2.2 Inductive and 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 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, -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. Do a couple of examples to find pattern and then write conjecture using phrase  = = 63  = = 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 = 84*6= 24 2*10 = 206*8 = 48  pattern: answers is positive Conjecture: The product of any two even integers is a positive answer. Test: 6*20 = 120

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

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, 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. 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 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