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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:
      1. Divide n by however many numbers are in the pattern.
      2. Identify the remainder, R.

Example

Rules

  1. If there is a blue marble, it is followed by 2 yellow marbles and a red marble.
  2. If there is a red marble, it is followed by a blue marble.
  3. 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.

GeoA01.02 DeductiveInductive

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?

GeoA01.02 Q1-04c

  • 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.

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