Algebra II

Units

1. 1

   Polynomials

   6 lessons

   1. Seeing structure: factoring higher-degree polynomials by patternFactor higher-degree polynomials completely by spotting chunks, choosing a pattern, and checking every factor.
   2. Polynomial arithmetic, closure and proving identitiesMultiply polynomials, prove an identity, generate Pythagorean triples, and explain closure.
   3. Polynomial division, the Remainder Theorem and the Factor TheoremDivide polynomials, find remainders by substitution, and decide when a linear expression is a factor.
   4. Zeros, multiplicity and end behavior: sketching polynomial graphsSketch a polynomial using its zeros, crossings or touches, end behavior, and y-intercept.
   5. The Binomial Theorem and Pascal's triangleUse Pascal's triangle to expand a binomial or find a specified term.
   6. Rewriting rational expressions: simplify, divide, add and multiplyRewrite rational expressions in four ways while keeping every original denominator restriction.
2. 2

   Rational and radical equations

   3 lessons

   1. Rational exponents and radicalsEvaluate rational powers, combine them using exponent laws, and simplify radicals.
   2. Solving radical equations and spotting extraneous solutionsSolve square-root equations and check every candidate in the original equation.
   3. Solving rational equationsSolve rational equations by clearing denominators, then reject answers that make an original denominator zero.
3. 3

   Complex numbers

   3 lessons

   1. The number i and arithmetic with complex numbersSimplify complex-number sums, products, powers of i, and quotients into a + bi form.
   2. Complex solutions of quadratic equationsSolve quadratics with non-real roots and connect those roots to graphs and factors.
   3. The Fundamental Theorem of Algebra and complex zeros of polynomialsFind and count complex zeros, identify real x-intercepts, and build real polynomials from zeros.
4. 4

   Systems of equations

   3 lessons

   1. Linear systems in two variables: graphing, substitution and eliminationSolve a pair of linear equations by graphing, substitution and elimination, and say when there is one solution, none or infinitely many.
   2. Linear systems in three variablesSolve three linear equations in three unknowns by eliminating one unknown twice, check the triple, and use the method to fit a parabola through three points.
   3. Systems with a line and a quadratic or a circleSolve a system made of a line and a parabola or a circle by substitution, give every solution as a point, and say how many real solutions a system can have.
5. 5

   Matrices

   2 lessons

   1. Matrices: adding, scaling and multiplyingName the size of a matrix, add and subtract matrices of one size, multiply by a scalar, and multiply two matrices when their inner sizes match.
   2. Determinants, inverses and solving systems with matricesFind the determinant and inverse of a 2 by 2 matrix, write a system as AX = B, solve it with the inverse, and explain what a determinant of zero means.
6. 6

   Inequalities

   3 lessons

   1. Linear, compound and absolute value inequalitiesSolve linear, compound and absolute value inequalities, graph each solution set on a number line, and write it in interval notation.
   2. Quadratic inequalitiesSolve a quadratic inequality by finding its zeros and the sign of the parabola between them, graph the solution on a number line and write it in interval notation.
   3. Systems of linear inequalities and linear programmingGraph a system of linear inequalities as a region, find its corner points, and maximise or minimise a linear quantity by comparing its value at the corners.
7. 7

   Exponential and logarithmic functions

   5 lessons

   1. Exponential growth and decay models, e and continuous growthChoose an exponential model, then calculate an amount from a given time.
   2. Logarithms as exponentsRewrite logarithms as powers, evaluate them, use log and ln, and identify invalid inputs.
   3. Properties of logarithmsExpand, condense, and evaluate logarithms using the product, quotient, power, and change-of-base properties.
   4. Solving exponential and logarithmic equationsSolve exponential and logarithmic equations, give exact log answers and calculator approximations, and check log inputs.
   5. Exponential and logarithmic functions as inverses: graphs and featuresGraph an exponential–logarithmic inverse pair, identify its features, and find an inverse after a shift.
8. 8

   Trigonometric functions

   5 lessons

   1. Radian measure: angles as arc length on the unit circleFind an angle from its arc, convert degrees and radians, and find arc length from an angle.
   2. The unit circle: trigonometric functions of any real numberFind sine and cosine for any real angle, and tangent where it is defined.
   3. Graphs of sine and cosine: amplitude, period and midlineGraph one cycle of a sine or cosine function, marking its midline, amplitude, period, and starting point.
   4. Modeling periodic phenomena with sinusoidal functionsBuild a sinusoidal model from a cycle’s heights and timing, then predict a value.
   5. The Pythagorean identity and finding trig valuesUse the unit-circle identity and a quadrant to find missing sine, cosine, and tangent values.
9. 9

   Building and modeling functions

   9 lessons

   1. Inverse functions: finding, graphing and restricting the domainDecide whether a function can be undone, find its inverse by swapping and solving, and restrict the domain when it cannot.
   2. Graphing square root, cube root, absolute value, piecewise and step functionsLocate familiar graph shapes, move them, and draw only the parts allowed by each input rule.
   3. Transformations across function familiesMove, stretch and reflect any parent function, track its key points and features, read a transformation from a formula, and test for even and odd symmetry.
   4. Key features and average rate of change across function familiesRead the key features of a function from its rule, graph or table, and compute and interpret its average rate of change.
   5. Solving f(x) = g(x) by graphs and tables across familiesSolve f(x) = g(x) for functions of different families by reading intersections, bracketing a sign change in a table, and checking each answer against its context.
   6. Rewriting functions to reveal features and comparing functions in different representationsRewrite functions to reveal rates, maxima, or zeros, then compare matching features across representations.
   7. Building functions by combining: sums, products, quotients and compositionBuild and evaluate function combinations, compare composition orders, and find allowed inputs.
   8. Modeling with nonlinear equations and rearranging formulasChoose the right family for a situation, build and solve its equation, rearrange formulas for any letter, and check every solution against the story.
   9. Arithmetic and geometric sequences: recursive and explicit rules, and finite geometric seriesWrite recursive and explicit rules, find a distant term, and sum a finite geometric series.
10. 10

    Variation

    2 lessons

    1. Direct and inverse variationRecognise direct variation y = kx and inverse variation y = k/x from words, tables and graphs, find the constant k from one case, and use it to predict.
    2. Joint and combined variationModel a quantity that varies with several others, such as y = kxz or y = kx/z², find k from one case, and predict how the quantity changes when the others change.
11. 11

    Conic sections

    4 lessons

    1. Parabolas: focus, directrix and vertex formDescribe a parabola as the points equally far from a focus and a directrix, find its focus and directrix from an equation x² = 4py, and write the equation in vertex form.
    2. Circles: centre-radius equations by completing the squareWrite and read the equation (x − h)² + (y − k)² = r² of a circle, complete the square to find the centre and radius from an expanded equation, and test whether a point is inside, on or outside.
    3. Ellipses: foci, axes and equationsRead the vertices, co-vertices, foci and centre of an ellipse from its equation, use c² = a² − b², and use the definition that the distances to the two foci add to a constant.
    4. Hyperbolas: vertices, foci and asymptotesRead the centre, vertices, foci and asymptotes of a hyperbola from its equation, use c² = a² + b², and tell the four conics apart from their equations.
12. 12

    Modeling with data

    2 lessons

    1. Regression lines, correlation and residualsFit a least-squares line with technology, read its slope and intercept in context, use r, r² and a residual plot to judge the fit, and predict without over-reaching.
    2. Choosing and fitting quadratic and exponential models to dataDecide from differences and ratios whether data are linear, quadratic or exponential, fit that model with technology, and use it, and its limits, to predict.
13. 13

    Statistics and inference

    6 lessons

    1. Statistics as inference: random samples, sampling variability and testing a model by simulationUse simulated samples to judge whether an observed result is unusual under a proposed model.
    2. Sample surveys, experiments and observational studiesClassify a study and explain whether it supports a population estimate, a causal claim, or neither.
    3. The normal distribution: fitting data, the empirical rule and z-scoresFit a normal model to bell-shaped data, use the 68-95-99.7 rule, compute z-scores, find areas and percentiles with technology, and check whether a normal model is appropriate.
    4. Estimating a population proportion or mean with a margin of errorCalculate and interpret approximate margin-of-error intervals for a population proportion or mean.
    5. Comparing two treatments with a randomization testCompare an observed treatment difference with shuffled differences and draw a justified conclusion.
    6. Evaluating reports based on dataEvaluate a report by checking its design, the claim it supports, and its display.