#### Inductive and Deductive reasoning with patterns and functions:

The most important thing about a function rule is that it must
always give you the right output (no matter what the input). When we
figure out what we think the function rule is, we generally use **inductive
reasoning**: we look at a pattern of numbers (often in a table)
and try to figure out an algebraic rule that would fit that pattern. In
order to make sense of a rule, however, and to prove that it is the
correct rule using **deductive reasoning, **you need to
know some properties of the function. I

A property of a function rule is sometimes the same as knowing where
it comes from. For example, if I know that the function rule is
describing the perimeter of a pattern block train, I can use properties
of pattern block trains and of perimeters to justify my function rule.
For example, with a triangle train, you have the picture:

the table:

n=# of triangles |
p = perimeter |

1 |
3 |

2 |
4 |

3 |
5 |

4 |
6 |

and the equation: p = n + 2 (pretty
easy to guess from table)

But what does it have to do with the perimeter? Well...

So the top and bottom sections of the perimeter are **n** long altogether, and the ends
are **2** long:

**p = n + 2**

See a similar example with the trapezoid
train

Another property might come out of the step size. A situation like:
John starts 4 feet from the tree. If he walks 3 feet every second (away
from the tree), how many feet is he from the tree after **n**
seconds? In this case part of the problem situation tells you that for
sure the step size is always 3. This is a property that you can use
deductively. (When you look at a table and figure out that the step
size is 3, you are infering that it will always be 3, and you are
inductively continuing that pattern. The difference between this and a
deductive property and hypothesis is that with a deductive hypothesis,
you are explicitly stating it as a condition of the problem).

The function rule is **d = 3n+4**

(d=distance
from tree, n=#
of seconds)

This rule makes sense because after **n** seconds, he has walked **3n** feet*. If he started out
**4** feet from the tree, that would make him **3n + 4** feet from the tree.

*When I say that after n seconds at 3 feet per second he walks 3n
feet, I am using the definition of multiplication as repeated
addition--after n
feet, he walks 3 feet n times, which
is written 3n.