Secondary Mathematics III (Utah Core Standards)

Number and Quantity

• SIII.N.CN.8 (+) Extend polynomial identities to complex numbers, e.g. x² + 4 = (x + 2i)(x − 2i).
• SIII.N.CN.9 Know the Fundamental Theorem of Algebra: a degree-n polynomial with real coefficients has n complex roots.

Algebra

• SIII.A.SSE.1 Interpret polynomial and rational expressions in terms of their context.
  • a. Interpret terms, factors and coefficients in an expression.
  • b. Interpret complex expressions by treating parts as single quantities, e.g. P(1 + r)ⁿ.
• SIII.A.SSE.2 Use an expression's structure to find ways to rewrite it (e.g. x⁴ − y⁴ as a difference of squares).
• SIII.A.SSE.4 Understand and use formulas for the sums of series.
  • a. Derive and use the arithmetic series sum formula to solve problems.
  • b. Derive and use the geometric series sum formula (e.g. for mortgages and infinite sums).
• SIII.A.APR.1 Know polynomials are closed under addition, subtraction and multiplication like the integers; add, subtract and multiply them.
• SIII.A.APR.2 Use the Remainder Theorem: dividing p(x) by x − a leaves p(a), so p(a) = 0 exactly when x − a is a factor.
• SIII.A.APR.3 Find zeros of factorable polynomials and use them to sketch a rough graph.
• SIII.A.APR.4 Prove polynomial identities and use them to show number relationships (e.g. Pythagorean triples).
• SIII.A.APR.5 Expand (x + y)ⁿ for a positive integer n using Pascal's Triangle.
• SIII.A.APR.6 Rewrite simple rational expressions as quotient plus remainder over divisor, by inspection, long division or technology.
• SIII.A.APR.7 (+) Know rational expressions are closed under the four operations (nonzero divisor) and add, subtract, multiply and divide them.
• SIII.A.CED.1 Create one-variable equations and inequalities from linear, quadratic, rational and exponential situations.
• SIII.A.CED.2 Create equations in two or more variables for relationships and graph them with labeled axes and scales.
• SIII.A.CED.3 Express constraints with equations, inequalities or systems and judge whether solutions are viable in context.
• SIII.A.CED.4 Rearrange formulas to isolate a quantity of interest (e.g. solve V = IR for R).
• SIII.A.REI.2 Solve simple rational and radical equations and show how extraneous solutions can appear.
• SIII.A.REI.11 Explain why intersections of y = f(x) and y = g(x) solve f(x) = g(x) and find them approximately with graphs, tables or technology.

Functions

• SIII.F.IF.4 Interpret key graph and table features in context (intercepts, increase/decrease, maxima, symmetry, end behavior) and sketch graphs.
• SIII.F.IF.5 Link a function's domain to its graph and to the situation it models.
• SIII.F.IF.6 Calculate and interpret average rate of change over an interval and estimate it from a graph.
• SIII.F.IF.7 Graph functions from their equations and show key features, by hand or with technology.
  • b. Graph square root, cube root and other radical functions, and step, piecewise and absolute value functions.
  • c. Graph polynomial functions showing zeros and end behavior.
  • d. (+) Graph rational functions showing zeros, asymptotes and end behavior.
  • e. Graph exponential, logarithmic and trigonometric functions showing intercepts, end behavior, period, midline and amplitude.
• SIII.F.IF.8 Rewrite a function in equivalent forms to reveal different properties.
• SIII.F.IF.9 Compare properties of two functions shown in different ways.
• SIII.F.BF.1 Write a function for a relationship between two quantities.
  • b. Combine standard function types with arithmetic operations to model a situation.
• SIII.F.BF.3 Describe how adding k, multiplying by k or replacing x by kx or x + k changes a graph; find k; recognize even and odd functions.
• SIII.F.BF.4 Find inverse functions.
  • a. Solve f(x) = c for simple invertible functions and write inverses (e.g. exponential, cube root).
• SIII.F.LE.3 Use graphs and tables to see that exponential growth eventually exceeds linear, quadratic or any polynomial growth.
• SIII.F.LE.4 Write solutions of ab^(ct) = d as logarithms (base 2, 10 or e) and evaluate them with technology.
• SIII.F.LE.5 Interpret the parameters of linear, quadratic and exponential functions in context.
• SIII.F.TF.1 Understand radian measure as arc length on the unit circle.
• SIII.F.TF.2 Explain how the unit circle extends trigonometric functions to all real numbers.
• SIII.F.TF.3 Find the sine, cosine and tangent of π/3, π/4 and π/6 using special triangles.
• SIII.F.TF.5 Choose trigonometric functions to model periodic situations with given amplitude, frequency and midline.
• SIII.F.TF.7 Use inverse trig functions to solve trig equations on a given interval.

Geometry

• SIII.G.SRT.9 (+) Derive the triangle area formula A = ½ab sin C.
• SIII.G.SRT.10 (+) Prove the Law of Sines and the Law of Cosines and apply them.
• SIII.G.SRT.11 (+) Apply the Laws of Sines and Cosines to find unknown sides and angles in any triangle.
• SIII.G.GMD.4 Identify 2-D cross-sections of 3-D objects and 3-D objects formed by rotating 2-D shapes.
• SIII.G.MG.1 Use shapes and their measures to describe real objects (e.g. a tree trunk as a cylinder).
• SIII.G.MG.2 Use density by area or volume in modeling (e.g. people per square mile).
• SIII.G.MG.3 Use geometric methods to solve design problems with physical limits or cost goals.

Statistics and Probability

• SIII.S.ID.4 Use mean and standard deviation to fit a normal distribution and estimate percentages, knowing when this is not appropriate.
• SIII.S.IC.1 Understand statistics as making inferences about a population from a random sample.
• SIII.S.IC.3 Know the purposes of sample surveys, experiments and observational studies and how randomization relates to each.
• SIII.S.IC.4 Estimate a population proportion from a sample survey and develop a margin of error by simulation.
• SIII.S.IC.6 Evaluate reports based on data.