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1.2 – Deduction: Making a Case
Objectives
 Use deductive methods to test whether a rule or pattern is correct.
 Identify conditional statements and use them to form logical statements.
Key Terms
 Deduction – a way of thinking that starts with a given set of rules and conditions and figures out what must be true based on what is given.
 Known as deductive reasoning.
 Conditional Statement – statement that has the form “If A, then B,” where A is what you assume is true and B is the conclusion.
 Consecutive – the next one in the pattern.
Notes
Patterns 
 The n in nth stands for any counting number (positive numbers).
 The nth number in a series…
 To find the nth term:
 Divide n by however many numbers are in the pattern.
 Identify the remainder, R.
Example
Rules
 If there is a blue marble, it is followed by 2 yellow marbles and a red marble.
 If there is a red marble, it is followed by a blue marble.
 What color is the 98th marble?
Answer
 Use the nth number pattern to solve this problem.
 We know there are 4 marbles in the pattern: blue, yellow, yellow, red.
 Divide 98 by 4, and find the remainder (R): 2
 The 2nd marble in the pattern is yellow, so the answer is yellow!

Logic: Deduction vs Induction 
 You CANNOT rely solely upon induction to prove that your conclusion is correct.
 You CAN rely solely upon deduction to prove that your conclusion is correct.
 In geometry, you can use deductive rules to PROVE conjectures.

Conditional Statements 
 If A, then B
 Means: If A is true, then B is always true.
 Math Example: If 2 + 2 = 4, then 4 – 2 = 2.
 Life Example: If I study, then I will learn.
 Syllogism – a chain of conditional statements.
 How to Use a Chain of Conditional Statements
 Step 1: Start with a chain of conditional statements.
 Step 2: Accept that all of those statements are true.
 Step 3: Make connections between statements in the chain.
 Step 4: Make a conclusion statement based on those connections.
Example 1
Example 2
 All squares are rectangles.
 All rectangles are shapes that have four sides
 All shapes with four sides are quadrilaterals.
 So, all squares are quadrilaterals!

Examples
 Ex 1. If the pattern below follows the rule “Starting with five, every consecutive line has a number that is four less than twice the previous line,” how many marbles must be in the seventh line?
 Note: Notice that there are only 4 lines of marbles, but you need to find the number in the 7th line.
 Starting with 5 is not important in this case, as we already have the first few lines.
 Four less means: minus 4 (from something)
 Twice the previous line means: 2L
 Pattern: 2L – 4
 Answer
 5th Line: 2(12) – 4 = 24 – 4 = 20
 6th Line: 2(20) – 4 = 40 – 4 = 36
 7th Line: 2(36) – 4 = 72 – 4 = 68

 Ex 2. If the statement “If it is midnight, then the I am not awake” is assumed to be true, is its reverse, “If I am not awake, then it is midnight,” also always true?
 No, it is false! I might be napping around 2pm.

 Ex 3. The statement “If A, then B” can best be described as: “If A is true, then B MUST be true!”

 Ex 4. In deductive thinking, you start with a given set of rules and conditions and determine what must be true as a consequence.

Important!
Practice
 Practice: Interactives on Pg 8, 16
 Practice: Challenge on Pg 20
 Review: Pg 7, 15
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