Name: ______________________ Date: ___________________

Foundations of Mathematics 11

Chapter 1- Inductive and Deductive Reasoning 1.2 Exploring the Validity of Conjectures

Some conjectures initially seem to be valid, but are shown not to be valid after more evidence

is gathered. To show that a conjecture is not valid, it is sufficient to show just one example

that is not true. We must be cautious, therefore, about reaching conclusions by inductive

reasoning. A conjecture may be revised, based on new evidence.

1. Optical illusions are useful examples to disprove initial conjectures Make a

conjecture about the lines in this picture.

2. Tomas gathered the following evidence and noticed a pattern.

17 (11) = 187 23 (11) = 253

41 (11) = 451 62 (11) = 682

Tomas made this conjecture: When you multiply a two-digit number by 11, the first and last digits of

the product are the digits of the original number. Is Tomas’s conjecture reasonable? Develop

evidence to test his conjecture and determine whether it is reasonable.

Optical Illusions and Forming Conjectures

Which line is the longest?

How can you check the validity of your conjecture?

Can you be certain that the evidence you collect leads to a correct conjecture?

Aoccdrnig to rscheearch at Cmabrigde Uinervtisy, it deosn't mttaer in waht order the ltteers in a wrod

are, the only iprmoatnt tihng is taht the frist and lsat ltteer be at the rghit pclae. The rset can be a toatl

mses and you can sitll raed it wouthit a porbelm. Tihs is bcuseae the huamn mnid deos not raed ervey

lteter by istlef, but the wrod as a wlohe.

If possible, find a counterexample for each of the following assumptions:

a) Every prime number is odd.

b) Multiplying leads to a larger number.

In Summary

Key Idea – Some conjectures may seem valid, but are shown to be invalid after more

evidence is gathered

Need to Know – All we can say about a conjecture reached through inductive

reasoning is that there is evidence either to support or deny it A conjecture may be

revised, based on new evidence

Assignment: p. 17 #1, 2, 3