Math II

 

Wayne County Public Schools

Math II Pacing Guide


North Carolina Math 2

Collaborative Pacing Guide

This pacing guide is the collaborative work of math teachers, coaches, and curriculum leaders from 38 NC public school districts. The teams worked through two face-to-face meetings and digitally to compile the information presented. NC Math 1, 2, and 3 standards were used to draft possible units of study for these courses. This is a first draft living document. Teams plan to meet throughout the year to continually tweak, update and refine these guides. Updates will be posted as available to this google document.

 

Please reference the NC Math 1, 2, or 3 standards for any questions or discrepancies. This document should be used only after reading the NC Math 1, 2, and 3 standards and instructional guides provided by NC DPI.

 

If you have suggestions or comments that you would like the collaborative writing team to consider for revisions, please email sdupree@wcpss.net or stefanie.buckner@bcsemail.org.

 

Units for NC Math 2

Number of Days (Block)

Number of Days (Traditional)

Unit 1: Transformations

10

20

Unit 2: Quadratics

20

40

Unit 3: Square Root & Inverse Variation Functions

15

30

Unit 4: Similarity & Congruence

12

24

Unit 5: Trigonometry (Solving Right Triangles)

13

26

Unit 6: Probability

10

20

Total (allowing for flex days)

80

160

 

Unit 1: Transformations

Estimated Days: 10 Semester or 20 Year Long

Standards

Learning Intentions

NC.M2.F-IF.1

NC.M2.F-IF.2

NC.M2.G-CO.2 NC.M2.G-CO.3

NC.M2.G-CO.4 NC.M2.G-CO.5 NC.M2.G-CO.6 NC.M2.G-SRT.1 NC.M2.G-SRT.1a

NC.M2.G-SRT.1b NC.M2.G-SRT.1c NC.M2.G-SRT.1d

  1. Understand rigid transformations, or series of rigid transformations, result in congruent geometric figures.
  2. Understand that transformations are two variable functions.
  3. Understand, explain and apply the properties of transformations.

 

 

Unit 2: Quadratics

Estimated Days: 20 Semester or 40 Year Long

Standards

Learning Intentions

NC.M2.A.SSE.1a.

NC.M2.A.SSE.1b

NC.M2.F.IF.4

NC.M2.F.IF.7

NC.M2.F.IF.9

NC.M2.A.REI.7

Interpret, compare, and analyze quadratics in different representations (tables, graphs, algebraic expressions, and verbal descriptions).

NC.M2.APR.1

NC.M2.N.CN.1

NC.M2.A.SSE.3

NC.M2.A.REI.4a

NC.M2.A.REI.4b

NC.M2.A.REI.1

NC.M2.F.IF.8

Solve Quadratics algebraically using/by:

  • Factoring (M1)
  • Square Root Method (M1)
  • Quadratic Formula
  • Completing the Square

 

NC.M2.F.BF.1

NC.M2.F.BF.3

NC.M2.A.CED.1

NC.M2.A.CED.2

NC.M2.A.CED.3

Transformations and Modeling of Quadratics

 

Unit 3: Square Root & Inverse Variation Functions

Estimated Days: 15 Semester or 30 Year Long

Standards

Learning Intentions

NC.M2.N-RN.1

NC.M2.N-RN.2

Extend the properties of exponents to rational exponents.

NC.M2.N-RN.3

Use properties of rational and irrational numbers.

NC.M2.A-SSE.1

Interpret the structure of expressions.

NC.M2.A-CED.1

NC.M2.A-CED.2

NC.M2.F-BF.3

NC.M2.A-CED.3

Create equations that describe numbers or relationships.

NC.M2.A-REI.1

NC.M2.A-REI.2

Understand solving equations as a process of reasoning and explain the reasoning.

NC.M2.A-REI.11

Represent and solve equations and inequalities graphically.

NC.M2.F-IF.4

 

Interpret functions that arise in application in terms of the context.

NC.M2.F-IF.7

Analyze functions using different representations.

 

Unit 4: Similarity & Congruence

Estimated Days: 12 Semester or 24 Year Long

Standards

Learning Intentions

NC.M2.G-CO.9 (first 3 bullets)

NC.M2.G-SRT.1b

NC.M2.G-SRT.1c

NC.M2.G-SRT.1d

NC.M2.G-SRT.2

NC.M2.G-SRT.3

NC.M2.G-SRT.4 (first bullet)

NC.M2.G-CO.10 (last bullet)

A. Understand that dilations, similarity, and the properties of similar triangles allow the usage of the features of one figure to solve problems about a similar figure.

NC.M2.G-CO.6

NC.M2.G-CO.7

NC.M2.G-CO.8

NC.M2.G-CO.9 (last 2 bullets)

NC.M2.G-CO.10 (third bullet)

B. Proving and applying congruence provides a basis for modeling situations geometrically.

           

 

Unit 5: Trigonometry

Estimated Days: 13 Semester or 26 Year Long

Standards

Learning Intentions

NC.M2.G.CO.10a

NC.M2.G.CO.10b

NC.M2.G.SRT.4b

A. Understand, explain and justify properties of triangles.

 

NC.M2.G.SRT.12

B. Understand the need for and application of special right triangles

NC.M2.G.SRT.6

NC.M2.A.SSE.1a

NC.M2.A.CED.1

NC.M2.G.SRT.8

C. Understand the unique ratios that exist in right triangles and apply these ratios to solve problems.

 

 

 

 

 

 

 

 

 

Unit 6: Probability

Estimated Days: 10 Semester or 20 Year Long

Standards

Learning Intentions

NC.M2.S.IC.2

NC.M2.S.CP.1

NC.M2.S.CP.3

NC.M2.S.CP.4

NC.M2.S.CP.5

NC.M2.S.CP.6

NC.M2.S.CP.7

NC.M2.S.CP.8

Understand, explain, and use

  1. Conditional Probabilities
  2. The Addition Rule for probabilities
  3. The Multiplication Rule for probabilities

           

 

 

 

 

 

Based on the NC StandardCourse of Study for Mathematics

                                                                                                                                            1st 9 WEEKS: UNITS 1 – 6

 

Unit

Days

Unit

Days

Unit

Days

1

     8

2

       8

3

     10

Functions

Polynomials

Quadratic Functions

F. IF.2

F.IF.4

F.IF.5

F. BF. 3

F.BF.1

 

    A.APR.1              A.REI.10

    A.SSE.1               A.REI.11

    A.SSE.2               F.IF.7

    A.CED.1

     A.CED.2              A.CED.3

     A.REI.4               A.REI.7

     A.APR.3              A.REI.10

     F.BF.3                A. REI.11

     F.IF.7                 F.IF.8

Unit

Days

Unit

Days

Unit

Days

4

        9

5

       5

6

       5

Rational Functions

Rational Exponents & Radicals

Exponential Functions

     A.CED.1               F.IF.11

     A.CED.4               A.REI.10

     A.REI.2                 A.REI.11

     N.RN.2                    F.IF.7

     A.REI.2

     A.REI.10

     A.REI.11

     A.SSE.1               A.REI.10

     A.SSE.3               A.REI.11

     F.BF.3                  F.IF.7

 

                                                                                                                                          2nd 9 WEEKS:  UNITS 7 – 12

 

 

Unit

Days

Unit

Days

Unit

Days

7

      6

8

           9

9

       5

Preliminary Geometry

Triangles

Transformations

Review of basic geometry – vocab,

Angles, bisectors, etc.

                      G. CO.10

                      G.CO.13

                      G.SRT.1

                      G.SRT.6

F.BF.3

G.CO.2

G.CO.3

G.CO.4

G.CO.5

G.CO.6

G.CO.7

G.CO.8

Unit

Days

Unit

Days

Unit

Days

10

     6

11

       7

12

        3

Trigonometric Functions

Probability

Perimeter, Area & Volume

A.CED.2

                        F.IF.4

                        F.IF.5

                        F.IF.7

                        F.IF.9

                        G.SRT.8

                        F.BF.3

                        G.GPE.6

                        G.SRT.7

                        G.SRT.9

                        G.SRT.11

 

 

                          S.CP.1

                          S.CP.2

                          S.CP.3

                          S.CP.4

                          S.CP.5

                          S.CP.6

                          S.CP.7

                          S.CP.8

                          S.CP.9

                          S.IC.2

                          S.IC.6

 

 

G.GMD.4

G.MG.1

G.MG.2

G.MG.3

                                                                                                                           

PowerStandards                     A.REI.1, N.Q.1, N.Q.2, and N.Q.3 will be reinforced throughout Math II

 

 Revised: August 2014       

 

Math II

 

The RealNumber System                                                                                                              N-RN

Extend the propertiesof exponents to rational exponents.

N-RN.2 Rewriteexpressions involving radicals and rational exponents using the properties ofexponents.

 

Quantities                                                                                                                                           N-Q

Reason quantitativelyand use units to solve problems.

N-Q.1 Use unitsas a way to understand problems and to guide the solution of multi-stepproblems; choose and interpret units consistently in formulas; choose andinterpret the scale and the origin in graphs and data displays.

 

N-Q.2 Defineappropriate quantities for the purpose of descriptive modeling.

 

N-Q.3 Choose alevel of accuracy appropriate to limitations on measurement when reportingquantities.

 

SeeingStructure in Expressions                                                                                                   A-SSE

Interpret the structure of expressions.

A-SSE.1 Interpretexpressions 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 ormore of their parts as a single entity. Forexample, interpret P(1+r)n as the product of P and a factor notdepending on P.

 

Note:  At this level include polynomialexpressions 

 

A-SSE.2 Use thestructure 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 squaresthat can be factored as (x2 – y2)(x2 + y2).

 

Write expressions in equivalent forms to solveproblems.

A-SSE.3 Chooseand produce an equivalent form of an expression to reveal and explainproperties of the quantity represented by the expression.

c.      Use the properties of exponents to transformexpressions for exponential functions. Forexample the expression 1.15t can be rewritten as (1.151/12)12t≈ 1.01212t to reveal the approximate equivalent monthly interestrate if the annual rate is 15%.

 

Arithmetic with Polynomials& Rational Expressions                                                               A-APR

Perform arithmetic operations on polynomials.

A-APR.1 Understandthat polynomials form a system analogous to the integers, namely, they areclosed under the operations of addition, subtraction, and multiplication; add,subtract, and multiply polynomials.

 

Note:  At this level, add and subtract anypolynomial and extend multiplication to as many as three linearexpressions. 

 

Understand the relationship between zeros and factorsof polynomials.

A-APR.3 Identifyzeros of polynomials when suitable factorizations are available, and use thezeros to construct a rough graph of the function defined by the polynomial.

 

Note:  At this level, limit to quadraticexpressions. 

 

 

 

CreatingEquations                                                                                                                      A-CED

Create equations that describe numbers orrelationships.

A-CED.1 Createequations and inequalities in one variable and use them to solve problems. Include equations arising from linear andquadratic functions, and simple rational and exponential functions.

 

Note:  At this level extend to quadratic and inversevariation (the simplest rational) functions and use common logs to solveexponential equations.  

 

A-CED.2 Createequations in two or more variables to represent relationships betweenquantities; graph equations on coordinate axes with labels and scales.

 

Note:  At this level extend to simple trigonometricequations that involve right triangle trigonometry.

 

A-CED.3 Representconstraints by equations or inequalities, and by systems of equations and/orinequalities, and interpret solutions as viable or nonviable options in amodeling context. For example, representinequalities describing nutritional and cost constraints on combinations ofdifferent foods.

 

Note:  Extend to linear-quadratic, andlinear–inverse variation (simplest rational) systems of equations.   

 

A-CED.4 Rearrangeformulas to highlight a quantity of interest, using the same reasoning as insolving equations. For example, rearrangeOhm’s law V = IR to highlight resistance R.

 

Note:  At this level, extend to compound variationrelationships.

 

Reasoning withEquations & Inequalities                                                                                    A-REI

Understand solvingequations as a process of reasoning and explain the reasoning.

A-REI.1 Explaineach step in solving a simple equation as following from the equality ofnumbers asserted at the previous step, starting from the assumption that theoriginal equation has a solution. Construct a viable argument to justify asolution method.

 

Note:  At this level, limit to factorablequadratics.

 

A-REI.2 Solvesimple rational and radical equations in one variable, and give examplesshowing how extraneous solutions may arise.

 

Note:  At this level, limit to inverse variation.

 

Solve equations and inequalities in one variable.

A-REI.4 Solvequadratic equations in one variable.

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 theequation. Recognize when the quadratic formula gives complex solutions andwrite them as a ± bi for real numbers a and b.

 

Note:  At this level, limit solving quadraticequations by inspection, taking square roots, quadratic formula, and factoringwhen lead coefficient is one.  Writingcomplex solutions is not expected; however recognizing when the formulagenerates non-real solutions is expected.   

 

Solve systems of equations.

A-REI.7 Solve a simple system consisting of a linear equation and aquadratic equation in two variables algebraically and graphically. For example,find the points of intersection between the line

y = –3x and the circle x2 + y2= 3.

 

 

Represent and solve equations and inequalitiesgraphically.

A-REI.10 Understandthat the graph of an equation in two variables is the set of all its solutionsplotted in the coordinate plane, often forming a curve (which could be a line).

 

Note:  At this level, extend to quadratics.

 

A-REI.11 Explainwhy the x-coordinates of the pointswhere 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 findsuccessive approximations. Include cases where f(x) and/or g(x)are linear, polynomial, rational, absolute value, exponential, and logarithmicfunctions.

 

Note:  At this level, extend to quadraticfunctions. 

 

InterpretingFunctions                                                                                                                       F-IF

Understand the concept of a function and use functionnotation.

F-IF.2 Usefunction notation, evaluate functions for inputs in their domains, andinterpret statements that use function notation in terms of a context.

 

Note:  At this level, extend to quadratic, simplepower, and inverse variation functions.

 

Interpret functions that arise in applications interms of the context.

F-IF.4 For afunction that models a relationship between two quantities, interpret keyfeatures of graphs and tables in terms of the quantities, and sketch graphsshowing key features given a verbal description of the relationship. Key features include: intercepts; intervalswhere the function is increasing, decreasing, positive, or negative; relativemaximums and minimums; symmetries; end behavior; and periodicity.

 

Note:  At this level, limit to simple trigonometricfunctions (sine, cosine, and tangent in standard position) with angle measuresof 180b.     Graph square root, cube root, and piecewise-definedfunctions, including step functions and absolute value functions.

e.      Graph exponential and logarithmic functions, showingintercepts and end behavior, and trigonometric functions, showing period,midline, and amplitude.

 

Note:  At this level, extend to simple trigonometricfunctions (sine, cosine, and tangent in standard position)

 

 

 

 

 

F-IF.8 Write afunction defined by an expression in different but equivalent forms to revealand explain different properties of the function.

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

 

Note:  At this level, completing the square is stillnot expected.

 

F-IF.9 Compareproperties 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 quadraticfunction and an algebraic expression for another, say which has the largermaximum.

 

Note:  At this level, extend to quadratic, simplepower, and inverse variation functions.

 

BuildingFunctions                                                                                                                           F-BF

Build a function that models a relationship betweentwo quantities.

F-BF.1 Write afunction that describes a relationship between two quantities.

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

 

Note:  Continue to allow informal recursive notationthrough this level.

 

b.     Combine standard function types using arithmeticoperations.  For example, build a function that models the temperature of a coolingbody by adding a constant function to a decaying exponential, and relate thesefunctions to the model.

 

Build new functions from existing functions.

F-BF.3 Identifythe effect on the graph of replacing f(x) by f(x) + k,k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find thevalue of k given the graphs.Experiment with cases and illustrate an explanation of the effects on the graphusing technology. Include recognizing even and odd functions from their graphsand algebraic expressions for them.

 

Note:  At this level, extend to quadratic functionsand, k f(x).

 

Congruence                                                                                                                                     G-CO

Experiment withtransformations in the plane

G-CO.2 Representtransformations in the plane using, e.g., transparencies and geometry software;describe transformations as functions that take points in the plane as inputsand give other points as outputs. Compare transformations that preservedistance and angle to those that do not (e.g., translation versus horizontalstretch).

 

G-CO.3 Given arectangle, parallelogram, trapezoid, or regular polygon, describe the rotationsand reflections that carry it onto itself.

 

G-CO.4 Developdefinitions of rotations, reflections, and translations in terms of angles,circles, perpendicular lines, parallel lines, and line segments.

 

G-CO.5 Given ageometric figure and a rotation, reflection, or translation, draw thetransformed figure using, e.g., graph paper, tracing paper, or geometrysoftware. Specify a sequence of transformations that will carry a given figureonto another.

 

 

 

 

 

Understand congruencein terms of rigid motions

G-CO.6 Usegeometric descriptions of rigid motions to transform figures and to predict theeffect of a given rigid motion on a given figure; given two figures, use thedefinition of congruence in terms of rigid motions to decide if they arecongruent.

 

G-CO.7 Use thedefinition of congruence in terms of rigid motions to show that two trianglesare congruent if and only if corresponding pairs of sides and correspondingpairs of angles are congruent.

 

G-CO.8 Explainhow the criteria for triangle congruence (ASA, SAS, and SSS) follow from thedefinition of congruence in terms of rigid motions.

 

Prove geometrictheorems

G-CO.10 Provetheorems about triangles. Theoremsinclude: measures of interior angles of a triangle sum to 180°; base angles ofisosceles triangles are congruent; the segment joining midpoints of two sidesof a triangle is parallel to the third side and half the length; the medians ofa triangle meet at a point.

 

Note:  At this level, include measures of interiorangles of a triangle sum to 180° and the segment joining midpoints of two sidesof a triangle is parallel to the third side and half the length.

 

Make geometricconstructions

G-CO.13 Constructan equilateral triangle, a square, and a regular hexagon inscribed in a circle.

 

Similarity,Right Triangles, & Trigonometry                                                                             G-SRT

Understand similarityin terms of similarity transformations

G-SRT.1 Verifyexperimentally the properties of dilations given by a center and a scale factor:

a.      A dilation takes a line not passing through the centerof the dilation to a parallel line, and leaves a line passing through thecenter unchanged.

b.     The dilation of a line segment is longer or shorter inthe ratio given by the scale factor.

 

Define trigonometricratios and solve problems involving right triangles

G-SRT.6Understand that by similarity, side ratios in right triangles are properties ofthe angles in the triangle, leading to definitions of trigonometric ratios foracute angles.

 

G-SRT.7 Explainand use the relationship between the sine and cosine of complementary angles.

 

G-SRT.8 Usetrigonometric ratios and the Pythagorean Theorem to solve right triangles inapplied problems.

 

Applytrigonometry to general triangles

G-SRT.9 (+) Derive the formula A = 1/2 absin(C) for the area of a triangle by drawing an auxiliary line from a vertexperpendicular to the opposite side.

 

G-SRT.11 (+) Understand and apply the Law of Sines and the Law of Cosines tofind unknown measurements in right and non-right triangles (e.g., surveyingproblems, resultant forces).

 

 

 

 

 

 

 

 

ExpressingGeometric Properties with Equations                                                                    G-GPE

Translate between thegeometric description and the equation for a conic section

G-GPE.1 Derivethe equation of a circle of given center and radius using the PythagoreanTheorem; complete the square to find the center and radius of a circle given byan equation.

 

Note:  At this level, derive the equation of thecircle using the Pythagorean Theorem.

 

G-GPE.6 Find the point on a directed line segmentbetween two given points that partitions the segment in a given ratio.

 

GeometricMeasurement and Dimension                                                                                  G-GMD

Visualizerelationships between two-dimensional and three-dimensional objects

G-GMD.4 Identifythe shapes of two-dimensional cross-sections of three-dimensional objects, andidentify three-dimensional objects generated by rotations of two-dimensionalobjects.

 

Modeling withGeometry                                                                                                               G-MG

Apply geometricconcepts in modeling situations

G-MG.1 Usegeometric shapes, their measures, and their properties to describe objects(e.g., modeling a tree trunk or a human torso as a cylinder).

 

G-MG.2 Applyconcepts of density based on area and volume in modeling situations (e.g.,persons per square mile, BTUs per cubic foot).

 

G-MG.3 Applygeometric methods to solve design problems (e.g., designing an object orstructure to satisfy physical constraints or minimize cost; working withtypographic grid systems based on ratios).

 

MakingInferences & Justifying Conclusions                                                                                 S-IC

Understand andevaluate random processes underlying statistical experiments

S-IC.2 Decide ifa specified model is consistent with results from a given data-generatingprocess, e.g., using simulation. Forexample, 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?

 

Make inferences andjustify conclusions from sample surveys, experiments, and observational studies

S-IC.6 Evaluatereports based on data.

 

ConditionalProbability and the Rules of Probability                                                                  S-CP

Understandindependence and conditional probability and use them to interpret data

S-CP.1 Describe events as subsets of asample space (the set of outcomes) using characteristics (or categories) of theoutcomes, or as unions, intersections, or complements of other events (“or,”“and,” “not”).

 

S-CP.2 Understand that two events A and B are independent if the probability of A and B occurringtogether is the product of their probabilities, and use this characterizationto determine if they are independent.

 

S-CP.3 Understand the conditionalprobability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying thatthe conditional probability of Agiven B is the same as theprobability of A, and the conditionalprobability of B given A is the same as the probability of B.

 

 

 

 

S-CP.4 Construct and interpret two-wayfrequency tables of data when two categories are associated with each objectbeing classified. Use the two-way table as a sample space to decide if eventsare independent and to approximate conditional probabilities. For example, collect data from a randomsample of students in your school on their favorite subject among math,science, and English. Estimate the probability that a randomly selected studentfrom your school will favor science given that the student is in tenth grade.Do the same for other subjects and compare the results.

 

S-CP.5 Recognize and explain the conceptsof conditional probability and independence in everyday language and everydaysituations. For example, compare thechance of having lung cancer if you are a smoker with the chance of being asmoker if you have lung cancer.

 

Use the rules ofprobability to compute probabilities of compound events in a uniformprobability model

S-CP.6 Find the conditional probability ofA given B as the fraction of B’soutcomes that also belong to A, andinterpret the answer in terms of the model.

 

S-CP.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.

 

S-CP.8 (+) Apply the generalMultiplication 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.

 

S-CP.9 (+) Use permutations andcombinations to compute probabilities of compound events and solve problems.

 

 

 

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