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Algebra I Related Domains



Interpret the structure of expressions. [Linear, exponential, and quadratic]  Interpret expressions that represent a quantity in terms of its context.
 Interpret parts of an expression, such as terms, factors, and coefficients.
 Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)^{n} as the product of P and a factor not depending on P.
 Use the structure of an expression to identify ways to rewrite it.
Write expressions in equivalent forms to solve problems. [Quadratic and exponential]  Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
 Factor a quadratic expression to reveal the zeros of the function it defines.
 Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
 Use the properties of exponents to transform expressions for exponential functions. For example, the expression 1.15^{t} can be rewritten as (1.15^{1/12})^{ 12t }≈ 1.012^{12t} to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.



Understand solving equations as a process of reasoning and explain the reasoning. [Master linear; learn as general principle.]  Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
Solve equations and inequalities in one variable. [Linear inequalities; literal equations that are linear in the variables being solved for; quadratics with real solutions]  Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
3.1
Solve onevariable equations and inequalities involving absolute value, graphing the solutions and interpreting them in context.  Solve quadratic equations in one variable.
 Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)^{2} = q that has the same solutions. Derive the quadratic formula from this form.
 Solve quadratic equations by inspection (e.g., for x^{2} = 49), taking square roots, completing the square, the quadratic formula, and factoring, 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.
Solve systems of equations. [Linearlinear and linearquadratic]  Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
Represent and solve equations and inequalities graphically. [Linear and exponential; learn as general principle.]  Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
 Explain why the xcoordinates 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 technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
 Graph the solutions to a linear inequality in two variables as a halfplane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding halfplanes.

Understand the concept of a function and use function notation. [Learn as general principle; focus on linear and exponential and on arithmetic and geometric sequences.]  Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n + 1) = f(n) + f(n − 1) for n ≥ 1.
Interpret functions that arise in applications in terms of the context. [Linear, exponential, and quadratic]  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.
 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h gives the number of personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
 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. Analyze functions using different representations. [Linear, exponential, quadratic, absolute value, step, piecewisedefined]  Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
 Graph linear and quadratic functions and show intercepts, maxima, and minima.
 Graph square root, cube root, and piecewisedefined functions, including step functions and absolute value functions.
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.  Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
 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.
 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}, and y = (1.2)^{t/10}, and classify them as representing exponential growth or decay.
 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.

Build a function that models a relationship between two quantities. [For F.BF.1, 2, linear, exponential, and quadratic]  Write a function that describes a relationship between two quantities.
 Determine an explicit expression, a recursive process, or steps for calculation from a context.
 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 exponential, and relate these functions to the model.
 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
Build new functions from existing functions. [Linear, exponential, quadratic, and absolute value; for F.BF.4a, linear only]  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.
 Find inverse functions.
 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.

Construct and compare linear, quadratic, and exponential models and solve problems.  Distinguish between situations that can be modeled with linear functions and with exponential functions.
 Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
 Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
 Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two inputoutput pairs (include reading these from a table).
 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
Interpret expressions for functions in terms of the situation they model.  Interpret the parameters in a linear or exponential function in terms of a context. [Linear and exponential of form f(x) = bx + k]
 Apply quadratic functions to physical problems, such as the motion of an object under the force of gravity.

Summarize, represent, and interpret data on a single count or measurement variable.  Represent data with plots on the real number line (dot plots, histograms, and box plots).
 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
Summarize, represent, and interpret data on two categorical and quantitative variables. [Linear focus; discuss general principle.]  Summarize categorical data for two categories in twoway frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
 Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.
 Informally assess the fit of a function by plotting and analyzing residuals.
 Fit a linear function for a scatter plot that suggests a linear association.
Interpret linear models.  Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
 Compute (using technology) and interpret the correlation coefficient of a linear fit.
 Distinguish between correlation and causation.
Algebra II Related Domains


Interpret the structure of expressions. [Polynomial and rational]  Interpret expressions that represent a quantity in terms of its context.
 Interpret parts of an expression, such as terms, factors, and coefficients.
 Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)^{n} as the product of P and a factor not depending on P.
 Use the structure of an expression to identify ways to rewrite it.
Write expressions in equivalent forms to solve problems.  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. For example, calculate mortgage payments.

Perform arithmetic operations on polynomials. [Beyond quadratic]  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. Understand the relationship between zeros and factors of polynomials.
 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).
 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. Use polynomial identities to solve problems.
 Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x^{2} + y^{2} ) ^{2} = (x^{2} – y^{2} ) ^{2} + (2xy)^{2} can be used to generate Pythagorean triples. 5. (+) Know and apply the Binomial Theorem for 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 for example by Pascal’s Triangle.
Rewrite rational expressions. [Linear and quadratic denominators]  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.
 (+) 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.


Understand solving equations as a process of reasoning and explain the reasoning. [Simple radical and rational]  Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Solve equations and inequalities in one variable.
3.1
Solve onevariable equations and inequalities involving absolute value, graphing the solutions and interpreting them in context.
Represent and solve equations and inequalities graphically. [Combine polynomial, rational, radical, absolute value, and exponential functions.]  Explain why the xcoordinates 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 technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

Interpret functions that arise in applications in terms of the context. [Emphasize selection of appropriate models.]
 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.
 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
 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.
Analyze functions using different representations. [Focus on using key features to guide selection of appropriate type of model function.]  Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
b. Graph square root, cube root, and piecewisedefined functions, including step functions and absolute value functions.
c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

Build a function that models a relationship between two quantities. [Include all types of functions studied.]  Write a function that describes a relationship between two quantities.
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 exponential, and relate these functions to the model.
Build new functions from existing functions. [Include simple radical, rational, and exponential functions; emphasize common effect of each transformation across function types.]  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.
 Find inverse functions.
 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. For example, f(x) =2x3 or f(x) = ^{(x + 1)}/_{(x − 1) }for x ≠ 1.






Geometry Related Domains

Experiment with transformations in the plane.  Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
Understand congruence in terms of rigid motions. [Build on rigid motions as a familiar starting point for development of concept of geometric proof.]  Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
Prove geometric theorems. [Focus on validity of underlying reasoning while using variety of ways of writing proofs.]  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.
 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.
 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.
Make geometric constructions. [Formalize and explain processes.]  Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Understand similarity in terms of similarity transformations.  Verify experimentally the properties of dilations given by a center and a scale factor:
 A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
 The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
 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.
 Use the properties of similarity transformations to establish the AngleAngle (AA) criterion for two triangles to be similar.
Prove theorems involving similarity.  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.
 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Define trigonometric ratios and solve problems involving right triangles.  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.
 Explain and use the relationship between the sine and cosine of complementary angles.
 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
8.1
Derive and use the trigonometric ratios for special right triangles (30°, 60°, 90°and 45°, 45°, 90°).
Apply trigonometry to general triangles.  (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
 (+) Prove the Laws of Sines and Cosines and use them to solve problems.
 (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and nonright triangles (e.g., surveying problems, resultant forces).



Explain volume formulas and use them to solve problems.  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.
 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
Visualize relationships between twodimensional and threedimensional objects.  Identify the shapes of twodimensional crosssections of threedimensional objects, and identify threedimensional objects generated by rotations of twodimensional objects.
 Know that the effect of a scale factor k greater than zero on length, area, and volume is to multiply each by k, k^{2} , and k^{3} , respectively; determine length, area and volume measures using scale factors.
 Verify experimentally that in a triangle, angles opposite longer sides are larger, sides opposite larger angles are longer, and the sum of any two side lengths is greater than the remaining side length; apply these relationships to solve realworld and mathematical problems.


Understand independence and conditional probability and use them to interpret data. [Link to data from simulations or experiments.]  Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).
 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that 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.
 Construct and interpret twoway frequency tables of data when two categories are associated with each object being classified. Use the twoway table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, 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.
 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.
Use the rules of probability to compute probabilities of compound events in a uniform probability model.  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 terms of the model.
 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.
 (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(BA) = P(B)P(AB), and interpret the answer in terms of the model.
 (+) Use permutations and combinations to compute probabilities of compound events and solve problems.
