Transcript 2-1 Inductive Reasoning and Conjecture

2-1 Inductive Reasoning and
Conjecture
Ms. Andrejko
Real - Life
Vocabulary
 Inductive Reasoning- reasoning that uses a
number of specific examples to arrive at your
conclusion
 Conjecture- a concluding statement reached using
inductive reasoning
 Counterexample- a false example that can be a
number, drawing, or a statement.
Steps to making a conjecture
 1. Find a pattern in the sequence that you are given, or write
out examples to find a pattern based on what you are given.
 2. Write the pattern that you have found in the form of a
conjecture.
 IF GIVEN A CONJECTURE: Find a counter example, to
prove that it is false
Examples
1.
Conjecture: Each figure grows by increasing by 2 shaded diamonds
and 1 shaded diamond. The next figure will have a total of 8 shaded
diamonds and 5 white diamonds.
2.
-4, -1, 2, 5, 8
Conjecture: Each number is increasing by 3 every time. The next
number will be 11.
3.
-2, -4, -8, -16, -32
Conjecture: Each number is two times as many as the previous
number. The next number will be -64.
PRACTICE
1.
Conjecture: Each figure grows by 2 dots each time. The next figure
will contain 12 dots (5 on each side, and one on the top and bottom)
2.
-5, -10, -15, -20
Conjecture: Each number is decreasing by 5 more each time. The
next number in the series will be -25.
3.
Conjecture: Each number is being multiplied by - ½ . The next
number will be 1/16.
Examples – Make a conjecture about
the geometric relationships
4. N is the midpoint of QP
Conjecture: QN ≅ NP
5.
<3 ≅ <4
Conjecture: < 3 and < 4 are vertical angles
6.
<1, <2, <3, <4 form 4 linear pairs
Conjecture: <1 and < 3 are vertical angles, <2
and angle 4 are vertical angles
4
1
3
2
PRACTICE– Make a conjecture about
the geometric relationships
4. <ABC is a right angle.
Conjecture: < ABC = 90°
5.
ABCD is a parallelogram
Conjecture: ABCD has 4 sides
6.
Conjecture: PQRS is a square
Examples – T or F - Counterexamples
7. If <ABC and <CBD form a linear pair, then <ABC ≅ <CBD
Counterexample:
160
20
8. If AB, BC, and AC are congruent, then A, B, and C are
collinear
A
Counterexample:
B
C
9. If AB + BC = AC, then AB = BC
---- 10 ---- -------- 20 -------B
A
C
Counterexample:
---------------- 30 --------------10 + 20 = 30 , but 10 ≠ 20.
PRACTICE– T or F - Counterexamples
7. If <1 and <2 are adjacent angles, then <1 and <2 form a
linear pair
Counterexample:
2 1
8. If S,T, and U are collinear, and ST = TU, then T is the
midpoint of SU
TRUE
9. If n is a real number, then n2 > n
Counterexample:
02= 0 ; 12 = 1 ; ( ½)2 = (¼)