Module 2 Summary

The ability to reason is very useful for developing personal strategies for winning games, figuring out puzzles, and solving problems related to design.

 

In this module you investigated the following inquiry questions:

• How is reasoning used to develop and analyze problem-solving strategies?
• How is reasoning used to verify aspects of a design?

Examples where inductive reasoning was used as part of the solution strategy were investigated in this module. These examples involved solving puzzles or winning games that required recognizing patterns, extending a pattern, or identifying a certain order. Games and puzzles were also presented that used deductive reasoning as part of the winning strategy. These games and puzzles were types that required inquiry and discovery to complete. Some games or puzzles required both types of reasoning.

 

A preferred strategy may be to begin by making a conjecture; this is usually based on observation. Inductive reasoning is then used to develop a rule or generalization from a conjecture—remember that inductive reasoning moves from the specific to the general. Deductive reasoning is then used to show that the generalization can be applied to all possible cases, which logically proves the conjecture. Deductive reasoning goes from the general to the specific.

 

Inductive reasoning can also be used to develop conjectures that explain many geometric relationships in design including relationships between parallel lines, transversals, angles, and triangles. Deductive reasoning can then be used to prove these conjectures. These proven conjectures are known as properties. Properties can be used to solve a variety of design problems and are essential when verifying that specific design components are accurate.

 

When deductive reasoning is used correctly, you can be sure that the conclusion you draw is valid. However, proofs arrived at by deductive reasoning can be invalid if the reasoning is flawed. Flaws in reasoning can be detrimental to the success of a design. Thus, it is important that designers have good reasoning skills and can apply properties appropriately.

 

The lines are parallel if a transversal intersects two lines and the following occurs:

• The corresponding angles are equal: ∠1 = ∠5, ∠2 = ∠6, ∠3 = ∠7, and ∠4 = ∠8.
  
  
• The alternate exterior angles are equal: ∠1 = ∠8 and ∠2 = ∠7.
  
  
• The alternate interior angles are equal: ∠3 = ∠6 and ∠4 = ∠5.
  
  
• The interior angles on the same side of the transversal are supplementary: ∠3 + ∠5 = 180° and ∠4 + ∠6 = 180°.