**1 :**How much would it cost to build a patio?

This is an example of a question that might occur to (or be asked of) you at some
point in your life. If this is all you are given, then it seems impossible to
answer the question. In reality it is most often the case that this is how
a problem is presented; someone will ask a generic question or hand you
a problem, with little information attached and tell you to “just figure it
out”. This means we must use our own problem solving skills to come to a
solution.

Question Phase: First we need to acquire information. In this case we might ask questions like:

- What do we want to make the patio out of?
- What size is the patio?
- What kind of expertise is required to build the patio?

Usually it’s not clear what information is useful and what information is not
(information that is not useful is often called *extraneous* information). One way to
determine this, and a way to figure out how to get to an answer to our original
question, is to try and convert the information we have into quantified information or
”data”. Once we have done that, we can create a model which relates the data
together, and then manipulate the model to arrive at our result (ie the answer to our
problem).

Modeling Phase: Thus, we might end our process with something that looks like this:

- = The number of bricks needed to build our patio
- = The cost of the patio (in dollars).
- = The area of the patio (in square feet).

At first glance the above may seem confusing, and in fact it should be. The problem
is that we’ve ”jumped to the end” and have just a bunch of equations and variables.
But, the intervening steps that let us go from the ”questioning” phase to
the ”modeling” phase are where the ”mathematical reasoning” comes into
play.

What kind of problems are most likely to involve/require ‘mathematical reasoning’?