Select one of the series of shapes from the
Examples menu.
The tool shows the first four shapes in the series (you may have to scroll laterally to see all four shapes).
Click the new button to add a fifth shape. Draw the shape in the window on the right,
then click the update button.
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Present the value table and graph of the counting function
for the series of shapes, as well as its change. Can you use this data
to predict the number of squares in the sixth shape in the series?
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Construct a correspondence rule for the
counting function. You can enter the correspondence rules at the
bottom of the tool, then present their graphs.
Use the correspondence rule to predict the number of squares in
the sixth shape in the series.
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Add a sixth shape and check your previous predictions.
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Describe interesting ideas that
came up in your work with series of shapes. Give
examples. Some questions to consider:
- What series of shapes define counting functions
with linear growth? When do you obtain quadratic growth? Are
there series of shapes for which the function is neither linear
nor quadratic?
- Relations between different series of shapes:
Are there cases in which you can use the correspondence rule
you constructed for one series of shapes to
construct a correspondence rule for a different series?
- Is it possible to construct two different correspondence rules
for the same series of shapes by using different approaches?
How can you prove that two different correspondence rules
define the same function?
- Is it possible to find different series of shapes that have
the same counting function?
- What questions about large shapes can you answer based on one of the
representations of the counting function?
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