Integrated Math 3 Math 3 Course Description:

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1 Course Description: Integrated Math 3 Math 3 Course Description: Integrated strands include algebra, functions, geometry, trigonometry, statistics, probability and discrete math. Scope and sequence includes reasoning and proof, inequalities and linear programming, similarity and congruence, samples and variation, polynomial and rational functions, circles and circular functions, recursion and iteration, and inverse functions and logarithms. A graphing calculator is highly recommended. Course Goals: Students will develop their ability to understand formal reasoning in geometric, algebraic, and statistical contexts and of basic principles that underlie those reasoning strategies. Students will develop their ability to reason both algebraically and graphically to solve inequalities in one and two variables, introduces systems of inequalities in two variables, and develops a strategy for optimizing a linear function in two variables within a system of linear constraints on those variables. Students will extend their understanding of similarity and congruence and their ability to use those relations to solve problems and to prove geometric assertions with and without the use of coordinates. Students will extend their understanding of the measurement of variability, develops student ability to use the normal distribution as a model of variation, introduces students to the binomial distribution and its use in decision making, and introduces students to the probability and statistical inference involved in control charts used in industry for statistical process control. Students will extend their ability to represent and draw inferences about polynomial and rational functions using symbolic expressions and manipulations. Students will develop their ability to an understanding of relationships among special lines, segments, and angles in circles and the ability to use properties of circles to solve problems; develops student understanding of circular functions and the ability to use these functions to model periodic change; and extends student ability to reason deductively in geometric settings. Students will extend their ability to represent, analyze, and solve problems in situations involving sequential and recursive change. Students will develop their understanding of inverses of functions with a focus on logarithmic functions and their use in modeling and analyzing problem situations and data patterns. Strands include advanced geometry and algebra topics. Advanced geometric topics include parallel lines, congruence, similarity, circle properties, trigonometry and proof. Advanced algebraic topics are polynomial functions, rational functions, exponential and logarithmic functions. This course extends the use of logical thinking to the deductive reasoning processes. Course Goals: Students will become problem solvers by reasoning abstractly and quantitatively, and by constructing viable arguments and critiquing the reasoning of others. Students will deepen their understanding of concepts by using appropriate tools and technology to analyze relationships mathematically to draw conclusions, discover patterns, interpret their mathematical results, and be precise in their calculations. 1

2 Course Lesson Objectives: Course Lesson Objectives: Recognize the role of inductive reasoning in main conjectures and recognize the limitation of inductive reasoning Recognize the need for proof and be able to create a simple deductive argument to prove a mathematical assertion Create a counterexample to prove a conjecture is false Write if-then statements and their converses and use if-then reasoning patterns in arguments Know and be able to use the angle relationship theorems involving two intersecting lines. Know and be able to use the theorems justifying the construction of a line perpendicular to a given line through a given point and construction of a line parallel to a given line through a given point Know and be able to use the angle relationship theorems involving two parallel lines cut by a transversal and their converses Know and be able to use the angle sum theorem and the exterior angle theorem for triangles Use algebraic notation-letters, expressions, equations, and inequalities-to represent general patterns and relationships among variables Use algebraic transformations of expressions, equations, and inequalities to establish general propositions about quantitative relationships Know the characteristics of a well-designed experiment Understand the importance of subject and evaluator blinding and the placebo effect Under the hypothesis of no treatment effect, construct an approximate sampling distribution for the difference of two means by re-randomizing; identify extreme events Use a randomization test to decide if an experiment provides statistically significant evidence that one treatment is more effective than another. Distinguish between three types of statistical studies - sample surveys, experiments and observational studies - and understand what inference can be made from each Write inequalities to express questions about functions of one or two variables Given a graph of one or more functions, solve inequalities related to the function(s) Solve quadratic inequalities in one variable by solving the corresponding equation algebraically and reasoning about the graph of the related function(s) Describe the solution set of an inequality in one variable symbolically, as a graph on a number line, and using interval notation Graph the solution set of a linear inequality in two variables Graph the solution set of a system of inequalities in two variables Solve linear programming problems involving two independent variables Identify similar polygons and determine the scale factor of Simplify and evaluate algebraic expressions Classify and use the properties of real numbers. Solve equations. Solve absolute value equations. Solve and graph inequalities. Analyze relations and functions Identify, graph, and write linear equations Find the slope of a line Draw scatterplots and find prediction equations Graph special functions, linear inequalities, and absolute value inequalities Solve systems of linear equations by graphing, by elimination, and by substitution. Solve systems of inequalities. Use linear programming to find maximum and minimum values of functions. Use matrices to solve systems of equations. Make conjectures, determine whether a statement is true or false, and find counterexamples for statements. Use deductive reasoning to reach valid conclusions. Verify algebraic and geometric conjectures using informal and formal proof. Write proofs involving segment and angle theorems. Add, subtract, multiply, divide, and factor polynomials. Simplify and solve equations involving roots, radicals, and rational exponents. Perform operations with complex numbers. Graph quadratic functions Solve quadratic equations Write quadratic equations and functions Analyze graphs of quadratic functions Graph and solve quadratic inequalities Use Midpoint and Distance Formulas Write and graph equations of parabolas and circles Find values of trigonometric functions. Solve problems by using right triangle trigonometry, the Law of Sines, and the Law of Cosines. Evaluate polynomial functions and solve polynomial equations. Graph polynomial and square root functions. Find factors and zeros of polynomial functions. Find the composition of functions. Determine the inverses of functions or relations. Simplify rational expressions. Graph rational functions. Solve direct, joint, and inverse variation problems. Identify graphs and equations as different types of functions. Solve rational equations and inequalities. Simplify exponential and logarithmic expressions. Solve exponential equations and inequalities. Solve logarithmic equations and inequalities. Solve problems involving exponential growth and decay. Identify angle relationships that occur with parallel lines and 2

3 similar polygons Review and extend understanding of the Laws of Sines and Cosines Know and be able use the three theorems providing sufficient conditions to prove triangles are similar (SSS, SAS, AAA) Continue to develop the ability to write both synthetic and analytic arguments Understand congruence of figures as a special case of similarity of figures Know and be able to use the four theorems providing sufficient conditions to prove triangles are congruent (SSS, SAS, AAS, ASA) Know and be able to use properties of the incenter, circumcenter, and centroid of a triangle Continue to develop the ability to write both synthetic and analytic arguments Know and be able to use both necessary and sufficient conditions for quadrilaterals to be (special) parallelograms Know and be able to use the Midpoint Connector Theorems for Triangles and Quadrilaterals Explore, prove, and apply properties of congruencepreserving transformations Describe characteristics of a normal distribution Understand that the number of standard deviations from the mean is a measure of location Use standardized values and a table of the normal distributions to find probabilities Use simulation to construct an approximate binomial distribution Predict the shape of a binomial distribution Use the formulas for the expected value and standard deviation of a binomial distribution Use standardized values to find probabilities of events in binomial situations Use a random sample to decide whether a given proportion p is plausible as the proportion of successes in the population from which the sample came Recognize when the mean and standard deviation change on a plot-over-time Use control charts and tests for out-of-control behavior Compute the probability of a false alarm on a set of readings, that is, the probability that a test will give an out-of-control signal for a process that is under control Understand the Central Limit Theorem and how it is applied to statistical process control. Model problem situations using polynomial functions Identify patterns relating rules and graphs of polynomial functions-connecting polynomial degree to local maximum and local minimum values and zeroes Add, subtract, and multiply polynomials- connecting degrees of component polynomials to degrees of sums, differences, and products. Find zeroes of polynomials and create polynomial functions with prescribed zeroes Express quadratic function rules in complete square or vertex form Use vertex form of quadratic expressions to solve quadratic a transversal, and identify and prove lines parallel from given angle relationships. Classify triangles Identify corresponding parts of congruent triangles. Prove triangle congruence using congruence postulates. Identify and use perpendicular bisectors, angle bisectors, medians, and altitudes of triangles. Apply properties of inequalities relating to the measures of angles and sides of triangles. Investigate interior and exterior angles of polygons. Recognize and apply the properties of parallelograms. Recognize and apply the properties of rectangles, rhombi, squares, and trapezoids. Use arithmetic and geometric sequences and series. Use special sequences and iterate functions. Expand powers by using the Binomial Theorem. Prove statements by using mathematical induction. Solve problems involving independent events, dependent events, permutations, and combinations. Find probability and odds. Find statistical measures. 3

4 equations and locate the vertex of parabolic graphs Use completing the square to prove the quadratic formula Use the quadratic formula to analyze solution possibilities for quadratic equations and indicate the rationale for extending the number system to include complex numbers Create rational functions to model problem situations Analyze graphs of rational functions and their vertical asymptotes. Simplify rational expressions. Add, subtract, multiply and divide rational expressions. Determine and prove that a line tangent to a circle is perpendicular to the radius at the point of tangency, and the tangent segments to a circle from an external point are congruent. State and prove the relationships among the measures of central angles, their chords, and their arcs. State and prove the properties relating a radius, a chord, and the midpoint and perpendicular bisector of the chord. State, prove, and apply the Inscribed Angle Theorem and the property that angles that intercept the same or congruent arcs are congruent. Analyze situations involving pulleys or sprockets to determine transmission factors, angular velocity, and linear velocity Use sines and cosines functions to describe rotations of circular objects Use radian and degree measures to measure angles and rotations Define sine and cosine as functions of real numbers and analyze the resulting periodic graphs Use the sine and cosine functions to model periodic patterns of change in various physical phenomena Understand arithmetic sequences and their connections to linear functions, using recursive formulas, function formulas, and applications Understand geometric sequences and their connections to exponential functions, using recursive formulas, function formulas, and applications Understand and apply arithmetic and geometric series (sums of sequences) Use finite differences tables to find function formulas for certain recursive formulas and to describe the connection between such tables and polynomial functions Use linear, exponential, and polynomial functions to model discrete situations Iterate functions and describe the resulting patterns, the longterm behavior in particular Describe the connection between function iteration and recursive formulas Analyze long-term behavior when iterating linear functions, using graphical iteration, numerical iteration, and algebraic methods, including fixed point analysis and connections to slope Solve problems involving direct and inverse variation Discover conditions that guarantee existence of an inverse for a given function Develop strategies for recognizing invertible functions from 4

5 study of tables of values and/or graphs of these functions Develop and use strategies for finding rules of inverses for linear and power functions Express a positive number as a power of 10 Define and evaluate common logarithms Use logarithms to solve exponential equations Develop and use basic properties of the logarithmic function Know and be able to use the definition of the inverse sine, inverse cosine, and inverse tangent functions Express the general solutions of a trigonometric equation in forms such as x = k + 2πn or x = k+ 360 o n Use trigonometric equation and their solutions to model and answer questions about periodic phenomena 5

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