Explain the Modeling reporting category (a cross-cutting score) and produce, interpret, evaluate and improve mathematical models that translate a real situation into equations, expressions or graphs.

What this topic is asking

Modeling is a reporting category that is not a separate block of questions. Instead it is scored across questions from the other categories whenever a question asks you to connect mathematics to a real situation. The category measures four moves: produce a model from a description, interpret the parts of a model, evaluate how well a model fits, and improve a model. Because modeling questions are spread throughout the test, strengthening this skill lifts your score broadly rather than on one topic.

The four modeling moves

Each modeling question is doing one of these four things.

The unifying idea is the dictionary between the real situation and the mathematics: a fee becomes a constant, a per-unit charge becomes a coefficient, "doubles each step" becomes a base of 2.

Producing a model

To build a model, classify each quantity by how it behaves.

The decision in step 1, constant versus rate, is the heart of producing a model, and it is the same decision whether the model is linear, like this one, or exponential.

Interpreting a model

Interpretation questions give you a model and ask what a part of it means. In a linear model $y = mx + b$, the slope $m$ is a rate of change (with units of output per input) and the intercept $b$ is the starting value. In an exponential model $y = a \cdot b^{x}$, $a$ is the initial value and $b$ is the growth or decay factor ($b > 1$ grows, $0 < b < 1$ decays). Naming the part with its units and direction is what the ACT rewards: not "18", but "the cost per month, in dollars".

Evaluating and improving a model

Evaluation questions use a model to predict (substitute an input and read the output) or ask whether a model is reasonable (does it match the data, does it predict sensibly outside the given range). Improvement questions change a condition, a new fee, a different rate, a cap on the output, and ask you to adjust the model. The skill is the same: keep the dictionary between situation and mathematics consistent as the situation changes.

Why modeling lifts the whole score

Because modeling is scored across the test, every word problem with an equation, every "what does this coefficient mean" question, and every "predict the value" question contributes to it. Practising the match-each-number-to-its-role habit therefore pays off on Number and Quantity, Algebra, Functions and Statistics questions alike. A student who can fluently turn a paragraph into an equation, and an equation back into a sentence, is doing exactly what this category measures.

Try this

Q1. A phone plan costs $20 per month plus$0.10 per gigabyte of data. Write a model for the monthly cost $C$ for $g$ gigabytes. [1 point]

• Cue. $C = 0.10g + 20$. The $0.10$ is a rate (times $g$); the $20$ is a fixed monthly charge.

Q2. In the model $V(t) = 5000(0.85)^{t}$ for a car's value after $t$ years, what does the $0.85$ represent? [1 point]

• Cue. A decay factor: the car keeps 85 percent of its value each year, a 15 percent annual loss.