gr/kmaragos/sketchpad/scetchpad.pdfwith sketchpad, students can...

Building Educational scenarios with

“Sketchpad”

Costantinos Maragos

Athens, December 2004

Building Educational scenarios with “Sketchpad”

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Contents

DYNAMIC GEOMETRY......................................................................................................................2

THE GEOMETER’S SKETCHPAD....................................................................................................4

INTRODUCTION ..................................................................................................................................6

THE WORK AREA...................................................................................................................................6 ANIMATION AND TRACING TECHNIQUES...............................................................................................8

Constructing Interiors ......................................................................................................................8 Animation .........................................................................................................................................8 Tracing .............................................................................................................................................9

SCRIPTS.................................................................................................................................................9 Recording a script ............................................................................................................................9 Making a script from a construction ..............................................................................................10

MEASURING AND CALCULATING.........................................................................................................11 Measuring angles ...........................................................................................................................11 Calculating .....................................................................................................................................11 Calculating lengths and distances ..................................................................................................11 Other calculations ..........................................................................................................................12

TRANFORMATIONS ..............................................................................................................................12 Reflections ......................................................................................................................................13 Rotations.........................................................................................................................................13 Dilations .........................................................................................................................................13 Translations....................................................................................................................................14

ELEMENTARY EDUCATIONAL SCENARIOS.............................................................................15

METRIC RELATIONS TO A CIRCLE.........................................................................................................15 Secants of a circle...........................................................................................................................15 Secant and tangent of a circle ........................................................................................................16 Power of a point to a circle ............................................................................................................17 Exercise ..........................................................................................................................................17

TRIANGLES OF EQUAL AREA ...............................................................................................................18

INTERMEDIATE EDUCATIONAL SCENARIOS .........................................................................19

THE PYTHAGOREAN THEOREM ............................................................................................................19 GRAPH OF SINE AND COSINE ................................................................................................................20 EXERCISE ............................................................................................................................................21

ADVANCED EDUCATIONAL SCENARIOS ..................................................................................22

LOCUS .................................................................................................................................................22 A FRACTAL – THE KOCH CURVE .........................................................................................................24

BIBLIOGRAPHY.................................................................................................................................25


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Dynamic Geometry Dynamic is the opposite of "static." Dynamic also connotes action, energy, even hype. Dynamic geometry is active, exploratory geometry carried out with interactive computer software. Mathematicians all know the power that a figure can provide--often a quick sketch or a diagram can make everything clear. We say "I see" and mean "I see and understand." In geometry, figures seem to be essential for most descriptions and proofs. Yet mathematicians also know the danger in relying on figures--inevitably, extra assumptions are made (suggested by a sketch), special cases are missed (omitted from a sketch), or absurd results are derived (from an inaccurate sketch). A classic case of the last occurence is the following oft-cited "theorem" and proof Dynamic geometry software has had a profound effect on classroom teaching wherever it has been introduced. Although not originally intended by its developers, it has also become an indispensible research tool for mathematicians and scientists. The basic question is: "What is dynamic geometry software good for?" • Accuracy of construction. Dynamic geometry software provides an accurate constructor

for any ruler-and-straightedge construction in Euclidean geometry, any configuration that can arise by applying affine transformations (isometries and dilations) to a Euclidean construction, or any locus of an object (or set of objects) that arises when part of a construction is moved along some path. Reliance on the accuracy of the software's geometric sketches and measurements is so basic that it is not discussed explicitly in any article, but it is implicit in every discussion. (Accuracy is, of course, limited to the tolerences of internal computation, screen display, allowed numerical display, and printer fidelity. And accuracy is occasionally diminished by the choice of heuristic for an algorithm or by pesky bugs that are bound to occur in such complex software.)

• Visualization. As a demonstration tool in the classroom, dynamic geometry software can help students see what is meant by a general fact. Students can construct, revise, and continuously vary geometric sketches. While visualization in itself is a powerful problem-solving tool, the capacity for students to make instantaneous and precise variations to their own visual representations adds a new dynamic dimension whose implications are only beginning to be understood. By allowing students to investigate continuous variation directly (without intermediary algebraic calculation), dynamic geometry environments can be used to help students build mental constructs that are useful (even prerequisite) skills for analytic thinking

• Exploration and Discovery. In a traditional geometry course, students are told definitions and theorems and assigned problems and proofs; they do not experience the discovery of geometric relationships, nor invent any mathematics. Dynamic geometry software is perfectly suited for exploration and discovery--either guided, or completely open-ended (a.k.a. research). The range of investigation and the amount of guidance


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provided varies greatly with the level and experience of the students (or researchers. Students using the software often discover surprising things that are not in any text, and not known to the teacher. Although it may seem a cliché, dynamic geometry software empowers. Students can get hooked pursuing open-ended problems

• Proof. While dynamic geometry software cannot actually produce proofs, the experimental evidence it provides produces strong conviction which can motivate the desire for proof. Conviction is necessary for undertaking the (often difficult) search for a proof, he contends, and in addition, the software may even give insight into geometric behavior that can help with a proof. Subtle geometric relationships may be not at all obvious, but be revealed in experimenting with dynamic figures. Some teachers have been reluctant to use the software because they fear that visually convincing evidence will replace proofs of theorems.

• Transformations. Dynamic geometry software can transform figures in front of your eyes. Isometries and similarities are important examples of functions. In witnessing the action of these transformations moving and scaling figures, students see that functions are not synonymous with symbolic formulas.

• Loci. It is virtually impossible for most people to imagine a point moving in a configuration (in which other several parts may also may be moving) and be able to describe the locus of the point's path as it travels. Dynamic geometry software, with its built-in feature to trace the locus of any specified object is ideally suited to show how a locus is generated and to reveal the shape of its traced path. Except for the most simple loci (circles, as the locus of points equidistant from a fixed center, and perhaps the conics), this rich subject has been avoided in most geometry texts. In fact, most of the classic curves arose as loci, and one must go to long out-of-print books to find these nonanalytic descriptions of them. Classic locus problems and intriguing generalizations of these, as well as surprising new loci, are discussed by our authors.

• Simulation. Dynamic geometry software's special features of dragging, animation (of points on line segments or on circles), tracing loci, and random point generation provide many opportunities to simulate a surprising variety of situations.

• Microworlds. Dynamic geometry software produces an environment in which Euclidean geometry can be explored. Jean-Marie Laborde (Cabri-géomètre) and Nick Jackiw (The Geometer's Sketchpad) each discuss ways in which their differing software can create a "Poincaré world" of hyperbolic geometry. Jackiw also discusses how other microworlds can be created through the use of scripts which produce new "tools" that replace the Euclidean tools and allow exploration fully within a new geometry. All these software developers acknowledge the challenges that dynamic geometry software presents--there is a delicate balance between what might be desired and what is feasible, and there is an ongoing dialogue as to what should be the design of software that can best enhance learning for all.

(From The Geometry turned on!: Dynamic Software in Learning, Teaching, and Research)


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THE GEOMETER’S SKETCHPAD

The Geometer’s Sketchpad has been developed by Key Curriculum Press (http://www.keypress.com), 2512 Martin Luther King Jr Way, Berkeley, California 94704. The software is designed for teaching geometry and is the outcome of a project headed by Drs Eugene Klotz and Doris Schattschneider. It was first released in 1991. • Mathematics Learning beyond Traditional Boundaries

With a scope that spans the mathematics curriculum from middle school to college, The Geometer's Sketchpad brings a powerful dimension to the study of mathematics. Sketchpad™ is a dynamic construction and exploration tool that enables students to explore and understand mathematics in ways that are simply not possible with traditional tools.

• Dynamic Visualization Is the Heart of the Program With Sketchpad, students can construct an object and then explore its mathematical properties by dragging the object with the mouse. All mathematical relationships are preserved, allowing students to examine an entire set of similar cases in a matter of seconds, leading them by natural course to generalisations. Sketchpad encourages a process of discovery in which students first visualise and analyse a problem, and then make conjectures before attempting a proof.

• Enhances Learning across the Mathematics Curriculum Effective with students from KS3 to A2 level, Sketchpad brings its full dynamic power to the study of Euclidean and non-Euclidean geometries, algebra, trigonometry, precalculus, and calculus. And Sketchpad's dynamic, visual approach allows younger students to develop the concrete foundation they need to move ahead to more advanced levels of study.

• Dynamic Mathematics Is Now More Powerful than Ever Before! Take a look at what's new in Sketchpad version 4:

• The capability to define, combine, evaluate, graph, and differentiate functions makes Sketchpad the perfect tool for algebra and calculus, as well as for geometry.

• Animation features are more powerful, more flexible, and easier to use. • Multi-page Sketchbooks make it easy to assemble related activities and package

them with activity-specific tools, to create electronic portfolios and presentations, to develop curriculum, and to design activities.

• Split/Merge and editing of calculations and functions allow easy modification of sketches.

• Styled text enhances professional appearance; mathematical text increases authenticity.

• User can integrate Web-based resources into Sketchpad activities and publish “live” Dynamic Geometry activities on the Web.

• Fully supports Windows® and Macintosh® on the same CD-ROM. • Easy-to-create Custom Tools extend Sketchpad's mathematics.


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• Enhanced graphics allow full-colour objects, text, and background for striking visual effect.

• Use of parametric colour adds an “extra dimension” in any visualization, ranging from simple mathematical finger-painting activities at the early grades to surface plots at the higher grades.

• Easier user interface gets beginners immediately up to speed. • Selecting multiple objects is easier than ever. • All of your Sketchpad skills are upwardly transferable to GSP 4.

• Amazing Sketchpad Tools Sketchpad gives you the tools to do the following: • Accomplish Euclidean constructions with the drawing tools in the Toolbox and with

commands in the Construct menu. • Go beyond Euclidean constructions using commands in the Transform menu to

construct translations, reflections, rotations, dilations, and iterations by fixed, computed, and dynamic quantities.

• Enter the realm of analytic geometry using the Graph menu to plot functions and work in rectangular or polar coordinate systems.

• Calculate and plot derivatives of functions; see derivatives both graphically and symbolically.

• Create animations that trace out sine waves and explore other trigonometric identities. • Encapsulate complex constructions in single steps with user-designed Custom Tools.


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Introduction

The Work area The SketchPad window can be resized to suit your convenience. Each sketch has its own window inside the main SketchPad window. These can also be resized and several sketches can be active at the same time. See a Window at the right. Menu items. They are File, Edit, Display, Construct, Transform Measure, Graph, Work, and Help listed across the top. Their meaning will change depending on the objects selected. Check each one to get an idea of the options available. Tools (on left side): The six basic tools do the following:

Pointer: use to select and move objects Dot: use to create points Circle: hold the left mouse button and drag to create a circle Segment: hold the left mouse button and drag to create a segment (or line or ray) Finger: selecting this allows labels and text boxes to be created Question mark: this tool gives the properties of the objects

Note: pressing the left mouse button while over the segment tool allows it to be changed to a line or ray tool. Next two simple figures will be created to illustrate the use of various tools and menu items. Example 1 Start with a new sketch. This example constructs a triangle, labels the parts and adds a text box. The resulting sketch is shown below. 1. After selecting the dot tool click the work area in 3 places creating 3 points. 2. Select all three points. [To do this use the arrow tool. Hold the shift key while clicking

each point. While it does not matter in this case, sometimes the order in which the points are selected are important; for example when selecting points to form an angle.]

3. With the points selected go to the Construct menu and select Segment. The three points should now be joined to form a triangle.

4. Click somewhere to deselect the sides of the triangle. With the arrow tool selected hold the mouse on different parts of the triangle and move.

5. Labels. Select the finger tool. The cursor will change to a finger. Click each vertex and side of the triangle to reveal the default label. Clicking an object a second time will hide


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the label. To change the name of a label, double click the label with the finger. A dialog box will appear in which the name can be changed.

6. Text boxes. With the finger tool selected move the mouse (finger) to an area where the text is to be placed. Press the left mouse button and drag out a rectangular area. When the mouse is released the cursor should appear in the box ready for text to be entered. Add to your sketch a box that says: This is my first sketch. To reposition or resize the text box use the arrow tool.

7. To delete an object or group of objects, select them and press the delete key. Be careful with delete. If you delete an object, its “children” (objects whose construction depends on them) will also go. If you make a mistake remember Undo is in the Edit menu. Instead of deleting an object it may be better to just hide it. This is illustrated in the next example.

Example 2 Start with a clean work area. In this example you will construct a line segment congruent to and perpendicular to a given line segment. 1. Select the Segment tool and draw a short line segment near the center of the work area.

Label the endpoints A and B. 2. Select the point A and the segment (see step 2 in Example 1 if you forgot how). From the

Construct menu choose Perpendicular line. This produces a line through A perpendicular to the segment.

3. Now select A and the segment again. This time choose Circle by Center and Radius from the Construct menu. This produces a circle with center A and radius AB.

4. Select the circle and the line. From the Construct menu choose Point at Intersection. The two points where the circle and line intersect are created.

5. Select one of the points created in Step 4 (call it C) and select A. Create the line segment AC using the Construct menu. You won’t be able to see it because the original line perpendicular to AB covers the segment.

6. Clean up time! Select the line created in Step 2 and the circle created in Step 3. From the

Display menu choose Hide Objects. The extra point created in Step 4 is still around. Hide it too.

A

B

CThis is my first sketch.

A B

C


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Here is the result after clean up.

Now we have a segment AC that is perpendicular to AB. Select various parts and move. Notice that moving an initial point (A or B) changes the size or position, but moving a child (C or a segment AB or AC) moves the whole configuration. Exercise: Continue the above construction to form a square that always retains its shape when moved.

Animation and Tracing Techniques We need something to work with. Construct a circle, an arbitrary point A on the circle object, and a point P interior to the circle. Then construct a circle by center (A) and point (P) as in the picture. Constructing Interiors First select the object, for example a circle or a series of vertices of a polygon. Then go to the Construct menu and select Circle Interior (or Polygon Interior). Under the Display menu choose Color to set a color for the interior if you do not like the default choice. Here is the result. Animation You must select two objects: the point to animate and the path (for example a circle or line segment) on which the animation is to run. Having made these selections go to Display>Animate. The following dialog box appears.

A B

C

AP


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Select the options you want in the box and click Animate. Click the screen to stop the animation. Tracing Animating an object is interesting but it would be nice to see the locus of points being generated. To do this you must turn on "Trace" before starting the animation. To turn the "Trace" feature on, select the object you want to trace. Then go to Display>Trace. That's it. Now when an animation (or any other movement) is performed the locus of the object will remain on the screen until the screen is clicked. Below is the result of animating A around the circle.

Scripts When investigating problems in SketchPad many constructions are performed over and over. We can use scripts to eliminate repetitive constructions. The script can be played whenever that construction is needed. There are two ways to create scripts: recording a new one from scratch or using an already made construction. You will make a script below using each method. More information can be found by going to the GSP Help item then selecting Scripts. Recording a script We will build a script to construct the centroid (intersections of the medians) of a triangle. From the File menu select New Script. This should create a small script window on the screen. Arrange the script window so you can see the Play, REC (recording), and Stop buttons of the script window as well as your sketch work area as in the image below.


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Now we’re ready to start. Click the REC button in the script window, then construct the centroid of a triangle. [Start by creating three points (the triangle vertices), construct the segments connecting the vertices, constructing the midpoints of each side, constructing the segment from each midpoint to the opposite vertex, and finally constructing the point of intersection of two of the medians (the third will fall on the intersection).] Clean up the construction by hiding the medians. This should leave the triangle and the point where the medians intersect. You can label the point G. Now you need to turn off the recording. Click the Stop button to do this. Here is the resulting sketch with script. In the script window you should see the Given conditions on which the construction is based. (In this case the three vertices of the triangle.) You should also see the construction steps. To test the script select three points in the work window and click the Play button. Step allows you to step through one click at a time and Fast does it instantly. When you are satisfied the script works correctly make the script window active and choose Save As from the File menu. Call this script centroid.gss. Notice that scripts have a .gss file extension. Making a script from a construction Suppose you created a nice construction and then wished that you had scripted it so that you could reuse it. No problem – we’ve got you covered! Let’s use this approach to construct the circumcenter of a triangle.


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First construct a triangle, the perpendicular bisectors of each side, and the point at which they intersect. Clean up the construction by labeling the circumcenter C and hiding the perpendicular bisectors. Now lets create the script. Use the arrow tool to select all the objects (or shift-click to select each point and segment individually). Then choose Make a Script from the Work menu. You have a script ready to test and save.

Measuring and Calculating In some geometry problems it is necessary to measure angles or lengths and to perform calculations with the results. Tools for doing this (and more) are found in the Measure menu. The following examples should get you started. The examples will use a triangle with vertices A, B, C. Start by constructing one.

Measuring angles Select (using the shift-click method) vertices A, B, and C in order. The select Measure > Angle. The value m∠ABC should be placed on the sketchpad. You can position the equation wherever you wish by clicking it and dragging it to another location. Measure ∠BCA and ∠CAB in the same way.

Calculating Ok, let’s find the sum of the angles. Make sure nothing is selected in the sketch. Go to Calculate in the Measure menu. Clicking this brings up a calculator like screen with a black square in the center. Click the m∠ABC equation. This moves m∠ABC to the calculation window. Now, click “+” on the calculator keypad, then the m∠BCA equation, the “+” again, and finally the m∠CAB equation. Clicking “OK” places the calculation on the sketchpad. Notice that resizing the triangle causes the angle measures to change, but the sum of the angle measures stays the same. Calculating lengths and distances

A

B

C

A

BC

m ABC = 110°

m BCA = 33°

m CAB = 37°


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To calculate the length of the segment AB select the segment, then select Length from the Measure menu. To measure the distance from A to B select the two points A and B, then choose the Distance option from the Measure menu. What’s the difference? For practice calculate the lengths of the segments AC and BC. Then go to the Calculate option and compute AB2+AC2 and BC2. Move the vertices of the triangle until AB2+AC2 = BC2. What’s the key? Other calculations Area and perimeter: Before using these an interior needs to be selected. To form the interior of a polygon select the vertices in order then go to Interior under the Construct menu. Slope: Select a segment or line in order to use this option of the Measure menu. Equation: Select a line before using this option. Before using either slope or equation it makes sense to select Show Axes from the Graph menu. This way you see the coordinate system. From the Graph menu you can also select either Rectangular or Polar coordinates and specify the form of the equation you desire.

Tranformations The Geometer’s SketchPad has the capability to transform objects by translations, dilations (scaling), reflections, and rotations. These are found under the Transform menu. A full explanation of the transformations can be found under Help > Transform menu. Below we illustrate reflections and rotations. To begin, construct a triangle with vertices A, B, C, two intersecting lines m and n, and the point H where the two lines intersect as pictured. (your labeling may be different.)

m

nA

B

C


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Reflections We want to reflect triangle ABC about line m. First select the line m (the axes). Choose Mark Mirror from the Transform menu. Now, select the triangle and choose Reflect from the Transform menu.You just performed your first transformation. Select the “finger” tool and identify the labels of the new triangle. They should be A’, B’ and C’. Now reflect the new triangle A’B’C’ about the line n to get a triangle A’’B’’C’’. You can see the results dynamically by moving one of the lines or the original triangle. Rotations Objects are rotated by a specified amount around a fixed point (the center of rotation). We will need to specify both of these. First check (and change if you so desire) the setting in the Preferences option of the Display menu to see whether the unit for angle measurement is degrees or radians. Select H as the center of rotation and choose Mark Center from the Transform menu. We want to mark an angle. Select a point on line m, H, and then a point on line n. Choose Mark Angle from the Transform menu. Now, we’re ready to rotate. Select the triangle ABC and choose Rotate from the Transform menu. The dialog box should confirm you wish to rotate an object about H by the angle you have marked. (If you did not mark an angle, you will get a different dialog box which lets you enter the number of units for a rotation.) Click OK. The results of the rotation should be selected (otherwise select it). Rotate the image one more time by the same amount around H. Where did the second image go? How does it compare with the two reflections? Move the lines and triangle to dynamically see if things change. Conjecture a result you think is true about reflections and rotations. Here is a quick introduction to dilations and translations. Dilations To dilate an object we need a center of the dilation (like an origin) and a scaling factor. The center is easy. Select a point and choose Mark Center from the Transform menu as you did with a rotation. The scale can be set in either of two ways.

Numerically: In this case select the object to be dilated and choose Dilation from the Transform menu. Then enter the numerical scaling factor in the dialog box and click OK.


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Ratio of segments: In this case first select two segments and choose Mark Ratio from the Transform menu. Then select the object to be dilated, choose Dilation from the Transform menu, and click OK to using the marked center and ratio.

Translations A translation is specified by either giving a vector or by specifying the direction and magnitude of the translation. Both options are available here.

Vector: Select the points that determine the tail and the head of the vector (in that order). Then choose Mark Vector from the Transform menu. Now select the object to be translated and choose Translate from the Transform menu. Direction and Magnitude: Select the object to be translated and choose Translate from the Transform menu. In the dialog box specify the direction (0° is the positive x-direction) and the magnitude. Then click OK.

There is another useful option in the Transform menu, namely, Define transformation. After constructing a series of transformations you may want to have a single mapping that performs the composition. That’s where Define Transformation comes in. Select the initial object and the final object (remember to clean up first by hiding the intermediate steps of the composition). Then choose Define Transformation from the Transform menu. Give the new mapping an easy to remember name and click OK. When you next select an object and go to the Transform menu the new transformation will be listed at the bottom of the menu.


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Elementary educational scenarios

Metric relations to a circle Secants of a circle • Construct a circle and let O its center • Take two points A and B on the circle • Construct the line AB • In similar way take two point C and D on the circle and construct line CD • Move the lines AB and CD so that their intersection point P to be outside the circle • Measure segments PA, PB, PC, PD and create the products PA * PB and PC * PD.

Arrive at a conclusion. Move point P to many places outside the circle. Is their any difference to the relation of the products PA * PB and PC * PD? Move point A so that point P lies inside the circle. Is their any difference to the relation of the products PA * PB and PC * PD? From your observation could you arrive at a conclusion? Can you prove this theoretically?

O

A

B

C

D

P

PA = 4.09 cm

PB = 0.75 cm

PD = 0.92 cm

PC = 3.33 cm

PA PB = 3.06 cm2

PC PD = 3.06 cm2


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Secant and tangent of a circle • Construct a circle with center O and let E a point of the circle • Construct radius OE and tangent on point E • Take two points of the circle A and B and construct segment AB • Find the intersection point P of the tangent and AB • Measure the segments PA, PB, PE. Can you write down a relation between these 3

segments? From your observation can you arrive at a conclusion? Can you prove this theoretically?

O

D

C

BA P

DP = 1.25 cm

CP = 4.05 cm

AP = 5.69 cm

PB = 0.89 cm

CP DP = 5.04 cm2

PB AP = 5.04 cm2

O

B

A

E

P

PA = 1.55 cm

PE = 2.68 cm

PB = 4.61 cm

PE2 = 7.17 cm2

PA PB = 7.17 cm2


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Power of a point to a circle • Construct a circle with center O. • Take a point E on the circle and construct segment OE. Label it R • Take two more points A and B on the circle and construct segment AB. • Take a random point P on segment AB. • Measure the segments PA, PB, PO and R and calculate PA * PB and PO2 – R2. • What is the relation between these two representations? • Move point P on segment AB. The above relation remains constant? The quantity PO2 – R2 is called the power of point P to circle (O,R), and is represented as DP(O,R). Complete the following propositions: If DP(O,R) < 0 then P ____________________________________________________ If DP(O,R) > 0 then P ____________________________________________________ If DP(O,R) = 0 then P ____________________________________________________ Exercise • Take 4 points A, B, C, D • Construct segments AC and BD • Construct the intersection point P, so that P lies between A and C as well as between B

and D • Measure PA, PB, PC, PD • Move one of the points A, B, C, D so that the expression PA * PC = PB * PD to be valid • Construct the quadrangle ABCD • Can you construct a circle that passes from every point A, B and D? • Is the point C on the circle? Can you arrive at a conclusion when a quadrangle is inscribed to a circle?

R

O

B

A

E

P

R = 1.65 cm

PO = 4.35 cm

PB = 2.98 cm

PA = 5.43 cm

PO2 - R2 = 16.18 cm2

PA PB = 16.18 cm2


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Triangles of equal Area • Construct a line j and take a segment CD on the line j • Take a point E outside the segment CD • Construct a parallel line i to segment CD from point E • Take a random point F on line i • Construct the triangle CDF • Holding shift select all the vertices of the triangle CDF • Select Polygon Interior from the menu Construct • Fill the interior of the triangle with color blue • Measure the area of the triangle CDF • Construct a random segment GH on line I • Select point F and shift+select line j and construct the perpendicular line p • Construct the intersection point I of the line j and line p • Construct the segment FI (height of the triangle) • Select hide line from the menu Display for the line p • Select the vertex F of the triangle and shift+select the height FI of the triangle • Select Animation from the menu Edit > Action Button and then click Animate Double click on button

A

B

C

D

P

M

N

H

PA = 2.81 cm

PB = 2.90 cm

PC = 0.96 cm

PD = 0.93 cm

PA PC = 2.69 cm2

PB PD = 2.69 cm2

Animate


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What you observe about the area of the rectangle CDF? Can you arrive at a conclusion? Can you prove it theoretically?

Intermediate educational scenarios

The Pythagorean theorem • Construct a segment j • Take a point D on segment j • Construct the perpendicular line from D to segment j • Construct a square DEHI (E lies on k and I lies on j) in one side of the line k • Fill the square with color blue • Construct another square DCJK (C lies on j and K lies on k) on the opposite side of line k • Fill the square with color pink • Construct segment CE and the square CEML. • Fill the square with color yellow • Measure the area of the 3 squares. • Use the calculator to add the measured areas of the squares DEHI and DCJK • Hold down the shift key and on point C, segment j, point E and line k (beware of that order) • Select Animation from the menu Edit > Action Button and then click Animate Double click on button Can you observe anything interesting? Can you conclude a theorem? Prove it theoretically.

Animate

j

i

C D

E FG H

I

Area FCD = 2,223 cm2

Animate


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Graph of sine and cosine • Select Arc Units = Radians from menu Display > Preferences • Select Create Axes from menu Graph • Label the center of the axes O • Select Plot points from menu Graph • Type 1 for x and 0 for y and press OK • Label the new created point A • Select the center of the axes O and shift+select point A • Select Circle By Center and Point from menu Construct to create a circle with radius 1 • Take a point B on circle • Holding shift select point A, circle and point B (in that order) • Select Arc on Circle from menu Construct • Select Arc Angle from menu Measure

j

k

DC

E

J

M

L

K

I

H

Animate

Pink area = 8,444 square cmBlue area = 2,620 square cm

Yellow area = 11,063 square cmPink area + Blue area = 11,063 square cm


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• Click on point B and select Coordinates from menu Measure • Double click on the coordinates of B to display the calculator • Click on B coordinates and select x • Click on B coordinates and select y • Select the measured Arc Angle and shift+select the calculated x coordinate of B • Select PlotAs (x,y) from menu Graph • Click on the constructed point and give a color pink and Trace Point from menu Display • Repeat the above 3 steps for the y coordinate of point B giving a green color • Click on point B and shift+select the circle • Select Animation from the menu Edit > Action Button and then click Animate Double click on button Which functions are being plotted? Why? What is the minimum and maximum value of these functions? As you realized the plotted functions are the sin and the cos of the angle <AOB Can you construct the graph of the function tan(<AOB)?

Exercise Develop an educational scenario for the sum of two functions r(x) and g(x) like the ones shown below. You may use your own functions.

4

2

-2 2 4 6

k

j B

AO

Animate

sin(<AOB) = 0,834

cos(<AOB)= 0,552

Arc angle c1CD = 0,987 radians

B: (0,552, 0,834)

OA = 1,000 cm

Animate


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Advanced educational scenarios

Locus • Construct a circle with center O • Construct 2 perpendicular diameters AF and EH of the circle • Take o point B on the circle and construct the triangle OAB • Construct the medians of the triangle OAB and let G their intersection point • Move B around the circle. What you observe for the point G? • Click on point G and select Trace Point form menu Display • Select a green color from the menu Display • Select point B and shift+select the circle • Select Animation from the menu Edit > Action Button and then click on Animate What is your observation about the point G as B moves around the circle? Move point B around the circle trying to find points of the above locus. • Select the vertices of the triangle OGA and select Polygon Interior from the menu

Construct • Select Area from the menu Construct • Repeat the above 2 steps for the triangle OAB giving that a different interior color • Use the calculator to compute the ratio (OAB) / (OGA)

4

2

-2

-4

4 -2 2 4 6 8 10

r

q

p

c

b

a

g(x) = p * sin( q * x - r)

r(x) = a * sin( b * x - c)

h(x) = r(x) + g(x) where

Animate


23

• Double click on button What is your observation about the ratio (OAB) / (OGA)? Can you prove that theoretically? • Select the center O of the circle • Select Define Origin from the menu Graph • Select point B and then click on Coordinates from menu Measure • Double click on B coordinates to show the calculator • Select B coordinates and click on x coordinate. Press OK • Select the x coordinate of point B and shift+select the area of triangle OAB • Select PlotAs(x,y) from the menu Graph • Click on the plotted point and select Trace Point from menu Display • Select a blue color from the menu Display • Double click on button The above steps give the graph of the area of the triangle OAB as B moves around the circle. What is the position of point B for the maximum area of triangle OAB? This also exist for the area of triangle OGA? Why? Can you plot the graph of the area OGA as B moves around the circle?

Animate

4

2

-2

-6 -4 -2 2

G

B

H

E

F AO

Area(Polygon OBA)Area(Polygon OGA) = 3,000

Area(Polygon OBA) = 3,738 square cm

Area(Polygon OGA) = 1,246 square cm

xB = 1,308

B: (1,308, 2,582)

Animate

Animate


24

A Fractal – The Koch curve In this scenario we’ re going to use the recursive scripts of Geometer’s Sketchpad. • Open a new sketch and a new script and click REC • In the sketch construct a horizontal segment AB. Drag from left to right • Select the right endpoint B and choose Mark Center from menu Transform • Select the left endpoint A and choose Dilate from menu Transform • Uncheck By Mark Ratio and type 1 in the top box and 3 in the bottom box. Press OK • A new point constructed 1/3 of the left point A distance from the marked center B • Repeat the above 3 steps to dilate point A now 2/3 distance from the marked center B • Mark the 2/3 point as Center from the Transform menu • Select the 1/3 point and choose Rotate from the Transform menu. Enter 60 and press OK • A new point appears • Construct segments to connect the 1/3 and 2/3 points to this new point • Hide the original segment • Construct segments from the original endpoints to the 1/3 and 2/3 points Your sketch now should look like this • Moving from left to right select the first two points and click LOOP in the script window • Repeat the above for every other pair of segments • Select the 3 inside vertices and choose Hide points from the Display menu • Click STOP in your script. Save your script with name Koch • Open a new sketch • Construct a new point, hold down shift and construct a second point • Click FAST in the script • Enter 3 in the highlighted box and click OK

A B


25

Bibliography (From Key Curriculum Press) Sketchpad & Primary/Middle School Mathematics

Battista, Michael T. Shape Makers: Developing Geometric Reasoning withThe Geometer's Sketchpad (Berkeley, CA: Key Curriculum Press, 1997).

Chu, Chi Wai. "Digital Manipulatives for Primary Mathematics" in Proceedings of Second East Asia Regional Conference on Mathematics Education, Vol. II, eds. Douglas Edge and Yeap Ban Har (Singapore: EARCOME2/SEACME9, 2002): 366-371.

Manouchehri, Azita, Mary C. Enderson, and Lyle A. Pugnucco "Exploring Geometry with Technology." Mathematics Teaching in the Middle School 3, no. 6 (March-April 1998): 436-442.

Taylor, Lyn. "Exploring Geometry with The Geometer's Sketchpad" Arithmetic Teacher 40, no. 2 (November 1992): 187-199.

Wyatt, Karen W., Ann Lawrence, and Gina M. Foletta, Geometry Activities for Middle School Students with The Geometer's Sketchpad (Berkeley, CA: Key Curriculum Press, 1997).

Yin, Ho Siew. "Using Geometer's Sketchpad with Primary Five Students" in Proceedings of Second East Asia Regional Conference on Mathematics Education, Vol. II, eds. Douglas Edge and Yeap Ban Har (Singapore: EARCOME2/SEACME9, 2002): 390-393.

Sketchpad & Secondary School Mathematics

Bennett, Dan. Pythagoras Plugged In: Proofs and Problems for The Geometer's Sketchpad (Berkeley, CA: Key Curriculum Press, 1995).

Bennett, Dan. Exploring Geometry with The Geometer's Sketchpad (Berkeley, CA: Key Curriculum Press, 1994).

Bennett, Dan. "Dynamic Geometry Renews Interest in an Old Problem" in Geometry Turned On!: Dynamic Software in Learning, Teaching, and Research, eds. James R. King and Doris Schattschneider (Washington, D.C.: The Mathematical Association of America, 1997): 25-28.

Boehm, Kathryn W. "Experiences with The Geometer's Sketchpad in the Classroom" in Geometry Turned On!: Dynamic Software in Learning, Teaching, and Research, eds. James R. King and Doris Schattschneider (Washington, D.C.: The Mathematical Association of America, 1997): 71-74.

Brumbaugh, Doug. "Animate Your Triangles...And Your Students!" Consortium 51 (Fall 1994): 8-9.


26

Brumbaugh, Doug. "Moving Triangles" in Geometry Turned On!: Dynamic Software in Learning, Teaching, and Research, eds. James R. King and Doris Schattschneider (Washington, D.C.: The Mathematical Association of America, 1997): 69-70.

Cuoco, Albert A., and E. Paul Goldenberg. "Dynamic Geometry as a Bridge from Euclidean Geometry to Analysis" in Geometry Turned On!: Dynamic Software in Learning, Teaching, and Research, eds. James R. King and Doris Schattschneider (Washington, D.C.: The Mathematical Association of America, 1997): 33-46.

Davis, Ben. "The Rise, Fall, and Possible Transfiguration of Triangle Geometry: A Mini-History" American Mathematical Monthly 102, no. 3 (March 1995).

De Villiers, Michael D. "Exploring Loci on Sketchpad" Pythagoras 46/47 (Mathematical Association of Southern Africa: August/December 1998): 71-73.

De Villiers, Michael D. Rethinking Proof with The Geometer's Sketchpad (Berkeley, CA: Key Curriculum Press, 1999).

Dywer, Marlene, and Richard E. Pfiefer. "Exploring Hyperbolic Geometry with The Geometer's Sketchpad" The Mathematics Teacher 92, no. 7 (October 1999): 632-637.

Finzer, Bill, and Dan Bennett. "From Drawing to Construction with The Geometer's Sketchpad" The Mathematics Teacher 88, no. 5 (May 1995): 428-431.

Erickson, Tim, and Steve Rasmussen. "Danny Vizcaino's Sketchpad 'Oval'" California Technology Project Quarterly 3, no. 2 (1993).

Feller, Leslie Chess. "The Eternal Challenge of Euclid's Geometry" The New York Times (March 7, 1999, print and CyberTimes editions).

Flores, Alfinio. "Curves as Envelopes with The Geometer's Sketchpad" Mathematics and Computer Education 31 (January 1997): 56-65.

Flores, Alfinio. "Mechanical Arguments in Geometry," Primus IX (September 1999): 241-250.

Flower, Jean. "Fitting from Function Families with CAS and DGS" in Borovcnik, Manfred and Kautschitsch, Hermann (eds): Technology in Mathematics Teaching, Proc. of ICTMT 5 in Klagenfurt 2001 (Vienna: Shriftenreihe Didaktik der Mathematik, 2002), 273-288.

Garry, Tim. "Geometer's Sketchpad in the Classroom" in Geometry Turned On!: Dynamic Software in Learning, Teaching, and Research, eds. James R. King and Doris Schattschneider (Washington, D.C.: The Mathematical Association of America, 1997): 55-62.

Gorini, Catherine A. "Dynamic Visualization in Calculus" in Geometry Turned On!: Dynamic Software in Learning, Teaching, and Research, eds. James R. King and Doris Schattschneider (Washington, D.C.: The Mathematical Association of America, 1997): 89-94.

Hurtado, Rolando. "A Student's Perspective" The Global Schoolhouse (1997).


27

Jackiw, R. Nicholas. "The Geometry of Computer Graphics" Proceedings of the National Conference on Inquiry-Based Geometry Throughout the Secondary Curriculum (Northfield, Minnesota: St. Olaf College, 1998): 35-52.

Keyton, Michael. "Students Discovering Geometry Using Dynamic Geometry Software" in Geometry Turned On!: Dynamic Software in Learning, Teaching, and Research, eds. James R. King and Doris Schattschneider (Washington, D.C.: The Mathematical Association of America, 1997): 63-68.

King, James R. Geometry Through the Circle with The Geometer's Sketchpad (Berkeley, CA: Key Curriculum Press, 1995).

King, James R. "Quadrilaterals Formed by Perpendicular Bisectors" in Geometry Turned On!: Dynamic Software in Learning, Teaching, and Research, eds. James R. King and Doris Schattschneider (Washington, D.C.: The Mathematical Association of America, 1997): 29-32.

Lichtfield, Dan, Dave Goldenheim, and Charles H. Dietrich. "Euclid, Fibonacci, and Sketchpad" The Mathematics Teacher 90, no. 1 (January 1997): 8-12.

Lichtfield, Dan, Dave Goldenheim, and Charles H. Dietrich. The GLaD Homepage [Online]. Available: http://www.gfacademy.org/GLaD (1996).

Lufkin, Daniel. "The Incredible 3x5 Card" The Mathematics Teacher 89, no. 2 (February 1996): 96-98.

Lufkin, Daniel. "A Golden Triangle...Sort Of!" The Mathematics Teacher 92, no. 2 (February 1999): 154.

McGehee, Jean. "Interactive Technology and Classic Geometry Problems." The Mathematics Teacher 91, no. 3 (March 1998): 204-208.

Meza Cascante, Luis Gerardo. Experiencias educativas con Geometer’s Sketchpad. En Memoria del "Primer Festival de Matemática". San José, Costa Rica.

Meza Cascante, Luis Gerardo. Programación de sesiones interactivas en Geometer’s Sketchpad. En Memoria del "Primer Congreso Internacional de Informática Educativa para Secundaria". San José, Costa Rica.

Meza Cascante, Luis Gerardo. Enseñanza y aprendizaje de funciones con apoyo de Geometer’s Sketchpad En libro de memorias del "Primer Congreso Internacional de Enseñanza de la matemática asistida por computadora". Cartago, Costa Rica. 1999.

Meza Cascante, Luis Gerardo. Enseñanza de la geometría en sétimo año con el programa Geometer’s Sketchpad En libro de memorias del "Primer Congreso Internacional de Enseñanza de la matemática asistida por computadora". Cartago, Costa Rica. 1999.

Meza Cascante, Luis Gerardo. Problemas de máximos y mínimos en la educación secundaria con Geometer’s Sketchpad. En libro de memorias del "Primer Congreso Internacional de Enseñanza de la matemática asistida por computadora". Cartago, Costa Rica. 1999.


28

Morrow, James. "Dynamic Visualization from Middle School through College" in Geometry Turned On!: Dynamic Software in Learning, Teaching, and Research, eds. James R. King and Doris Schattschneider (Washington, D.C.: The Mathematical Association of America, 1997): 47-54.

Naik, Gautam. "Teen Math Whizzes Go Euclid One Better" The Wall Street Journal (December 9, 1996).

Neiss, Margaret L. "Lines and Angles: Using Geometer's Sketchpad to Construct Geometric Knowledge" Learning and Leading with Technology (December-January 1996-97): 27-31.

Olive, John. "Opportunities to Explore and Integrate Mathematics with The Geometer's Sketchpad" in Designing Learning Environments for Developing Understanding of Geometry and Space, eds. Richard Lehrer and Daniel Chazan (Mahwah, New Jersey: Lawrence Erlbaum Associates, 1998): 395-418.

Pereira, Peter. "Notes on a Square: Explorations Off and On a Computer" Proceedings of the National Conference on Inquiry-Based Geometry Throughout the Secondary Curriculum (Northfield, Minnesota: St. Olaf College, 1998): 10-23.

Peterson, Blake. "A New Angle on Stars" The Mathematics Teacher 90, no. 8 (November 1997): 634-639.

Purdy, David C. "Using The Geometer's Sketchpad to Visualize Maximum-Volume Problems" The Mathematics Teacher 93, no. 3 (March 2000): 224-228.

Rich, Eric. "Elegance of Euclid" The Hartford Courant (March 13, 1997): E1, E4.

Saunders, Cathi. Perspective Drawing with The Geometer's Sketchpad (Berkeley, CA: Key Curriculum Press, 1994).

Saunders, Cathleen. "Geometric Constructions: Visualizing and Understanding Geometry" The Mathematics Teacher 91, no. 7 (October 1998): 554-556 .

Scher, Daniel. Exploring Conic Sections with The Geometer's Sketchpad (Berkeley, CA: Key Curriculum Press, 1995).

Scher, Daniel. "Problem Solving and Proof in the Age of Dynamic Geometry" Micromath 51, no. 1 (Spring 1999): 24-30.

Schumann, Heinz. "Interactive Calculations on Geometric Figures," in Proceedings of Technology in Mathematics Teaching '93 (1993).

Shaffer, David. Exploring Trigonometry with The Geometer's Sketchpad (Berkeley, CA: Key Curriculum Press, 1995).

Shilgalis, Thomas W. "Finding Buried Treasures: An Application of The Geometer's Sketchpad" The Mathematics Teacher 91, no. 2 (February 1998): 162-165.

Stone, Michael. "Teaching Relationships Between Area and Perimeter with The Geometer's Sketchpad" The Mathematics Teacher 87, no. 8 (November 1994).


29

Wagner, Betsy. "The Ninth Graders Who Answered Euclid" US News and World Report (December 23, 1996): 20.

Wilson, Jim. Comments on the GLaD Construction [Online]. Available: http://jwilson.coe.uga.edu/ Texts.Folder/GLaD/GLaD.Comments.html (1997).

Sketchpad in Undergraduate and Research Mathematics, or Math History

Demirköz, Bilge. "'N' Noktada Uzaklik Toplamari ve Egriler Ailesi" Matematik Dünyasi 6, sayi 3 (Türk Matematik Dernegi Tarafindan Iki Ayda Bir Yayinlanir: Haziran 1996): 7-10.

Dennis, David. "René Descartes Curve-Drawing Devices: Experiments in the Relations Between Mechanical Motion and Symbolic Language" Mathematics Magazine 70, no. 3 (June 1997): 163-174.

Dennis, David, and Jere Confrey. "Drawing Logarithmic Curves with Geometer's Sketchpad: A Method Inspired by Historical Sources" in Geometry Turned On!: Dynamic Software in Learning, Teaching, and Research, ed. James R. King and Doris Schattschneider (Washington, D.C.: The Mathematical Association of America, 1997): 147-156.

Dwyer, Marlene. "Exploring the Poincaré Models of Hyperbolic Geometry Using Geometer's Sketchpad." (Thesis, San Jose State University, 1996).

Hampson, Tony. "Beginning Geometry at College" in Geometry Turned On!: Dynamic Software in Learning, Teaching, and Research, eds. James R. King and Doris Schattschneider (Washington, D.C.: The Mathematical Association of America, 1997): 95-104.

Hoehn, Larry. "A Concurrency Theorem and Geometer's Sketchpad" The College Mathematics Journal 28, no. 2 (March 1997): 129-132.

Hofstadter, Douglas R. "Discovery and Dissection of a Geometric Gem" in Geometry Turned On!: Dynamic Software in Learning, Teaching, and Research, eds. James R. King and Doris Schattschneider (Washington, D.C.: The Mathematical Association of America, 1997): 3-14.

Juraschek, Bill. "Investigating Circles in the Poincaré Disk Using GSP" The College Mathematics Journal 25, no. 2 (March 1994): 145-154.

Kimberling, Clark. Triangle Centers and Central Triangles, Congressus Numerantium volume 129 (Winnipeg: Utilitas Mathematica Publishing, 1998).

King, James R. "An Eye for Similarity Transformations" in Geometry Turned On!: Dynamic Software in Learning, Teaching, and Research, eds. James R. King and Doris Schattschneider (Washington, D.C.: The Mathematical Association of America, 1997): 109-120.

"Mathematics Profiles: The University of Georgia" Syllabus 4, no. 28 (May-June 1993): 15-16.


30

Meza Cascante, Luis Gerardo. "Enseñanza del cálculo diferencial e integral con apoyo del programa Geometer’s Sketchpad". Revista "Comunicación" 11, no. 1. Año 20. Cartago, Costa Rica. 1999.

Olive, John. "Creating Airfoils from Circles: The Joukowski Transformation" in Geometry Turned On!: Dynamic Software in Learning, Teaching, and Research, eds. James R. King and Doris Schattschneider (Washington, D.C.: The Mathematical Association of America, 1997): 169-177.

Parks, James M. "Identifying Transformations by their Orbits" in Geometry Turned On!: Dynamic Software in Learning, Teaching, and Research, eds. James R. King and Doris Schattschneider (Washington, D.C.: The Mathematical Association of America, 1997): 105-108.

Schattschneider, Doris. "Visualization of Group Theory Concepts with Dynamic Geometry Software" in Geometry Turned On!: Dynamic Software in Learning, Teaching, and Research, eds. James R. King and Doris Schattschneider (Washington, D.C.: The Mathematical Association of America, 1997): 121-128.

Scher, Daniel. "Demystifying ei*pi" International Journal of Computers for Mathematical Learning, 3 (1999): 275-278.

Smith, Alvy Ray, "Infinite Regular Hexagon Sequences on a Triangle", Experimental Mathematics 9, No. 3 (November 2000): 397-406.

Sketchpad & Curriculum Design, Pre-Service, In-Service, and Assessment

Chazan, Daniel and Michal Yerushalmy. "Charting a Course for Secondary Geometry" in Designing Learning Environments for Developing Understanding of Geometry and Space, eds. Richard Lehrer and Daniel Chazan (Mahwah, New Jersey: Lawrence Erlbaum Associates, 1998): 67-90.

Flewelling, Gary, with William Higginson. A Handbook on Rich Learning Tasks Kingston, Ontario: Queens University Press, 2000.

Foletta, Gina. "One Approach to Integrating Dynamic Geometry into a Geometry Course for Middle-Grades Teachers: Transformational Perspectives" Proceedings of the National Conference on Inquiry-Based Geometry Throughout the Secondary Curriculum (Northfield, Minnesota: St. Olaf College, 1998): 53-59.

Galindo, Enrique. "Assessing Justification and Proof in Geometry Classes Taught Using Dynamic Software" The Mathematics Teacher 91, No. 1 (January 1998): 76-82.

Goldenberg, E. Paul, Albert A. Cuoco, and June Mark. "A Role for Geometry in General Education" in Designing Learning Environments for Developing Understanding of Geometry and Space, eds. Richard Lehrer and Daniel Chazan (Mahwah, New Jersey: Lawrence Erlbaum Associates, 1998): 3-44.

Gorini, Catherine A. "Implications of Technology for Teaching Geometry" Proceedings of the National Conference on Inquiry-Based Geometry Throughout the Secondary Curriculum (Northfield, Minnesota: St. Olaf College, 1998): 60-68.


31

Jiang, Zhonghong and Edwin McClintock. "Using The Geometer's Sketchpad with Preservice Teachers" in Geometry Turned On!: Dynamic Software in Learning, Teaching, and Research, eds. James R. King and Doris Schattschneider (Washington, D.C.: The Mathematical Association of America, 1997): 129-136.

Langford, William, Robert Long, and Douglas McDougall. Mathematics Education for the 21st Century: A Fields-Nortel White Paper (Toronto, Ontario: The Fields Institute, 1997).

Olive, John. "Teaching and Learning Mathematics with The Geometer's Sketchpad" New Directions for Learning and Teaching Geometry (Lawrence Erlbaum Associates: 1992).

Orton, Anthony, and Jean Orton. "Introducing Teachers to The Geometer's Sketchpad" Micromath (Autumn 1995): 28-32.

Roulet, Geoffrey, Peter Harrison, and Peter Taylor, Tomorrow's Mathematics Classroom (Grades 10-12): A Vision of Mathematics Education in Canada (Toronto, Ontario: Ontario Mathematics Coordinator's Association, 1997).

Rowling, Dawn. "Introducing The Geometer's Sketchpad to the Classroom" The Global Schoolhouse (1997).

Veloso, Eduardo. Geometria: Temas Actuais (Lisbon: Instituto de Inovacao Educacional, 1998).

Wallace, Edward C. and Stephen E. West, Roads to Geometry 2nd edition (Upper Saddle River, NJ: Prentice Hall, 1998).

Sketchpad & Mathematics Education Research

De Villiers, Michael. "An Alternative Approach to Proof in Dynamic Geometry" in Designing Learning Environments for Developing Understanding of Geometry and Space, eds. Richard Lehrer and Daniel Chazan (Mahwah, New Jersey: Lawrence Erlbaum Associates, 1998): 369-394.

De Villiers, Michael D. "The Role of Proof in Investigative, Computer-Based Geometry: Some Personal Reflections" in Geometry Turned On!: Dynamic Software in Learning, Teaching, and Research, eds. James R. King and Doris Schattschneider (Washington, D.C.: The Mathematical Association of America, 1997): 69-70.

Dixon, Juli. "English Language Proficiency and Spatial Visualization in Middle School Students' Construction of the Concepts of Reflection and Rotation using The Geometer's Sketchpad" (University of Florida, 1995).

Elchuck, Larry. "The Effects of Software Type...on Geometric Conjecturing Ability" (Thesis, Pennsylvania State University, 1992).

Goldenberg, E. Paul, and Albert A. Cuoco. "What is Dynamic Geometry?" in Designing Learning Environments for Developing Understanding of Geometry and Space, eds.


32

Richard Lehrer and Daniel Chazan (Mahwah, New Jersey: Lawrence Erlbaum Associates, 1998): 351-368.

Hanna, Gila. "Proof, Explanation and Exploration: An Overivew," Educational Studies in Mathematics 44, Nos. 1-2 (2000): 5-23.

Hoyles, Celia, and Richard Noss. "Dynamic Geometry Environments: What's the Point?" The Mathematics Teacher 87, no. 9 (December 1994): 176-117.

Jackiw, R. Nicholas. "Drawing Worlds: Scripted Exploration Environments in The Geometer's Sketchpad" in Geometry Turned On!: Dynamic Software in Learning, Teaching, and Research, eds. James R. King and Doris Schattschneider (Washington, D.C.: The Mathematical Association of America, 1997): 179-184.

Koedinger, Kenneth R. "Conjecturing and Argumentation in High-School Geometry Students" in Designing Learning Environments for Developing Understanding of Geometry and Space, eds. Richard Lehrer and Daniel Chazan (Mahwah, New Jersey: Lawrence Erlbaum Associates, 1998): 319-348.

Leong, Yew Hoong & Lim-Teo, Suat Khoh. "Guided-Inquiry with the use of The Geometer's Sketchpad" in Proceedings of Second East Asia Regional Conference on Mathematics Education, Vol. II, eds. Douglas Edge and Yeap Ban Har (Singapore: EARCOME2/SEACME9, 2002): 427-432.

Leong, Yew Hoong & Lim-Teo, Suat Khoh. "Effects of The Geometer's Sketchpad on Spatial Ability and Achievement in Transformation Geometry among Secondary Two Students in Singapore" in Proceedings of Second East Asia Regional Conference on Mathematics Education, Vol. II, eds. Douglas Edge and Yeap Ban Har (Singapore: EARCOME2/SEACME9, 2002): 433-439.

Scher, Daniel. "Lifting the Curtain: The Evolution of The Geometer's Sketchpad" The Mathematics Educator, 10, no. 2 (Summer 2000): 42-48.

Sketchpad & Science Education

Almgren, Robert. "Geometric Biology for the Chicago Public Schools" Proceedings of the National Conference on Inquiry-Based Geometry Throughout the Secondary Curriculum (Northfield, Minnesota: St. Olaf College, 1998): 24-34.

Backus, Benjamin T. "The Use of Dynamic Geometry Software in Teaching and Researching Optometry and Vision Science" in Geometry Turned On!: Dynamic Software in Learning, Teaching, and Research, eds. James R. King and Doris Schattschneider (Washington, D.C.: The Mathematical Association of America, 1997): 161-168.

Van Brummelen, Glen. "Computer Animations of Ptolemy's Models of the Motions of the Sun, Moon and Planets" Journal for History of Astronomy 29 (1998): 271-274.

Von Friedrich Wilhelm Dustmann. "Elektronische Arbeitsblätter im Physikunterricht" Unterricht Physik 13, no. 69 (2002): 23-26.

Sketchpad & Educational Technology


33

Finzer, Bill. "Network Neighbors" The Mathematics Teacher 88, no. 6 (September 1995): 475-477.

Jackiw, R. Nicholas, and William F. Finzer. "The Geometer's Sketchpad: Programming by Geometry" in Watch What I Do: Programming by Demonstration (Cambridge, MA: The MIT Press, 1993): 293-308.

National Council for Educational Technology. Dynamic Geometry (Coventry, UK: 1996).

Ritter, Steven, and Kenneth Koedinger. "An Architecture for Plug-in Tutor Agents" Journal of Artificial Intelligence in Education 7, no. 3/4 (1996): 315-347.

Roanes-Lozano, Eugenio. "Boosting the Geometrical Possibilities of Dynamic Geometry Systems and Computer Algebra Systems through cooperation" in Borovcnik, Manfred and Kautschitsch, Hermann (eds): Technology in Mathematics Teaching, Proc. of ICTMT 5 in Klagenfurt 2001 (Vienna: Shriftenreihe Didaktik der Mathematik, 2002), 335-348.

Schupp, Hans. "Problem Solving via Computer," in Proceedings of the Technology in Math Teaching '93 (1993).

Wingfield, Nick. "The Big Game" The Wall Street Journal (November 17, 1997, print and on-line editions), Special Report on Educational Techology.

Sketchpad Software Reviews

DeTurck, Dennis. "The Geometer's Sketchpad and Cabri-Géometre" The College Mathematics Journal 24, no. 4 (September 1993): 370-376.

Devaney, Robert L. "The Geometer's Sketchpad (review)" UME Trends (May 1992).

De Villiers, Michael D. "The Geometer's Sketchpad (review)" Pythagoras 30 (Mathematical Association of Southern Africa: December 1992): 38-40.

Field, Cynthia E. "The Geometer's Sketchpad (review)" InCider/A+ (November 1991).

Hertzberg, Lanny. "Exploring Geometry: Software with an Inductive Approach to Geometry" Electronic Learning Special Edition (September 1992): 23-24.

Hinders, Duane. "The Geometer's Sketchpad (review)" The Mathematics Teacher no. 5 (May 1992): 392-393.

"Im Dreieck krabbeln: Dynamische Software für Geometrie" MAC Up (November 1991): 64-66.

Kantrowitz, Betty. "The Geometer's Sketchpad (review)" The Computing Teacher (February 1992): 45-49.

Levy, Benjamin. "Software Twins" Consortium 47 (Fall 1993).

Martin, Mike. "Window on Resources: The Geometer's Sketchpad (review)" Mathematics Teaching in the Middle School (May 1996): 836-838.


34

Olive, John. "The Geometer's Sketchpad Version 2.0 (review)" The Mathematics Educator 4, no. 1 (University of Georgia Mathematics Education Student Association: Winter 1993): 21-23.

Rothstein, Edward. "Technology Connections: Software Puts Joy Back in Mathematics" The New York Times (November 10, 1997, print and CyberTimes editions).

Seiter, Charles and Mary S. Toth. "The Geometer's Sketchpad (review)" Macworld (November 1991).

Stanton, David. "Software Tools For Exploring Math" Electronic Learning (November-December 1991): 34.

Statt, Paul. "The Old Geometer" InCider/A+ (July 1991): 22.

St. Onge, Julie. "A Dynamic Way of Learning" Curriculum Product News (January 1992): 28-29.

Troy, Beth. "The Geometer's Sketchpad Update (review)" The Computing Teacher (December/January 1993-94): 45-48.

with “sketchpad”hermis.di.uoa.gr/kmaragos/sketchpad/scetchpad.pdfwith sketchpad, students can...

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