1. Week 2--Glad you came back! Please take out your homework. Staple it if necessary. Your homework and exploration should be turned in separately. Make sure your name is on it. Write who helped you on any problems.

2. Agenda for today! Exploration 1.4 (Darts) discussion. Collect homework More with problem solving and patterns Assign homework for week Number systems--Egyptian, Babylonian, Mayan, and Roman

3. Exploration 1.4 Darts Discuss strategies you would use to find a solution to the assigned darts problem.

4. Problem Solving Strategies

6. 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?

7. Patterns… the beginnings of reasoning 1 + 22 termssum = 3 1 + 2 + 33 termssum = 6 1 + 2 + 3 + 44 termssum = 10 1 + 2 + 3 + 4 + 55 termssum = 15 1 + 2 + 3 + … + nn termssum = ?

8. More Patterns Find the next term in the sequence. Describe the pattern in a sentence. J, F, M, A, M, J, … 1, 3, 6, 10, 15, … 25, 5, 1, 0.2, 0.04, … 1, 11, 21, 1211, 111221, 312211, … 1, 2, 4, 8, 16, … 1, 1, 2, 3, 5, 8, … 1, 0, -1, 0, 1, 0, -1, …

9. 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, …

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

11. Reasoning and Proof if x = 10 and y = 10, then 2 10 + 5 10 = 70 this is too small. If x = 12 and y = 8, then 2 12 + 5 8 = 64 making y smaller decreases the amount of money raised. Since my first guess was y = 10, I will try y > 10 for my next guess. If x = 6 and y = 14, then 2 6 + 5 14 = 82. Still too small. I will try y > 14 next. If x = 5 and y = 15, then 2 5 + 5 15 = 85! Five $2 tickets and fifteen $5 tickets = $85 and 5 + 15 = 20.

12. Homework For class Friday: in textbook, begin reading Section 2.3 (no farther than p. 111) Due Friday: in textbook, p. 54-57 # 11, 12, 30a, True or False and explain (1) sum of 3 consecutive numbers is divisible by 3 (2) sum of 4 consecutive numbers is divisible by 4