Reading a Modeling Problem

In this lesson you will: Name what changes, what it changes with respect to, and the units of each; Pull a starting value and a rate of change out of a sentence or out of two data points; Decide which inputs the model is allowed to take, and what a sensible answer looks like.

Three questions, in this order

A modeling problem is a paragraph pretending to be an equation. Before you write anything algebraic, answer three questions about the paragraph. They take about twenty seconds and they decide everything that follows.

Students who skip them end up with a correct-looking equation that describes a different situation — and no amount of careful algebra afterward repairs that.

A variable without a unit is half a variable

Write the definition line. t is the number of hours of video played; B is the battery reading in percent. It looks like busywork until the moment you have two quantities in play and cannot remember which one you called x.

A phone battery reading 86 percent that loses about 7 percentage points an hour gives B(t)=86-7t. The 86 is where the graph starts; the -7 is how steeply it falls.

Two data points are enough

Often the problem hands you no starting value at all — only two readings. Subtract to get the change in output, subtract to get the change in input, divide. That is the rate.

Then walk back to the input value 0. This is the step that gets skipped, and it is the step that ruins the model.

Common mistake — Treating the first number you meet as the starting value

A tank holds 52 gallons after 3 hours of draining and 36 gallons after 7 hours. A student finds the rate correctly — (36-52)/(7-3)=-4 gallons an hour — and then writes V(t)=52-4t.

The pull is strong. 52 is the first number in the sentence, it is a volume, and every model the student has ever written started with a number just like it.

But 52 is the volume at t=3, not at t=0. Three hours earlier the tank held more: 52+3(4)=64 gallons. The model is V(t)=64-4t.

The wrong version is off by 12 gallons at every single input, and it will still look tidy, still graph as a line, and still produce a confident answer.

How to catch it: Feed the model an input the problem already answered. Here t=3 should return 52. The wrong model gives 52-12=40; the right one gives 64-12=52. One substitution, ten seconds.