Microworld: How
to draw a star
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to visit the Microworld in your Browser.
Author:
Dan
Kalman
User's Guide for How to draw a star
Overview
The
Star mathwright microworld provides opportunities to explore the geometry
of stars. There are three activities pages. On the first you can explore
the relationship between the number of points in a star and the size of the
angle formed at each point. The second allows you to construct stars by
drawing polygons with vertices regularly spaced around a circle. The third
automates this process according to a user defined pattern for drawing the
sides of the polygon. Specific suggestions for using each page are included
in this User's Guide.
Page 1
This is the title page.
Page 2
This is an overview about stars, and an introduction to the following activity page.
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Page 3
On
this page you specify the number of points and the size in degrees of the
angle for each point. Click the Draw button to see the result. If
that result is a star, then a point is added to the graph showing angle (y)
versus number of points (x). A proposed equation relating those variables
can be entered to the right of the graph. Use the Graph Equation button
to see how the proposed equation relates to the data points.
You can control the speed at which the figure is drawn with the Faster and Slower buttons.
Here
are two suggestions for using this page.
First,
try to draw a star with a specified number of points. For example, try to
make a nine pointed star. Enter 9 in the box labeled How many points?
and guess an angle for each point. If you are successful, try again with
a different number of points. On the other hand, if your first guess does
not create a nine pointed star, can you tell whether you should make the angle
smaller or larger? Try to refine the angle through a process of trial and
error, until you find the correct angle to make a none pointed star.
Second,
put a very large number in for the number of points, say 90 or 100. Now try
various angles. The computer will start drawing lines and making angles,
and if a closed star results, the drawing will stop. In this way you can
discover many different kinds of stars.
Each
time you find a star, a new data point will be added to the graph in the lower
part of the screen. One goal here is to find an equation that relates the
number of points (x) and the angle at each point (y). If you think you see
a pattern that fits the data (or part of the data), enter an equation for
your pattern in the white equation box and then use the Graph Equation
button.
Page 4
On
this page you can draw stars with the computer's help. First decide how many
points you would like, and enter that number in the white box. Next, click
Mark Points to see where the points should be. Finally, to draw a
star, click points in the graph. As you click the points, the computer will
draw lines between them, creating a figure. The goal is to click the points
to produce a star. The colored block in the lower right corner of the graph
shows the color that will be used to draw the lines. Use the Change Color
button before you start drawing the lines to select the color for the star.
Specific activies:
1. Try drawing stars with 11 points. How many different stars can you draw? Try again for 7 points. How many stars can you draw? How about for 13 points.
2. Try drawing stars with 6, 8, 10, or 12 points. Can you create stars like the ones shown below?
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Page 5
This
page is similar to the preceding page. However, on this page, the computer
will draw the lines for you. You have to tell the total number of points,
and must also indicate the pattern for drawing the lines. For example, if
you say to draw the lines skipping ahead 3 points each time, the lines will
be drawn as if you had used the mouse to click every third point. This way
you can easily create star patterns without actually doing all the clicking.
As before, you can speed up or slow down the drawing, and you can change the
color.
Specific Questions to consider:
1. For 10 pointed stars, how many different stars can you find? Which skip ahead numbers lead to 10 pointed stars and which do not?
2. Repeat question 1 for 12 pointed stars. Repeat again for 15 pointed stars.
3. Can you guess the general rule which tells when a star will be formed, and when it will not?
4. Can you see a pattern that indicates how many different stars are possible for a given number of points?
5. Try combining different stars with different colors to create interesting patterns. Here is an example:
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| - James E. White, Ph.D. , Library Director, | ||
| author of this website, Mathwright 2000, MindScapes, | ||
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