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The focus of Mathematics II is on quadratic expressions, equations, and functions; comparing their characteristics and behavior to those of linear and exponential relationships from Mathematics I as organized into 6 critical areas, or units. The need for extending the set of rational numbers arises and real and complex numbers are introduced so that all quadratic equations can be solved. The link between probability and data is explored through conditional probability and counting methods, including their use in making and evaluating decisions. The study of similarity leads to an understanding of right triangle trigonometry and connects to quadratics through Pythagorean relationships. Circles, with their quadratic algebraic representations, round out the course. 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.
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UnitsIncludes Standards and ClustersMathematical Practices
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Quadratic Functions and Modeling
ch. 13		•Interpret functions that arise in applications in terms of a context.
•Analyze functions using different representations.
•Build a function that models a relationship between two quantities.
•Build new functions from existing functions.
•Construct and compare linear, quadratic, and exponential models and solve problems. 		Make sense of problems and persevere in solving them.


Reason abstractly and quantitatively.
What is the difference between exerimental probability and theoretical probability?
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Similarity, Right Triangle Trigonometry, and Proof•Understand similarity in terms of similarity transformations.
•Prove geometric theorems.
•Prove theorems involving similarity.
•Use coordinates to prove simple geometric theorems algebraically.
•Define trigonometric ratios and solve problems involving right triangles.
•Prove and apply trigonometric identities.		Look for and express regularity in repeated reasoning.
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Circles•Understand and apply theorems about circles.
•Find arc lengths and areas of sectors of circles.
•Translate between the geometric description and the equation for a conic section.
•Use coordinates to prove simple geometric theorem algebraically.
•Explain volume formulas and use them to solve problems.
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*In some cases clusters appear in more than one unit within a course or in more than one course.
Instructional notes will indicate how these standards grow over time.
In some cases only certain standards within a cluster are included in a unit.
†Note that solving equations follows a study of functions in this course.
To examine equations before functions, this unit could come before Unit 2.
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Critical Area 1: Students extend the laws of exponents to rational exponents and explore distinctions between rational and irrational numbers by considering their decimal representations. In Unit 3,
students learn that when quadratic equations do not have real solutions the number system must be extended so that solutions exist, analogous to the way in which extending the whole numbers to the negative numbers allows x+1 = 0 to have a solution. Students explore relationships between number systems: whole numbers, integers, rational numbers, real numbers, and complex numbers. The guiding principle is that equations with no solutions in one number system may have solutions in a larger number system.
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Quadratic Functions and Modeling
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Interpret functions that arise in appli-cations in terms of a context. Focus on quadratic functions; compare with linear and exponential functions studied in Mathematics I.F.IF.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.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.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.
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Analyze functions using different representations for F.IF.7b, compare and contrast absolute value, step and piecewise-defined functions with linear, quadratic, and exponential functions. Highlight issues of domain, range and usefulness when examining piecewise-defined functions. Note that this unit, and in particular in F.IF.8b, extends the work begun in Mathematics I on exponential functions with integer exponents. For F.IF.9, focus on expanding the types of functions considered to include, linear, exponential, and quadratic.Extend work with quadratics to include the relationship between coefficients and roots, and that once roots are known, a quadratic equation can be factored.F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★a.Graph linear and quadratic functions and show intercepts, maxima, and minima.b.Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
F.IF.8 Write a functiondefined by an expression in different but equivalent forms to reveal and explain different properties of the function. a.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.★b.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.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.
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Build a function that models a relation-ship between two quantities.Focus on situations that exhibit a quadratic or exponential relationshipF.BF.1 Write a function that describes a relationship between two quantities.★ a.Determine an explicit expression, a recursive process, or steps for calculation from a context.b.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 expo-nential, and relate these functions to the model
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Build new functions from existing functions. For F.BF.3, focus on quadratic functions and consider including absolute value functions.. For F.BF.4a, focus on linear functions but consider simple situations where the domain of the function must be restricted in order for the inverse to exist, such as f(x) = x2, x>0.F.BF.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.4 Find inverse functions. a.Solve an equation of the form f(x) = c for a simple function fthat 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
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Construct and compare linear, quadrat-ic, and exponential models and solve problems.Compare linear and exponential growth studied in Mathematics I to quadratic growth.F.LE.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.
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Similarity, Right Triangle Trigonometry, and Proof
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Understand similarity in terms of simi-larity transformations.G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor.a.A dilation takes a line not passing through the center of the dila-tion to a parallel line, and leaves a line passing through the center unchanged.b.The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
G.SRT.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.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
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Prove geometric theorems.Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of G.CO.10 may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for G.C.3 in Unit 6.G.CO.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.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.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.
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Prove theorems involving similarity.G.SRT.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.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
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Use coordinates to prove simple geo-metric theorems algebraically.G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
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Define trigonometric ratios and solve problems involving right triangles.G.SRT.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.7 Explain and use the relationship between the sine and cosine of complementary angles.
G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems
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Prove and apply trigonometric identi-ties.In this course, limit θ to angles between 0 and 90 degrees. Connect with the Pythagorean theorem and the distance formula. A course with a greater focus on trigonometry could include the (+) standard F.TF.9: Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. This could continue to be limited to acute angles in Mathematics II. Extension of trigonometric functions to other angles through the unit circle is included in Mathematics III.F.TF.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.
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Circles With an Without Coordinates
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Understand and apply theorems about circles.G.C.1 Prove that all circles are similar.
G.C.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.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
G.C.4 (+) Construct a tangent line from a point outside a given circle to the circle.
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Find arc lengths and areas of sectors of circles.Emphasize the similarity of all circles. Note that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course.G.C.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.
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Translate between the geometric de-scription and the equation for a conic section.Connect the equations of circles and parabolas to prior work with quadratic equations. The directrix should be parallel to a coordinate axisG.GPE.1 Derive the equation of a circleof 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.2 Derive the equation of a parabola given a focus and directrix.
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Use coordinates to prove simple geo-metric theorems algebraically.Include simple proofs involving circles.G.GPE.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).
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Explain volume formulas and use them to solve problems.Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k2 times the area of the first. Similarly, volumes of solid figures scale by k3 under a similarity transformation with scale factor k.G.GMD.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.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
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