2013-2014 Curriculum Blueprint Grade: 9-12 Course: Analytical Geometry Unit 1: Equations and Analyzing Functions and Their Graphs Trad 7 weeks 4 x 4 4 weeks 1 st Quarter CPALMS site that contains the Specifications to include the Content limits, Attributes/Stimulus, and additional information. Date Range: Given during the instruction per the outline in this section Key Vocabulary: Function, Zero of a Function, Interval Notation, Composite Function, Decomposing a Function, Inverse Function, Smooth, Continuous, End Behavior, Asymptote, Oblique Students will be able to solve multi-step linear, polynomial, rational, absolute value, and radical equations, graph and analyze key, polynomial, and rational functions, and solve their real-world applications. 1. Students will solve multi-step linear equations, polynomial equations, rational equations, absolute value equations, and radical equations and their real-world applications. 2. Recognize, graph, analyze, and transform key functions, polynomial functions, rational functions, with and without graphing calculators. 3. Define, graph, and check inverse functions. 4. Perform basic operations of functions and compositions of functions and solve real-world applications of functions. MA.912.A.4.5: Graph polynomial functions with and without technology and describe end behavior. MA.912.A.4.8: Describe the relationships among the solutions of an equation, the zeros of a function, the x- intercepts of a graph, and the factors of a polynomial expression with and without technology. MA.912.A.4.9: Use graphing technology to find approximate solutions for polynomial equations. MA.912.A.5.6: Identify removable and non-removable discontinuities and vertical, horizontal, and oblique asymptotes of a graph of a rational function, find the zeros, and graph the function. A. Solve equations (multi-step linear equations, rational equations, quadratic equations, polynomial equations, radical equations, absolute value equations) 1. Equation balancing using properties of equality 2. Factoring and using the zero-product property 3. Completing the square 4. Using the quadratic formula 5. Using roots and powers B. Analyze and graph functions 1. Recognize and graph key functions, polynomial functions, and rational functions, with and without graphing calculators a. Identify domain and range b. Locate x and y intercepts c. Determine increasing, decreasing, or constant intervals d. Determine relative maxima and relative minima e. Locate vertical, horizontal, and oblique asymptotes f. Identify discontinuities in graphs g. Describe symmetry of functions graphs h. Determine if functions are even, odd, or neither i Apply transformations to graphing functions j. Describe the end behavior of polynomial functions 2. Define, graph, and check inverses of functions 3. Determine sums, differences, products, and quotients of functions 4. Determine compositions of functions. Solve real-world problems using equations and functions 1. What is analytical geometry? 2. What are the techniques commonly used to solve equations? 3. What steps are needed to solve word problems? 4. What determines a function? 5. How can a function and its graph be analyzed? 6. How can a function and its graph be transformed? 7. How can the key features of a graph be predicted based on a function s algebraic form? 8. How can a function s inverse be determined and graphed? 9. How can functions be combined? 10. What strategies can be used to graph polynomial functions and rational functions? 11. What real-world situations could be analyzed and solved using functions? http://www.cpalms.org/courses Remediation & Enrichment Resource:
2013-2014 Curriculum Blueprint Grade: 9-12 Course: Analytical Geometry Unit 2: Exponential and Logarithmic Functions Trad 4 weeks 4 x 4 2 weeks 1 st & 2 nd Quarters CPALMS site that contains the Specifications to include the Content limits, Attributes/Stimulus, and additional information. Date Range: Given during the instruction per the outline in this section Key Vocabulary: Exponential Function, e, Natural Exponential Function, Logarithmic Function, Natural Logarithmic Function, Model Students will be able to model, graph, and analyze exponential and logarithmic functions and solve their real-world applications. 1. Evaluate exponential and logarithmic expressions. 2. Recognize, graph, analyze, and transform exponential and logarithmic functions, with and without graphing calculators. 3. Solve exponential and logarithmic equations. 4. Evaluate, analyze, model, and solve real-world applications of exponential and logarithmic functions. MA.912.A.8.7: Solve applications of exponential growth and decay. A. Evaluate exponential and logarithmic functions, with and without graphing calculators 1. Convert between exponential and logarithmic forms 2. Use properties of logarithms to expand and condense logarithms 3. Use change of base formula B. Analyze and graph functions 1. Recognize and graph exponential and logarithmic functions, with and without graphing calculators a. Identify domain and range b. Locate x and y intercepts c. Determine conditions that result in increasing or decreasing intervals d. Locate vertical or horizontal asymptotes e. Apply transformations to graphing functions C. Solve exponential and logarithmic equations 1. How can exponential and logarithmic functions be analyzed and graphed? 2. What are the similarities and differences between exponential and logarithmic functions and their graphs? 3. What strategies are used in solving exponential and logarithmic equations? 4. How can functions be created to model real-world data? 5. What real-world issues could be analyzed and solved using exponential and logarithmic functions? http://www.cpalms.org/courses D. Solve real-world problems using exponential and logarithmic functions 1. Interest 2. Exponential growth and decay
2013-2014 Curriculum Blueprint Grade: 9-12 Course: Analytical Geometry Unit 3: Conic Sections Trad 4 weeks 4 x 4 2 weeks 2 nd Quarter CPALMS site that contains the Specifications to include the Content limits, Attributes/Stimulus, and additional information. Date Range: Given during the instruction per the outline in this section Key Vocabulary: Conic Section, Circle, Ellipse, Hyperbola, Parabola, Focus, Vertex, Branches, Directrix, Latus Rectum, Degenerate Conic Sections, Eccentricity Students will be able to model, graph, and analyze equations of conic sections and solve their real-world applications. 1. Write equations for conic sections given key features. 2. Recognize, graph, analyze, and transform equations of conic sections, with and without graphing calculators. 3. Evaluate, analyze, model, and solve real-world applications of conic sections. MA.912.A.9.1: Write the equations of conic sections in standard form and general form, in order to identify the conic section and to find its geometric properties (foci, asymptotes, eccentricity, etc.). MA.912.A.9.2: Graph conic sections with and without using graphing technology. MA.912.A.9.3: Solve real-world problems involving conic sections. MA.912.A.D.6.4: Use methods of direct and indirect proof and determine whether a short proof is logically valid. A. Analyze and graph equations of conic sections, with and without graphing calculators B. Write the equations of conic sections in standard form and general form 1. Completing the square C. Identify type of conic section from its equation D. Prove the standard form of conic sections using the distance formula and definition of conic sections E. Identify the geometric properties of a conic section using its equation and graph 1. Circle: center, radius, intercepts 2. Ellipse: center, foci. vertices, major axis, minor axis 3. Hyperbola: center, foci, vertices, asymptotes, transverse axis, conjugate axis 4. Parabola: Focus, vertex, directrix, axis of symmetry, latus rectum F. Solve real-world problems involving conic sections How are the shapes, known as conic sections, created? How are equations of conic sections analyzed and graphed? What are the similarities and differences in the equations of conic sections and the key features of their graphs? How can the general form of a conic section equation be converted to standard form? What real-world issues could be analyzed and solved using equations and graphs of conic sections? http://www.cpalms.org/courses Remediation & Enrichment Resource:
2013-2014 Curriculum Blueprint Grade: 9-12 Course: Analytical Geometry Unit 4: Parametric Equations and Their Graphs Trad 2 weeks 4 x 4 1 week 2 nd Quarter CPALMS site that contains the Specifications to include the Content limits, Attributes/Stimulus, and additional information. Date Range: Given during the instruction per the outline in this section Key Vocabulary: Parametric Equations, Parameter, Velocity, Representations Students will be able to model, graph, and analyze parametric equations and solve their real-world applications. 1. Recognize, graph, and analyze parametric functions, with and without graphing calculators. 2. Convert parametric equations from parametric form to rectangular form and vice versa. 3. Analyze, model, and solve real-world applications of parametric equations. MA.912.D.10.1: Sketch the graph of a curve in the plane represented parametrically, indicating the direction of motion. MA.912.D.10.2: Convert from a parametric representation of a plane curve to a rectangular equation and vice-versa. MA.912.D.10.3: Use parametric equations to model applications of motion in the plane. MA.912.A.D.6.4: Use methods of direct and indirect proof and determine whether a short proof is logically valid. A. Create a table of values and graph parametric relationships, with and without graphing calculators 1. Identify direction with arrow, as appropriate B. Determine rectangular equations 1. Determine if parametric equations are equivalent to a rectangular equation by examining a short proof 2. Determine a rectangular equation for parametric forms of conic sections by eliminating parameters 3. Graph parametric equations using rectangular form C. Solve real-world problems involving parametric equations What alternate methods for graphing exist in mathematics? What are the advantages of using parametric equations rather than rectangular equations? What information is needed to model parametric equations? What real-world issues could be analyzed and solved using parametric equations? http://www.cpalms.org/courses
2013-2014 Curriculum Blueprint Grade: 9-12 Course: Trigonometry Unit 5: The Unit Circle and Trigonometric Ratios Trad 4 weeks 4 x 4 2 weeks 3 rd Quarter CPALMS site that contains the Specifications to include the Content limits, Attributes/Stimulus, and additional information. Date Range: Given during the instruction per the outline in this section Key Vocabulary: Sine, Cosine, Tangent, Cotangent, Secant, Cosecant, Reference Angle, Unit Circle, Quadrantal Angle, Radian, Arc Length, Coterminal Angles Students will be able to sketch angles, define and use radian measures, explain characteristics of trigonometric functions, and evaluate trigonometric ratios and their inverses using special right triangles and the unit circle. 1. Illustrate angles and trigonometric ratios in the rectangular coordinate system. 2. Evaluate trigonometric ratios and their inverses using right triangles and the unit circle, with and without graphing calculators. 3. Explain characteristics of trigonometric functions using the unit circle. MACC.912.F-TF.1.1: Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. MACC.912.F-TF.1.2: Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. MACC.912.F-TF.1.3: Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π x, π+x, and 2π x in terms of their values for x, where x is any real number. MACC.912.F-TF.1.4: Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. A. Sketch angles in standard position using protractors 1. Define terminal and initial sides 2. Identify the direction for angles B. Define the radian measure of an angle on the unit circle 1. Convert between degree and radian measures 2. Explain the relationship between radian measure of an angle along the unit circle, terminal coordinate of that angle, and the associated real number C. Explain how the unit circle allows the extension of trigonometric functions to all real numbers D. Identify and use reference angles and reference triangles and their quadrants to evaluate trigonometric ratios E. Determine sine, cosine, and tangent of angles using the unit circle F. Find and use exact values of trigonometric functions for special angles using degree and radian measures G. Evaluate inverse trigonometric ratios with and without graphing calculators H. Explain symmetry (odd and even) and the periodicity of trigonometric functions using the unit circle What is trigonometry? How are angles illustrated and measured? What is meant by one radian? What is the difference between positive and negative measures of angles? How can trigonometric ratios evaluated? How can diagrams be drawn to determine values of trigonometric ratios? What real-world issues might be analyzed using trigonometric ratios? http://www.cpalms.org/courses/publicpreviewcourse24.aspx
2013-2014 Curriculum Blueprint Grade: 9-12 Course: Trigonometry Unit 6: Triangles and Trigonometric Ratios Trad 2 weeks 4 x 4 1 week 3 rd Quarter CPALMS site that contains the Specifications to include the Content limits, Attributes/Stimulus, and additional information. Date Range: Given during the instruction per the outline in this section Key Vocabulary: Cofunctions, Angle of Depression, Angle of Elevation, Oblique, Representations, Ambiguous Case Students will be able to prove the Pythagorean Identity, the Law of Sines, and the Law of Cosines and use them in solving right and oblique triangles and their real-world applications. 1. Prove the Pythagorean Identity. 1. Formulate strategies and use them to solve right and oblique triangles. 2. Prove the Laws of Sines and Cosines and derive the sine area of a triangle formula. 3. Solve real-world applications of right and oblique triangles. MACC.912.F-TF.3.8: 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. MACC.912.G-SRT.3.7: Explain and use the relationship between the sine and cosine of complementary angles. MACC.912.G-SRT.3.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. MACC.912.G-SRT.4.9: Derive the formula A = 1/2 ab sin(c) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. MACC.912.G-SRT.4.10: Prove the Laws of Sines and Cosines and use them to solve problems. A. Prove the Pythagorean Identity B. Use trigonometric ratios to solve triangles 1. Right triangles a. Special right triangles b. Two sides given c. One side and one acute angle given 2. Oblique triangles a. Using Law of Sines b. Using Law of Cosines C. Derive the area of a triangle formula which uses the sine ratio D. Prove the Law of Sines and Cosines E. Solve real-world problems involving right and oblique triangles, involving but not limited to area of triangles and similar triangles What are the similarities and differences between the sine and the cosine ratios? What are trigonometric identities and how are they proven? How can the Pythagorean Identity be manipulated to include other versions? What methods can be used to solve right and oblique triangles? What real-world problems can be analyzed and solved using right and oblique triangles? http://www.cpalms.org/courses/publicpreviewcourse24.aspx
2013-2014 Curriculum Blueprint Grade: 9-12 Course: Trigonometry Unit 7: Trigonometric Functions and Their Graphs Trad 4 weeks 4 x 4 2 weeks 3 rd & 4 th Quarters CPALMS site that contains the Specifications to include the Content limits, Attributes/Stimulus, and additional information. Date Range: Given during the instruction per the outline in this section Key Vocabulary: Sinusoidal Graphs, Period, Frequency, Amplitude, Phase Shift, Vertical Shift, Midline, Scale, Oscillate, Inverse Trigonometric Functions Students will be able to model, graph, and analyze real-world and mathematical periodic phenomena using trigonometric functions and their inverses, with and without graphing calculators. 1. Sketch all six parent trigonometric functions, including transformations, with and without graphing calculators, and analyze their key features. 2. Model periodic phenomena with trigonometric functions and solve their real-world applications. 3. Understand why domains of trigonometric functions must be restricted in order to construct inverses and sketch their graphs. MACC.912.F-TF.2.5: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. MACC.912.F-TF.2.6: Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed. A. Graph trigonometric functions on graph paper and identify and interpret key features, with and without graphing calculators 1. Domain and range 2. Intercepts 3. Period 4. Frequency 5. Amplitude 6. Midline 7. Phase shift 8. Vertical shift 9. Vertical asymptotes 10. Setting the appropriate window on the graphing calculator B. Solve real-world problems involving applications of trigonometric functions C. Explain why real-world or mathematical phenomena exhibit periodicity and model that behavior with appropriate trigonometric functions D. Sketch graphs for inverse trigonometric functions and understand how the domains of the functions must be restricted What are the key characteristics of the sine curve? What is the relationship between the sine function and the cosine function? How are the graphs of sine and cosine functions used to obtain graphs of cosecant and secant functions? Why can trigonometric functions be used to model periodic behavior? Why must the domains of trigonometric functions be restricted to allow trigonometric functions to have inverses? What real-world issues can be analyzed and solved using trigonometric functions? http://www.cpalms.org/courses/publicpreviewcourse24.aspx
2013-2014 Curriculum Blueprint Grade: 9-12 Course: Trigonometry Unit 8: Trigonometric Identities and Equations Trad 3 weeks 4 x 4 2 weeks 4 th Quarter CPALMS site that contains the Specifications to include the Content limits, Attributes/Stimulus, and additional information. Date Range: Given during the instruction per the outline in this section Key Vocabulary: Identity, Verifying an Identity, Equivalent Expressions, Substitution Students will be able to verify trigonometric identities, prove addition and subtraction formulas, use formulas to evaluate trigonometric functions, and solve trigonometric equations and their real-world applications. 1. Create logically valid algebraic steps for verifying trigonometric identities and proving trigonometric formulas. 2. Evaluate sine, cosine, and tangent functions using special angle formulas. 3. Solve trigonometric equations and their real-world applications, with and without graphing calculators. MACC.912.F-TF.3.9: Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. MACC.912.F-TF.2.7: Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context. A. Use the basic trigonometric identities and algebraic methods to verify other identities 1. Rewrite as equivalent expressions 2. Prove the sum and difference formulas for sine, cosine, and tangent B. Evaluate sine, cosine, and tangent functions using trigonometric formulas using known angles or given one trigonometric relationship and its quadrant 1. Sum and difference formulas 2. Double angle formulas C. Solve trigonometric equations and their real-world applications, with and without graphing calculators 1. Using algebraic methods 2. Using substitution of identities and formulas 3. Using inverse trigonometric functions What strategies are used in verifying an identity? What are similarities and differences between verifying an identity and writing a formal proof? How can the double angle formulas be derived? What are solutions to trigonometric equations? What methods used in verifying an identity can be used to solve a trigonometric equation? http://www.cpalms.org/courses/publicpreviewcourse24.aspx
2013-2014 Curriculum Blueprint Grade: 9-12 Course: Trigonometry Unit 9: Polar Coordinates and Complex Numbers Trad 2 weeks 4 x 4 1 week 4 th Quarter The below benchmark(s) is linked to the CPALMS site that contains the Specifications to include the Content limits, Attributes/Stimulus, and additional information. Date Range: Given during the instruction per the outline in this section Key Vocabulary: Modulus, Argument, Polar Form Students will be able to represent points and equations in both rectangular and polar forms with and without graphing calculators, represent complex numbers in both rectangular and polar forms, and operate with complex numbers in polar form. 1. Convert points and equations from rectangular to polar form and vice versa. 2. Graph polar equations, with and without graphing calculators. 2. Represent complex numbers in rectangular forms and polar forms, and explain why they represent the same number. 3. Represent operations of complex numbers geometrically on the complex plane and use them to operate with complex numbers. 4. Calculate the midpoint and distance between complex numbers. 5. Apply DeMoivre s Theorem to operations with complex numbers. MACC.912.N-CN.1.3: Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. MACC.912.N-CN.2.4: Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number. MACC.912.N-CN.2.5: Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, ( 1 + 3 i)³ = 8 because ( 1 + 3 i) has modulus 2 and argument 120. MACC.912.N-CN.2.6: Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints. Essential Content & Understanding: A. Define polar coordinates and relate them to Cartesian coordinates B. Represent equations given in rectangular coordinates in terms of polar coordinates and vice versa C. Graph equations in the polar coordinate plane, with and without graphing calculators D. Convert complex numbers written in rectangular form to trigonometric form and vice versa E. Represent operations of complex numbers geometrically on the complex plane and use those representations for computation F. Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of the segment as the average of the numbers at its endpoints G. Apply DeMoivre s Theorem to operations with complex numbers How are the coordinates of points written in rectangular form converted to polar form? How are equations written in rectangular form converted to polar form? How is graphing in the polar coordinates different from graphing in the rectangular system? How are complex numbers converted to polar form? How do you operate on complex numbers written in polar form? http://www.cpalms.org/courses/publicpreviewcourse24.aspx
2013-2014 Curriculum Blueprint Grade: 9-12 Course: Trigonometry Unit 10: Vectors Trad 2 weeks 4 x 4 1 week 4 th Quarter CPALMS site that contains the Specifications to include the Content limits, Attributes/Stimulus, and additional information. Date Range: Given during the instruction per the outline in this section Key Vocabulary: Vector, Magnitude, Position Vector, Zero Vector, Unit Vector, Scalar, Resultant Vector, Velocity Vector Students will be able to represent vectors and their operations using sketches, symbols, rectangular components, and in terms of its magnitude and direction, determine magnitude of vectors and their sums and scalar multiples, and solve real-world applications of vectors. 1. Represent vectors using sketches, symbols, rectangular components, and in terms of its magnitude and direction. 2. Add and subtract vectors using their components, their geometric representations, and the parallelogram rule. 3. Multiply vectors by scalars. 4. Calculate magnitudes for vectors, sums of vectors, and scalar multiples of vectors. 5. Solve real-world applications of velocity and other quantities that can be modeled by vectors. MACC.912.N-VM.1.1: Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, v, v, v). MACC.912.N-VM.1.2: Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. MACC.912.N-VM.1.3: Solve problems involving velocity and other quantities that can be represented by vectors. MACC.912.N-VM.2.4: Add and subtract vectors. a. Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. b. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. A. Sketch vectors using directed line segments and appropriate symbols for them and their magnitudes B. Find the components of vectors C. Perform vector operations 1. Add vectors a. Represent sums graphically by placing vectors end-to-end b. Find sums using components c. Find sums using the parallelogram rule d. Determine the magnitude and direction of the sum of vectors 2. Subtract vectors a. Represent differences graphically by connecting the tips in the appropriate order b. Find differences using components 3. Multiply a vector by a scalar a. Represent scalar multiplication graphically b. Perform scalar multiplication using components c. Compute the magnitude of a scalar multiple D. Solve real-world problems involving velocity and other quantities that can be modeled with vectors How are vectors represented and graphed? How can vectors be added? Why is the magnitude of a sum of two vectors typically not the sum of the two magnitudes? How can a vector written in rectangular components be written in terms of its magnitude and direction? What real-world issues can be analyzed and solved using vectors? http://www.cpalms.org/courses/publicprevie wcourse24.aspx
2013-2014 Curriculum Blueprint Grade: 9-12 Course: Trigonometry Unit 10: Vectors Trad 2 weeks 4 x 4 1 week 4 th Quarter c. Understand vector subtraction v w as v + ( w), where w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise. MACC.912.N-VM.2.5: Multiply a vector by a scalar. a. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c =. b. Compute the magnitude of a scalar multiple cv using cv = c v. Compute the direction of cv knowing that when c v 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).
MATHEMATICS PRACTICE STANDARDS: MACC.K12.MP.1.1: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. MACC.K12.MP.2.1: Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. MACC.K12.MP.3.1: Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. MACC.K12.MP.4.1: Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. MACC.K12.MP.5.1: Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.
MACC.K12.MP.6.1: Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. MACC.K12.MP.7.1: Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered 7 5 + 7 3, in preparation for learning about the distributive property. In the expression x² + 9x + 14, older students can see the 14 as 2 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y)² as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. MACC.K12.MP.8.1: Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y 2)/(x 1) = 3. Noticing the regularity in the way terms cancel when expanding (x 1)(x + 1), (x 1)(x² + x + 1), and (x 1)(x³ + x² + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. ANALYTICAL GEOMETRY: LACC.910.RST.2.4: Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific scientific or technical context relevant to grades 9 10 texts and topics. LACC.1112.RST.2.4: Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific scientific or technical context relevant to grades 11-12 texts and topics. LACC.910.RST.3.7: Translate quantitative or technical information expressed in words in a text into visual form (e.g., a table or chart) and translate information expressed visually or mathematically (e.g., in an equation) into words. LACC.1112.RST.3.7: Integrate and evaluate multiple sources of information presented in diverse formats and media (e.g., quantitative data, video, multimedia) in order to address a question or solve a problem. TRIGONOMETRY: LACC.910.RST.1.3: Follow precisely a complex multistep procedure when carrying out experiments, taking measurements, or performing technical tasks, attending to special cases or exceptions defined in the text. LACC.910.RST.2.4: Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific scientific or technical context relevant to grades 9 10 texts and topics. LACC.910.RST.3.7: Translate quantitative or technical information expressed in words in a text into visual form (e.g., a table or chart) and translate information expressed visually or mathematically (e.g., in an equation) into words. LACC.910.SL.1.1: Initiate and participate effectively in a range of collaborative discussions (one-on-one, in groups, and teacher-led) with diverse partners on grades 9 10 topics, texts, and issues, building on others ideas and expressing their own clearly and persuasively.
a. Come to discussions prepared, having read and researched material under study; explicitly draw on that preparation by referring to evidence from texts and other research on the topic or issue to stimulate a thoughtful, well-reasoned exchange of ideas. b. Work with peers to set rules for collegial discussions and decision-making (e.g., informal consensus, taking votes on key issues, presentation of alternate views), clear goals and deadlines, and individual roles as needed. c. Propel conversations by posing and responding to questions that relate the current discussion to broader themes or larger ideas; actively incorporate others into the discussion; and clarify, verify, or challenge ideas and conclusions. d. Respond thoughtfully to diverse perspectives, summarize points of agreement and disagreement, and, when warranted, qualify or justify their own views and understanding and make new connections in light of the evidence and reasoning presented. LACC.910.SL.1.2: Integrate multiple sources of information presented in diverse media or formats (e.g., visually, quantitatively, orally) evaluating the credibility and accuracy of each source. LACC.910.SL.1.3: Evaluate a speaker s point of view, reasoning, and use of evidence and rhetoric, identifying any fallacious reasoning or exaggerated or distorted evidence. LACC.910.SL.2.4: Present information, findings, and supporting evidence clearly, concisely, and logically such that listeners can follow the line of reasoning and the organization, development, substance, and style are appropriate to purpose, audience, and task. LACC.910.WHST.1.1: Write arguments focused on discipline-specific content. a. Introduce precise claim(s), distinguish the claim(s) from alternate or opposing claims, and create an organization that establishes clear relationships among the claim(s), counterclaims, reasons, and evidence. b. Develop claim(s) and counterclaims fairly, supplying data and evidence for each while pointing out the strengths and limitations of both claim(s) and counterclaims in a discipline-appropriate form and in a manner that anticipates the audience s knowledge level and concerns. c. Use words, phrases, and clauses to link the major sections of the text, create cohesion, and clarify the relationships between claim(s) and reasons, between reasons and evidence, and between claim(s) and counterclaims. d. Establish and maintain a formal style and objective tone while attending to the norms and conventions of the discipline in which they are writing. e. Provide a concluding statement or section that follows from or supports the argument presented. LACC.910.WHST.2.4: Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience. LACC.910.WHST.3.9: Draw evidence from informational texts to support analysis, reflection, and research.