Exploration 1.1 With 24 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.

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?

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 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

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.

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?

Exploration 1.7 This problem explores both representations and connections. Read this together as a class. Model the first one. 10 7 13

Exploration 1.7 This problem explores both representations and connections. Read this together as a class. Model the first one. 10 5 8 2 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.

Homework Due Thursday 1/22 Complete and write up Exploration 1.4: Problem 1 Darts, parts 1-4 Read Textbook sections 1.2, 1.3, 1.4 Do Textbook problems pp. 26 - 29: 3, 5,14, 21a, 22d,f, 25, 30c, 39.

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.