Defining Quantities for Descriptive Modeling - Algebra

Volume. Volume changes predictably regardless of tank shape.

$\frac{\text{mass}}{\text{volume}}$. Density is defined as mass divided by volume.

The input quantity that is chosen or controlled. The independent variable is the input you can change.

$\text{number of tickets}$. Number of tickets is the controllable input variable.

The measurement scale, such as dollars, seconds, or meters. Units specify how the quantity is measured or expressed.

$\text{height in centimeters}$. Continuous quantities can take any value within a range.

$\text{number of students}$. Count quantities take only whole number values.

$\text{miles per hour}$. A rate expresses one quantity per unit of another quantity.

Identify the question being answered by the model. Understanding the goal helps select relevant quantities to measure.

Choose a variable that matches the context, unit, and purpose. The quantity must be relevant, measurable, and serve the model's goal.

$\text{total cost}$. Total cost changes based on the number of tickets bought.

Use magnitudes and units that fit the situation and data range. Scale should match the context and make calculations practical.

Minutes. Minutes are more practical for a 2-hour duration.

Meters. Meters are the appropriate scale for room-sized measurements.

State what $0$ means and what positive values represent. Clear reference points prevent ambiguity in interpretation.

A starting (initial) temperature. Temperature change needs a baseline for comparison.

Change in the output quantity $y$. Delta notation indicates the change in a variable's value.

$\frac{\text{dollars}}{\text{hour}}$. Slope units come from dividing output units by input units.

Choose variables with an approximately constant rate of change. Linear models work best with quantities that change at constant rates.

$y = \text{total taxi cost in dollars}$. Total cost is the output quantity being modeled.

$x = \text{miles traveled}$. Miles traveled determines the variable portion of taxi cost.

Percent correct. Percent correct allows fair comparison across different tests.

$\frac{\text{liters}}{\text{minute}}$. Rate units are output units divided by input units.

Each part of the situation is represented once in the model. Each aspect of the situation should be counted only once.

$170$ cm. Practical height measurements don't need extreme precision.

Round to a place value that matches measurement accuracy. Precision should reflect the accuracy of your measurements.

$t \ge 0$. Time cannot be negative when measuring elapsed duration.

$p$ is a whole number and $p \ge 0$. People must be counted in non-negative whole numbers.

State allowable values, such as $x \ge 0$ or whole numbers only. Constraints define the realistic range of variable values.