Common Mistakes
Frequent Errors
1. Stopping after naming the type (1.13) without writing $f(x)=\ldots$. 2. Sign errors in vertex form $a(x-h)^2+k$. 3. Wrong piece of a piecewise model. 4. Boundary values assigned to both pieces inconsistently. 5. Regression equation copied without answering the context question. 6. Inverse model with wrong power ($1/d$ vs $1/d^2$). 7. Extrapolating far beyond data as if certainty were unchanged. 8. Forgetting units when interpreting slope or $k$.
Why These Happen
Construction feels like “the hard algebra,” so students skip the check step. Piecewise and inverse models are less drilled than lines and parabolas.
How to Fix Them
- Always: givens → equation → plug-back check → apply.
- Highlight active piecewise interval in the question.
- Say the proportion out loud: “inversely proportional to distance squared.”
- Separate “build” points from “predict” points.
Avoidance Scripts
- “If I can’t check a given point, I’m not done.”
- “Which interval? Then which formula?”
- “Inverse to $d^2$ means $k/d^2$, not $k/d$.”
Quick Self-Check
- Does my equation satisfy the given conditions?
- Did I apply the correct piece?
- Did I answer the contextual question, not only write $y=ax+b$?