 # TEKS Standards Alignment 6-12

## HS Geometry

2. Numbers and Operations. The student applies mathematical process standards to represent and use rational numbers in a variety of forms. The student is expected to:

Standard Description
2A classify whole numbers, integers, and rational numbers using a visual representation such as a Venn diagram to describe relationships between sets of numbers
2B identify a number, its opposite, and its absolute value
2C locate, compare, and order integers and rational numbers using a number line
2D order a set of rational numbers arising from mathematical and real-world contexts
2E extend representations for division to include fraction notation such as a/b represents the same number as a ÷ b where b ≠ 0

3. Number and operations. The student applies mathematical process standards to represent addition, subtraction, multiplication, and division while solving problems and justifying solutions. The student is expected to:

Standard Description
3A recognize that dividing by a rational number and multiplying by its reciprocal result in equivalent values
3B determine, with and without computation, whether a quantity is increased or decreased when multiplied by a fraction, including values greater than or less than one
3C represent integer operations with concrete models and connect the actions with the models to standardized algorithms
3D add, subtract, multiply, and divide integers fluently
3E multiply and divide positive rational numbers fluently

4. Proportionality. The student applies mathematical process standards to develop an understanding of proportional relationships in problem situations. The student is expected to:

Standard Description
4A compare two rules verbally, numerically, graphically, and symbolically in the form of y = ax or y = x + a in order to differentiate between additive and multiplicative relationships
4B apply qualitative and quantitative reasoning to solve prediction and comparison of real- world problems involving ratios and rates
4C give examples of ratios as multiplicative comparisons of two quantities describing the same attribute
4D give examples of rates as the comparison by division of two quantities having different attributes, including rates as quotients
4E represent ratios and percents with concrete models, fractions, and decimals
4F represent benchmark fractions and percents such as 1%, 10%, 25%, 33 1/3%, and multiples of these values using 10 by 10 grids, strip diagrams, number lines, and numbers
4G generate equivalent forms of fractions, decimals, and percents using real-world problems, including problems that involve money
4H convert units within a measurement system, including the use of proportions and unit rates

5. Proportionality. The student applies mathematical process standards to solve problems involving proportional relationships. The student is expected to

Standard Description
5A represent mathematical and real-world problems involving ratios and rates using scale factors, tables, graphs, and proportions
5B solve real-world problems to find the whole given a part and the percent, to find the part given the whole and the percent, and to find the percent given the part and the whole, including the use of concrete and pictorial models
5C use equivalent fractions, decimals, and percents to show equal parts of the same whole

6. Expressions, equations, and relationships. The student applies mathematical process standards to use multiple representations to describe algebraic relationships. The student is expected to

Standard Description
6A identify independent and dependent quantities from tables and graphs
6B write an equation that represents the relationship between independent and dependent quantities from a table
6C represent a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b

7. Expressions, equations, and relationships. The student applies mathematical process standards to develop concepts of expressions and equations. The student is expected to:

Standard Description
7A generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization
7B distinguish between expressions and equations verbally, numerically, and algebraically
7C determine if two expressions are equivalent using concrete models, pictorial models, and algebraic representations
7D generate equivalent expressions using the properties of operations: inverse, identity, commutative, associative, and distributive properties

8. Expressions, equations, and relationships. The student applies mathematical process standards to use geometry to represent relationships and solve problems. The student is expected to:

Standard Description
8A extend previous knowledge of triangles and their properties to include the sum of angles of a triangle, the relationship between the lengths of sides and measures of angles in a triangle, and determining when three lengths form a triangle
8B model area formulas for parallelograms, trapezoids, and triangles by decomposing and rearranging parts of these shapes
8C write equations that represent problems related to the area of rectangles, parallelograms, trapezoids, and triangles and volume of right rectangular prisms where dimensions are positive rational numbers
8D determine solutions for problems involving the area of rectangles, parallelograms, trapezoids, and triangles and volume of right rectangular prisms where dimensions are positive rational numbers

9. Expressions, equations, and relationships. The student applies mathematical process standards to use equations and inequalities to represent situations. The student is expected to:

Standard Description
9A write one-variable, one-step equations and inequalities to represent constraints or conditions within problems
9B represent solutions for one-variable, one-step equations and inequalities on number lines
9C write corresponding real-world problems given one-variable, one-step equations or inequalities

10. Expressions, equations, and relationships. The student applies mathematical process standards to use equations and inequalities to solve problems. The student is expected to:

Standard Description
10A model and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts
10B determine if the given value(s) make(s) one-variable, one-step equations or inequalities true

12. Measurement and data. The student applies mathematical process standards to use numerical or graphical representations to analyze problems. The student is expected to:

Standard Description
12A represent numeric data graphically, including dot plots, stem-and-leaf plots, histograms, and box plots
12B use the graphical representation of numeric data to describe the center, spread, and shape of the data distribution
12C summarize numeric data with numerical summaries, including the mean and median (measures of center) and the range and interquartile range (IQR) (measures of spread), and use these summaries to describe the center, spread, and shape of the data distribution
12D summarize categorical data with numerical and graphical summaries, including the mode, the percent of values in each category (relative frequency table), and the percent bar graph, and use these summaries to describe the data distribution

13. Measurement and data. The student applies mathematical process standards to use numerical or graphical representations to solve problems. The student is expected to:

Standard Description
13A interpret numeric data summarized in dot plots, stem-and-leaf plots, histograms, and box plots
13B distinguish between situations that yield data with and without variability

14. Personal financial literacy. The student applies mathematical process standards to develop an economic way of thinking and problem solving useful in one's life as a knowledgeable consumer and investor. The student is expected to:

Standard Description
14A compare the features and costs of a checking account and a debit card offered by different local financial institutions
14B distinguish between debit cards and credit cards
14C balance a check register that includes deposits, withdrawals, and transfers
14D explain why it is important to establish a positive credit history
14E describe the information in a credit report and how long it is retained
14F describe the value of credit reports to borrowers and to lenders
14G explain various methods to pay for college, including through savings, grants, scholarships, student loans, and work-study
14H compare the annual salary of several occupations requiring various levels of post- secondary education or vocational training and calculate the effects of the different annual salaries on lifetime income

3. Number and operations. The student applies mathematical process standards to add, subtract, multiply, and divide while solving problems and justifying solutions. The student is expected to:

Standard Description
3A add, subtract, multiply, and divide rational numbers fluently
3B apply and extend previous understandings of operations to solve problems using addition, subtraction, multiplication, and division of rational numbers

4. Proportionality. The student applies mathematical process standards to develop an understanding of proportional relationships in problem situations. The student is expected to:

Standard Description
4A represent constant rates of change in mathematical and real-world problems given pictorial, tabular, verbal, numeric, graphical, and algebraic representations, including d = rt
4B calculate unit rates from rates in mathematical and real-world problems
4C determine the constant of proportionality (k = y/x) within mathematical and real-world problems
4D solve problems involving ratios, rates, and percents, including multi-step problems involving percent increase and percent decrease, and financial literacy problems
4E convert between measurement systems, including the use of proportions and the use of unit rates

5. Proportionality. The student applies mathematical process standards to use geometry to describe or solve problems involving proportional relationships. The student is expected to:

Standard Description
5A generalize the critical attributes of similarity, including ratios within and between similar shapes
5B describe π as the ratio of the circumference of a circle to its diameter
5C solve mathematical and real-world problems involving similar shape and scale drawings

6. Proportionality. The student applies mathematical process standards to use probability and statistics to describe or solve problems involving proportional relationships. The student is expected to:

Standard Description
6A represent sample spaces for simple and compound events using lists and tree diagrams
6B select and use different simulations to represent simple and compound events with and without technology
6C make predictions and determine solutions using experimental data for simple and compound events
6D make predictions and determine solutions using theoretical probability for simple and compound events
6E find the probabilities of a simple event and its complement and describe the relationship between the two
6F use data from a random sample to make inferences about a population
6G solve problems using data represented in bar graphs, dot plots, and circle graphs, including part-to-whole and part-to-part comparisons and equivalents
6H solve problems using qualitative and quantitative predictions and comparisons from simple experiments
6I determine experimental and theoretical probabilities related to simple and compound events using data and sample spaces

8. Expressions, equations, and relationships. The student applies mathematical process standards to develop geometric relationships with volume. The student is expected to:

Standard Description
8A model the relationship between the volume of a rectangular prism and a rectangular pyramid having both congruent bases and heights and connect that relationship to the formulas
8B explain verbally and symbolically the relationship between the volume of a triangular prism and a triangular pyramid having both congruent bases and heights and connect that relationship to the formulas
8C use models to determine the approximate formulas for the circumference and area of a circle and connect the models to the actual formulas

9. Expressions, equations, and relationships. The student applies mathematical process standards to solve geometric problems. The student is expected to:

Standard Description
9A solve problems involving the volume of rectangular prisms, triangular prisms, rectangular pyramids, and triangular pyramids
9B determine the circumference and area of circles
9C determine the area of composite figures containing combinations of rectangles, squares, parallelograms, trapezoids, triangles, semicircles, and quarter circles
9D solve problems involving the lateral and total surface area of a rectangular prism, rectangular pyramid, triangular prism, and triangular pyramid by determining the area of the shape's net

10. Expressions, equations, and relationships. The student applies mathematical process standards to use one-variable equations and inequalities to represent situations. The student is expected to:

Standard Description
10A write one-variable, two-step equations and inequalities to represent constraints or conditions within problems
10B represent solutions for one-variable, two-step equations and inequalities on number lines
10C write a corresponding real-world problem given a one-variable, two-step equation or inequality

11. Expressions, equations, and relationships. The student applies mathematical process standards to solve one-variable equations and inequalities. The student is expected to:

Standard Description
11A model and solve one-variable, two-step equations and inequalities
11B determine if the given value(s) make(s) one-variable, two-step equations and inequalities true
11C write and solve equations using geometry concepts, including the sum of the angles in a triangle, and angle relationships

12. Measurement and data. The student applies mathematical process standards to use statistical representations to analyze data. The student is expected to:

Standard Description
12A compare two groups of numeric data using comparative dot plots or box plots by comparing their shapes, centers, and spreads
12B use data from a random sample to make inferences about a population
12C compare two populations based on data in random samples from these populations, including informal comparative inferences about differences between the two populations

13. Personal financial literacy. The student applies mathematical process standards to develop an economic way of thinking and problem solving useful in one's life as a knowledgeable consumer and investor. The student is expected to:

Standard Description
13A calculate the sales tax for a given purchase and calculate income tax for earned wages
13B identify the components of a personal budget, including income; planned savings for college, retirement, and emergencies; taxes; and fixed and variable expenses, and calculate what percentage each category comprises of the total budget
13C create and organize a financial assets and liabilities record and construct a net worth statement
13D use a family budget estimator to determine the minimum household budget and average hourly wage needed for a family to meet its basic needs in the student's city or another large city nearby
13E calculate and compare simple interest and compound interest earnings
13F analyze and compare monetary incentives, including sales, rebates, and coupons

2. Number and operations. The student applies mathematical process standards to represent and use real numbers in a variety of forms. The student is expected to:

Standard Description
2A extend previous knowledge of sets and subsets using a visual representation to describe relationships between sets of real numbers
2B approximate the value of an irrational number, including π and square roots of numbers less than 225, and locate that rational number approximation on a number line
2C convert between standard decimal notation and scientific notation
2D order a set of real numbers arising from mathematical and real-world contexts

3. Proportionality. The student applies mathematical process standards to use proportional relationships to describe dilations. The student is expected to:

Standard Description
3A generalize that the ratio of corresponding sides of similar shapes are proportional, including a shape and its dilation
3B compare and contrast the attributes of a shape and its dilation(s) on a coordinate plane
3C use an algebraic representation to explain the effect of a given positive rational scale factor applied to two-dimensional figures on a coordinate plane with the origin as the center of dilation

4. Proportionality. The student applies mathematical process standards to explain proportional and non-proportional relationships involving slope. The student is expected to:

Standard Description
4A use similar right triangles to develop an understanding that slope, m, given as the rate comparing the change in y-values to the change in x-values, (y2 - y1)/ (x2 - x1), is the same for any two points (x1, y1) and (x2, y2) on the same line
4B graph proportional relationships, interpreting the unit rate as the slope of the line that models the relationship
4C use data from a table or graph to determine the rate of change or slope and y-intercept in mathematical and real-world problems

5. Proportionality. The student applies mathematical process standards to use proportional and non- proportional relationships to develop foundational concepts of functions. The student is expected to:

Standard Description
5A represent linear proportional situations with tables, graphs, and equations in the form of y = kx
5B represent linear non-proportional situations with tables, graphs, and equations in the form of y = mx + b, where b ≠ 0
5C contrast bivariate sets of data that suggest a linear relationship with bivariate sets of data that do not suggest a linear relationship from a graphical representation
5D use a trend line that approximates the linear relationship between bivariate sets of data to make predictions
5E solve problems involving direct variation
5F distinguish between proportional and non-proportional situations using tables, graphs, and equations in the form y = kx or y = mx + b, where b ≠ 0
5G identify functions using sets of ordered pairs, tables, mappings, and graphs
5H identify examples of proportional and non-proportional functions that arise from mathematical and real-world problems
5I write an equation in the form y = mx + b to model a linear relationship between two quantities using verbal, numerical, tabular, and graphical representations

6. Expressions, equations, and relationships. The student applies mathematical process standards to develop mathematical relationships and make connections to geometric formulas. The student is expected to:

Standard Description
6A describe the volume formula V = Bh of a cylinder in terms of its base area and its height
6B model the relationship between the volume of a cylinder and a cone having both congruent bases and heights and connect that relationship to the formulas
6C use models and diagrams to explain the Pythagorean theorem

7. Expressions, equations, and relationships. The student applies mathematical process standards to use geometry to solve problems. The student is expected to:

Standard Description
7A solve problems involving the volume of cylinders, cones, and spheres
7B use previous knowledge of surface area to make connections to the formulas for lateral and total surface area and determine solutions for problems involving rectangular prisms, triangular prisms, and cylinders
7C use the Pythagorean Theorem and its converse to solve problems
7D determine the distance between two points on a coordinate plane using the Pythagorean Theorem

8. Expressions, equations, and relationships. The student applies mathematical process standards to use one-variable equations or inequalities in problem situations. The student is expected to:

Standard Description
8A write one-variable equations or inequalities with variables on both sides that represent problems using rational number coefficients and constants
8B write a corresponding real-world problem when given a one-variable equation or inequality with variables on both sides of the equal sign using rational number coefficients and constants
8C model and solve one-variable equations with variables on both sides of the equal sign that represent mathematical and real-world problems using rational number coefficients and constants
8D use informal arguments to establish facts about the angle sum and exterior angle of triangles, the angles created when parallel lines are cut by a transversal, and the angle- angle criterion for similarity of triangles

10. Two-dimensional shapes. The student applies mathematical process standards to develop transformational geometry concepts. The student is expected to:

Standard Description
10A generalize the properties of orientation and congruence of rotations, reflections, translations, and dilations of two-dimensional shapes on a coordinate plane
10B differentiate between transformations that preserve congruence and those that do not
10C explain the effect of translations, reflections over the x- or y-axis, and rotations limited to 90°, 180°, 270°, and 360° as applied to two-dimensional shapes on a coordinate plane using an algebraic representation
10D model the effect on linear and area measurements of dilated two-dimensional shapes

11. Measurement and data. The student applies mathematical process standards to use statistical procedures to describe data. The student is expected to:

Standard Description
11A construct a scatterplot and describe the observed data to address questions of association such as linear, non-linear, and no association between bivariate data
11B determine the mean absolute deviation and use this quantity as a measure of the average distance data are from the mean using a data set of no more than 10 data points
11C simulate generating random samples of the same size from a population with known characteristics to develop the notion of a random sample being representative of the population from which it was selected

12. Personal financial literacy. The student applies mathematical process standards to develop an economic way of thinking and problem solving useful in one's life as a knowledgeable consumer and investor. The student is expected to:

Standard Description
12A solve real-world problems comparing how interest rate and loan length affect the cost of credit
12B calculate the total cost of repaying a loan, including credit cards and easy access loans, under various rates of interest and over different periods using an online calculator
12C explain how small amounts of money invested regularly, including money saved for college and retirement, grow over time
12D calculate and compare simple interest and compound interest earnings
12E identify and explain the advantages and disadvantages of different payment methods
12F analyze situations to determine if they represent financially responsible decisions and identify the benefits of financial responsibility and the costs of financial irresponsibility
12G estimate the cost of a two-year and four-year college education, including family contribution, and devise a periodic savings plan for accumulating the money needed to contribute to the total cost of attendance for at least the first year of college

## High School

### Algebra 1

2. Linear functions, equations, and inequalities. The student applies the mathematical process standards when using properties of linear functions to write and represent in multiple ways, with and without technology, linear equations, inequalities, and systems of equations. The student is expected to:

Standard Description
2A determine the domain and range of a linear function in mathematical problems; determine reasonable domain and range values for real-world situations, both continuous and discrete; and represent domain and range using inequalities
2B write linear equations in two variables in various forms, including y = mx + b, Ax + By = C, and y - y1 = m(x - x1), given one point and the slope and given two points
2C write linear equations in two variables given a table of values, a graph, and a verbal description
2D write and solve equations involving direct variation
2E write the equation of a line that contains a given point and is parallel to a given line
2F write the equation of a line that contains a given point and is perpendicular to a given line
2G write an equation of a line that is parallel or perpendicular to the X or Y axis and determine whether the slope of the line is zero or undefined
2H write linear inequalities in two variables given a table of values, a graph, and a verbal description
2I write systems of two linear equations given a table of values, a graph, and a verbal description

3. Linear functions, equations, and inequalities. The student applies the mathematical process standards when using graphs of linear functions, key features, and related transformations to represent in multiple ways and solve, with and without technology, equations, inequalities, and systems of equations. The student is expected to:

Standard Description
3A determine the slope of a line given a table of values, a graph, two points on the line, and an equation written in various forms, including y = mx + b, Ax + By = C, and y - y1 = m(x - x1)
3B calculate the rate of change of a linear function represented tabularly, graphically, or algebraically in context of mathematical and real-world problems
3C graph linear functions on the coordinate plane and identify key features, including x- intercept, y-intercept, zeros, and slope, in mathematical and real-world problems
3D graph the solution set of linear inequalities in two variables on the coordinate plane
3E determine the effects on the graph of the parent function f(x) = x when f(x) is replaced by af(x), f(x) + d, f(x - c), f(bx) for specific values of a, b, c, and d
3F graph systems of two linear equations in two variables on the coordinate plane and determine the solutions if they exist
3G estimate graphically the solutions to systems of two linear equations with two variables in real-world problems
3H graph the solution set of systems of two linear inequalities in two variables on the coordinate plane

4. Linear functions, equations, and inequalities. The student applies the mathematical process standards to formulate statistical relationships and evaluate their reasonableness based on real- world data. The student is expected to:

Standard Description
4A calculate, using technology, the correlation coefficient between two quantitative variables and interpret this quantity as a measure of the strength of the linear association
4B compare and contrast association and causation in real-world problems
4C write, with and without technology, linear functions that provide a reasonable fit to data to estimate solutions and make predictions for real-world problems

5. Linear functions, equations, and inequalities. The student applies the mathematical process standards to solve, with and without technology, linear equations and evaluate the reasonableness of their solutions. The student is expected to:

Standard Description
5A solve linear equations in one variable, including those for which the application of the distributive property is necessary and for which variables are included on both sides
5B solve linear inequalities in one variable, including those for which the application of the distributive property is necessary and for which variables are included on both sides
5C solve systems of two linear equations with two variables for mathematical and real-world problems

6. Quadratic functions and equations. The student applies the mathematical process standards when using properties of quadratic functions to write and represent in multiple ways, with and without technology, quadratic equations. The student is expected to:

Standard Description
6A determine the domain and range of quadratic functions and represent the domain and range using inequalities
6B write equations of quadratic functions given the vertex and another point on the graph, write the equation in vertex form (f(x) = a(x - h)2+ k), and rewrite the equation from vertex form to standard form (f(x) = ax2+ bx + c)
6C write quadratic functions when given real solutions and graphs of their related equations

7. Quadratic functions and equations. The student applies the mathematical process standards when using graphs of quadratic functions and their related transformations to represent in multiple ways and determine, with and without technology, the solutions to equations. The student is expected to:

Standard Description
7A graph quadratic functions on the coordinate plane and use the graph to identify key attributes, if possible, including x-intercept, y-intercept, zeros, maximum value, minimum values, vertex, and the equation of the axis of symmetry
7B describe the relationship between the linear factors of quadratic expressions and the zeros of their associated quadratic functions
7C determine the effects on the graph of the parent function f(x) = x2 when f(x) is replaced by af(x), f(x) + d, f(x - c), f(bx) for specific values of a, b, c, and d

8. Quadratic functions and equations. The student applies the mathematical process standards to solve, with and without technology, quadratic equations and evaluate the reasonableness of their solutions. The student formulates statistical relationships and evaluates their reasonableness based on real-world data. The student is expected to:

Standard Description
8A solve quadratic equations having real solutions by factoring, taking square roots, completing the square, and applying the quadratic formula
8B write, using technology, quadratic functions that provide a reasonable fit to data to estimate solutions and make predictions for real-world problems

9. Exponential functions and equations. The student applies the mathematical process standards when using properties of exponential functions and their related transformations to write, graph, and represent in multiple ways exponential equations and evaluate, with and without technology, the reasonableness of their solutions. The student formulates statistical relationships and evaluates their reasonableness based on real-world data. The student is expected to:

Standard Description
9A determine the domain and range of exponential functions of the form f(x) = abx and represent the domain and range using inequalities
9B interpret the meaning of the values of a and b in exponential functions of the form f(x) = abx in real-world problems
9C write exponential functions in the form f(x) = abx (where b is a rational number) to describe problems arising from mathematical and real-world situations, including growth and decay
9D graph exponential functions that model growth and decay and identify key features, including y-intercept and asymptote, in mathematical and real-world problems
9E write, using technology, exponential functions that provide a reasonable fit to data and make predictions for real-world problems

10. Number and algebraic methods. The student applies the mathematical process standards and algebraic methods to rewrite in equivalent forms and perform operations on polynomial expressions. The student is expected to:

Standard Description
10A add and subtract polynomials of degree one and degree two
10B multiply polynomials of degree one and degree two
10C determine the quotient of a polynomial of degree one and polynomial of degree two when divided by a polynomial of degree one and polynomial of degree two when the degree of the divisor does not exceed the degree of the dividend
10D rewrite polynomial expressions of degree one and degree two in equivalent forms using the distributive property
10E factor, if possible, trinomials with real factors in the form ax2 + bx + c, including perfect square trinomials of degree two
10F decide if a binomial can be written as the difference of two squares and, if possible, use the structure of a difference of two squares to rewrite the binomial

11. Number and algebraic methods. The student applies the mathematical process standards and algebraic methods to rewrite algebraic expressions into equivalent forms. The student is expected to:

Standard Description
11A simplify numerical radical expressions involving square roots
11B simplify numeric and algebraic expressions using the laws of exponents, including integral and rational exponents

12. Number and algebraic methods. The student applies the mathematical process standards and algebraic methods to write, solve, analyze, and evaluate equations, relations, and functions. The student is expected to:

Standard Description
12A decide whether relations represented verbally, tabularly, graphically, and symbolically define a function
12B evaluate functions, expressed in function notation, given one or more elements in their domains
12C identify terms of arithmetic and geometric sequences when the sequences are given in function form using recursive processes
12D write a formula for the nth term of arithmetic and geometric sequences, given the value of several of their terms
12E solve mathematic and scientific formulas, and other literal equations, for a specified variable

## High School

### Algebra 2

2. Attributes of functions and their inverses. The student applies mathematical processes to understand that functions have distinct key attributes and understand the relationship between a function and its inverse. The student is expected to:

Standard Description
2A graph the functions f(x)=√x, f(x)=1/x, f(x)=x3, f(x)= 3√x, f(x)=bx, f(x)=|x|, and f(x)=logb (x) where b is 2, 10, and e, and, when applicable, analyze the key attributes such as domain, range, intercepts, symmetries, asymptotic behavior, and maximum and minimum given an interval
2B graph and write the inverse of a function using notation such as f -1 (x)
2C describe and analyze the relationship between a function and its inverse (quadratic and square root, logarithmic and exponential), including the restriction(s) on domain, which will restrict its range
2D use the composition of two functions, including the necessary restrictions on the domain, to determine if the functions are inverses of each other

3. Systems of equations and inequalities. The student applies mathematical processes to formulate systems of equations and inequalities, use a variety of methods to solve, and analyze reasonableness of solutions. The student is expected to:

Standard Description
3A formulate systems of equations, including systems consisting of three linear equations in three variables and systems consisting of two equations, the first linear and the second quadratic
3B solve systems of three linear equations in three variables by using Gaussian elimination, technology with matrices, and substitution
3C solve, algebraically, systems of two equations in two variables consisting of a linear equation and a quadratic equation
3D determine the reasonableness of solutions to systems of a linear equation and a quadratic equation in two variables
3E formulate systems of at least two linear inequalities in two variables
3F solve systems of two or more linear inequalities in two variables
3G determine possible solutions in the solution set of systems of two or more linear inequalities in two variables

4. Quadratic and square root functions, equations, and inequalities. The student applies mathematical processes to understand that quadratic and square root functions, equations, and quadratic inequalities can be used to model situations, solve problems, and make predictions. The student is expected to:

Standard Description
4A write the quadratic function given three specified points in the plane
4B write the equation of a parabola using given attributes, including vertex, focus, directrix, axis of symmetry, and direction of opening
4C determine the effect on the graph of f(x) = x when f(x) is replaced by af(x), f(x) + d, f(bx), and f(x - c) for specific positive and negative values of a, b, c, and d
4D transform a quadratic function f(x) = ax2 + bx + c to the form f(x) = a(x - h)2 + k to identify the different attributes of f(x)
4E formulate quadratic and square root equations using technology given a table of data
4F solve quadratic and square root equations
4G identify extraneous solutions of square root equations

5. Exponential and logarithmic functions and equations. The student applies mathematical processes to understand that exponential and logarithmic functions can be used to model situations and solve problems. The student is expected to:

Standard Description
5A determine the effects on the key attributes on the graphs of f(x) = bx and f(x) = logb (x) where b is 2, 10, and e when f(x) is replaced by af(x), f(x) + d, and f(x - c) for specific positive and negative real values of a, c, and d
5B formulate exponential and logarithmic equations that model real-world situations, including exponential relationships written in recursive notation
5C rewrite exponential equations as their corresponding logarithmic equations and logarithmic equations as their corresponding exponential equations
5D solve exponential equations of the form y = abx where a is a nonzero real number and b is greater than zero and not equal to one and single logarithmic equations having real solutions
5E determine the reasonableness of a solution to a logarithmic equation

6. Cubic, cube root, absolute value and rational functions, equations, and inequalities. The student applies mathematical processes to understand that cubic, cube root, absolute value and rational functions, equations, and inequalities can be used to model situations, solve problems, and make predictions. The student is expected to:

Standard Description
6A analyze the effect on the graphs of f(x) = x3 and f(x) = 3√x when f(x) is replaced by af(x), f(bx), f(x - c), and f(x) + d for specific positive and negative real values of a, b, c, and d
6B solve cube root equations that have real roots
6C analyze the effect on the graphs of f(x) = |x| when f(x) is replaced by af(x), f(bx), f(x-c), and f(x) + d for specific positive and negative real values of a, b, c, and d
6D formulate absolute value linear equations
6E solve absolute value linear equations
6F solve absolute value linear inequalities
6G analyze the effect on the graphs of f(x) = 1/x when f(x) is replaced by af(x), f(bx), f(x-c), and f(x) + d for specific positive and negative real values of a, b, c, and d
6H formulate rational equations that model real-world situations
6I solve rational equations that have real solutions
6J determine the reasonableness of a solution to a rational equation
6K determine the asymptotic restrictions on the domain of a rational function and represent domain and range using interval notation, inequalities, and set notation
6L formulate and solve equations involving inverse variation

7. Number and algebraic methods. The student applies mathematical processes to simplify and perform operations on expressions and to solve equations. The student is expected to:

Standard Description
7A add, subtract, and multiply complex numbers
7B add, subtract, and multiply polynomials
7C determine the quotient of a polynomial of degree three and of degree four when divided by a polynomial of degree one and of degree two
7D determine the linear factors of a polynomial function of degree three and of degree four using algebraic methods
7E determine linear and quadratic factors of a polynomial expression of degree three and of degree four, including factoring the sum and difference of two cubes and factoring by grouping
7F determine the sum, difference, product, and quotient of rational expressions with integral exponents of degree one and of degree two
7G rewrite radical expressions that contain variables to equivalent forms
7H solve equations involving rational exponents
7I write the domain and range of a function in interval notation, inequalities, and set notation

8. Data. The student applies mathematical processes to analyze data, select appropriate models, write corresponding functions, and make predictions. The student is expected to:

Standard Description
8A analyze data to select the appropriate model from among linear, quadratic, and exponential models
8B use regression methods available through technology to write a linear function, a quadratic function, and an exponential function from a given set of data
8C predict and make decisions and critical judgments from a given set of data using linear, quadratic, and exponential model

## High School

### Geometry

2. Coordinate and transformational geometry. The student uses the process skills to understand the connections between algebra and geometry and uses the one- and two-dimensional coordinate systems to verify geometric conjectures. The student is expected to:

Standard Description
2A determine the coordinates of a point that is a given fractional distance less than one from one end of a line segment to the other in one- and two-dimensional coordinate systems, including finding the midpoint
2B derive and use the distance, slope, and midpoint formulas to verify geometric relationships, including congruence of segments and parallelism or perpendicularity of pairs of lines
2C determine an equation of a line parallel or perpendicular to a given line that passes through a given point

3. Coordinate and transformational geometry. The student uses the process skills to generate and describe rigid transformations (translation, reflection, and rotation) and non-rigid transformations (dilations that preserve similarity and reductions and enlargements that do not preserve similarity). The student is expected to:

Standard Description
3A describe and perform transformations of figures in a plane using coordinate notation
3B determine the image or pre-image of a given two-dimensional figure under a composition of rigid transformations, a composition of non-rigid transformations, and a composition of both, including dilations where the center can be any point in the plane
3C identify the sequence of transformations that will carry a given pre-image onto an image on and off the coordinate plane
3D identify and distinguish between reflectional and rotational symmetry in a plane figure

4. Logical argument and constructions. The student uses the process skills with deductive reasoning to understand geometric relationships. The student is expected to:

Standard Description
4A distinguish between undefined terms, definitions, postulates, conjectures, and theorems
4B identify and determine the validity of the converse, inverse, and contrapositive of a conditional statement and recognize the connection between a biconditional statement and a true conditional statement with a true converse
4C verify that a conjecture is false using a counterexample
4D compare geometric relationships between Euclidean and spherical geometries, including parallel lines and the sum of the angles in a triangle

5. Logical argument and constructions. The student uses constructions to validate conjectures about geometric figures. The student is expected to:

Standard Description
5A investigate patterns to make conjectures about geometric relationships, including angles formed by parallel lines cut by a transversal, criteria required for triangle congruence, special segments of triangles, diagonals of quadrilaterals, interior and exterior angles of polygons, and special segments and angles of circles choosing from a variety of tools
5B construct congruent segments, congruent angles, a segment bisector, an angle bisector, perpendicular lines, the perpendicular bisector of a line segment, and a line parallel to a given line through a point not on a line using a compass and a straightedge
5C use the constructions of congruent segments, congruent angles, angle bisectors, and perpendicular bisectors to make conjectures about geometric relationships
5D verify the Triangle Inequality theorem using constructions and apply the theorem to solve problems

6. Proof and congruence. The student uses the process skills with deductive reasoning to prove and apply theorems by using a variety of methods such as coordinate, transformational, and axiomatic and formats such as two-column, paragraph, and flow chart. The student is expected to:

Standard Description
6A verify theorems about angles formed by the intersection of lines and line segments, including vertical angles, and angles formed by parallel lines cut by a transversal and prove equidistance between the endpoints of a segment and points on its perpendicular bisector and apply these relationships to solve problems
6B prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Side-Side-Side, Angle-Angle-Side, and Hypotenuse-Leg congruence conditions
6C apply the definition of congruence, in terms of rigid transformations, to identify congruent figures and their corresponding sides and angles
6D verify theorems about the relationships in triangles, including proof of the Pythagorean Theorem, the sum of interior angles, base angles of isosceles triangles, midsegments, and medians, and apply these relationships to solve problems
6E prove a quadrilateral is a parallelogram, rectangle, square, or rhombus using opposite sides, opposite angles, or diagonals and apply these relationships to solve problems

7. Similarity, proof, and trigonometry. The student uses the process skills in applying similarity to solve problems. The student is expected to:

Standard Description
7A apply the definition of similarity in terms of a dilation to identify similar figures and their proportional sides and the congruent corresponding angles
7B apply the Angle-Angle criterion to verify similar triangles and apply the proportionality of the corresponding sides to solve problems

8. Similarity, proof, and trigonometry. The student uses the process skills with deductive reasoning to prove and apply theorems by using a variety of methods such as coordinate, transformational, and axiomatic and formats such as two-column, paragraph, and flow chart. The student is expected to:

Standard Description
8A prove theorems about similar triangles, including the Triangle Proportionality theorem, and apply these theorems to solve problems
8B identify and apply the relationships that exist when an altitude is drawn to the hypotenuse of a right triangle, including the geometric mean, to solve problems

9. Similarity, proof, and trigonometry. The student uses the process skills to understand and apply relationships in right triangles. The student is expected to:

Standard Description
9A determine the lengths of sides and measures of angles in a right triangle by applying the trigonometric ratios sine, cosine, and tangent to solve problems
9B apply the relationships in special right triangles 30°-60°-90° and 45°-45°-90° and the Pythagorean theorem, including Pythagorean triples, to solve problems

10. Two-dimensional and three-dimensional figures. The student uses the process skills to recognize characteristics and dimensional changes of two- and three-dimensional figures. The student is expected to:

Standard Description
10A identify the shapes of two-dimensional cross-sections of prisms, pyramids, cylinders, cones, and spheres and identify three-dimensional objects generated by rotations of two- dimensional shape
10B determine and describe how changes in the linear dimensions of a shape affect its perimeter, area, surface area, or volume, including proportional and non-proportional dimensional change

11. Two-dimensional and three-dimensional figures. The student uses the process skills in the application of formulas to determine measures of two- and three-dimensional figures. The student is expected to:

Standard Description
11A apply the formula for the area of regular polygons to solve problems using appropriate units of measure
11B determine the area of composite two-dimensional figures comprised of a combination of triangles, parallelograms, trapezoids, kites, regular polygons, or sectors of circles to solve problems using appropriate units of measure
11C apply the formulas for the total and lateral surface area of three-dimensional figures, including prisms, pyramids, cones, cylinders, spheres, and composite figures, to solve problems using appropriate units of measure
11D apply the formulas for the volume of three-dimensional figures, including prisms, pyramids, cones, cylinders, spheres, and composite figures, to solve problems using appropriate units of measure

12. Circles. The student uses the process skills to understand geometric relationships and apply theorems and equations about circles. The student is expected to:

Standard Description
12A apply theorems about circles, including relationships among angles, radii, chords, tangents, and secants, to solve non-contextual problems
12B apply the proportional relationship between the measure of an arc length of a circle and the circumference of the circle to solve problems
12C apply the proportional relationship between the measure of the area of a sector of a circle and the area of the circle to solve problems
12D describe radian measure of an angle as the ratio of the length of an arc intercepted by a central angle and the radius of the circle
12E show that the equation of a circle with center at the origin and radius r is x2 + y2 = r2 and determine the equation for the graph of a circle with radius r and center (h, k), (x - h)2 + (y - k)2 =r2

13. Probability. The student uses the process skills to understand probability in real-world situations and how to apply independence and dependence of events. The student is expected to:

Standard Description
13A develop strategies to use permutations and combinations to solve contextual problems
13B determine probabilities based on area to solve contextual problems
13C identify whether two events are independent and compute the probability of the two events occurring together with or without replacement
13D apply conditional probability in contextual problems
13E apply independence in contextual problems