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Patterns, Functions, and Algebra
Session 3 Part A Part B Part C Part D Part E Homework
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Session 3 Materials:



Notes for Session 3, Part E

Note 6

The final part of this session introduces a more general notation of function, rather than just algorithmic functions with numeric inputs and outputs.

Groups: Work in pairs to describe what you think a function is. Some people may recall struggling with learning or teaching about functions using diagrams like these:

example of a function

<< back to Part E: Other Kinds of Functions


Note 7

Read the definition of a function and take a look at the examples in the course text.

Groups: Work on Problems E1-E4 in small groups, then as a whole group. Discuss the other examples of functions and non-functions before moving on.

<< back to Part E: Other Kinds of Functions


Note 8

Read about the "Prime?" function and review the definition of a prime. Think about some examples of primes and non-primes and how you could test to see if a number is prime if you aren't sure.

Work on Problems E5-E8. These problems address common confusion about both prime numbers and functions.

Groups: Summarize these problems in a discussion as everyone completes their work.

Here are some points to consider:


2 is a prime number. It is the only even prime.


1 is not a prime. This is a convention. The number 1 fits the definition of prime we have given, since it is only divisible by itself -- one -- and one. However, an important theorem in mathematics, called the fundamental theorem of arithmetic, says that every integer greater than one is either prime or can be expressed as a unique product of prime numbers. If 1 is considered a prime, this would no longer be the case. Consider: 10 = 2 x 5. But 10 = 1 x 2 x 5. But 10 = 1 x1 x 1 x 1 x 1 x 2 x 5.


The fundamental theorem of arithmetic is essential for proving many mathematical results, so it would never do to allow 1 to be a prime!


Two inputs to a function may give the same output. In this case, many numbers produce the output "yes," and many will produce the output "no."


Not every function can be undone. In this case, if the output is "yes," for example, there's no way of knowing what the input was. (You may want to discuss how this is related to the point above.)

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Note 9

Read about the "3" function and work through Problems E9-E12. These problems may reinforce many of the points in Note 8.

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Note 10

Look at how Algorithm M works by going through the steps with a pentagon. After finishing the drawing, consider if there was any other way you could have followed the directions. For example, you could connect the midpoints in a different order. No matter how you connect the midpoints, however, the output will be the same. Once this is clear, work on Problems E13-E16.

Groups: If there is time, compare results of this geometric algorithm.

There are several surprising things that some people may notice:


No matter what shape triangle you start with, you end up with four identical triangles inside your original triangle. Three are oriented the same way as the original triangle, and one is upside down.


The four triangles are all similar to the original. For example, if you connect the midpoints of a right triangle, you will end up with right triangles inside.


The areas of each of the triangles are 1/4 the area of the original (since there are 4 of them, and they are identical).


The inside figure of the quadrilateral is a parallelogram. That is, opposite sides are parallel. It doesn't necessarily resemble the outer figure at all.


The inside figure of a quadrilateral contains half the original area (this may be more difficult to see).

<< back to Part E: Other Kinds of Functions


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