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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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Presentation Transcript
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.
polya1
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.

approach to constructing a proof
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
polya2
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.

prove that
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

what we want to prove
What we want to prove:

Written in positive terms...

slide7

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