Dynamic Geometry Software

in the Classroom

 

A Workshop by

 

Dr. Gian Mario Besana     &               Dr. Vesna Kilibarda

gbesana@cti.depaul.edu                             vkilibar@iun.edu

 

 

Sponsored by

IU SBC FELLOWS PROGRAM

http://www.amfellow.iu.edu

WORKSHOP SCHEDULE

8:30 ñ 9:00 Hawthorn Hall 436

Registration, coffee, and pastries

 

9:00 -- 10:00 Hawthorh Hall 454 and 428

Introduction to the Geometerís Sketchpad: SQUARE DANCE, KALEIDOSCOPE, p. 3 - 7

 

10:00 ñ 11:00 Hawthorn Hall 454

Hands-on group work: AT THE CENTER OF THE CENTERS, p. 8 - 15

 

11:00 ñ 11:15 Hawthorn Hall 436

Coffee break

 

11:15 ñ 12:15 HH 454 and 428

Using Sketchpad in Algebra classes (HH 454): A MOOSE'S DIET, NOW I KNOW MY a, b, cÖ, TREE HOUSE ON FIRE, p. 16 - 25

Using Sketchpad in Calculus classes (HH 428): RATE OF CHANGE, MYSTERY RATIO, MRS. E IN THE LIBRARY WITH THE CANDLESTICK, p. 26 - 32

 

12:15 ñ 1:45 Savannah Center Art Gallery

LUNCH

 

1:45 ñ 2:45 HH 454 and 428

Transformations with Sketchpad: MIRROR, MIRROR, ON THE WALL, p. 33 - 40

 

2:45 ñ 3:00 Hawthorn Hall 436

Coffee Break

 

3:00 ñ 4:00 HH454

Hands-on group work: YOUR TURN, NOW, p. 41

 

4:00 ñ 5:00 HH454

Discussion, Q/A, Survey

In this workshop we will use dynamic software Geometerís Sketchpad to investigate geometry by discovery. If you want to explore problems using the software in the future, the demo version of the Geometerís Sketchpad is available for downloading on the site: http://www.keypress.com/catalog/products/software/Prod_GSP.html

 

 

 

Activity 1   SQUARE DANCE

 

What is a square?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

What do you know about its diagonals?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

We will construct a square using the property that its diagonals are perpendicular and bisect each other.

 

Step 1.            Using the segment tool construct a segment AB. You may label the endpoints of your segment using the Labeling tool.

 

Step 2.            Select the segment AB. Under the Construct menu choose construct Point at Midpoint. Label it C.

 

Step 3.            Select both (by holding the shift key) the segment AB and the midpoint C. Under the construct menu choose construct the Perpendicular Line. Label the line k.

 

Step 4.            Select in order (by holding the shift key) the point C (as the center of a circle) and the point A (as a point on a circle). Under the Construct menu choose Construct Circle by Center and Point.

 

Step 5.            By bringing the Selection tool close to the points of intersection of the circle and the line k, click on the points of intersection (in the left lower corner you can read Select Point at Intersection) and label them D and E.

 

Step 6.            Select in order points A, D, B, and E as vertices of your square (by holding the shift key.) Under the Construct menu choose Segment. Under the Display menu choose the Line Style Thick and choose the Color you like.

 

Step 7.            Select (by holding the shift key) the circle, the segment AB, the point C, and the line k. Under the Display menu choose the Hide Objects option.

 

Step 8.            Drag the point A in the plane to see what happens. Does the figure stay a square?

 

Step 9.            Using the Circle Construction Tool construct a circle close to the point A. Select the circle and the point A (by holding the shift key.) Under the Edit menu choose the Action Button, Animation which will create the new window. Click OK in the window and the Animate button will apear. Double click with the Labeling Toll on the button. The window will appear in which you should replace the word Animate with Square Dance. Click OK.

 

Step 10.          Select the circle and its center (by holding the shift key.) Under the Display menu choose the Hide Objects option. Now click the Square Dance button and watch your square dance.

 

 

Follow the directions below to construct a Sketchpad kaleidoscope. Whenever needed, use the detailed instructions marked by} .Make sure you did each step correctly before you go on to the next step.

Activity 2        KALEIDOSCOPE

St~p 1 :

Open a new sketch and construct a many-sided polygon.

 

a. Go to the File menu and choose New Sketch.

b. Use the Segment tool [2] to construct a polygon with many sides (make it long and somewhat slender).

Step 2:

 

Construct several polygon interiors within your polygon. Shade them different colors.

a. Click on the Selection Arrow tool ~. Click in any blank space to deselect objects.

 

b. Hold down the Shift key. Select three or four points in clockwise or counter- clockwise order.

c. Go to the Construct menu and choose

 

Polygon Interior.

d. While the polygon interior is still selected, go to the Display menu and choose a shade and/or color for your polygon interior.

e. Click in any blank space to deselect objects. Repeat steps b, c, and d

until you have constructed several polygon interiors with different colors

or shades.

 

Step 3:

Mark the bottom vertex point of your polygon as the center. Hide the points and rotate the polygon by an angle of 60ƒ,

a. Click in any blank space to deselect objects.

b. Select the bottom vertex point. Go to the Transform menu and choose Mark Center ,

 

c. Click on the Point tool 0. Go to the Edit menu and choose Select All

Points. Go to the Display menu and choose Hide Points.

5

 

 

 

 

Constructing a Sketchpad Kaleidoscope (continued)

 

d. Click on the Selection Arrow tool ~. . Use a selection marquee to select

your polygon. Go to the Transform menu and choose Rotate.

 

e. Choose By Fixed Angle. Enter 60 and then click on OK. (Pick a

different factor of 360 if you wish.

Step 4: Continue to rotate the new rotated images

until you have completed your kaleidoscope.

a. While the new rotated image is still selected, go to the Transform menu and rotate this image by an angle of 60ƒ. Remember to click on OK.

 

 

b. When the newer rotated image appears, and while it is still selected, go to the Transform menu and rotate this image by an angle of 60ƒ. Remember to click on OK.

 

c. Repeat this process until you have constructed your complete kaleidoscope.

d. Go to the Display menu and choose Show All Hidden. You should see the points on the original arm reappear.

 

Step 5: Construct circles with their centers at

the center of your kaleidoscope.

a. Click in any blank space to deselect objects.

b. Click on the Circle tool ~. Press on the center point of your kaleidoscope and

drag a circle with a radius a little larger than the outside edge of your kaleidoscope.

 

 

6

 

 

 

Constructing a Sketchpad Kaleidoscope (continued)

c. Using the Circle tool construct another circle with its center at the center of your kaleidoscope, but this time let the radius be about half the radius of your kaleidoscope. Repeat for a circle with a radius about one-third the radius of your kaleidoscope.

 

Note." Make sure you release your mouse in a blank space between two arms of your kaleidoscope. You do not want the outside control points of your circles to be constructed on any part of your kaleidoscope.

 

Step 6: Animate points of your kaleidoscope on the three circles.

a. Click on the Selection Arrow tool. Click in any blank space to deselect objects.

b. Hold down the Shift key. Select one point

on the original polygon near the outside

circle and select the outside circle (do

not click on one of the control points of

the circle).

c. While you continue to hold down the Shift key. select a point near the middle circle and then select the middle circle. Select a point near the smallest circle and select the smallest circle.

 

d. Go to the Edit menu, choose Action Button, and

drag to the right and choose Animation. Click on Animate in the Animate dialog box.

e. When the Animate button appears, double

click on it to start the animation. Watch

your kaleidoscope turn!

f. Click in any blank space to stop the

animation. To hide all the points, click on

the Point tool. Go to the Edit menu

and choose Select All Points. Go to the

Display menu and choose Hide Points. Click on

the Circle tool , select all the circles, and hide them.

 

7

Activity 3            AT THE CENTER OF THE CENTERS

 

 

Exploring altitudes.

 

 

PROBLEM 1:    Discuss in your group the definition of altitudes of a triangle. Once you have reached a consensus, write down a clear, concise definition.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

PROBLEM 2:    Do the three altitudes of any triangle ABC meet at a point?

 

Even if you remember this well, go with your group to one of the machines. Construct a triangle ABC and its three altitudes. The definition you wrote above and the Construct menu should help your group do the construction with no problem. Experiment with sketchpad by changing your triangle.

 

 

The point where the three altitudes meet is called the Orthocenter. Go ahead, if you have not done it yet, and construct this point on your sketch. Be careful: sketchpad does not let you construct the intersection of three objectsÖ.

 

 

 

 

PROBLEM 3:    Where does the orthocenter fall with respect of your triangle ABC? Is it always inside the triangle? Can it be outside? Can it be on one of the sides of the triangle? Experiment with sketchpad, discuss in your group and formulate a conjecture. Write down your conjecture in a clear and concise manner.

 

 

PROBLEM 4:    What happens to the orthocenter if we keep the vertices B and C of your triangle fixed and we let A move on a line parallel to BC? Does the orthocenter describe any familiar curve?

 

 

Try to think about this question first with pencil and paper in your group. If you reach any kind of temporary agreement, or even if you just have ìguessesî, write them down.

 

 

 

 

 

 

 

 

Now move to one of the machines. To explore problem 4 with sketchpad, you might want to use a new sketch. Remember that here you want A to be forced to be on a line, which is parallel to BC. Discuss with your group a construction strategy. Your sketch should look like the one you see at http://www.indiana.edu/~geometry/altitudes1.html

In particular, make sure that the freedom of movement of your point A agrees with the given sketch.

 

 

 

 

 

 

 

Now construct the orthocenter O. Move A on the line. What does O do?

 

 

 

 

 

 

 

 

To see the locus described by O, as A moves, you can use sketchpad in two different ways:

 

TRACING A POINT:          Select O and under the Display menu, select Trace Point. Now move A.

 

 

CONSTRUCTING A LOCUS:        Select O and A. Under the Construct menu, select Locus.

 

Reconsider your conjectures about the type of curve that O describes.

Write down a clear and concise final conjecture. Try to address in your conjecture as many essential features of the curves that O describes as you can (for example, if this curve is a circle, can you say where its center is, what its radius is Ö.)

 

PROBLEM 5:    What happens to the orthocenter if we keep the vertices B and C of your triangle fixed and we let A move on a circle centered at B, with radius BC? Does the orthocenter describe any familiar curve?

NINE POINTS IN SEARCH OF A COMMON HOME

 

STEP 1:     Open a new sketch.

 

STEP 2:     Construct a triangle ABC and the midpoints of its three sides. Label these points H, J, and K.

 

STEP 3:     Construct the three altitudes, the orthocenter O and the three points where the altitudes meet the sides (or their prolongments !!!). Label these three new points L, M, and N.

 

STEP 4:     Connect O with A, B, and C and find the midpoints of these three segments. Label these three new points P, Q, and R.

 

PROBLEM 6:    Consider the nine points H, J, K, P, Q, R, L, M, N. Can you see a familiar curve on which they all live? Can you construct this curve?

Do you remember what the Circumcenter of a triangle is? Discuss it in your group. If you need help, ask one of the mentors in the room.

 

 

 

 

 

 

 

 

 

 

 

 

How do you construct the circumcenter of a triangle ?

 

 

 

 

 

 

 

 

 

 

 

 

 

On the same sketch used for problem 6 construct the circumcenter of the triangle ABC. Hide all unnecessary objects on your screen.

PROBLEM 7: The circle you constructed as a solution to problem 5, is called the Nine Points Circle. What is the relationship between the orthocenter, the circumcenter and the center of the nine points circle?

 

USING SKETCHPAD IN ALGEBRA CLASSES

Activity 4   A MOOSE'S DIET

 

A moose feeding primarily on tree leaves and aquatic plants is capable on digesting no more than 33 kilograms of these foods daily. Although the aquatic plants are lower in energy content, the animal must eat at least 17 kilograms to satisfy its sodium requirement. A kilogram of leaves provides four times as much energy as a kilogram of aquatic plants. Find the combination of foods that maximizes the daily energy intake.

 

Form groups of (ideally) four people. Analyze the problem.


We will use the sketchpad to help us solve the problem.

Step 1.           Open a new sketch and choose Show Grid from the Graph menu.

 

Step 2.            Plot Points (under the Graph menu) that will help you graph the boundary of the desired region. You will be able to enter several points (using their coordinates) in the window that opens up.

 

Step 3.            Have a line selected on the side palette. With two selected points highlighted Construct the Line you want. Repeat the process and change the Line Style and the Color of your lines using the Display menu. Construct Point At Intersection wherever you need one.

 

Step 4.            Select (in order) the corner points of the feasible region and under the Construct menu choose the Polygon Interior option.

 

Step 5.            Using the Point tool on the side of the sketch construct an arbitrary point in the region and label it G. Select point G and Measure its Coordinates. Under the Measure menu, open the Calculate option and type the function that you want maximized over the region. If you want to select a coordinate of point G, you can do it by highlighting its measurement, selecting the Calculate option in the Measure menu, and making the choice of x or y-coordinate under Values in the window that opens.

 

Step 6.            Move the point G throughout the region and find the maximum of your function. Record your result.

 

If you set x = # of kilograms of leaves and y = # of kilograms of aquatic plants your sketch will look like this:

 

 

What happens in sketches where variables are switched?

 

From the relationship you found in Step 5, you may have energy = (or. You may want to graph the line + energy (or the alternate line.) Point G is one point on the line. To create another point on the line, say H( follow the Steps 7-9.

 

Step 7.            Under the Measure menu pull Calculate and type in the window shown. You created the x-coordinate of the point H. Under the Measure menu pull Calculate and type . You created the y-coordinate of the point H.

 

Step 8.            Select, in this order, x-coordinate and y-coordinate just created. Under the Graph menu Plot as (x,y). Label the point that you just created H.

 

Step 9.            Select the points G and H only. Under the Construct menu select the Line option. The y-intercept of this line is the energy. You may want to move the coordinate system origin to the left lower corner of your screen and reduce the unit length to be able to see both the y-intercept abd the region at once on your screen.

 

Step 10.          Move the point G throughout the region and find the maximum energy.

 

Does this confirm the result you found in the Step 6?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Activity 5   NOW I KNOW MY a, b, c...Ö

 

In this activity you will construct a dynamic sketch that will allow the user to explore the effect of changing the coefficients a, b and c in the equation y=ax2+bx+c of a parabola.

 

Building the control sliders.

 

You might have learned earlier how to build a slider. If you feel comfortable, open a new sketch and build three parallel sliders, naming them a, b, c. Otherwise, follow the step-by-step (1 through 6) instructions given below:

 

STEP 1:     Construct a vertical line on the far left side of the sketch. We will call this line r (you do not need to label it in your sketch).

 

STEP 2:     Construct three different points on r. We will call these points A, B, C, although you do not need to label them in your sketch. Through each of these three points, construct a line perpendicular to r. To do this, select the three points and the line r. Under the Construct menu, select Perpendicular Line. Weíll refer to the three lines you just constructed as l, m, n. Again, you do not need to label them.

 

STEP 3:     Construct an additional point on each of the lines l, m, n (to the right of the line r.) We will call these points, in order, D, E, F. Hide the lines r, l, m, n.

 

STEP 4:     Making sure that the Segment tool is selected in the palette, construct the three segments AD, BE, CF.

 

STEP 5:     Hide A, B, C and label the segments a, b, c.

 

STEP 6:     Select the three segments a, b, c. Under the Measure menu, measure their length. If you have followed the instructions, your sketch should look like this:

 

 

 

Building the Parabola

 

STEP 7:     Under the Graph menu, select Create Axis.

 

STEP 8:     Construct a point on the x-axis. Label this point X. Be careful to avoid the unit-control point on the x-axis. Select X and under the Measure menu select Coordinates. Sketchpad will create the coordinates of your point somewhere on the sketch.

 

STEP 9:     Go under the Measure menu and select Calculate. Click on the coordinates of X. Sketchpad will offer you the x-coordinate of X in the screen of the calculator. Give it your OK. You will now have the xX value somewhere on your screen. This is what we need. You can now hide the coordinates of X. If you followed the instructions, your sketch at this point should look like this:

 

 

STEP 10:   Under the Measure menu, select Calculate. In the dialog box that opened up, you will now have to type the function ax2+bx+c, using the following instructions:

 

i)               Every time you need to type x, just select the measure of the x-coordinate of X that sketchpad put on your sketch in STEP 9.

 

ii)             The parameters a, b, c must be entered by selecting the measures of the sliders a, b, c that you computed in STEP 6.

 

 

iii)            Click on OK when you are done. Sketchpad will create the algebraic expression for your function, somewhere in the upper left part of your sketch.

 

STEP 11:   Now to the plotting: select (IN THIS EXACT ORDER!) the x-coordinate of X and the function you just typed. Under the Graph menu, select Plot as (x, y). Sketchpad will create a point; letís call it P, so that we can refer to it. If you canít see this point, stop and think. What is this point? Why canít you see it? What is the y-coordinate of this point? Play with the unit-control point on the x-axis and/or a, b, c until you actually see the point P.

 

STEP 12:   To see the whole graph of your parabola, select P and X. Under the Construct menu, select Locus.

 

STEP 13:   We are now going to add the vertex and the axis of our parabola to the sketch. Under the Measure menu, select Calculate. Type ñb/(2a), where b and a must be entered as in STEP 10, ii), and click OK. Under the Measure menu, select Calculate again and compute the y-coordinate of the vertex by computing a(-b/(2a))2+b(-b/(2a))+c. As usual, every time you want to enter

-b/(2a) you must select the already calculated value for this expression. When you are done, click OK.

 

STEP 14:   Somewhere on your sketch now you should have the x and y coordinates of the vertex. Select them both, in the x-y order. Under the Graph menu, select Plot as (x, y). The Vertex of your parabola should now be plotted. Change the color of this point to make it more visible, and label it vertex.

 

STEP 15:   Select the vertex and the x-axis. Construct a perpendicular line. This is the axis of your parabola. Change its color and label it axis. If you have followed the instructions correctly, the graph you just created should look similar to the one you see at http://www.indiana.edu/~geometry/parabppp.html

 

 

Exploring the roles of the coefficients

 

You are now ready to play with your sketch, and explore the role of the coefficients a, b, c. Pull on the sliders and observe. You can now explore a myriad questions. Here are some:

 

…      If you keep b and c fixed, and you move a, what happens to the vertex? Does it describe a familiar figure? Can you find its equation?

 

…      If you keep a and c fixed and move b, what happens to the vertex? Does it describe a familiar figure? Can you find its equation?

 

…      Which of the coefficients is responsible for how wide or narrow the parabola is?

 

…      What happens if b=0?

 

…      What happens if a=0?

 

…      What happens if c=0?

 

Exploring negative values for a, b, c.

 

The above construction allows the user to explore only parabolas in which a, b, c are all positive. To allow for negative values of the coefficients, we need to modify the construction slightly.

The following steps show you how to construct a sketch to investigate parabolas with a negative first coefficient, and a positive second and third.

We leave to you to construct all possible combinations of positive and negative coefficients (How many different possibilities are there?)

 

STEP 16:   We need to modify STEP 10 above. Instead of the equation given there, we need to type (-a)x2+bx+c. To change the sign you must use the +- option on the calculator keypad. The instructions i) ii) iii) of STEP 10 still apply.

 

STEP 17:   Now follow every step from STEP 11 to 16, being careful to remember that now your first coefficient is no longer a, but ña.

 

A sketch with a possible combination of signs of the coefficients can be found at http://www.indiana.edu/~geometry/parabnnp.html

 

 

 

 

 

 

Activity 6   TREE HOUSE ON FIRE

 

You are on an expedition with your friends. You have just built a tree house and are on a hike toward it. You are using a positioning device which locates the river running along an x-axis. You are at the position (0,5) when you spot your tree house at the position (10,3) on fire. You happen to carry several buckets. You want to run to the river, fetch water, and then run to the tree house, using the shortest path. Assume that the terrain is flat. Use the Sketchpad to help you find the shortest path.

 

PART I

 

Step 1. Under the File menu open the New Sketch.

 

Step 2. Under the Graph menu choose the Show Grid option.

 

Step 3. Under the Graph menu choose the Plot Points option. The window will appear in which you should enter your position (0,5) as one point and the tree house position (10,3) as the other point. Click OK. Using the Labeling tool label your position point as S and the tree house as T. You will have to change the labels the Sketchpad assigned by double clicking on the labels themselves. In the windows that appear do the necessary changes.

 

Step 4. Select the x-axis (by clicking on it while you have the select tool highlighted) and under the Construct menu choose the Point on Object. Label the point R.

 

Step 5. Select (by holding the shift key) the point R and the point T. Under the Measure menu choose the Distance option. Repeat the procedure for points R and S.

 

Step 6. To calculate the total distance select the Measure menu at the top of the screen. Choose Calculate option. The window will appear. Click on the recorded distance from R to T, then on the " + " sign on the calculator, and finally on the recorded distance from R to S. Click OK in the calculator window. The sum of distances will appear on your sketch.

 

Step 7. Select the point R on the river and move it right and left until RT + RS is minimized.

 

Step 8. To see the graph of total distance RT + RS as a function of the x-coordinate of R, select the point R. Open the Measure menu and select Coordinates. Now open the Calculate under the Measure menu and highlight the coordinates measure just found. You created the x-coordinate xof point R.

 

 

Step 9. Now to the plotting : select (IN THIS EXACT ORDER !) the x-coordinate x you found in 8 and the distance RT + RS. (Hold the shift key.) Under the Graph menu choose Plot as (x,y). This will create a point, that we will label P. If you do not see the point you just created on your sketch, move the origin down until you spot the point right above the point R.

 

Step 10. To see the whole graph of your function, select P and R. Under the Construct menu select the Locus option.

 

Does the minimum of the function you just created coincide with the minimum found in Step 7?

_______________________________________________________________________

 

PART II

 

You are encouraged at this point to get together in groups of four people to keep working on this problem.

 

Step 1. If we call R = (x,0) the point on the river where you fill your buckets, can you express the total distance you covered from (0,5) to (x,0) and from (x,0) to (10,3) as a function of x ? Find the algebraic expression of this function. Mr. Pythagoras might helpÖÖÖ.

 

 

 

 

 

 

Step 2. We are now going to plot the function you found in Step 1, with the help of Sketchpad. Go back to your computer, as a group or by yourself. Work on the same sketch you constructed in part I. You may want to change the color to keep the two activities separate.

.

Step 3. Under the Measure menu, select Calculate. In the dialog box that opened up, you will now have to type the function you found in Step 1, using the following instructions:

i)    Every time you need to type x, just select the measure of the x-coordinate of R that sketchpad put on your sketch in Step 8, Part I.

ii)   If you need any particular function, say for example a square root, you will find it by clicking on the Functions button in the dialog box. The square root is called SQRT. (Pay attention to the instructions that sketchpad gives you right under the yellow box)

iii)  Click on OK when you are done. Sketchpad will create the algebraic expression for your function, somewhere in the upper left part of your sketch.

 

Step 4. Now to the plotting : select (IN THIS EXACT ORDER !) the x-coordinate of R you found in Step 8, Part I the function you just typed in (recorded somewhere on your sketch).

 

Step 5. Under the Graph menu, select Plot As (x,y). Sketchpad will create a point. What is this point? Move R back and forth to understand better.

 

 

What is the locus of this point?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

If you have found the right function in Step 1 and followed the instructions correctly, the locus should be the same as the one you created with sketchpad in Step 10, PART I .

 

USING SKETCHPAD IN CALCULUS CLASSES

Activity 7  RATE OF CHANGE

 

In this activity we will graph several functions. We will create a function whose graph represents an average rate of change of the given function. We will compare this graph with the graph of the derivative to visualize how the average rate of change approximates the instantaneous rate of change.

 

Step 1.           Open a new sketch and choose Create axis from the Graph menu.

 

Step 2.            Construct a point C on the x-axis (use the point tool and bring it close to x-axis) and measure its coordinates (while the point C is highlighted select Coordinates command under the Measure menu.)

 

Step 3.            Use the Calculator under the Measure menu, highlight the measured coordinates of point C and select the x-coordinate. Drag the point C and observe how you vary . At this point your sketch should look like this:

 

 

Step 4.            Use the Functions command in the Calculator to create sin().

 

Step 5.            Select in order the x-coordinate and sin(). Under the Graph menu select Plot as (x,y) command. The plotted point will appear on your screen. Label it D. Drag point C and watch what happens to the point D.

 

Step 6.            Select points C and D and under the Construct menu choose Locus. Under the Display menu choose the Line Style Thick and choose the Color you like.

 

Step 7.            Using the Point tool come close to the curve you created until you read Point on locus D in the left lower corner. Construct a point E close to the point D on the locus.

 

Step 8.            Construct a segment DE. (Select the two points and choose Segment under the Construct menu.) Measure the slope of the segment.

 

Step 9.            Select in order the x-coordinate and the Slope DE. Under the Graph menu select Plot as (x,y) command. The plotted point will appear on your screen. Label it F. Drag the point C and watch what happens to the point F.

 

Step 10.          Select points C and F and under the Construct menu choose Locus. Under the Display menu choose the Line Style Thick and choose the Color you like.

 

What is the locus of point F?

 

 

 

 

 

 

 

 

How does the locus F change when you move the point E closer to or further away from the point D?

 

 

 

 

 

 

 

 

You may want to graph some additional functions that you know to test your conjectures.

 

 

 

 

 

 

 

 

 

Activity 8   MYSTERY RATIO

 

Following the slope of the tangent line to the graph of an exponential function, students can easily see that the slope itself behaves very much like an exponential.

 

Is the slope of an exponential function the same as the starting exponential function?

In other words, is the derivative of a^x always a^x ?

 

This activity guides you through setting up a sketch that allows your students to easily investigate this question and enrich their understanding of the fundamental role of the magic number e.

 

Step 1.           Open a new sketch and choose Create axis from the Graph menu.

 

Step 2.            Construct a point C on the x-axis (use the point tool and bring it close to x-axis) and measure its coordinates (while the point C is highlighted select coordinates command under the Measure menu.)

 

Step 3.            Use the Calculator under Measure menu, highlight the measured coordinates of point C and select the x-coordinate.

 

Step 4.            Use the commands in the Calculator window to create 2^().

 

Step 5.            Select in order the x-coordinate and 2^(). Under the Graph menu select Plot as (x,y) command. The plotted point will appear on your screen. Label it D.

 

Step 6.            Select points C and D and under the Construct menu choose Locus. Under the Display menu choose the Line Style Thick and choose the Color you like. This is a graph of a doubling growth function.

 

We will use calculus to construct the tangent line to the curve at an arbitrary point. The equation of the tangent line will be y ñ 2^() = ln2 * 2^() (x - ), so its y-intercept is 2^() - ln2 * 2^()*.

 

Step 7.            Construct a point on the y-axis and measure its coordinates. Label it F.

 

Step 8.            Use the calculator under Measure menu, highlight the measured coordinates of point F and select its x-coordinate.

 

Step 9.            Use the Calculator to create the y-intercept of a tangent line, namely type 2^() - ln2 * 2^()*.

 

Step 10.          Select in order the x-coordinate and 2^() - ln2 * 2^()*. Under the Graph menu select Plot as (x,y) command. The plotted point will appear on y-axis. Label the point H.

 

Step 11.          Make sure the infinite line option is selected in the tool palette. Highlight points E and H. Construct a line through them. Move the point C and check that the line stays tangent to the curve.

 

Step 12.          Hide measurements of point F, its x-coordinate , and 2^() - ln2 * 2^()*.

 

The sketch is ready for your students to discover the mystery ratio of the slope of the tangent line and the value of doubling growth function.

 

Step 13.          Select the tangent line and Measure its Slope.

 

Step 14.          Using Calculator create a ratio of the Slope HE and the function value 2^()

 

Step 15.          Slide the point C along the x-axis and watch what is happening to your measurements.

 

What can you conclude about the slope of the tangent line?

 

 

 

 

 

 

 

 

 

 

What is the ratio?

 

 

 

 

 

 

 

 

 

 

The next activity will provide the answer to the last question.

 

Activity 9   MRS. E, IN THE LIBRARY, WITH THE CANDLESTICK

 

For this activity we will use an already prepared sketch. Open the file Slope(a^x).gsp or Slope(a^x). Vary values of a and watch what is happening to the graphs of a^x and Slope(a^x).

 

Which value of a makes the graphs coincide?

 

 

 

 

 

 

 

 

 

 

 

 

 

Do you know this number?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

For an arbitrary value of a what is the ratio of the Slope(a^x) to a^x?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

To test your conjecture follow the next activity in which you will discover the relation of the ratio and the value a:

 

Step 1.           Select in this order, the measure of a and the measure of ratio. Under the Graph menu Plot as (x,y). The point will appear on your screen. Label it M. Change values of a and watch what is happening to the point M. Does it trace a familiar function? Which one?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Step 2.            Select points C and M and under the Construct menu choose Locus. Under the Display menu choose the Line Style Thick and choose the Color you like. Does the curve that appears confirm your conjecture from Step 1?

 

You might want to try to create the sketch Slope(a^x) from the beginning. As a guide you may adapt parts of activity 7 to the general case.

Activity 10 MIRROR, MIRROR, ON THE WALL

 

 

 

Felix Klein (look him up!) was appointed to a professorship at the German university of Erlangen in 1872. As was customary at the time, the newly appointed professor delivered an address to the faculty, in which he discussed the state of the art of his discipline, and his indications for the future. Kleinís address contained the first formal formulation of what is considered today the true essence of geometry.

Here are Klein's words:

 

As a generalization of Geometry, the following comprehensive problem arises:

 

Given a variety and a group of transformations on it, study the shapes belonging to the variety for what concerns those properties which are not altered by the transformations of the given group.

 

What Klein is saying is essentially the following.

 

  1. Choose the place (variety) where you want to do geometry: the plane, the sphere, three-dimensional spaceÖ

 

  1. Choose the rules of the game. Choose what kind of transformations are allowed in your variety to move shapes around. For example, you might decide you want your transformations not to alter distances, or you might want your transformations just not to alter angles, or you might want your transformation just to be continuousÖ

 

  1. You are ready to play geometry. Pick different shapes, move them around your variety, and look for properties of your shapes that do NOT change by the action of the transformations.

 

We are now going to play Euclidean Geometry in the plane. Our variety is the plane and our transformations are the so-called isometries.

 

An isometry is a transformation from the plane to the plane that preserves distances.

 

Basic examples of isometries are: reflections, rotations, and translations.

Reflection

 

A reflection across a line l is a transformation that takes a point P to a point Pí, so that l is the perpendicular bisector of the segment PPí.

 

Reflection with Sketchpad:

 

STEP1: Construct a line (in Sketchpad this can be a line segment, an infinite line, or a ray; it does not matter.) To tell sketchpad that you want this line to be your mirror of reflection, select it and pull down the Transform menu. Highlight the Mark Mirror option. You can achieve the same by double clicking on the line. Sketchpad lets you know that it understood your intentions by flashing the selection squares on the line.

 

STEP2: Now create an object, any object, in your sketch. Select it, go to the Transform menu and select Reflect. Sketchpad will reflect all the items that were selected in your sketch.

 

STEP 3:Take a minute to observe how Sketchpad keeps track of labels in a reflection. As usual, if you move either the mirror or the object you reflected, sketchpad will adjust the whole sketch accordingly.


 

 

Rotation.

 

A rotation around a point C (called the center of rotation), by an oriented angle a, is a transformation that takes a point P to a point Pí so that CP is the same length as CPí and the oriented angle ­ PCPí has the same size and orientation as a.

Rotation with Sketchpad

 

STEP 1: Create a point. Label it C. To tell sketchpad that you want this point to be your center of rotation, select it and pull down the Transform menu. Highlight the Mark Center option.

 

There are two ways to tell sketchpad which oriented angle you want to use as your angle of rotation:

            i) Through a dialog box.

            ii) Creating an angle in your sketch.

STEP 2 shows you how to use the dialog box. STEP 3 shows you the more dynamic way of creating a modifiable angle.

 

STEP 2: Create an object in your sketch and select it. In the Transform menu select Rotate. You will be prompted with a dialog box in which you can specify the angle of rotation. Sketchpad follows the usual trigonometry convention of considering positive angles as oriented counterclockwise and negative angles as oriented clockwise. When you click on OK, Sketchpad will perform the rotation for you.

 

STEP 3: Create an angle with three points either free in the sketch or on objects, it does not matter. Select the angle by selecting the three points in the following order: point on the initial side, vertex, point on the terminal side. Be careful that the order in which you select the points changes the angle and the angle orientation. If you want to use the angle ­ PQR, oriented counterclockwise, you MUST select the three points in the order P, Q, R. Selecting R, Q, P will give you the same angle but oriented clockwise.

 

STEP 4: Under the Transform menu select Mark Angle. Notice that Sketchpad lets you know that it understood your intentions by flashing an arc through the angle, showing you which angle and with which orientation it is looking at.

 

STEP 5: Create an object and select it. Under the Transform menu, choose Rotate. In the dialog box now you will see the option by Marked Angle automatically selected. Choose OK (or hit return) and watch sketchpad rotating your object.

 

STEP 6: Exploit fully the dynamic nature of Sketchpad by playing with your angle.

Translation

 

A translation given by a vector v is a transformation that takes a point P to a point Pí so that the vector with tail at P and tip at Pí is equal to v (it has the same length, slope, and orientation).

 

 

Translation with Sketchpad

 

STEP 1: Create a vector by creating a line segment or just two points. Label the vertices of the segment or your two points. To tell sketchpad that this is the vector you want to use to translate, select the vertices of your segment or your two points. Be careful that the order in which you select the points is read by sketchpad as tail, tip for your vector, so a different order changes the direction of the vector. Go under the Transform menu and select Mark Vector. Sketchpad will flash the vector, from tail to tip, to confirm your intentions.

                       

STEP 2: Create an object and select it. Go under the Transform menu and select Translate. The dialog box will propose by default to translate by the vector you have marked in STEP 1. Click on OK or hit return.

 

STEP 3: Exploit fully the dynamic nature of Sketchpad by playing with your vector.

 

 

 

 

 

 

 

 

 

 

 

 

 

PROBLEM 1: What kind of isometry do I get if I perform two consecutive reflections?

 

 

 

Think about the possible mutual positions of two lines in the plane.

Experiment with Sketchpad. Talk to your neighbor.

 

 

We suggest that you use triangles or quadrilaterals as objects to reflect. Concave quadrilaterals are particularly effective since they easily reveal changes in orientation. Reflecting a single point or two points will NOT be sufficient.

 

 

When you have a clear conjecture, write it down as concisely as you can.

 

 

 

PROBLEM 2: Suppose you are reflecting consecutively across two mirrors that meet at a point. From Problem 1, you believe that the resulting isometry is a rotation. Can you find the angle of rotation and the center of rotation knowing only where the mirrors are?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

PROBLEM 3: What happens if you pivot your two mirrors in Problem 2 around their point of intersection, keeping the angle between them fixed? Does the final resulting isometry change? Why ?

 

To explore problem 3 with Sketchpad follow the steps below:

 

STEP 1: Mark the point of intersection of the two mirrors as center by double clicking on it.

 

STEP 2: Hold down the mouse button while you are clicking on the Select and Translate(Arrow) tool on the tool palette. A sub-menu will spring out to the right. Select the Rotate tool.

 

STEP 3: Select both mirrors in your sketch and drag them around their point of intersection.

 

 

PROBLEM 4:            Suppose you are reflecting across two parallel mirrors. You believe, from Problem 1, that the resulting isometry is a translation. Can you find the vector (its slope, direction and length) knowing the position of the mirrors ?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

PROBLEM 5: What happens if you move your two mirrors in Problem 4 around the plane, keeping the distance between them fixed? Does the final resulting isometry change? Why?

 

To explore Problem 5 with Sketchpad follow the steps below:

 

 

STEP 1: Make sure that the Select and Translate tool is selected at the top of the tool palette.

 

STEP 2: Select both mirrors in your sketch and drag them around the plane.


Activity 11           YOUR TURN, NOW

 

 

STEP 1: Get together with a group of (ideally) four people who are interested in the same type of activities for this afternoon session (Geometry, Algebra, or Calculus.)

 

 

 

 

 

 

 

STEP 2: With the optional help of the textbook that you should have brought along, choose with your group an activity (a problem, a class presentation, a topic, a concept) that you think could greatly benefit and be enriched by the use of the Geometerís Sketchpad.

 

 

 

 

 

 

 

 

 

STEP 3: Start developing the activity and all the necessary sketches. This process will probably take more than the allotted hour. Donít worry about how far you will go. It is important that you start working independently. Feel free to ask for help and assistance from the mentors in the room at any time.