Distinguish proportional relationships from linear but non-proportional relationships.
Sort mixed representations by proportional structure
Problem
Identify \(\text{every}~\text{representation}\) that is proportional.
A. Equation: \(y~=~4x\) B. Table: \(x~=~1,~2,~3,~4~\text{and}~y~=~5,~10,~15,~20\) C. Context: A gym charges a \(\$12~\text{sign}\text{-}\text{up}~\text{fee}\) and then \(\$8~\text{per}~\text{visit}\). D. Points: \((0,~0),~(2,~6),~(5,~15)\) E. Equation: \(y~=~3x~+~7\)
Apply one structural test across equations, tables, contexts, and point sets: proportional output is a constant multiple of input with a zero start. Look for no added term, a constant nonzero y-to-x ratio, no initial fee, or an origin point as appropriate. A table may imply the origin even when its displayed inputs begin above zero.
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