Residuals and Goodness of Fit
I can make and read a residual plot, and decide from it whether a line is the right model.
The problem this sheet solves
A high correlation gets read as proof that a line is the right model. It is not: r measures closeness to a straight line, and gently curving data can hug a line closely enough to score 0.97 while being the wrong shape everywhere. The rule also runs against a year of training, because a GOOD residual plot is shapeless and students have been rewarded for spotting patterns since September. Fitting a line to the curved set gives residuals 7, 1, minus 3, minus 5, minus 5, minus 3, 1, 7 — a perfect symmetric U that can be read off the SIGNS before any picture is drawn. One item is worth slowing down for: both sets of residuals add to zero, including the wrong model's, because a least-squares line makes that happen every time. The sheet keeps size and shape apart throughout — shapeless but large means right model, poor predictor.
The key rule
A residual plot should show no pattern. A curve in it means the line is the wrong model, no matter how high the correlation is.
Look for shape, not size.
Watch out
The mistake: A student sees r = 0.976 and reports that a line fits this data well.
The fix: r only measures closeness to a straight line, and it can be high for data that bends. The residual plot is a symmetric U, so the line is the wrong model.
What is included
- Reference page — student-friendly definitions, a worked example, a Watch Out error pair, and the key rule
- Practice page — six problems, building from foundational to multi-step
- Apply page — a word problem, a reasoning prompt, and an error-analysis task
- Assess — a half-page exit ticket, two to a sheet, with a score box
- Full answer key — every item worked, plus a grading guide that says what each wrong answer tells you
- Ink-saver edition of all 7 pages
- 14-slide teacher deck (PowerPoint) with presenter notes on every slide
Standard codes refer to the Common Core State Standards for Mathematics. Math Class 678 is an independent publisher, not affiliated with, sponsored by, or endorsed by NGA Center or CCSSO. Every product is sold on Teachers Pay Teachers.