Task Description: This task asks students to recognize geometric patterns, visualize and extend the pattern, generate a non-linear sequence, develop and algebraic generalization that models the growth of a quadratic function and verify the inverse relationship of the quadratic relationship. The Aussie Fir Tree task is a culminating task for a 2-3 week unit on algebra that uses the investigation of growing patterns as a vehicle to teach students to visualize, identify and describe real world mathematical relationships. Students who demonstrate mastery of the unit are able to solve the Aussie Fir Tree task in one class period.
1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
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4. Solve quadratic equations in one variable.
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b. Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
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3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
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1. Write a function that describes a relationship between two quantities ?
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a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
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Make sense of problems and persevere in solving them.
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Construct viable arguments and critique the reasoning of others.
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Model with mathematics.
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Attend to precision.
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Look for and make use of structure.
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