1. 2.2 Inductive and Deductive Reasoning

2. What We Will Learn
Use inductive reasoning
Use deductive reasoning

3. 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.

4. 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

5. 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
5+6+7 = = 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

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

7. 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 x2 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

8. Ex. 4 Law of Detachment Must be told hypothesis is true.
If two segments have the same length, then they are congruent. You know that BC = XY.
What can you conclude?
Are you told hypothesis is true?
BC ≅ XY
If you pass the final exam, then you pass the class. You pass the final exam.
You pass the class

9. 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

10. 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.

11. 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.

12. 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