MATH FLOW CHART - Board Docs

Updated November 12, 2014

MAP GLA – Missouri Assessment Program Grade Level Assessment; MAP EOC – Missouri Assessment Program End-‐of-‐Course Exam

MATH FLOW CHART

Grade 6 Grade 7 Grade 8 Grade 9 Grade 10 Grade 11 Grade 12

Advanced Algebra I

MAP GLA-‐Grade 8

Advanced Geometry

Advanced Algebra II

MAP EOC-‐Algebra I

Math

Analysis MAP -‐ ACT

Calculus AB or BC

AP Statistics

Math 6 Extension

MAP GLA-‐Grade 6 or

Math 6 MAP GLA-‐Grade 6

Pre-‐Algebra Extension

MAP GLA-‐Grade 7 or

Pre-‐Algebra MAP GLA-‐Grade 7

Algebra I

MAP GLA-‐Grade 8

Geometry

Algebra II

MAP EOC-‐Algebra I

College Algebra/

Trigonometry MAP -‐ ACT

Calculus

AP Statistics

Statistics

Foundations of Algebra

MAP GLA-‐Grade 8

Algebra I

Geometry

Geometry Concepts

Algebra II MAP EOC-‐Algebra I

MAP -‐ ACT Algebra II Concepts

MAP EOC-‐Algebra I MAP -‐ ACT

College Algebra/ Trigonometry

Statistics

Discovery Algebra

Algebra I

Geometry MAP -‐ ACT

Geometry Concepts MAP -‐ ACT

Algebra II MAP EOC-‐Algebra I

Algebra II Concepts

MAP EOC-‐Algebra I Additional Full Year Courses: General Math, Applied Technical Mathematics, and AP Computer Science. Calculus III is available for students who complete Calculus BC before grade 12.

Mathematics – Algebra II Concepts Pending Board Approval, May 13, 2015

Honoring Tradition ~ Continuing Excellence

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Critical Areas of Instruction Algebra II Concepts Overview Building on their work with linear, quadratic, and exponential functions, students extend their repertoire of functions to include polynomial, rational, and radical functions. Students work closely with the expressions that define the functions, and continue to expand and hone their abilities to model situations and to solve equations, including solving exponential equations using the properties of logarithms. The Mathematical Practice Standards apply throughout each course and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations. The critical areas for this course are as follows: Critical Area 1: The first critical area develops the structural similarities between the system of polynomials and the system of integers. Students draw on analogies between polynomial arithmetic and base-ten computation, focusing on properties of operations, particularly the distributive property. Students connect multiplication of polynomials with multiplication of multi-digit integers, and division of polynomials with long division of integers. Students identify zeros of polynomials and make connections between zeros of polynomials and solutions of polynomial equations. The unit culminates with the fundamental theorem of algebra. A central theme of this unit is that the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers. Critical Area 2: Building on their previous work with functions, and on their work with trigonometric ratios and circles in Geometry, students now use the coordinate plane to extend trigonometry to model periodic phenomena. Critical Area 3: Students synthesize and generalize what they have learned about a variety of function families. They extend their work with exponential functions to include solving exponential equations with logarithms. They explore the effects of transformations on graphs of diverse functions, including functions arising in an application, in order to abstract the general principle that transformations on a graph always have the same effect regardless of the type of the underlying function. They identify appropriate types of functions to model a situation, they adjust parameters to improve the model, and they compare models by analyzing appropriateness of fit and making judgments about the domain over which a model is a good fit. The description of modeling as “the process of choosing and using mathematics and statistics to analyze empirical situations, to understand them better, and to make decisions” is at the heart of this unit. The narrative discussion and diagram of the modeling cycle should be considered when knowledge of functions, statistics, and geometry is applied in a modeling context. Critical Area 4: Students see how the visual displays and summary statistics they learned in earlier grades relate to different types of data and to probability distributions. They identify different ways of collecting data— including sample surveys, experiments, and simulations—and the role that randomness and careful design play in the conclusions that can be drawn.

Common Core State Standards for Mathematics – Appendix A, pg. 36

NUMBER AND QUANTITY

The Real Number System A. Extend the properties of exponents to rational exponents. The Complex Number System A. Perform arithmetic operations with complex numbers. B. Use complex numbers in polynomial identities and equations.

ALGEBRA

Seeing Structure in Expressions A. Interpret the structure of expressions. B. Write expressions in equivalent forms to solve problems. Arithmetic with Polynomials and Rational Expressions A. Perform arithmetic operations on polynomials. B. Understand the relationship between zeros and factors of polynomials. D. Rewrite rational expressions. Creating Equations★ A. Create equations that describe numbers or relationships. Reasoning with Equations and Inequalities A. Understand solving equations as a process of reasoning and explain the reasoning. B. Solve equations and inequalities in one variable. C. Solve systems of equations. D. Represent and solve equations and inequalities graphically.

FUNCTIONS

Interpreting Functions B. Interpret functions that arise in applications in terms of the context. C. Analyze functions using different representations. Building Functions B. Build new functions from existing functions. Linear and Exponential Models A. Construct and compare linear, quadratic, and exponential models and solve problems. B. Interpret expressions for functions in terms of the situation they model.

GEOMETRY

Similarity, Right Triangles, and Trigonometry C. Define trigonometric ratios and solve problems involving right triangles.

STATISTICS AND PROBABILITY

Interpreting Categorical and Quantitative Data A. Summarize, represent, and interpret data on a single count or measurement variable. C. Interpret linear models Making Inferences and Justifying Conclusions A. Understand and evaluate random processes underlying statistical experiments. B. Make inferences and justify conclusions from sample surveys, experiments, and observational studies. Conditional Probability and the Rules of Probability B. Use the rules of probability to compute probabilities of compound events in a uniform probability model. Common Core State Standards for Mathematics – Appendix A, pgs. 82-91



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Mathematical Practices The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy). Overall Habits of Mind of a Productive Mathematics Thinker 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. 6. 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.

Reasoning and Explaining Modeling and Using Tools Seeing Structure and Generalizing 2. 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. 3. 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.

4. 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. 5. 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.

7. 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. 8. 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.



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★Modeling★

Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol ★ . Modeling links classroom mathematics and statistics to everyday life, work, and decision-making. Modeling is the process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions. Quantities and their relationships in physical, economic, public policy, social, and everyday situations can be modeled using mathematical and statistical methods. When making mathematical models, technology is valuable for varying assumptions, exploring consequences, and comparing predictions with data. A model can be very simple, such as writing total cost as a product of unit price and number bought, or using a geometric shape to describe a physical object like a coin. Even such simple models involve making choices. It is up to us whether to model a coin as a three-dimensional cylinder, or whether a two-dimensional disk works well enough for our purposes. The basic modeling cycle is summarized in the diagram. It involves: (1) identifying variables in the situation and selecting those that represent essential features, (2) formulating a model by creating and selecting geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between the variables, (3) analyzing and performing operations on these relationships to draw conclusions, (4) interpreting the results of the mathematics in terms of the original situation, (5) validating the conclusions by comparing them with the situation, and then either improving the model or, if it is acceptable, (6) reporting on the conclusions and the reasoning behind them. Choices, assumptions, and approximations are present throughout this cycle.



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NUMBER and QUANTITY - The Real Number System N-RN A. Extend the properties of exponents to rational exponents. Missouri Learning Standards Ladue Objectives

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

N-RN.A.1.i N-RN.A.1.ii

Describe the rules of integer exponents and properties of exponents. Simplify exponential expressions using the properties of exponents.

MP.1 MP.2

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

N-RN.A.2.i Simplify expressions involving radicals and integer exponents using the properties of exponents.

MP.2 MP.7

NUMBER and QUANTITY - The Real Number System N-RN B. Use properties of rational and irrational numbers. Missouri Learning Standards Ladue Objectives

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.

N-RN.B.3.i Justify with examples the relationship between the number system being used (natural numbers, whole numbers, integers, rational numbers, and irrational numbers) and the question of whether or not an equation has a solution in that number system.

MP.2 MP.3 MP.6



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NUMBER and QUANTITY - The Complex Number System N-CN

A. Perform arithmetic operations with complex numbers. Missouri Learning Standards Ladue Objectives

1. Know there is a complex number I such that i2 = –1, and every complex number has the form a + bi with a and b real.

N-CN.A.1.i N-CN.A.1.ii

Define a pure imaginary number in the form bi. Define complex numbers in the form a + bi.

MP.1 MP.7

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

N-CN.A.2.i Perform operations and simplify complex number expressions. MP.2 MP.7

NUMBER and QUANTITY - The Complex Number System N-CN C. Use complex numbers in polynomial identities and equations (polynomials with real coefficients). Missouri Learning Standards Ladue Objectives

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

N-CN.C.7.i Solve quadratic equations with real coefficients that have complex solutions. MP.1 MP.2 MP.7



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ALGEBRA - Seeing Structure in Expressions A-SSE A. Interpret the structure of expressions (polynomial and rational). Missouri Learning Standards Ladue Objectives

1. Interpret expressions that represent a quantity in terms of its context.★

1a. Interpret parts of an expression, such as terms, factors, and coefficient

A-SSE.A.1a.i Analyze a rational expression based on its factors and coefficients.★ MP.2 MP.7

1b. 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.

A-SSE.A.1b.i Construct models and solve problems involving exponential growth and decay. MP.2 MP.4 MP.7

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

A-SSE.A.2.i Describe and use algebraic manipulations, including factoring and rules of integer exponents and apply properties of exponents to simplify and rewrite expressions.

MP.1 MP.7 MP.8

ALGEBRA - Seeing Structure in Expressions A-SSE B. Write expressions in equivalent forms to solve problems. Missouri Learning Standards Ladue Objectives

3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.★

3a. Factor a quadratic expression to reveal the zeros of the function it defines.

A-SSE.B.3a.i Factor a quadratic expression to reveal the zeros of the function it defines. MP.1 MP.2 MP.7

3c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

A-SSE.B.3c.i Use properties of exponents (such as power of a power, product of powers, power of a product, and rational exponents, etc.) to write an equivalent form of an exponential function to reveal and explain specific information about its approximate rate of growth or decay.

MP.2 MP.4 MP.7



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ALGEBRA - Arithmetic with Polynomials and Rational Expressions A-APR A. Perform arithmetic operations on polynomials (beyond quadratic). Missouri Learning Standards Ladue Objectives

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.

A-APR.A.1.i A-APR.A.1.ii A-APR.A.1.iii

Understand the definition of a polynomial of degree greater than 2. Understand the concepts of combining like terms and closure. Add, subtract, and multiply polynomials and understand how closure applies under these operations.

MP.7 MP.8

ALGEBRA - Arithmetic with Polynomials and Rational Expressions A-APR B. Understand the relationship between zeros and factors of polynomials. Missouri Learning Standards Ladue Objectives

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.

A-APR.B.3.i A-APR.B.3.ii

Identify the zeros of a polynomial when the polynomial is factored. Draw a graph of the function using the zeros of a function.

MP.5 MP.7

ALGEBRA - Arithmetic with Polynomials and Rational Expressions A-APR D. Rewrite rational expressions (linear and quadratic denominators). Missouri Learning Standards Ladue Objectives

6. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

A-APR.D.6.i Rewrite rational expressions,(a(x))/(b(x)), in the form (q(x)) + ((r(x))/(b(x))) by using factoring, long division, or synthetic division. Use a computer algebra system for complicated examples to assist with building a broader conceptual understanding.

MP.5 MP.7

7. Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

A-APR.D.7.i Add, subtract, multiply, and divide rational expressions.

MP.2 MP.3 MP.6 MP.7 MP.8



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ALGEBRA - Creating Equations A-CED A. Create equations that describe numbers or relationships (equations using all available types of expressions, including simple root functions). Missouri Learning Standards Ladue Objectives


A-CED.A.1.i Create linear, quadratic, rational and exponential equations and inequalities in one variable and use them in a contextual situation to solve problems.★

MP.1 MP.2 MP.4 MP.5

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

A-CED.A.2.i A-CED.A.2.ii

Create equations in two or more variables to represent relationships between quantities.★ Graph equations in two variables on a coordinate plane and label the axes and scales.★

MP.1 MP.2 MP.4 MP.5

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


Write and use a system of equations and/or inequalities to solve a real world problem.★ Recognize that the equations and inequalities represent the constraints of the problem.★

MP.1 MP.2 MP.4 MP.5



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ALGEBRA - Reasoning with Equations and Inequalities A-REI A. Understand solving equations as a process of reasoning and explain the reasoning (simple radical and rational). Missouri Learning Standards Ladue Objectives

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

A-REI.A.2.i A-REI.A.2.ii

Solve simple rational and radical equations in one variable. Write examples of how extraneous solutions arise.

MP.1 MP.2 MP.5

ALGEBRA - Reasoning with Equations and Inequalities A-REI B. Solve equations and inequalities in one variable. Missouri Learning Standards Ladue Objectives

4. Solve quadratic equations in one variable

4b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

A-REI.B.4b.i A-REI.B.4b.ii

Find roots of quadratic functions by graphing, factoring, completing the square, and using the quadratic formula. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

MP.5 MP.7



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ALGEBRA - Reasoning with Equations and Inequalities A-REI C. Solve systems of equations. Missouri Learning Standards Ladue Objectives

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.C.5.i A-REI.C.5.ii

Solve systems of equations using the elimination method (sometimes called linear combinations). Solve systems of equations by substitution (solving for one variable in the first equation and substitution it into the second equation).

MP.1 MP.5

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

A-REI.C.6.i Solve systems of equations by graphing.

ALGEBRA - Reasoning with Equations and Inequalities A-REI D. Represent and solve equations and inequalities graphically (combine polynomial, rational, radical, absolute value, and exponential functions). Missouri Learning Standards Ladue Objectives

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.★

A-REI.D.11.i Explain why the intersection of y = f(x) and y = g(x) is the solution of f(x) = g(x) for any combination of linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Find the solution(s) by using technology to graph the equations and determine their point of intersection, or using tables of values.★

MP.3 MP.4 MP.5 MP.6



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FUNCTIONS - Interpreting Functions F-IF B. Interpret functions that arise in applications in terms of the context (emphasize selection of appropriate models). Missouri Learning Standards Ladue Objectives

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.★

F-IF.B.5.i Given the graph of a function, determine the practical domain of the function as it relates to the numerical relationship it describes.★


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

F-IF.B.6.i F-IF.B.6.ii F-IF.B.6.iii

Calculate the average rate of change over a specified interval of a function presented algebraically or in a table.★ Estimate the average rate of change over a specified interval of a function from the function’s graph.★ Explain, in context, the average rate of change of a function over a specified interval.★




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FUNCTIONS - Interpreting Functions F-IF C. Analyze functions using different representations (focus on using key features to guide selection of appropriate type of model function). Missouri Learning Standards Ladue Objectives

7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★

7a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

F-IF.C.7a.i Graph quadratic functions showing intercepts, maxima, or minima.★ MP.1 MP.2 MP.4 MP.5 MP.6

7e. Graph exponential and logarithmic functions, showing intercepts and end behavior.

F-IF.C.7e.i

Draw graphs of logarithmic and exponential functions showing characteristics of the functions.★




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

8a. Use the process of factoring in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

F-IF.C.8a.i F-IF.C.8a.ii

Use the process of factoring in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Use the properties of exponents to interpret expressions for percent rate of change, and classify them as growth or decay.


8b. 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.C.8b.i Identify rate of change of exponential functions in order to determine whether it represents growth or decay.

MP.2 MP.4 MP.5 MP.6

FUNCTIONS - Building Functions F-BF B. Build new functions from existing functions (include all types of functions studied). Missouri Learning Standards Ladue Objectives

4. Find inverse functions. 4a. 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.

F-BF.B.4a.i Define the inverse of a given function.

MP.4 MP.5 MP.7

4c. Read values of an inverse function from a graph or a table, given that the function has an inverse.

F-BF.B.4c.i F-BF.B.4c.ii

Evaluate the values of an inverse function from a graph or table of a given function. Understand vertical line test.

MP.4 MP.5



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FUNCTIONS - Linear and Exponential Models F-LE A. Construct and compare linear, quadratic, and exponential models and solve problems (logarithms as solutions for exponentials). Missouri Learning Standards Ladue Objectives

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.

F-LE.A.4.i F-LE.A.4.ii

Solve exponential equations by rewriting as a logarithmic equation. Evaluate logarithmic expressions using technology.


FUNCTIONS - Linear and Exponential Models F-LE B. Interpret expressions for functions in terms of the situation they model. Missouri Learning Standards Ladue Objectives

5. Interpret the parameters in a linear or exponential function in terms of a context.

F-LE.B.5.i

Interpret and understand quantities, rates of change, and other values of linear and exponential functions in the context of real world scenarios.


GEOMETRY - Similarity, Right Triangles, and Trigonometry G-SRT C. Define trigonometric ratios and solve problems involving right triangles. Missouri Learning Standards Ladue Objectives

8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★

G-SRT.C.8.i Use trigonometric relationships with right triangles to determine lengths and angle measures.★




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STATISTICS and PROBABILITY - Interpreting Categorical and Quantitative Data S-ID A. Summarize, represent, and interpret data on a single count or measurement variable. Missouri Learning Standards Ladue Objectives

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

S-ID.A.3.i S-ID.A.3.ii

Analyze multiple data sets using shape, center, spread, and standard deviation. Determine five-number summary when creating box plots.

MP.5 MP.6 MP.7 MP.8

4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population 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-ID.A.4.i S-ID.A.4.ii S-ID.A.4.iii S-ID.A.4.iv

Evaluate the standard deviation of a data set. Display a data set using appropriate graphical representation. Use the mean and standard deviation of a data set to fit to a normal distribution curve. Use the three-sigma rule to calculate the percent of a normal population that lines within three deviations of the mean.

MP.5 MP.6 MP.7

STATISTICS and PROBABILITY - Interpreting Categorical and Quantitative Data S-ID C. Interpret linear models. Missouri Learning Standards Ladue Objectives

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

S-ID.C.8.i Evaluate the correlation coefficient of a line of best fit using technology. MP.2 MP.5 MP.6

9. Distinguish between correlation and causation. S-ID.C.9.i Explain the difference between causation and correlation using data sets. MP.2 MP.3



16

STATISTICS and PROBABILITY - Making Inferences and Justifying Conclusions S-IC A. Understand and evaluate random processes underlying statistical experiments. Missouri Learning Standards Ladue Objectives

1. Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

S-IC.A.1.i

Make conclusions about a population based on a random sample from that population.

MP.2

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

S-IC.A.2.i Analyze consistency of data and expected probability. MP.3

STATISTICS and PROBABILITY - Making Inferences and Justifying Conclusions S-IC B. Make inferences and justify conclusions from sample surveys, experiments, and observational studies. Missouri Learning Standards Ladue Objectives

3. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

S-IC.B.3.i S-IC.B.3.ii

Compare the methodologies of sample surveys, experiments, and observational studies. Explain the purpose of randomization in the various methodologies.

MP.3

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.


Make an estimate of a population mean using a sample survey. Predict a margin of error using a simulation model for random sampling.

MP.4 MP.7

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


Compare two treatments using data from a randomized experiment. Analyze the difference between parameters using simulations.


6. Evaluate reports based on data. S-IC.B.6.i Make conjectures about the relationship between characteristics of a sample using data.




17

STATISTICS and PROBABILITY - Conditional Probability and the Rules of Probability S-CP B. Use the rules of probability to compute probabilities of compound events in a uniform probability model (include more complex situations). Missouri Learning Standards Ladue Objectives

6. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.

S-CP.B.6.i Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.

MP.1 MP.4 MP.7

7. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.

S-CP.B.7.i Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.

MP.1 MP.4 MP.7

Mathematics – High School Algebra I Pending Board Approval, May 13, 2015


1

Critical Areas of Instruction Essential Contents and Skills

The fundamental purpose of this course is to formalize and extend the mathematics that students learned in the middle grades. Because it is built on the middle grades standards, this is a more ambitious version of Algebra I than has generally been offered. The critical areas deepen and extend understanding of linear and exponential relationships by contrasting them with each other and by applying linear models to data that exhibit a linear trend, and students engage in methods for analyzing, solving, and using quadratic functions. The Mathematical Practice Standards apply throughout the course and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations. Critical Area 1: By the end of eighth grade, students have learned to solve linear equations in one variable and have applied graphical and algebraic methods to analyze and solve systems of linear equations in two variables. Now, students analyze and explain the process of solving an equation. Students develop fluency writing, interpreting, and translating between various forms of linear equations and inequalities, and using them to solve problems. They master the solution of linear equations and apply related solution techniques and the laws of exponents to the creation and solution of simple exponential equations. Critical Area 2: In earlier grades, students define, evaluate, and compare functions, and use them to model relationships between quantities. In this unit, students will learn function notation and develop the concepts of domain and range. They explore many examples of functions, including sequences; they interpret functions given graphically, numerically, symbolically, and verbally, translate between representations, and understand the limitations of various representations. Students build on and informally extend their understanding of integer exponents to consider exponential functions. They compare and contrast linear and exponential functions, distinguishing between additive and multiplicative change. Students explore systems of equations and inequalities, and they find and interpret their solutions. They interpret arithmetic sequences as linear functions and geometric sequences as exponential functions. Critical Area 3: This critical area builds upon prior students’ prior experiences with data, providing students with more formal means of assessing how a model fits data. Students use regression techniques to describe approximately linear relationships between quantities. They use graphical representations and knowledge of the context to make judgments about the appropriateness of linear models. With linear models, they look at residuals to analyze the goodness of fit. Critical Area 4: Students build on their knowledge from Critical Area 2, where they extended the laws of exponents to rational exponents. Students apply this new understanding of number and strengthen their ability to see structure in and create quadratic and exponential expressions. They create and solve equations, inequalities, and systems of equations involving quadratic expressions. Critical Area 5: Students consider quadratic functions, comparing the key characteristics of quadratic functions to those of linear and exponential functions. They select from among these functions to model phenomena. Students learn to anticipate the graph of a quadratic function by interpreting various forms of quadratic expressions. In particular, they identify the real solutions of a quadratic equation as the zeros of a related quadratic function. Students expand their experience with functions to include more specialized functions—absolute value, step, and those that are piecewise-defined. Common Core State Standards for Mathematics – Appendix A, pg. 15

NUMBER AND QUANTITY

The Real Number System A. Extend the properties of exponents to rational exponents. B. Use properties of rational and irrational numbers.

Quantities A. Reason quantitatively and use units to solve problems B. Use properties of rational and irrational numbers

ALGEBRA

Seeing Structure in Expressions A. Interpret the structure of expressions B. Write expressions in equivalent forms to solve problems

Arithmetic with Polynomials and Rational Expressions A. Perform arithmetic operations on polynomials B. Rewrite rational expressions

Creating Expressions A. Create equations that describe numbers or relationships

Reasoning with Equations and Inequalities A. Understand solving equations as a process of reasoning and explain the reasoning B. Solve equations and inequalities in one variable. C. Solve systems of equations D. Represent and solve equations and inequalities graphically

FUNCTIONS

Interpreting Functions A. Understand the concept of a function and use function notation B. Interpret functions that arise in applications in terms of the context C. Analyze functions using different representations

Building Functions A. Build a function that models a relationship between two quantities B. Build new functions from existing functions

Linear, (Quadratic), and Exponential Models A. Construct and compare linear, quadratic, and exponential models and solve problems. B. Interpret expressions for functions in terms of the situation they model


Interpreting Categorical and Quantitative Data A. Summarize, represent, and interpret data on a single count or measurement variable. B. Summarize, represent, and interpret data on two categorical and quantitative variables. C. Interpret linear models.



2

Mathematical Practices The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy). Overall Habits of Mind of a Productive Mathematics Thinker MP.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. MP.6 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.

Reasoning and Explaining Modeling and Using Tools Seeing Structure and Generalizing MP.2 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. MP.3 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.

MP.4 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. MP.5 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.

MP.7 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. MP.8 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.



3

★Modeling★




4

NUMBER AND QUANTITY - The Real Number System N-RN A. Extend the properties of exponents to rational exponents Missouri Learning Standards Ladue Objectives


N-RN.A.1.i Describe the rules of rational exponents and how this informs notation of radicals in terms of exponents.

MP.2 MP.7 MP.8


N-RN.A.2.i Simplify expressions involving radicals and integer exponents using the properties of exponents.

MP.6 MP.7

NUMBER AND QUANTITY - The Real Number System N-RN B. Use properties of rational and irrational numbers Missouri Learning Standards Ladue Objectives



MP.7 MP.8



5

NUMBER AND QUANTITY - Quantities★ N-Q A. Reason quantitatively and use units to solve problems (foundation for work with expressions, equations, and functions).

Missouri Learning Standards Ladue Objectives 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.A.1.i N-Q.A.1.ii N-Q.A.1.iii N-Q.A.1.iv

Solve multi-step problems where quantities are given in two different units of measure.★ Interpret units in the context of the problem.★ Use units to evaluate the appropriateness of the solution when solving a multi-step problem.★ Choose the appropriate units for a specific formula and interpret the meaning of the unit in that context.★ Choose and interpret the scale and the origin in graphs and data displays.★

MP.1 MP.2 MP.4 MP.6

2. Define appropriate quantities for the purpose of descriptive modeling.

N-Q.A.2.i Determine and interpret appropriate quantities when using descriptive modeling.★ MP.1 MP.3 MP.4 MP.6 MP.8

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

N-Q.A.3.i N-Q.A.3.ii N-Q.A.3.iii

Judge the reasonableness of numerical computations and their results in the context of the situation.★ Use estimation to determine the reasonableness of a solution (e.g., parabolas). ★ Attend to precision by rounding to the appropriate unit.★

MP.2 MP.4 MP.5 MP.6



6

ALGEBRA - Seeing Structure in Expressions A-SSE A. Interpret the structure of expressions (linear, exponential, quadratic). Missouri Learning Standards Ladue Objectives

1.

Interpret expressions that represent a quantity in terms of its context.★

1a.

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

A-SSE.A.1a.i Interpret parts of an expression, including the terms, factors, and coefficients.★ MP.2 MP.4 MP.7

1b.

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.

A-SSE.A.1b.i Decompose expressions and make sense of the multiple factors and terms by explaining the meaning of the individual parts.★

MP.2 MP.4 MP.7

2.

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

A-SSE.A.2.i A-SSE.A.2.ii A-SSE.A.2.iii

Rewrite algebraic expressions in different equivalent forms such as factoring or simplifying. Use factoring techniques such as common factors, grouping, difference of squares, and perfect square trinomials. Simplify expressions including combining like terms using distributive property and other operations with polynomials.




7

ALGEBRA - Seeing Structure in Expressions A-SSE B. Write expressions in equivalent forms to solve problems (quadratic and exponential). Missouri Learning Standards Ladue Objectives



A-SSE.B.3a.i Factor a quadratic expression to find the zeros of the function it defines.★ MP.2 MP.4 MP.6 MP.7 MP.8

3b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

A-SSE.B.3b.i A-SSE.B.3b.ii

Complete the square in a quadratic expression.★ Find the minimum or maximum (vertex) of the function.★



A-SSE.B.3c.i Use properties of exponents (such as power of a power, product of powers, power of a product, and negative exponents) to write an equivalent form of an exponential function to reveal and explain specific information about its approximate rate of growth or decay.★


ALGEBRA - Arithmetic with Polynomials and Rational Expressions A-APR A. Perform arithmetic operations on polynomials (linear and quadratic). Missouri Learning Standards Ladue Objectives



Describe the definition of a polynomial. Apply the concepts of combining like terms and closure. Add, subtract, and multiply polynomials and understand how closure applies under these operations.

MP.4 MP.7 MP.8



8

ALGEBRA - Arithmetic with Polynomials and Rational Expressions A-APR D. Rewrite rational expressions. Missouri Learning Standards Ladue Objectives


A-APR.D.6.i

Multiply and divide monomial expressions with integer exponents.

MP.2 MP.6 MP.7 MP.8

ALGEBRA - Creating Equations★ A-CED A. Create equations that describe numbers or relationships (linear, quadratic, and exponential - integer inputs only). Missouri Learning Standards Ladue Objectives

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.

A-CED.A.1.i

Create linear, quadratic, and exponential equations and inequalities in one variable and use them in a contextual situation to solve a problem.★

MP.1 MP.2 MP.4 MP.8



Create and use equations, graphs, tables, descriptions, or sets of ordered pairs to express a relationship between two variables.★ Graph equations in two variables on a coordinate plane and label the axis and scales.★

MP.1 MP.2 MP.4 MP.5 MP.6 MP.8

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


Write and use a system of equations and/or inequalities to solve a real world problem. Recognize constraints of the problem by applying the definitions of domain and range of a function as it applies to the real world situation.★

MP.1 MP.2 MP.6

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

A-CED.A.4.i Solve multi-variable formulas or equations for a specific variable.★ MP.2 MP.4 MP.6 MP.8

.



9

ALGEBRA - Reasoning with Equations and Inequalities A-REI A. Understand solving equations as a process of reasoning and explain the reasoning (master linear, learn as general principle). Missouri Learning Standards Ladue Objectives

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.

A-REI.A.1.i

Construct a convincing argument that justifies each step in the solution process, assuming an equation has a solution.

MP.1 MP.3 MP.6

ALGEBRA - Reasoning with Equations and Inequalities A-REI B. Solve equations and inequalities in one variable (linear inequalities; literal equations that are linear in the variables being solved for; quadratics with real solutions). Missouri Learning Standards Ladue Objectives

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

A-REI.B.3.i A-REI.B.3.ii

Solve linear equations in one variable, including equations with coefficients represented by letters. Solve linear inequalities in one variable, including inequalities with coefficients represented by letters.


4. Solve quadratic equations in one variable.

4a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x-p)2=q that has the same solutions. Derive the quadratic formula from this form.

A-REI.B.4a.i Describe and use algebraic rules and rules of exponents to transform quadratics. Use algebraic rules to explain each step in the derivation of the quadratic formula.

MP.1 MP.2 MP.3 MP.4 MP.6 MP.7 MP.8



Solve quadratic equations in one variable by simple inspection, taking the square root, factoring, and completing the square, and using the quadratic formula. Explain why taking the square root of both sides of an equation yields two solutions.




10

ALGEBRA - Reasoning with Equations and Inequalities A-REI C. Solve systems of equations (linear-linear and linear-quadratic). Missouri Learning Standards Ladue Objectives


A-REI.C.5.i

Prove that, given a system of two equations in two variables, the sum of one equation and a multiple of the other produces the same solutions.

MP.3 MP.6 MP.8


A-REI.C.6.i A-REI.C.6.ii A-REI.C.6.iii A-REI.C.6.iv A-REI.C.6.v A-REI.C.6.vi

Solve systems of equations using the elimination method. Solve systems of equations using the elimination with multiplication method. Solve a system of equations by substitution (solving for one variable in the first equation and substituting it into the second equation). Solve systems of equations using graphs. Explain how the point of intersection of two graphs will represent the solution to the system of two linear equations. Solve real-world problems and mathematical problems dealing with systems of linear equations and interpret the solution in the context of the problem.


7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically, for example, find the points of intersection between the line y = -3x and the circle x2 + y2 = 3


Solve a system of linear-quadratic equations using the substitution method, finding all solutions. Use a graphing calculator to find points of intersection of a system consisting of a linear equation and a quadratic equation.


.



11

ALGEBRA - Reasoning with Equations and Inequalities A-REI D. Represent and solve equations and inequalities graphically (linear and exponential; learn as a general principle). Missouri Learning Standards Ladue Objectives 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.D.10.i

Illustrate that all solutions to an equation in two variables are contained on the graph of that equation.

MP.3 MP.7

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.★

A-REI.D.11.i A-REI.D.11.ii

Explain that a point of intersection on the graph of a system of equations, y = f(x) and y = g(x), is a solution to both equations.★ Explain that since y = f(x) and y = g(x), then f(x) = g(x) by the substitution property.★

MP.3 MP.4 MP.5 MP.6

12. Graph 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.

A-REI.D.12.i A-REI.D.12.ii A-REI.D.12.iii A-REI.D.12.iv A-REI.D.12.v A-REI.D.12.vi

Define linear inequality, half-plane, and boundary. Graph a linear inequality on a coordinate plane, resulting in a boundary line (solid or dashed) and a shaded half-plane. Graph a system of linear equations on a coordinate plane. Graph a system of linear inequalities on a coordinate plane. Explain that the solution set for a system of linear inequalities is the intersection of the shaded regions (half-planes) of both inequalities. Verify the points in the intersection of the half-planes represent a solution to the system.

MP.4 MP.5 MP.6



12

FUNCTIONS - Interpreting Functions F-IF A. Understand the concept of a function and use function notation. Missouri Learning Standards Ladue Objectives

1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

F-IF.A.1.i F-IF.A.1.ii F-IF.A.1.iii F-IF.A.1.iv F-IF.A.1.v

Define relation, domain, and range. Define a function as a relation in which each input (domain) has exactly one output (range). Determine if a graph, table, or set of ordered pairs represents a function. Determine if stated rules (both numeric and non-numeric) produce ordered pairs that represent a function. Explain that when x is an element of the input of a function, f(x) represents the corresponding output of the function.

MP.2 MP.4 MP.7

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

F-IF.A.2.i F-IF.A.2.ii F-IF.A.2.iii

When a relation is determined to be a function, use f(x) notation. Evaluate functions for inputs in their domain. Decode function notation and explain how the output of a function is matched to its input.


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.

F-IF.A.3.i F-IF.A.3.ii F-IF.A.3.iii F-IF.A.3.iv

Convert a list of numbers (sequence) into a function by making the whole numbers (0, 1, 2, etc.) the input and the elements of the sequence the outputs. Explain that a recursive formula tells how a sequence starts and tells how to use the previous value to generate the next element of the sequence. Explain that an explicit formula finds any element of a sequence without knowing the element before it. Distinguish between explicit and recursive formulas for sequences.

MP.4 MP.6 MP.7 MP.8



13

FUNCTIONS - Interpreting Functions F-IF B. Interpret functions that arise in applications in terms of the context. Missouri Learning Standards Ladue Objectives

4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★

F-IF.B.4.i F-IF.B.4.ii F-IF.B.4.iii F-IF.B.4.iv

Identify key features in graphs and tables including: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries.★ Sketch the graph, given the key features of a function.★ Interpret the meaning of an ordered pair.★ Determine if negative inputs and outputs make sense in the problem situation.★

MP.1 MP.2 MP.4


F-IF.B.5.i F-IF.B.5.ii

Explain how the domain of a function is represented in its graph.★ State the appropriate domain of a function that represents a problem situation, defend the choice, and explain why other numbers might be excluded from the domain.★




Define interval, rate of change, and average rate of change.★ Estimate the average rate of change over a specified interval of a function from the function’s graph.★ Interpret, in context, the average rate of change of a function over a specified interval.★

MP.2 MP.4 MP.6



14

FUNCTIONS - Interpreting Functions F-IF C. Analyze functions using different representations. Missouri Learning Standards Ladue Objectives



F-IF.C.7a.i F-IF.C.7a.ii F-IF.C.7a.iii F-IF.C.7a.iv F-IF.C.7a.v F-IF.C.7a.v

Explain why the equation y = mx + b represents a linear function and interpret the slope (m) and y-intercept (b) in relation to the function. ★ Give examples of relationships that are non-linear functions in the form of a graph or table.★ Graph linear functions showing intercepts.★ Explain why the equation y = ax2 + bx + c represents a quadratic function and interpret the y-intercept (c) in relation to the function.★ Graph quadratic functions showing intercepts, maxima, or minima (vertex). * Graph linear and quadratic functions using a graphing calculator. Find intercepts, maxima and minima using the calculator.


7b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

F-IF.C.7b.i Graph absolute value functions.★

MP.4

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

F-IF.C.7e.i Graph exponential functions showing intercepts.★

MP.4


8a. 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.

F-IF.C.8a.i 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.

MP.1 MP.2 MP.3

8b. 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.C.8b.i Use the properties of exponents to interpret expressions for percent rate of change, and classify them as growth or decay.

MP.1 MP.2 MP.3MP.8

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.

F-IF.C.9.i Compare the key features of two functions represented in different ways.

MP.2 MP.3



15

FUNCTIONS - Building Functions F-BF A. Build a function that models a relationship between two quantities. Missouri Learning Standards Ladue Objectives

1. Write a function that describes a relationship between two quantities.★

1a. Determine an explicit expression, a recursive process, or steps for calculation from a context.

F-BF.A.1a.i F-BF.A.1a.ii F-BF.A.1a.iii F-BF.A.1a.iv F-BF.A.1a.v F-BF.A.1a.vi

Identify the quantities being compared in a real-world problem.★ Write a linear function that models a given situation as a table of x and y values or as a graph.★ Define the rate of change in relation to the situation.★ Define the y-intercept in relation to the situation.★ Explain any constraints on the domain in relation to the situation.★ Write an explicit and/or recursive expressions of a function to describe a real-world problem.★


1b. 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.

F-BF.A.1b.i Combine standard function types, such as linear and exponential, using arithmetic operations.★

MP.1 MP.2 MP.4 MP.7

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

F-BF.A.2.i F-BF.A.2.ii

Write arithmetic sequences recursively and explicitly, use the two forms to model a situation, and translate between the two forms.★ Write geometric sequences recursively and explicitly, use the two forms to model a situation, and translate between the two forms.★

MP.4 MP.5 MP.6



16

FUNCTIONS - Building Functions F-BF B. Build new functions from existing functions. Missouri Learning Standards Ladue Objectives

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.

F-BF.B.3.i F-BF.B.3.ii

Identify, through experimenting with technology, the effect on the graph of a function by replacing f(x) with f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative). Determine the value of (k), given the graphs of the original function and a transformation.

MP.4 MP.5MP.7 MP.8

4. Find inverse functions.

4a. 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) = 2x3 jor f(x) = (x+1/(x-1) for x≠1.

F-BF.B.4a.i 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 (linear only).



17

FUNCTIONS - Linear, (Quadratic), and Exponential Models★ F-LE A. Construct and compare linear, quadratic, and exponential models and solve problems. Missouri Learning Standards Ladue Objectives

1. Distinguish between situations that can be modeled with linear functions and with exponential functions.

1a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

F-LE.A.1a.i Explain why linear functions change at the same rate over time and why exponential functions change by equal factors over time.★

MP.1 MP.2 MP.3 MP.4 MP.5 MP.6 MP.7MP.8

1b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

F-LE.A.1b.i Describe situations where one quantity changes at a constant rate per unit interval as compared to another.★

MP.2 MP.4MP.7MP.8

1c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

F-LE.A.1c.i Describe situations where a quantity grows or decays at a constant percent rate per unit interval as compared to another.★

MP.1 MP.2 MP.4 MP.7MP.8

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.A.2.i Create linear and exponential functions given the following situations: arithmetic and geometric sequences, a graph, a description of a relationship, or two points, which can be read from a table.★


3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

F-LE.A.3.i Explain why, using graphs and tables, that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or as a polynomial function.★




18

FUNCTIONS - Linear, (Quadratic), and Exponential Models★ F-LE B. Interpret expressions for functions in terms of the situation they model. Missouri Learning Standards Ladue Objectives


F-LE.B.5.i Explain the meaning of the coefficients, factors, exponents, and/or intercepts in a linear or exponential function, based on the context of a situation.★


STATISTICS AND PROBABILITY - Interpreting Categorical and Quantitative Data S-ID A. Summarize, represent, and interpret data on a single count or measurement variable. Missouri Learning Standards Ladue Objectives

1. Represent data with plots on the real number line (dot plots, histograms, and box plots).

S-ID.A.1.i Construct dot plots, histograms, and box plots for data on a real number line.

MP.5

2. Use 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.


Describe the center of the data distribution (mean or median). Use the correct measure of center and spread (interquartile range) to describe a distribution that is symmetric or skewed. Identify outliers (extreme data points) and their effects on data sets. Compare the distributions of two or more data sets by examining their shapes, centers, and spreads when drawn on the same scale.

MP.2 MP.3 MP.4


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

MP.2 MP.3 MP.4



19

STATISTICS AND PROBABILITY - Interpreting Categorical and Quantitative Data S-ID B. Summarize, represent, and interpret data on two categorical and quantitative variables. Missouri Learning Standards Ladue Objectives

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.B.5.i S-ID.B.5.ii S-ID.B.5.iii

Summarize categorical data for two categories in two-way frequency tables. Interpret and describe relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

MP.2 MP.4 MP.7

6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

6a. 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, quadratic, and exponential models.

S-ID.B.6a.i S-ID.B.6a.ii S-ID.B.6a.iii S-ID.B.6a.iv S-ID.B.6a.v

Categorize data as linear, nonlinear, quadratic, or exponential. Use algebraic methods or technology to fit a function to the data. Use the function to predict values. Explain the meaning of the slope and y-intercept in context of the data. Explain the meaning of the constant and coefficients in context of the data.

MP.2 MP.4 MP.5

6b. Informally assess the fit of a function by plotting and analyzing residuals.

S-ID.B.6b.i S-ID.B.6b.ii

Calculate a residual (observed value minus predicted value). Create and analyze a residual plot.

MP.3 MP.4 MP.5

6c. Fit a linear function for a scatter plot that suggests a linear association.

S-ID.B.6c.i S-ID.B.6c.ii

Sketch a line of best fit on a scatter plot that appears linear. Write the equation of the line of best fit (y = mx + b) using technology or by using two points on the best fit line.

MP.2 MP.4 MP.5 MP.8



20

STATISTICS AND PROBABILITY - Interpreting Categorical and Quantitative Data S-ID C. Interpret linear models. Missouri Learning Standards Ladue Objectives

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

S-ID.C.7.i Interpret the meaning of the slope and y-intercept in terms of the units stated in the data.

MP.2 MP.3


S-ID.C.8.i Using a graphing calculator, determine the line of best fit for a given set of data using a linear regression function. Determine the correlation coefficient and interpret.

MP.2 MP.5

9. Distinguish between correlation and causation. S-ID.C.9.i Given various situations, determine if they represent correlation or causation and explain.

MP.1 MP.3

Mathematics – Algebra II/Advanced Algebra II Pending Board Approval, May 13, 2015


1

Critical Areas of Instruction Algebra II Overview

Building on their work with linear, quadratic, and exponential functions, students extend their repertoire of functions to include polynomial, rational, and radical functions. Students work closely with the expressions that define the functions, and continue to expand and hone their abilities to model situations and to solve equations, including solving quadratic equations over the set of complex numbers and solving exponential equations using the properties of logarithms. The Mathematical Practice Standards apply throughout each course and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations. The critical areas for this course are as follows: Critical Area 1: The first critical area develops the structural similarities between the system of polynomials and the system of integers. Students draw on analogies between polynomial arithmetic and base-ten computation, focusing on properties of operations, particularly the distributive property. Students connect multiplication of polynomials with multiplication of multi-digit integers, and division of polynomials with long division of integers. Students identify zeros of polynomials, including complex zeros of quadratic polynomials, and make connections between zeros of polynomials and solutions of polynomial equations. The unit culminates with the fundamental theorem of algebra. A central theme of this unit is that the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers. Critical Area 2: Building on their previous work with functions, and on their work with trigonometric ratios and circles in Geometry, students now use the coordinate plane to extend trigonometry to model periodic phenomena. Critical Area 3: Students synthesize and generalize what they have learned about a variety of function families. They extend their work with exponential functions to include solving exponential equations with logarithms. They explore the effects of transformations on graphs of diverse functions, including functions arising in an application, in order to abstract the general principle that transformations on a graph always have the same effect regardless of the type of the underlying function. They identify appropriate types of functions to model a situation, they adjust parameters to improve the model, and they compare models by analyzing appropriateness of fit and making judgments about the domain over which a model is a good fit. The description of modeling as “the process of choosing and using mathematics and statistics to analyze empirical situations, to understand them better, and to make decisions” is at the heart of this unit. The narrative discussion and diagram of the modeling cycle should be considered when knowledge of functions, statistics, and geometry is applied in a modeling context. Critical Area 4: Students see how the visual displays and summary statistics they learned in earlier grades relate to different types of data and to probability distributions. They identify different ways of collecting data— including sample surveys, experiments, and simulations—and the role that randomness and careful design play in the conclusions that can be drawn. Common Core State Standards for Mathematics – Appendix A, pg. 36

NUMBER AND QUANTITY

The Real Number System A. Extend the properties of exponents to rational exponents. B. Use properties of rational and irrational numbers.

The Complex Number System A. Perform arithmetic operations with complex numbers. C. Use complex numbers in polynomial identities and equations. Vector and Matrix Quantities C. Perform operations on matrices and use matrices in applications.

ALGEBRA Seeing Structure in Expressions A. Interpret the structure of expressions. B. Write expressions in equivalent forms to solve problems. Arithmetic with Polynomials and Rational Expressions A. Perform arithmetic operations on polynomials. B. Understand the relationship between zeros and factors of polynomials. C. Use polynomial identities to solve problems. D. Rewrite rational expressions. Creating Equations★ A. Create equations that describe numbers or relationships. Reasoning with Equations and Inequalities A. Understand solving equations as a process of reasoning and explain the reasoning. B. Solve equations and inequalities in one variable. C. Solve systems of equations. D. Represent and solve equations and inequalities graphically.

FUNCTIONS Interpreting Functions B. Interpret functions that arise in applications in terms of the context. C. Analyze functions using different representations. Building Functions A. Build a function that models a relationship between two quantities. B. Build new functions from existing functions. Linear, Quadratic, and Exponential Models A. Construct and compare linear, quadratic, and exponential models and solve problems. B. Interpret expressions for functions in terms of the situation they model. Trigonometric Functions A. Extend the domain of trigonometric functions using the unit circle. B. Model periodic phenomena with trigonometric functions. C. Prove and apply trigonometric identities

GEOMETRY Similarity, Right Triangles, and Trigonometry C. Define trigonometric ratios and solve problems involving right triangles.

STATISTICS AND PROBABILITY Interpreting Categorical and Quantitative Data A. Summarize, represent, and interpret data on a single count or measurement variable. B. Summarize, represent, and interpret data on two categorical and quantitative variables. C. Interpret linear models Making Inferences and Justifying Conclusions A. Understand and evaluate random processes underlying statistical experiments. B. Make inferences and justify conclusions from sample surveys, experiments, and observational studies. Conditional Probability and the Rules of Probability B. Use the rules of probability to compute probabilities of compound events in a uniform probability model.

Common Core State Standards for Mathematics – Appendix A, pgs. 82-91



2

Mathematical Practices

The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy). Overall Habits of Mind of a Productive Mathematics Thinker MP.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. MP.6 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.






3

★Modeling★




4

NUMBER and QUANTITY - The Real Number System N-RN A. Extend the properties of exponents to rational exponents. Missouri Learning Standards Ladue Objectives


N-RN.A.1.i N-RN.A.1.ii

Describe the rules of integer and rational exponents and properties of exponents. Simplify exponential expressions using the properties of exponents.

MP.1 MP.2


N-RN.A.2.i Simplify expressions involving radicals, integer, and rational exponents using the properties of exponents.

MP.2 MP.7

NUMBER and QUANTITY - The Real Number System N-RN B. Use properties of rational and irrational numbers. Missouri Learning Standards Ladue Objectives



MP.2 MP.3 MP.6

NUMBER and QUANTITY - The Complex Number System N-CN A. Perform arithmetic operations with complex numbers.

Missouri Learning Standards Ladue Objectives 1. Know there is a complex number I such that i2 = –1, and every

complex number has the form a + bi with a and b real. N-CN.A.1.i N-CN.A.1.ii

Define a pure imaginary number in the form bi. Define complex numbers in the form a + bi.

MP.1 MP.7

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

N-CN.A.2.i Perform operations and simplify complex number expressions, using conjugates, algebraic properties, and properties of exponents.

MP.2 MP.7



5

NUMBER and QUANTITY - The Complex Number System N-CN C. Use complex numbers in polynomial identities and equations (polynomials with real coefficients). Missouri Learning Standards Ladue Objectives

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

N-CN.C.7.i Solve quadratic equations with real coefficients that have complex solutions. MP.1 MP.2 MP.7

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

N-CN.C.9.i N-CN.C.9.ii

Determine the number and type of roots for a polynomial equation. Find the zeros of a polynomial function.

MP.2 MP.3 MP.7

NUMBER and QUANTITY - Vector and Matrix Quantities N-VM C. Perform operations on matrices and use matrices in applications. Missouri Learning Standards Ladue Objectives

6. Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

N-VM.C.6.i Use matrices to represent and manipulate data. MP.6 MP.7

8. Add, subtract, and multiply matrices of appropriate dimensions. N-VM.C.8.i Perform matrix operations (addition, subtraction, and multiplication). MP.1 MP.7

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

N-VM.C.9.i N-VM.C.9.ii

Recognize that the Commutative Property of Multiplication does not hold for matrix multiplication. Recognize that the Associative and Distributive Properties do hold for matrix multiplication.

MP.1 MP.7

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

N-VM.C.10.i Calculate determinants of square matrices (Advanced Algebra II only). MP.1 MP.7



6

ALGEBRA - Seeing Structure in Expressions A-SSE A. Interpret the structure of expressions (polynomial and rational). Missouri Learning Standards Ladue Objectives

1. Interpret expressions that represent a quantity in terms of its context.★

1a. Interpret parts of an expression, such as terms, factors, and coefficient.

A-SSE.A.1a.i Analyze a rational expression based on its factors and coefficients.★ MP.2 MP.7

1b. 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.

A-SSE.A.1b.i Construct models and solve problems involving exponential growth and decay. MP.2 MP.4 MP.7

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

A-SSE.A.2.i Describe and use algebraic manipulations, including factoring and rules of integer exponents and apply properties of exponents to simplify and rewrite expressions.

MP.1 MP.7 MP.8



7

ALGEBRA - Seeing Structure in Expressions A-SSE B. Write expressions in equivalent forms to solve problems. Missouri Learning Standards Ladue Objectives



A-SSE.B.3a.i Factor a quadratic expression to reveal the zeros of the function it defines. MP.1 MP.2 MP.7

3b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

A-SSE.B.3b.i A-SSE.B.3b.ii

Use the process of completing the square to write equations of parabolas in standard form. Determine the maximum or minimum value (i.e. the vertex) of an equation of a parabola written in standard form.

MP.2 MP.4 MP.7


A-SSE.B.3c.i Use properties of exponents (such as power of a power, product of powers, power of a product, and rational exponents, etc.) to write an equivalent form of an exponential function to reveal and explain specific information about its approximate rate of growth or decay.

MP.2 MP.4 MP.7

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.

A-SSE.B.4.i A-SSE.B.4.i

Develop the formula for the sum of a finite geometric series when the ratio is not 1. Use the formula for the sum of a finite geometric series to solve problems (e.g., calculate mortgage payments).




8

ALGEBRA - Arithmetic with Polynomials and Rational Expressions A-APR A. Perform arithmetic operations on polynomials (beyond quadratic). Missouri Learning Standards Ladue Objectives



Understand the definition of a polynomial of degree greater than 2. Understand the concepts of combining like terms and closure. Add, subtract, and multiply polynomials and understand how closure applies under these operations.

MP.7 MP.8

ALGEBRA - Arithmetic with Polynomials and Rational Expressions A-APR B. Understand the relationship between zeros and factors of polynomials. Missouri Learning Standards Ladue Objectives

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).

A-APR.B.2.i Recognize and apply the Remainder Theorem. MP.3 MP.8

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.

A-APR.B.3.i A-APR.B.3.ii

Identify the zeros of a polynomial when the polynomial is factored. Draw a graph of the function using the zeros of a function.

MP.5 MP.7



9

ALGEBRA - Arithmetic with Polynomials and Rational Expressions A-APR C. Use polynomial identities to solve problems. Missouri Learning Standards Ladue Objectives

4. Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples.

A-APR.C.4.i A-APR.C.4.ii A-APR.C.4.iii

Recognize that polynomial identities include but are not limited to the product of the sum and difference of two terms, the difference of two squares, the sum and difference of two cubes, the square of a binomial, etc. Prove polynomial identities by showing steps and providing reasons. llustrate how polynomial identities are used to determine numerical relationships such as 252 = (20 + 5)2 = 202 + 2(20 x 5) + 52.

MP.2 MP.3 MP.6 MP.7

ALGEBRA - Arithmetic with Polynomials and Rational Expressions A-APR D. Rewrite rational expressions (linear and quadratic denominators). Missouri Learning Standards Ladue Objectives


A-APR.D.6.i Rewrite rational expressions,(a(x))/(b(x)), in the form (q(x)) + ((r(x))/(b(x))) by using factoring, long division, or synthetic division. Use a computer algebra system for complicated examples to assist with building a broader conceptual understanding.

MP.5 MP.7


A-APR.D.7.i Add, subtract, multiply, and divide rational expressions.




10

ALGEBRA - Creating Equations A-CED A. Create equations that describe numbers or relationships (equations using all available types of expressions, including simple root functions). Missouri Learning Standards Ladue Objectives


A-CED.A.1.i Create linear, quadratic, rational and exponential equations and inequalities in one variable and use them in a contextual situation to solve problems.★

MP.1 MP.2 MP.4 MP.5



Create equations in two or more variables to represent relationships between quantities.★ Graph equations in two variables on a coordinate plane and label the axes and scales.★

MP.1 MP.2 MP.4 MP.5

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


Write and use a system of equations and/or inequalities to solve a real world problem.★ Recognize that the equations and inequalities represent the constraints of the problem.★

MP.1 MP.2 MP.4 MP.5

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

A-CED.A.4.i Solve multi-variable formulas or literal equations, for a specific variable.★ MP.1 MP.2 MP.4 MP.5



11

ALGEBRA - Reasoning with Equations and Inequalities A-REI A. Understand solving equations as a process of reasoning and explain the reasoning (simple radical and rational). Missouri Learning Standards Ladue Objectives

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

A-REI.A.2.i A-REI.A.2.ii

Solve simple rational and radical equations in one variable. Write examples of how extraneous solutions arise.

MP.1 MP.2 MP.5

ALGEBRA - Reasoning with Equations and Inequalities A-REI B. Solve equations and inequalities in one variable. Missouri Learning Standards Ladue Objectives

4. Solve quadratic equations in one variable.

4a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.

A-REI.B.4a.i A-REI.B.4a.ii

Transform a quadratic equation written in standard form to an equation in vertex form (x – p)2 = q by completing the square. Derive the quadratic formula by completing the square on the standard form of a quadratic equation. (Advanced Algebra II only).

MP.3 MP.5



Find roots of quadratic functions by graphing, factoring, completing the square, and using the quadratic formula. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

MP.5 MP.7



12




Solve systems of equations using the elimination method (sometimes called linear combinations). Solve systems of equations by substitution (solving for one variable in the first equation and substituting it into the second equation).

MP.1 MP.5


A-REI.C.6.i Solve systems of equations by graphing. MP.5

7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3.


Solve a system containing a linear equation and a quadratic equation in two variables (conic sections possible) graphically and algebraically. (Advanced Algebra II) Recognize the difference between linear and quadratic equations.

MP.2 MP.5



13

ALGEBRA - Reasoning with Equations and Inequalities A-REI D. Represent and solve equations and inequalities graphically (combine polynomial, rational, radical, absolute value, and exponential functions). Missouri Learning Standards Ladue Objectives 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.★

A-REI.D.11.i Explain why the intersection of y = f(x) and y = g(x) is the solution of f(x) = g(x) for any combination of linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Find the solution(s) by using technology to graph the equations and determine their point of intersection, or using tables of values.★

MP.3 MP.4 MP.5 MP.6

12. Graph the solutions to a linear inequality in two variables as a halfplane (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.

A-REI.D.12.i Graph solutions to linear inequalities in two variables and systems of linear inequalities in two variables.

MP.5 MP.6



14

FUNCTIONS - Interpreting Functions F-IF B. Interpret functions that arise in applications in terms of the context (emphasize selection of appropriate models). Missouri Learning Standards Ladue Objectives

4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★

F-IF.B.4.i F-IF.B.4.ii

Given a function, identify key features in graphs and tables including: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★ Given the key features of a function, sketch the graph.★



F-IF.B.5.i Given the graph of a function, determine the practical domain of the function as it relates to the numerical relationship it describes.★




Calculate the average rate of change over a specified interval of a function presented algebraically or in a table.★ Estimate the average rate of change over a specified interval of a function from the function’s graph.★ Explain, in context, the average rate of change of a function over a specified interval.★




15




F-IF.C.7a.i Graph quadratic functions showing intercepts, maxima, or minima.★ MP.1 MP.2 MP.4 MP.5 MP.6

7b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

F-IF.C.7b.i F-IF.C.7b.ii F-IF.C.7b.iii F-IF.C.7b.iv F-IF.C.7b.v F-IF.C.7b.vi

Graph functions expressed algebraically and show key features of the graph. Graph simple cases by hand, and use technology to show more complicated cases.★ Graph linear functions showing intercepts. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.★ Graph rational functions, identifying zeros and asymptotes when factorable, and showing end behavior.★ (Advanced Algebra II) Graph exponential and logarithmic functions, showing intercepts and end behavior.★ Graph trigonometric functions, showing period, midline, and amplitude.★


7c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

F-IF.C.7c.i Graph polynomial functions, identifying zeros when factorable, and showing end behavior.★


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

F-IF.C.7e.i F-IF.C.7e.ii

Draw graphs of logarithmic and exponential functions showing characteristics of the functions.★ Draw graph of trigonometric function showing period, midline, and amplitude.★




16



F-IF.C.8.i F-IF.C.8.ii F-IF.C.8.iii

Know standard and factored form of quadratics and identify key characteristics of each. Know that quadratic functions result in parabolas. Interpret quadratic functions in real world problems.

MP.1 MP.4 MP.6

8a. 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.

F-IF.C.8a.i F-IF.C.8a.ii

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. Use the properties of exponents to interpret expressions for percent rate of change, and classify them as growth or decay.


8b. 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.C.8b.i Identify rate of change of exponential functions in order to determine whether it represents growth or decay.

MP.2 MP.4 MP.5 MP.6

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.

F-IF.C.9.i Compare functions given various representations (algebraically, graphically, numerically, table of values, or verbal description).

MP.2 MP.3 MP.4 MP.5



17

FUNCTIONS - Building Functions F-BF A. Build a function that models a relationship between two quantities. Missouri Learning Standards Ladue Objectives 1b. 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.

F-BF.A.1b.i

Combine standard function types, such as linear and exponential, using arithmetic operations.


FUNCTIONS - Building Functions F-BF B. Build new functions from existing functions (include all types of functions studied). Missouri Learning Standards Ladue Objectives

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.

F-BF.B.3.i F-BF.B.3.ii

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 (include all types of functions studied) and use technology to illustrate. Identify even and odd functions algebraically and from a graph. (Advanced Algebra II)

MP.4 MP.5 MP.7 MP.8

4. Find inverse functions. 4a. 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.

F-BF.B.4a.i F-BF.B.4a.ii

Define the inverse of a given function. Verify that one function is the inverse of another using composition.

MP.4 MP.5 MP.7


F-BF.B.4c.i F-BF.B.4c.ii

Evaluate the values of an inverse function from a graph or table of a given function. Understand horizontal and vertical line test.

MP.4 MP.5



18

FUNCTIONS – Linear, Quadratic, and Exponential Models F-LE A. Construct and compare linear, quadratic, and exponential models and solve problems (logarithms as solutions for exponentials). Missouri Learning Standards Ladue Objectives

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.

F-LE.A.4.i F-LE.A.4.ii

Solve exponential equations by rewriting as a logarithmic equation. Evaluate logarithmic expressions using technology.


FUNCTIONS – Linear, Quadratic, and Exponential Models F-LE B. Interpret expressions for functions in terms of the situation they model. Missouri Learning Standards Ladue Objectives


F-LE.B.5.i

Interpret and understand quantities, rates of change, and other values of linear and exponential functions in the context of real world scenarios.

MP.1 MP.4 MP.6



19

FUNCTIONS - Trigonometric Functions F-TF A. Extend the domain of trigonometric functions using the unit circle. Missouri Learning Standards Ladue Objectives

1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

F-TF.A.1.i Write the radian measure of an angle given the angle measure in degrees.

MP.6

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.

F-TF.A.2.i F-TF.A.2.ii

Evaluate trigonometric values of angles given in radian measure using the unit circle. Identify and define coterminal angles.

MP.6 MP.7

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.

F-TF.A.3.i Determine sine, cosine, and tangent of π/4, π/6, π/3, using right triangles and the unit circle.

MP.6 MP.7

FUNCTIONS - Trigonometric Functions F-TF B. Model periodic phenomena with trigonometric functions.

Missouri Learning Standards Ladue Objectives 5. Choose trigonometric functions to model periodic phenomena

with specified amplitude, frequency, and midline.★ F-TF.B.5.i Write a trigonometric function with specified amplitude, frequency, and midline.★

(Advanced Algebra II). MP.2 MP.3 MP.4 MP.5 MP.6 MP.7



20

FUNCTIONS - Trigonometric Functions F-TF C. Prove and apply trigonometric identities. Missouri Learning Standards Ladue Objectives

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.

F-TF.C.8.i F-TF.C.8.ii F-TF.C.8.iii

Write a proof of the Pythagorean Identity. (Advanced Algebra II) Evaluate trigonometric values of angles using the Pythagorean Identity. (Advanced Algebra II) Identify the quadrant of an angle given the trigonometric values of the angle. (Advanced Algebra II)

MP.3 MP.6 MP.7

GEOMETRY - Similarity, Right Triangles, and Trigonometry G-SRT C. Define trigonometric ratios and solve problems involving right triangles. Missouri Learning Standards Ladue Objectives


G-SRT.C.8.i G-SRT.C.8.ii

Use trigonometric relationships with right triangles to determine lengths and angle measures.★ Identify patterns of exact values for general angles and apply to evaluate the values of trigonometric functions (including circular functions) in radians and degrees.




21

STATISTICS and PROBABILITY - Interpreting Categorical and Quantitative Data S-ID A. Summarize, represent, and interpret data on a single count or measurement variable.

Missouri Learning Standards Ladue Objectives 1. Represent data with plots on the real number line (dot plots,

histograms, and box plots). S-ID.A.1.i Determine five-number summary when creating box plots.


S-ID.A.3.i

Analyze multiple data sets using shape, center, spread, and standard deviation of outliers.

MP.5 MP.6 MP.7 MP.8

4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population 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.


Evaluate the standard deviation of a data set. Display a data set using appropriate graphical representation. Use mean and standard deviation of a data set to fit it to a normal distribution curve. Use three-sigma rule to calculate the percent of a normal population that lies within three deviations of the mean.

MP.5 MP.6 MP.7

STATISTICS and PROBABILITY - Interpreting Categorical and Quantitative Data S-ID B. Summarize, represent, and interpret data on two categorical and quantitative variables.

Missouri Learning Standards Ladue Objectives 6. Represent data on two quantitative variables on a scatter plot,

and describe how the variables are related.

6a. 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, quadratic, and exponential models.

S-ID.B.6a.i

MP.5 MP.6 MP.7 MP.8

6b. Informally assess the fit of a function by plotting and analyzing residuals.

S-ID.B.6b.i Calculate and plot residuals for the data set and use to analyze fit of a function. (Advanced Algebra II).

MP.5 MP.6 MP.7

6c. Fit a linear function for a scatter plot that suggests a linear association.

S-ID.B.6c.i Represent data on a scatter plot and find line of best fit algebraically and using technology.



22

STATISTICS and PROBABILITY - Interpreting Categorical and Quantitative Data S-ID C. Interpret linear models. Missouri Learning Standards Ladue Objectives


S-ID.C.8.i Evaluate the correlation coefficient of a line of best fit using technology. MP.2 MP.5 MP.6

9. Distinguish between correlation and causation. S-ID.C.9.i Explain the difference between causation and correlation using data sets. MP.2 MP.3

STATISTICS and PROBABILITY - Making Inferences and Justifying Conclusions S-IC A. Understand and evaluate random processes underlying statistical experiments. Missouri Learning Standards Ladue Objectives

1. Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

S-IC.A.1.i

Make conclusions about a population based on a random sample from that population.

MP.2

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

S-IC.A.2.i Analyze consistency of data and expected probability. MP.3



23

STATISTICS and PROBABILITY - Making Inferences and Justifying Conclusions S-IC B. Make inferences and justify conclusions from sample surveys, experiments, and observational studies. Missouri Learning Standards Ladue Objectives

3. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.


Compare the methodologies of sample surveys, experiments, and observational studies. Explain the purpose of randomization in the various methodologies.

MP.3

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.


Make an estimate of a population mean using a sample survey. Predict a margin of error using a simulation model for random sampling.

MP.4 MP.7

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


Compare two treatments using data from a randomized experiment. Analyze the difference between parameters using simulations.


6. Evaluate reports based on data. S-IC.B.6.i Make conjectures about the relationship between characteristics of a sample using data.


STATISTICS and PROBABILITY - Conditional Probability and the Rules of Probability S-CP B. Use the rules of probability to compute probabilities of compound events in a uniform probability model (include more complex situations). Missouri Learning Standards Ladue Objectives

6. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.


MP.1 MP.4 MP.7

7. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.

S-CP.B.7.i Apply the Addition Rule, P(A or B) = P(A) + P(B) – _P(A and B), and interpret the answer in terms of the model.

MP.1 MP.4 MP.7

Mathematics – College Algebra

Pending Board Approval, May 13, 2015


1

Course Rationale and Course Description College Algebra Overview

Course Rationale Research consistently finds that taking mathematics above the Algebra II level highly corresponds to many measures of student success. In his groundbreaking report Answers in the Toolbox, Clifford Adelman found that the strongest predictor of postsecondary success is the highest level of mathematics completed (Executive Summary). ACT has found that taking more mathematics courses correlates with greater success on their college entrance examination. Of students taking (Algebra I, Geometry and Algebra II and no other mathematics courses), only thirteen percent of those students met the benchmark for readiness for college algebra. One additional mathematics course greatly increased the likelihood that a student would reach that benchmark, and three-fourths of students taking Calculus met the benchmark (ACTb 13). Students should be encouraged to select from a range of high quality mathematics options. STEM-intending students should be strongly encouraged to take Precalculus and Calculus (and perhaps a computer science course). A student interested in psychology may benefit greatly from a course in discrete mathematics, followed by AP Statistics. A student interested in starting a business after high school could use knowledge and skills gleaned from a course on mathematical decision-making. Mathematically-inclined students can, at this level, double up on courses—a student taking college calculus and college statistics would be well-prepared for almost any postsecondary career. Taken together, there is compelling rationale for urging students to continue their mathematical education throughout high school, allowing students several rich options once they have demonstrated mastery of core content.


Course Description College Algebra is a college-level math course, which builds upon skills introduced in Algebra II and Geometry. Topics emphasized are functions and graphs, polynomial and rational functions, and exponential and logarithmic functions, and sequences.

Ladue Horton Watkins High School Scheduling Handbook, 2015-16

NUMBER AND QUANTITY

The Complex Number System A. Perform arithmetic operations with complex numbers. C. Use complex numbers in polynomial identities and equations.

ALGEBRA

Arithmetic with Polynomials and Rational Expressions D. Rewrite rational expressions.

FUNCTIONS

Interpreting Functions C. Analyze functions using different representations. Building Functions A. Build a function that models a relationship between two quantities B. Build new functions from existing functions

GEOMETRY

Expressing Geometric Properties with Equations A. Translate between the geometric description and the equation for a

conic section.




2








3

★Modeling★





4

NUMBER AND QUANTITY - The Complex Number System N-CN A. Perform arithmetic operations with complex numbers. Missouri Learning Standards Ladue Objectives

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

N-CN.A.3.i

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

MP.6 MP.8

NUMBER AND QUANTITY - The Complex Number System N-CN C. Use complex numbers in polynomial identities and equations. Missouri Learning Standards Ladue Objectives


N-CN.C.9.i Know the Fundamental Theorem of Algebra; show that it is true for nth degree polynomials.

MP.4




5



A-APR.D.7.i A-APR.D.7.ii

Show that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression. Add, subtract, multiply, and divide rational expressions.

MP.2 MP.6 MP.7 MP.8




6



F-IF.C.7.i Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★

MP.2 MP.4 MP.5 MP.6

7d. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

F-IF.C.7d.i Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

MP.2 MP.6 MP.7 MP.8




7

FUNCTIONS - Building Functions F-BF A. Build a function that models a relationship between two quantities. Missouri Learning Standards Ladue Objectives

1. Write a function that describes a relationship between two quantities.★

1c. Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.

F-BF.A.1c.i Compose and decompose functions.

MP.2 MP.7

FUNCTIONS - Building Functions F-BF B. Build new functions from existing functions. Missouri Learning Standards Ladue Objectives


4b. Verify by composition that one function is the inverse of another.

F-BF.B.4b.i Verify by composition that one function is the inverse of another.

MP.1 MP.2 MP.6 MP.7


F-BF.B.4c.i Read values of an inverse function from a graph or a table, given that the function has an inverse.

MP.4 MP.6 MP.7

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

F-BF.B.5.i Identify the relationship between exponents and logarithms as inverses and use this relationship to solve problems involving logarithms and exponents.

MP.2 MP.4 MP.6 MP.7




8

GEOMETRY - Expressing Geometric Properties with Equations G-GPE A. Translate between the geometric description and the equation for a conic section. Missouri Learning Standards Ladue Objectives

3. Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

G-GPE.A.3.i Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

MP.1 MP.2 MP.3 MP.4

Mathematics – Geometry Concepts Pending Board Approval, May 13, 2015


1

Critical Areas of Instruction Geometry Overview The fundamental purpose of the course in Geometry is to formalize and extend students’ geometric experiences from the middle grades. Students explore more complex geometric situations and deepen their explanations of geometric relationships, moving towards formal mathematical arguments. Important differences exist between this Geometry course and the historical approach taken in Geometry classes. For example, transformations are emphasized early in this course. Close attention should be paid to the introductory content for the Geometry conceptual category found in the high school CCSS. The Mathematical Practice Standards apply throughout each course and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations. The critical areas, organized into six units are as follows. Critical Area 1: In previous grades, students were asked to draw triangles based on given measurements. They also have prior experience with rigid motions: translations, reflections, and rotations and have used these to develop notions about what it means for two objects to be congruent. In this unit, students establish triangle congruence criteria, based on analyses of rigid motions and formal constructions. They use triangle congruence as a familiar foundation for the development of formal proof. Students prove theorems—using a variety of formats—and solve problems about triangles, quadrilaterals, and other polygons. They apply reasoning to complete geometric constructions and explain why they work. Critical Area 2: Students apply their earlier experience with dilations and proportional reasoning to build a formal understanding of similarity. They identify criteria for similarity of triangles, use similarity to solve problems, and apply similarity in right triangles to understand right triangle trigonometry, with particular attention to special right triangles and the Pythagorean theorem. Students develop the Laws of Sines and Cosines in order to find missing measures of general (not necessarily right) triangles, building on students’ work with quadratic equations done in the first course. They are able to distinguish whether three given measures (angles or sides) define 0, 1, 2, or infinitely many triangles. Critical Area 3: Students’ experience with two-dimensional and three-dimensional objects is extended to include informal explanations of circumference, area and volume formulas. Additionally, students apply their knowledge of two-dimensional shapes to consider the shapes of cross-sections and the result of rotating a two-dimensional object about a line. Critical Area 4: Building on their work with the Pythagorean theorem in 8th grade to find distances, students use a rectangular coordinate system to verify geometric relationships, including properties of special triangles and quadrilaterals and slopes of parallel and perpendicular lines, which relates back to work done in the first course. Students continue their study of quadratics by connecting the geometric and algebraic definitions of the parabola. Critical Area 5: In this unit students prove basic theorems about circles, such as a tangent line is perpendicular to a radius, inscribed angle theorem, and theorems about chords, secants, and tangents dealing with segment lengths and angle measures. They study relationships among segments on chords, secants, and tangents as an application of similarity. In the Cartesian coordinate system, students use the distance formula to write the equation of a circle when given the radius and the coordinates of its center. Given an equation of a circle, they draw the graph in the coordinate plane, and apply techniques for solving quadratic equations, which relates back to work done in the first course, to determine intersections between lines and circles or parabolas and between two circles. Critical Area 6: Building on probability concepts that began in the middle grades, students use the languages of set theory to expand their ability to compute and interpret theoretical and experimental probabilities for compound events, attending to mutually exclusive events, independent events, and conditional probability. Students should make use of geometric probability models wherever possible. They use probability to make informed decisions.


GEOMETRY Congruence

A. Experiment with transformations in the plane B. Understand congruence in terms of rigid motions C. Prove geometric theorems D. Make geometric constructions

Similarity, Right Triangles, and Trigonometry A. Understand similarity in terms of similarity transformations B. Prove theorems involving similarity C. Define trigonometric ratios and solve problems involving right triangles D. Apply trigonometry to general triangles

Circles A. Understand and apply theorems about circles B. Find arc lengths and areas of sectors of circles

Expressing Geometric Properties with Equations A. Translate between geometric description and the equation for a conic

section B. Use coordinates to prove simple geometric theorems algebraically

Geometric Measurement and Dimension A. Explain volume formulas and use them to solve problems B. Visualize relationships between two-dimensional and three-dimensional

objects Modeling with Geometry

A. Apply geometry concepts in modeling situations

STATISTICS AND PROBABILITY Conditional Probability and the Rules of Probability

A. Understand independence and conditional probability and use them to interpret data

B. Use the rules of probability to compute probabilities of compound events in a uniform probability model

Common Core State Standards for Mathematics, pg. 75



2







3

★Modeling★




4

GEOMETRY - Congruence G-CO A. Experiment with transformations in the plane. Missouri Learning Standards Ladue Objectives

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.

G-CO.A.1.i Define 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.

MP.6

2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

G-CO.A.2.i G-CO.A.2.ii G-CO.A.2.iii

Illustrate and identify transformations in the plane, geometry software may be used. Describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus dilations).


3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. (MP1, MP4, MP5,MP6, MP7)


Given a rectangle, parallelogram, trapezoid, or regular polygon, describe and identify any rotational symmetries that exist. Given a rectangle, parallelogram, trapezoid, or regular polygon, draw any lines of reflection that exist. Given a rectangle, parallelogram, trapezoid, or regular polygon, draw the image across a point/line of reflection.

MP.4 MP.6 MP.7

4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. (MP3, MP4, MP6, MP7, MP8)

G-CO.A.4.i Describe rotations, reflections, and translations in terms of angles, perpendicular lines, and line segments.

MP.3

5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. (MP1, MP3, MP4, MP5, MP6

G-CO.A.5.i G-CO.A.5.ii

Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

MP.3 MP.5 MP.6 MP.8



5

GEOMETRY - Congruence G-CO B. Understand congruence in terms of rigid motions (build on rigid motions as a familiar starting point for development of concept of geometric proof). Missouri Learning Standards Ladue Objectives

6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

G-CO.B.6.i G-CO.B.6.ii

Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure. Given two figures, use the definition of congruence in terms of rigid motions to determine whether they are congruent.

MP.3

7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

G-CO.B.7.i Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

MP.3

8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

G-CO.B.8.i Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Students may also apply right triangle congruence theorems (HL, LL HA) to prove right triangles congruent.

MP.1 MP.3 MP.7 MP.8



6

GEOMETRY - Congruence G-CO C. Prove geometric theorems (focus on validity of underlying reasoning while using variety of ways of writing proofs). Missouri Learning Standards Ladue Objectives

9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

G-CO.C.9.i

Prove and apply theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.


10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

G-CO.C.10.i G-CO.C.10.ii G-CO.C.10.iii

Prove and apply theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Construct the medians, angle bisectors, altitudes, and perpendicular bisectors of a triangle. Prove and apply the triangle inequality theorem.

MP.1 MP.3 MP.7 MP.8

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.

G-CO.C.11.i

Prove and apply 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.

MP.1 MP.3 MP.7



7

GEOMETRY - Congruence G-CO D. Make geometric constructions (formalize and explain processes). Missouri Learning Standards Ladue Objectives 12. Make formal geometric constructions with a variety of tools and

methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

G-CO.D.12.i Make formal geometric constructions with a variety of tools and methods (compass and straightedge, paper folding, dynamic geometric software, etc.).

MP.4 MP.5 MP.6

13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

G-CO.D.13.i G-CO.D.13.ii G-CO.D.13.iii

Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Determine the sum of the interior and exterior angles of a given convex polygon. Given the measure of an interior or exterior angle of a regular polygon determine the number of sides of the polygon.

MP.4



8

GEOMETRY - Similarity, Right Triangles, and Trigonometry G-SRT A. Understand similarity in terms of similarity transformations. Missouri Learning Standards Ladue Objectives

1. Verify experimentally the properties of dilations given by a center and a scale factor

1a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

G-SRT.A.1a.i Verify experimentally that a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

MP.1 MP.2 MP.3 MP.7

1b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

G-SRT.A.1b.ii Verify experimentally that the dilation of a line segment is longer or shorter in the ratio given by the scale factor.

MP.2 MP.7 MP.8

2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

G-SRT.A.2.i

Verify experimentally that the dilation of a line segment is longer or shorter in the ratio given by the scale factor.

MP.1 MP.3 MP.6

3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

G-SRT.A.3.i Verify experimentally that the dilation of a line segment is longer or shorter in the ratio given by the scale factor.

MP.1 MP.3



9

GEOMETRY - Similarity, Right Triangles, and Trigonometry G-SRT B. Prove theorems involving similarity. Missouri Learning Standards Ladue Objectives 4. Prove theorems about 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.

G-SRT.B.4.i

Explain and apply theorems about 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.

MP.3

5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Define trigonometric ratios and solve problems involving right triangles.

G-SRT.B.5.i Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

MP.1 MP.3

GEOMETRY - Similarity, Right Triangles, and Trigonometry G-SRT C. Define trigonometric ratios and solve problems involving right triangles. Missouri Learning Standards Ladue Objectives 6. Understand that by similarity, side ratios in right triangles are

properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

G-SRT.C.6.i G-SRT.C.6.ii G-SRT.C.6.iii G-SRT.C.6.iv

Derive trigonometric ratios from the measures of the sides of right triangles. Recognize that trigonometric ratios remain constant across similar triangles. Calculate the geometric mean of two given numbers. Derive the properties of 30-60-90 and 45-45-90 special right triangles to find missing side and angle lengths.

MP.3 MP.8

7. Explain and use the relationship between the sine and cosine of complementary angles.

G-SRT.C.7.i Derive that the sine of an acute angle is equivalent to the cosine of its complement.

MP.3 MP.8


G-SRT.C.8.i Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★

MP.4 MP.7



10

GEOMETRY - Circles G-C A. Understand and apply theorems about circles. Missouri Learning Standards Ladue Objectives 1. Prove that all circles are similar G-C.A.1.i Prove that all circles are similar. MP.5

MP.8

2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

G-C.A.2.i G-C.A.2.ii G-C.A.2.iii G-C.A.2.iv G-C.A.2.v

Identify and describe relationships among inscribed angles, radii, arcs, and chords. Describe the relationships between central, inscribed, and circumscribed angles and the arcs formed from them. Explain why that inscribed angles on a diameter are right angles. Construct and apply that the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply relationships of angles and arcs formed by tangents and secants to circles.

MP.2 MP.7

3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

G-C.A.3.i G-C.A.3.ii

Construct the inscribed and circumscribed circles of a triangle. Prove and apply properties of angles for a quadrilateral inscribed in a circle.

MP.4 MP.5 MP.6

4. Construct a tangent line from a point outside a given circle to the circle.

G-C.A.4.i Construct a tangent line from a point outside a given circle to the circle. MP.5

GEOMETRY - Circles G-C B. Find arc lengths and areas of sectors of circles (radian introduced as unit of measure). Missouri Learning Standards Ladue Objectives 5. Derive using similarity the fact that the length of the arc

intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

G-C.B.5.i G-C.B.5.ii G-C.B.5.iii

Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius. Define the radian measure of the angle as the constant of proportionality. Derive the formula for the area of a sector.

MP.2 MP.3 MP.5



11

GEOMETRY - Expressing Geometric Properties with Equations G-GPE A. Translate between the geometric description and the equation for a conic section. Missouri Learning Standards Ladue Objectives 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 given by an equation.

G-GPE.A.1.i G-GPE.A.1.ii

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 given by an equation.

MP.1 MP.2

2. Derive the equation of a parabola given a focus and directix. G-GPE.A.2.ii Derive the equation of a parabola given a focus and directix. MP.4 MP.7

GEOMETRY - Expressing Geometric Properties with Equations G-GPE B. Use coordinates to prove simple geometric theorems algebraically (include distance formula; relate to Pythagorean theorem). Missouri Learning Standards Ladue Objectives 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).

G-GPE.B.4.i Use coordinates to prove simple geometric theorems and formulas algebraically.

MP.2 MP.3

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.B.5.i G-GPE.B.5.ii G-GPE.B.5.iii G-GPE.B.5.iv G-GPE.B.5.v

Prove that the slopes of parallel lines are equivalent. Prove that the slopes of perpendicular lines are opposite reciprocals. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems. Find the equation of a line parallel or perpendicular to a given line that passes through a given point. Find the distance between a point and a line, and between two parallel lines.

MP.2 MP.3

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

G-GPE.B.6.i Determine the point on a line segment between two given points that partitions the segment in a given ratio.

MP.2

7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.★

G-GPE.B.7.i G-GPE.B.7.ii

Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Compute perimeters of polygons and areas of triangles and rectangles.

MP.4 MP.6 MP.7



12

GEOMETRY - Geometric Measurement and Dimension G-GMD A. Explain volume formulas and use them to solve problems. Missouri Learning Standards Ladue Objectives 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.

G-GMD.A.1.i G-GMD.A.1.ii G-GMD.A.1.iii G-GMD.A.1.iv G-GMD.A.1.v

Derive and apply formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Derive and apply formulas for areas of polygons and irregular figures. Calculate lateral area and surface area of three-dimensional figures, including prisms, cylinders, cones, pyramids, and spheres. Apply formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Apply formulas for areas of polygons and irregular figures.

MP.4 MP.7

3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★

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

MP.4 MP.6

GEOMETRY - Geometric Measurement and Dimension G-GMD B. Visualize relationships between two-dimensional and three-dimensional objects. Missouri Learning Standards Ladue Objectives 4. Identify the shapes of two-dimensional cross-sections of three-

dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

G-GMD.B.4.i G-GMD.B.4.ii G-GMD.B.4.iii G-GMD.B.4.iv

Identify the shapes of two-dimensional cross-sections of three-dimensional objects. Identify three-dimensional objects generated by rotations of two-dimensional objects. Draw 2-dimensional models for 3-dimensional figures. Construct representations of 3-dimensional geometric objects from different perspectives.

MP.2 MP.4



13

GEOMETRY - Modeling with Geometry G-MG A. Apply geometric concepts in modeling situations. Missouri Learning Standards Ladue Objectives 1. Use geometric shapes, their measures, and their properties to

describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).★

G-MG.A.1.i

Use geometric shapes, their measures, and their properties to describe objects. ★ MP.4 MP.7

2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).★

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

MP.4 MP.7

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).★

G-MG.A.3.i Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost).★

MP.1 MP.3



14

STATISTICS and PROBABILITY - Conditional Probability and the Rules of Probability S-CP A. Understand independence and conditional probability and use them to interpret data (link to data from simulations or experiments). Missouri Learning Standards Ladue Objectives 1. Describe events as subsets of a sample space (the set of

outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).

S-CP.A.1.i

Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).

MP.2

2. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

S-CP.A.2.i Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

MP.2

3. Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

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

MP.2

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

S-CP.A.4.i Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.

MP.3

5. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.

S-CP.A.5.i Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

MP.3



15

STATISTICS and PROBABILITY - Conditional Probability and the Rules of Probability S-CP B. Use the rules of probability to compute probabilities of compound events in a uniform probability model. Missouri Learning Standards Ladue Objectives 6. Find the conditional probability of A given B as the fraction of B’s

outcomes that also belong to A, and interpret the answer in terms of the model.


MP.2

7. Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model.

S-CP.B.6.ii Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model.

MP.3

Mathematics – Geometry/Advanced Geometry Pending Board Approval, May 13, 2015


1

Critical Areas of Instruction Geometry Overview The fundamental purpose of the course in Geometry is to formalize and extend students’ geometric experiences from the middle grades. Students explore more complex geometric situations and deepen their explanations of geometric relationships, moving towards formal mathematical arguments. Important differences exist between this Geometry course and the historical approach taken in Geometry classes. For example, transformations are emphasized early in this course. Close attention should be paid to the introductory content for the Geometry conceptual category found in the high school CCSS. The Mathematical Practice Standards apply throughout each course and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations. The critical areas, organized into six units are as follows. Critical Area 1: In previous grades, students were asked to draw triangles based on given measurements. They also have prior experience with rigid motions: translations, reflections, and rotations and have used these to develop notions about what it means for two objects to be congruent. In this unit, students establish triangle congruence criteria, based on analyses of rigid motions and formal constructions. They use triangle congruence as a familiar foundation for the development of formal proof. Students prove theorems—using a variety of formats—and solve problems about triangles, quadrilaterals, and other polygons. They apply reasoning to complete geometric constructions and explain why they work. Critical Area 2: Students apply their earlier experience with dilations and proportional reasoning to build a formal understanding of similarity. They identify criteria for similarity of triangles, use similarity to solve problems, and apply similarity in right triangles to understand right triangle trigonometry, with particular attention to special right triangles and the Pythagorean theorem. Students develop the Laws of Sines and Cosines in order to find missing measures of general (not necessarily right) triangles, building on students’ work with quadratic equations done in the first course. They are able to distinguish whether three given measures (angles or sides) define 0, 1, 2, or infinitely many triangles. Critical Area 3: Students’ experience with two-dimensional and three-dimensional objects is extended to include informal explanations of circumference, area and volume formulas. Additionally, students apply their knowledge of two-dimensional shapes to consider the shapes of cross-sections and the result of rotating a two-dimensional object about a line. Critical Area 4: Building on their work with the Pythagorean theorem in 8th grade to find distances, students use a rectangular coordinate system to verify geometric relationships, including properties of special triangles and quadrilaterals and slopes of parallel and perpendicular lines, which relates back to work done in the first course. Students continue their study of quadratics by connecting the geometric and algebraic definitions of the parabola. Critical Area 5: In this unit students prove basic theorems about circles, such as a tangent line is perpendicular to a radius, inscribed angle theorem, and theorems about chords, secants, and tangents dealing with segment lengths and angle measures. They study relationships among segments on chords, secants, and tangents as an application of similarity. In the Cartesian coordinate system, students use the distance formula to write the equation of a circle when given the radius and the coordinates of its center. Given an equation of a circle, they draw the graph in the coordinate plane, and apply techniques for solving quadratic equations, which relates back to work done in the first course, to determine intersections between lines and circles or parabolas and between two circles. Critical Area 6: Building on probability concepts that began in the middle grades, students use the languages of set theory to expand their ability to compute and interpret theoretical and experimental probabilities for compound events, attending to mutually exclusive events, independent events, and conditional probability. Students should make use of geometric probability models wherever possible. They use probability to make informed decisions. Common Core State Standards for Mathematics – Appendix A, pg. 27

GEOMETRY Congruence

A. Experiment with transformations in the plane B. Understand congruence in terms of rigid motions C. Prove geometric theorems D. Make geometric constructions

Similarity, Right Triangles, and Trigonometry

A. Understand similarity in terms of similarity transformations B. Prove theorems involving similarity

Circles

A. Understand and apply theorems about circles B. Find arc lengths and areas of sectors of circles

Expressing Geometric Properties with Equations

A. Translate between geometric description and the equation for a conic section

B. Use coordinates to prove simple geometric theorems algebraically Geometric Measurement and Dimension

A. Explain volume formulas and use them to solve problems B. Visualize relationships between two-dimensional and three-dimensional

objects Modeling with Geometry

A. Apply geometry concepts in modeling situations

STATISTICS AND PROBABILITY Conditional Probability and the Rules of Probability

A. Understand independence and conditional probability and use them to interpret data

B. Use the rules of probability to compute probabilities of compound events in a uniform probability model

Common Core State Standards for Mathematics, pg. 75



2







3

★Modeling★




4

GEOMETRY - Congruence G-CO A. Experiment with transformations in the plane. Missouri Learning Standards Ladue Objectives

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.

G-CO.A.1.i Define 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.

MP.6

2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).


Illustrate and identify transformations in the plane, geometry software may be used. Describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus dilations).


3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. (MP1, MP4, MP5,MP6, MP7)


Given a rectangle, parallelogram, trapezoid, or regular polygon, describe and identify any rotational symmetries that exist. Given a rectangle, parallelogram, trapezoid, or regular polygon, draw any lines of reflection that exist. Given a rectangle, parallelogram, trapezoid, or regular polygon, draw the image across a point/line of reflection.

MP.4 MP.6 MP.7

4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. (MP3, MP4, MP6, MP7, MP8)

G-CO.A.4.i Describe rotations, reflections, and translations in terms of angles, perpendicular lines, and line segments.

MP.3

5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. (MP1, MP3, MP4, MP5, MP6

G-CO.A.5.i G-CO.A.5.ii

Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

MP.3 MP.5 MP.6 MP.8



5

GEOMETRY - Congruence G-CO B. Understand congruence in terms of rigid motions (build on rigid motions as a familiar starting point for development of concept of geometric proof). Missouri Learning Standards Ladue Objectives

6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

G-CO.B.6.i G-CO.B.6.ii

Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure. Given two figures, use the definition of congruence in terms of rigid motions to determine whether they are congruent.

MP.3

7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

G-CO.B.7.i Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

MP.3

8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

G-CO.B.8.i Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Students may also apply right triangle congruence theorems (HL, LL HA) to prove right triangles congruent.

MP.1 MP.3 MP.7 MP.8



6

GEOMETRY - Congruence G-CO C. Prove geometric theorems (focus on validity of underlying reasoning while using variety of ways of writing proofs). Missouri Learning Standards Ladue Objectives

9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

G-CO.C.9.i

Prove and apply theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.


10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

G-CO.C.10.i G-CO.C.10.ii G-CO.C.10.iii

Prove and apply theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Construct the medians, angle bisectors, altitudes, and perpendicular bisectors of a triangle. Prove and apply the triangle inequality theorem.

MP.1 MP.3 MP.7 MP.8

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.

G-CO.C.11.i

Prove and apply 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.

MP.1 MP.3 MP.7



7

GEOMETRY - Congruence G-CO D. Make geometric constructions (formalize and explain processes). Missouri Learning Standards Ladue Objectives 12. Make formal geometric constructions with a variety of tools and

methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

G-CO.D.12.i Make formal geometric constructions with a variety of tools and methods (compass and straightedge, paper folding, dynamic geometric software, etc.).

MP.4 MP.5 MP.6

13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

G-CO.D.13.i G-CO.D.13.ii G-CO.D.13.iii

Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Determine the sum of the interior and exterior angles of a given convex polygon. Given the measure of an interior or exterior angle of a regular polygon determine the number of sides of the polygon.

MP.4



8

GEOMETRY - Similarity, Right Triangles, and Trigonometry G-SRT A. Understand similarity in terms of similarity transformations. Missouri Learning Standards Ladue Objectives

1. Verify experimentally the properties of dilations given by a center and a scale factor

1a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

G-SRT.A.1a.i Verify experimentally that a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

MP.1 MP.2 MP.3 MP.7

1b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

G-SRT.A.1b.ii Verify experimentally that the dilation of a line segment is longer or shorter in the ratio given by the scale factor.

MP.2 MP.7 MP.8

2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

G-SRT.A.2.i

Given two figures, determine whether they are similar and explain their similarity based on the equality of corresponding angles and the proportionality of corresponding sides.

MP.1 MP.3 MP.6

3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

G-SRT.A.3.i Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

MP.1 MP.3



9

GEOMETRY - Similarity, Right Triangles, and Trigonometry G-SRT B. Prove theorems involving similarity. Missouri Learning Standards Ladue Objectives 4. Prove theorems about 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.

G-SRT.B.4.i

Explain and apply theorems about 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.

MP.3

5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Define trigonometric ratios and solve problems involving right triangles.

G-SRT.B.5.i Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

MP.1 MP.3

GEOMETRY - Similarity, Right Triangles, and Trigonometry G-SRT C. Define trigonometric ratios and solve problems involving right triangles. Missouri Learning Standards Ladue Objectives 6. Understand that by similarity, side ratios in right triangles are

properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

G-SRT.C.6.i G-SRT.C.6.ii G-SRT.C.6.iii G-SRT.C.6.iv

Derive trigonometric ratios from the measures of the sides of right triangles. Recognize that trigonometric ratios remain constant across similar triangles. Calculate the geometric mean of two given numbers. Derive the properties of 30-60-90 and 45-45-90 special right triangles to find missing side and angle lengths.

MP.3 MP.8

7. Explain and use the relationship between the sine and cosine of complementary angles.

G-SRT.C.7.i Derive that the sine of an acute angle is equivalent to the cosine of its complement.

MP.3 MP.8


G-SRT.C.8.i Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★

MP.4 MP.7



10

GEOMETRY - Circles G-C A. Understand and apply theorems about circles. Missouri Learning Standards Ladue Objectives 1. Prove that all circles are similar G-C.A.1.i Prove that all circles are similar. MP.5

MP.8

2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

G-C.A.2.i G-C.A.2.ii G-C.A.2.iii G-C.A.2.iv G-C.A.2.v

Identify and describe relationships among inscribed angles, radii, arcs, and chords. Describe the relationships between central, inscribed, and circumscribed angles and the arcs formed from them. Explain why that inscribed angles on a diameter are right angles. Construct and apply that the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply relationships of angles and arcs formed by tangents and secants to circles.

MP.2 MP.7

3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

G-C.A.3.i G-C.A.3.ii

Construct the inscribed and circumscribed circles of a triangle. Prove and apply properties of angles for a quadrilateral inscribed in a circle.

MP.4 MP.5 MP.6

4. Construct a tangent line from a point outside a given circle to the circle.

G-C.A.4.i Construct a tangent line from a point outside a given circle to the circle.

MP.5

GEOMETRY - Circles G-C B. Find arc lengths and areas of sectors of circles (radian introduced as unit of measure). Missouri Learning Standards Ladue Objectives 5. Derive using similarity the fact that the length of the arc

intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

G-C.B.5.i G-C.B.5.ii G-C.B.5.iii

Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius. Define the radian measure of the angle as the constant of proportionality. Derive the formula for the area of a sector.

MP.2 MP.3 MP.5



11

GEOMETRY - Expressing Geometric Properties with Equations G-GPE A. Translate between the geometric description and the equation for a conic section. Missouri Learning Standards Ladue Objectives 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 given by an equation.

G-GPE.A.1.i G-GPE.A.1.ii

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 given by an equation.

MP.1 MP.2

2. Derive the equation of a parabola given a focus and directix. G-GPE.A.2.ii Derive the equation of a parabola given a focus and directix MP.4 MP.7

GEOMETRY - Expressing Geometric Properties with Equations G-GPE B. Use coordinates to prove simple geometric theorems algebraically (include distance formula; relate to Pythagorean theorem). Missouri Learning Standards Ladue Objectives 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).

G-GPE.B.4.i Use coordinates to prove simple geometric theorems and formulas algebraically. MP.2 MP.3

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.B.5.i G-GPE.B.5.ii G-GPE.B.5.iii G-GPE.B.5.iv G-GPE.B.5.v

Prove that the slopes of parallel lines are equivalent. Prove that the slopes of perpendicular lines are opposite reciprocals. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems. Find the equation of a line parallel or perpendicular to a given line that passes through a given point. Find the distance between a point and a line, and between two parallel lines.

MP.2 MP.3

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

G-GPE.B.6.i Determine the point on a line segment between two given points that partitions the segment in a given ratio.

MP.2

7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.★

G-GPE.B.7.i G-GPE.B.7.ii

Use coordinates to compute perimeters of polygons and areas of triangles and rectangles (e.g., using the distance formula). Compute perimeters of polygons and areas of triangles and rectangles.

MP.4 MP.6 MP.7



12

GEOMETRY - Geometric Measurement and Dimension G-GMD A. Explain volume formulas and use them to solve problems. Missouri Learning Standards Ladue Objectives 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.

G-GMD.A.1.i G-GMD.A.1.ii G-GMD.A.1.iii G-GMD.A.1.iv G-GMD.A.1.v

Derive and apply formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Derive and apply formulas for areas of polygons and irregular figures. Calculate lateral area and surface area of three-dimensional figures, including prisms, cylinders, cones, pyramids, and spheres. Apply formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Apply formulas for areas of polygons and irregular figures.

MP.4 MP.7

2. Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

G-GMD.A.2.i Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

MP4 MP6

3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★

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

MP.4 MP.6

GEOMETRY - Geometric Measurement and Dimension G-GMD B. Visualize relationships between two-dimensional and three-dimensional objects. Missouri Learning Standards Ladue Objectives 4. Identify the shapes of two-dimensional cross-sections of three-

dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

G-GMD.B.4.i G-GMD.B.4.ii G-GMD.B.4.iii G-GMD.B.4.iv

Identify the shapes of two-dimensional cross-sections of three-dimensional objects. Identify three-dimensional objects generated by rotations of two-dimensional objects. Draw 2-dimensional models for 3-dimensional figures. Construct representations of 3-dimensional geometric objects from different perspectives.

MP.2 MP.4



13

GEOMETRY - Modeling with Geometry G-MG A. Apply geometric concepts in modeling situations. Missouri Learning Standards Ladue Objectives 1. Use geometric shapes, their measures, and their properties to

describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).★

G-MG.A.1.i

Use geometric shapes, their measures, and their properties to describe objects. ★ MP.4 MP.7

2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).★

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

MP.4 MP.7

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).★

G-MG.A.3.i Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost.)★

MP.1 MP.3



14

STATISTICS and PROBABILITY - Conditional Probability and the Rules of Probability S-CP A. Understand independence and conditional probability and use them to interpret data (link to data from simulations or experiments). Missouri Learning Standards Ladue Objectives 1. Describe events as subsets of a sample space (the set of

outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).

S-CP.A.1.i

Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).

MP.2

2. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

S-CP.A.2.i Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

MP.2

3. Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

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

MP.2

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

S-CP.A.4.i Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.

MP.3

5. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.

S-CP.A.5.i Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

MP.3



15

STATISTICS and PROBABILITY - Conditional Probability and the Rules of Probability S-CP B. Use the rules of probability to compute probabilities of compound events in a uniform probability model. Missouri Learning Standards Ladue Objectives 6. Find the conditional probability of A given B as the fraction of B’s

outcomes that also belong to A, and interpret the answer in terms of the model.


MP.2

7. Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model.

S-CP.B.6.ii Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model.

MP.3

Mathematics – Math Analysis Pending Board Approval on May 13, 2015


1

Course Rationale and Course Description Math Analysis Overview



Course Description Math Analysis prepares students for calculus. There is a strong emphasis on functions and their graphs. Quadratic, polynomial, logarithmic, exponential, trigonometric, circular functions, and limits are treated extensively. Additional topics include vectors, conic sections, sequences, and series.


NUMBER AND QUANTITY

The Complex Number System A. Perform arithmetic operations with complex numbers. B. Represent complex numbers and their operations on the complex plane. C. Use complex numbers in polynomial identities and equations. Vector and Matrix Quantities A. Represent and model with vector quantities. B. Perform operations on vectors. C. Perform operations on matrices and use matrices in applications.

ALGEBRA Arithmetic with Polynomials and Rational Expressions C. Use polynomial identities to solve problems. D. Rewrite rational expressions. Reasoning with Equations and Inequalities C. Solve systems of equations.

FUNCTIONS

Interpreting Functions C. Analyze functions using different representations. Building Functions A. Build a function that models a relationship between two quantities.. Trigonometric Functions A. Extend the domain of trigonometric functions using the unit circle. B. Model periodic phenomena with trigonometric functions C. Prove and apply trigonometric identities



2


The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy). Overall Habits of Mind of a Productive Mathematics Thinker 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. 6. 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.






3

★Modeling★




4

NUMBER AND QUANTITY - The Complex Number System N-CN A. Perform arithmetic operations with complex numbers. Missouri Learning Standards Ladue Objectives

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

N-CN.A.3.i

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

MP.6 MP.8

NUMBER AND QUANTITY - The Complex Number System N-CN B. Represent complex numbers and their operations on the complex plane. Missouri Learning Standards Ladue Objectives

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.

N-CN.B.4.i

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.

MP.4

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°.

N-CN.B.5.i

Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.

MP.4 MP.6

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.

N-CN.B.6.i

Calculate the distance between numbers in the complex plane as the modulus of the difference. Calculate the midpoint of a segment as the average of the numbers at its endpoints.

MP.3 MP.6



5

NUMBER AND QUANTITY - The Complex Number System N-CN C. Use complex numbers in polynomial identities and equations. Missouri Learning Standards Ladue Objectives

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

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

MP.6 MP.7


N-CN.C.9.i Know the Fundamental Theorem of Algebra; show that it is true for nth degree polynomials.

MP.4

NUMBER AND QUANTITY - Vector and Matrix Quantities N-VM

A. Represent and model with vector quantities. Missouri Learning Standards Ladue Objectives

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).

N-VM.A.1.i 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).

MP.4 MP.6

2. Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

N-VM.A.2.i Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

MP.4 MP.6

3. Solve problems involving velocity and other quantities that can be represented by vectors.

N-VM.A.3.i Model and solve problems involving velocity and other quantities that can be represented by vectors.

MP.2 MP.4 MP.6



6

NUMBER AND QUANTITY - Vector and Matrix Quantities N-VM B. Perform operations on vectors. Missouri Learning Standards Ladue Objectives

4. Add and subtract vectors.

4a. 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.

N-VM.B.4a.i 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.

MP.4 MP.5 MP.6 MP.7

4b. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

N-VM.B.4b.i Determine the magnitude and direction of their sum, given two vectors in magnitude and direction form.

MP.2 MP.4 MP.6

4c. 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.

N-VM.B.4c.i Demonstrate that vector subtraction v – w is the same 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.

MP.2 MP.4 MP.6

5. Multiply a vector by a scalar.

5a. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy).

N-VM.B.5a.i Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy).

MP.4 MP.6

5b. 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).

N-VM.B.5b.i 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).

MP.4 MP.6



7

NUMBER AND QUANTITY - Vector and Matrix Quantities N-VM C. Perform operations on matrices and use matrices in applications. Missouri Learning Standards Ladue Objectives

6. Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

N-VM.C.6.i Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

MP.4

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

N-VM.C.7.i Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

MP.6

8. Add, subtract, and multiply matrices of appropriate dimensions. (partially addressed in Algebra II)

N-VM.C.8.i Add, subtract, and multiply matrices of appropriate dimensions. MP.6

9. Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties. (partially addressed in Algebra II)

N-VM.C.9.i Demonstrate that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

MP.7 MP.8

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

N-VM.C.10.i Demonstrate that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. Demonstrate that the determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

MP.7 MP.8

11. Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.

N-VM.C.11.i Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.

MP.6

12. Work with 2 x 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

N-VM.C.12.i Work with 2 x 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

MP.4



8

ALGEBRA - Arithmetic with Polynomials and Rational Expressions A-APR C. Use polynomial identities to solve problems. Missouri Learning Standards Ladue Objectives

5. Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle. (Note: The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.)

A-APR.C.5.i Apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle. (Note: The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.)

MP.4 MP.6 MP.7 MP.8



A-APR.D.7.i A-APR.D.7.ii

Show that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression. Add, subtract, multiply, and divide rational expressions.

MP.2 MP.6 MP.7 MP.8


8. A-REI8 (+M) Represent a system of linear equations as a single matrix equation in a vector variable.

A-REI.C.8.i Represent a system of linear equations as a single matrix equation in a vector variable.

MP.4 MP.6

9. A-REI9 (+M) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 x 3 or greater).

A-REI.C.9.i Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 x 3 or greater).

MP.6 MP.7 MP.8



9



F-IF.C.7.i Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★

MP.2 MP.4 MP.5 MP.6

7d. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

F-IF.C.7d.i Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

MP.2 MP.6 MP.7 MP.8



10

FUNCTIONS - Building Functions F-BF K.CC A. Build a function that models a relationship between two quantities.

Common Core Content Standards Ladue Objectives 1. Write a function that describes a relationship between two

quantities.★

1c. Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.

F-BF.A.1c.i Compose and decompose functions.

MP.2 MP.7


4b. Verify by composition that one function is the inverse of another.

F-BF.A.4b.i Verify by composition that one function is the inverse of another.

MP.1 MP.2 MP.6 MP.7


F-BF.A.4c.i Read values of an inverse function from a graph or a table, given that the function has an inverse.

MP.4 MP.6 MP.7

4d. Produce an invertible function from a non-invertible function by restricting the domain.

F-BF.A.4d.i Produce an invertible function from a non-invertible function by restricting the domain.

MP.1 MP.2 MP.4

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

F-BF.A.5.i Identify the relationship between exponents and logarithms as inverses and use this relationship to solve problems involving logarithms and exponents.

MP.2 MP.4 MP.6 MP.7



11

FUNCTIONS - Trigonometric Functions F-TF A. Extend the domain of trigonometric functions using the unit circle. Common Core Content Standards Ladue Objectives


F-TF.A.3.i 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.

MP.2 MP.6 MP.7

4. Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

F-TF.A.4.i Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

MP.4 MP.6 MP.7

FUNCTIONS - Trigonometric Functions F-TF B. Model periodic phenomena with trigonometric functions. Common Core Content Standards Ladue Objectives

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.

F-TF.B.6.i Restrict the domain of a trigonometric function on which it is always increasing or always decreasing in order construct its inverse.

MP.1 MP.2 MP.4

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.★

F-TF.B.7.i 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.★

MP.4 MP.5 MP.6



12

FUNCTIONS - Trigonometric Functions F-TF C. Prove and apply trigonometric identities. Common Core Content Standards Ladue Objectives

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

F-TF.C.9.i Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

MP.1 MP.2 MP.3

Mathematics – Statistics Pending Board Approval on May 13, 2015


1

Course Rationale and Course Description Statistics Overview



Course Description Statistics is a course, which deals with applied statistics in a variety of settings.



Conditional Probability and the Rules of Probability B. Use the rules of probability to compute probabilities of compound events in a uniform probability model. Using Probability to Make Decisions A. Calculate expected values and use them to solve problems. B. Use probability to evaluate outcomes of decisions.



2







3

★Modeling★




4

STATISTICS AND PROBABILITY - Conditional Probability and the Rules of Probability S-CPC B. Use the rules of probability to compute probabilities of compound events in a uniform probability model. Missouri Learning Standards Ladue Objectives

8. Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.

S-CP.B.8.i Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.

MP.3 MP.4 MP.6

9. Use permutations and combinations to compute probabilities of compound events and solve problems.

S-CP.B.9.i Use permutations and combinations to compute probabilities of compound events and solve problems.

MP.5 MP.6



5

STATISTICS AND PROBABILITY - Using Probability to Make Decisions S-MD A. Calculate expected values and use them to solve problems. Missouri Learning Standards Ladue Objectives

1. Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.

S-MD.A.1.i Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.

MP.2 MP.4 MP.7 MP.8

2. Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

S-MD.A.2.i Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

MP.5 MP.6 MP.8

3. Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.

S-MD.A.3.i Create a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.

MP.5 MP.8

4. Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.

S-MD.A.4.i Create a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.




6

STATISTICS AND PROBABILITY - Using Probability to Make Decisions S-MD B. Use probability to evaluate outcomes of decisions. Missouri Learning Standards Ladue Objectives

5. Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

5a. Find the expected payoff for a game of chance.

S-MD.B.5a.i Find the expected payoff for a game of chance.

MP.1 MP.4 MP.5 MP.8

5b. Evaluate and compare strategies on the basis of expected values.

S-MD.B.5b.i Evaluate and compare strategies on the basis of expected values.

MP.1 MP.5 MP.8

6. Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

S-MD.B.6.i Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).


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

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


Mathematics – Trigonometry Pending Board Approval on May 13, 2015


1

Course Rationale and Course Description Trigonometry Overview



Course Description Trigonometry is a college-level math course, which builds upon skills introduced in Algebra II, Geometry, and College Algebra. Topics emphasized are circular functions, right and oblique triangles, identities and equations, and complex numbers.


NUMBER AND QUANTITY

Vector and Matrix Quantities A. Represent and model with vector quantities. B. Perform operations on vectors.

FUNCTIONS Trigonometric Functions A. Extend the domain of trigonometric functions using the unit circle. C. Prove and apply trigonometric identities.

GEOMETRY

Similarity, Right Triangles, and Trigonometry D. Apply trigonometry to general triangles.



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The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy). Overall Habits of Mind of a Productive Mathematics Thinker 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. 6. 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.






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★Modeling★




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NUMBER AND QUANTITY - Vector and Matrix Quantities N-VM A. Represent and model with vector quantities. Missouri Learning Standards Ladue Objectives

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).

N-VM.A.1.i N-VM.A.1.ii N-VM.A.1.iii

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). Find angular speed and relate it to linear speed in a variety of applications.

MP.4 MP.6

2. Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

N-VM.A.2.i Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

MP.4 MP.6

NUMBER AND QUANTITY - Vector and Matrix Quantities N-VM B. Perform operations on vectors. Missouri Learning Standards Ladue Objectives

4. Add and subtract vectors.

4b. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

N-VM.B.4b.i Determine the magnitude and direction of their sum, given two vectors in magnitude and direction form.

MP.2 MP.4 MP.6

5. Multiply a vector by a scalar.

5a. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy).

N-VM.B.5a.i Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy).

MP.4 MP.6



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FUNCTIONS - Trigonometric Functions F-TF A. Extend the domain of trigonometric functions using the unit circle. Missouri Learning Standards Ladue Objectives


F-TF.A.3.i 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.

MP.2 MP.6 MP.7

4. Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

F-TF.A.4.i Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

MP.4 MP.6 MP.7

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.

F-TF.A.6.i Restrict the domain of a trigonometric function on which it is always increasing or always decreasing in order construct its inverse.

MP.1 MP.2 MP.4

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.★

F-TF.A.7.i 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.★

MP.4 MP.5 MP.6

FUNCTIONS - Trigonometric Functions F-TF C. Prove and apply trigonometric identities. Missouri Learning Standards Ladue Objectives

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

F-TF.C.3.i Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

MP.1MP.3 MP.6 MP.7



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GEOMETRY - Similarity, Right Triangles, and Trigonometry G-SRT D. Apply trigonometry to general triangles. Missouri Learning Standards Ladue Objectives

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.

G-SRT.D.9.i 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.

MP.1 MP.2 MP.3

10. Prove the Laws of Sines and Cosines and use them to solve problems.

G-SRT.D.10.i Prove the Laws of Sines and Cosines and use them to solve problems.

MP.1 MP.2 MP.3 MP.6

11. Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

G-SRT.D.11.i Distinguish when to use the Law of Sines and the Law of Cosines in finding unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). Apply the laws when appropriate.

MP.4 MP.5 MP.6

Documents

MATH FLOW CHART - Board Docs