TL;DR:
A free-response answer is read as evidence of reasoning. A correct number with no setup can lose the credit that the steps would have earned.
Define your variables with units, write the relationship, differentiate, substitute, then state the answer in a full sentence with units.
Check the official scoring guidelines for the current year to see exactly what each part is worth, since rubrics change.
Many students lose points on free-response questions without making an arithmetic error. They write a few lines, jump to a number, and box it. A grader can't see how the number was reached, so the reasoning earns nothing. The fix is to write the answer as a short, readable argument where each line follows from the one before.
What graders look for
Graders look for the reasoning that a correct solution must contain. For a related rates problem that means defined variables, a relationship between them, a derivative with respect to time, a substitution of the given values, and a final answer with units. Your job is to show each of these clearly, in order, so that someone can follow your work without guessing.
Read the scoring guidelines for the current exam year before you practice. They describe what counts as a complete justification for each part of a question, and they are the best guide to how much detail is enough.
Worked example: a related rates free-response question
Here is a question written for this article in the style of a related rates problem. A spherical balloon is being inflated. Its radius is increasing at 0.2 centimeters per second. At the moment when the radius is 5 centimeters, how fast is the volume increasing?
Step 1: define the variables with units
Let r be the radius of the balloon, in centimeters, at time t, in seconds. Let V be the volume, in cubic centimeters. The question gives dr/dt = 0.2 cm/s, and asks for dV/dt when r = 5 cm.
Step 2: write the relationship
The volume of a sphere is V = (4/3)πr³. This equation holds at every moment, so you can differentiate both sides with respect to t.
Step 3: differentiate with the chain rule
Differentiate each side with respect to t. The left side gives dV/dt. On the right, the chain rule gives (4/3)π · 3r² · dr/dt, which simplifies to 4πr² · dr/dt.
Step 4: substitute the values
Substitute r = 5 and dr/dt = 0.2 into dV/dt = 4πr² · dr/dt.
dV/dt = 4π(5)²(0.2)
= 4π(25)(0.2)
= 20π
≈ 62.8 cm³/sCheck the arithmetic: 4 × 25 = 100, and 100 × 0.2 = 20, so the result is 20π. Since π is about 3.14, 20π is about 62.8. The units check too: centimeters squared times centimeters per second gives cubic centimeters per second, which is the unit of a rate of change of volume.
Step 5: state the answer in a sentence
A complete final statement reads: when the radius is 5 cm, the volume of the balloon is increasing at 20π ≈ 62.8 cubic centimeters per second. Writing it as a sentence makes the meaning of your number clear, even to a grader who skims.
Why the setup earns points even when the arithmetic slips
Suppose you forget to square the radius and write 4π(5)(0.2) = 4π. The arithmetic is wrong, but the defined variables, the volume formula, the chain rule step and the substitution are all correct. A grader can still give credit for most of the reasoning, which is why the setup lines matter so much. A bare final answer with an error gives them nothing to credit.
Justification is more than a final number
Some parts ask you to justify a conclusion, such as why a function is increasing or why a value must exist. For those, cite the theorem or the test you used and show that its conditions hold. Write what you checked, such as continuity on the interval or the sign of the derivative, not just the word because.
Tip: If a question says to justify your answer, the answer alone is not enough. Name the reasoning tool, state the condition it needs, and show that the condition is met.
Mistakes to avoid
Writing numbers without defining what they stand for. Always say which quantity each symbol is.
Substituting values before you differentiate. Differentiate the general relationship first, then plug in.
Dropping units or writing the wrong ones. A rate of volume change must have cubic units per time.
Leaving the answer as a fragment, such as 20π alone. Finish with a sentence that names the quantity and the moment.
Your plan for this week
Pick three past free-response questions: Use the official released questions and a current set of scoring guidelines.
Write the setup before any calculation: Define variables with units, write the relationship and the derivative, and only then substitute.
Compare with the guidelines: Mark where your work matches each line of the rubric, and where you left something out, such as a unit or a justification.
Rewrite the weakest answer: Write it again from scratch with the missing parts, then check the arithmetic a second time.
After a few rounds, the setup becomes a habit. You will find that the points you were losing were in the words around the numbers, not in the numbers.
Frequently asked questions
Do I need to write every algebra step?
Show enough steps that a reader can follow each line. Skipping a step that a grader would expect can cost credit, so check the scoring guidelines for what they reward.
Can I use a calculator for the final value?
Check the current exam rules for calculator use on each part. Writing the exact form, such as 20π, and then the decimal approximation is a safe way to show the value.
What if I can't finish a problem?
Write the setup you know is right. Partial credit is awarded for correct reasoning, so a clear setup is worth more than a guess presented as a final answer.
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