1. Lesson 4

1.1. Discover

Mathematics 20-2 M2 Lesson 4

Module 2: Logic and Geometry

 
Discover
 

In order for a proof to be valid, the arguments used to support the generalization or conclusion must be valid.


textbook

Read “Example 5: Communicating reasoning about a divisibility rule” on page 30 of your textbook. As you examine Lee’s proof, consider why his proof is valid based on what you have learned about deductive reasoning in Lesson 3.

 

Try This 1
 

Consider the following sets of integers:

 

 
10, 11, 12 5, 6, 7, 8, 9 1, 2, 3, 4, 5, 6, 7
  1. Show that the sum of each set of consecutive integers divided by the number of integers in the set is equal to the middle term in each set of numbers. For example,

    This is graphic showing that the sum of 10 + 11 + 12 is 33 and when the sum is divided by the number of terms, which is 3, the middle term is 11.

  2. Form a deductive proof that the sum of any set of 5 consecutive integers divided by the number of integers in the set (5) is equal to the middle term in each set.

This is a photo of a hand holding a light bulb.

iStockphoto/Thinkstock

Share 1
 

Having a discussion with a partner or group will help to identify alternative strategies that may be effective in developing a deductive proof. As well, you may discover a way to revise your personal strategy to make it more efficient.

 

Compare your answers from Try This 1 with a partner or in a group. In your discussion, discuss the following questions:

  • When you divided the sum of each set of numbers by the number of integers in the set in question 1, did you get the middle term of the set?
  • What strategy did you each use to write a deductive proof for question 2? What did your proof look like?
  • Did the strategies used result in a valid proof? If not, can you identify any errors in the proof?

After completing the preceding activities, Joel decides that he wants to look at an even number of consecutive integers. He decides to prove that the sum of 4 consecutive integers is evenly divisible by the number of integers in the set (4).

 

You can watch the animation titled Joel’s Solution.

 

 

This is a screenshot for Joel’s Solution.

 

A copy of Joel’s Solution: Print Version is also available. You may find it helpful to have a copy of his solution handy when you work on the Share 2 questions.

 

Share 2
 

After examining Joel’s proof, answer the following questions with a partner or group:

  1. Is Joel’s proof valid? Why or why not? If the proof is incorrect, identify the error.

  2. How could a counterexample be used in this situation? Provide an example.