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AP Precalculus centers on functions modeling dynamic phenomena. Research indicates that deep understanding of functions and their graphs as embodying dynamic covariation of quantities best supports student preparation for calculus.
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Learning Progression 1: Change in Tandem
Changes in tandem sets the stage for covariation, which is integrated throughout the course. Given a scenario describing covariation, students produce graphical representations. Students begin to understand a graph not as something static but something embodying change. This unit is based on research on best practices in preparation for calculus (see Marilyn Carlson et. al.).
Learning Intention 1.1.A: I am learning to describe how the input and output values of a function vary together by comparing function values.
Learning Intention 1.1.A: I am learning to describe how the input and output values of a function vary together by comparing function values.
Success Criteria:
Learning Intention 1.1B: I am learning to construct a graph representing two quantities that vary with respect to each other in a contextual scenario.
Learning Intention 1.1B: I am learning to construct a graph representing two quantities that vary with respect to each other in a contextual scenario.
Success Criteria:
Learning Progression 2: Rates of Change
Rates of change, rates of change in linear & quadratic functions, and polynomial functions & rates of change.
Include changing rates of change and connection to concavity.
Learning Intention 1.2.A: I am learning to compare the rates of change at two points using average rates of change near the points.
Learning Intention 1.2.A: I am learning to compare the rates of change at two points using average rates of change near the points.
Success Criteria:
Learning Intention 1.2.B I am learning to describe how two quantities vary together at different points and over different intervals of a function.
Learning Intention 1.2.B I am learning to describe how two quantities vary together at different points and over different intervals of a function.
Success Criteria:
Learning Intention 1.3.A: I am learning to determine the average rates of change for linear and quadratic sequences and functions.
Learning Intention 1.3.A: I am learning to determine the average rates of change for linear and quadratic sequences and functions.
Success Criteria:
Learning Intention 1.3.B: I am learning to determine the change of average rates of change for linear and quadratic functions.
Learning Intention 1.3.B: I am learning to determine the change of average rates of change for linear and quadratic functions.
Success Criteria:
Learning Intention 1.4.A: I am learning to identify key characteristics of polynomial functions related to rates of change.
Learning Intention 1.4.A: I am learning to identify key characteristics of polynomial functions related to rates of change.
Success Criteria:
Learning Progression 3: Polynomial Functions and Complex Zeros
Include non-real roots. Examine complex numbers as zeroes of polynomials and interpretation them.
This progression is not intended to review all the factoring techniques of Algebra .
Learning Intention 1.5.A: I am learning to identify key characteristics of a polynomial function related to its zeros when suitable factorizations are available or with technology.
Learning Intention 1.5.A: I am learning to identify key characteristics of a polynomial function related to its zeros when suitable factorizations are available or with technology.
Success Criteria:
Success Criteria:
Learning Intention 1.5.A and/or 1.5.B: Optional Day - The goal is to complete 1.5 in 2-3 days.
Learning Intention 1.5.A and/or 1.5.B: Optional Day - The goal is to complete 1.5 in 2-3 days.
Success Criteria: See 1.5.a and/or 1.5.B.
Learning Progression 4: Polynomial & Rational Functions
Limits are not formally defined (saved for Calculus). This is an informal introduction to limits, unbounded behavior and end behavior with limit notation to summarize. The idea of point discontinuity without formalization required to define continuous functions.
Success Criteria:
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Success Criteria:
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Learning Intention 1.7.A-1.10.A: Optional Day - The goal is to complete 1.7-1.10 in 4-7 days.
Learning Intention 1.7.A-1.10.A: Optional Day - The goal is to complete 1.7-1.10 in 4-7 days.
Success Criteria: See 1.7.a, 1.8.A, 1.9.A and/or 1.10.A.
Learning Intention 1.7.A-1.10.A: Optional Day - The goal is to complete 1.7-1.10 in 4-7 days.
Learning Intention 1.7.A-1.10.A: Optional Day - The goal is to complete 1.7-1.10 in 4-7 days.
Success Criteria: See 1.7.a, 1.8.A, 1.9.A and/or 1.10.A.
Learning Intention 1.7.A-1.10.A: Optional Day - The goal is to complete 1.7-1.10 in 4-7 days.
Learning Intention 1.7.A-1.10.A: Optional Day - The goal is to complete 1.7-1.10 in 4-7 days.
Success Criteria: See 1.7.a, 1.8.A, 1.9.A and/or 1.10.A.
Learning Progression 5: Equivalent Representations of Polynomial and Rational Expressions
This progression includes polynomial long division (synthetic is not included in the Framework nor is it assessed).
This progression also includes the Binomial Theorem .
Learning Intention 1.11.A: I am learning to rewrite polynomial and rational expressions in equivalent forms.
Learning Intention 1.11.A: I am learning to rewrite polynomial and rational expressions in equivalent forms.
Success Criteria:
Learning Intention 1.11.B: I am learning to determine the quotient of two polynomials using long division.
Learning Intention 1.11.B: I am learning to determine the quotient of two polynomials using long division.
Success Criteria:
Learning Intention 1.11.C: I am learning to rewrite the repeated product of binomials using the binomial theorem.
Learning Intention 1.11.C: I am learning to rewrite the repeated product of binomials using the binomial theorem.
Success Criteria:
Learning Intention 1.11: Optional Day - The goal is to complete 1.11 in 3-4 days.
Learning Intention 1.11: Optional Day - The goal is to complete 1.11 in 3-4 days.
Success Criteria: See 1.11.A, 1.11.B and/or 1.11.C.
Learning Progression 6: Transformations of Functions
This progression includes translations (additive) & dilations (multiplicative).
Learning Intention 1.12.A: I am learning to construct a function that is an additive and/or multiplicative transformation of another function.
Learning Intention 1.12.A: I am learning to construct a function that is an additive and/or multiplicative transformation of another function.
Success Criteria:
Success Criteria:
Success Criteria:
Learning Progression 7: Modeling
Three types: deterministic, matching criteria, bivariate data. Includes piecewise linear.
Describe assumptions and contextual restrictions for a given model. Apply solutions in a context.
Learning Intention 1.13.A: I am learning to identify an appropriate function type to construct a function model for a given scenario.
Learning Intention 1.13.A: I am learning to identify an appropriate function type to construct a function model for a given scenario.
Success Criteria:
Learning Intention 1.13.B: I am learning to describe assumptions and restrictions related to building a function model.
Learning Intention 1.13.B: I am learning to describe assumptions and restrictions related to building a function model.
Success Criteria:
Learning Intention 1.13.A and/or 1.13.B: Optional Day - The goal is to complete 1.13 in 2-3 days.
Learning Intention 1.13.A and/or 1.13.B: Optional Day - The goal is to complete 1.13 in 2-3 days.
Success Criteria: See 1.13.A and/or 1.13.B.
Learning Intention 1.14.A: I am learning to construct a linear, polynomial, or related piecewise defined function model.
Learning Intention 1.14.A: I am learning to construct a linear, polynomial, or related piecewise defined function model.
Success Criteria:
Success Criteria:
Learning Intention 1.14.C: I am learning to apply a function model to answer questions about a data set or contextual scenario.
Learning Intention 1.14.C: I am learning to apply a function model to answer questions about a data set or contextual scenario.
Success Criteria:
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and embedded common formative assessments.
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