Problem Solving & Dimensional Analysis

1. What the "show your work" space reveals

This test has no answer key, by design. The diagnostic value is not in the letter a student circled — it is in the reasoning written on the Show your work / why lines beneath each question.

• A correct guess and a correct deduction look identical on the answer line but completely different in the work space. If the work is blank or doesn't match the method, treat the item as a gap even when the letter is right.
• Dimensional-analysis errors are visible here: look for whether the student wrote the fraction chain so units cancel, or just multiplied numbers. The setup, not the final number, is what you are grading.
• Watch for the direction of an operation — multiplying when they should divide (or vice versa) shows up as a reasonableness failure (e.g., 5.5 lb becoming 250 g). The written work is where you catch it.
• Ask students to always jot why they chose their method, not just the arithmetic. That one sentence tells you whether the concept is secure.

2. Running the session

• Suggested pacing: Stimulus 1 + Q1–6, break; Stimulus 2 + Q7–12, break; Stimulus 3 + Q13–18; Stimulus 4 + Q19–24; then the constructed response (which can be homework).
• Before Q1, walk through the fuel-economy table together: cover the Range column and ask students to predict which vehicle goes farthest, then reveal — this builds table-reading confidence.
• For Stimulus 3, model one full fraction chain on the board (the 3 ft → inches example) before students begin, so the "units cancel" idea is concrete.
• A calculator is allowed. Time is a guideline, not a limit — accuracy and clear reasoning come first; speed follows.
• This is a college-level course: hold students to the expectation that answers require written explanation, not just a number.

3. Using the nudges

Each question carries a Nudge — a printed hint about how to attack the problem (name the method, set up the fraction chain, find the scale factor first). A nudge never states or implies the answer; it hands over a first step so a stuck student gets unstuck by reasoning. Three ways to run them:

• Open (first exposure): student reads the nudge freely whenever stuck. Best for a first pass through unfamiliar material.
• Delayed (default): student attempts the question first, then reads the nudge only if they cannot start. Ask them to tick which questions needed a nudge — that tick list is your diagnostic.
• Closed (exam simulation): fold or cover the nudges, then review them afterwards while going over the work. Use this once the skills are solid.

Over time the nudges should become the student's own internal checklist, so they can be withdrawn.

4. Reviewing the work — without an answer key

There is no key on purpose. The goal is not to produce a score — it is to understand how the student is thinking. Rework each miss together, checking the method:

1. Ask first, tell second. Before settling on the right answer, ask "How did you approach this?" or "What did you try first?"
2. Rework each miss together. Return to the stimulus, table, or given formula and rebuild the solution step by step with the student — don't just mark it. Verify the answer by recomputing together rather than reading it off a key.
3. Correct answers still need explanation. If the student can't say why, treat it as a gap — a right answer for the wrong reason is a future wrong answer.
4. Name the specific error. Not "review unit conversion," but "you multiplied when you needed to divide, because the unit you wanted to remove should go in the denominator." Dimensional-analysis errors almost always come from one of two places: not knowing which direction to convert, or not setting up the fraction chain so units cancel. Identify which it was.
5. When a pattern appears across several questions, change the approach. Re-explaining the same way rarely works — try a physical analogy ("units cancel like fractions"), a sketch, or a simpler version of the problem first.
6. Always ask students to say what their numerical answer means in plain English — MA 321 emphasizes interpretation as much as computation.

5. Constructed response — scoring guidance

Score the road-trip response holistically. A response need not meet every bullet at a level to earn that score. Give credit for correct reasoning even when arithmetic errors are present; Parts B and D depend on Part A, so trace an early error through and credit sound reasoning relative to the student's own Part A value. Part D's plain-English interpretation is a graded component, not optional. For Part C, require units labeled at each step of the chain.

Common part-specific errors to probe: Part A — dividing tank size by MPG instead of total miles by MPG ("Are we finding gallons for the whole trip, or miles per gallon?"). Part B — forgetting to double for the round trip. Part C — stopping at a decimal number of hours without converting to minutes, or dividing distance by 60 (confusing speed with time). Part D — applying the 15% increase to the total bill rather than to the original gas price; require the student to show the new per-gallon price before multiplying by gallons.