Mixtures and Weighted Averages
▶︎ Watch it animatedinteractive step-through · ~3 min · optionalWhen unequal groups combine, the bigger one pulls the average its way, and the only honest path runs through totals: each amount times its own average or concentration. The points leak in the weights, two averages met at their midpoint regardless of group size, in the bookkeeping, concentrations added or weights swapped between groups, and at the end, the wrong family member: the total volume reported when the pure amount was asked.
These patterns aren't really about whether you can compute an average. They're about whether every average got multiplied by its own group before combining, whether each percent stayed glued to its own batch, and whether the number you reported carries the exact name, the total, the part, or the percent, that the question used.
The work
4 ways in · any order
Lesson
Mixtures and Weighted Averages
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Combine averages through their totals, weight grades and blend prices by what the syllabus and the scale say, and track the pure substance through every mixture. The lesson works the method and the three patterns that derail it, and it saves the trap for last: the right mixture with the wrong member of its family reported.
Diagnostic
10-item topic check
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Ten items across the three patterns: averaging the averages without weights, building the mixture on the wrong quantity, and reporting a component when the mixture was asked. A mix of combined classes, weighted grades, blend prices, and dilution targets. 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.