Grade 11 · printable course

Algebra II: Functions, Structure, and Modeling

Grade 11 · 509 pages · 74 lessons · PDF, print at home

A full-year Algebra II course for grade 11 that extends the linear and quadratic function work of Algebra I into polynomial, rational, radical, exponential, logarithmic, and trigonometric functions, with a statistics unit grounding inference in the algebra of distributions. The course treats "function" as the unifying object of study: every unit asks how a family of functions is built, what its graph and equation reveal about each other, and where it models a real quantitative situation. Algebraic structure (factoring, completing the square, rewriting expressions) is taught as a tool for seeing hidden structure, not as isolated procedure. Aligned to the CCSS conceptual categories for high school mathematics.

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Every unit and lesson in this book, in order. The same lessons are taught online at lettersanddevotion.com, so a family can move between paper and screen.

Algebra II: Functions, Structure, and Modeling

74 lessons · try a lesson online free

Unit 1: Quadratic Functions Revisited: Structure and Complex Numbers10 lessons
  1. 1.Completing the square as a rewriting procedure on x^2+bx+c
  2. 2.The discriminant as a predictor of root type
  3. 3.The derivation of the quadratic formula from completing the square
  4. 4.Representation choice among standard, factored, and vertex form of a quadratic
  5. 5.Arithmetic of complex numbers in a+bi form
  6. 6.Complex numbers as a closure extension of the real number system
  7. 7.Quadratics disguised in non-standard form (substitution-reducible or literal-coefficient equations)
  8. 8.Reconstructing multiple equivalent forms of a quadratic from a graph, including the ambiguity of scale
  9. 9.The definition of the imaginary unit i
  10. 10.The three algebraic forms of a quadratic and their visible features
Unit 2: Polynomial Functions: Zeros, Behavior, and the Structure of Degree n11 lessons
  1. 11.Multiplicity of a zero as the exponent on its linear factor
  2. 12.Polynomial long division algorithm
  3. 13.The logical equivalence between a zero of p and (x-a) being a factor (Remainder/Factor Theorem)
  4. 14.The Rational Root Theorem as a search-space-narrowing tool
  5. 15.Factoring by grouping for higher-degree polynomials
  6. 16.The graphical signature of even vs. odd multiplicity
  7. 17.End behavior as a function of degree parity and leading coefficient sign
  8. 18.A polynomial's graph as the joint product of zeros-with-multiplicity and end behavior
  9. 19.Reverse-construction of a polynomial equation from graphical/verbal constraints
  10. 20.Expansion as a check on a factored-form construction
  11. 21.Factored form vs. standard form as different windows onto the same polynomial
Unit 3: Rational Expressions and Functions: When Division Doesn't Finish9 lessons
  1. 22.Domain restrictions of a rational expression, found before simplifying
  2. 23.The distinction between a hole (removable discontinuity) and a vertical asymptote in a rational function
  3. 24.Multiplication and division of rational expressions
  4. 25.Addition and subtraction of rational expressions with unlike denominators
  5. 26.The relationship between algebraic equivalence of two rational expressions and their domains
  6. 27.Solving rational equations and identifying extraneous solutions
  7. 28.Horizontal asymptote behavior as a ratio of leading-term end behaviors
  8. 29.Structural justification of hole versus asymptote using factored form, independent of graphical appearance
  9. 30.Construction of a rational function from specified structural features
Unit 4: Radical Functions, Rational Exponents, and Inverse Functions10 lessons
  1. 31.The definition of a rational exponent as an extension of integer exponent rules
  2. 32.Simplification of radical/rational-exponent expressions and denominator rationalization
  3. 33.Radical equations and the extraneous-solution check
  4. 34.The non-reversibility of squaring as the mechanism producing extraneous solutions
  5. 35.The horizontal line test and the formal definition of a one-to-one function
  6. 36.The structural distinction between one-to-one and many-to-one functions as it determines invertibility
  7. 37.The algebraic procedure for finding and verifying an inverse function by composition
  8. 38.Domain restriction as the method for creating an invertible piece of a non-one-to-one function
  9. 39.Invertibility determined from algebraic structure alone, generalized beyond graphed function families
  10. 40.Discriminant interpretation and rational-equation extraneous solutions, as long-retention targets
Unit 5: Exponential and Logarithmic Functions: Two Kinds of Change8 lessons
  1. 41.The distinguishing tests for linear, quadratic, and exponential tables (constant difference, constant second difference, constant ratio)
  2. 42.Exponential growth/decay models built from a stated percent rate
  3. 43.The logarithm as the inverse function of an exponential
  4. 44.Logarithm properties (product, quotient, power) as restatements of exponent rules
  5. 45.Exponential equations solved via logarithms
  6. 46.Logarithmic equations and their domain restrictions
  7. 47.Model selection and construction among linear, quadratic, and exponential families for an unfamiliar context
  8. 48.The long-run divergence of linear versus exponential processes despite short-run similarity
Unit 6: Trigonometric Functions: Modeling Periodic Behavior8 lessons
  1. 49.Radian measure as arc length on the unit circle
  2. 50.Unit circle definition of sine and cosine as coordinates
  3. 51.Periodicity as a structural property, contrasted against monotonic exponential behavior
  4. 52.Amplitude, midline, and period as vertical stretch, vertical shift, and horizontal stretch of sine/cosine
  5. 53.Fitting a sine/cosine model to features of periodic data
  6. 54.Extrapolation from a periodic model versus a monotonic model
  7. 55.The Pythagorean identity derived from the unit circle
  8. 56.Recognizing and modeling an unfamiliar periodic context using the unit-circle-derived sine/cosine schema
Unit 7: Sequences and Series: Discrete Growth as a Special Case8 lessons
  1. 57.The structural distinction between arithmetic (constant additive difference) and geometric (constant multiplicative ratio) sequences
  2. 58.The explicit formula for arithmetic and geometric sequences
  3. 59.The correspondence between sequence formulas and their parent continuous functions (linear and exponential)
  4. 60.The conversion between recursive and explicit representations of arithmetic and geometric sequences
  5. 61.The finite geometric series sum formula and the multiply-and-subtract structure that produces it
  6. 62.Arithmetic vs. geometric sequence models applied to unfamiliar real-world contexts
  7. 63.Convergence of an infinite geometric series when |r| < 1 and the sum formula S = a/(1-r)
  8. 64.Discrimination between arithmetic and geometric sequence structure under confusable surface conditions
Unit 8: Statistics: Modeling and Trusting Data10 lessons
  1. 65.The empirical (68-95-99.7) rule for normal distributions
  2. 66.Appropriateness conditions for the normal-distribution model
  3. 67.The constant-difference/constant-ratio diagnostic for choosing a regression family
  4. 68.Residuals and residual plots as evidence of model fit
  5. 69.Function-family selection and extrapolation risk for a novel data set
  6. 70.Sampling method and its relationship to bias
  7. 71.The distinction between correlation and causation
  8. 72.Conditional probability from two-way tables
  9. 73.Evaluation criteria for reports based on data
  10. 74.Structural discrimination between quadratic and exponential fit given similar-looking scatter data

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