fchsmath.weebly.comfchsmath.weebly.com/uploads/8/3/6/8/8368274/fcs_algeb… · Web view2018. 10. 10. · Timeline: 1st Trimester. 3 weeks. Vocabulary: Quadratic Function. Factor

Algebra 2 Pacing Guide

Timeline:1st Trimester

3 weeks

Vocabulary:Quadratic FunctionFactorGCFMonomialBinomialTrinomialPerfect Square TrinomialDifference of SquaresComplete the squareComplex numberQuadratic FormulaDiscriminantZerosRadicalsSquare rootsStandard formConjugate

Unit 1: Solving Quadratic EquationsState Standards:Number and Quantity(The Complex Number System)HSN-CN.APerform arithmetic operations with complex numbers.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.

2. Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

Number and Quantity(The Complex Number System)HSN-CN.CUse complex numbers in polynomial identities and equations.7. Solve quadratic equations with real coefficients that have complex solutions

8. (+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i).

9. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

Algebra(Seeing Structure in Expressions)HSA-SSE.AInterpret the structure of expressions.1. Interpret expressions that represent a quantity in terms of its context.*

a. Interpret parts of an expression, such as terms, factors, and coefficients.b. Interpret complicated expressions by viewing one or more of their parts as a

single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.

2. Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2

+ y2).

Algebra(Seeing Structure in Expressions)HSA-SSE.BWrite expressions in equivalent forms to solve problems.3. Choose and produce an equivalent form of an expression to reveal and explain properties of

the quantity represented by the expression.*a. Factor a quadratic expression to reveal the zeros of the function it defines.b. Complete the square in a quadratic expression to reveal the maximum or

minimum value of the function it defines.

Activities:Solving Quadratic Equations: Cutting Corners FAL

Concepts and Skills:Students will: Solve by factoring Solve by finding square

roots Solving by graphing Simplify radicals using i Perform operations on

complex numbers (including rationalize the denominator)

Factor using imaginary numbers

Finding complex solutions Solve perfect square

trinomial equations Complete the square to get

vertex form Solve by completing the

square Derive the quadratic

formula Find complex solutions Find x-intercepts where

leading coefficient not equal to 1

Rewrite quadratics in vertex from

Use the Quadratic Formula Find the discriminant Use the discriminant to

determine number and type of solutions

Resources:

Strategies:


Algebra(Creating Equations)HSA-CED.ACreate equations that describe numbers or relationships.2. Create equations in two or more variables to represent relationships between quantities;

graph equations on coordinate axes with labels and scales.

Algebra(Reasoning with Equations and Inequalities)HSA-REI.AUnderstand solving equations as a process of reasoning and explain the reasoning.1. Explain each step in solving a simple equation as following from the equality of numbers

asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Algebra(Reasoning with Equations and Inequalities)HSA-REI.BSolve equations and inequalities in one variable.4. Solve quadratic equations in one variable.

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.

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.

Algebra(Reasoning with Equations and Inequalities)HSA-REI.DRepresent and solve equations and inequalities graphically.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).

Quality Core:A1C: Factor trinomials in the form ax^2 + bx+ cA1D: Solve single-step and multistep equations and inequalities in one variableA1J: Use inductive reasoning to make conjectures and deductive reasoning to arrive at valid conclusionsB1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problemsC1A: Identify complex numbers and write their conjugatesC1B: Add, subtract, and multiply complex numbersC1C: Simplify quotients of complex numbersE1A: Solve quadratic equations and inequalities using various techniques, including completing the square and using the quadratic formulaE1B: Use the discriminant to determine the number and type of roots for a given quadratic equation


E1C: Solve quadratic equations with complex number solutionsE1D: Solve quadratic systems graphically and algebraically with and without technologyF1B: Factor polynomials using a variety of methods (e.g., factor theorem, synthetic division, long division, sums and differences of cubes, grouping)


1.5 weeks

Vocabulary:Quadratic FunctionFactorGCFMonomialBinomialTrinomialPerfect Square TrinomialDifference of Squares

Unit 2: Factoring Quadratics

New State Standards:Algebra( Seeing Structure in Expressions)HSA-SSE.AInterpret the structure of expressions.1. Interpret expressions that represent a quantity in terms of its context.*




+ y2).

Quality Core:A1C: Factor trinomials in the form ax^2 + bx + cB1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problemsC1A: Identify complex numbers and write their conjugatesC1C: Simplify quotients of complex numbersF1B: Factor polynomials using a variety of methods (e.g., factor theorem, synthetic division, long division, sums and differences of cubes, grouping)

Activities:

Concepts and Skills:Students will: Define parts of a quadratic

function Find greatest common

factors Factor quadratics Include graphical

representations of factoring

Resources:

Strategies:


2.5 weeks

Vocabulary:Quadratic FunctionParabolaFactorMonomialBinomialTrinomialZerosStandard FormVertex Form

Unit 3a: Properties and Graphs of Quadratics

New State Standards:Number and Quantity(Quantities)HSN-Q.AReason quantitatively and use units to solve problems.1. Use units as a way to understand problems and to guide the solution of multi-step problems;

choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

Number and Quantity(The Complex Number System)HSN-CN.CUse complex numbers in polynomial identities and equations.

Activities:Functions in Everyday Situations FAL

Forming Quadratics FAL

Concepts and Skills:Students will: Graph quadratic functions

in vertex, intercept, and standard form

Given any quadratic equation, write equation in other forms

Graph using

Resources:

Strategies:


Intercept Formx-intercepty-interceptVertexDomainRange

7. Solve quadratic equations with real coefficients that have complex solutions






+ y2).


the quantity represented by the expression.★a. Factor a quadratic expression to reveal the zeros of the function it defines.b. Complete the square in a quadratic expression to reveal the maximum or


Algebra(Arithmetic with Polynomials and Rational Expressions)HSA-APR.BUnderstand the relationship between zeros and factors of polynomials.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.




asserted at the previous step, starting from the assumption that the original equation has a

transformations Determine characteristics

based on form including shape, direction, vertex, symmetry, and intercepts

Write a quadratic based on zeros

Describe the domain and range of quadratic

Understand the concepts are similar given x = y^2

Graph quadratic inequalityGraph system of quadratic inequalities


solution. Construct a viable argument to justify a solution method.

Algebra(Reasoning with Equations and Inequalities)HSA-REI.BSolve equations and inequalities in one variable.4. Solve quadratic equations in one variable.

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.

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.



Functions(Interpreting Functions)HSF-IF.AUnderstand the concept of a function and use function notation.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).

Functions(Interpreting Functions)HSF-IF.BInterpret functions that arise in applications in terms of the context.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.

Functions(Interpreting Functions)HSF-IF.CAnalyze functions using different representations.7. Graph functions expressed symbolically and show key features of the graph, by hand in

simple cases and using technology for more complicated cases.*

a. Graph linear and quadratic functions and show intercepts, maxima, and minima.


8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

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.

Quality Core:A1B: Multiply monomials and binomialsA1C: Factor trinomials in the form ax^2 + bx+ cA1D: Solve single-step and multistep equations and inequalities in one variableA1J: Use inductive reasoning to make conjectures and deductive reasoning to arrive at valid conclusionsB1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problemsC1B: Add, subtract, and multiply complex numbersE1A: Solve quadratic equations and inequalities using various techniques, including completing the square and using the quadratic formulaE1B: Use the discriminant to determine the number and type of roots for a given quadratic equationE1C: Solve quadratic equations with complex number solutionsE1D: Solve quadratic systems graphically and algebraically with and without technologyE2A: Determine the domain and range of a quadratic function; graph the function with and without technologyE2C: Graph a system of quadratic inequalities with and without technology to find the solution set to the systemF1B: Factor polynomials using a variety of methods (e.g., factor theorem, synthetic division, long division, sums and differences of cubes, grouping)


1 week

Vocabulary:Parent FunctionAmplitudeMaximumMinimumTransformationHorizontal ShiftVertical ShiftHorizontal StretchVertical Stretch

Unit 3b: Transforming Functions

New State Standards:Number and Quantity(Quantities)HSN-Q.AReason quantitatively and use units to solve problems.3. Choose a level of accuracy appropriate to limitations on measurement when reporting

quantities.


Activities:

Concepts and Skills:Students will: Graph main parent

functions Apply transformations to

parent functions Determine domain/ range

for parent functions Determine domain/ range

for transformed functions Graphically represent

transformations Verbally describe

Resources:

Strategies:


DomainRangeExponential






Functions(Building Functions)HSF-BF.BBuild new functions from existing functions.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.

Quality Core:B1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problemsC1D: Perform operations on functions, including function composition, and determine domain and range for each of the given functionsE2A: Determine the domain and range of a quadratic function; graph the function with and without technologyE2B: Use transformations (e.g., translation, reflection) to draw the graph of a relation and determine a relation that fits a graph

transformations Graph transformations

based on a graph of f(x) that is not a defined function

Write equation based on transformations on a defined function


4 weeks

Vocabulary:ConjugatesFactor TheoremFundamental Theorem of

Unit 4: Polynomials and Polynomial Functions

New State Standards:Number and Quantity(The Complex Number System)HSN-CN.APerform arithmetic operations with complex numbers.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.

Activities:Representing PolynomialsFAL

Manipulating Polynomials FAL

Concepts and Skills:Students will:

Resources:

Strategies:


AlgebraImaginary Root TheoremIrrational Root TheoremMultiplicityPolynomial FunctionRational Root TheoremRemainder TheoremStandard form of PolynomialLong DivisionSynthetic DivisionDegreeDegree of PolynomialDifference of CubesMultiple ZeroPolynomialRelative MaximumRelative MinimumSum of CubesEnd BehaviorIncreasingDecreasing

2. Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

3. (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

Number and Quantity(The Complex Number System)HSN-CN.CUse complex numbers in polynomial identities and equations.7. Solve quadratic equations with real coefficients that have complex solutions

8. (+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i).






+ y2).


the quantity represented by the expression.★a. Factor a quadratic expression to reveal the zeros of the function it defines.b. Complete the square in a quadratic expression to reveal the maximum or


Algebra(Arithmetic with Polynomials and Rational Expressions)HSA-APR.APerform arithmetic operations on polynomials.1. Understand that polynomials form a system analogous to the integers, namely, they are

closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Classify polynomials Write a polynomial in

standard form write a polynomial in

factored form Find zeros of polynomial

functions Write a polynomial

function from its zeros Find and use the

multiplicity of a zero Perform polynomial long

division Use synthetic division Check factors Evaluate polynomials

using synthetic division Solve polynomial

equations by factoring Factor sum/difference of

cubes Solve a polynomial

equation Factor by using a quadratic

pattern Solve higher-degree

polynomial equations Find rational roots Use Rational Root

Theorem Find irrational roots Find imaginary roots Write polynomials

equations from roots Use Fundamental Theorem

of Algebra Use zero’s and end

behavior to create a graphical representation of a polynomial

Identify key features of polynomial graphs (including domain, range, roots, relative max, relative min, increasing and decreasing intervals, positive or negative, symmetries, and end behavior)


Algebra(Arithmetic with Polynomials and Rational Expressions)HSA-APR.BUnderstand the relationship between zeros and factors of polynomials.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).

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.

Algebra(Arithmetic with Polynomials and Rational Expressions)HSA-APR.CUse polynomial identities to solve problems.4. Prove polynomial identities and use them to describe numerical relationships. For example,

the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples.

5. (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle.







Functions(Interpreting Functions)HSF-IF.AUnderstand the concept of a function and use function notation.


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).

2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.



6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.*



b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.


Quality Core:A1B: Multiply monomials and binomialsA1C: Factor trinomials in the form ax2 + bx+ cA1D: Solve single-step and multistep equations and inequalities in one variableA1J: Use inductive reasoning to make conjectures and deductive reasoning to arrive at valid conclusionsB1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problemsE1A: Solve quadratic equations and inequalities using various techniques, including completing the square and using the quadratic formulaE1D: Solve quadratic systems graphically and algebraically with and without technologyF1A: Evaluate and simplify polynomial expressions and equationsF1B: Factor polynomials using a variety of methods (e.g., factor theorem, synthetic division,


long division, sums and differences of cubes, grouping)F2A: Determine the number and type of rational zeros for a polynomial functionF2B: Find all rational zeros of a polynomial functionF2C: Recognize the connection among zeros of a polynomial function, x-intercepts, factors of polynomials, and solutions of polynomial equationsF2D: Use technology to graph a polynomial function and approximate the zeros, minimum, and maximum; determine domain and range of the polynomial function

Timeline:2nd Trimester

2 weeks

Vocabulary:InequalitiesExpressionsEquationsSubstitutionOne step equationsTwo step equationsLike termsTermsFactorsCoefficientsDegreeDistributiveClosureCommutativeAssociativeIdentitiesInverse PropertiesLeading CoefficientLiteral EquationConstraintsLinear ProgrammingObjective FunctionFeasible RegionVerticesOptimizationMaximumMinimumCompound InequalityAbsolute ValueIntersectionInterval NotationSet Notation

Unit 5: Equations, Inequalities & Systems

New State Standards:Algebra(Seeing Structure in Expressions)HSA-SSE.AInterpret the structure of expressions.1. Interpret expressions that represent a quantity in terms of its context.*

a. Interpret parts of an expression, such as terms, factors, and coefficients.

Algebra(Creating Equations)HSA-CED.ACreate equations that describe numbers or relationships.1. Create equations and inequalities in one variable and use them to solve problems

3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.



Algebra(Reasoning with Equations and Inequalities)HSA-REI.BSolve equations and inequalities in one variable.3. Solve linear equations and inequalities in one variable, including equations with coefficients

represented by letters.

Algebra

Activities:Solving Linear Equations in Two Variables FAL

Concepts and Skills:Students will: Find distance and midpoint Solve equations and

inequalities for word problems

Write and solve equations and inequalities for word problems

Simplify expressions and interpret their parts

Solve literal equations (for a specified variable)

Solve system of equation and inequalities using various methods (including graphing)

Solve linear programming problems

Solve absolute value inequalitiesMultiple forms for linear equations including point slope

Resources:

Strategies:


(Reasoning with Equations and Inequalities)HSA-REI.DRepresent and solve equations and inequalities graphically.6. Explain why the x-coordinates 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 cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.*

Quality Core:A1D: Solve single-step and multistep equations and inequalities in one variableA1E: Solve systems of two linear equations and inequalities in one variableA1F: Write linear equations in standard form and slope-intercept form when given two points, a point and the slope, or the graph of the equationA1G: Graph a linear equation using a table of values, x- and y-intercepts, or slope-intercept formA1H: Find the distance and midpoint between two points in the coordinate planeA1J: Use inductive reasoning to make conjectures and deductive reasoning to arrive at valid conclusionsB1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problemsD1A: Solve linear inequalities containing absolute valueD1B: Solve compound inequalities containing “and” and “or” and graph the solution setD1C: Solve algebraically a system containing three variablesD2A: Graph a system of linear inequalities in two variables with and without technology to find the solution set to the systemD2B: Solve linear programming problems by finding maximum and minimum values of a function over a region defined by linear inequalities


2.5 weeks3 weeks

Vocabulary:CenterCircleConic SectionCo-verticesDirectrixEllipseFocus of ParabolaFocus of EllipseFocus of HyperbolaHyperbolaMajor Axis

Unit 6: Conics


quantities.



Algebra(Reasoning with Equations and Inequalities)

Activities:

Concepts and Skills:Students will: Graph a Circle Graph an ellipse Graph a hyperbola Identify the graphs of

conic sections Use the definition of a

parabola Write the equation of a

parabola Identify focus and directrix

of a parabola Graph using the equation

of a parabola Write the equation of a

circle Use translations to write an

Resources:

Strategies:


Minor AxisRadiusStandard Form of CircleTransverse AxisVertices of EllipseVertices of Hyperbola

HSA-REI.AUnderstand solving equations as a process of reasoning and explain the reasoning.1. Explain each step in solving a simple equation as following from the equality of numbers













equation of a circle Find the center and radius

of a circle Graph circle using center

and radius Write the equation of an

ellipse Graph hyperbola Write the equation of a

translated ellipse Write the equation of a

translated hyperbola Write the equation of a

translated parabola Identify translated conic

sections




Quality Core:B1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problemsC1D: Perform operations on functions, including function composition, and determine domain and range for each of the given functionsE2A: Determine the domain and range of a quadratic function; graph the function with and without technologyE2B: Use transformations (e.g., translation, reflection) to draw the graph of a relation and determine a relation that fits a graphE3A: Identify conic sections (e.g., parabola, circle, ellipse, hyperbola) from their equations in standard formE3B: Graph circles and parabolas and their translations from given equations or characteristics with and without technologyE3C: Determine characteristics of circles and parabolas from their equations and graphsE3D: Identify and write equations for circles and parabolas from given characteristics and graphs


2 weeks

Vocabulary:Like RadicalsNth RootPrincipal RootRadical EquationRadical FunctionRadicandRational ExponentRationalize DenominatorSquare Root EquationSquare Root FunctionExtraneous Solution

Unit 7: Radical Functions and Rational Exponents


quantities.

Number and Quantity(The Real Number System)HSN-RN.AExtend the properties of exponents to rational exponents.1. Explain how the definition of the meaning of rational exponents follows from extending the

properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.

Activities:

Concepts and Skills:Students will: Find all real roots Simplify radical

expressions Multiply radicals Divide radicals Rationalize the

denominator Add and subtract radical

expressions Multiply binomial radical

expressions Multiplying conjugates Rationalize binomial

radical denominators

Resources:

Strategies:


2. Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Number and Quantity(The Real Number System)HSN-RN.BUse properties of rational and irrational numbers.3. Explain why the sum or product of two rational numbers is rational; that the sum of a

rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.


the quantity represented by the expression.*



2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.






Convert between radical and rational exponents

Simplify expressions with rational exponents

Simplify numbers with rational exponents

Write rational exponents in simplest form

Solve square root equations

Solve radical equations with rational exponents

Check for extraneous solutions

Solve quadratic equations that lead to extraneous solutions






b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

Quality Core:B1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problemsG1B: Simplify radicals that have various indicesG1C: Use properties of roots and rational exponents to evaluate and simplify expressionsG1D: Add, subtract, multiply, and divide expressions containing radicalsG1E: Rationalize denominators containing radicals and find the simplest common denominatorG1F: Evaluate expressions and solve equations containing nth roots or rational exponentsG1G: Evaluate and solve radical equations given a formula for a real-world situationE2B: Use transformations (e.g., translation, reflection) to draw the graph of a relation and determine a relation that fits a graph


2 weeks

Vocabulary:Rational ExpressionSimplest FormComplex FractionRational Equations

Unit 8: Rational Functions


quantities.



Activities:

Concepts and Skills:Students will: Simplify rational

expressions Add, Subtract, Multiply,

and Divide rational expressions

Find Least Common Multiples

Solve rational equations

Resources:

Strategies:






+ y2).


the quantity represented by the expression.*



Algebra(Arithmetic with Polynomials and Rational Expressions)HSA-APR.DRewrite rational expressions.

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.

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.


asserted at the previous step, starting from the assumption that the original equation has a


solution. Construct a viable argument to justify a solution method.




Functions(Interpreting Functions)HSF-IF.CAnalyze functions using different representations.8. Write a function defined by an expression in different but equivalent forms to reveal and

explain different properties of the function.


Quality Core:A1A: Identify properties of real numbers and use them and the correct order of operations to simplify expressionsA1B: Multiply monomials and binomialsA1C: Factor trinomials in the form ax2 + bx+ cB1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problemsE1A: Solve quadratic equations and inequalities using various techniques, including completing the square and using the quadraticformulaF1A: Evaluate and simplify polynomial expressions and equationsF1B: Factor polynomials using a variety of methods (e.g., factor theorem, synthetic division, long division, sums and differences of cubes, grouping)G1A: Solve mathematical and real-world rational equation problems (e.g., work or rate problems)


3 weeksVocabulary:

Unit 9: Basic Trigonometry

New State Standards:Functions(Trigonometric Functions)HSF-TF.A

Activities:

Concepts and Skills:Students will: Convert between radians

and degrees

Resources:

Strategies:


Standard PositionInitial SideTerminal SideCoterminal AnglesUnit CircleCosineSineTangentSecantCosecantCotangentCentral AngleReference AngleRadianDegreesRatioAngleOppositeAdjacentRight TriangleSpecial Right TrianglesHypotenuseArc LengthProportion

Extend the domain of trigonometric functions using the unit circle.1. Understand radian measure of an angle as the length of the arc on the unit circle subtended

by the angle.

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.

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, cosines, and tangent for x, π + x, and 2π – x in terms of their values for x, where x is any real number.

Quality Core:A1I: Use sine, cosine, and tangent ratios to find the sides or angles of right trianglesB1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problemsG3A: Use the law of cosines and the law of sines to find the lengths of sides and measures of angles of triangles in mathematical andreal-world problemsG3B: Use the unit-circle definition of the trigonometric functions and trigonometric relationships to find trigonometric values for general anglesG3C: Measure angles in standard position using degree or radian measure and convert a measure from one unit to the other

Find coterminal angles Find reference angles Work with special right

triangles Construct the unit circle

using special right triangles

Measure an angle in standard position

Sketch an angle in standard position

Find the cosine and sine of an angle

Find exact value of sine and cosine

Find coordinates of points on the unit circle

Use radian measures for angles

Find the length of an arc of a circle

Find cosine and sine of radian measures

Find all six trig values for an angle

Find all six trig values for a point not on the unit circle

Timeline:3rd Trimester

2 weeks

Vocabulary:Sine FunctionSine CurveCosine FunctionCosine CurveTangent FunctionTangent CurveAmplitudePeriodMaximumMinimumPhase ShiftsTransformationHorizontal ShiftVertical Shift

Unit 10: Trigonometric Functions

New State Standards:Functions(Trigonometric Functions)HSF-TF.AExtend the domain of trigonometric functions using the unit circle.4. Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric

functions.

Functions(Trigonometric Functions)HSF-TF.BModel periodic phenomena with trigonometric functions.5. Choose trigonometric functions to model periodic phenomena with specified amplitude,

frequency, and midline.

Functions(Trigonometric Functions)HSF-TF.C

Activities:Ferris Wheel (FAL)

Concepts and Skills:Students will: Estimate sine, cosine, and

tangent values in radians and degrees

Find the period, amplitude, domain, and range of the sine and cosine curves

Sketch the graph of the sine and cosine curves

Graph sine from an equation

Graph and find the domain and range of the tangent function

Identify phase shifts Graph translations Graph a combined

Resources:

Strategies:


DomainRange

Prove and apply trigonometric identities.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.

9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Quality Core:B1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problemsG3D: Graph the sine and cosine functions with and without technologyG3E: Determine the domain and range of the sine and cosine functions, given a graphG3F: Find the period and amplitude of the sine and cosine functions, given a graphG2G: Use sine, cosine, and tangent functions, including their domains and ranges, periodic nature, and graphs, to interpret and analyze relations

translationWrite an equation based on transformations of a graph and or points


1.5 weeks

Vocabulary:AsymptoteChange of Base FormulaCommon LogarithmExponential EquationExponential FunctionLogarithmLogarithmic EquationLogarithmic FunctionNatural Logarithmic Function

Unit 11: Exponential and Logarithmic Functions

New State Standards:Number and Quantity(Quantities)HSN-Q.AReason quantitatively and use units to solve problems.1. Use units as a way to understand problems and to guide the solution of multi-step problems;

choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.





the quantity represented by the expression.*c. Use the properties of exponents to transform expressions for exponential

functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈

Activities:

Concepts and Skills:Students will: Graph exponential function Translate exponential

functions Evaluate e Convert between

exponentials and logarithmic forms

Evaluate logarithms Graph logarithmic

functions Translate logarithmic

functions Identify properties of

logarithms Expand and simplify

logarithms Solve exponential

equations Solve logarithmic

equations Use logarithmic properties

to solve an equation

Resources:

Strategies:


1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

Algebra(Creating Equations)HSA-CED.ACreate equations that describe numbers or relationships.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.

2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.









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.*

5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*


simple cases and using technology for more complicated cases.★e. Graph exponential and logarithmic functions, showing intercepts and end

behavior, and trigonometric functions, showing period, midline, and amplitude.


a. 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.


Functions(Building Functions)HSF-BF.ABuild a function that models a relationship between two quantities.1. Write a function that describes a relationship between two quantities.★

a. Determine an explicit expression, a recursive process, or steps for calculation from a context.



4. Find inverse functions.c. (+) Read values of an inverse function from a graph or a table, given that the

function has an inverse.


5. Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Functions(Linear, Quadratic, and Exponential Models)HSF-LE.AConstruct and compare linear, quadratic, and exponential models and solve problems.1. Distinguish between situations that can be modeled with linear functions and with

exponential functions.a. Prove that linear functions grow by equal differences over equal intervals, and

that exponential functions grow by equal factors over equal intervals.b. Recognize situations in which one quantity changes at a constant rate per unit

interval relative to another.c. Recognize situations in which a quantity grows or decays by a constant percent

rate per unit interval relative to another.

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).

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.

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.

Functions(Linear, Quadratic, and Exponential Models)HSF-LE.BInterpret expressions for functions in terms of the situation they model.5. Interpret the parameters in a linear or exponential function in terms of a context.

Quality Core:A1A: Identify properties of real numbers and use them and the correct order of operations to simplify expressionsA1J: Use inductive reasoning to make conjectures and deductive reasoning to arrive at valid conclusionsB1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problemsC1D: Perform operations on functions, including function composition, and determine domain and range for each of the given functionsE2B: Use transformations (e.g., translation, reflection) to draw the graph of a relation and determine a relation that fits a graphG2A: Graph exponential and logarithmic functions with and without technologyG2B: Convert exponential equations to logarithmic form and logarithmic equations to exponential form



1 Week

Vocabulary:Direct VariationInverse VariationJoint VariationInverseComposition of FunctionsHorizontal Line Test

Unit 12: Function Operations


quantities.



Functions(Building Functions)HSF-BF.BBuild new functions from existing functions.4. Find inverse functions.

a. Solve an equation of the form f(x) = c for a simple function that has an inverse and write an expression for the inverse. For example, f(x) = 2 x3 or f(x) = (x+1)/(x-1) for x ≠ 1.

b. (+) Verify by composition that one function is the inverse of another.c. (+) Read values of an inverse function from a graph or a table, given that the

function has an inverse.d. (+) Produce an invertible function from a non-invertible function by restricting

the domain.





Activities:

Concepts and Skills:Students will: Model inverse variation Identify and solve

problems using direct, joint, and inverse variation

Find an inverse from a graph or table

Decide if a function has an inverse

Verify inverse by composition

Resources:

Strategies:



Functions(Interpreting Functions)HSF-IF.CAnalyze functions using different representations.8. Write a function defined by an expression in different but equivalent forms to reveal and

explain different properties of the function.


Quality Core:A1A: Identify properties of real numbers and use them and the correct order of operations to simplify expressionsA1B: Multiply monomials and binomialsA1C: Factor trinomials in the form ax2 + bx+ cB1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problemsC1D: Perform operations on functions, including function composition, and determine domain and range for each of the given functionsF1A: Evaluate and simplify polynomial expressions and equationsG1A: Solve mathematical and real-world rational equation problems (e.g., work or rate problems)


1 week

Vocabulary:Arithmetic MeanArithmetic SequenceArithmetic SeriesCommon DifferenceCommon RatioExplicit FormulaGeometric MeanRecursive FormulaSequenceSeriesTermSummation Notation

Unit 13: Sequences and Series

New State Standards:Algebra(Seeing Structure in Expressions)HSA-SSE.AInterpret the structure of expressions.1. Interpret expressions that represent a quantity in terms of its context.*

a. Interpret parts of an expression, such as terms, factors, and coefficients.

Algebra(Seeing Structure in Expressions)HSA-SSE.BInterpret the structure of expressions.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. For example, calculate mortgage payments.*

Functions(Interpreting Functions)

Activities:

Concepts and Skills:Students will: Generate a sequence Identify an arithmetic

sequence Identify a geometric

sequence Write and evaluate

arithmetic and geometric series

Write a series in summation notation

Find the sum of a finite arithmetic series

Find the sum of a finite geometric series

Find the sum of an infinite

Resources:

Strategies:


Infinite Series HSF-IF.AUnderstand the concept of a function and use function notation.3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a

subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) +f(n-1) for n ≥ 1.



Functions(Building Functions)HSF-BF.ABuild a function that models a relationship between two quantities.1. Write a function that describes a relationship between two quantities.*

a. Determine an explicit expression, a recursive process, or steps for calculation from a context.

2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.*

Functions(Linear, Quadratic, and Exponential Models)HSF-LE.AConstruct and compare linear, quadratic, and exponential models and solve problems.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).

Quality CoreA1A: Identify properties of real numbers and use them and the correct order of operations to simplify expressionsA1D: Solve single-step and multistep equations and inequalities in one variableA1J: Use inductive reasoning to make conjectures and deductive reasoning to arrive at valid conclusionsB1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problemsH2A: Find the nth term of an arithmetic or geometric sequenceH2B: Find the position of a given term of an arithmetic or geometric sequenceH2C: Find sums of a finite arithmetic or geometric seriesH2D: Use sequences and series to solve real-world problemsH2E: Use sigma notation to express sums

arithmetic series Find the sum of an infinite

geometric series



2 weeks

Vocabulary:Binomial ProbabilityBox and Whisker PlotConditional ProbabilityCumulative ProbabilityInterquartile RangeMeasures of Central TendencyMeasures of VariationNormal DistributionOutlierPercentileProbability DistributionQuartilesSampleSample SpaceStandard DeviationStandard Normal CurveZ-Score

Unit 14: Data Analysis and Probability

New State Standards:Statistics and Probability(Interpreting Categorical and Quantitative Data)HSS-ID.ASummarize, represent, and interpret data on a single count or measurement variable.4. Use the mean and standard deviation of a dataset to fit it to a normal distribution and to

estimate population percentages. Recognize that there are datasets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

Statistics and Probability(Making Inferences and Justifying Conclusions)HSS-IC.AUnderstand and evaluate random processes underlying statistical experiments.1. Understand statistics as a process for making inferences about population parameters based

on a random sample from that population.

2. Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?

Statistics and Probability(Making Inferences and Justifying Conclusions)HSS-IC.BMake inferences and justify conclusions from sample surveys, experiments, and observational studies.3. Recognize the purposes of and differences among sample surveys, experiments, and

observational studies; explain how randomization relates to each.

4. Use data from a sample survey to estimate population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

5. Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

6. Evaluate reports based on data.

Statistics and Probability(Using Probability to Make Decisions)HSS-MD.BUse probability to evaluate outcomes of decisions.6. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number

generator).

7. (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

Activities:

Concepts and Skills:Students will: Make a frequency table Calculate probability

distributions Find conditional

probability Construct tree diagrams Compute Combinations

and Permutations Find measures of central

tendency Construct box-and-whisker

plots Construct and use

percentiles Identify an outlier Find standard deviation of

set of data Find and interpret z-scores Determine sample sizes Work with binomial

distributions Work with standard

normal curve and normal distributions

Resources:

Strategies:


Quality Core:H1A: Use the fundamental counting principle to count the number of ways an event can happenH1B: Use counting techniques, like combinations and permutations, to solve problems (e.g., to calculate probabilities)H1C: Find the probability of mutually exclusive and nonmutually exclusive eventsH1D: Find the probability of independent and dependent eventsH1E: Use unions, intersections, and complements to find probabilitiesH1F: Solve problems involving conditional probability


2 Weeks

Vocabulary:Augmented MatrixDeterminantEqual MatricesMatrixMatrix AdditionMatrix ElementMatrix EquationMatrix MultiplicationRow OperationsScalar MultiplicationVariable MatrixZero MatrixSquare MatrixInverse Matrix

Unit 15: Matrices

New State Standards:Number and Quantity(Vector and Matrix Quantities)HSN-VM.CPerform operations on matrices and use matrices in applications.7. (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence

relationships in a network.

8. (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

9. (+) Add, subtract, and multiply matrices of appropriate dimensions.

10. (+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

11. (+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.



Algebra(Reasoning with Equations and Inequalities)HSA-REI.CSolve systems of equations.5. Prove that, given a system of two equations in two variables, replacing one equation by the

sum of that equation and a multiple of the other produces a system with the same solutions.

Activities:

Activities:

Concepts and Skills:Students will: Write the dimensions of a

matrix Identify a matrix element Use identity and inverse

matrices Subtract matrices Determine equal matrices Find unknown matrix

elements Use scalar products Multiply matrices Determine if matrix

multiplication is defined Evaluate determinant of

2X2 matrix Find an inverse matrix Solve a matrix equation Evaluate determinant of

3X3 matrix Use technology to solve

matrix problems Write a system as a matrix

equation Solve a system of two

equations Solve a system of three

equations Write an augmented matrix Write a system from and

augmented matrix

Resources:

Strategies:


8. (+) Represent a system of linear equations as a single matrix equation in a vector variable.

9. (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

Quality Core:A1A: Identify properties of real numbers and use them and the correct order of operations to simplify expressionsA1D: Solve single-step and multistep equations and inequalities in one variableA1E: Solve systems of two linear equations using various methods, including elimination, substitution, and graphingB1: Mathematical processes learned in the context of increasingly complex mathematical and real-world problemsD1C: Solve algebraically a system containing three variablesF1A: Evaluate and simplify polynomial expressions and equationsI1A: Add, subtract, and multiply matricesI1B: Use addition, subtraction, and multiplication of matrices to solve real-world problemsI1C: Calculate the determinant of 2 × 2 and 3 × 3 matricesI1D: Find the inverse of a 2 × 2 matrixI1E: Solve systems of equations by using inverses of matrices and determinantsI1F: Use technology to perform operations on matrices, find determinants, and find inverses

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fchsmath.weebly.comfchsmath.weebly.com/uploads/8/3/6/8/8368274/fcs_algeb… · Web view2018. 10. 10. · Timeline: 1st Trimester. 3 weeks. Vocabulary: Quadratic Function. Factor