1. Math 302A section 5 Spring, 2009 Instructor: Laurie Varecka

2. January 15, 2009 Class policies About the course Sec 1.1: NCTM Principles and Standards Sec 1.2: Problem solving Sec 1.3: Patterns (Exploration 1.1) Sec 1.4: Representations

3. NCTM Principles & Standards for School Mathematics Process standards: Problem Solving Reasoning & Proof Communication Connections Representation

4. Sec 1.2 – Problem Solving Polya’s Four Steps: 1.Understand the problem 2.Devise a plan 3.Monitor your plan 4.Look back at your work

5. 1.2 (cont’d) An example and discussion of homework expectations. Ex: A school play charges $2 for students and $5 for adults. For the three days of the play, 20 tickets were sold and $85 was raised. How many student tickets were sold?

6. 1.2 - Homework expectations The following solutions to the “school play” problem are unacceptable. Why? 15 tickets for $5, 5 tickets for $2 The school sold fifteen $5 tickets and five $2 tickets. I did the work in my head. I did the work on my calculator.

7. 1.2 - Homework expectations More unacceptable solutions: 2x + 5y = 85; x + y = 2 so x = 5 I tried different numbers until I got the answer. This problem is too basic to bother with an explanation.

8. 1.2 - Homework expectations A Good solution might look like: There are 20 tickets total, so I need to find two numbers that add to 20. If I have 10 students and 10 adults, that gives me 20 tickets, but $2 * 10 students and $5 * 10 adults gives me only $70. Since the adult tickets are more expensive, …

9. 1.2 - Homework expectations … I need more than 10 adults. In fact, each time I trade a student for an adult, the total ticket sales go up by $3. So, I’ll try adding 5 adults (and subtracting 5 students): 5 students + 15 adults = 20 tickets sold 5 students * $2 + 15 adults * $5 = $85 Therefore, 5 student tickets were sold.

10. 1.2 - Homework expectations Another Good solution: If we let S = the number of student tickets and A = the number of adult tickets, then S + A = 20 is the equation for the number of tickets and 2S + 5A = 85 is the equation for the amount of money raised.…

11. 1.2 - Homework expectations … I can rewrite S + A = 20 as A = 20 – S and then substitute into the other equation: 2S + 5(20 – S) = 85 2S + 100 – 5S = 85 100 – 3S = 85 15 = 3S Therefore, S = 5, and there were 5 student tickets sold.

12. 1.2 - Homework expectations In general, Explain what you did. Explain why you did it. Be sure to check that your answer really does answer the question. Be sure to check for arithmetic errors.

13. 1.2 (cont’d.) Ex: A farmer looks out his window and sees pigs and chickens. He says to his daughter, “I count 14 heads and 48 feet. How many pigs and how many chickens are out there?” (Introduce yourself to your table, then work on this together.)

14. What approach did you take?

15. Problem Solving Strategies Guess & check Solve simpler problem Solve similar problem Draw diagram Find pattern Find counter-example Induction Work backwards Make table Make graph Write equation(s) Estimate Act it out Organized list (try all possibilities)

16. 1.3 - Patterns Ex: Find the next term in the sequence. Describe the pattern you found (use a sentence). a) 1, 2, 4, 8, 16, … b) 25, 5, 1, 0.2, 0.04, … c) 1, 3, 6, 10, 15, … d)1, 0, -1, 0, 1, 0, -1, …

17. Homework Read sections 1.1 – 1.4 Write down answers to the questions in section 1.1 in your notebook (not to hand in).