1.13.A Identify an appropriate function type to construct a function model for a given scenario.
1.13.B Describe assumptions and restrictions related to building a function model.

Ideas

Identifying the problem

Identifying the problem

When we model real-world contexts we can use the modeling cycle:

A modeling cycle. Starting with the phrase Identify the problem inside a circle. An arrow pointing to the right where the phrase Create a model is inside a rectangle. Next is an arrow pointing downward where the phrase Apply and analyze is inside a rectangle. Then, an arrow pointing to the right where the phrase Interpret results is inside a rectangle. Next is an arrow pointing upward where the phrase Verify the model is inside a triangle. Then, an arrow pointing to the right where the phrase Report findings is inside a circle. There is an arrow pointing to the left from the phrase Verify the model to Create a model.

Situations that require modeling are often broad and complex. There may be multiple ways to solve the problems and multiple solutions to the problems. Before we can determine a path to a solution, we often need to clarify the problem that needs to be solved.

To clarify the problem, we need to do the following:

Restate the problem to clarify what the model intends to measure, predict, and/or solve
Identify questions that need to be considered prior to answering the given problem
List factors that will affect the outcome
Research information needed to answer the problem
State assumptions

By making assumptions, you are narrowing the focus of the problem you are trying to solve and thus, simplifying the process for solving the problem.

For example, if you are asked to identify the best noise-canceling headphones on the market today, you will need to make assumptions about what the word "best" refers to. You may make an assumption that "best" refers to the clarity of the sound produced and the level of outside noise that can be heard. Someone else may assume it refers to the value for money in terms of sound quality, price, durability, and/or battery life. Various assumptions lead to different but valid outcomes.

Examples

Example 1

Design a model for a roller coaster that contains at least 3 peaks.

Identify the important factors that need to be considered when answering this question.

Worked Solution

Create a strategy

Important factors affect the decision being made but they may not be explicitly stated in the problem. Our goal is to think of questions that need to be answered and identify these underlying aspects.

Apply the idea

Some important factors are:

What will the independent variable represent?
What will the dependent variable represent?
Will there be any restrictions on the variables?
How long are average roller coasters?
How tall are average roller coasters?
How long do average roller coasters last?
Can roller coasters begin and end at different heights?

Reflect and check

There may be other factors that are important, but you may not think of them until later in the modeling cycle. This list will be continuously revised, and the model may need to change too. That is normal and expected while working through the modeling cycle.

List assumptions that need to be made.

Worked Solution

Create a strategy

Because the question is broad and there are many factors that could affect our decision, we need to make assumptions to narrow our focus.

Apply the idea

A roller coaster's speed is variable throughout the ride. The ride moves slowly as it climbs a hill, and it moves quickly as it descends a hill. We do not have the skills or mechanics to calculate the speeds of the coaster, so we can assume that the plan for the roller coaster is focused more on its shape and design.

Reflect and check

Similar to the previous part, there may be other assumptions that need to be made later in the modeling cycle. It is important to continue writing down each assumption that is made throughout the process.

State a recommended plan for solving the problem, including any research that needs to be done to further clarify the problem.

Worked Solution

Create a strategy

To propose a plan, we need to address each of the important factors we previously listed by conducting research to clarify the factor or by creating an action item to account for the factor.

Apply the idea

A proposed plan for solving the problem may be as follows:

To focus on the shape and design of the coaster, the independent variable should represent length. We still need to decide on the unit of measure, so we can research how other coasters are measured.
The dependent variable will represent the height of the coaster. It will need the same unit of measure as the independent variable.
Since the variables are lengths, the domain is restricted to positive values. We still need to decide if the x-axis represents ground level or the height at which the riders board the coaster.
We need to research the average length and heights of roller coasters.

Reflect and check

Ideally, this plan will give us the information we need to create a good model for a roller coaster.

Example 2

What does it cost to feed a family in the United States?

Brainstorm a list of questions that can be answered to help solve this problem.

Worked Solution

Apply the idea

A possible list of brainstormed ideas might be:

What are the different food requirements based on size or age?
How many people are in a family?
What are the current food costs in my area?

Reflect and check

The brainstorming will help identify assumptions that need to be made in order to create a model.

Research and determine how a model can be used to solve the problem.

Worked Solution

Create a strategy

In order to answer the questions in our brainstorm, there are a few things we need to decide:

How the dietary needs of people differ.
How our model can apply to different types of families.
How food costs are calculated.
What span of time we are using: daily cost, weekly cost, monthly cost, or other.

Apply the idea

One possible approach to this problem could be:

We will create a model that outputs the daily cost to feed a single person based on their age. The cost to feed a family can then be determined by combining the costs of each individual in the family.

The caloric needs will come from dietaryguidelines.gov and we will find the cost of a meal plan for a single day that meets the main nutrition requirements (calories, proteins, carbohydrates, and fats) for the different age groups. Instead of creating different models for different genders, we will use the average nutritional needs between males and females of the same age group. We will research the average cost of each food on the meal plan, paying attention to cost-effective choices, since we are trying to calculate the minimum costs.

Reflect and check

Some of the assumptions made in this problem statement include:

Age is the main influence in caloric need.
One meal plan can be used to calculate the costs for every day.
The average nutritional needs of males and females will make a good representation of the nutritional needs of any person.

As we build and test our model, we can revisit these assumptions to see if they should be changed or updated to better reflect the problem.

Idea summary

A modeling problem is often open-ended with many unknowns. Our job is to research and brainstorm the problem, determine how a model can solve the problem, and make necessary assumptions to support our model.

Outcomes

1.13.A

Identify an appropriate function type to construct a function model for a given scenario.

1.13.B

Describe assumptions and restrictions related to building a function model.

1.13 Function model selection and assumption articulation

Introduction

Ideas

Identifying the problem

Examples

Example 1

Example 2

Outcomes

1.13.A

1.13.B

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