1. Teach Me to Think: Developing Thinking Skills, It’s what sets us apart.

2. What is Critical Thinking?
Focused thinking
Thinking with a definite purpose (goal)
Can be a complex & involved process
An active process that involves constant questioning

3. Why we need Critical Thinking students
“The significant problems we face cannot be solved at the same level of thinking we were at when we created them.”
An Albert Einstein Quote on Creativity

4. What are the Goals of Critical Thinking?
Finding Meaning
Seeking Logic
Searching for reason
Looking answers
Developing facts and opinions
Appreciating different points of view

5. How Can I develop my Critical Thinking Skills?
Think about your thinking
Think about why you make your choices and decisions
Think about why the world is the way it is
Practice every day!
Word problems
Math problems
Puzzles
Games of strategy

6. Critical Thinking and Reasoning
Chapter 2 introduces INDUCTIVE and DEDUCTIVE Reasoning These types of Reasoning are essential to CRITICAL THINKING

7. Lesson 2.1 Inductive Reasoning in Geometry
HOMEWORK: 2.1/1-15 odds
Objectives:
Use inductive reasoning to find the next term in a number or picture pattern
To use inductive reasoning to make conjectures.

8. Vocabulary Inductive reasoning:
make conclusions based on patterns you observe
Conjecture:
conclusion reached by inductive reasoning based on evidence
Geometric Pattern:
arrangement of geometric figures that repeat

9. Ex. 1: Find the next term in the sequence:
Inductive Reasoning – Is reasoning that is based on patterns you observe.
If you observe data, then recognize a pattern (the rule) in a sequence you can use inductive reasoning to find the next term.
Ex. 1: Find the next term in the sequence:
A) 3, 6, 12, 24, ___, ___
B) 1, 2, 4, 7, 11, 16, 22, ___, ___ C) 48 96
Rule: x2 29 37
Rule: +1, +2, +3, +4, …
Rule: divide each section by half

10. Inductive Reasoning
1. Process of observing data
2. Recognizing patterns
3. Making generalizations based on those patterns

11. An example of inductive reasoning
Suppose your history teacher likes to give “surprise” quizzes. You notice that, for the first four chapters of the book, she gave a quiz the day after she covered the third lesson. Based on the pattern in your observations, you might generalize that you will have a quiz after the third lesson of every chapter.

12. Identifying a Pattern Find the next item in the pattern.
January, March, May, ...
Observe the data..
identify the pattern..
state the pattern.
Alternating months of the year make up the pattern.
(skip every other month)
The next month is July.

13. Identifying a Pattern Find the next item in the pattern.
7, 14, 21, 28, …
Observe the data..
identify the pattern..
state the pattern.
Multiples of 7 make up the pattern.
(add 7 to each term to get the next)
The next multiple is 35.

14. Identifying a Pattern Find the next item in the pattern.
In this pattern, the figure rotates 90°
counter-clockwise each time.
The next figure is

15. Inductive reasoning can be used to make a conjecture about a number sequence
Consider the sequence 10, 7, 9, 6, 8, 5, 7, Make a conjecture about the rule for generating the sequence. Then find the next three terms.

16. Solution
10, 7, 9, 6, 8, 5, 7, . . Look at how the numbers change from term to term The 1st term in the sequence is 10. You subtract 3 to get the 2nd term. Then you add 2 to get the 3rd term. You continue alternating between subtracting 3 and adding 2 to generate the remaining terms. The next three terms are 4, 6, and 3.

17. Identifying a Pattern Find the next item in the pattern
0.4, 0.04, 0.004, …
Rules & descriptions can be stated in many different ways:
Multiply each term by 0.1 to get the next.
Divide each term by 10 to get the next.
When reading the pattern from left to right, the next item in the pattern has one more zero after the decimal point.
The next item would have 3 zeros after the decimal point, or

18. Writing Conjectures Complete the conjecture.
The sum of two positive numbers is ? .
List some examples and look for a pattern.
1 + 1 = = 3.15
3, ,000,017 = 1,003,917
The sum of two positive numbers is positive.

19. Writing Conjectures Complete the conjecture.
The product of two odd numbers is ? .
(what kind of numbers are ‘odd numbers’?)
List some examples and look for a pattern.
1  1 =  3 =  7 = -35
The product of two odd numbers is odd.

20. Use inductive reasoning to find the next two
Geometric Patterns
Arrangement of geometric figures that repeat
Use inductive reasoning and make conjecture as to the next figure in a pattern
Use inductive reasoning to find the next two
figures in the pattern.

21. Geometric Patterns
Use inductive reasoning to find the next two figures in the pattern.

22. Geometric Patterns Describe the figure that goes in the missing boxes.
Describe the next three figures in the pattern below.

23. Lesson 2.1 – Inductive Reasoning
Objectives:
Use inductive reasoning to find the next term in a number or picture pattern
To use inductive reasoning to make conjectures.
Homework: WS 2.1

24. We will be doing this ALOT this year!!
Mathematicians use Inductive Reasoning to find patterns which will then allow them to conjecture.
We will be doing this ALOT this year!!

25. A generalization made with inductive reasoning (Drawing conclusions)
Conjectures
A generalization made with inductive reasoning (Drawing conclusions)
EXAMPLES:
Bell rings M, T, W, TH at 7:40 am
Conjecture about Friday?
The bell will ring at 7:40 am on Friday
Chemist puts NaCl on flame stick and puts into flame and sees an orange-yellow flame. Repeats for 5 other substances that also contain NaCl also producing the same color flame.
Conjecture?
All substances containing NaCl will produce an orange-yellow flame

26. Finding Patterns
2, 4, 7, 11, ...
Rule? Add the next consecutive integer
Next 3 terms?
16, 22, 29
1, 1, 2, 3, 5, 8, 13, ...
Rule? Add previous two terms
(Fibonacci Sequence)
21, 34, 55
1, 4, 9, 16, 25, 36, ...
Rule? Add consecutive odd numbers OR
the perfect squares
49, 64, 81

27. Make a conjecture about the sum of the
first 30 odd numbers.
= 12
= 22
= 32
= 42
= 52 ..
= 302
1 = 1
= 4
= 9
= 16
= 25 .. =
900

28. cont.: Make a conjecture about the sum of the first 30 odd numbers.
𝒏 = the amount of numbers added
Sum of the first 𝒏 odd numbers = 𝒏 𝟐

30. Truth in Conjectures
To show that a conjecture is always true, you must prove it.
To show that a conjecture is false, you have to find only one example in which the conjecture is not true.
This case is called a counterexample.
A counterexample can be a drawing, a statement, or a number.

31. Inductive Reasoning assumes that an observed pattern will continue.
This may or may not be true.
Ex: x = x • x
This is true only for x = 0 and x = 1
Conjecture – A conclusion you reach using inductive reasoning.

32. The first 3 odd prime numbers are
Counter Example – To a conjecture is an example for which the conjecture is incorrect.
The first 3 odd prime numbers are
3, 5, 7. Make a conjecture about the 4th.
3, 5, 7, ___
One would think that the rule is add 2, but that gives us 9 for the fourth prime number.
Is that true?
What is the next odd prime number? 11 No

33. Finding a Counterexample
Show that the conjecture is false by finding a counterexample.
For every integer n, n3 is positive.
Pick integers and substitute them into the expression to see if the conjecture holds.
Let n = 1. Since n3 = 1 and 1 > 0, the conjecture holds.
Let n = –3. Since n3 = –27 and –27  0, the conjecture is false.
n = –3 is a counterexample.

34. Finding a Counterexample
Show that the conjecture is false by finding a counterexample.
Two complementary angles are not congruent.
45° + 45° = 90°
If the two congruent angles both measure 45°,
the conjecture is false.

35. Monthly High Temperatures (ºF) in Abilene, Texas
Finding a Counterexample
Show that the conjecture is false by finding a counterexample.
The monthly high temperature in Abilene is never below 90°F for two months in a row.
Monthly High Temperatures (ºF) in Abilene, Texas
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec 88 89 97 99
107
109
110
106
103 92
The monthly high temperatures in January and February were 88°F and 89°F, so the conjecture is false.

36. For any real number x, x2 ≥ x.
Finding a Counterexample
Show that the conjecture is false by finding a counterexample.
For any real number x, x2 ≥ x.
Let x = .
1 2
Since = , ≥
1 2 2
1 4
The conjecture is false.

37. Supplementary angles are adjacent.
Finding a Counterexample
Show that the conjecture is false by finding a counterexample.
Supplementary angles are adjacent.
23°
157°
The supplementary angles are not adjacent, so the conjecture is false.

38. Planets’ Diameters (km)
Finding a Counterexample
Show that the conjecture is false by finding a counterexample.
The radius of every planet in the solar system is less than 50,000 km.
Planets’ Diameters (km)
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
4880
12,100
12,800
6790
143,000
121,000
51,100
49,500
Since the radius is half the diameter, the radius of Jupiter is 71,500 km and the radius of Saturn is 60,500 km. The conjecture is false.