Standards: Connecticut has adopted the Common Core State Standards (CCSS) as the Connecticut State Standards (CSS).
PREREQUISITE SKILLS AND UNDERSTANDING
STANDARDS
Represent and solve equations and inequalities graphically.
HSA.REI.D.10 - Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
Understand the concept of a function and use function notation.
HSF.IF.A.1 - Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
HSF.IF.A.2 - Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context
Interpret functions that arise in applications in terms of the context.
HSF.IF.B.5 - Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if thefunction h(n) gives the number of person-hours it takes to assemble n engines in a factory, then thepositive integers would be an appropriate domain for the function.
Analyze functions using different representations.
HSF.IF.C.7 - Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
HSF.IF.C.7b - Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
HSF.IF.C.9 - Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.
Build a function that models a relationship between two quantities.
HSF.BF.A.1b - Combine standard function types using arithmetic operations.
HSF.BF.A.1c - Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.
Build new functions from existing functions.
HSF.BF.B.3 - Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
HSF.BF.B.4 - Find inverse functions.
HSF.BF.B.4a - Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) = 2(x3) for x > 0 or f(x) = (x + 1)/(x – 1) for x ≠ 1 (x not equal to 1).
HSF.BF.B.4b - Verify by composition that one function is the inverse of another.
HSF.BF.B.4c - Read values of an inverse function from a graph or a table, given that the graph has an inverse.
OBJECTIVES Students will learn to . . .
PREREQUISITE SKILLS AND UNDERSTANDING
STANDARDS
Represent and solve equations and inequalities graphically.
HSA.REI.D.11 - Explain why the x-coordinate of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include where f(x) and/or g(x) are linear, rational, absolute value, exponential and logarithmic functions.
HSA.REI.D.12 - Graph the solutions to a linear inequality in two variables as a half plane (excluding the boundary in the case of a strict inequality) and graph the solution set of a system of linear inequalities as the intersection of the corresponding half-planes.
OBJECTIVES Students will learn to . . .
PREREQUISITE SKILLS AND UNDERSTANDING
STANDARDS
Write expressions in equivalent forms to solve problems.
HSA.SSE.B.3 - Choose and produce an equivalent from of an expression to reveal and explain properties of the quantity represented by the expression.
HSA.SSE.B.3a - Factor a quadratic expression to reveal zeros of the function it defines.
Understand solving equations as a process of reasining 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.4 - Solve quadratic equations in one variable.
HSA.REI.B.4b - 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 for real numbers a and b.
Build a function that models a relationship between two quanitities.
HSA.BF.A.1 - Write a function that describes a relationship between two quantities.
Create equations that describe numbers or relationships.
HSA.CED.A.1 - Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
HSA.CED.A.2 - Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
Interpret functions that arise in applications in terms of the context.
HSF.IF.B.4 - For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
Analyze functions using different representations.
HSF.IF.C.7 - Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
HSF.IF.C.7a - Graph linear and quadratic functions and show intercepts, maxima, and minima.
Perform arithmetic operations with complex numbers.
HSN.CN.A.1 - Know there is a complex number i such that , 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 distributiveproperties to add, subtract, and multiply complex numbers.
Use complex numbers in polynomial identities adn equations.
HSN.CN.C.7 - Solve quadratic equations with real coefficients that have complex solutions.
HSN.CN.C.9 - Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
OBJECTIVES Students will learn to . . .
PREREQUISITE SKILLS AND UNDERSTANDING
STANDARDS
Interpret functions that arise in applications in terms of the context.
HSF.IF.B.4 - For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maxima and minima; multiplicity of roots; symmetries; end behavior; and periodicity.
Analyze functions using different representations.
HSF.IF.C.7 - Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
HSF.IF.C.7c - Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
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.
Interpret the structure of expressions.
HSA.SSE.A.2 - Use the structure of an expression to identify ways to rewrite it. For example, see x4– y4as (x2)2– (y2)2, thus recognizing it as a difference of squares that can be factored as (x2–y2) (x2+ y2).
Create equations that describe numbers or relationships.
HSA.CED.A.2 - Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
OBJECTIVES Students will learn to . . .
PREREQUISITE SKILLS AND UNDERSTANDING
STANDARDS
Understand and evaluate random processes underlying statistical experiments.
HSS.IC.A.1 - Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
Summarize, represent, and interpret data on a single count or measurement variable.
HSS.ID.A.4 - Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
Make inferences and justify conclusions from sample surveys, experiments, and observational studies.
HSS.IC.B.3 - Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
Calculate expected values and use them to solve problems.
HSS.MD.A.1 - Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.
HSS.MD.A.2 - Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
HSS.MD.A.3 - Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.
HSS.MD.A.4 - Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.
CONTENT TOPICS
PREREQUISITE SKILLS AND UNDERSTANDING
STANDARDS
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 roots may arise.
OBJECTIVES Students will learn to . . .
PREREQUISITE SKILLS AND UNDERSTANDING
STANDARDS
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 more complicated examples, a computer algebra system.
Understand solving equations as a process of reasining and explain the reasoning.
HSA.REI.A.2 - Solve simple rational and radical equations in one variable, and give examples showing how extraneous roots may arise.
OBJECTIVES Students will learn to . . .
PREREQUISITE SKILLS AND UNDERSTANDING
STANDARDS
Interpret the structure of expressions.
HSA.SSE.A.1 - Interpret expressions that represent a quantity in terms of its context.
Create equations that describe numbers or relationships.
HSA.CED.A.1 - Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
HSA.CED.A.2 - Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
Analyze functions using different representations.
HSF.IF.C.7e - Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
HSF.IF.C.8 - Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
Build a function that models a relationship between two quantities.
HSF.BF.A.1 - Write a function that describes a relationship between two quantities.
Build new functions from existing functions.
HSF.BF.B.3 - Identify the effect on the graph of replacing f(x) by fk, kf(x), f(kx), and f(x+k) for specific values of (both positive and negative); find the values of kgiven the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
HSF.BF.B.5 - Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
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 where a, cand 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 situations they model.
HSF.LE.B.5 - Interpret the parameters of an exponential function in terms of a context.
OBJECTIVES Students will learn to . . .
PREREQUISITE SKILLS AND UNDERSTANDING
STANDARDS
Interpret the structure of expressions.
HSA.SSE.A.1 - Interpret expressions that represent a quantity in terms of its context.
Create equations that describe numbers or relationships.
HSA.CED.A.1 - Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponentialfunctions.
HSA.CED.A.2 - Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
Analyze functions using different representations.
HSF-IF.C.7e - Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
HSF.IF.C.8 - Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
Build a function that models a relationship between two quantities.
HSF.BF.A.1 - Write a function that describes a relationship between two quantities.
Build new functions from existing functions.
HSF.BF.B.3 - Identify the effect on the graph of replacing f(x) by fk, kf(x), f(kx), and f(x+k) for specific values of (both positive and negative); find the values of kgiven the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
HSF.BF.B.5 - Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
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 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 teh situation they model.
HSF.LE.B.5 - Interpret the parameters of an exponential function in terms of a context.
OBJECTIVES Students will learn to . . .