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Problem Solving

Problem Solving. Ch 1. “The Handshake Problem”. Shake Hands with Everyone. Some things to think about: How many handshakes occurred? How did you keep track that you didn’t shake someone’s hand twice? How do you know you shook everyone’s hand?. Is Your Friend a Cheater?.

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Problem Solving

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  1. Problem Solving • Ch 1 “The Handshake Problem”

  2. Shake Hands with Everyone • Some things to think about: • How many handshakes occurred? • How did you keep track that you didn’t shake someone’s hand twice? • How do you know you shook everyone’s hand?

  3. Is Your Friend a Cheater? • You’re friend invited you to play a game • You get 2 darts where there are six sections scored 1, 2, 3, 4, 5, and 6. • The rule is you throw both darts. If the second score is greater than the first, you win. Otherwise you loose. • Is the game fair?

  4. Review • There are many different approaches to a problem. And there is not one that is better than another. This chapter will focus on trying different methods. • The first part of your homework will focus on solving problems similar to the handshake problem. • The second part of you homework will focus on solving problems with a graphical model.

  5. Day 2 - Geometric Modeling • Let’s look at the handshake problem again. • Can anyone think of how to draw a geometric figure to represent our class size and the number of hand shakes? draw a shape with the # of sides = to the # of students. vertices = ppl total # of diag + the # of sides = # of handshakes

  6. Practice - on your own • How many diagonals are there in an octogon? • An icosagon (20 sides) has how many diagonals? • If there were 35 people in this class, how many handshakes would be required if everyone were to shake hands?

  7. One of the most important mathematical tools: Always draw a picture!

  8. Finding a Pattern • We’re in Algebra now right?! • So let’s look at the handshake problem algebraically... Who knows what sigma notation is?

  9. Sigma Notation

  10. Let’s find a pattern for the total number of handshakes for different class sizes. • 2 people = 1 handshake • 3 people = 3 handshakes • 4 people = 6 handshakes • 5 people = 10 handshakes ... • What is the pattern in the total number of handshakes?

  11. Pattern: • Below is the sequence representing the total number of handshakes: • 1, 3, 6, 10, ... Now here is the series of how the handshakes are increasing when another person is added to the class... 1 + 2 + 3 + 4 + ...

  12. Sigma Notation • This needs to represent a sum = the total number of handshakes. • Let n = the number of people in the class. • i.e. when n = 2, there was 1 handshake. when n = 3, there was 2 handshakes added to the previous amount. • Is there an expression representing the number of people and the number of handshakes added to the total? n - 1

  13. What would the sigma notation look like for 8 people in the class? Then evaluate it. • Remember: (2-1) + (3-1) + (4-1) + (5-1) + (6-1) + (7-1) + (8-1) = 28

  14. Day 3 - A Recursive Solution • Recursive - a rule in which determining a certain term in a sequence relies on the previous term(s).

  15. Handshake Problem (again) • Yesterday, we looked at how the total number of handshakes were increasing with each person added to the room: • 1, 2, 3, 4, ... • Let’s think of it a different way:

  16. Handshake Problem (again) • With 2 people, there is 1 handshake • With 3 people, 2 more handshakes were added • With 4 people, 3 more handshakes were added • Therefore:

  17. Handshake Problem (again) • This is a recursive sequence. • Can you think of a way to write it algebraically? Something important is missing!

  18. Checkpoint • Generate the first five terms of the sequence that has the recursive definition • Find a recursive formula that will generate the sequence 1, 2, 6, 24, 120, 720

  19. Demonstrations • 1st - Think of an expression that represents an even number • 2nd - Think of an expression that represents an odd number

  20. Even and Odd Integers • The even and odd integers can be divided into two sets with no intersection, called disjoint sets. • a) Draw a picture to represent this situation • Keep your answer on a whiteboard and wait to share with other groups.

  21. Even and Odd Integers • Prove it! • I will give each group one of the problems (b - g) to prove algebraically. • Hint: even integers are represented as 2n and odd integers are represented as 2n - 1

  22. Even and Odd Integers • Which set of integers contain the answer to each of the following operations: • b) Add two even integers • c) Add two odd integers • d) Add an even and an odd integer • e) Multiply two even integers • f) Multiply two odd integers • g) Multiply an odd and an even integer • Keep your answer on one of your whiteboards

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