A sufficient condition related to mistaken intuition about adjusted sums-of-squares in linear regression

We consider a misconception common among students of statistics involving "adjusted" and "unadjusted" sums-of-squares. While the presence of misconception has been noted before (e.g. Hamilton (1986)), we argue that it may be related to the language we use in describing the meaning of sums-of-squares. For linear regression with two independent variables, we then present a sufficient condition for SSR( X1 | X2 ) > SSR( X1 ) in terms of the signs of the sample correlations between pairs of predictor and response variables, and note how this sufficient condition may also be related to misconceptions held by some students of statistics.