Modeling

The One Thing

Mathematical modeling translates a real-world situation into a mathematical representation that can be used to calculate, estimate, predict, compare, or make decisions. The result must then be interpreted in context, checked against evidence or reasonable expectations, and revised if necessary.

Definitions and Results

• Mathematical model: A mathematical representation of a real situation, such as an equation, function, graph, table, geometric figure, or simulation. A model is a useful simplification of reality rather than an exact copy of every detail.
• Variable: A quantity that can change or whose value is unknown, such as time, distance, cost, or population.
• Assumption: A statement accepted as reasonable in order to simplify a situation, such as assuming a constant speed or ignoring air resistance. Changing an assumption can change the model and its conclusions.
• Parameter: A fixed value in a model that describes important conditions, such as a starting amount, rate, or fixed fee.
• Independent variable: The input or quantity chosen or controlled, often represented on the horizontal axis of a graph.
• Dependent variable: The output or quantity that changes in response to the independent variable, often represented on the vertical axis.
• Function: A rule that assigns exactly one output to each allowed input and can describe relationships between quantities.
• Linear model: A model with a constant rate of change, commonly written as

$$y=mx+b,$$ where $m$ is the rate of change and $b$ is the initial value.

• Exponential model: A model in which a quantity changes by a constant percentage or multiplication factor over equal intervals, commonly written as

$$y=a(b{)}^x,$$ where $a$ is the initial value and $b$ is the growth or decay factor. For growth rate $r$, $b=1+r$; for decay rate $r$, $b=1-r$.

• Quadratic model: A model involving a squared variable, commonly written as

$$y=a{x}^2+bx+c.$$ It is often used for projectile motion or area problems. Its vertex occurs at $$x=-\frac{b}{2a},$$ and may represent a maximum or minimum.

• Rate of change: The amount one quantity changes compared with another. For a line,

$$m=\frac{y_2-y_1}{x_2-x_1}.$$ Its units may be dollars per item, miles per hour, or people per year.

• Average rate of change: Over the interval from $a$ to $b$,

$$\frac{f(b)-f(a)}{b-a}.$$

• Constraint: A condition or limit restricting possible values, such as nonnegative time, a budget, or a maximum capacity.
• Error: The difference between a model’s result and an observed or accepted value.
• Residual: The difference between an observed value and the value predicted by a model, usually written as

$$\text{residual}=\text{observed value}-\text{predicted value}.$$

• Validation: The process of checking whether a model gives reasonable results by comparing predictions with data, graphs, calculations, or real-world expectations.
• Percent error: When the observed value is used as the reference,

$$\frac{|\text{observed value}-\text{predicted value}|}{|\text{observed value}|}\times 100\%.$$

• Domain: The set of input values that make sense in the context.
• Range: The set of possible output values.
• Dimensional analysis: Using units to check calculations and confirm that quantities are combined appropriately.

Worked Methods

1. Constructing and applying a mathematical model

• Identify the real-world problem and determine what must be calculated, estimated, predicted, compared, or decided.
• Define the variables and distinguish the independent variable from the dependent variable.
• State the assumptions used to simplify the situation, such as assuming a constant speed or ignoring air resistance.
• Collect or organize relevant information using tables, graphs, equations, technology, or simulations.
• Choose a suitable model based on the situation, available data, variables, and purpose.
• Solve or calculate using the model.
• Interpret the result in the original context, including its units, meaning, domain, range, and constraints.
• Validate the result by comparing it with real data, graphs, calculations, or reasonable expectations.
• Revise the data, assumptions, or model if the prediction is unreasonable.

A model can be useful without being perfectly accurate, but its accuracy, assumptions, and appropriate range of use should be stated. More complicated models are not automatically better; a simpler model may be preferable if it explains the important behavior accurately and clearly.

2. Using a linear model

• Identify two data points $(x_1,y_1)$ and $(x_2,y_2)$.
• Calculate the constant rate of change:

$$m=\frac{y_2-y_1}{x_2-x_1}.$$

• Determine $b$, the value of $y$ when $x=0$.
• Write the model:

$$y=mx+b.$$

• Use the equation to calculate or predict values.
• Interpret $m$ with appropriate units and interpret $b$ as the initial value.

3. Using an exponential model

• Identify the initial value $a$.
• Determine whether the quantity grows or decays by a constant percentage or multiplication factor over equal intervals.
• Convert the rate $r$ into a factor:

- Growth: $b=1+r$ - Decay: $b=1-r$

• Write the model:

$$y=a(b{)}^x.$$

• Substitute the relevant value of $x$.
• Interpret the result in context, including the units and the time or interval represented by $x$.

4. Using a quadratic model

• Represent the situation using

$$y=a{x}^2+bx+c.$$

• Identify the coefficients $a$, $b$, and $c$.
• Find the vertex’s $x$-coordinate:

$$x=-\frac{b}{2a}.$$

• Substitute this value into the model to determine the corresponding output.
• Interpret the vertex as a maximum or minimum where appropriate, such as in projectile motion or area problems.
• Apply the context’s domain and constraints before accepting the result.

5. Calculating average rate of change

• Identify the endpoints $a$ and $b$ of the interval.
• Find the corresponding function values $f(a)$ and $f(b)$.
• Substitute into

$$\frac{f(b)-f(a)}{b-a}.$$

• State the result with units and interpret it as the average change in output per unit change in input.

6. Calculating percent error and residuals

• Identify the observed value and the predicted value.
• Calculate the residual:

$$\text{observed value}-\text{predicted value}.$$

• For percent error, calculate:

$$\frac{|\text{observed value}-\text{predicted value}|}{|\text{observed value}|}\times 100\%.$$

• Interpret the size and significance of the error in the original context.

7. Checking a model using dimensional analysis

• Write the units of each quantity before calculating.
• Confirm that quantities are combined appropriately.
• Check that the final units match the required quantity.
• Use the units to interpret the result; for example, a slope may represent dollars per item, miles per hour, or people per year.

Where It Goes Wrong

• Variables are not defined clearly, or the independent and dependent variables are confused.
• Assumptions such as constant speed or ignoring air resistance are used without being stated.
• A model is applied outside its appropriate domain or without considering constraints such as nonnegative time, a budget, or a maximum capacity.
• A numerical answer is given without units or without explaining its meaning in the original situation.
• Growth and decay factors are confused: growth uses $b=1+r$, whereas decay uses $b=1-r$.
• A prediction is accepted without validation; signs, size, units, consistency with the situation, observed data, and reasonable expectations should be checked. Correlation alone does not prove that one quantity causes another.

What Gets Asked

• Define and distinguish a mathematical model, variable, assumption, parameter, independent variable, dependent variable, function, constraint, error, residual, validation, and dimensional analysis.
• Describe the stages of the mathematical modeling process.
• Select an appropriate model for a situation and justify the choice.
• Calculate the slope of a linear model using

$$m=\frac{y_2-y_1}{x_2-x_1},$$ then interpret the rate and initial value in context.

• Construct or use an exponential model $y=a(b{)}^x$, including growth and decay factors.
• Construct or analyze a quadratic model $y=a{x}^2+bx+c$, including its vertex and possible maximum or minimum.
• Calculate and interpret an average rate of change.
• Determine a model’s domain and range from the real-world context.
• Calculate residuals and percent error.
• Use units and dimensional analysis to check a calculation.
• Evaluate whether a model is reasonable by comparing its predictions with data or common sense.
• Explain why assumptions, accuracy, limitations, and the appropriate range of use must be stated.
• Explain why a simpler model may be preferable to a more complicated model when it represents the important behavior accurately and clearly.
• Explain why correlation in data does not, by itself, establish causation.