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CURRICULUM MAP: ALGEBRA 2 and TRIG. RCSD- Department of Mathematics 2010 - 2011 Note for teachers: The Pacing below is a general guideline on how much time you need to spend on each unit. Feel free to adjust according to your students needs. We chose to start with 3 Trig. units since it is a new and interesting topic that will capture student interest and attention at the start of the school year. However, you may choose to move the units around as it serves you best. Make sure you refer to the Performance Indicators as you go along. The primary text is Algebra 2 by Pearson. Refer to http://www.jmap.org and http://emathinstruction.com for additional practice Pacing Sept 3rd – Sept 14th Unit/Essential Questions Unit 1: Intro to Trig What are the six trigonometric ratios in relation to right triangles? What is the unit circle and how is it used in trigonometry? How do we find the values of the six trigonometric functions? What is radian measure and how do we convert between radians and degrees? Essential KnowledgeContent/Performance Indicators (What students must learn) A2.A.55 Express and apply the six trigonometric functions as ratios of the sides of a right triangle Essential Skills (What students will be able to do) Students will be able to - A2.A.57 Sketch and use the reference angle for angles in standard position - - - A2.A.59 Use the reciprocal and cofunction relationships to find the value of the secant, cosecant, and cotangent of 0º, 30º, 45º, 60º, 90º, 180º, and 270º angles - - A2.A.60 Sketch the unit circle and represent angles in standard position A2.A.61 Determine the length of an arc of a circle, given its radius and the measure of its central angle A2.A.62 Find the value of trigonometric functions, if given a point on the terminal Resources Pearson Algebra 2 A2.A.56 Know the exact and approximate values of the sine, cosine, and tangent of 0º, 30º, 45º, 60º, 90º, 180º, and 270º angles A2.A.58 Know and apply the co-function and reciprocal relationships between trigonometric ratios Vocabulary - Find missing angle using inverse trig functions Understand the concept of the unit circle and its relation to trigonometry Sketch a given angle on the unit circle Find both negative and positive coterminal angles Find the sine and cosine of an angle on the unit circle Distinguish between exact and approximate values of trig. functions Find the exact value of a sine/cosine function Convert between radians and degrees Find the length of the intercepted arc Find the value of trig. function given a point on the unit circle Find the terminal point on the unit circle given a trig. angle. Trig. Ratios Inverse Trig functions Unit Circle Standard side Initial side Terminal side Coterminal angle Exact value Central angle Intercepted arc Radian 14-3 Right triangles and Trig Ratios 13-2 Angles and Unit Circle 13-3 Radian measure side of angle θ A2.A.64 Use inverse functions to find the measure of an angle, given its sine, cosine, or tangent A2.A.66 Determine the trigonometric functions of any angle, using technology A2.M.1 Define radian measure A2.M.2 Convert between radian and degree measures Sept 15th – Sept 28th Unit 2 Trig Functions and Graphing What are the characteristics of the graphs of the trigonometric functions? A2.A.63 Restrict the domain of the sine, cosine, and tangent functions to ensure the existence of an inverse function A2.A.65 Sketch the graph of the inverses of the sine, cosine, and tangent functions A2.A.69 Determine amplitude, period, frequency, and phase shift, given the graph or equation of a periodic function Students will be able to - - - How do you write a trigonometric equation represented by a graph? How do you sketch the graphs of the six trigonometric functions? A2.A.70 Sketch and recognize one cycle of a function of the form y = Asin Bx or y = Acos Bx A2.A.71 Sketch and recognize the graphs of the functions y = sec(x) , y = csc(x), y = tan(x), and y = cot(x) - - A2.A.72 Write the trigonometric function that is represented by a given periodic graph - Find the amplitude, frequency, period and phase shift of a sine curve given its equation or graph Find the amplitude, frequency, period and phase shift of a cosine curve given its equation or graph Graph a sine or cosine curve given its equation Write the trig. Function given its graph Recognize and sketch the inverse trig. Functions (know its domain and range). Recognize and sketch the reciprocal trig. Functions (know its domain and range) Graph all trig function with a graphing calculator Solve trig. functions graphically using a graphing calculator by finding the points of intersection Periodic function Cycle Period Amplitude Frequency Phase shift Domain Range Sine curve Cosine curve 13-1 Exploring Periodic Functions 13-4 The Sine Function 13-5 The Cosine Function 13-6 the Tangent Function 13-7 Translating Sine and cosine Function 13-8 Reciprocal Trigonometric Functions Sept 29th – Oct 8th Unit 3 Trig Applications (Laws) How do you use the Law of Sines to find missing parts of oblique triangles? How do you use the Law of Cosines to find missing parts of oblique triangles? How do you use the trigonometry to find the area of oblique triangles? How many distinct triangles are possible given certain parts of oblique triangles? A2.A.73 Solve for an unknown side or angle, using the Law of Sines or the Law of Cosines Students will be able to - A2.A.74 Determine the area of a triangle or a parallelogram, given the measure of two sides and the included angle - A2.A.75 Determine the solution(s) from the SSA situation (ambiguous case) - - - Use the Law of Sines to find a missing angle or missing side Use the Law of Cosines to find a missing angle or missing side Find the area of a triangle or a parallelogram Find the possible number of triangles given an angle and two sides Apply the Law of Sines and Law of Cosines to word problems Law of Sine Law of Cosine Oblique triangle 14-4 Area and the Law of Sines 14-5 The Law of Cosines Page 927 The Ambiguous Case Oct 12th – Oct 21st Unit 4: Equations and Inequalities How do you solve absolute value equation/inequality and plot on the number line? A2.A.1 Solve absolute value equations and inequalities involving linear expressions in one variable Review of Algebra Topics Student will be able to - simplify expressions write and evaluate algebraic expressions - represent mathematical phrases and real world quantities using algebraic expressions - solve multi step equations and check - distinguish between solution, no solution and identity - solve literal equations - solve multi step inequalities and graph them - write inequality from a sentence using key word at least, at most, fewer, less, more … Algebra 2 and Trig. Topics Students will be able to - solve absolute value equations and check solve absolute value inequalities and check for extraneous solution distinguish between an “and” problem and an “or” problem and accordingly write the solution Term Constant term Like terms Coefficient Expression Equation Literal Equation Inequality Absolute Value Extraneous solution 1-3: Algebraic Expressions 1-4: Solving equations. Supplement with additional worksheets on equations with fractional coefficients 1-5: Solving Inequalities 1-6 Absolute Value Equations and inequalities Oct 22nd – Oct 27th Unit 5: Linear Equations and Functions A2.A.5 Use direct and inverse variation to solve for unknown values A2.A.37 Define a relation and function How do you distinguish between Direct and Inverse variation? How do you distinguish between a relation and a function? A2.A.38 Determine when a relation is a function Review of Algebra Topics Student will be able to - A2.A.39 Determine the domain and range of a function from its equation - A2.A.40 Write functions in functional notation - How do you find the domain and range of a function? A2.A.41 Use functional notation to evaluate functions for given values in the domain How do you transformation with functions? A2.A.46 Perform transformations with functions and relations: f(x + a) , f(x) + a, f(−x), − f(x), af(x) Algebra 2 and Trig. Topics Student will be able to - A2.A.52 Identify relations and functions, using graphs A2.S.8 Interpret within the linear regression model the value of the correlation coefficient as a measure of the strength of the relationship Determine if a function is linear Graph a linear function with/without a calculator. Find the Slope of a linear function given an equation, graph or 2 points Find the equation for a linear function given two points or a point and a graph. Draw a scatter plot and find the line of best fit - - - - Distinguish between a relation and a function. Determine if a relation is a function given a set of ordered pair, mapping diagram, graph or table of values Distinguish between direct and indirect variation Determine if a given function is direct given a function rule, graph or table of values Solve word problems related to direct and indirect variation (ref. to regents questions from jmap.org) Distinguish between parallel and perpendicular lines. Do linear regression using a graphing calculator Determine the correlation between the data sets by viewing or plotting a scatter-plot. Perform vertical and horizontal translations Graph absolute value equations and Relation Function Vertical line test Function Rule Function notation Domain Range Direct Variation Constant of Variation Linear function Linear equation x-intercept y-intercept Slope Standard form of linear function Slope intercept form of linear function Point slope form of linear function Line of best fit Scatter plot Correlation Correlation coefficient Regression Absolute value Overview of Chapter 2 with special emphasis on transformation. Go over review questions on page 80 perform related translations Oct 28th – Dec 3rd Unit 6: Quadratic Equations and functions How do you perform transformations of functions? How do you factor completely all types of quadratic expressions? How do you use the calculator to find appropriate regression formulas? How do you use imaginary numbers to find square roots of negative numbers? How do you solve quadratic equations using a variety of techniques? How do you determine the kinds of roots a quadratic will have from its equation? How do you find the solution set for quadratic inequalities? How do you solve systems of linear and A2.A.46 Perform transformations with functions and relations: f (x + a) , f(x)+ a, f (−x), − f (x), af (x) A2.A.40 Write functions in functional notation Students will be able to - A2.A.39 Determine the domain and range of a function from its equation - A2.A.7 Factor polynomial expressions completely, using any combination of the following techniques: common factor extraction, difference of two perfect squares, quadratic trinomials A2.S.7 Determine the function for the regression model, using appropriate technology, and use the regression function to interpolate and extrapolate from the data A2.A.20 Determine the sum and product of the roots of a quadratic equation by examining its coefficients A2.A.21 Determine the quadratic equation, given the sum and product of its roots A2.A.13 Simplify radical expressions A2.A.24 Know and apply the technique of completing the square A2.A.25 Solve quadratic equations, using the quadratic formula A2.A.2 Use the discriminant to determine the nature of the roots of a quadratic equation - - - - - perform horizontal and vertical translations of the graph of y = x2 graph a quadratic in vertex form: f(x) =a(x - h)2 + k identify and label the vertex as ( h , k) identify and label the axis of symmetry of a parabola graph parabolas in the form of y = a x2 with various values of a graph a quadratic in vertex form: f(x) = ax2+bx+c find the axis of symmetry algebraically using the standard form of the equation identify the y-intercept as ( 0, c ) find the vertex of a parabola algebraically using the standard form of the equation identify the range of parabolas sketch a graph of a parabola after finding the axis of symmetry, the vertex, and the y-intercept use the calculator to find a quadratic regression equation factor using “FOIL” finding a GCF perfect square trinomials difference of two squares zero product property finding the sum and product of roots writing equations knowing the roots or knowing the sum and product of the roots solve by taking square roots solve by completing the square solve by using the quadratic formula use the discriminant to find the Parabola Quadratic function Vertex form Axis of symmetry Vertex of the parabola Maximum Minimum Standard form Domain and Range Regressions Factoring Greatest Common Factor Perfect square trinomial Difference of two squares Zero of a function (root) Discriminan t Imaginary numbers Complex numbers Conjugates 4-1 Quadratic functions and transformations 4-2 Standard form of a quadratic function 4-3 Modeling with quadratic functions 4-4 Factoring quadratic expressions 4-5 Quadratic equations 4-6 Completing the square 4-7 Quadratic Formula 4-8 Complex Numbers Additional resources at www.emathinstruction.com Quadratic Inequalities Page 256-257 Powers of complex numbers Page 265 4-9 Quadratic Systems 10-3 Circles quadratic equations graphically and algebraically? A2.A.4 Solve quadratic inequalities in one and two variables, algebraically and graphically A2.A.3 Solve systems of equations involving one linear equation and one quadratic equation algebraically Note: This includes rational equations that result in linear equations with extraneous roots. - - A2.N6 Write square roots of negative numbers in terms of i A2.N7 Simplify powers of i A2.N8 Determine the conjugate of a complex number A2.N9 Perform arithmetic operations on complex numbers and write the answer in the form a+bi A2.A47 Determine the equation of a circle A2.A48 Write the equation of a circle given a point A2.A49 Write the equation of a circle from its graph nature of the roots simplify expressions containing complex numbers (include rationalizing the denominator) solve quadratic inequalities solve systems of quadratics algebraically Determine the equation of a circle given the center and the radius, a point and the radius, the center and a point Determine the equation of a circle in center-radius form by completing the square of the equation in standard form Dec 6th – Dec 17th Unit 7: Polynomials How do you perform arithmetic operations with polynomial expressions? How do you factor polynomials? How do you solve polynomial equation? How do you expand a polynomial to the nth Order? How do you find the nth term of a binomial expansion? A2.N.3 Perform arithmetic operations with polynomial expressions containing rational coefficients A2.A.7 Factor polynomial expressions completely, using any combination of the following techniques: common factor extraction, difference of two perfect squares, quadratic trinomials A2.A.26 Find the solution to polynomial equations of higher degree that can be solved using factoring and/or the quadratic formula Student will be able to - - A2.A.50 Approximate the solution to polynomial equations of higher degree by inspecting the graph - A2.A.36 Apply the binomial theorem to expand a binomial and determine a specific term of a binomial expansion - - combine like terms subtract polynomial expressions multiply monomials, binomials and trinomials recognize and classify polynomials factor polynomials using common factor extraction, difference of two perfect squares and or trinomial factoring. Write a polynomial function given its roots. Solve polynomial equations /find the roots graphically. Divide polynomials by factoring, long division or synthetic division Apply the Binomial Theorem to expand a binomial expression Find a specific term of a binomial expansion. Polynomial Monomial Binomial Trinomial Degree Root Solution Zero Property 5-1 Polynomial Functions 5-2 Polynomials, Linear Factors and Zeros 5-3 Solving Polynomial Equations 5-4 Dividing Polynomials 5-7 The Binomial Theorem Jan 3rd – Jan 15th Unit 8: Radical Functions, Rational Exponents, Function Operations How do you write algebraic expressions in simplest radical form? How do you simplify by rationalizing the denominator? A2.N.1 Evaluate numerical expressions with negative and/or fractional exponents, without the aid of a calculator (when the answers are rational numbers A2.N.2 Perform arithmetic operations with expressions containing irrational numbers in radical form A2.N.4 Perform arithmetic operations on irrational expressions A2.A.8 Use rules of exponents to simplify expressions involving negative and/or rational exponents How do you express sums and differences of radical expressions in simplest form? A2.A.9 Rewrite expressions that contain negative exponents using only positive exponents How do you write radicals with fractional exponents? A2.A.10 Rewrite algebraic expressions with fractional exponents as radical expressions How do you change an expression with a fractional exponent into a radical expression? A2.A.11 Rewrite radical expressions as algebraic expressions with fractional exponents How do you solve radical equations? How do you add, subtract, multiply, and divide functions? How do you perform composition of functions? How do you find the A2.A.12 Evaluate exponential expressions A2.A.13 Simplify radical expressions A2.A.14 Perform basic operations on radical expressions A2.N.5 Rationalize a denominator containing a radical expression A2.A.15 Rationalize denominators of algebraic radical expressions Review of Algebra Topics Student will be able to - - Use rules of positive and negative exponents in algebraic computations Use squares and cubes of numbers Know square roots of perfect squares from 1-15 Algebra 2 and Trig Topics Students will be able to - Simplify radical expressions Multiply and divide radical expressions Add and subtract radical expressions Use rational exponents Solve radical equations and check for extraneous roots Add, subtract, multiply, and divide functions Find composition of functions Find inverses of functions Determine if a function is one to one or onto or both Exponents Conjugates Radicals Rationalize the denominato r Extraneous roots f- 1(x) inverse of a function one to one onto Page 360 Properties of exponents 6-1 Simplify radical expressions 6-2 Multiply and divide radical expressions 6-3 Binomial Radical Expressions 6-4 Rational Exponents 6-5 Solve radical equations 6-6 Function operations 6-7 Inverse relations and functions (the text does not cover “onto” so this will have to be supplemented with Ch 4-1 of AMSCO) inverse of a function? A2.A.22 Solve radical equations How do you determine if a function is 1 to 1 or onto? A2.A.40 Write functions using function notation A2.A.41 Use function notation to evaluate functions for given values in the domain A2.A.42 Find the composition of functions A2.A.43 Determine if a function is 1 to 1, onto, or both A2.A.44 Define the inverse of a function A2.A.45 Determine the inverse of a function and use composition to justify the result Jan 18th – Jan 24th MIDTERM REVIEW Jan 31st – Feb 19th Unit 9:Exponential and Logarithmic Functions How do you model a quantity that changes regularly over time by the same percentage? How are exponents and logarithms related? How are exponential functions and logarithmic functions related? Which type of function models the data best? A2.A.6 Solve an application with results in an exponential function. A2.A.12 Evaluate exponential expressions, including those with base e. Students will be able to: - model exponential growth and decay explore the properties of functions of the form y ab graph exponential functions that have base e write and evaluate logarithmic expressions graph logarithmic functions derive and use the properties of logarithms to simplify and expand logarithms. solve exponential and logarithmic equations evaluate and simplify natural logarithmic expressions solve equations using natural logarithms x A2.A.53 Graph exponential functions of the form. y b for positive values of b, including b = e. x - A2.A.18 Evaluate logarithmic expressions in any base - A2.A.54 Graph logarithmic functions, using the inverse of the related exponential function. - A2.A.51 Determine the domain and range of a function from its graph. A2.A.19 Apply the properties of logarithms to rewrite logarithmic expressions in equivalent forms. A2.A. 27 Solve exponential equations with and without common bases. A2.A. 28 Solve a logarithmic equations by rewriting as an exponential equation. A2.S.6 Determine from a scatter plot whether a linear, logarithmic, exponential, or power regression model is most appropriate. - asymptote change of base formula common logarithm exponential equation exponential function exponential decay exponential growth logarithm logarithmic equation logarithmic function natural logarithmic function 7 -1 Exploring Exponential Models 7 - 2 Properties of Exponential functions 7 – 3 Logarithmic Functions as Inverses - Fitting Curves to Data Page 459 7 - 4 Properties of Logarithms 7 - 5 Exponential and Logarithmic Equations 7 - 6 Natural Logarithms NOTE- the text only does problems compounding interest continuously. You will need to supplement to do problems that compound quarterly, monthly, etc.) Ch 7-7 AMSCO Feb 28th – March 15th Unit 10: Rational Expressions and Functions How do we perform arithmetic operations on rational expressions? How do we simplify a complex fraction? How do we solve a rational equation? A2.A.17 Simplify complex fractional expressions Review of Algebra Topics All topics in this unit except complex fractions are taught in Integrated Algebra. In Algebra most problems involve monomials and simple polynomials. In Algebra 2 factoring becomes more complex and may require more than one step to factor completely. A2.A.23 Solve rational equations and inequalities Algebra 2 Topics Students will be able to A2.A.5 Use direct and inverse variation A2.A.16 Perform arithmetic operations with rational expressions and rename to lowest terms - - Identify from tables, graphs and models direct and inverse variation Solve algebraically and graph inverse variation Graph rational functions with vertical and horizontal asymptotes Simplify a rational expression to lowest terms by factoring and reducing State any restrictions on the variable Multiply and divide rational expressions Add and subtract rational expressions Simplify a complex fraction Solve rational equations Solve rational inequalities Inverse Variation Asymptotes Simplest form Rational Expression Common factors Reciprocal Least Common Multiple Lowest Common Denominato r Common factors Complex Fraction Rational equation 8-1 Inverse Variation(omit combined and joint variation) 8-2 Reciprocal functions and transformations 8-3 Rational functions and their graphs 8-4 Rational Expressions 8-5 Adding and Subtracting Rational Expressionsincludes simplifying complex fractions 8-6 Solving Rational Equations NOTE: Teachers must supplement for solving rational inequalities (Ch. 2-8 of AMSCO) March 16th – March 25th Unit 11: Probability How do you calculate the probability of an event? A2.S.9 Differentiate between situations requiring permutations and those requiring combinations Students will be able to - A2.S.10 Calculate the number of possible permutations (nPr) of n items taken r at a time A2.S.11Calculate the number of possible combinations (nCr) of n items taken r at a time. A2.S.12 Use permutations, combinations, and the Fundamental Principle of Counting to determine the number of elements in a sample space and a specific subset (event) A2.S.13 Calculate theoretical probabilities, including geometric applications A2.S.14 Calculate empirical probabilities A2.S.15 Know and apply the binomial probability formula to events involving the terms exactly, at least, and at most - Use permutations, combinations, and the Fundamental Principle of Counting to determine the number of elements in a sample space and a specific subset (event) Determine theoretical and experimental probabilities for events, including geometric applications Find the probability of the event A and B Find the probability of event A or B Know and apply the binomial probability formula to events involving the terms exactly, at least, and at most Permutation Combinatio n Factorial Counting Principle Event Outcome Sample Space Theoretical probability Experimenta l Probability Dependent events Independent events Mutually exclusive 11-1 Permutations and Combinations 11-2 Probability 11-3 Probability of Multiple Events 11-8 Binomial Distributions You may wish to supplement the text using additional resources from www.emathinstruction.com March 28th – April 8th Unit 12: Statistics What methods are there for analyzing data? A2.S.1 Understand the differences among various kinds of studies (e.g., survey, observation, controlled experiment) A2.S.2 Determine factors which may affect the outcome of a survey Students will be able to - A2.S.3 Calculate measures of central tendency with group frequency distributions A2.S.4 Calculate measures of dispersion (range, quartiles, interquartile range, standard deviation, variance) for both samples and populations A2.S.5 Know and apply the characteristics of the normal distribution A2.S.16 Use the normal distribution as an approximation for binomial probabilities - Calculate measures of central tendency given a frequency table Calculate measures of dispersion (range, quartiles, interquartile range, standard deviation, variance) for both samples and populations (standard deviation & variance using graphing calculator) Calculate probabilities using the normal distribution (use the normal curve given on the Algebra 2 reference sheet) Survey Experiment Bias Sample Population Standard deviation Variance Central tendency Outlier Frequency distribution Dispersion Quartiles Interquartile range Binomial probability Normal Distribution 11-5 Analyzing Data 11-6 Standard Deviation 11-7 Samples and Surveys 11-9 Normal Distributions P 741 Approximating a Binomial Distribution April 11th – April 15th Unit 13: Sequences and Series What is the difference between arithmetic and a geometric sequence? How do you find an explicit formula? How do you write a recursive definition for a sequence? How do you find the common difference and the nth term of an arithmetic sequence? How do you find the common ratio and the nth term of a geometric sequence? How do you find the sum of a finite series using the formulas? A2.A.29 Identify an arithmetic or geometric sequence and find the formula for its nth term A2.A.30 Find the common difference in an arithmetic sequence Students will be able to - A2.A.31 Determine the common ratio of a geometric sequence A2.A.32 Determine a specified term of an arithmetic or a geometric sequence - A2.A.33 Specify terms of a sequence given its recursive definition - A2.N.10 Know and apply sigma notation - A2.A.34 Represent the sum of a series using sigma notation - A2.A.35 Determine the sum of the first n terms of an arithmetic or a geometric series - Use patterns to find subsequent terms of a sequence Use explicit formulas to find terms of sequences Find a recursive definition for a sequence Find an explicit formula to define a sequence Tell whether a sequence is arithmetic, geometric, or neither Find the common difference of an arithmetic sequence Find the nth term of an arithmetic sequence Find the common ratio in a geometric sequence Find the nth term of a geometric sequence Find the sum of a finite arithmetic series Write a series using sigma notation Find the sum of a finite geometric series Sequence Arithmetic sequence Geometric sequence Explicit formula Recursive definition Finite Series Sigma notation 9-1 Mathematical patterns 9-2 Arithmetic Sequences 9-3 Geometric Sequence 9-4 Arithmetic Series 9-5 Geometric series April 25th – May 13th Unit 14: Solving Trig Equations How do you verify a trigonometric identity? How do you solve trigonometric equations? How do you use the trigonometric angle formulas to find values for trig functions? A2.A.67 Justify the Pythagorean identities A2.A.68 Solve trigonometric equations for all values of the variable from 0º to 360º A2.A.59 Use the reciprocal and cofunction relationships to find the value of the secant, cosecant, and cotangent of 0º, 30º, 45º, 60º, 90º, 180º, and 270º angles A2.A.76 Apply the angle sum and difference formulas for trigonometric functions A2.A.77 Apply the double-angle and halfangle formulas for trigonometric functions Students will be able to - - Identify reciprocal identities Verify trig equations using trig identities Verify Pythagorean identities Simplify trig expressions using identities Solve linear and quadratic trig equations within the given domain Verify an angle identity Use the angle sum and difference formulas to evaluate a trig expression or verify a trig. equation Use the angle double angle and half angle formulas to evaluate a trig expression or verify a trig. equation CATCH-UP, REVIEW AND FINALS Trig. Identities Reciprocal Trig. Function Pythagorea n identities Negative angle identity Cofunction identity Angle sum formula Angle difference formula Double angle formula Half angle formula 14-1 Trigonometric Identities 14-6 Angle Identities 14-7 Double Angle and Half Angle Identities 14-2 Solving Trigonometric Equations