Integrated Math III

Units

1. 1

   Inferences and conclusions from data

   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.
2. 2

   Polynomial, rational and radical relationships

   11 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.
   7. Solving radical equations and spotting extraneous solutionsSolve square-root equations and check every candidate in the original equation.
   8. Solving rational equationsSolve rational equations by clearing denominators, then reject answers that make an original denominator zero.
   9. The Fundamental Theorem of Algebra and complex zeros of polynomialsFind and count complex zeros, identify real x-intercepts, and build real polynomials from zeros.
   10. 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.
   11. 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.
3. 3

   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.
4. 4

   Trigonometry of general triangles

   1 lesson

   1. Trigonometry in general triangles: area formula, Law of Sines, Law of CosinesFind a triangle’s area and missing parts; choose a law and check SSA for multiple triangles.
5. 5

   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.
6. 6

   Mathematical modeling

   17 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. Cross-sections of solids and solids of revolutionIdentify cube cross-sections and predict the solids and volumes made by rotating flat regions about specified axes.
   11. Modeling real objects with geometric shapesEstimate a tree trunk’s volume from tape measurements and judge whether taper matters.
   12. Density based on area and volumeCalculate and compare amounts per unit area or volume, matching units before calculating.
   13. Design problems: satisfying constraints and minimising costKeep a design constraint fixed, compare feasible dimensions, and justify the best design.
   14. 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.
   15. 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.
   16. 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.
   17. 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.