Indiana College and Career Ready Standards Academic Standards Comments

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1 Trigonometry / Precalculus Indiana Academic Standards Crosswalk The Process Standards demonstrate the ways in which students should develop conceptual understanding of mathematical content and the ways in which students should synthesize and apply mathematical skills. These process standards should be embedded in and taught with all content standards. See pages 9 and 10 for more information. Indiana College and Career Ready Standards Academic Standards Comments Content Standard Standard/Indicator TR.CO.1: Determine how the graph of a parabola IAS PC.1.10: Write the equations of conic sections in There is partial alignment with IAS PC Additionally changes if a, b and c changes in the equation y = a(x standard form (completing the square and using an emphasis is placed on describing how the graph of a b)² + c. Find an equation for a parabola when given translations as necessary), in order to find the type of parabola changes based on given information. sufficient information. conic section and to find its geometric properties (foci, asymptotes, eccentricity, etc.). Conics TR.CO.2: Derive the equation of a parabola given a focus and directrix. TR.CO.3: 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. TR.CO.4: 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. TR.CO.5: Graph conic sections. Identify and describe features like center, vertex or vertices, focus or foci, directrix, axis of symmetry, major axis, minor axis, and eccentricity. IAS PC.1.10: Write the equations of conic sections in standard form (completing the square and using translations as necessary), in order to find the type of conic section and to find its geometric properties (foci, asymptotes, eccentricity, etc.). IAS PC.1.10: Write the equations of conic sections in standard form (completing the square and using translations as necessary), in order to find the type of conic section and to find its geometric properties (foci, asymptotes, eccentricity, etc.). IAS PC.1.10: Write the equations of conic sections in standard form (completing the square and using translations as necessary), in order to find the type of conic section and to find its geometric properties (foci, asymptotes, eccentricity, etc.). IAS PC.1.10: Write the equations of conic sections in standard form (completing the square and using translations as necessary), in order to find the type of conic section and to find its geometric properties (foci, asymptotes, eccentricity, etc.). There is partial alignment with IAS PC Additionally an emphasis is placed on deriving the equation of a parabola. There is partial alignment with IAS PC Additionally, an emphasis is placed on deriving the equation of a circle. There is partial alignment with IAS PC Additionally, an emphasis is placed on deriving the equations of ellipses and hyperbolas. There is partial alignment with IAS PC Additionally, an emphasis is placed on graphing conic sections and describing the resulting features. TR.CO.6: 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. Indianapolis Public Schools Curriculum and Instruction Page 1 of 11

2 TR.UC.1: Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. Unit Circles TR.UC.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. TR.UC.3: Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. IAS PC.4.3: Learn exact sine, cosine, and tangent values for 0, 2, 3, 4, 6, and multiples of π. Use those values to find other trigonometric values. IAS PC.4.6: Find domain, range, intercepts, periods, amplitudes, and asymptotes of trigonometric functions. There is partial alignment with IAS PC 4.3 and APC.4.6. An emphasis is placed on using the unit circle to explain symmetry and periodicity. Geometry TR.G.1: Solve real-world problems with and without technology that can be modeled using right triangles, including problems that can be modeled using trigonometric ratios. Interpret the solutions and determine whether the solutions are reasonable. TR.G.2: Explain and use the relationship between the sine and cosine of complementary angles. TR.G.3: Use special triangles to determine the values of sine, cosine, and tangent for π/3, π/4, and π/6. Apply special right triangles to the unit circle and use them to express the values of sine, cosine, and tangent for x, π + x, and 2π x in terms of their values for x, where x is any real number. TR.G.4: Prove the Laws of Sines and Cosines and use them to solve problems. IAS PC.3.1: Solve word problems involving right and oblique triangles. IAS PC.4.4: Solve word problems involving applications of trigonometric functions. IAS PC.4.11: Make connections between right triangle ratios, trigonometric functions, and circular functions. IAS PC.5.6: Solve word problems involving applications of trigonometric equations. IAS PC.4.3: Learn exact sine, cosine, and tangent values for 0, 2, 3, 4, 6, and multiples of π. Use those values to find other trigonometric values.. IAS PC.3.2: Apply the laws of sines and cosines to solving problems. There is partial alignment with IAS PC.3.1, PC.4.4, PC 4.11, and PC 5.6. Students are required to solve problems with and without technology, to interpret solutions, and to determine reasonableness. There is partial alignment with IAS PC.4.3. Additionally, an emphasis is placed on using special right triangles to determine the trigonometric values. Right triangles must also be applied to the unit circle to express values of trig functions. There is partial alignment with IAS.PC.3.2. An emphasis is placed on proving the laws. Indianapolis Public Schools Curriculum and Instruction Page 2 of 11

3 TR.G.5: Understand and apply the Laws of Sines and Cosines to solve real-world and other mathematical problems involving right and non-right triangles. IAS PC.3.1: Solve word problems involving right and oblique triangles. The skills mentioned in PC 3.1 and PC 3.2 have merged into the language of TR.G.5. IAS PC.3.2: Apply the laws of sines and cosines to solving problems. TR.G.6: Derive the formula A = 1/2 ab sin(c) for the area of a triangle by drawing an auxiliary line. Use the formula to find areas of triangles. IAS PC.3.3: Find the area of a triangle given two sides and the angle between them. In addition to using the formula to find areas of triangles, students must also derive the formula referenced in TR.G.6. TR.PF.1: Find a sinusoidal function to model a data set and explain the parameters of the model. IAS PC.4.4: Solve word problems involving applications of trigonometric functions. IAS PC.8.3: Find a quadratic, exponential, logarithmic, power, or sinusoidal function to model a data set and explain the parameters of the model. There is partial alignment with IAS PC.4.4 and PC 8.3 in terms of word problems and in terms of sinusoidal functions. Periodic Functions TR.PF.2: Graph trigonometric functions with and without technology. Use the graphs to model and analyze periodic phenomena, stating amplitude, period, frequency, phase shift, and midline (vertical shift). TR.PF.3: Construct the inverse trigonometric functions of sine, cosine, and tangent by restricting the domain. IAS PC.4.5: Define and graph trigonometric functions (i.e., sine, cosine, tangent, cotangent, secant, cosecant). IAS PC.4.6: Find domain, range, intercepts, periods, amplitudes, and asymptotes of trigonometric functions. IAS PC.4.7: Draw and analyze graphs of translations of trigonometric functions, including period, amplitude, and phase shift. IAS PC.4.8: Define and graph inverse trigonometric functions. IAS PC.4.9: Find values of trigonometric and inverse trigonometric functions. Many of the skills from IAS PC.4.5, PC 4.6, and PC.4.7 have merged into the content of TR.PF.2. A further emphasis is placed on students being proficient graphing with and without technology. There is partial alignment with IAS PC.4.8. Additionally, there is an emphasis on students being proficient with restricting the domain in order to construct the functions. TR.PF.4: 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. IAS PC.5.5: Solve trigonometric equations. There is partial alignment with IAS PC.5.5. The language of TR.PF.4 specifically references the use of inverse functions when solving trig. equations and requires students to evaluate solutions using technology and to interpret solutions. Indianapolis Public Schools Curriculum and Instruction Page 3 of 11

4 TR.PF.5: Prove the addition and subtraction formulas for sine, cosine, and tangent. Use the formulas to solve problems. IAS PC.5.3: Understand and use the addition formulas for sines, cosines, and tangents. There is partial alignment with IAS PC.5.3. In addition to understanding and using the addition formulas, students must be able to prove them. TR.PF.6: Prove the double- and half-angle formulas for sine, cosine, and tangent. Use the formulas to solve problems. IAS PC.5.4: Understand and use the half-angle and double-angle formulas for sines, cosines, and tangents. There is partial alignment with IAS PC.5.4. In addition to understanding and using the half-angle formulas, students must be able to prove them. TR.PF.7: Define and use the trigonometric ratios (sine, cosine, tangent, cotangent, secant, cosecant) in terms of angles of right triangles and the coordinates on the unit circle. IAS PC.4.1: Define sine and cosine using the unit circle. Identities TR.ID.1: Prove the Pythagorean identity sin^2(x) + cos^2(x) = 1 and use it to find trigonometric ratios, given sin(x), cos(x), or tan(x), and the quadrant of the angle. TR.ID.2: Verify basic trigonometric identities and simplify expressions using these and other trigonometric identities. IAS PC.5.1: Know the basic trigonometric identity cos2x + sin2x = 1 and prove that it is equivalent to the Pythagorean Theorem. IAS PC.5.2: Use basic trigonometric identities to verify other identities and simplify expressions There is partial alignment with IAS PC.5.1. There is an additional emphasis placed on using this identity to find trig ratios. There is close alignment with IAS PC.5.2 Polar Coordinates TR.PC.1: Define polar coordinates and relate polar coordinates to Cartesian coordinates. TR.PC.2: Translate equations from rectangular coordinates to polar coordinates and from polar coordinates to rectangular coordinates. Graph equations in the polar coordinate plane. IAS PC.6.1: Define polar coordinates and relate polar coordinates to Cartesian coordinates. IAS PC.6.2: Represent equations given in rectangular coordinates in terms of polar coordinates. IAS PC.6.3: Graph equations in the polar coordinate plane. The skills from IAS PC.6.2 and IAS PC.6.3 have merged into the language of TR.PC.2. TR.V.1: Solve problems involving velocity and other quantities that can be represented by vectors. Vectors TR.V.2: 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). Indianapolis Public Schools Curriculum and Instruction Page 4 of 11

5 TR.V.3: 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). PC.PCN.1: 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. Polar Coordinates and Complex Numbers PC.PCN.2: Understand and use complex numbers, including real and imaginary numbers, on the complex plane in rectangular and polar form, and explain why the rectangular and polar forms of a given complex number represent the same number. PC.PCN.3: Understand and use addition, subtraction, multiplication, and conjugation of complex numbers, including real and imaginary numbers, on the complex plane in rectangular and polar form. IAS PC.6.3: Graph equations in the polar coordinate plane. IAS PC.6.4: Define complex numbers, convert complex numbers to trigonometric form, and multiply complex numbers in trigonometric form. IAS PC.6.3: Graph equations in the polar coordinate plane. IAS PC.6.4: Define complex numbers, convert complex numbers to trigonometric form, and multiply complex numbers in trigonometric form. There is partial alignment with IAS PC.6.3 and PC 6.4. Students must be able to understand and use complex number on the complex plane and explain why both forms of a given number represent the same number. There is partial alignment with IAS PC.6.3 and PC 6.4. Addition, subtraction, multiplication, and conjugation of complex numbers are specifically referenced in PC.PCN.3. Students must also include real and imaginary numbers on the complex plane in both rectangular and polar form. PC.PCN.4: State, prove, and use DeMoivre s Theorem. IAS PC.6.5 State, prove, and use De Moivre s Theorem. The language of IAS PC.6.5 and PC.PCN.4 is identical. Functions PC.F.1: 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. IAS PC.1.5: Describe the symmetry of the graph of a function. There is partial aligment with IAS PC.1.5. In addition to describing the symmetry, students must be able to interpret and sketch graphs containing the key features specifically referenced in PC.F.1. Indianapolis Public Schools Curriculum and Instruction Page 5 of 11

6 PC.F.2: Find linear models by using median fit and least squares regression methods. Decide which among several linear models gives a better fit. Interpret the slope and intercept in terms of the original context. PC.F.3: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. IAS PC.8.1: Find linear models using the median fit and least squares regression methods. Decide which model gives a better fit. There is partial alignment with IAS.PC.8.1. Additionally, there is an emphasis on interpreting the slope and intercept in terms of the original context. PC.F.4: Determine if a graph or table has an inverse, and justify if the inverse is a function, relation, or neither. Identify the values of an inverse function/relation from a graph or a table, given that the function has an inverse. Derive the inverse equation from the values of the inverse. IAS PC.1.4: Define, find, and check inverse functions. There is partial alignment with IAS PC.1.4. Additionally, students must justify if the inverse is a function, relation, or neither. They must also derive the inverse equation from the values of the inverse. Functions PC.F.5: Produce an invertible function from a noninvertible function by restricting the domain. PC.F.6: Describe 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 graph f(x) and the graph of f(x) + k, k f(x), f(kx), or f(x + k). Experiment with cases and illustrate an explanation of the effects on the graph using technology. Recognize even and odd functions from their graphs and algebraic expressions. IAS PC.1.6: Decide if functions are even or odd. There is partial alignment with IAS PC.1.6 and PC 1.7. Additionally, students are required to understand the effects of placing an additional value, k, in various locations within the value of the given function and to understand the effect of that value on the resulting IAS PC.1.7: Apply transformations to functions. graph. PC.F.7: Decide if a function is continuous at a point. Find the types of discontinuities of a function and relate them to finding limits of a function. Use the concept of limits to describe discontinuity and endbehavior of the function. IAS PC.1.2: Find domain, range, intercepts, zeros, asymptotes, and points of discontinuity of functions. Use paper and pencil methods and graphing calculators. There is partial alignment with IAS PC.1.2. Iin addition to determining points of discontinuity, students must relate them to finding limits. Additionally, they must use the concepts of limits to describe end-behavior. Indianapolis Public Schools Curriculum and Instruction Page 6 of 11

7 Functions PC.F.8: Define arithmetic and geometric sequences recursively. Use a variety of recursion equations to describe a function. Model and solve word problems involving applications of sequences and series, interpret the solutions and determine whether the solutions are reasonable. PC.F.9: Use iteration and recursion as tools to represent, analyze, and solve problems involving sequential change. PC.F.10: Describe the concept of the limit of a sequence and a limit of a function. Decide whether simple sequences converge or diverge. Recognize an infinite series as the limit of a sequence of partial sums. IAS PC.7.4: Use recursion to describe a sequence. IAS PC.7.6: Solve word problems involving applications of sequences and series. IAS PC.7.4: Use recursion to describe a sequence. IAS PC.7.5: Understand and use the concept of limit of a sequence or function as the independent variable approaches infinity or a number. Decide whether simple sequences converge or diverge. There is partial alignment with IAS PC.7.4 and PC.7.6. They must also interpret solutions and determine reasonableness. There is partial alignment with IAS PC.7.4. Additionally, students must represent, analyze, and solve problems involving sequential change. Quadratic, Polynomial, and Rational Equations and Functions PC.QPR.1: Use the method of completing the square to transform any quadratic equation into an equation of the form (x p)^2 = q that has the same solutions. Derive the quadratic formula from this form. PC.QPR.2: Graph rational functions with and without technology. Identify and describe features such as intercepts, domain and range, and asymptotic and end behavior. PC.QPR.3: 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). PC.QPR.4: Understand the Fundamental Theorem of Algebra. Find a polynomial function of lowest degree with real coefficients when given its roots. IAS PC.1.1: Recognize and graph various types of functions, including polynomial, rational, algebraic, and absolute value functions. Use paper and pencil methods and graphing calculators. Example: Draw the graphs of the functions y = x5 2x3 5x2, y =, and y =. IAS PC.1.2: Find domain, range, intercepts, zeros, asymptotes, and points of discontinuity of functions. Use paper and pencil methods and graphing calculators. The skills of IAS PC.1.1 and PC.1.2. have merged into the language of P.QPR.2. Indianapolis Public Schools Curriculum and Instruction Page 7 of 11

8 Exponential and Logarithmic Functions and Equations PC.EL.1: Use the definition of logarithms to convert logarithms from one base to another and prove simple laws of logarithms. PC.EL.2: Use the laws of logarithms to simplify logarithmic expressions and find their approximate values. PC.EL.3: Graph and solve real-world and other mathematical problems that can be modeled using exponential and logarithmic equations and inequalities; interpret the solution and determine whether it is reasonable. PC.EL.4: Use technology to find a quadratic, exponential, logarithmic, or power function that models a relationship for a bivariate data set to make predictions; compute (using technology) and interpret the correlation coefficient. IAS PC.2.3: Draw and analyze graphs of logarithmic and exponential functions. IAS PC.2.3: Draw and analyze graphs of logarithmic and exponential functions. IAS PC.8.2: Calculate and interpret the correlation coefficient. Use the correlation coefficient and residuals to evaluate a best-fit line. There is partial alignment with IAS PC.2.3. Additionally, students must interpret the solution and determine reasonableness. The skills in PC.2.3 and PC.8.2 have merged into PC.EL.4. There is a specific reference on using technology to make predictions. Parametric Equations PC.PE.1: Convert between a pair of parametric equations and an equation in x and y. Model and solve problems using parametric equations. PC.PE.2: Analyze planar curves, including those given in parametric form. IAS PC.1.8: Understand curves defined parametrically and draw their graphs. IAS PC.1.8: Understand curves defined parametrically and draw their graphs. There is partial alignment with IAS PC.1.8. Students must also convert between a pair of parametric equations and an equation in x and y. There is partial alignment with IAS PC.1.8. Additionally, there is an emphasis placed on analyzing planar curves. Indianapolis Public Schools Curriculum and Instruction Page 8 of 11

9 PROCESS STANDARDS FOR MATHEMATICS The Process Standards demonstrate the ways in which students should develop conceptual understanding of mathematical content, and the ways in which students should synthesize and apply mathematical skills. PS.1: Make sense of problems and persevere in solving them. PS.2: Reason abstractly and quantitatively. PS.3: Construct viable arguments and critique the reasoning of others. PROCESS STANDARDS FOR MATHEMATICS 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. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? and "Is my answer reasonable?" They understand the approaches of others to solving complex problems and identify correspondences between different approaches. Mathematically proficient students understand how mathematical ideas interconnect and build on one another to produce a coherent whole. 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. 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 analyze situations by breaking them into cases and recognize and use counterexamples. They organize their mathematical thinking, justify their conclusions and 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. They justify whether a given statement is true always, sometimes, or never. Mathematically proficient students participate and collaborate in a mathematics community. They listen to or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. Indianapolis Public Schools Curriculum and Instruction Page 9 of 11

10 PS.4: Model with mathematics. PS.5: Use appropriate tools strategically. PS.6: Attend to precision. PS.7: Look for and make use of structure. PS.8: Look for and express regularity in repeated reasoning. Mathematically proficient students apply the mathematics they know to solve problems arising in everyday life, society, and the workplace using a variety of appropriate strategies. They create and use a variety of representations to solve problems and to organize and communicate mathematical ideas. Mathematically proficient students apply what they know and 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 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. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Mathematically 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. Mathematically proficient students identify relevant external mathematical resources, such as digital content, and use them to pose or solve problems. They use technological tools to explore and deepen their understanding of concepts and to support the development of learning mathematics. They use technology to contribute to concept development, simulation, representation, reasoning, communication and problem solving. Mathematically proficient students communicate precisely to others. They use clear definitions, including correct mathematical language, 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 express solutions clearly and logically by using the appropriate mathematical terms and notation. They specify units of measure and label axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently and check the validity of their results in the context of the problem. They express numerical answers with a degree of precision appropriate for the problem context. Mathematically proficient students look closely to discern a pattern or structure. They step back for an overview and shift perspective. They recognize and use properties of operations and equality. They organize and classify geometric shapes based on their attributes. They see expressions, equations, and geometric figures as single objects or as being composed of several objects. Mathematically proficient students notice if calculations are repeated and look for general methods and shortcuts. They notice regularity in mathematical problems and their work to create a rule or formula. Mathematically proficient students maintain oversight of the process, while attending to the details as they solve a problem. They continually evaluate the reasonableness of their intermediate results. Indianapolis Public Schools Curriculum and Instruction Page 10 of 11

11 Indicators that No Longer Need to Be Taught In This Course Content Standard IAS PC.1.3: Model and solve word problems using functions and equations. Comments This skill is now taught in Algebra Two (Standard AII.Q.1). IAS PC.1.9: Compare relative magnitudes of functions and their rates of change. IAS PC.2.2: Find the domain, range, intercepts, and asymptotes of logarithmic and exponential functions. IAS PC.2.4: Define, find, and check inverse functions of logarithmic and exponential functions. This skill is now taught in Algebra Two (Standard AII.EL.2). This skill is now taught in Algebra Two (Standard AII.EL.5). IAS PC.4.2: Convert between degree and radian measures. No longer taught in Indiana Academic Standards 2014 IAS PC.4.10: Know that the tangent of the angle that a line makes with the x-axis is No longer taught in Indiana Academic Standards 2014 equal to the slope of the line. IAS PC.7.1: Understand and use summation notation. No longer taught in Indiana Academic Standards 2014 IAS PC.7.2: Find sums of infinite geometric series. IAS PC.7.3: Prove and use the sum formulas for arithmetic series and for finite and infinite geometric series. No longer taught in Indiana Academic Standards 2014 No longer taught in Indiana Academic Standards 2014 Indianapolis Public Schools Curriculum and Instruction Page 11 of 11