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1/7/2009 Section 1.2 Reasoning Mathematically a) Add the following numbers. 1+3 3+7 35 + 31 b) Does every squared number equal itself? 0² = 1² = 2² = c) The product of two odd numbers is what type of number? 5x9= 3x7= 11 x 13 = 1) Inductive Reasoning ` involves the use of information from specific examples to draw a general conclusion (generalization). A pattern or relationship is detected through observation. ` is a central process in elementary school math since children form most of their general mathematics ideas by seeing pattern in examples ` ` A counterexample is an example that shows a generalization to be false. What was the counterexample used to disprove the generalization in part b of the previous example? d) The three times I went on a picnic it rained. Sequences: patterns involving ordered arrangements of numbers, geometric figures, letters, etc. The numbers, figures, etc. are called the terms of the sequence. ◦ Arithmetic sequence: numeric sequence in which each term is obtained from the previous term by adding ddi a fi fixed d number b ((common diff difference). ) 2, 4, 6, 8, 10, 12, … ◦ Geometric sequence: numeric sequence in which each term is obtained from the previous term by multiplying a fixed number (common ratio). 1, 3, 9, 27, 81, 243, … ◦ Fibonacci sequence: each term is the sum of the two preceding terms 1, 1, 2, 3, 5, 8, 13, 21, … ◦ Column extension patterns ◦ Row relationship patterns efficient) (usually more 1 1/7/2009 ` x 64, 32, 16, 8 x 16, 17, 19, 23, 31 x 1, 2, 5, 14 x 1 4 1, 4, 9 9, 16 x 2, 5, 10, 17 x 1,8, 27, 64 x 2, 10, 42, 170 ` ` ` ` Counting numbers ` Even numbers ` Odd numbers ` 2, 6, 10, 14, … Consider 4, 8, 12, 16, 20, … nth term = Find the 7th term. ` ` The rules for some sequences can be stated algebraically. Finding the rule can sometimes be simple, sometimes challenging. 1, 2, 3, 4, … ` What is the first term? The next three terms? In many everyday situations a rule relates two sets of numbers. This relationship can often be represented with a formula. Consider the following. A 10-inch candle burns according to the information in the table below. Time 0 1 (min.) Height 10 8 (in.) 2, 4, 6, 8, … 1, 3, 5, 7, … ` ` ` 2) Deductive Reasoning ` the process of reaching a necessary conclusion from given facts or hypothesis ` involves drawing conclusions from given true statements by using rules of logic ` is used by detectives solving crimes and by philosophers thinking logically Suppose the rule for the nth term of a sequence is 2n+1. ` ` ` ` ` ` ` 2 3 4 5 6 What is the height of the candle after burning 3 minutes After 5 minutes? Write a formula that represents the relationship between the height of the candle and the time spent burning. The painting “By Numbers” was stolen from Jane Dough’s Illinois home last night. Today the painting was found in an alley. Who was the thief? The only suspects are Jane Dough (the owner of the house), Jeeves (the butler), Sharky (the pool man), and Fluffy (the pet Doberman). Doberman) The police have gathered the following evidence. The painting was stolen between 8 p.m. and 9 p.m. Fluffy can’t carry a painting. Jeeves has been in Liverpool all week. Sharky’s and Jane’s fingerprints were on the canvas. Jane was with three friends at the movie from 7:30 p.m. to 10:00 p.m. last night. 2 1/7/2009 ` Understanding if-then statements and deciding when they are true. ` ` ` If-then statements are called conditionals. If p, then q. p: hypothesis ` ` ` ` q: conclusion ` Consider the following statement made by your basketball coach: “If you play well in practice, then you will start in tomorrow’s game.” In which of the following cases would you feel that you were being treated unfairly and the coach didn’t tell the truth? Case 1: You play well in practice. You start the game. (Hypothesis true) (Conclusion true) Case 2: You play well in practice. You do not start the game game. (Hypothesis true) (Conclusion false) Case 3: You do not play well in practice. (Hypothesis false) You start the game. (Conclusion true) Case 4: You do not play well in practice. (Hypothesis false) You do not start the game. (Conclusion false) ` Using rules of logic -- Determine whether each of the following is TRUE or FALSE. ` If you live in Tuscaloosa, then you live in Alabama. ` ` Given: You live in Tuscaloosa. C Conclusion: l i You Y li live iin Al Alabama. b This is called affirming the hypothesis. Given: You do not live in Alabama. Conclusion: You do not live in Tuscaloosa. This is called denying the conclusion. 3) Proportional Reasoning ` involves drawing conclusions or solving problems with either the formal or informal use of proportions (statements which assert that two ratios are equal). ` A dog d owner gives i each h off her h three th dogs d a puppy biscuit twice a day. How many puppy biscuits does the dog owner give out in one week? Given: You live in Alabama Conclusion: You live in Tuscaloosa. This is called assuming the converse. If you live in Alabama, then you live in Tuscaloosa. ` Given: You do not live in Tuscaloosa. Conclusion: You do not live in Alabama. This is called assuming the inverse. If you do not live in Tuscaloosa, then you do not live in Alabama. 4) Spatial Reasoning ` involves visualizing 3-D geometric figures or objects to draw conclusions about properties or relationships involving these figures or objects. ` A Rubik’s Cube™ is a 3x3x3 cube that is comprised of 27 smaller cubes. Each of the six faces of the large cube is painted a different color (red, (red blue, blue green, green yellow, yellow orange, orange and white). ` How many of the smaller cubes are painted on only one face? 3