1. Math is Empirical? “empirical” = “needing evidence, usually through experience”

2. A posteriori Something that needs to be experienced to be understood? Imagine trying to teach a child arithmetic. How do you start? Imagine trying to teach a child arithmetic. How do you start? At what point does a child learn a “level of abstraction?” Can abstraction exist independent of experience? At what point does a child learn a “level of abstraction?” Can abstraction exist independent of experience?

3. Generalizations? Why do we feel more certain about a mathematical statement (2 + 2 = 4) and less about a scientific statement (“All metals expand when heated”)?

4. Is this the same? Imagine there are two hungry lions in a cage. You open the door and throw in two lambs. How many animals are in the cage when you return the next day? Does this do anything to convince you that 2 + 2 is not always equal to 4? Can you imagine a world in which 2 + 2 = 5? For example, what if every time you brought two pairs of objects, a fifth one appeared?

5. Triangle Sum Conjecture Objective: Write a conjectures about the triangle angle measures

6. 1.Draw 3 different triangles on your paper, using a straightedge Include at least one acute and one obtuse triangle 2.Measure and label each angle as accurately as possible, using your protractor 3.Find the sum of all angles in each triangle 4.Is there a pattern? What type of reasoning are we using? Inductive

7. a b c The sum of the measures of the angles in every triangle is ______. Triangle Sum Conjecture 180 0 a+ b+ b+ c+ c= 180 0

8. The sum of the measures of the angles in every triangle is ______. Triangle Sum Conjecture Proof 180 0 Parallel Lines Conjecture: Alternate Interior Angles Substitution Linear Angles Draw line ReasonStatement Proof 3 2 1 54 CE B A Auxiliary Line (line that helps with proof)

9. If two angles of one triangle are congruent to two angles of another triangle, then the third angle in each triangle __________________________. must be congruent. Third Angle Conjecture ab c ab x Pg 201 #2-10, 17-22

10. Math is Elegant? “elegant” = simple, unusually effective, sublime, maybe even “beautiful!”

11. “Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.” Bertrand Russell, philosopher & mathematician

12. “Why are numbers beautiful? It's like asking why is Beethoven's Ninth Symphony beautiful. If you don't see why, someone can't tell you. I know numbers are beautiful. If they aren't beautiful, nothing is.” Paul Erdős, mathematician and eccentric

14. 1 st round = 512 games remaining 2 nd round = 256 3 rd round = 128 4 th round = 64 5 th round = 8 6 th round = 4 7 th round = 2 5122561286484 + 2 + 2-------

15. 1 st round = 512 games remaining 2 nd round = 256 3 rd round = 128 4 th round = 64 5 th round = 8 6 th round = 4 7 th round = 2 5122561286484 + 2 + 2-------1,023

16. OR: Consider: If every game results in one winner and one loser, there must be at least 1,023 losers at the end of the tournament. After all, there can only be 1 winner! Therefore, since there is a one-to- one correspondence between losers & games played, 1,023 games must be played.

18. 1 2 3 4 5 … 46 47 48 49 50 100 99 98 97 96 … 55 54 53 52 51 ----------------------------------------------------------- 101 101 101 101 … 101 101 101 101 UGH!! TOO MUCH ADDITION! OR: Consider: How many pairs of integers? 50! So the sum of the first 100 integers is 50 x 101 = 5,050 OR: Consider: How can we simplify/generalize? ½ n (n + 1)