Welcome to MATH 302A Please find the index card with your name on it and sit there. On the other side of the index card, write: Name as you wish to be called Where you are from Year at UA Teaching interests Other interests

Principles and Standards of School Mathematics (NCTM, 2000) Website: http://www.nctm.org/standards/ Five process standards: problem solving reasoning and proof communication connections representations

Exploration 1.1 With 25 people in class, including yourself, if each person shakes hands with every person, how many handshakes will there be?

General Solution How many handshakes will there be in a group of n people, if everyone shakes each other person’s hand once?

How to Solve a Problem Read--understand EVERY ASPECT of the problem. What is given, what is to be found, what can be assumed, what should not be assumed, all vocabulary, what the final answer should look like, etc. Plan--ways to get at the final answer Find the answer. Check and extend.

Solve this problem Pretend that you do not know algebra. 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?

Using random trial and error

Guess-check-revise Organized trial and error.

Make a diagram Use pictures to clarify and help solve the problem.

Organizing the information in a table

Use algebra There are 20 tickets total. Let a represent the number of adult tickets and s represent the number of student tickets. Then, a + s = 20 because the sum of the number of adult and student tickets is 20. Since each adult ticket costs $5, all of the adult tickets sold have a value of 5a. And, since each student ticket costs $2, all of the student tickets have a value of 2s. The total value is $85, so 5a + 2s = 85. The number of adult tickets is 20 – s. Substitute this value into the second equation to find the value of s. 5(20 – s) + 2s = 85. Show and explain the algebra steps. Answer the question in the problem. There were 15 adult tickets and 5 student tickets sold for the play.

Warm up Use four 4s and the arithmetic operations (+, -, x, ÷) plus grouping symbols to create each of the counting numbers from 0 to 10. Use exactly four 4s for each number.

Problem Solving Strategies Guess and check • Work backwards Solve a simpler problem • Draw a picture Solve a similar problem • Make a table Draw a diagram • Make a graph Find a pattern • Write an equation Find a counter-example • Estimate Solve by induction • Act it out Organized List (Proof by exhaustion) Other?

Things to Remember Explain what you did. Explain why you did it. Be sure you check to see that the answer does really answer the question asked. Check to make sure you have not made arithmetic errors.

Exploration 1.7 This problem explores both representations and connections. Put 10 7 13

More Patterns Try to figure out the next number in the sequence--explain how you got it in words that a 3rd grader would understand. 1, 3, 6, 10, … 1, 4, 9, 16, … 2, 4, 8, 16, … 1, 1, 2, 3, 5, 8, …

Exploration 1.4 Problem 1 Darts We will start this in class, and you will finish it for homework. The purpose of this exploration is to focus on having a solution strategy, instead of trying random things. Instead, we will try to think of strategies that work for solving different problems.