5.NF.1: Methods for operations with fractions with …



North Carolina Math 3 Sampson County Schools Pacing GuideAdopted 2020-2021Math 3 Standards Document Math 3 Unpacking DocumentMath 3 Test Specifications Math 3 Released ItemsOverview of Math 1-3NC HS Collaborative Instructional FrameworkPacing Guide 2019-2020Units for NC Math 3Number of Days SemesterUnit 1: Functions and Their Inverses10 - 13Unit 2: Exponential and Logarithmic Functions10 - 12Unit 3: Polynomial Functions10 - 12Unit 4: Rational Functions15 - 18Unit 5: Trigonometric Functions 8 - 11Unit 6: Modeling with Geometry7 - 9Unit 7: Reasoning with Geometry10 - 13Unit 8: Statistics5 - 7Total (range allows for flex days)Fall - 75 + 5 flex + 5 testing = 85 Spring – 75 + 20 flex + 5 testing = 100***Note: Highlighted Standards are identified as gap standards for students who took Math 1 and Math 2 in the Spring of 2020*** Math 3 Pre-Assessment Window – First 10 days of semesterUnit 1: Functions and Their InversesEstimated Days: 10 – 13 daysOverarching Standards - NC.M3.A-SSE.1b, NC.M3.A-SSE.2, NC.M3.A-REI.1, NC.M3.F-BF.1b StandardsLearning Intentions NC.M3.F-IF.4NC.M3.F-IF.9NC.M3.F-IF.7NC.M3.F-IF.2NC.M3.F-BF.1bNC.M3.F-BF.3Compare functions (complex functions, no limitations in F-IF.4, F-IF.7: piecewise, absolute value, polynomials, exponential, rational, and trigonometric functions (sine and cosine) this standard is repeated in later units) using multiple representations of and understand key features (domain/range, increasing/decreasing intervals, positive/negative intervals, max/min, and end behavior) to interpret, analyze, and find solutions. Use compound inequalities as well as interval notation to describe key features.Understand and recognize piecewise defined relationships. Focus on evaluating given the function over evaluating with graphs. Build a new function by combining functions using addition, subtraction, and multiplication.Transform functions and determine the effects on the graph. (k*fx, fx+k, fx+k, f(k*x) .NC.M3.F-BF.4aNC.M3.F-BF.4bNC.M3.F-BF.4cUnderstand inverse relationships, describe them algebraically and graphically, and use these relationships to solve, analyze and interpret. Find the inverse of a given function. NC.M3.A-CED.1NC.M3.A-CED.2NC.M3.A-CED.3NC.M3.A-SSE.1NC.M3.A-REI.11Understand and solve absolute value equations and inequalities. Create and graph equations in one and two variables using absolute value equations and inequalities. Solve systems of equations including all function types learned in the course for more complex functions using technology, graphs, and tables. Build an understanding of why the x-coordinate is the solution of the equation fx=g(x) and approximate solutions using technology.Identify key parts in expressions and equations. Unit 1 Foundational StandardsNC.M2.F-IF.7 Analyze quadratic, square root, and inverse variation functions by generating different representations, by hand in simple cases and using technology for more complicated cases, to show key features, including: domain and range; intercepts; intervals where the function is increasing, decreasing, positive, or negative; rate of change; maximums and minimums; symmetries; and end behavior. NC.M2.F-IF.4 Interpret key features of graphs, tables, and verbal descriptions in context to describe functions that arise in applications relating two quantities, including: domain and range, rate of change, symmetries, and end behavior.NC.M2.A-REI.2: Solve and interpret one variable inverse variation and square root equations arising from a context and explain how extraneous solutions may be produced. This is added again in the Rational Unit can be covered in both or either place. Unit 1 Honors ExtensionsFor any standard, increase DOK of questions.NC.M3.F-IF.2: Evaluate Piecewise functions given graphs not just in function notation.NC.M3.F-BF.1b: Compose functions.NC.M3.F-BF.4: Prove two functions are inverses by using a composition of functions. NC.M3.A-CED.3: Solve systems with more complex function types algebraically.Unit 2: Exponential and Logarithmic FunctionsEstimated Days: 10 – 12 daysOverarching Standards - NC.M3.A-SSE.1b, NC.M3.A-SSE.2, NC.M3.A-REI.1, NC.M3.F-BF.1bStandardsLearning Intentions NC.M3.A-CED.1NC.M3.A-CED.2MC.M3.A-SSE.1NC.M3.A-SSE.3cNC.M3.F-LE.4Understand how to create exponential equations and graphs with one or two variables, and be able to identify the different parts of an exponential equation and relate them to the real world.Build an understanding of why the x-coordinate is the solution of the equation fx=g(x) and approximate solutions using technology.Write an equivalent form of an exponential based on the properties of exponents.Solve exponential and log equations. NC.M3.F-BF.3NC.M3.F-IF.4NC.M3. F-IF.7NC.M3. F-IF.9NC.M3.F-BF.4NC.M3.F-BF.1aTransform functions and determine the effects on the graph. (k*fx, fx+k, fx+k, f(k*x)) .Recognize the relationship between exponential and logarithmic equations as inverses using multiple representations, interpret the key features (domain/range, increasing/decreasing intervals, positive/negative intervals, asymptotes, and end behavior) of the pare functions (complex functions, no limitations in F-IF.4, F-IF.7: piecewise, absolute value, polynomials, exponential , rational, and trigonometric functions (sine and cosine) this standard is repeated in later units) using multiple representations of and understand key features (domain/range, increasing/decreasing intervals, positive/negative intervals, max/min, and end behavior) to interpret, analyze, and find solutions. Use compound inequalities as well as interval notation to describe key features.Model real world phenomena with exponential functions including compound interest, continuous relationships involving e, and doubling time/half-life. Unit 2 Foundational StandardsIntroduce the need for logarithms by giving students 2^x=8 vs 2^x=10.8.EE.A.1: Students need to know their exponent rules and how to evaluate an expression with exponent.NC.M1.F-LE.3 Students need to know the form of an exponential equation and how to recognize growth or decay.NC.M1.S-ID.6c: Fit a function to exponential data using technology. Use the fitted function to solve problems.Unit 2 Honors ExtensionsFor any standard, increase DOK of questions.NC.M3.F-BF.4: Prove two functions are inverses by using a composition of functions involving solving an exponential or log equation.NC.M3.F-BF.1a: Real-world extension project involving compound interest.NC.M3.F-IF.4: Graph log functions by hand by creating a table of values.NC.M3.F-LE.4: Properties of Logs, Condense and Expand to solve problems.Unit 3: Polynomial FunctionsEstimated Days: 10 - 12 daysOverarching Standards - NC.M3.A-SSE.1b, NC.M3.A-SSE.2, NC.M3.A-REI.1, NC.M3.F-BF.1bStandardsLearning Intentions NC.M3.N-CN.9NC.M3.A-SSE.1NC.M3.A-APR.2NC.M3.A-APR.3NC.M3.A-CED.1NC.M3.A-CED.2NC.M3.F-BF.1aNC.M3.F-IF.4NC.M3.F-IF.7NC.M3.F-IF.9NC.M3.F-BF.1aNC.M3.F-BF.1bNC.M3.F-BF.3NC.M3.F-LE.3Use the Fundamental Theorem of Algebra to determine the number and potential types of solutions for polynomial functions.Identify the terms, factors, coefficients, and exponents in piecewise, absolute value, polynomial, exponential and rational expressions.Factor and find zeros of higher degree polynomial functions.Build a function from the factors or zeros given the leading coefficient of 1.The relationship between multiplicity and quadratics with a one solution could be made, and then further polynomial functions can be formed with even/odd multiplicity. Include Desmos in graphing for students to prove the rule of multiplicity. Identify the intercepts, maximum/minimum values, domain, range, end behavior, increasing and decreasing intervals, positive and negative intervals given different types of graphs. Note the different types of notation that can be pare functions (complex functions, no limitations in F-IF.4, F-IF.7: piecewise, absolute value, polynomials, exponential, rational, and trigonometric functions (sine and cosine) this standard is repeated in later units) using multiple representations of and understand key features (domain/range, increasing/decreasing intervals, positive/negative intervals, max/min, and end behavior) to interpret, analyze, and find solutions. Use compound inequalities as well as interval notation to describe key features.Find the rate of change for specific intervals of the polynomial function, so an exponential function will eventually exceed a polynomial function.Unit 3 Foundational StandardsNC.M1.A-SSE.3: Factor and find the zeros of a quadratic. NC.M2.A-SSE.3: Write an equivalent form of a quadratic expression by completing the square.NC.M1.A-APR.3: Understand the relationships among the factors of a quadratic expression, the solutions of a quadratic, ad the zeros. NC.M2.A-APR.1: Extend the understanding that operations with Polynomials are comparable to operations with integers by adding, subtracting, and multiplying Polynomials. NC.M2.F-BF.3: Understand the effects of transformations on functions and be able to write equations of functions based off of a graph. NC.M1.F-IF.2: Evaluate a function for inputs in their domain and interpret in context.NC.M2.A-REI.7: Use tables, graphs, and algebraic methods to approximate or find exact solutions of systems of linear and quadratic equations and interpret the solutions in terms of a context.Unit 3 Honors ExtensionsFor any standard, increase DOK of questions.NC.M3.A-APR.3: Extend on imaginary roots of polynomials using synthetic division. NC.M3.F-LE.3: Find Average rate of change and discuss second differences for quadratic functions.Unit 4: Rational FunctionsEstimated Days: 15 - 18 daysRationale: This unit is intended to develop students’ understanding of rational functions. It is suggested to be taught in close proximity to the polynomials unit because of the connection of rational expressions to the division of polynomials. This unit should begin with reviewing both simplification of fractions and all arithmetic operations to help students understand the similarities and differences between rational numbers and expressions. Overarching Standards - NC.M3.A-SSE.1b, NC.M3.A-SSE.2, NC.M3.A-REI.1, NC.M3.F-BF.1bStandardsLearning Intentions NC.M3.A-SSE.1NC.M3.A-APR.6NC.M3.A-APR.7NC.M3.A-CED.1NC.M3.A-CED.2NC.M3.A-CED.3NC.M3.A-REI.2NC.M3.F-IF.4NC.M3.F-IF.7NC.M3.F-IF.9Simplify a rational expression (by factoring to cancel common factors).Multiply, divide (with and without remainder) rational expressions. **Include long division of polynomials. Add and subtract rational expressions with unlike linear denominators.Solve rational equations and check for extraneous solutions. Understand key components of functions and their graphs. (End behavior, intercepts, maximums and minimums, decreasing and increasing intervals) Understand key components of graphs of rational functions. Recognize a hole in the graph of a rational function and also be able to see algebraically why that hole exists. Recognize that a hole and an asymptote are both discontinuities. Compare components of different rational functions and their pare functions (complex functions, no limitations in F-IF.4, F-IF.7: piecewise, absolute value, polynomials, exponential, rational, and trigonometric functions (sine and cosine) this standard is repeated in later units) using multiple representations of and understand key features (domain/range, increasing/decreasing intervals, positive/negative intervals, max/min, and end behavior) to interpret, analyze, and find solutions. Use compound inequalities as well as interval notation to describe key features.Solve systems of equations including all function types learned in the course for more complex functions using technology, graphs, and tables. Unit 4 Foundational Standards5.NF.1: Methods for operations with fractions with numbers and not expressions. (*/+-)NC.M1.A-SSE.3: Write an equivalent form of a quadratic expression by factoring.NC.M2.A-REI.2: Solve and interpret one variable inverse variation and square root equations arising from a context and explain how extraneous solutions may be produced. Unit 4 Honors ExtensionsFor any standard, increase DOK of questions.NC.M3.A-CED.1: Solve inequalities involving Polynomials with degree higher than 2, tie into PC.A.1.1. NC.M3.A-CED.3: Solve systems with more complex function types algebraically.PC.A.1.1: Implement algebraic (sign analysis) methods to solve rational and polynomial inequalities. Math 3 Benchmark Window Unit 5: Trigonometric FunctionsEstimated Days: 8 – 11 days Rationale: This unit should immediately follow the Reasoning with Geometry unit. Students’ understanding of radians and the idea of circular motion are connections that can help students better understand trigonometric functions. Overarching Standards - NC.M3.A-SSE.1b, NC.M3.A-SSE.2, NC.M3.A-REI.1, NC.M3.F-BF.1bStandardsLearning Intentions NC.M3.F-TF.1NC.M3.F-TF.2NC.M3.F-IF.1NC.M3.F-IF.4NC.M3.F-IF.7NC.M3.F-IF.9Understand radian measure of an angle by dividing arc length by the radius. Understand sine and cosine values are the y and x values of angle measures on the unit circle in the coordinate plane, use this to solve for exact values of trig ratios. Use the coordinates of the Unit Circle to graph trigonometric functions where the domain is the measure of the angle and the range is the value of the associated trig ratio. Understand and interpret the key features (domain/range, max/min, midline, period, discontinuity), uses and limitations of multiple representations of trigonometric functions that model real world periodic pare functions (complex functions, no limitations in F-IF.4, F-IF.7: piecewise, absolute value, polynomials, exponential, rational, and trigonometric functions (sine and cosine) this standard is repeated in later units) using multiple representations of and understand key features (domain/range, increasing/decreasing intervals, positive/negative intervals, max/min, and end behavior) to interpret, analyze, and find solutions. Use compound inequalities as well as interval notation to describe key features.NC.M3.F-TF.5Use technology to investigate the parameters a, b, and h of a sine function in the form .Note: Phase shifts and transformations of trigonometric functions are NOT required in Math 3. Unit 5: Foundational Standards7.RP.A.2: Solve proportional equations.NC.M2.G-SRT.8: Use trig ratios and the Pythagorean Theorem to solve problems involving right triangles in terms of a context.NC.M2.G-SRT.12: Develop properties of special right triangles (45-45-90 and 30-60-90) and use them to solve problems.Unit 5: Honors ExtensionsFor any standard, increase DOK of questions.NC.M3.F-TF.1: Solve for values of other trig functions, tangent, cotangent, secant, cosecant. NC.M3.F-IF.1: Explore graphs of tangent and cotangent functions and discuss asymptotes that exist in their graphs, connect to knowledge of rational functions, where values are excluded from the domain.NC.M3.F-BF.3: Transform trig functions by changing the period and midline of the graph. NC.M3.F-TF.5: Understand that sine and cosine functions graph the same wave, but cosine is a phase shift of sine.Unit 6: Modeling with GeometryEstimated Days: 7 - 9 daysRationale: This unit transitions from polynomial work to geometric concepts that require the use of algebra. It is intentionally placed after the polynomials unit because the polynomials unit is suggested to begin with geometric modeling that results in a polynomial. Teaching this unit right after the conclusion of polynomials, allows you to circle back to the geometric modeling concept and study it to its full depth. Overarching Standards - NC.M3.A-SSE.1b, NC.M3.A-SSE.2, NC.M3.A-REI.1, NC.M3.F-BF.1bStandardsLearning Intentions NC.M3.G-GMD.3NC.M3.G-GMD.4NC.M3.G-MG.1Implement surface area and volume of geometric figures and model using polynomial functions, then solve for volume.Identify two-dimensional cross sections of three-dimensional objects and identify three-dimensional generated by rotations of two-dimensional objects.Apply geometric and algebraic concepts to model real-world three-dimensional figures, model relationships, determine density based on area or volume, solve and design optimization problems.Unit 6 Foundational Standards7.G.B.6 & 6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.Unit 6 Honors ExtensionsFor any standard, increase DOK of questions.NC.M3.G-GMD.4: Calculate the volume of a three-dimensional object generated by rotating a two-dimensional object across an axis.Unit 7: Reasoning with GeometryEstimated Days: 10 – 13 daysRationale: This unit transitions into geometric concepts with an emphasis on reasoning, justification, and formalizing proof. Students will extend upon their work with proof in Math 2 (NC.M2.G.CO.9 and NC.M2.G.CO.10) focusing on both paragraph and flow proofs. Students are familiar with the properties of parallelograms from middle school and have categorized parallelograms and informally verified parallelogram properties through coordinate geometry in Math 1. Students will prove more theorems about triangles including the centers of triangles. This concept can be used as a transition into reasoning with circles. The Reasoning with Geometry Unit purposefully concludes with circles. In students’ work with circles, they will develop their understanding of radian measure through proportions in circles. This sets up a connection of circular motion to trigonometric functions in the next unit. Overarching Standards - NC.M3.A-SSE.1b, NC.M3.A-SSE.2, NC.M3.A-REI.1, NC.M3.F-BF.1bStandardsLearning Intentions NC.M3.G-CO.10 NC.M3.G-CO.11 NC.M3.G-CO.14NC.M3.G-GPE.1 Demonstrate an understanding of the properties of three of a triangle’s points of concurrency: centroid, incenter, and circumcenter. Verify properties of each point of concurrency. (See unpacking for properties to be taught.)Construct logical arguments and explain reasoning with two-dimensional figures to prove geometric theorems about parallelograms. Apply properties of triangle points of concurrency and parallelograms and use them to solve problems.Use the Pythagorean Theorem to derive the equation of a circle and complete the square to find the center and radius of a given circle in conic form. NC.M3.G-C.2NC.M3.G-C.5Understand properties of circles and how to apply them algebraically and geometrically. Demonstrate understanding that within circles, segments, lines, and angles create special relationships and use these to solve geometric problems.Solve for the length of arcs and areas of sectors within circles.Unit 7 Foundational Standards8.G.7: Students will have prior knowledge of pythagorean theorem.Unit 7 Honors ExtensionsFor any standard, increase DOK of questions.NC.M3.G-CO.14: Define and explore the orthocenter of a triangle.NC.M3.G-CO.14: Explore properties of a Kite and use them to solve problems. Math 3 MOCK EOC WindowUnit 8: StatisticsEstimated Days: 5 – 7 daysStandardsLearning IntentionsNC.M3.S-IC.1NC.M3.S-IC.3Understand statistics as a process of making inferences about a population (parameter) based on results from a random sample (statistic). B. Acknowledge the role of randomization in using sample surveys, experiments, and observational studies to collect data and understand the limitations of generalizing results to populations (related to randomization).NC.M3.S-IC.4NC.M3.S-IC.5C. Understand simulation is useful for using data to make decisions. D. Understand that samples can differ by chance.NC.M3.S-IC.6E. Understand not all data that is reported is valid. Reports should be evaluated based on source, design of the study, and data displays.Unit 8 Foundational Standards6.SP.A.2/6.SP.B.5.C: Students have an understanding of mean, median, mode. Center, spread and overall shape of data points.Unit 8 Honors ExtensionsFor any standard, increase DOK of questions.NC.M3.S-IC.6: Students can analyze data from their own sampling project and choose the best method to display their data. ................
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