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Patterns, Functions, and Algebra
Session 8 Part A Part B Part C Homework
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Notes for Session 8, Part B

Note 4

The two ways to think of inverse proportions are as input * output = constant and as output = constant / input. Both are useful for different situations.

<< back to Part B: Inverse Proportions


Note 5

Groups: Work through Problems B1-B3, and then discuss the terms "inverse proportion" and "inverse variation." Move on to Problems B4-B12. Discuss these problems as a group, especially how inverse proportions are different from other functions we've seen.

Unlike other functions we've seen, increasing the input (in magnitude) decreases the output by the same factor (in magnitude). As long as the numerator is positive, as x gets bigger, y gets smaller. This is a striking difference from other functions.

<< back to Part B: Inverse Proportions


Note 6

Groups: Discuss Problem B6, particularly what's going on in the table. Some observations that may arise are, "As x increases, y decreases, but by less and less each time." Or, "As y decreases, so does the amount by which it decreases." Or, "As y gets smaller, it does so at a slower and slower rate."

It's also important to think about the graph and whether it will cross either of the axes. Going back to the contexts of the problems (sharing money among several people, for example) should help make sense of the asymptotic nature of the graphs.

<< back to Part B: Inverse Proportions


Note 7

Groups: Take 10 minutes to jot down answers to Problems B11 and B12, then share answers.

Take time to think about the how the word "reciprocal" describes the relationship (x) (y) = 1. There are some important and often confused ideas here. For example, 1 has a reciprocal (itself). The fraction 1/5 also has a reciprocal: 5. Too often, students and teachers think of whole numbers as the "regular" numbers and the "reciprocals" as 1/n. Reciprocals can be used in solving equations. When you solve 3x = 9 by "dividing each side by 3," another description is "multiplying each side by the reciprocal of 3." This method generalizes to other systems (equations involving matrices or equations involving modular systems, which we will see in Session 9), whereas the "divide by 3" method is specific to the real numbers.

Groups: End this part by adding inverse proportions to the list of functions.

<< back to Part B: Inverse Proportions


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