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Polya

Polya. A teacher of mathematics has a great opportunity. If he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity. Polya.

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Polya

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  1. Polya A teacher of mathematics has a great opportunity. If he fills his allotted time with drilling his students in routine operations he • kills their interest, • hampers their intellectual development, and • misuses his opportunity.

  2. Polya But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a taste for, and some means of, independent thinking.

  3. Approach to constructing a proof • Work examples to understand problem • Write what is given in usable form • Write what must be shown in usable form • Think about approach to problem • Possibly repeat steps 2 and 3. • Construct informal proof • Construct formal proof

  4. Polya Incomplete understanding of the problem … is perhaps the most widespread deficiency in solving problems.... Some students rush into calculations and constructions without any plan or general idea; others wait clumsily for some idea to come and cannot do anything that would accelerate its coming.

  5. Prove that Let’s look at some examples!!! SHOCKING!!! It worked! i = 3 j = 5 k = 7 i2 + j2 + k2 = 8 x 10 + 3 = 83 i = 2 j = 4 k = 6 i2 + j2 + k2 = 8 x 7 = 56 i = 3 j = 4 k = 5 i2 + j2 + k2 = 8 x 6 + 2 = 50 i = 4 j = 5 k = 6 i2 + j2 + k2 = 8 x 9 + 5 = 77

  6. What we want to prove: Written in positive terms...

  7. Cases: i j k Can we avoid doing 8 cases? 1. Even Even Even 2. Even Odd Even 3. Even Even Odd 4. Even Odd Odd 5. Odd Even Even 6. Odd Odd Even 7. Odd Even Odd 8. Odd Odd Odd

  8. First case: i, j, k are even m=0, or 4

  9. Second case: i even, j odd, k even

  10. Third case: i even, j odd, k odd m=2 or 6

  11. Fourth case: i, j, k odd

  12. All four cases lead to m Therefore...

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