INDUCTIVE REASONING. b) Is this a good example of inductive reasoning? c) What counterexample would disprove her conjecture?

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1 INDUCTIVE REASONING Inductive Reasoning--- sometimes called inductive logic, is the process to make a generalization on a few recurring patterns or observations. 1. Brenda has just gotten a job as the plumber s assistant. Her first task is to open all the water valves to release the pressure on the lines. The first four valves she discovered opened when turning counterclockwise. a) What is her conjecture? b) Is this a good example of inductive reasoning? c) What counterexample would disprove her conjecture? 2. Wile E. Coyote is setting up a trap for the roadrunner. Every previous trap has destroyed some part of his body. However, after he blows him self up he seems to be healed by the next seen. a) What are two possible conjectures? b) Is this a good example of inductive reasoning? c) What counterexample would disprove your conjectures? 3. Ralphy noticed that of the last four days he ate bacon for breakfast Lisa ran from him. Ralphy concluded that if he had cereal for breakfast, then Lisa would not run from him. a) Is this a good example of inductive reasoning? Why? b) What would be a good conjecture?

2 2.1 Number patterns Use inductive reasoning to find the next two terms. 1) 2, 20, 200, 2000, 2) 2, 5, 10, 17, 26, 37, 3) a, 6, c, 12, e, 18, 4) 1, 4, 9, 16, 25, 5) 0, 10, 21, 33, 46, 6) 3, 5, 11, 29, 83, 245 7) Fill in the blank to complete the pattern. a) b) Picture patterns 1) Find the next picture in the pattern a) c) c)

3 2.2 Finding the nth term n n-3 n n-1 n [(n-1)+n] 1. Find the formula for the nth term, then find the 15th term. 2. Square in a cross pattern. 3.

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5 Figurate numbers-numbers that correspond to geometric figures. 1. Find the differences between the numbers in the series. 2. Sequences that have a constant difference of this type have there patterns in the factors of each term. See if you can find a pattern in the factors. First factor Second factor 3. This gives us two new sequences. The first factors The second factors 4. Put them together this is your nth term. Now find the 20th term

6 Doubling Since we doubled it, we know have to do the opposite and dive the nth term.

7 2.3 Mathematical Modeling Creating the sequences from events. 1. How many ways can you connect 30 dots on a piece of paper if no three are in a line? Don't try to draw it, try drawing smaller versions and find the pattern. n If there are 20 people sitting around a table, how many different pairs of people can have conversations during dinner? What geometric figures can you use to represent this? 3.If a polygon has 20 diagonals leaving a vertex, then how many sides does it have. Hint: a polygon has at least 3 sides. 4. Area of a square

8 2.4 Deductive Reasoning Deductive Reasoning-- is the process of showing that certain statements follow logically from agreed-upon assumptions and proven facts. Using true statements/properties to prove an argument. EX/ It is a fact that Jimbo the mechanic is always the cheapest mechanic in town. Using deductive reasoning: "If I go to see a mechanic other than Jimbo, then I will pay more than I would with Jimbo." EX/ It is agreed upon that every Monday it rains. Wrong use: "If it is raining, than it is Monday." Give a counter example. Using deductive reasoning you could say, "It is Monday, so it will rain." When you use reasoning, you are generalizing from careful observations/patterns that something is probably going to happen or is true. When you use reasoning, you are using established/accepted properties to show something is true.

9 2.5 Angle relationships Two angles are COMPLEMENTARY ANGLES if Draw an example: m m Two angles are SUPPLEMENTARY ANGLES if Draw an example: m m VERTICAL ANGLES are angles that are opposite each other when two lines cross. Draw an example: m m and m m Two angles form a LINEAR PAIR if Draw an example: m m The sum of a Linear pair of angles is

10 1. Find the missing angles

11 2.6 Properties of Parallel Lines Two or more lines are if and only if they are coplanar and do not intersect A line intersecting two or more coplanar lines is called a Using the picture below give one pair of: Corresponding angels Alternate interior angles Alternate exterior angles If two parallel lines are cut by a transversal, then corresponding angles are, alternate interior angles are, and alternate exterior angles are Complete each statement. 1. If two angles are vertical angles, then they are. 2. If two angles are a linear pair of angles, then they are. 3. If two angles are both equal in measure and then each angle measures Find the measure of each angle indicated in the figure below. a = b = c = d = e = f = g = h = j = d c j b a f g h e r s t w

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