Geometry: Lesson 1.3 â Conditional Statements
Geometry: Lesson 1.3 – Conditional Statements Geometry Oklahoma Academic Standards:
G.RL.1.2 Analyze and draw conclusions based on a set of conditions using inductive and deductive reasoning. Recognize the logical relationships between a conditional statement and its inverse, converse, and contrapositive. G.RL.1.3 Assess the validity of a logical argument and give counterexamples to disprove a statement.
Lesson Objectives:
1. To understand/develop the various conditional statements. 2. To determine validity of conditional statements. 3. To use conditional statements in deductive reasoning.
Introduction: So far, we’ve really only focused on inductive reasoning. But since we are already masters at inductive reasoning, let’s shift our focus on the harder to master deductive
reasoning. Because we are still not at the age of proper, complete development of deductive
reasoning, we will need some tools to help us process our thinking. These tools we will develop rely heavily on our knowledge of conditional statements.
Vocabulary: Definition – Conditional Statement A Conditional Statement is a statement written in “if-then” form.
The phrase after the word “if” is called the hypothesis (condition). The phrase after the word “then” is called the conclusion (result).
Example 1: If Samantha works at Wal-Mart, then she will get a paycheck.
In this example, the hypothesis (or condition to be met) is “Samantha works at Wal-Mart”. The conclusion (or result of the condition being met) is “she will get a paycheck”.
You have probably heard the terminology of hypothesis and conclusion in other classes (most likely a science class). Remember inductive and deductive reasoning is also a STEM process. Geometry: Lesson 1.3
1
Example 2: Circle the hypothesis and underline the conclusion in each of the following conditional statements.
a.) If I give you a dollar, then you give me a hamburger. b.) If I got my phone back, then I did the dishes.
c.) If there is a choice, you must decided between them. d.) We will go to Disney Land if you get straight A’s.
It turns out every proper sentence can be turned into a conditional statement.
For example, the sentence “ A bumble bee flies into a wall.” can be turned into the conditional statement: _______________________________________________________________.
Here’s another one: “Superman wears Batman pajamas.” can be turned into the conditional statement: ________________________________________________________________.
Remember, the goal of inductive and deductive reasoning is to create and test conjectures.
Since conditional statements are easy to check to see if the condition has been met, we can
easily verify conjectures are true or show (using a counterexample) the conjectures are false. Example 3: Determine if the following conditionals are true or false. If they are false, show they are false using a counterexample.
a.) If it rains in Oklahoma, then it is December. TRUE / FALSE
Counterexample: ______________________________________________________________. b.) If x2 = 4, then x = 2. TRUE / FALSE
Counterexample: ______________________________________________________________. Geometry: Lesson 1.3
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Assignment 1.3a You have the choice of the following:
1. Throughout the day, record at least 10 instances of conditional statements you or someone you have talked to have said. For each conditional statement, circle the hypothesis and underline the conclusion. Then determine for every conditional statement if the conditional was true or false. If it was false, write a counterexample showing the statement was false. 2. Complete the worksheet: “Geometry: Handout 1.3a” and turn it in.
Sometimes conditional statements can be confusing to understand if it is true or false because the sentence is structured poorly. There is a way to unscramble the poorly constructed
statement to make it more clear. This is done by using the alternate forms of conditional statements.
Vocabulary: Definition – Alternate Form of Conditional Statements – Converse A Converse is a conditional statement written with the hypothesis and conclusion switched places.
Instead of “If hypothesis, then conclusion”, the converse is “if conclusion, then hypothesis”.
For example, the sentence “If you got your phone back, then you did the dishes.” can be turned into the converse statement of
________________________________________________________________________________. Geometry: Lesson 1.3
3
Now, it should be clear what the conditional statement was intending to say. Sometimes
looking at the converse statement doesn’t make the meaning clear either. We will then have to look at another alternate form.
Vocabulary: Definition – Alternate Form of Conditional Statements – Inverse An Inverse is a conditional statement written with the hypothesis and conclusion negated.
Instead of “If hypothesis, then conclusion”, the inverse is “if NOT hypothesis, then NOT conclusion”.
For example, the sentence “ If we open a window, then the cold air will get in.” can be turned into the inverse statement of
___________________________________________________________. If we were to combine the idea of the converse and inverse statements, we will make the last alternate form called the contrapositive.
Vocabulary: Definition – Alternate Form of Conditional Statements – Contrapositive A contrapositive is a conditional statement written with the hypothesis and conclusion switched and negated.
Instead of “If hypothesis, then conclusion”, the inverse is “if NOT conclusion, then NOT hypothesis”.
Geometry: Lesson 1.3
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For example, the sentence “ If Humpty Dumpty sits on a wall, then he will take a great fall.” can be turned into the contrapositive statement of
___________________________________________________________. Example 4: Write out the alternate forms of each conditional statement. Determine if each one
is true or false.
a.) Conditional Statement: If you do your work, then you will get an A. TRUE / FALSE
Converse: _________________________________________________________ TRUE / FALSE
Inverse: ___________________________________________________________ TRUE / FALSE Contra.: ___________________________________________________________ TRUE / FALSE b.) Conditional Statement: If you live in Paris, then you live in France. TRUE / FALSE
Converse: _________________________________________________________ TRUE / FALSE
Inverse: ___________________________________________________________ TRUE / FALSE Contra.: ___________________________________________________________ TRUE / FALSE c.) Conditional Statement: If a bird can fly, then it has feathers. TRUE / FALSE
Converse: _________________________________________________________ TRUE / FALSE
Inverse: ___________________________________________________________ TRUE / FALSE Contra.: ___________________________________________________________ TRUE / FALSE Can you make a conjecture about how the truth values of each conditional is related?
______________________________________________________________________________ ______________________________________________________________________________ Geometry: Lesson 1.3
5
This shortcut allows us to quickly determine if something is true or false. Being able to
unscramble a conditional statement and determine its truth value will be crucial in using our
deductive reasoning skills. In the next lesson, we will use the conditional statements to help us practice deductive reasoning.
Assignment 1.3b You have the choice of the following:
1. Throughout the day, record at least 3 instances of conditional statements you or someone you
have talked to have said. For each conditional statement, write each of the alternate forms of conditional statements (converse, inverse, and contrapositive). Determine the truth value of each.
2. Complete the worksheet: “Geometry: Handout 1.3b” and turn it in.
Geometry: Lesson 1.3
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G.RL.1.2 Analyze and draw conclusions based on a set of conditions using inductive and deductive reasoning. Recognize the logical relationships between a conditional statement and its inverse, converse, and contrapositive. G.RL.1.3 Assess the validity of a logical argument and give counterexamples to disprove a statement.
Lesson Objectives:
1. To understand/develop the various conditional statements. 2. To determine validity of conditional statements. 3. To use conditional statements in deductive reasoning.
Introduction: So far, we’ve really only focused on inductive reasoning. But since we are already masters at inductive reasoning, let’s shift our focus on the harder to master deductive
reasoning. Because we are still not at the age of proper, complete development of deductive
reasoning, we will need some tools to help us process our thinking. These tools we will develop rely heavily on our knowledge of conditional statements.
Vocabulary: Definition – Conditional Statement A Conditional Statement is a statement written in “if-then” form.
The phrase after the word “if” is called the hypothesis (condition). The phrase after the word “then” is called the conclusion (result).
Example 1: If Samantha works at Wal-Mart, then she will get a paycheck.
In this example, the hypothesis (or condition to be met) is “Samantha works at Wal-Mart”. The conclusion (or result of the condition being met) is “she will get a paycheck”.
You have probably heard the terminology of hypothesis and conclusion in other classes (most likely a science class). Remember inductive and deductive reasoning is also a STEM process. Geometry: Lesson 1.3
1
Example 2: Circle the hypothesis and underline the conclusion in each of the following conditional statements.
a.) If I give you a dollar, then you give me a hamburger. b.) If I got my phone back, then I did the dishes.
c.) If there is a choice, you must decided between them. d.) We will go to Disney Land if you get straight A’s.
It turns out every proper sentence can be turned into a conditional statement.
For example, the sentence “ A bumble bee flies into a wall.” can be turned into the conditional statement: _______________________________________________________________.
Here’s another one: “Superman wears Batman pajamas.” can be turned into the conditional statement: ________________________________________________________________.
Remember, the goal of inductive and deductive reasoning is to create and test conjectures.
Since conditional statements are easy to check to see if the condition has been met, we can
easily verify conjectures are true or show (using a counterexample) the conjectures are false. Example 3: Determine if the following conditionals are true or false. If they are false, show they are false using a counterexample.
a.) If it rains in Oklahoma, then it is December. TRUE / FALSE
Counterexample: ______________________________________________________________. b.) If x2 = 4, then x = 2. TRUE / FALSE
Counterexample: ______________________________________________________________. Geometry: Lesson 1.3
2
Assignment 1.3a You have the choice of the following:
1. Throughout the day, record at least 10 instances of conditional statements you or someone you have talked to have said. For each conditional statement, circle the hypothesis and underline the conclusion. Then determine for every conditional statement if the conditional was true or false. If it was false, write a counterexample showing the statement was false. 2. Complete the worksheet: “Geometry: Handout 1.3a” and turn it in.
Sometimes conditional statements can be confusing to understand if it is true or false because the sentence is structured poorly. There is a way to unscramble the poorly constructed
statement to make it more clear. This is done by using the alternate forms of conditional statements.
Vocabulary: Definition – Alternate Form of Conditional Statements – Converse A Converse is a conditional statement written with the hypothesis and conclusion switched places.
Instead of “If hypothesis, then conclusion”, the converse is “if conclusion, then hypothesis”.
For example, the sentence “If you got your phone back, then you did the dishes.” can be turned into the converse statement of
________________________________________________________________________________. Geometry: Lesson 1.3
3
Now, it should be clear what the conditional statement was intending to say. Sometimes
looking at the converse statement doesn’t make the meaning clear either. We will then have to look at another alternate form.
Vocabulary: Definition – Alternate Form of Conditional Statements – Inverse An Inverse is a conditional statement written with the hypothesis and conclusion negated.
Instead of “If hypothesis, then conclusion”, the inverse is “if NOT hypothesis, then NOT conclusion”.
For example, the sentence “ If we open a window, then the cold air will get in.” can be turned into the inverse statement of
___________________________________________________________. If we were to combine the idea of the converse and inverse statements, we will make the last alternate form called the contrapositive.
Vocabulary: Definition – Alternate Form of Conditional Statements – Contrapositive A contrapositive is a conditional statement written with the hypothesis and conclusion switched and negated.
Instead of “If hypothesis, then conclusion”, the inverse is “if NOT conclusion, then NOT hypothesis”.
Geometry: Lesson 1.3
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For example, the sentence “ If Humpty Dumpty sits on a wall, then he will take a great fall.” can be turned into the contrapositive statement of
___________________________________________________________. Example 4: Write out the alternate forms of each conditional statement. Determine if each one
is true or false.
a.) Conditional Statement: If you do your work, then you will get an A. TRUE / FALSE
Converse: _________________________________________________________ TRUE / FALSE
Inverse: ___________________________________________________________ TRUE / FALSE Contra.: ___________________________________________________________ TRUE / FALSE b.) Conditional Statement: If you live in Paris, then you live in France. TRUE / FALSE
Converse: _________________________________________________________ TRUE / FALSE
Inverse: ___________________________________________________________ TRUE / FALSE Contra.: ___________________________________________________________ TRUE / FALSE c.) Conditional Statement: If a bird can fly, then it has feathers. TRUE / FALSE
Converse: _________________________________________________________ TRUE / FALSE
Inverse: ___________________________________________________________ TRUE / FALSE Contra.: ___________________________________________________________ TRUE / FALSE Can you make a conjecture about how the truth values of each conditional is related?
______________________________________________________________________________ ______________________________________________________________________________ Geometry: Lesson 1.3
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This shortcut allows us to quickly determine if something is true or false. Being able to
unscramble a conditional statement and determine its truth value will be crucial in using our
deductive reasoning skills. In the next lesson, we will use the conditional statements to help us practice deductive reasoning.
Assignment 1.3b You have the choice of the following:
1. Throughout the day, record at least 3 instances of conditional statements you or someone you
have talked to have said. For each conditional statement, write each of the alternate forms of conditional statements (converse, inverse, and contrapositive). Determine the truth value of each.
2. Complete the worksheet: “Geometry: Handout 1.3b” and turn it in.
Geometry: Lesson 1.3
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