Common Mistakes

Frequent Errors

1. Defaulting to linear when rates of change are not roughly constant. 2. Choosing quadratic without symmetry / unique extremum / 2nd-difference support. 3. Using degree $n$ for $n$ points instead of degree at most $n-1$ (for $n$ points, degree ≤ $n-1$). 4. Ignoring piecewise when the story has a clear switch. 5. Writing overlapping domain pieces. 6. Skipping assumptions (“the model just is”). 7. Omitting domain limits from context (negative time, negative length). 8. Omitting range limits (fractional people, unrounded currency when required).

Why These Happen

Students rush to formulas (Topic 1.14) and treat selection as obvious. Finite-difference and “$n+1$ points” rules are easy to invert. Assumptions feel non-mathematical.

How to Fix Them

• Always answer: type? why (CED)? assumption? restriction?
• Count points → max degree = points − 1.
• Sketch interval boundaries for piecewise before writing formulas.
• Keep a bank of assumption sentence starters.

Avoidance Scripts

• “Constant first differences → linear; constant second → quadratic.”
• “$n+1$ distinct inputs → polynomial degree ≤ $n$.”
• “Different behaviors → piecewise on nonoverlapping intervals.”
• “Say what must stay true and where the model stops.”

Quick Self-Check

• Did I name a type with a CED clue?
• Is my degree bound correct for the number of points?
• Did I state assumption + domain/range restriction?