Friday, February 26, 2010

Math problems of the week: 1900's algebra vs. College Preparatory Mathematics

I. Two of the problems in the second chapter of Wentworth's New School Algebra, published in 1898 (written by G.A. Wentworth), p. 22:


Write in symbols:

A man has x dollars y dimes and z cents. If he spends a half-dollars and b quarters, how many cents has he left?

A rectangular floor is a feet long and b feet wide. In the middle of the floor there is a square carpet c feet on a side. How many square yards of the floor are bare?

II. Two of the problems in the final chapter of College Preparatory Mathematics Algebra Connections, Volume 1, (written by  Leslie Dietiker, Evra Baldinger, Carlos Cabana, John Cooper, Mark Cote, Joanne da Luz, David Gulick, Patricia King, Lara Lomac, Bob Petersen, Ward Qincey, Babara Shreve, and Michael Titelbaum) p. 266:

Using the variable x, write an equation that has no solution. Explain how you know it has no solution.

Given the hypothesis that 2x + 3y = 6 and x = 0, what can you conclude?  Justify your conclusion.

III. Extra Credit

The following is the final problem in CPM's Algebra Connections, Volume 1.  Discuss.

HOW AM I THINKING?

This course focuses on five different Ways of Thinking: reverse thinking, justifying, generalizing, making connections, and applying and extending understanding These are some of the ways in which you think while trying to make sense of a concept of to solve a problem (even outside of math class). During the chapter, you have probably used each Way of Thinking multiple times without realizing it!

Review each of the Ways of Thinking that are described in the closure sections of Chapters 1 through 5. Then choose three of these Ways of Thinking that you remember using while working in this chapter. For each Way of Thinking that you choose, show and exlain where yo used it and how you used it. Describe why thinking in this way helped you solve a particular problem or understand something new. (For instance, explain why you wanted to generalize in this particular case or why it was useful to see these particular connections). Be sure to include examples to demonstrate your thinking.

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