2 2 Inductive and Deductive Reasoning What We

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2. 2 Inductive and Deductive Reasoning

2. 2 Inductive and Deductive Reasoning

What We Will Learn �Use inductive reasoning �Use deductive reasoning

What We Will Learn �Use inductive reasoning �Use deductive reasoning

Needed Vocab. �Conjecture: an unproven statement that is based on observations �Inductive reasoning: find

Needed Vocab. �Conjecture: an unproven statement that is based on observations �Inductive reasoning: find a pattern and write a conjecture �Counterexample: specific case for which the conjecture is false �Deductive reasoning: uses facts, definitions, accepted properties, and laws of logic to form a logical argument �Law of Detachment: if the hypothesis of a true conditional is true, then the conclusion is true �Law of syllogism: If hypothesis p, then conclusion q and if hypothesis q and conclusion r. Therefore; if hypothesis p, then conclusion r.

Ex. 1 Writing Conjectures �Write the conjecture and write next two terms. � 1,

Ex. 1 Writing Conjectures �Write the conjecture and write next two terms. � 1, -2, 3, -4, 5, … Alternating negative and going up by 1 -6, 7 �z, y, w, x, v, … Alphabet backwards u, t �o, t, t, f, f, s, s, … First letter of numbers e, n

Ex. 2 Making and Testing a Conjecture �the sum of any three consecutive integers.

Ex. 2 Making and Testing a Conjecture �the sum of any three consecutive integers. �Do a couple of examples to find pattern and then write conjecture using phrase 10+11+12 = 33 5+6+7 = 18 20+21+22 = 63 8+9+10 = 27 � pattern: answer is three times middle number Conjecture: The sum of any three consecutive integers is three times the middle number. Then test conjecture for accuracy • If wrong, rethink conjecture

Your Practice �The product of any two even integers. � 2*4 = 8 �

Your Practice �The product of any two even integers. � 2*4 = 8 � 2*10 = 20 4*6= 24 6*8 = 48 pattern: answers is even �Conjecture: The product of any two even integers is an even answer. �Test: 6*20 = 120

Ex. 3 Finding a Counterexample �The sum of two numbers is always more than

Ex. 3 Finding a Counterexample �The sum of two numbers is always more than the greater number. Find counterexample if one. Only need one � -2 + (-4) = -6 �The value of x 2 is always greater than the value of x. (0)2 = 0 �If two angles are supplements of each other, then one of the angles must be acute. Right angles

Ex. 4 Law of Detachment �

Ex. 4 Law of Detachment �

Your Practice �If a quadrilateral is a square, then it has four right angles.

Your Practice �If a quadrilateral is a square, then it has four right angles. Quadrilateral QRST has four right angles. �Hypothesis is not told true, so cannot make a conclusion

Ex. 5 Law of Syllogism �If hypothesis p, then conclusion q. If hypothesis q,

Ex. 5 Law of Syllogism �If hypothesis p, then conclusion q. If hypothesis q, then conclusion r. If conclusion of one is hypothesis of other, then use law of syllogism �Syllogism say: If hypothesis p, then conclusion r. �If a polygon is regular, then all angles in the interior of the polygon are congruent. If all the angles in the interior of a polygon are congruent, then the sides of the polygon are congruent. If a polygon is regular, then the sides of the polygon are congruent.

Your Practice �If a figure is a rhombus, then the figure is a parallelogram.

Your Practice �If a figure is a rhombus, then the figure is a parallelogram. If a figure is a parallelogram, then the figure has two pairs of opposite sides that are parallel. �If a figure is a rhombus, then the figure has two pairs of opposite sides that are parallel.

Ex. 7 Inductive or Deductive �Inductive based on patterns. �Deductive based on definitions and

Ex. 7 Inductive or Deductive �Inductive based on patterns. �Deductive based on definitions and properties. �Each time Monica kicks a ball into the air, it returns to the ground. Next time Monica kicks a ball up in the air, it will return to the ground. Which is it? � Inductive because observable pattern