Rates of Change in Linear and Quadratic Functions
▶︎ Watch it animatedinteractive step-through · ~3 min · optionalLines and parabolas are identified by their rate signatures. A linear function changes at one constant rate, so a table with equally spaced inputs shows constant first differences. A quadratic function’s rate drifts at a steady pace, so its first differences change but its second differences are constant. Two rows of subtraction in the margin of a table settle the model.
The mistakes live one layer off: reading any steady-looking pattern as linear, expecting a quadratic’s first differences to be constant, or letting a function’s rising values stand in for its rate. The lesson drills the differences procedure and the rate language for concavity until the signatures are automatic.
The work
3 ways in · any order
Lesson
Rates of Change in Linear and Quadratic Functions
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Constant first differences make a line; constant second differences make a parabola. The lesson turns tables into rate stories, walks the thrown-ball classic, then closes with ten scenarios reading difference patterns, rates, and concavity without mixing the layers.
Diagnostic
10-item topic check
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Ten items spanning the Topic 1.3 misconceptions: difference-pattern misreads, average rates treated as pointwise, and rate layers collapsed into value layers. Results route you to the drills that fix what fired.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears it for now and moves you to the next. Two in a row is a checkpoint, not proof: if the error resurfaces later, the misconception comes back.