1. Discuss your strategies to solve #3 with your neighbors. Use a different strategy to solve the following problem: Problem: 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? Welcome Back!

2. A New Way of Looking at Teaching and Learning Mathematics Teaching mathematics is NOT just about teaching students a specific set of mathematical skills.

3. A New Way of Looking at Teaching and Learning Mathematics The process of learning mathematics is a vehicle through which students learn to solve problems, period (mathematical or otherwise).

4. A New Way of Looking at Teaching and Learning Mathematics It is through mathematics that you can really focus on teaching how to organize your thoughts – to plan – to strategize how to approach a situation you have never faced before.

5. Unacceptable Answers  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.

6. More Unacceptable Answers  2x + 5y = 85; x + y = 20 so x = 5 and y = 15. I subtracted y from both sides in the second equation then substituted x = 20 – y into the second and solved for y. Then I solved for x.  I just tried a bunch of things. Aren’t I lucky--I got it on the first try.  This is a stupid problem and I am not going to waste your time explaining it to you.

7. So, what is a good answer? I let x represent the number of student tickets and y represent the number of adult tickets sold. Then I wrote down equations showing how they are related to one another. To solve these equations I used informed guessing to find out numbers that add to 20 that also made 2x + 5y = 85 dollars.

8. So, what is a good answer? Continued… 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.

10. Exploration 1.1 Handshakes  1.Work on this problem alone for a few minutes. Can you apply ideas discussed in the preface to find patterns in this problem? Can you use what you see to help you plan a solution?

11. Exploration 1.1 2.Discuss your ideas with everyone at your table. Describe new ideas that you like that arose from the discussion. (Note: if you have already solved the problem, do not just tell the answer. Tell IDEAS that will allow your peers to get the answer on their own.) Solve the problem together.

13. Homework For class Wednesday: in textbook, Read Section 1.7 Due Wednesday: in textbook, p. 26-29 # 5, 21a, 22df, 25, 30c, 39 in Explorations, 1.1 Handshakes #3 (you can use your solution from class), 5 Use the formula you obtain and show that it works for 22 students. Be sure to explain what each part of your formula means. Due Friday: 1.4 Darts #1, 4, 5 (We will discuss our answers in class on Wed., Jan. 21)