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Logical Reasoning Algebra 1

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Algebraic Logic

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Logical Reasoning Algebra 1

1. 1. Logical Reasoning Chapter 1 section 7
2. 2. Riddles <ul><li>Farmer Brown told Bob and Sue that they could pick apples from his tree, but that neither of them could take more than 20. They worked for a while, and then Bob asked Sue, &quot;Have you picked your limit yet?&quot; Sue replied, &quot;Not yet. But if I had twice as many as I have now, plus half as many as I have now, I would have my limit.&quot; How many did Sue have? </li></ul>
3. 3. There’s Algebra in a Riddle???? <ul><li>How is this algebraic? Can you come up with a reasonable algebra equation to solve this riddle? </li></ul><ul><li>How about this: . Let x = the number of apples she had.  </li></ul><ul><li>                  2x + 1/2 x = 20 </li></ul>
4. 4. How many does she have?
5. 5. Now try this one: <ul><li>A little boy was told not to eat the grapes from the vine for fear that he would eat too many and get a stomachache. Sneaking out to the grape arbor when his mother wasn't looking, the little boy ate grapes for five days, each day eating 6 more than the day before. In fact, after five days, the little boy was so sick that he had to confess to his mother that he had eaten 100 grapes. How many grapes did the little boy eat on EACH of the five days? </li></ul>
6. 6. How can you represent that one algebraically?
7. 7. Does your guess look anything like this??? <ul><li>Let x = number of grapes the little boy ate the first day  </li></ul><ul><li>                  x + 6= number of grapes eaten the second day </li></ul><ul><li>                  x + 6 + 6 = number of grapes eaten the third day </li></ul><ul><li>                  x + 6 + 6 + 6 = number of grapes eaten the fourth day </li></ul><ul><li>                  x + 6 + 6 + 6 + 6 = number of grapes eaten the fifth day </li></ul><ul><li>Five days' worth of grapes = 100 in all. Therefore, the equation to set up is: </li></ul><ul><li>  x + (x + 6) + (x + 6 + 6) + (x + 6 + 6 + 6) + (x + 6 + 6 + 6 + 6) = 100 </li></ul>
8. 8. So, how many did he eat on each day???
9. 9. How about this one? <ul><li>How high is a tree that is 15 feet shorter than a pole three times as tall as the tree?   </li></ul>
10. 10. Did you come up with this??? <ul><li>Let x = number of feet in height of tree </li></ul><ul><li>3x would equal the number of feet in height of the pole </li></ul><ul><li>    </li></ul><ul><li>               Set up the equation: </li></ul><ul><li>                  x = 3x - 15 </li></ul>
11. 11. Well, what did you get???
12. 12. Logical reasoning is like solving riddles: <ul><li>It is also like science experiments! </li></ul><ul><li>You will have a hypothesis, and a conclusion! </li></ul>
13. 13. If, then statements: <ul><li>Check out the following statement: </li></ul><ul><li>If it is raining, then we will get wet. </li></ul><ul><li>Hypotheses: it is raining </li></ul><ul><li>Conclusion: we will get wet </li></ul>
14. 14. You try: <ul><li>If we are tired, then we take a nap. </li></ul><ul><li>What is the hypothesis? </li></ul><ul><li>What is the conclusion? </li></ul>
15. 15. Can you do that in reverse??? <ul><li>Hypothesis: we are bored </li></ul><ul><li>Conclusion: we will read a book </li></ul><ul><li>Form an if-then statement: </li></ul>
16. 16. Pretty good, how bout this one? <ul><li>Hypothesis: She is angry </li></ul><ul><li>Conclusion: she will yell at me! </li></ul><ul><li>What would that look like in if-then form? </li></ul>
17. 17. Finding Counterexamples: <ul><li>(that’s just a big word for figuring out how to prove that a statement could be false in certain situations!) </li></ul><ul><li>Consider this statement: </li></ul><ul><li>If you play a musical instrument then you are in the school band. </li></ul>
18. 18. Is this an “always” true statement? <ul><li>Think about it!! </li></ul><ul><li>If you play a musical instrument then you are in the school band. </li></ul><ul><li>Can you prove it wrong? </li></ul>
19. 19. What about proving this one wrong? <ul><li>If 2x > 10, then x > 4 </li></ul><ul><li>Can you think of a counterexample? </li></ul>
20. 20. Slideshow by: Ms. K. Williams
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