Section 1.1: Making Conjectures: Inductive Reasoningmrmcdonaldshomepage.weebly.com/uploads/1/3/9/2/13921997/...Chapter 1: Inductive and Deductive Reasoning Section 1.1 4. Make a conjecture

Chapter 1: Inductive and Deductive Reasoning Section 1.1

Chapter 1: Inductive and Deductive Reasoning

Section 1.1: Making Conjectures: Inductive Reasoning

Terminology:

Conjecture:

A testable expression that is based on available evidence but is not yet proven. A

conjecture can be a true conclusion or it can be false!

Inductive Reasoning:

A form of reasoning in which a conclusion is reached based on a pattern present

in numerous observations. The premises makes the conclusion likely, but does

not guarantee it to be true.

𝑻𝑹𝑼𝑬 𝑷𝑹𝑬𝑴𝑰𝑺𝑬 + 𝑻𝑹𝑼𝑬 𝑷𝑹𝑬𝑴𝑰𝑺 = 𝑷𝑹𝑶𝑩𝑨𝑩𝑳𝒀 𝑻𝑹𝑼𝑬 𝑪𝑶𝑵𝑪𝑳𝑼𝑺𝑰𝑶𝑵

Deductive Reasoning:

The process of coming up with a conclusion based on facts that have already been

shown to be true. The facts that can be used to prove your conclusion deductively

may come from accepted definitions, properties, laws or rules. The truth of the

premises guarantees the truth of the conclusion.

𝑻𝑹𝑼𝑬 𝑷𝑹𝑬𝑴𝑰𝑺𝑬 + 𝑻𝑹𝑼𝑬 𝑷𝑹𝑬𝑴𝑰𝑺𝑬 = 𝑻𝑹𝑼𝑬 𝑪𝑶𝑵𝑪𝑳𝑼𝑺𝑰𝑶𝑵

MAKING CONJECTURES

For each situation below, make a conjecture based on your observations

1. Determine the pattern and make a conjecture about which shape comes next.

CONJECTURE: The next shape in the pattern is:


2. Complete the conjecture below that is true for all the equations:

3 + 7 = 10 11 + 5 = 16 9 + 13 = 22 7 + 11 = 18

CONJECTURE: The sum of two odd natural numbers is always __________.

3. A pattern is created between the figure number, n, and the number of triangles, t,

required to make it. Make a conjecture between the figure number and the

number of triangles creating it.

Figure number (n) 1 2 3

Numbers of Triangles Making up the Figure (t)

1 4 9

CONJECTURE:

_______________________________________________________

_______________________________________________________

_______________________________________________________


4. Make a conjecture about the number of sides in the polygon and the number of

interior regions formed by connecting opposite vertices to one point.

Number of Sides (n)

3 4 5 6

Number of Regions

Formed (r) 1 2 3 4

CONJECTURE:

___________________________________________________

___________________________________________________

___________________________________________________


INDUCTIVE REASONING

When we make a conjecture, we often use inductive reasoning. In these scenarios, we

use a series of true statements and make a conjecture based on these statements. The

conjecture that we make is based solely on the statements and is not necessarily true.

Ex. Make a conjecture based on inductive reasoning:

1. Emily is between 15 and 17 years old

Makenzie is between 15 and 17 years old

Isaiah is between 15 and 17 years old

CONJECTURE:

___________________________________________________

___________________________________________________

2. A dog is a mammal with 4 legs

A cat is a mammal with 4 legs

A horse is a mammal with 4 legs

A mouse is a mammal with 4 legs

CONJECTURE:

___________________________________________________

___________________________________________________

3. The sum of the interior angles of an equilateral triangle is 180°

The sum of the interior angles of an isosceles triangle is 180°

The sum of the interior angles of a scalene triangle is 180°

CONJECTURE:

___________________________________________________

___________________________________________________


4. Use inductive reasoning to make a conjecture about the sum of any two

consecutive numbers.


Section 1.2: Validity of Conjectures and Counterexamples

Terminology:

Counter Example:

An example that shows that a conjecture is not always true, thus proving a

conjecture to be invalid.

Ex. Conjecture: All prime numbers are odd.

Counter Example: 2 is a prime number, 2 is an even number, therefore

not all prime numbers are odd.

Using Counter Examples to Invalidate Conjectures

Ex. In each situation, use a counter example to prove the conjecture to be invalid.

Conjecture 1: The product of two integers is a positive number.

Conjecture 2: Every student at Dorset Collegiate takes the bus to school.

Conjecture 3: Every quadrilateral with four right angles is a square.

Conjecture 4: Adam scored twice as well as John.


Section 1.3: Proving Conjectures using Deductive Reasoning

Terminology:

Deductive Reasoning:

The process of coming up with a conclusion based on facts that have already been

shown to be true. The facts that can be used to prove your conclusion deductively

may come from accepted definitions, properties, laws or rules. The truth of the

premises guarantees the truth of the conclusion.

𝑻𝑹𝑼𝑬 𝑷𝑹𝑬𝑴𝑰𝑺𝑬 + 𝑻𝑹𝑼𝑬 𝑷𝑹𝑬𝑴𝑰𝑺𝑬 = 𝑻𝑹𝑼𝑬 𝑪𝑶𝑵𝑪𝑳𝑼𝑺𝑰𝑶𝑵

Proof:

A mathematical proof is an argument that shows that a statement is always valid

and that no counterexample exists.

Transitive Property:

If two quantities are equal to the same quantity, then they are equal to each other.

In other words, if A implies B and B implies C, than A also implies C.

𝑰𝒇 𝑨 → 𝑩 𝒂𝒏𝒅 𝑩 → 𝑪, 𝒕𝒉𝒆𝒏 𝑨 → 𝑪

𝑶𝑹

𝑰𝒇 𝑨 = 𝑩 𝒂𝒏𝒅 𝑩 = 𝑪, 𝒕𝒉𝒆𝒏 𝑨 = 𝑪

Properties of Equality:

If two quantities are equal, then whenever you perform any operation to one of

the quantities, you must also perform the same operation to the other quantity.

Ex. If 𝐴 = 𝐵, 𝑡ℎ𝑒𝑛:

𝑨 + 𝟐 = 𝑩 + 𝟐

𝑨 − 𝟓 = 𝑩 − 𝟓

𝑨 × 𝟔 = 𝑩 × 𝟔 𝑨

𝟕=

𝑩

𝟕


Using Deductive Reasoning to Prove Conjectures

When using deductive reasoning we can employ a variety of strategies to prove a

conjecture. Some that we will use in this chapter include:

1. Visual Representations

2. Algebraic Expressions and Equations

3. Two-Column Proof

Visual Representations

Visual Representations involve using Venn Diagrams and/or Tables to help support

your conjecture.

Ex. All dogs are mammals. All mammals are vertebrates. Shaggy is a dog. What can be

deduced about Saggy?

Ex. Science is considered a Core subject. Chemistry is a science course. All Core courses

in grade 12 have public exams. What can be deduced about the grade 12 chemistry class?


Algebraic Expressions and Equations

Algebraic expressions and equations are often used to prove questions involving

patterns and numbers in which we use variables in the place of numbers, therefore

generalizing the context of a conjecture and proving it true for all settings.

Some useful information for using Algebra and Deductive Reasoning include:

o If 𝒏 is any integer, than 𝟐𝒏 must be an even integer

o If 𝒏 is any integer, than 𝟐𝒏 + 𝟏 must be an odd integer

o If 𝒏 is any integer, than 𝒏, 𝒏 + 𝟏, 𝒂𝒏𝒅 𝒏 + 𝟐 are three consecutive

integers.

o A number with digits 𝒂𝒃𝒄 can be written using place value such

that 𝒂𝒃𝒄 = 𝟏𝟎𝟎𝒂 + 𝟏𝟎𝒃 + 𝒄

Ex. In each situation prove the conjecture is true using only deductive reasoning:

Conjecture 1: The sum of two consecutive numbers is an odd number

Conjecture 2: The product of any two perfect squares is another perfect square


Conjecture 3: The sum of any two digit number with another two digit number with

its digits being the reverse of the first, will be divisible by 11.

Conjecture 4: The product of any fraction and its reciprocal will equal 1.


Two Column Proof

A two column proof is a form of deductive reasoning in which the statements of the

argument are written in one column and the justifications for the statements are written

in the second column.

Ex. Prove that the measure of the exterior angle of a triangle is equal to the sum of the

two non-adjacent interior angles.

STATEMENTS JUSTIFICATIONS


Proof by Inductive and Deductive Reasoning

Sometimes a conjecture will need to be proven by both inductive and deductive

reasoning. In such situations we use inductive reasoning to provide several examples

where the conjecture is true, then we use deductive reasoning to prove the conjecture

true in all settings.

Ex. Chuck made the conjecture that the sim of any five consecutive integers is equal to 5

times the median. Use inductive and deductive reasoning to prove the conjecture.

Ex. Minnie made a conjecture that the sum of any three consecutive integers is divisible

by three. Use inductive and deductive reasoning to prove the conjecture.

Ex. Kenny made a conjecture that the difference between the square of any two

consecutive numbers is equal to an odd number. Use inductive and deductive reasoning

to prove the conjecture.


Proving Mathematical Tricks

Ex. Isaac created a step-by-step number trick:

Choose any number

Multiply it by 4

Add 10

Divide by 2

Subtract 11

Divide by 2

Add 3

The answer is the number you started with.

Use inductive and deductive reasoning to prove that the result will always be the

number you started with


Ex. Jamie created the following step-by-step math trick:

Choose any positive number

Double it

Subtract 2

Double it again

Add 8

Divide by 4

Subtract the number you started with

Your answer is 1

Use inductive and deductive reasoning to prove this trick is true for any number


Section 1.4: Proofs that are Not Valid

Terminology:

Invalid Proof:

A proof that contains an error in reasoning or contains an invalid assumption.

Premise:

A statement that is assumed to be true.

Circular Reasoning:

An argument that is incorrect because it makes use of the conclusion that is

intended to be proved.

Invalid Proofs

A single error in reasoning will break down the logical argument of a deductive proof.

This will result in an invalid conclusion or a conclusion that is not supported by the

proof.

Division by zero always creates an error in a proof, leading to an invalid conclusion.

Circular reasoning must be avoided. Be careful not to assume a result that follows

from what you are trying to prove.

The reason you are writing a proof is so that others can read and understand it. After

you write a proof, have someone else who has not seen your proof read it. If this

person gets confused, your proof may need to be clarified.


Ex. Brie says that she can prove that 2 = 0. Here is her proof.

Let 𝑎 and 𝑏 both be equal to 1.

𝑎 = 𝑏

𝑎2 = 𝑏2

𝑎2 − 𝑏2 = 0

(𝑎 + 𝑏)(𝑎 − 𝑏) = 0 (𝑎 + 𝑏)(𝑎 − 𝑏)

(𝑎 − 𝑏)=

0

(𝑎 − 𝑏)

𝑎 + 𝑏 = 0

1 + 1 = 0

2 = 0

Transitive Property

Square Both Sides

Subtract 𝑏2 from both sides

Factor using difference of squares

Divide both sides by 𝑎 − 𝑏

Simplification

Substitution (since 𝑎 = 𝑏 = 1)

Explain whether each statement in Brie’s proof is valid.


Ex. Lee claims to have a proof that shows −5 = 5.

Lee’s Proof

I assumed that −5 = 5

Then I squared both sides: (−5)2 = 52

I got a true statement:25 = 25

This means that my assumption that −5 = 5 must be correct

Where is the error in Lee’s Proof?

Ex. Brad says he can prove that $1 = 1¢

Brad’s Proof

$1 𝑐𝑎𝑛 𝑏𝑒 𝑐𝑜𝑛𝑣𝑒𝑟𝑡𝑒𝑑 𝑡𝑜 100¢

100 𝑐𝑎𝑛 𝑏𝑒 𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑒𝑑 𝑎𝑠 (10)2

10¢ 𝑖𝑠 𝑜𝑛𝑒 − 𝑡𝑒𝑛𝑡ℎ 𝑜𝑓 𝑎 𝑑𝑜𝑙𝑙𝑎𝑟 𝑠𝑜 𝑡ℎ𝑖𝑠 𝑏𝑒𝑐𝑜𝑚𝑒𝑠 (0.1)2

(0.1)2 = 0.01

𝑂𝑛𝑒 𝑜𝑛𝑒 − ℎ𝑢𝑛𝑑𝑟𝑒𝑡ℎ 𝑜𝑓 𝑎 𝑑𝑜𝑙𝑙𝑎𝑟 𝑖𝑠 𝑜𝑛𝑒 𝑐𝑒𝑛𝑡, 𝑠𝑜 $1 = 1¢

Where is the problem is Brad’s reasoning?

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Section 1.1: Making Conjectures: Inductive Reasoningmrmcdonaldshomepage.weebly.com/uploads/1/3/9/2/13921997/...Chapter 1: Inductive and Deductive Reasoning Section 1.1 4. Make a conjecture