Math III Pacing Guide 2013-2014

Brunswick County Schools

Math III Pacing Guide 2013-2014 Brunswick County Schools Math III Pacing Guide 2013-2014

Topic Days Basics of Geometry 3 Congruence 4 Triangles 3 Similarity 6 Trig and Trigonometric Functions 4 Quadrilaterals 3 Circles 8 Concepts of Algebra 5 Benchmark & Review 2 Equations and Functions 6 Systems of Equations 4 Polynomials 9 Radicals & Complex Numbers 3 Conics 6 Rational Expressions & Equations 6 Exponential & Logarithmic Functions 6 Statistics & Probability 5 EOC Review 2 Final Exams 6 Total: 91

Math Resources are located at the Brunswick Count High School Mathematics Webpage: http://www.bcswan.net/education/components/scrapbook/default.php?sectiondetailid=17599

Assessments: Teachers should monitor and assess students daily. Suggestions for Summative Formative Assessments: Projects, Tests, Quizzes…Suggestions for Formative Assessments: Exit slip, group work, labs, drills, discussions, journal entries/student reflection, Study Island……

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COMMON CORE STATE STANDARDS FOR MATHEMATICS Standards for Mathematical Practice

Mathematical Practice DO STUDENTS: MP 1: Make sense of problems and persevere  Use multiple representations (verbal descriptions, symbolic, tables, in solving them. graphs, etc.)?  Check their answers using different methods?  Continually ask, “Does this make sense?”  Understand the approaches of others and identify correspondences between different approaches? MP 2: Reason abstractly and quantitatively.  Make sense of quantities and their relationships in problem situations?  Decontextualize a problem?  Contextualize a problem?  Create a coherent representation of the problem, consider the units involved, and attend to the meaning of quantities? MP 3: Construct viable arguments and critique  Make conjectures and build a logical progression of statements to the reasoning of others. explore the truth of their conjectures?  Analyze situations and recognize and use counter examples?  Justify their conclusions communicate them to others, and respond to arguments of others?  Hear or read arguments of others, and decide whether they make sense, and ask useful questions to clarify or improve the argument? MP 4: Model with mathematics.  Apply the mathematics they know to solve problems in everyday life?  Apply what they know and make assumptions and approximations to simplify a complicated situation as an initial approach?  Identify important quantities in a practical situation?  Analyze relationships mathematically to draw conclusions?  Interpret their mathematical results in the context of the situation and reflect on whether the results make sense? MP 5: Use appropriate tools strategically.  Consider the available tools when solving mathematical problems?  Know the tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful? Assessments: Teachers should monitor and assess students daily. Suggestions for Summative Formative Assessments: Projects, Tests, Quizzes…Suggestions for Formative Assessments: Exit slip, group work, labs, drills, discussions, journal entries/student reflection, Study Island……

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 Identify relevant external mathematical resources and use them to pose or solve problems?  Use technological tools to explore and deepen their understanding of concepts? MP 6: Attend to precision.  Communicate precisely to others?  Use clear definitions?  Use the equal sign consistently and appropriately?  Calculate accurately and efficiently? MP 7: Look for and make use of structure.  Look closely to determine a pattern or structure?  Utilize properties?  Decompose and recombine numbers and expressions?  Understand equivalence? MP 8: Look for and express regularity in  Notice if calculations are repeated, and look both for general methods repeated reasoning. and for shortcuts?  Maintain oversight of the process, while attending to the details?  Continually evaluate the reasonableness of their intermediate result?

7/10/12 Page 4 of 44 Brunswick County Schools Math III Pacing Guide 2013-2014 Math III CCSS/NCDPI Standard Course of Study The Real Number System N-RN Use properties of rational and irrational numbers. N-RN.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. Quantities N-Q Reason quantitatively and use units to solve problems. N-Q.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.

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

N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. Seeing Structure in Expressions A-SSE Interpret the structure of expressions. A-SSE.1Interpret 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. Note: At this level, limit to linear expressions, exponential expressions with integer exponents and quadratic expressions.

A-SSE.2Use 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).

Write expressions in equivalent forms to solve problems. A-SSE.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.

Note: At this level, the limit is quadratic expressions of the form ax2 + bx + c. Arithmetic with Polynomials & Rational Expressions A-APR Perform arithmetic operations on polynomials.

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A-APR.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.

Note: At this level, limit to addition and subtraction of quadratics and multiplication of linear expressions.

Creating Equations A-CED Create equations that describe numbers or relationships. A-CED.1Create 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.

Note: At this level, focus on linear and exponential functions.

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

Note: At this level, focus on linear, exponential and quadratic. Limit to situations that involve evaluating exponential functions for integer inputs.

A-CED.3Represent 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.

Note: At this level, limit to linear equations and inequalities.

A-CED.4Rearrange 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.

Note: At this level, limit to formulas that are linear in the variable of interest, or to formulas involving squared or cubed variables.

Reasoning with Equations & Inequalities A-REI Understand solving equations as a process of reasoning and explain the reasoning. Assessments: Teachers should monitor and assess students daily. Suggestions for Summative Formative Assessments: Projects, Tests, Quizzes…Suggestions for Formative Assessments: Exit slip, group work, labs, drills, discussions, journal entries/student reflection, Study Island……

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A-REI.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.

Solve equations and inequalities in one variable. A-REI.3Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Solve systems of equations. A-REI.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.

A-REI.6Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

Represent and solve equations and inequalities graphically. A-REI.10Understand 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).

Note: At this level, focus on linear and exponential equations.

A-REI.11 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.

Note: At this level, focus on linear and exponential functions.

A-REI.12Graph 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 to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. Interpreting Functions F-IF Understand the concept of a function and use function notation. F-IF.1Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly

7/10/12 Page 7 of 44 Brunswick County Schools Math III Pacing Guide 2013-2014 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).

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

Note: At this level, the focus is linear and exponential functions.

F-IF.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.

Interpret functions that arise in applications in terms of the context. F-IF.4For 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.

Note: At this level, focus on linear, exponential and quadratic functions; no end behavior or periodicity.

F-IF.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.

Note: At this level, focus on linear and exponential functions.

F-IF.6Calculate 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.

Note: At this level, focus on linear functions and exponential functions whose domain is a subset of the integers.

Analyze functions using different representations. Assessments: Teachers should monitor and assess students daily. Suggestions for Summative Formative Assessments: Projects, Tests, Quizzes…Suggestions for Formative Assessments: Exit slip, group work, labs, drills, discussions, journal entries/student reflection, Study Island……

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F-IF.7Graph 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. . Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

Note: At this level, for part e, focus on exponential functions only.

F-IF.8Write 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.

Note: At this level, only factoring expressions of the form ax2 + bx +c, is expected. Completing the square is not addressed at this level.

b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.

F-IF.9Compare 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.

Note: At this level, focus on linear, exponential, and quadratic functions

Building Functions F-BF Build a function that models a relationship between two quantities. F-BF.1Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

Note: At this level, limit to addition or subtraction of constant to linear, exponential or quadratic functions or addition of linear functions to linear or Assessments: Teachers should monitor and assess students daily. Suggestions for Summative Formative Assessments: Projects, Tests, Quizzes…Suggestions for Formative Assessments: Exit slip, group work, labs, drills, discussions, journal entries/student reflection, Study Island……

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

Note: At this level, formal recursive notation is not used. Instead, use of informal recursive notation (such as NEXT = NOW + 5 starting at 3) is intended.

Build new functions from existing functions. F-BF.3 Identify the 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 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.

Note: At this level, limit to vertical and horizontal translations of linear and exponential functions. Even and odd functions are not addressed. Linear, Quadratic, & Exponential Models F-LE Construct and compare linear and exponential models and solve problems. F-LE.1Distinguish 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.

F-LE.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).

F-LE.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.

Note: At this level, limit to linear, exponential, and quadratic functions; general polynomial functions are not addressed.

Interpret expressions for functions in terms of the situation they model. Assessments: Teachers should monitor and assess students daily. Suggestions for Summative Formative Assessments: Projects, Tests, Quizzes…Suggestions for Formative Assessments: Exit slip, group work, labs, drills, discussions, journal entries/student reflection, Study Island……

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F-LE.5Interpret the parameters in a linear or exponential function in terms of a context. Congruence G-CO Experiment with transformations in the plane. G-CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

Note: At this level, distance around a circular arc is not addressed. Expressing Geometric Properties with Equations G-GPE Use coordinates to prove simple geometric theorems algebraically. G-GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

Note:Conics is not the focus at this level, therefore the last example is not appropriate here.

G-GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

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

Note: At this level, focus on finding the midpoint of a segment.

G-GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Geometric Measurement & Dimension G-GMD Explain volume formulas and use them to solve problems. G-GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

Note: Informal limit arguments are not the intent at this level.

G-GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.*

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Note: At this level, formulas for pyramids, cones and spheres will be given. Interpreting Categorical & Quantitative Data S-ID Summarize, represent, and interpret data on a single count or measurement variable. S-ID.1Represent data with plots on the real number line (dot plots, histograms, and box plots).

S-ID.2Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

S-ID.3Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

Summarize, represent, and interpret data on two categorical and quantitative variables. S-ID.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

S-ID.6Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear and exponential models. b. Informally assess the fit of a function by plotting and analyzing residuals.

Note: At this level, for part b, focus on linear models.

c. Fit a linear function for a scatter plot that suggests a linear association.

Interpret linear models. S-ID.7Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

S-ID.8Compute (using technology) and interpret the correlation coefficient of a linear fit.

S-ID 9Distinguish between correlation and causation.

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7/10/12 Page 13 of 44 Brunswick County Schools Math III Pacing Guide 2013-2014 The pacing guide should be used along with NCSCOS and the Geometry Crosswalks To be Addressed Through-out the Course When Appropriate  N-Q.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.  N-Q.2 Define appropriate quantities for the purpose of descriptive modeling.  N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. Basics of Geometry - 3 days Standards: G.CO.1 Learning Targets Glencoe Geometry Vocabulary Sample Questions G.CO.1 - Know precise definitions of G-CO.1 Definitions are used to begin building G.CO.1 angle, circle, perpendicular line, parallel blocks for proof. Infuse these definitions into proofs Given a picture identify and name line, and line segment, based on the and other problems. Pay attention to Mathematical various geometric figures. undefined notions of point, line, distance practice 3 “Construct viable arguments and critique along a line, and distance around a circular the reasoning of others: Understand and use stated Solve problems involving Segment, arc. assumptions, definitions and previously established results in constructing arguments.” Also Angle, and Arc Addition  Identify, name, and illustrate the mathematical practice number six says, “Attend to undefined terms - point, line, precision: Communicate precisely to others and use and plane, line segments, rays, clear definitions in discussion with others and in their and angles, perpendicular and own reasoning.” parallel lines, circles and arcs. Know that a point has position, no thickness or  Use appropriate tools distance. A line is made of infinitely many points, (ruler/protractor) discuss and a line segment is a subset of the points on a line measures of segments and with endpoints. A ray is defined as having a point on angles, and classify angles as one end and a continuing line on the other. acute, obtuse, or right. An angle is determined by the intersection of two rays.  Identify vertical angles and A circle is the set of infinitely many points that are linear pairs the same distance from the center forming a circular  Use segment and angle addition are, measuring 360 degrees. to find missing measures. Perpendicular lines are lines in the interest at a point Include to form right angles. complementary/supplementary Parallel lines that lie in the same plane and are lines angles and definitions of in which every point is equidistant from the Assessments: Teachers should monitor and assess students daily. Suggestions for Summative Formative Assessments: Projects, Tests, Quizzes…Suggestions for Formative Assessments: Exit slip, group work, labs, drills, discussions, journal entries/student reflection, Study Island……

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midpoint and bisect corresponding point on the other line.  Compare/contrast distance along an arc with distance along a line (no formulas) Congruence – 4 days Standards: G.CO.9, G.CO.12 Learning Targets Glencoe Vocabulary Sample Questions G.CO.12 - Make formal geometric Geometry G.CO.9 constructions with a variety of tools and G.CO.12 Prove that HIB DJG, given that AB // DE . methods (compass and straightedge, string, reflective devices, paper folding, dynamic Copy a segment: geometric software, etc.). Copying a Pg. 15 segment; copying an angle; bisecting a segment; bisecting an angle; constructing Bisect a segment: perpendicular lines, including the pg. 24 perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Copy an angle:  Make formal geometric pg. 31 constructions of the following using a variety of tools and Bisect an angle: methods (compass and pg. 33 straightedge, string, reflective devices, paper folding, Construct a Geometer’s Sketchpad, etc.). perpendicular Copy a segment; copy an angle; line: bisect a segment; bisect an angle. pg. 44 Verify constructions using the definition of congruence in terms G.CO.9 of rigid transformations Vertical Angles: Section 1.5 & G.CO.9 - Prove theorems about lines and pg. 109 & 110 angles. Theorems include: vertical angles are congruent; when a transversal crosses Angle Assessments: Teachers should monitor and assess students daily. Suggestions for Summative Formative Assessments: Projects, Tests, Quizzes…Suggestions for Formative Assessments: Exit slip, group work, labs, drills, discussions, journal entries/student reflection, Study Island……

7/10/12 Page 15 of 44 Brunswick County Schools Math III Pacing Guide 2013-2014 parallel lines, alternate interior angles are Relationships: congruent and corresponding angles are Section 2.8 (Use congruent; points on a perpendicular bisector of a line segment are exactly those these proofs to equidistant from the segment’s endpoints. introduce proof  Prove vertical angles are methods) congruent. Verify using the definition of congruence in terms Parallel Lines & of rigid transformations Angles: Section  Identify transversal, alternate 3.1 & 3.2 (Focus interior angles, corresponding on proofs) angles, especially in the context of parallel lines  Prove alternate interior and Hypothesis/Concl corresponding angles are usion: Section 2.3 congruent when a transversal crosses parallel lines. Verify using the definition of congruence in terms of rigid transformations  Prove additional theorems about lines and angles.  Review and Assessment

Triangles - 2 days (End Unit w/ Review and Test) – 2 Days Standards: G.CO.10 Learning Targets Glencoe Vocabulary Sample Questions G.CO.10 - Prove theorems about Geometry G-CO.10 triangles. Theorems include: measures of Using any method you choose, construct the medians of interior angles of a triangle sum to 180°; a triangle. Each median is divided up by the centroid. base angles of isosceles triangles are Investigate the relationships of the distances of these Assessments: Teachers should monitor and assess students daily. Suggestions for Summative Formative Assessments: Projects, Tests, Quizzes…Suggestions for Formative Assessments: Exit slip, group work, labs, drills, discussions, journal entries/student reflection, Study Island……

7/10/12 Page 16 of 44 Brunswick County Schools Math III Pacing Guide 2013-2014 congruent; the segment joining midpoints segments. Can you create a deductive argument to justify of two sides of a triangle is parallel to the why these relationships are true? Can you prove why the third side and half the length; the medians medians all meet at one point for all triangles? of a triangle meet at a point. Extension:  Prove points on a perpendicular using coordinate geometry, how can you calculate the bisector of a line segment are the coordinate of the centroid? Can you provide an algebraic same distance from the endpoints argument for why this works for any triangle? using congruent triangles and Using Interactive Geometry Software or tracing paper, CPCTC investigate the relationships of sides and angles when  Prove the measures of interior you angles of a triangle have a sum of connect the midpoints of the sides of a triangle. Using 180 coordinates can you justify why the segment that connects  Review triangles by sides and the midpoints of two of the sides is parallel to the angles. opposite side. If you have not done so already, can you  Define congruence and congruent generalize your argument and show that it works for all triangles cases? Using coordinates justify that the segment that  Identify base angles of an connects the midpoints of two of the sides is half the isosceles triangle and prove they length of the opposite side. If you have not done so are congruent already, G.SRT.5 - Use congruence and similarity can you generalize your argument and show that it works criteria for triangles to solve problems for all cases? and to prove relationships in geometric G.CO.10 figures. Given that ΔABC is isosceles, prove that ABCACB .  Prove additional theorems about triangles.  Prove triangle congruence  Combine Basics of Geometry, Congruence, and Triangles for Review/Test

7/10/12 Page 17 of 44 Brunswick County Schools Math III Pacing Guide 2013-2014 Similarity – 6 days Standards: G-SRT.2, G-SRT.3, G-SRT.4, G-SRT.5, G-CO.10 Learning Targets Glencoe Vocabulary Sample Questions  Review rational expressions, Geometry G-SRT.2 Use the idea of geometric transformations to proportions and scale factor develop the definition of similarity. G.SRT.2 G-SRT.2 Given two figures determine whether they are similar and explain their similarity based on the G.SRT.2 - Given two figures, use the Similar Polygons: congruency of corresponding angles and the proportionality definition of similarity in terms of Section 6.2 pg. of corresponding sides. similarity transformations to decide if they 289 Instructional note: The ideas of congruency and similarity are are similar; explain using similarity related. It is important for students to connect that transformations the meaning of similarity Similar Triangles: congruency is a special case of similarity with a scale factor for triangles as the equality of all Section 6.3 (AA of one. Therefore these similarity rules can be corresponding pairs of angles and the only) expanded to work for congruency in triangles. AA similarity proportionality of all corresponding pairs of is the foundation for ASA and AAS congruency sides. G.SRT.4 theorems.  Describe the differences Proportional Knowing from the definition of a dilation, angle measures are between rigid and similarity Parts: preserved and sides change by a multiplication of scale factor k. transformations, and develop a Section 6.4 (Proof Using a document camera and an LCD projector, Vladimir definition of similar figures pg. 307) (corresponding angles are projects the following image the wall in his classroom. congruent, corresponding sides He wants to know if the original image is similar to the one. G.SRT.5 How could he verify that they are similar. If the are proportional according to Section 6. 3 preimage and image scale factor) (Use Geometer’s aren’t similar, what Sketchpad) could be the reason?  Extend the definition of If the preimage and similarity in terms of similarity image are similar, transformations to explain that how could he triangles are similar if and only find the scale factor? if all corresponding pairs of sides are proportional and all corresponding pairs of angles are congruent

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 Use proportionality of sides and congruence of angles to determine if two figures are similar, given varied examples of pre-image and images (including triangles)

G.SRT.3 - Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. G.SRT.2  Create triangles of varied size given two angles and make Are these two figures similar? Explain why or why not. conclusions about similarity in these examples  Reason that two pairs of congruent corresponding angles is sufficient to determine similar triangles (AA~) Are they similar? G.SRT.4 - Prove theorems about http://illustrativemathematics.org/illustrations/603 triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. How far can you go in a new York minute:  Prove triangles similar http://illuminations.nctm.org/LessonDetail.aspx?  Prove a line parallel to one side id=L848 of a triangle divides the other G-SRT.3 Use the properties of similarity transformations to two proportionally develop the criteria for proving similar triangles by  Prove the segment joining AA, SSS, and SAS. midpoints of two sides of a Connect this standard with standard G-SRT.4 triangle is parallel to the third side and half the length  Prove additional theorems about Assessments: Teachers should monitor and assess students daily. Suggestions for Summative Formative Assessments: Projects, Tests, Quizzes…Suggestions for Formative Assessments: Exit slip, group work, labs, drills, discussions, journal entries/student reflection, Study Island……

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similar figures  Prove the Pythagorean Theorem using similar triangles (geometric mean)

G.SRT.5 - Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.  Solve problems using congruency and similarity (Level III) theorems Given that VMNP is a dilation of VABC with scale factor k, use properties of dilations to show that the AA criterion  Prove relationships in geometric figures using congruency and is sufficient to prove similarity. G.SRT.3 similarity theorems Are all right triangles similar to one another? How do  Review/Test you know?

G.SRT.4 Prove that if two triangles are similar, then the ratio of corresponding altitudes is equal to the ratio of corresponding sides.

Trigonometry and Trigonometry Functions – 4 days Standards: F-TF.1, F-TF.2, F-TF.5, F-TF.8, F-IF.7e Learning Targets Glencoe Vocabulary Sample Questions F-TF.1 Understand radian measure of an Geometry angle as the length of the arc on the unit  13.1, page circle subtended by the angle. 699, 820 F-TF.5-As the Wheel Turns  Arc Length and  13.2 http://illustrativemathematics. radian/degree conversions  13.2 org/illustrations/595 F-TF.2 Explain how the unit circle in the  13.3-13.5 coordinate plane enables the extension of trigonometric functions to all real numbers, Assessments: Teachers should monitor and assess students daily. Suggestions for Summative Formative Assessments: Projects, Tests, Quizzes…Suggestions for Formative Assessments: Exit slip, group work, labs, drills, discussions, journal entries/student reflection, Study Island……

7/10/12 Page 20 of 44 Brunswick County Schools Math III Pacing Guide 2013-2014 interpreted as radian measures of angles 13.6-13.7 traversed counterclockwise around the unit circle  The Unit Circle (extend past one revolution) F-TF.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.  Trigonometric Identities – prove and apply ○ Sine, cosine, tangent, sin2x+cos2x=1 F-TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. F-IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in 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 Graph trigonometric functions – sine, cosine, and tangent (simple translations: model Review/Assessment Quadrilaterals - 3 days Standards: G.CO.11 Learning Targets Glencoe Vocabulary Sample Questions  Review quadrilaterals and their Geometry  Focus on parallelograms (rectangles, and squares, properties rhombi)

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G-CO.11 - Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.  Prove theorems of parallelograms o Opposite sides are congruent o Opposite angles are congruent o Diagonals of parallelograms bisect each other o Conversely, rectangles are parallelograms with congruent diagonals. Circles – 8 days Standards: G.C.1, G.C.2, G.C.3, G.C.5 Learning Targets Glencoe Vocabulary Sample Questions Review definition and Geometry G.C.1 characteristics of circles G.C.1 Draw or find examples of Chapter 10 several different circles. In G.C.1 - Prove that all circles are similar. what ways are they related?  Informally prove that all circles Activity: pg. 524 How can you describe this are similar based on the scale relationship in terms of factor π G.C.2 geometric ideas? Form a Section 10.2 hypothesis and prove it. G.C.5 - Derive using similarity the fact Angles Only that the length of the arc intercepted by an

7/10/12 Page 22 of 44 Brunswick County Schools Math III Pacing Guide 2013-2014 angle is proportional to the radius, and Section 10.4 define the radian measure of the angle as Angles Only the constant of proportionality; derive the formula for the area of a sector.  Review formulas for the area Section 10.5 and circumference of a circle Theorem 10.9  Find the length of an intercepted only arc as a fraction of the circumference  Identify that radii and lengths of arc intercepted by congruent central angles are proportional  Relate that the constant of this G.C.5 proportion is the radian measure Two Wheels and a Belt: of the central angle http://illustrativemathematics. G.C.2 – Identify and describe org/illustrations/621 relationships among inscribed angles, Setting Up Sprinklers: radii, and chords. Include the relationship between central, inscribed, and http://illustrativemathematics. circumscribed angles; inscribed angles on org/illustrations/607 a diameter are right angles; the radius of a circle is perpendicular to the tangent G.C.2 where the radius intersects the circle. power of points  Define a central angle (formed http://illuminations.nctm.org/ by 2 radii) and determine its LessonDetail.aspx?id=L700 measure is equal to measure of intercepted arc  Define an inscribed angle (formed by 2 chords with Given the circle below with common endpoint) and radius of 10 and chord length determine its measure is equal to of 12, find the distance from half the measure of the the chord to the center of the intercepted arc circle.  Derive a formula for the area of a sector Assessments: Teachers should monitor and assess students daily. Suggestions for Summative Formative Assessments: Projects, Tests, Quizzes…Suggestions for Formative Assessments: Exit slip, group work, labs, drills, discussions, journal entries/student reflection, Study Island……

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 Identify the inscribed angle formed on a diameter (intercepts a semicircle) as a right angle  Define tangent and secant of a circle  Identify the angle formed by the intersection of a tangent and a Find the unknown length in radius as a right angle the picture below. (Perpendicular)  Define a circumscribed angle (formed by 2 intersecting tangents) and determine its measure is equal to half the difference of the intercepted arcs  Compare and contrast central, inscribed, and circumscribed How does the angle between a angles and calculate their tangent to a circle and the line measures connecting the point of  Identify and describe tangency and the center of the relationships among inscribed circle change as you move the angles, radii, and chords as they tangent point? relate to central, inscribed, and circumscribed angles

G.C.3 - Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. (Use Geometer’s Sketchpad and by hand)  Prove the properties of the

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[opposite] angles in an inscribed quadrilateral.  Find the measures of missing angles in an inscribed quadrilateral using the fact that opposite angles are supplementary.  Review/Test

Concepts of Algebra – 5 days Standards: A-APR.1, A-CED.1, A-CED.4, A-REI.1, A-REI.10, A-REI.11, F-IF.7c, N-RN.3 Learning Targets Glencoe Algebra II Vocabulary Sample Questions  Review number system and all N-RN.3 Know and justify that properties 1.2, 1.3 when adding or multiplying two rational numbers the result is a A-APR.1 - Understand that polynomials 1.2 rational number form a system analogous to the integers, 2.1, 2.6, 6.1, 5.8, 9.3, follow-up (Level III) namely, they are closed under the 10-4 (page 552), 8.3 Ex: What set(s) of numbers in the operations of addition, subtraction, and Real Number System is the sum multiplication; add, subtract, and multiply of and an element of? What polynomials. set(s) of numbers N-RN.3 Explain why the sum or product of in the Real Number System is the two rational numbers is rational; that the product of and an element of? sum of a rational number and an irrational Explain how you know whether number is irrational; and that the product of the solution a nonzero rational number and an irrational would be rational or irrational number is irrational. without computing.  Closure Property N-RN.3 Know and justify that when adding a rational number A-REI.1 - Explain each step in solving a and an irrational number the simple equation as following from the result is irrational. (Level III) equality of numbers asserted at the previous Ex: What set(s) of numbers in the step, starting from the assumption that the Real Number System is the sum Assessments: Teachers should monitor and assess students daily. Suggestions for Summative Formative Assessments: Projects, Tests, Quizzes…Suggestions for Formative Assessments: Exit slip, group work, labs, drills, discussions, journal entries/student reflection, Study Island……

7/10/12 Page 25 of 44 Brunswick County Schools Math III Pacing Guide 2013-2014 original equation has a solution. Construct a of and an element of? Explain viable argument to justify a solution how you method. know whether the solution would A-CED.1 - Create equations and be rational or irrational without inequalities in one variable and use them to computing. solve problems. Include equations arising N.RN.3 Know and justify that from linear and quadratic functions, and when multiplying a nonzero simple rational and exponential functions. rational number and an irrational ● Solve one-variable, multi- number the result is step equations and irrational. application problems. Ex: What set(s) of numbers in the (Review) Real Number System is the product of and an element of? A-CED.4 - Rearrange formulas to Explain how you know whether the solution highlight a quantity of interest, using would be rational or irrational the same reasoning as in solving without computing. equations. For example, rearrange Ex: Can two unique irrational Ohm’s law V = IR to highlight numbers have a rational number resistance R. as their product? Justify your ● Solve literal equations for answer. specified variable. ○ Use sector of circle, circle formulas, science formulas, geometry formulas

A-REI.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). A-REI.11 - Explain why the x- coordinates of the points where the graphs of the equations y = f(x) and Assessments: Teachers should monitor and assess students daily. Suggestions for Summative Formative Assessments: Projects, Tests, Quizzes…Suggestions for Formative Assessments: Exit slip, group work, labs, drills, discussions, journal entries/student reflection, Study Island……

7/10/12 Page 26 of 44 Brunswick County Schools Math III Pacing Guide 2013-2014 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. F-IF.7 - Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. ● Sketch with and without calculator (linear, absolute value, quadratic, exponential, radical, step and piecewise defined, circle) F-IF.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. ● Evaluate for given Domain/Range Assessments: Teachers should monitor and assess students daily. Suggestions for Summative Formative Assessments: Projects, Tests, Quizzes…Suggestions for Formative Assessments: Exit slip, group work, labs, drills, discussions, journal entries/student reflection, Study Island……

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● Interpret the Domain/Range & intercepts in the context of the problem of linear, absolute value, step and piecewise defined, quadratic, exponential, radical, and logarithmic functions (introduce end behavior) ● Solve equations by graphing (linear, absolute value, quadratic, polynomial, exponential, radical, rational, log, and trig) ● Review/Assessment Benchmark Review & Benchmark – 2 days Equations & Functions – 7 days Standards: A-SSE.4, F-IF.2, F-IF.5, F-IF.7c&e, F-IF.9, F-BF.1a&b, F-BF.2, F-BF.3, F-BF.4 Learning Targets Glencoe Algebra II Vocabulary Sample Questions F-IF.2 Use function notation, evaluate 2.6, 5.8, 6.1, 9.3, pg. 593, functions for inputs in their domains, and 7.1, 7.9 Julie saves $5.00 and each week after she interpret statements that use function 2.3-2.5 adds $2. notation in terms of a context.  Use table to find arithmetic 7.8 sequence F-BF.2 Write arithmetic and geometric 2.1, 7.7  Take arithmetic sequence to sequences both recursively and with an 6.6, 2.2-2.4, 2.6 y=mx+b explicit formula, use them to model  Take to NEXT=NOW form situations, and translate between the two  forms.  Lead into geometric yields F-BF.1 Write a function that describes a exponential equations. relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context.

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b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. A-SSE.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. ● Write arithmetic and geometric sequences with a recursive and explicit formula. Use them to model situations. ● Derive the formula for the sum of finite geometric series F-BF.4 Find inverse functions. a. 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 or f(x) = (x+1)/(x–1) for x ≠ 1. ● Evaluate a function and composition of functions ● Find inverse functions F-IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal

7/10/12 Page 29 of 44 Brunswick County Schools Math III Pacing Guide 2013-2014 descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. ● Compare properties of two functions each represented in a different way (equation, table, or verbal) (use upper level functions such as logarithmic, polynomial, and circles) F-BF.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. ● Identify the 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 the value of k given the graphs. (use upper level functions such as logarithmic, polynomial, and circles) F-IF.7 Graph functions expressed symbolically and show key features of the

7/10/12 Page 30 of 44 Brunswick County Schools Math III Pacing Guide 2013-2014 graph, by hand in simple cases and using technology for more complicated cases. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. ● Recognize even and odd functions from their graphs and algebraic expressions for them. (upper level functions) ● Review/Assessment

Systems of Equations & Inequalities – 5 days Standards: A-CED.2, A-REI .10, G-MG.3 Learning Targets Glencoe Algebra II Vocabulary Sample Questions A-CED.2 Create equations in two or more variables to represent relationships between 3.1-3.2 A-REI.7 quantities; graph equations on coordinate 8.7, pg. 268, http://illustrativemathematics.org/illus axes with labels and scales. pg. 552 trations/223 ● Solve systems of linear 3.1-3.2, 3.5 equations by graphing, 3.3, 3.4 substitution, and elimination A-REI.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). ● Solve systems of equations Assessments: Teachers should monitor and assess students daily. Suggestions for Summative Formative Assessments: Projects, Tests, Quizzes…Suggestions for Formative Assessments: Exit slip, group work, labs, drills, discussions, journal entries/student reflection, Study Island……

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involving other functions (linear, polynomial, rational, radical, absolute value, exponential) A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. G-MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios. ● Solve applications of systems (2x2 and 3x3) by hand and calculator. (Linear Programming) (may use with and without geometric shapes, looking for vertices of polygons to get maximum or minimum points) ● Review/Assessment

Polynomials – 10 days Standards: A-SSE.2, A-APR.1, A-ARP.2, A-APR.3, N-CN.9, F-IF.7c&e Learning Targets Glencoe Algebra II Vocabulary Sample Questions A-APR.1 Understand that polynomials form a system analogous to the integers, 5.1-5.2 N-CN.9 Understand The Fundamental namely, they are closed under the 11.7 Theorem of Algebra, which says that the operations of addition, subtraction, and 5.4 number of complex solutions to a multiplication; add, subtract, and multiply 5.3, 7.4 polynomial equation is the same as the polynomials. 9.1-9.2 degree of the polynomial. Show that this ● Add/Subtract and 9.1 is true for any quadratic

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Multiply/Divide Monomial, ● 7.1-7.3 polynomial. Binomial, and Polynomial ● use handout link to the (Level III) expressions (Review). right Ex. If a function has a degree of 5, how Include rational and negative ● 7.5 many solutions would that function exponents. ● 7.2, page 359 have? Why? Ex. Solve the general form of a quadratic A-SSE.2 Use the structure of an expression 7.1-7.3 equation for x. What do the solutions to identify ways to rewrite it. For example, show in relation to the see x4 – y4 as (x2)2 – (y2)2, thus recognizing Fundamental Theorem of Algebra for it as a difference of squares that can be 2 2 2 2 quadratic equations? factored as (x – y )(x + y ). A-SSE.1-3-Seeing Dots ● Factor – GCF, difference of http://illustrativemathematics.org/illus squares, sum/difference of trations/21 cubes, trinomials, grouping (2 binomials and a A-APR.2-The Missing Coefficient monomial and a trinomial) http://illustrativemathematics.org/illus (Use higher degrees ○ trations/592 with squares and cubes) ○ Include x4-y4 as examples A-APR.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). ● Divide polynomials– Synthetic and Long ○ Remainder Theorem F-IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. c. Graph polynomial functions,

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identifying zeros when suitable factorizations are available, and showing end behavior. N-CN.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. ● Solve polynomial equations by factoring/graphing (higher than quadratic) *Know the Fundamental Theorem of Algebra; limit to polynomials with real coefficients ● Write equations from given zeros (Real & Imaginary)

A-APR.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. ● Given f(x)…state where f(x)=0, f(x)>0, f(x)<0, increasing/decreasing, end behavior (Set and Interval Notation) ● Review/Assessment

Radicals & Complex Numbers – 3 days Standards: N-CN.1, N-CN.2, N-CN.7, N-RN.3, A-REI.2 Learning Targets Glencoe Algebra II Vocabulary Sample Questions N-CN.1 Know there is a complex number i such that i2 = –1, and every complex N-CN.1 Every number is a complex Assessments: Teachers should monitor and assess students daily. Suggestions for Summative Formative Assessments: Projects, Tests, Quizzes…Suggestions for Formative Assessments: Exit slip, group work, labs, drills, discussions, journal entries/student reflection, Study Island……

7/10/12 Page 34 of 44 Brunswick County Schools Math III Pacing Guide 2013-2014 number has the form a + bi with a andb ● 5.6 number or the form a + bi, where a and b real. ● 5.9 are elements of the Real Numbers N-CN.2 Use the relation i2 = –1 and the ● 6.4-6.5 and bi is an element of the Pure commutative, associative, and distributive ● 5.8, page 268 Imaginary Numbers. Students should properties to add, subtract, and multiply ● 5.6, 5.8 know the sets and subsets of the complex numbers. Complex ● Simplify and perform Number System. The identity, i= −1, is operations with complex not only used to identify non-real numbers solutions for particular functions, but N-CN.7 Solve quadratic equations with is also be used to find the identity, i2= – real coefficients that have complex 1, which is used to simplify expressions. solutions. (Level III) ● Solve quadratic equations Ex. Is it possible to simplify using the with the quadratic formula Real Number System? Justify your reasoning. and completing the square – N.CN.2 When adding, subtracting, or focus on non-real and multiplying Complex Numbers and i2 irrational solutions remains in the expression, use the A-REI.2 Solve simple rational and radical equations in one variable, and give identity 2 1 i − and the commutative, associative, and distributive properties to examples showing how extraneous simplify the expression further. solutions may arise. (Level III) ● Solve radical equations with Ex. The voltage E, current I, and and without calculator resistance R in an electrical circuit are N-RN.3 Explain why the sum or product of related by Ohm’s Law: E=IR. two rational numbers is rational; that the Respectively, these quantities are sum of a rational number and an irrational measured in volts, amperes, and ohms, number is irrational; and that the product of respectively. Find the voltage in an a nonzero rational number and an irrational electrical circuit with current (2 + 4i) number is irrational. amperes and resistance (5-4i) ohms. ● The sum or product of two N-CN.7 Extend strategies for solving (ir)rational numbers is quadratics such as; taking the square root (ir)rational... and applying the quadratic formula, to find solutions of the form, a + ● Review/Assessment bi, for quadratic equations. This Assessments: Teachers should monitor and assess students daily. Suggestions for Summative Formative Assessments: Projects, Tests, Quizzes…Suggestions for Formative Assessments: Exit slip, group work, labs, drills, discussions, journal entries/student reflection, Study Island……

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extension is made when the identity i= −1is used to simplify radicals having negative numbers under the radical. (Level III) Ex. Describe how and explain why the solution(s) to a quadratic equation may be: a. One Real solution b. Two Complex solutions c. Two Real solutions Ex. Your class is given the quadratic equation ax2 + bx + c = 0, knowing that one solution is 2 + 3i. The class came up with several different possibilities for the second solution; (3 + 2i), (2 – 3i), (-2 – 3i), and (-2 + 3i). Are either of these proposed solutions correct? Why, or why not? N-CN http://illustrativemathematics.org/illus trations/617

A-REI.2-Radical equations http://illustrativemathematics.org/illus trations/39 Conics – 6 days Standards: A-SSE.3b, A-CED.1, A-REI.4a&b, F-IF.8a, G-GPE.1, G-GPE.2 Learning Targets Glencoe Algebra II Vocabulary Sample Questions G-GPE.2 Derive the equation of a parabola given a focus and directrix. 6.2-6.5 ● Derive the equation of a 6.1, pg. 320-321, 6.6 parabola given the focus and 6.1-6.6

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directrix using the distance pg. 300 formula ● 6.7

A-SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. A-REI.4 Solve quadratic 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 the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a andb. ● Solve quadratic equations – graphing, factoring, quadratic formula, completing the square (standard and vertex form) F-IF.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

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the graph, and interpret these in terms of a context. ● Find axis of symmetry, (review domain/range with extreme values) A-CED.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. ● Model and solve applications of quadratic equations – Egg Toss (1 day) ● Solve and graph quadratic inequalities – factoring/graphing (bring in transformations) F-IF.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. ● Quadratic regression – interpret coefficients in context of problem (residuals) G-GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle Assessments: Teachers should monitor and assess students daily. Suggestions for Summative Formative Assessments: Projects, Tests, Quizzes…Suggestions for Formative Assessments: Exit slip, group work, labs, drills, discussions, journal entries/student reflection, Study Island……

7/10/12 Page 38 of 44 Brunswick County Schools Math III Pacing Guide 2013-2014 given by an equation. ● Derive the equation of a circle given its center and radius using Pythagorean Theorem and identify the center and radius of a circle given its equation using completing the square (standard and center-radius form) ● Review/Assessment

Rational Expression & Equations – 6 days Standards: A-APR.6, A-APR.7, A-REI.2, F-BF.1b Learning Targets Glencoe Algebra II Vocabulary Sample Questions A-REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous ● page 471 solutions may arise. ● 9.1-9.2 ● Add/Subtract/Multiply/Divi ● 9.3, page 491 de rational expressions – ● 9.1-9.2 like and unlike denominators ● 9.6, page 512 A-APR.6 Rewrite simple rational ● 9.1-9.3, 9.6 a(x) expressions in different forms; write /b(x) r(x) in the form q(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. ● Graph rational functions – state equations of

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asymptotes (horizontal, vertical, holes) ● Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x)+r(x)/b(x) F-BF.1 Write a function that describes a relationship between two quantities. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. ● Simplify complex fractions ● Solve rational equations ● Rational function applications Review/Assessment

Exponents & Logarithmic Functions – 7 days Standards: A-SSE.1a, A-SSE.3, A-CEDI.1, F-BF.4, F-LE.4 Learning Targets Glencoe Algebra II Vocabulary Sample Questions A-SSE.3 Choose and produce an ● 5.7, 7.3 equivalent form of an expression to reveal ● 10.1,10.5, 10.6 and explain properties of the quantity ● 10.1, page 539-540 represented by the expression. ● 10.2 F-LE.3 Observe using graphs and tables ● 10.2, 10.5, 10.1 that a quantity increasing exponentially ● 10.3, 10.4 eventually exceeds a quantity increasing ● 10.1-10.4, 10.6 linearly, quadratically, or (more generally) ● 10.5 as a polynomial function. Assessments: Teachers should monitor and assess students daily. Suggestions for Summative Formative Assessments: Projects, Tests, Quizzes…Suggestions for Formative Assessments: Exit slip, group work, labs, drills, discussions, journal entries/student reflection, Study Island……

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● Graph/solve exponential applications (growth/decay, population, interest) ● Find exponential regression models – interpret coefficients in context of the problem (Residuals) (For residuals use: A-SSE.1 Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients.) ● Write exponentials as logs – vice versa F-BF.4 Find inverse functions. a. 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 or f(x) = (x+1)/(x–1) for x ≠ 1. ● Review functions before finding inverses. ● Find the inverse of exponential/logarithmic functions algebraically and sketch F-LE.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.

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● Switch from exponential to logarithmic form ● Solve using technology ● Review/Assessment

Statistics & Probability – 7 days Standards: S-ID.4, S-IC.1, S-IC.3, S-IC.4, S-IC.5, S-IC.6 Learning Targets Glencoe Algebra II Vocabulary Sample Questions S-ID.4 Use the mean and standard pg. 824-827 deviation of a data set to fit it to a normal 12.6-12.7 S-IC.1-Why Randomize? distribution and to estimate population http://illustrativemathematics.org/illus percentages. Recognize that there are data trations/191 sets for which such a procedure is not appropriate. Use calculators, spreadsheets, S-IC.2-Block Scheduling and tables to estimate areas under the 12.9, pg. 856 http://illustrativemathematics.org/illus normal curve. trations/125 ● Use the mean and standard pg. 681, 12.8-12.9 deviation of a data set to fit 12.9, pg. 686, 681 S-IC.1 and S-IC.3-Strict Parents it to a normal distribution http://illustrativemathematics.org/illus and to estimate population trations/122 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 S-IC.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population ● Understand statistics as a

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process for making inferences about population parameters based on a random sample from that population S-IC.6 Evaluate reports based on data. ● Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation S-IC.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling. ● Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each S-IC.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant ● Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling

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● Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant Exam Review – 3 days Exam

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