Standards: Connecticut has adopted the Common Core State Standards (CCSS) as the Connecticut State Standards (CSS).
Course Levels: Intermediate Geometry, College Prep Geometry, Honors Geometry - Topics are explored in greater depth at higher course levels.
Define trigonometric rations and solve problems involving right triangles.
HSG.SRT.C.6 - Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
HSG.SRT.C.7 - Explain and use the relationship between the sine and cosine of complementary angles.
HSG.SRT.C.8 - Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Extend the domain of trigonometric functions using the unit circle.
HSF.TF.A.1 - Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
HSF.TF.A.2 - Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
HSF.TF.A.3 - Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for x, π + x, and 2π - x in terms of their values for x, where x is any real number.
HSF.TF.A.4 - Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
Model periodic phenomena with trigonometric functions.
HSF.TF.B.5 - Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
HSF.TF.B.7 - Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.
Prove and apply trigonometric identities.
HSF.TF.C.8 - Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
Trigonometry
Write expressions in equivalent forms to solve problems.
HSA.SSE.B.3 - Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.*
HSA.SSE.B.3.a - Factor a quadratic expression to reveal the zeros of the function it defines.
HSA.SSE.B.3.b - Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
Create equations that describe numbers or relationships.
HSA.CED.A.1 - Create equations and inequalities in one variable and use them to solve problems.
HSA.CED.A.4 - Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
Understand solving equations as a process of reasoning and explain the reasoning.
HSA.REI.A.2 - Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
Solve equations and inequalities in one variable.
HSA.REI.B.3 - Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
HSA.REI.B.4 - Solve quadratic equations in one variable.
HSA.REI.B.4.a - Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)2 = q that has the same solutions. Derive the quadratic formula from this form.
HSA.REI.B.4.b - Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
Perform arithmetic operations with complex numbers.
HSN.CN.A.1 - Know there is a complex number i such that i2 = -1, and every complex number has the form a + bi with a and b real.
HSN.CN.A.2 - Use the relation i2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
HSN.CN.A.3 - Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
Use complex numbers in polynomial identities and equations.
HSN.CN.C.7 - Solve quadratic equations with real coefficients that have complex solutions.
Interpret the structure of expressions.
HSA.SSE.A.2 - Use the structure of an expression to identify ways to rewrite it.
Solving linear equations
Modeling linear equations
Solving Quadratic equations and Modeling with quadratics
Complex Numbers and imaginary solutions
Other Equations
Understand solving equations as a process of reasoning and explain the reasoning.
HSA.REI.A.2 - Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
Analyze functions using different representations.
HSF.IF.C.7.d - Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
Arithmetic with Polynomials and Rational Expressions.
HSA.APR.D.7 - Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
Understand the relationship between zeros and factors of polynomials.
HSA.APR.B.2 - Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x).
HSA.APR.B.3 - Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
Rewrite rational expressions.
HSA.APR.D.6 - Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
Analyze functions using different representations.
HSF.IF.C.7.C - Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
Build a function that models a relationship between two quantities.
HSF.BF.A.1 - Write a function that describes a relationship between two quantities.
HSF.BF.A.1.a - Determine an explicit expression, a recursive process, or steps for calculation from a context.
HSF.BF.A.2 - Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
Construct and compare linear, quadratic, and exponential models and solve problems.
HSF.LE.A.1 - Distinguish between situations that can be modeled with linear functions and with exponential functions.
HSF.LE.A.1.a - Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
HSF.LE.A.1.b - Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
HSF.LE.A.1.c - Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
HSF.LE.A.2 - Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
HSF.LE.A.3 - Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
Understand the concept of a function and use function notation.
HSF.IF.A.3 - Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
Write expressions in equivalent forms to solve problems.
HSA.SSE.B.4 - Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems
Construct and compare linear, quadratic, and exponential models and solve problems.
HSF.LE.A.4 - For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
Interpret expressions for functions in terms of the situation they model.
HSF.LE.B.5 - Interpret the parameters in a linear or exponential function in terms of a context.
Build new functions from existing functions.
HSF.BF.B.5 - Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
Analyze functions using different representations.
HSF.IF.C.8.b - Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)ᵗ, y = (0.97)ᵗ, y = (1.01)12ᵗ, y = (1.2)ᵗ/10, and classify them as representing exponential growth or decay.
Understand independence and conditional probability and use them to interpret data.
HSS.CP.A.1 - Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not").
HSS.CP.A.2 - Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
HSS.CP.A.3 - Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.
HSS.CP.A.4 - Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.
HSS.CP.A.5 - Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.
Use the rules of probability to compute probabilities of compund events.
HSS.CP.B.6 - Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model.
HSS.CP.B.7 - Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model.
HSS.CP.B.8 - Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
HSS.CP.B.9 - Use permutations and combinations to compute probabilities of compound events and solve problems.
Summarize, represent, and interpret data on two categorical and quantitative variables.
HSS.ID.B.5 - Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
HSS.ID.B.6 - Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
HSS.ID.B.6.a - Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.
HSS.ID.B.6.c - Fit a linear function for a scatter plot that suggests a linear association.
Interpret linear models.
HSS.ID.C.7 - Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
HSS.ID.C.8 - Compute (using technology) and interpret the correlation coefficient of a linear fit.
HSS.ID.C.9 - Distinguish between correlation and causation.