Ratios and Rates
▶︎ Watch it animatedinteractive step-through · ~3 min · optionalA ratio hands you two or three small numbers, and everything turns on what each one counts. The arithmetic is short: size one part, then scale. The points leak in the reading, a part treated as the whole or a part-to-whole ratio split into two parts, in the setup, a rate or proportion built upside down, and at the end, the right split reported as the wrong quantity.
These patterns aren't really about whether you can divide and multiply. They're about whether you decided what each ratio number counts before computing, whether the units in your rate cancel toward the answer, and whether the number you reported carries the exact name the question asked for.
The work
4 ways in · any order
Lesson
Ratios and Rates
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Read what each ratio number counts, split a whole by sizing one part, and keep rates and proportions right-side up by their units. The lesson works the method and the three patterns that derail it, and it saves the trap for last: the right split reported as the wrong quantity.
Diagnostic
10-item topic check
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Ten items across the three patterns: treating a part as a whole, setting a rate or proportion upside down, and reporting the wrong quantity of the ratio. A mix of splits, scaling, unit rates, and one ratio-change item. 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.