Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Math 302A section 5 0011 0010 1010 1101 0001 0100 1011 Spring, 2009 Instructor: Laurie Varecka 1 2 4 January 15, 2009 0011 0010 1010 1101 0001 0100 1011 • • • • • • 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 1 2 4 NCTM Principles & Standards for School Mathematics 0011 0010 1010 1101 0001 0100 1011 Process standards: • Problem Solving • Reasoning & Proof • Communication • Connections • Representation 1 2 4 Sec 1.2 – Problem Solving 0011 0010 1010 1101 0001 0100 1011 Polya’s Four Steps: 1. Understand the problem 2. Devise a plan 3. Monitor your plan 4. Look back at your work 1 2 4 1.2 (cont’d) 0011 0010 1010 1101 0001 0100 1011 An example and discussion of homework expectations. 1 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? 2 4 1.2 - Homework expectations 0011 0010 1010 1101 0001 0100 1011 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. 1 2 4 1.2 - Homework expectations 0011 0010 1010 1101 0001 0100 1011 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. 1 2 4 1.2 - Homework expectations 0011 0010 1010 1101 0001 0100 1011 A Good solution might look like: 1 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, … 2 4 1.2 - Homework expectations 0011 0010 1010 1101 0001 0100 1011 … 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. 1 2 4 1.2 - Homework expectations 0011 0010 1010 1101 0001 0100 1011 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.… 1 2 4 1.2 - Homework expectations 0011 0010 1010 1101 0001 0100 1011 … 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. 1 2 4 1.2 - Homework expectations 0011 0010 1010 1101 0001 0100 1011 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. 1 2 4 1.2 (cont’d.) 0011 0010 1010 1101 0001 0100 1011 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?” 1 2 4 (Introduce yourself to your table, then work on this together.) What approach did you take? 0011 0010 1010 1101 0001 0100 1011 1 2 4 Problem Solving Strategies 0011 0010 1010 1101 0001 0100 1011 • • • • • • • 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) 1 2 4 1.3 - Patterns 0011 0010 1010 1101 0001 0100 1011 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, … 1 2 4 Homework 0011 0010 1010 1101 0001 0100 1011 • Read sections 1.1 – 1.4 • Write down answers to the questions in section 1.1 in your notebook (not to hand in). 1 2 4