Proportional Relationships
Two quantities can hold a constant ratio (direct), a constant product (inverse), or neither, when a flat fee rides along. The arithmetic is one constant and one multiplication. The points leak when the two kinds trade places and the answer moves the wrong way, when a total hiding a fee gets scaled, and when the constant itself gets reported instead of the value it produces.
These patterns aren't really about whether you can divide two numbers. They're about whether you asked which way the second quantity moves before computing, whether you tested the origin before scaling a total, and whether you applied the constant at the asked input instead of handing the constant in.
The work
4 ways in · any order
Lesson
Proportional Relationships
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Tell direct from inverse by how the quantities move, find the constant and apply it at the asked input, and refuse to scale any total that hides a flat fee. The lesson works the method and the three patterns that derail it, and it saves the trap for last: the constant of proportionality reported as the answer.
Diagnostic
10-item topic check
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Ten items across the three patterns: treating an inverse relationship as direct, assuming proportionality where a flat fee breaks it, and reporting the constant instead of the value asked. A mix of symbolic pairs, crew-and-time problems, tables, and fee-plus-rate setups. Take it cold to surface the ones still catching you, or after the lesson to confirm they are gone.
Grid-in Check
Student-produced response
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About a quarter of SAT math answers are typed, not chosen, with no options to react to. These grid-in items diagnose by the value you enter, then route into the same drills the multiple-choice check feeds.
Targeted Practice
Drill a single pattern
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Pick one of the failure modes you've missed and grind it on its own. The round is adaptive: two correct in a row clears the pattern and you move on.