Accelerated Geometry

MAGC: Accelerated Geometry

MAGC.A: Algebra

MAGC.B: Function

MAGC.C: Geometry

MAGC.D: Numbers

MAGC.E: Statistics and Probability

MAGC.A.1: interpret expressions that represent a quantity in terms of its context

MAGC.A.2: interpret parts of an expression such as terms, factors, and coefficients, in context

MAGC.A.3: given situations which utilize formulas or expressions with multiple terms and/or factors, interpret the meaning (in context) of individual terms or factors

MAGC.A.4: use the structure of an expression to rewrite it in different equivalent forms (e.g., recognize x4 - y4  as (x²)² - (y²)², thus recognizing it as a difference of squares that can be factored as (x² - y²)(x² + y²))

MAGC.A.5: choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression

MAGC.A.6: use the properties of exponents to transform expressions for exponential functions

MAGC.A.7: 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 (e.g., calculate mortgage payments)

MAGC.A.8: 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

MAGC.A.9: 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)

MAGC.A.10: 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

MAGC.A.11: prove polynomial identities and use them to describe numerical relationships (e.g., the polynomial identity (x² + y²)² = (x² - y²)² + (2xy)² can be used to generate Pythagorean triples)

MAGC.A.12: know and apply that the Binomial Theorem gives 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 using Pascal’s Triangle

MAGC.A.13: 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

MAGC.A.14: create equations and inequalities in one variable and use them to solve problems (i.e., create equations in one variable that describes simple rational functions, exponential functions, etc.)

MAGC.A.15: create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales (i.e., create and graph equations in two variables to describe radical functions, rational functions, etc.)

MAGC.A.16: 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 (e.g., represent inequalities describing nutritional and cost constraints on combinations of different foods)

MAGC.A.17: rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations

MAGC.A.18: solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise

MAGC.A.19: solve quadratic equations in one variable

MAGC.A.20: solve quadratic equations by inspection (e.g.,x² = -49), taking square roots, factoring, completing the square, and using the quadratic formula, 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

MAGC.A.21: 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

MAGC.A.22: 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 a graph, find the solution to a system of equations where f(x) and/or g(x) are rational functions)

MAGC.B.23: using tables, graphs, equations, and verbal descriptions, interpret the key characteristics of a function which models the relationship between two quantities; sketch a graph showing key features, including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior

MAGC.B.24: relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes (e.g., 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)

MAGC.B.25: 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

MAGC.B.26: graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases

MAGC.B.27: graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions

MAGC.B.28: graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior

MAGC.B.29: graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior

MAGC.B.30: graph exponential and logarithmic functions, showing intercepts and end behavior

MAGC.B.31: write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function

MAGC.B.32: use the properties of exponents to interpret expressions for exponential functions

MAGC.B.33: compare properties of two functions each represented algebraically, graphically, numerically in tables, and/or by a verbal description

MAGC.B.34: write a function that describes a relationship between two quantities (e.g., quadratic, polynomial, rational, radical, exponential, logarithmic)

MAGC.B.35: combine standard function types using arithmetic operations

MAGC.B.36: compose functions

MAGC.B.37: identify the effect on the graph of replacing f(x) by f(x) + k, kf(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

MAGC.B.38: find inverse functions

MAGC.B.39: 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 (e.g., f(x) = 2(x^3) or f(x) = (x+1)/(x-1) for x?1)

MAGC.B.40: verify by composition that one function is the inverse of another

MAGC.B.41: read values of an inverse function from a graph or a table, given that the function has an inverse

MAGC.B.42: understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents

MAGC.B.43: express as a logarithm the solution to ab^(ct) = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology

MAGC.C.44: 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

MAGC.C.45: identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects

MAGC.C.46: give informal arguments for geometric formulas (i.e., informal arguments for the formulas of the circumference of a circle and area of a circle using dissection arguments and informal limit arguments; informal arguments for the formula of the volume of a cylinder, pyramid, and cone using Cavalieri's principle)

MAGC.C.47: use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder)

MAGC.C.48: give an informal argument using Cavalieri's principle for the formulas of the volume of a sphere and other solid figures

MAGC.C.49: apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot)

MAGC.C.50: use volume formulas for cylinders, pyramids, cones, and spheres to solve problems

MAGC.C.51: 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)

MAGC.C.52: use coordinates to prove simple geometric theorems algebraically (i.e., prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that a point lies on a circle centered at the origin and containing a given point), including quadrilaterals, circles, right triangles, and parabolas

MAGC.C.53: prove that all circles are similar (i.e., using transformations; ratio of circumference to the diameter is a constant)

MAGC.C.54: identify and describe relationships among inscribed angles, radii, chords, tangents, and secants (i.e., 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)

MAGC.C.55: construct the inscribed and circumscribed circle of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle

MAGC.C.56: construct a tangent line from a point outside a given circle to the circle

MAGC.C.57: 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

MAGC.C.58: 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)

MAGC.C.59: find the point on a directed line segment between two given points that partitions the segment in a given ratio

MAGC.C.60: use coordinates to compute perimeters of polygons and areas of triangles and rectangles (e.g., using the distance formula)

MAGC.D.61: 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

MAGC.D.62: rewrite expressions involving radicals and rational exponents using the properties of exponents

MAGC.D.63: 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

MAGC.D.64: use the relation i² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers

MAGC.D.65: find the conjugate of a complex number; use conjugates to find quotients of complex numbers

MAGC.D.66: solve quadratic equations with real coefficients that have complex solutions

MAGC.D.67: extend polynomial identities to the complex numbers (e.g., rewrite x² + 4 as (x + 2i)(x - 2i))

MAGC.D.68: know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials

MAGC.E.69: describe categories of events as subsets of a sample space using unions, intersections, or complements of other events (i.e., "or," "and," "not")

MAGC.E.70: understand that if two events A and B are independent, the probability of A and B occurring together is the product of their probabilities, and that if the probability of two events A and B occurring together is the product of their probabilities, the two events are independent

MAGC.E.71: understand the conditional probability of A given B as P(A and B)/P(B); interpret independence of A and B in terms of conditional probability (i.e., 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)

MAGC.E.72: 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 (e.g., 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)

MAGC.E.73: recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations (e.g., compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer)

MAGC.E.74: 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 context

MAGC.E.75: apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in context