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Welcome to Math 302B Fall 2009

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Agenda Go over syllabus, class procedures, etc. How to answer a homework problem Begin algebraic thinking Assign homework

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A good answer… Shows how you got the answer (all the work). Explains why you did what you did. Where appropriate, explains what a child did correctly/incorrectly, and what mathematics the child did/did not understand.

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Here’s a problem. The first five terms of a sequence are 2, 5, 10, 17, 26, … What is the 10th term? What is the nth term? –Introduce yourselves at your table. –In your groups, try to figure out the tenth term. Then, write a sentence or two that you think is perfect for a homework answer.

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Unacceptable Answers 10th term is 101 2, 5, 10, 17, 26, … 101, … ? 2, 5, 10, 17, 26, 37, 50, 65, 82, 101, … This is a trick question. There is no answer.

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More unacceptable answers The 10th term is 101. I did the math in my head. The 10th term is 101. I did the math on my calculator. The 10th term is 101. I just did trial and error--lucky me! I got it on the first try.

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Acceptable answer I see the pattern 2, 5, 10, 17, 26, …. So I made a table: Term: 1 2 3 4 5 6 7 8 9 10 Pattern: 2 5 10 17 26 37 50 65 82 101 Then, I saw that 5 - 2 = 3; 10 - 5 = 5; 17 - 10 = 7; 26 - 17 = 9. The differences increase by 2, so to continue, I would add 11, then 13, then 15, and so forth. So the 10th term is 101.

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Acceptable answer I see the pattern 2, 5, 10, 17, 26, …. By itself, there is nothing that leaps out at me. But, if I subtract 1 from each number, I get 1, 4, 9, 16, 25, … which are the perfect squares. So, each number is n 2 + 1. So the 10th term is 10 2 + 1 = 100 + 1 = 101.

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Algebraic Reasoning Organizing data and generalizing data: –Mathematical modeling through graphs –Mathematical modeling through tables. Generalizing patterns Relationships involving variables--things that vary. We use the rules of algebra to express algebraic reasoning.

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Does a variable “vary”? 5x = 60 y = 3x - 4 C = π d n/n = 1 if n ≠ 0. y = kx sin 2 (x) + cos 2 (x) = 1

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The Equal Sign = Signifies that the expressions on each side are equivalent. If 3x + 4 = 12, then 3x + 4 - 4 = 12 - 4. Look back at the properties: commutative, associative, distributive, identity, inverses Now look at algebraic “formulas” (x + 3)(x - 4) = 0 means either x + 3 = 0 or x - 4 = 0. Why?

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Function A special relationship found in algebra is the function. A function is a relation where every value of input has just one value of output. Your age today: your height today. Students name: Students ID number Number of hours worked: Paycheck Diameter of a circle: area of the circle

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These are not functions Shoe size: number of shoes you own. Your height: Your age Date: price of gasoline in the area Length of a football field: Year of Superbowl Number of courses you are registered for: number of students in your largest class

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In any function The input is called the independent variable. We usually put this on the x-axis. The output is called the dependent variable. That is, it depends on the independent variable. We usually put this on the y-axis. Example: age vs. height; typically, your height depends on your age.

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