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Patterns, Functions, and Algebra
Session 9 Part A Part B Part C Part D Part E Homework
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Notes for Session 9, Part C

Note 4

We are all familiar with our real number system. It's one thing to be handed a system and to learn how to work in it; it's another to see how a system evolves and then work in it. In this session, we'll develop an algebraic system and work in it.

Suppose you're interested in looking at units digits of whole numbers. Think about this problem:

Find the units digit of 364 * 123 + 48 * 135.

Groups: Take five minutes or so to come up with a solution. (You may have to review order of operations.) Share solutions, along with how you thought about the problem. In particular, discuss whether there is some way to do this problem without doing all the actual calculations. Have the whole group suggest possibilities. Be sure to discuss why you can just look at units digits.

<< back to Part C: Algebraic Structures


Note 5

Groups: Work on Problems C1-C5.

What you have developed in Problems C1-C5 is a "units digit arithmetic," an arithmetic whose objects are the digits 0 through 9, and whose operations are "add and take the units digit" and "multiply and take the units digit." The whole system can be captured in the two tables provided. These tables define an algebraic structure on the set of numbers {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.

<< back to Part C: Algebraic Structures


Note 6

Groups: Read the text before Problems C6-C19, and then work on these problems in small groups. Share what patterns you see in the table.

<< back to Part C: Algebraic Structures


Note: Optional Problems

This part introduces some key mathematical terms involved in current work in algebraic structures. All groups share these properties, and all groups share certain methods for solving equations (specifically, solutions involving inverting operations). It is as important to show examples of algebraic structures which are not groups in order to understand the relationship between structures which are groups.

Groups: Work on Problems C20-C22 in small groups.

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