1. Homework Explanations
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?

2. 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.

3. More Unacceptable Answers
2x + 5y = 85; x + y = 20.
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.

4. So, what is a good answer?
I know there are 20 tickets. So I tried to find out numbers that add to 20 that also made 85 dollars.
if x = 10 and y = 10, then 2 • • 10 = 70
If x = 12 and y = 8, then 2 • • 8 = 64 this is too small. So, try y > 10.
If x = 6 and y = 14, then 2 • • 14 = 82. Still too small.
If x = 5 and y = 15, then 2 • • 15 = 85!
The school sold 5 tickets for $2 and 15 tickets for $5.

5. Another good answer
If we let x = number of $2 tickets, and if we let y = number of $5 tickets, then
x + y = 20 is the equation for the number of tickets and 2x + 5y = 85 is the equation for the amount of money.
We can rewrite x + y = 20 as x = 20 - y.
Substitute: 2(20 - y) + 5y = 85.
40 - 2y + 5y = 85
40 + 3y = 85.
Subtract 40 from both sides of the equation: 3y = 45 Divide both sides by 3
y = 15, Therefore, because y represents the number of $5, there were fifteen $5 tickets and then, because x + y = 20, there were 5 $2 tickets sold.

6. 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.

7. Exploration 1.1
Goal of the problem: with (24) students in class, including yourself, if each student shakes hands with every student, how many handshakes will there be?

8. 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.

9. Exploration 1.1
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?

10. 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.)

11. Exploration 1.1 3. Now solve the problem on your own.
4. With those at your table, compare your solution strategies. Which one(s) do you like best? Why?

12. Exploration 1.1
4. Continued. As a table, write an explanation that shows WHAT you did and WHY you did it. Be ready to read this.
5. Can you explain how to find the number of handshakes if there are “n” students in class today? Work on this with your table-mates.

13. Pigs and Chickens
A farmer’s daughter likes working math problems so he gives her this problem to work on:
We have pigs and chickens in our barnyard. I count 24 heads and 80 feet. How many pigs and how many chickens are out there?

14. Using random trial and error

15. Guess-check-revise
Organized trial and error.

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

17. Use algebra
There are 24 animals total: p + c = 24 p represents the number of pigs and c represents the number of chickens.
There are 48 feet. Each pig has 4 feet and each chicken has 2 feet, so 4p + 2c = 80

18. Four 4s
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.

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

20. Homework
Due Mon 8/27 Complete and write up Exploration 1.1: how many handshakes? pp : 3, 5,14, 21a, 22d,f, 25, 30c, 39.
Due Wed 8/29 Complete and write up Exploration 1.4: darts pp : 11, 12a, 3a
Extra practice: pp. 59: 2, 4, 5, 8, 9, 12