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Transformations:
Open a new sketch. Draw a triangle and place a single point somewhere outside triangle.
1. Describe how you constructed the triangle:
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Select only the single point and choose "Mark Center" under the TRANSFORM menu.
2. What is the keyboard shortcut for marking a center?
Change the select tool to the rotation tool. Select all parts of the triangle. (You may select each object separately, or you may click to the upper left outside the triangle and drag a box down and to the right to surround the triangle. In drawing this box, you select all parts of the triangle.) Now, with the entire triangle selected, choose "Trace Objects" from the DISPLAY menu, and then drag the triangle.
3. What happens?
Choose "Erase Traces" from the DISPLAY menu. Change the rotation tool to the dilation tool. Now, with the entire triangle selected, choose "Trace Objects" again from the DISPLAY menu, and repeat your dragging of the triangle.
4. What happens when you try to move the triangle closer to and then farther away from the point?
The Select tool is sometimes called the translate tool. Define the three words translate, rotate and dilate.
5. Translate:
6. Rotate:
7. Dilate:
Start with a new sketch. On the left half of the sketch draw a smiley face, a horizontal segment just below it, and a vertical segment just to the right of it. (To draw the smile, construct a circle and with the circle selected, construct three points, two for the corners of the mouth and one that is between the two corners. Select those three points, in order from left to right. Then from the Construct menu choose Arc Through 3 Points. Adjust the three points so as to make a nice smile, and, after you are happy with your smile, hide the three points on the arc.) Select the vertical segment and choose "Mark Mirror" under the TRANSFORM menu. Select your entire smiley face and choose "Reflect" under the TRANSFORM menu. Now try moving one of the eyes on the left smiley face.
8. What happens to the smiley face at the right?
9. Mark the horizontal segment as a mirror
and perform a similar experiment on the smiley face. Describe
new mirror image and the effect of moving an eye.
10. Now try dragging the vertical segment to
the left to where it "hits" the nose. What does this
move tell you about your original smiley face? Of what property
of your original smiley face does this move provide a qualitative
measure?
Creating New Tools: The Script Tools
Start with a new sketch. Draw a large circle in the center of
your sketch. Hide the point that is on the circle. Now select
the circle (not the center) and choose "Point On Circle"
from the CONSTRUCT menu. A new point should appear on the circle.
Find its label and change it to "Flora." Construct one
more point now that is exterior to the circle and label it "Fauna."
Select Flora and Fauna and have the computer construct a segment
connecting the two by choosing Segment from the CONSTRUCT menu.
With the segment (and only the segment) still selected choose
Midpoint from the CONSTRUCT menu. With the midpoint still selected,
select the segment, too, and choose Perpendicular Line from the
CONSTRUCT menu. Change the color of the line to red.
Now select all the objects in the sketch plane, except the circle center, and press the bottom button, the Custom Tool Button, on the tool bar and choose Create New Tool. Click on Show Script View and press OK.
You've now created a new tool, which is of the type called a script. The script should also appear on your monitor. It should have two objects as Given and 4 Steps that construct other objects.
Most importantly, the script constructs the perpendicular bisector of the segment joining a randomly chosen point (in the plane) to a randomly chosen point on the circle.
The idea is that from now on, you can select two of the appropriate kind of objects as given, and let the script do the rest for you by "playing" the script. This script is pretty simple and the objects it constructs wouldn't be too hard to redo on your own many times, but when you make a complex diagram that you may want to construct many times, you can save a lot by having a script to do the job. Here's how to play this script:
Open a new sketch. (Don't close the one you just made with the script tool, however.) You need to construct the givens, which are a path object and a point. If you want the results of playing the script to be pretty much the same as before, you should have the path object be a circle and the point be exterior to that circle. But you could have the point anywhere and the path object could be straight instead of being a circle. On your new sketch, construct a point and a path (circle or line). Follow the instructions as they appear at the bottom in your script view.
11. What happens?
More on Locus:
First try dragging the point on the circle around that circle,
watching the constructed perpendicular as you drag. The object
of creating the locus of the perpendicular as the point traverses
the circle is to see the totality of locations for that perpendicular.
Make a prediction, in your mind, of what the locus should look
like. To construct the locus, you need to select the point that
is moving on the circle, the circle, and the perpendicular. With
that combination of objects selected, choose Locus from the CONSTRUCT
menu.
12. Describe the resulting locus.
13. Put your name on and print your sketch and script. (To print the script, on a Macintosh, press and hold the control key while clicking on any of the objects in the script view. Select Print Script View from the menu that pops up. In Windows, right click to get the print option.) Print several different versions of the sketch if you like.
Be sure to try dragging the given point around, both outside and inside the circle, to see the effect that such dragging has on the shape of the locus.
Measurements:
Start with a new sketch. Draw a segment, line and circle. Select
the segment, but not the endpoint. What about the segment can
you measure and what is its measurement? (do not include calculate)
14. What can be measured? What are those measurements?
Now drag one of the endpoints of the segment.
15. What happens to the measurement as you drag the endpoint?
Now select just the endpoints but not the actual segment. What about the segment can you measure and what are the measurements (again ignore calculate)? (Again, after making the measurements, try dragging an endpoint and note the effect.)
16. What can be measured? What are those measurements?
Select just the circle. What about the circle can you measure and what are the measurements (ignore calculate)?
17. What can be measured? What are those measurements?
Try dragging the point on the circle (the "sizing" point - the one Sketchpad constructs for you when you draw the circle with the circle tool.).
18. Which measurements change as you drag this point?
Now drag the circle itself.
19. Which measurements change as you drag the circle?
20. Be sure to use the Equation option under MEASURE to get an equation for the circle. What remains unchanged and what changes about the equation when the sizing point is dragged?
What remains unchanged and what changes about the equation when the circle itself is dragged?
What's the Point of a Circle Anyway:
Open a new sketch. Draw a segment and a point, and select both. From the Construct
menu select Circle by Center+Radius.
21. Does the circle have a sizing point? _________ By experimenting, determine what controls the size of the circle. The size is controlled by ______________
In a sense, the circle constructed above is non-messupable,
but its size
can be changed by dragging the segment that controls it.
A circle seems like such a fundamental geometric
object, but an object that folding has some difficulty with. Another view
of a circle
is that it “finds all
the points at a fixed distance from a fixed point,” hence Euclid’s 4th
postulate: To describe a circle with any center and radius.
So, replacing the idea of measuring a certain distance given a prescribed
number, one can construct a point at a distance that it is precisely the
length of a segment, by “describing
a circle with any center and radius.”
22. Looking at the Construct menu, in what other way is it possible to construct
a circle?
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