• Use inductive reasoning to identify patterns and make predictions based on data
  • Use inductive reasoning to provide evidence that conjectures are true or provide counterexamples to disprove them

Assignment

  • Three vocabulary definitions
  • p33 7–23 (17 problems, PDF link)

The remaining sections in the chapter deal with logic and going about proving things to be true (or proving them to be false). The first one covers inductive reasoning. Simply put, it’s looking for patterns and coming up with a conclusion based on that pattern. For example, take the first few numbers in a sequence.

\[\begin{align} 88, 82, 76, 70, 64, ... \end{align}\]

It appears that each term is six less than the one before. We can now write a conjecture, or a statement based on our inductive reasoning. Each term is six less than the one before it.

The catch with a conjecture is that it is unproven, and there is a good chance you are missing information. For example, some of you haven’t missed a day of school yet. Based on that pattern, I could conjecture that you have never missed a day of school. That’s likely untrue, and you can easily prove me wrong with a single counterexample. Just one day—one example—of where you did not make it to school.

A more “mathier” example of a counterexample is for the conjecture that all numbers are either positive or negative. This is a common thought since it is true for every single number, except for zero. We can then amend it so that it reads all numbers are positive or negative, aside from zero which has no sign.

Example: Statistics

Based on the table, how many residents can be expected to vote in year 7?

Year Total Residents Voters
1 3511 386
2 3790 414
3 4085 451
4 4907 544
5 5562 623
6 7014 767
7 7786 ?

SOLUTION Some trial and error might be necessary in problems like these. You can try using just the voter column, seeing if there is a pattern to the increase, but the results will be inconsistent.

Instead, if you look at the voters as a percentage of the total residents you can something you can work with.

\[\begin{align} \frac{386}{3511} &\approx 0.110 \\[1em] \frac{414}{3790} &\approx 0.109 \\[1em] \frac{451}{4085} &\approx 0.110 \\[1em] \frac{544}{4907} &\approx 0.111 \\[1em] \frac{623}{5562} &\approx 0.112 \\[1em] \frac{767}{7014} &\approx 0.109 \end{align}\]

Voter turnout seems to be about 11%, meaning we can now make a guess at the number of voters in year 7.

\[\begin{align} 0.11 \cdot 7786 &\approx 856 \end{align}\]

$\blacksquare$