11.1 Modeling Real-World Contexts Across Subject Areas
Key Takeaways
- Mathematical modeling begins by identifying the varying quantity, naming it with a variable, and stating the units before any equation is written.
- A constant that does not change with the variable becomes a constant term, while a per-unit rate becomes a coefficient.
- Science contexts supply the most common cross-subject models, including density, speed, and unit-rate relationships.
- The structure of the situation determines the function family: constant change is linear, constant percent change is exponential, and area-type relationships are quadratic.
- A completed model must be checked against the context, because an algebraically correct answer can be contextually impossible.
11.1 Modeling Real-World Contexts Across Subject Areas
Competency 5 carries 20% of test 025, and its items are scenario-based. Skill 1 opens the competency with modeling: turning a described situation into an expression or equation. Note the phrase "across subject areas" — expect science, social studies, and personal finance contexts, not only pure mathematics.
A repeatable modeling procedure
- Identify what varies. Which quantity changes? That becomes the independent variable.
- Name it with units. "Let t = number of hours" is a complete definition; "let t = time" is not, because the units are undetermined and a later answer of 2 could mean hours or minutes.
- Identify what stays fixed. Fixed amounts become constant terms.
- Identify per-unit rates. These become coefficients multiplying the variable.
- Assemble and check units. Every term in a sum must carry the same units.
- Test with a known value. Substitute a case you can verify by common sense.
A gym charges a $60 enrollment fee plus $28 per month. Let m = number of months. Fixed: 60. Rate: 28 per month. C(m) = 28m + 60 Check: after 3 months, C(3) = 84 + 60 = $144 — enrollment plus three payments. ✓
Mapping context structure to function family
+---------------------------------------------------------------------------+
| "increases by the same AMOUNT each period" -> LINEAR y = mx + b |
| "increases by the same PERCENT each period" -> EXPONENTIAL a(1+r)^x |
| "product of two changing dimensions" -> QUADRATIC |
| "total is fixed and shared among a group" -> INVERSE y = k/x |
| "within a tolerance of a target" -> ABSOLUTE VALUE |
| "at most / at least / no more than" -> INEQUALITY |
+---------------------------------------------------------------------------+
The wording distinction between amount and percent is the highest-value cue in the whole skill. "Loses $2,000 each year" is linear; "loses 12% of its value each year" is exponential. Items place both phrasings in adjacent answer choices.
Cross-subject contexts
Science — density. Density is mass per unit volume, D = m/V. A sample with mass 47.5 g and volume 5 cm³ has density 9.5 g/cm³. Rearranged forms m = DV and V = m/D are the same relationship solved for a different quantity, which is itself an assessed idea: one relationship, three equations.
Science — motion. d = rt, with r = d/t and t = d/r. A car traveling 55 mph for 3.5 hours covers 192.5 miles. Watch unit consistency: a rate in miles per hour paired with a time in minutes must have one of them converted first.
Science — temperature conversion. F = (9/5)C + 32 is linear, with slope 9/5 and intercept 32. Interpreting the slope — "each Celsius degree corresponds to 1.8 Fahrenheit degrees" — is a standard interpretation item.
Social studies — population and rates. Population growth at a constant percent is exponential; per-capita quantities are unit rates. A country with 4.2 million people and 630,000 students has a student rate of 0.15 students per person, or 15%.
Personal finance — simple versus compound. Simple interest I = Prt grows linearly; compound interest A = P(1 + r/n)^(nt) grows exponentially. Contrasting the two is a common item, and the structural difference is exactly the amount-versus-percent distinction above.
Multi-quantity models
When two unknowns vary together, a system results, as developed in section 5.4.
A field trip needs buses seating 48 and vans seating 12. Twelve vehicles carry 400 students. Let b = buses and v = vans. Then b + v = 12 and 48b + 12v = 400. Substituting v = 12 − b: 48b + 144 − 12b = 400 → 36b = 256 → b ≈ 7.1.
The non-integer result is itself informative: no combination of whole vehicles carries exactly 400 students, so the realistic answer is 8 buses and 4 vans, which seats 432. Recognizing that a model's exact solution may be contextually impossible — and adjusting rather than reporting 7.1 buses — is precisely the reasoning the skill targets.
Checking a model against its context
An algebraically correct answer can still be wrong for the situation. Run these checks:
- Sign. Can the quantity be negative? Lengths, counts, and elapsed times cannot.
- Domain. Should the variable be a whole number? You cannot buy 3.7 buses or seat 12.4 students.
- Magnitude. Is the size plausible? A monthly phone bill of $18,400 signals a units error.
- Units. Does the answer carry the units the question asked for?
A model predicts a plant's height as h = 3.5w + 12 centimeters after w weeks. What does it predict for w = 200? h = 712 cm, over 23 feet — implausible for most plants. The model was fit over a few weeks and extrapolating it to four years is invalid.
That last check connects directly to section 10.3: models are trustworthy inside the range where they were built and unreliable outside it. Items ask candidates to identify the limitation of a model, and "the model was extended beyond the data that supports it" is the expected reasoning.
A science teacher describes a bacteria population that starts at 500 and increases by 20% every hour. Which model is correct?
A model for a rental truck's cost is C(m) = 0.89m + 49.95, where m is miles driven. A student computes C(−20) = 32.15 and reports it as a valid cost. What is the best critique?
A social studies class compares two towns. Town A gains 400 residents each year; Town B grows 4% each year. Which pair of models is correct?