**1 :**What is the point of the first phase of modeling? (Select all that apply)

This section analyzes the previous example in detail to develop a three phase deductive process to develop a mathematical model.

Let’s revisit the previous example and work through it to get a solution. This will highlight the reasoning process, and in the following lectures we will work through each of the reasoning steps in more detail.

##### Phase One: Statement and clarification of the problem

*asks*of you, and what they

*want*of you, are often not the same. So, even though it seems like the first phase is given for free (that is, that you have to be asked a problem in order to start problem solving) the truth is that often the first step is clarifying what someone has asked of you until you can concretely describe what they want. Let’s look at our patio example.

The original question was simply “How much would it cost to build a patio?” However, this is entirely too vague. The desired answer is clearly quantifiable (they want cost, a number, as an answer after all), but the given information “a patio” isn’t quantifiable. Before we can give a quantified answer then, we need to clarify the question into something we can assign numbers to.

How one goes about this can vary, but at the very least we should ask the most basic questions, such as:

- What is the patio made of?
- What
*is*a patio, or more specifically, what is a patio to the person asking the question? (Pro-tip: Never assume the person asking a question is using the same precise vocabulary you are). - How big does the person want the patio? Specific dimensions (numbers!) are preferable.

After some back and forth questioning you determine that the patio is going to be made of cement paving stones, and encompass a flat area of between 15 and 20 feet on each side.

##### Phase Two: Quantifying the situation, ie turning Information into Data

We now know that the patio is made of cement paving stones, and that we want a surface between 15 and 20 feet on each side to be covered with them. Clearly, if we are trying to determine cost, we should know the cost of something. Hopefully it’s clear that, since we are using cement paving stones to cover the surface, we need to know the cost of those paving stones, but that’s not quite enough. We also need to know how many paving stones it will take to cover the given surface. After a quick trip to the local building supply store, you determine that cement paving stones are around $ each, and are about a foot and a half long and a half foot wide.

##### Phase Three: Developing your (numeric) answer

It would take ten pavers stacked end to end (the long way) to cover fifteen feet, and then it would take thirty of them stacked side by side to attain fifteen feet. Thus to completely cover a fifteen by fifteen foot patio with cement pavers, we would need pavers at $ each, for a total cost of $. Using similar calculations we find that we need approximately 520 cement pavers to cover a twenty by twenty foot patio, for a total cost of $.

##### Is that it?

The above is a basic example of using mathematical reasoning to answer a problem. But it can be used to do much more than that. To do so, we will introduce the idea of Modeling in the next lecture, and see how mathematical reasoning can be used to build a more general answer (after all; where did those equations come from in the example box?)

**2 :**What is the point of the second phase of modeling? (Select all that apply)

**3 :**What is the point of the third phase of modeling? (Select all that apply)

**4 :**If you cannot get a numeric answer after going through these three phases of modeling, what does that mean? (Select all that apply)