4.1 Patterns. Example 1 Find the patterns:

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1 4.1 Patterns One definition of mathematics is the study of patterns. In this section, you will practice recognizing mathematical patterns in various problems. For each example, you will work with a partner and try to find the pattern. Then you can confirm it by using a calculator if possible, or by checking with another group. Then we ll discuss it as a class. Example 1 Find the patterns: Example 2 Sixty-four teams are in the NCAA baskeball championship. It is a single elimination tournament, meaning if you lose, you are out. How many games are played in the tournament? Explain how you found an answer. Example 3 Draw the next shape in the pattern and count the line segments used to make it. Then predict how many line segments are in the next two shapes after without drawing them. Explain the pattern:

2 Example 4 Find the next three numbers in the following patterns: a) 3, 6, 8, 16, 18, 36, 38,,, Explain: b) 2, 4, 5, 10, 12, 24, 27,,, Explain: Example 5 Which of the number(s) in the last group are zuts? Explain how you know: Example 6 Complete the pattern: 1, 1, 2, 3, 5, 8, 13, 21,,,, Explain: Example 7 Can you find a way to add without adding them up one at a time?

3 4.2 Inductive Reasoning When we reach a conclusion by observing a pattern or trend and exploiting it, this is called inductive reasoning. Often, you can use inductive reasoning to build a general equation for a pattern. Example 1 Predict the n th term of the pattern 3, 6, 12, 24,... (This is another way of saying: Find a general equation for the pattern.) 1 st term 2 nd term 3 rd term 4 th term The general equation for the n th term is: Example 2 Predict the n th term of the pattern 2, 8, 14, 20,... 1 st term 2 nd term 3 rd term 4 th term Develop a general formula for the n th term: Example 3 Predict the n th term of the pattern 2, 6, 12, 20, 30, 42,... 1 st term 2 nd term 3 rd term 4 th term 5 th term 6 th term Develop a general formula for the n th term:

4 Example 4 A Lazy S has a vertical line drawn through it. How many pieces will form due to the number of cuts? Fill in the blanks by developing a general formula as an aid: What if the formula developed using inductive reasoning isn t true? To show a formula isn t true, only one counterexample must be provided. Example 5 Here are some Prime Numbers: 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47,... Can we make an assumption (inductively reason) about prime numbers? Example 6 Is the following assumption true, or can you provide a counterexample? If two double digit numbers have the same tens column digit, and one number has a 9 in the ones column, and the other has a 7, the number with the 9 in the ones column will always be larger.

5 4.3 Deductive Reasoning Inductive Reasoning is when you reach a conclusion based on an observed pattern. A counterexample can destroy conclusions reached by inductive reasoning. Deductive Reasoning is the process of reaching a conclusion based only on accepted facts. If the process is error-free, no counterexample can exist, and the conclusion can be taken as fact. We ve already used deductive reasoning in this course, when we were doing geometric proofs in Ch 2. Example: Example 1 Can you draw a conclusion using deductive reasoning? Premise: All the planets revolve around the sun in elliptical orbits. Mars is a planet. Example 2 Can you draw a conclusion using deductive reasoning? Premise: All English teachers like to read. Sam does not like to read. Example 3 Can you draw a conclusion using deductive reasoning? Premise: If n is a prime number greater than 3, then (n 1)(n + 1) is divisible by is a prime number greater than 3.

6 Example 4 Can you draw a conclusion using deductive reasoning? Premise: If a quadrilateral is a square, it is a regular polygon. A regular polygon has all sides and angles equal. Example 5 Can you draw a conclusion using deductive reasoning? Premise: Every number divisible by 6 is divisible by is divisible by 6. Example 6 What deduction, if possible, can be made about each person? Premise: A person must be at least 16 years old to have a driver s license. a) Tom has a driver s license b) Sue drives a car. c) Sally is 20 years old. d) Bill is 12 years old. e) Len does not drive a car. Example 7 Which of the following statement(s) are true? a) If two angles are right angles, then they are equal. b) If two angles are equal, then they are right angles. c) If two angles are not right angles, then they are not equal. d) If two angles are not equal, then they are not right angles.

7 4.4 Puzzles and Games of Strategy When solving mathematical puzzles, or employing mathematics in games of strategy, you may use both inductive and deductive reasoning. Some puzzles and games are very easy (or easy for some), while others can be rather difficult. If you are struggling to solve a puzzle or succeed at a game, try to approach the problem from a different standpoint. Collaborating with others can be helpful for this. Example 1 Put the numbers 1, 2, 3, 4, 5, 6, 7, 8, and 9 in the circles so that each edge adds up to the same number (there is more than one answer). Example 2 What do the numbers have in common? Example 3 Jack and Jill fetched two pails of water. One was a 7L pail and the other was 5L. How can Jack and Jill use the pails to get exactly 4L into one of the pails?

8 Example 4 Using a table to solve a Logic Problem: Example 5 Sudoku Puzzle Game

9 Example 6 Picross Puzzle Game Example 7 BATMAN!