High School Math South Dakota Standards

Bivariate Data

Numerical Data

Statistics and Probability

Statistical Relationships

Graphing

Interpret Key Features

Create and Solve

Exponential Functions and Equations

Graphing and Transformations

Interpret Key Features

Create and Solve

Quadratic Functions and Equations

Statistical Relationships

Systems of Equations and Inequalities

Interpret Key Features

Create and Solve

Linear Functions, Equations, and Inequalities

Construct and Compare

Domain and Range, Function Notation

Algebraic Functions

Polynomials, Radical Expressions and Rational Exponents

Rational and Irrational Numbers

Expressions

Use function notation, evaluate functions, and interpret statements that use function notation in terms of a context.

Determine whether a relationship is a function given a graph, an equation, or a table of values.

Compare linear, quadratic, and exponential growth using tables and graphs to show that exponential growth eventually exceeds others

Differentiate between real-world scenarios that can be modeled by exponential or linear functions by determining whether the relationship has a common difference or a common ratio.

Explain why the sum or product of two rational numbers is rational; the sum of a rational and an irrational number is irrational; and the product of a nonzero rational and an irrational number is irrational.

Understand polynomials to be a sum of algebraic terms having variables, coefficients, exponents, and/or constants.

Add, subtract and multiply polynomials.

Understand rational exponents as a way to represent roots as powers.

Simplify numerical expressions containing exponents and/or roots, including negative and rational exponents.

Simplify algebraic expressions using the laws of exponents

Construct exponential equations from geometric sequences with and without context.

Use properties of exponents to write equivalent expressions for exponential functions.

Write geometric sequences using explicit and recursive formulas.

Determine the domain and range of exponential functions

Determine reasonable domain and range values of exponential functions representing real-world situations.

Interpret the key features of an exponential function that models a relationship between two quantities in a given context.

Graph exponential functions that model real-world problems (growth, decay, and compound interest), showing key attributes.

Interpret the quantities in an exponential equation in the context of a real-world problem, including growth, decay, and compound interest.

Represent and solve real-world problems, using linear expressions, equations, and inequalities in one variable. Interpret the solution as reasonable or unreasonable in context.

Translate between equivalent forms of equations for linear functions, including standard, point-slope, and slope intercept forms; recognize that each form reveals key features in a given context.

Estimate the solution of a system of linear equations by graphing the equations on a coordinate plane

Solve a system of linear equations with integer coefficients algebraically (substitution and elimination).

Solve linear inequalities and systems of linear inequalities in two variables by graphing.

Explain why a solution to the equation 𝑓(𝑥) = 𝑔(𝑥) is the x-coordinate where the y-coordinate of 𝑓(𝑥) and 𝑔(𝑥) are the same using graphs, tables, or approximations. Include cases where 𝑓(𝑥) and/or 𝑔(𝑥) are linear, quadratic, and exponential.

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.

Solve linear equations and linear inequalities in one variable, including those with rational number coefficients, variables on both sides of the equal or inequality sign, and literal equations, explaining the process used.

Construct linear functions from arithmetic sequences with and without context.

Write arithmetic sequences using explicit and recursive formulas.

Identify the parts of expressions such as terms, factors, variables, constants, and coefficients.

Determine reasonable domain and range values of linear functions representing real-world situations, both continuous and discrete.

Interpret the key features of linear functions that model a relationship between two quantities in a given context.

Use different representations of a linear function, including graphs, tables, and equations.

Calculate and interpret the rate of change of a linear function represented in a table, graph, or as an equation in context of real-world and mathematical problems.

Solve quadratic equations with real number solutions, containing one variable, including those with variables on both sides of the equal sign. Equations should be solved by: graphing, factoring, completing the square (leading coefficient is one, 𝑏 value is even), taking the square root and using the quadratic formula.

Interpret the solutions for quadratic equations as reasonable or unreasonable in context.

Graph quadratic functions in standard and vertex form

Determine the domain and range of quadratic functions.

Determine reasonable domain and range values of quadratic functions representing real-world situations.

Interpret the key features of a quadratic function (direction, roots, zeros, x-intercepts, and maximum or minimum values) that models a relationship between two quantities in a given context.

Graph and describe how transformations (stretches, translations, and reflections) affect linear, absolute value, and quadratic functions.

Use box plots and histograms to determine the statistics appropriate to the shape of the data distribution; compare the center (mean and median) and spread (standard deviation and IQR) of two or more data sets

Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points.

Interpret relative frequencies and associations in two-way tables.

Statistical Experiments and Studies

Statistics and Probability

Systems of Equations

Rational Expressions

Factor Polynomials

Graph and Key Features

Create and Solve

Polynomial, Rational, and Other Functions and Equations

Unit Circle

Graphs and Key Features

Create and Solve

Trigonometric Functions and Equations

Graph and Key Features

Logarithms

Create and Solve

Exponential and Logarithmic Functions and Equations

System of Equations

Graph and Key Features

Create and Solve

Quadratic Functions, Equations, and Inequalities

Systems of Equations

Matrices

Arithmetic Sequences

Linear Functions, Equations and Inequalities

Inverses

Transformations

Graph and Key Features

Compositions

Algebraic Functions

Complex Number

Radical Expressions and Rational Exponents

Expressions

Compose functions and evaluate their composition

Using ordered pairs, determine the inverse of a function given a graph or table

Graph rational functions, identifying zeros and asymptotes (vertical and horizontal) when suitable factorizations are available and showing end behavior, with or without the appropriate technology.

Compare properties of graphs, tables, equations, and verbal descriptions of two functions.

Calculate and interpret the average rate of change of a function, both symbolically and from a table over a specified interval. Estimate the rate of change from a graph.

Given a graph, explain the effects of the transformation from the parent function including square root and cubic functions, rational, and absolute value functions.

Describe the transformation of functions in the coordinate plane including translation, reflection, and dilation.

Explain how restricting the domain of a function allows for the creation of its inverse.

rite and graph the inverse of a given function; understand that the graph of an inverse function is a reflection of the function over the line 𝑦 = 𝑥.

Verify if two functions are inverses of each other using composition of functions

Apply the properties of exponents to translate between radical and exponential forms of expressions.

Simplify and perform operations with radical expressions with and without variables; rationalizing denominators should include conjugates.

Understand an imaginary number to be a product of any real number and the imaginary unit 𝑖 where 𝑖^2 = −1.

Understand a complex number to be a number of the form 𝑎 + 𝑏i where 𝑎𝑎 and 𝑏 are real numbers and 𝑖 is the imaginary unit.

Add, subtract, multiply, and divide complex numbers.

Use the properties of exponents to find equivalent expressions and to solve equations, including those involving rational exponents.

Interpret the solution of a logarithmic equation as reasonable or unreasonable in context.

Translate between logarithmic and exponential forms of an equation.

Use properties of logarithms to simplify and evaluate logarithmic expressions, with or without technology.

Use the inverse relationship between exponents and logarithms to solve problems.

Graph logarithmic functions, showing intercepts and end behavior.

Write and use arithmetic sequences recursively and explicitly to model situations; translate between the two forms when given a graph, a description of the relationship, or two input-output pairs.

Multiply a matrix by a scalar

Add and subtract matrices.

Solve systems of linear equations in three variables

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.

Construct a possible polynomial given a graph.

Factor polynomials using polynomial identities, including difference of squares, sum and difference of cubes, and the square of a sum or difference.

Divide polynomial expressions using factorization, long division, and synthetic division, with and without a remainder.

Explain why a solution to the equation 𝑓(𝑥) = 𝑔(𝑥) is the x-coordinate where the y-coordinate of 𝑓(𝑥) and 𝑔(𝑥) are the same using graphs, tables, or approximations, include cases where 𝑓(𝑥) and/or 𝑔(𝑥) are linear, polynomial, exponential, or rational and where at least one of the functions is not linear.

Create equations and inequalities with one variable and use them to solve problems, including absolute value functions

Solve formulas for a specific variable.

Solve rational and radical equations containing one variable specifying extraneous solutions.

Explain how the multiplicity of zeros affects the shape of a polynomial graph, using reasoning and visual patterns to justify the behavior at each intercept.

Interpret the key features of polynomial functions that model a relationship between two quantities in a given context; translate between different representations of the function, especially graphs, tables, and equations.

Determine the domain and range of polynomial and rational functions.

Graph polynomial functions, identifying zeros when suitable factorizations are available and showing end behavior, with or without the appropriate technology.

Apply the remainder and factor theorems to identify factors and find solutions to polynomial equations of degree greater than 2 and explain how each theorem supports the reasoning process.

Select, justify and apply appropriate methods to solve quadratic equations in one variable. Recognize complex solutions and write them as a +/- bi for real numbers a and b.

Represent and solve real-world problems using quadratic equations and inequalities.

Use the discriminant to determine the number and type of solutions of a quadratic equation.

Sketch the graph of a quadratic function given a verbal description and show key features.

Solve a system of equations consisting of a linear equation and a nonlinear equation in two variables algebraically or graphically with or without technology.

Represent and use mathematical models for bivariate data sets to answer questions, draw conclusions, and make decisions.

Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate

Distinguish between sample surveys, experiments, and observational studies and explain the purpose of randomization in statistical studies.

Use data from a random sample to make inferences about a population.

Compare theoretical and experimental probabilities using simulations.

Read and explain, in context, the validity of data from outside reports by identifying the variables as quantitative or categorical and describing how the data was collected.

Indicate any potential biases or flaws and identifying inferences the author of the report made from sample data.

Select trigonometric functions that model real-world contexts.

Develop the Pythagorean identity, sin^2(𝜃) + cos^2(𝜃) = 1.

Apply the Pythagorean identity to find the remaining trigonometric functions when given sin (𝜃), cos (𝜃), or tan (𝜃) and the quadrant of the angle.

Explain how changes in amplitude, period, and midline affect the graph of sine and cosine functions, using transformations and real-world contexts to support understanding.

Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

Explain how the unit circle can be used to model sine, cosine, tangent, secant, cosecant, and cotangent for all real numbers

Conditional and Joint Probability

Statistics and Probability

Triangle Congruency

Similarity

Similarity and Congruency

Plane

Coordinate Plane

Transformations

Parallel and Perpendicular Lines

Coordinate Geometry

Define and Construct

Lines and Angles

Geometric Probability

Three-Dimensional

Two-Dimensional

Geometric Figures

Equations of Circles

Circle Relationships

Circles

Trigonometry Ratios

Special Right Triangles and Pythagorean Theorem

Right Triangles

Recognize and apply relationships between angles, radii, chords, tangents, and secants, including: the relationship between central, inscribed, and circumscribed angles; that inscribed angles intersecting a diameter are right angles; and the radius of a circle intersecting a tangent line at the point of tangency forms a right angle

Explain how arc length relates to the whole circle using proportional reasoning and visual models to justify solutions.

Use the proportional relationship between the measure of the area of a sector of a circle and the area of the circle to solve problems.

Write the equation of a circle, given the radius and center, where the center is at the origin or another point.

Identify the center and radius of a circle, given the equation of a circle, where the center is at the origin or another point.

Understand and apply the fundamental geometric elements—points, lines, line segments, rays, planes, angles, and circles—by describing their properties and using them to model and solve real-world and mathematical problems

Identify two-dimensional cross sections of three-dimensional objects.

Calculate probabilities as a proportion of area in a geometric context

Apply and prove theorems about triangles including: the isosceles triangle Theorem and its converse; the triangle midsegment theorem; the proportionality theorem; the triangle inequality theorem and its converse.

Apply and prove theorems about quadrilaterals

Apply theorems about polygons including interior angle sum and exterior angle theorems

Construct an equilateral triangle, square, and a regular hexagon.

Find the volume and surface area of complex three-dimensional figures composed of prisms, pyramids, cones, cylinders, and spheres.

Give an informal argument for the formulas for the volume of a cylinder, pyramid, sphere, and cone. Use dissection arguments, and informal limit arguments

Identify the three-dimensional figure generated by rotating a two-dimensional figure about a fixed axis.

Use three-dimensional geometric figures and their measures to model real-world objects and solve problems.

Make geometric constructions with a variety of tools and methods, including: congruent segments and angles; segment and angle bisectors; perpendicular lines; and the perpendicular bisector of a line segment.

Determine the point that cuts a line segment into a specified ratio on a number line and a coordinate plane, including finding the midpoint.

Derive the distance and midpoint formulas and use the formulas, including the slope formula, to verify geometric relationships on a coordinate plane.

Calculate the perimeter of polygons when given the vertices, including using the distance formula.

Use coordinates to prove geometric relationships algebraically

Prove and apply slope criteria of parallel and perpendicular lines to solve problems.

Prove and apply theorems about lines and angles including vertical angles, angles formed by parallel lines cut by a transversal, and points on a perpendicular bisector

Write an equation of a line that is parallel or perpendicular to a given line and passing through a given point.

Apply the properties of special right triangles (30°−60°−90° and 45°−45°−90°) to solve real-world and mathematical problems

Prove and apply the Pythagorean Theorem and its converse.

Define that side ratios in right triangles are related to the angles in the triangle, leading to definitions of trigonometric ratios (sine, cosine, and tangent) for acute angles.

Explain the relationship between the sine and cosine of complementary angles and use them to solve problems.

Use the definition of the trigonometric ratios (sine, cosine, tangent, secant, cosecant, cotangent) as ratios of side in a right triangle to solve problems about lengths of sides and measurements of angles.

Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles.

Given two figures, apply the definition of similarity in terms of a dilation to identify similar figures, proportional sides, and corresponding congruent angles and finding and using the scale factor of the dilation that maps one figure to the other.

Determine whether figures are similar, using the definition of similarity and using similarity transformations.

Verify experimentally and apply the properties of dilations as determined by a center and a scale factor

Develop, apply and prove the criteria of similarity for triangles (AA~, SAS~, and SSS~) to solve problems and prove geometric relationships.

Use transformations to prove all circles are similar

Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Explain, using rigid motion transformations, why two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

Develop and apply criteria for triangle congruence (ASA, SAS, AAS, SSS, HL) and similarity to solve problems and prove relationships within triangles and other geometric figures.

Describe events as subsets of a sample space or as unions, intersections, or complements of other events.

Determine whether two events A and B are independent.

Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the result, including everyday language and situations. Construct and interpret two-way frequency tables to represent data and use them to determine conditional probabilities and assess independence of events.

Apply the Addition Rule, 𝑝(𝐴 or 𝐵), and interpret the result.

Apply the general Multiplication Rule, 𝑝(𝐴and 𝐵), and interpret the result

Describe rotations, reflections, and translations as functions that take points in the coordinate plane as inputs and give other points as outputs; write in prime notation.

Compare transformations that preserve distance and angle (rotation, reflections, and translations) to those that do not (dilations) to develop definitions of congruence and similarity

Given a rectangle, parallelogram, trapezoid, or a regular polygon, describe the rotations and/or reflections that map the figure onto itself.

Given two congruent figures, identify the sequence of transformations that maps one figure to another.

Apply understanding of angles, circles, perpendicular lines, parallel lines, and line segments to develop definitions for rotations, reflections, and translations.

Identify whether a figure has reflectional (line) symmetry. If so, identify the lines of symmetry and determine how many lines of symmetry the given figure has.

Identify whether a figure has rotational symmetry. If so, state the angle of rotational symmetry, and the number of times the figure can be rotated onto itself (between 0° and 360°).

Define sequences.

Sequences

Define limits.

Define a continuous function.

Limits

Define parametric equations.

Parametric Equations

Define polar coordinates and the relationship between polar coordinates and Cartesian coordinates.

Polar Coordinates

Use probability to evaluate outcomes of decisions.

Calculate expected values and use them to solve problems.

Using Probability to Make Decisions

Use the rules of probability to compute probabilities of compound events in a uniform probability model.

Conditional Probability and the Rules of Probability

Translate between the geometric description and the equation for a conic section.

Expressing Geometric Properties

Apply trigonometry to general triangles.

Similarity, Right Triangles and Trigonometry

Prove and apply trigonometric identities.

Model periodic phenomena with trigonometric functions.

Extend the domain of trigonometric functions using the unit circle.

Trigonometric Functions

Build a function that models a relationship between two quantities.

Building Functions

Analyze functions using different representations.

Interpreting Functions

Write expressions in equivalent forms to solve problems.

Seeing Structure in Expressions

Solve systems of equations.

Reasoning with Equations and Inequalities

Rewrite rational expressions.

Use polynomial identities to solve problems.

Arithmetic with Polynomials and Rational Expressions

Perform operations on matrices and use matrices in applications.

Perform operations on vectors.

Represent and model with vector quantities.

Vector and Matrix Quantities

Represent complex numbers and their operations on the complex plane.

Perform arithmetic operations with complex numbers.

The Complex Number System

High School - 4th Year Mathematics

Make interferences and justify conclusions from sample surveys, experiments and observational studies.

Understand and evaluate random processes underlying statistical experiments.

Making Inferences and Justifying Conclusions

Summarize, represent and interpret data on a single count or measurement variable.

Interpreting Categorical and Quantitative Data

Use complex numbers in polynomials identities and equations.

Perform arithmetic operations with complex numbers.

The Complex Number System

Prove and apply trigonometric identities.

Model periodic phenomena with trigonometric functions.

Extend the domain of trigonometric functions using the unit circle.

Trigonometric Functions

Construct and compare linear and exponential models and solve

Linear, Quadratic and Exponential Models

Build new functions from existing functions.

Building Functions

Analyze functions using different representations.

Interpret functions that arise in applications in terms of the context.

Interpreting Functions

Represent and solve equations and inequalities graphically.

Solve equations and inequalities in one variable.

Understand solving equations as a process of reason and explain the reasoning.

Reasoning with Equations and Inequalities

Create equations that describe numbers or relationships.

Creating Equations

Rewrite rational expressions.

Understand the relationship between zeros and factors of polynomials.

Arithmetic with Polynomials and Rational Expressions

Interpret the structure of expressions.

Seeing Structure in Expressions

High School - Algebra II

Understand independence and conditional probability and use them to interpret data.

Statistics and Probability- Conditions Probability and Rules of Probability

Applying geometric concepts in modeling situations.

Modeling with Geometry

Visualize relationships between two-dimensional and three- dimensional objects.

Explain volume and surface area formulas and use them to solve problems.

Geometric Measurement and Dimension

Use coordinates to prove simple geometric theorems algebraically.

Translate between the geometric description and the equation for a conic section.

Expressing Geometric Properties with Equations

Find arc lengths and areas of sectors of circles.

Understand and apply theorems about circles.

Circles

Define trigonometric ratios and solve problems involving right triangles.

Prove theorems involving similarity.

Understand similarity in terms of similarity transformations.

Similarity, Right Triangles and Trigonometry

Make geometric constructions.

Prove geometric theorems.

Understand congruence in terms of rigid motions.

Experiment with transformations in the plane.

Congruence

High School - Geometry

Interpret linear models.

Summarize, represent and interpret data on two categorical and quantitative variables.

Summarize, represent and interpret data on a single count or measurement variable.

Interpreting Categorical and Quantitative Data

Interpret expressions for functions in terms of the situation they model.

Construct and compare linear and exponential models and solve problems.

Linear, Quadratic and Exponential Models

Build new functions from existing functions.

Build a function that models a relationship between two quantities.

Building Functions

Analyze functions using different representations.

Interpret functions that arise in applications in terms of the context.

Understand the concept of a function and use functions notation.

Interpreting Functions

Represent and solve equations and inequalities graphically.

Solve systems of equations.

Solve equations and inequalities in one variable.

Understand solving equations as a process of reasoning.

Reasoning With Equations and Inequalities

Create equations that describe numbers or relationships.

Creating Equations

Perform arithmetic operations on polynomials.

Arithmetic with Polynomials and Rational Expressions

Write expressions in equivalent forms to solve problems.

Interpret the structure of expressions.

Seeing Structure in Expression

Reason quantitatively and use units to solve problems.

Quantities

Use properties of rational and irrational numbers.

Extend the properties of exponents to rational exponents.

The Real Number System

High School - Algebra I

Standards for Mathematical Practice

Understand that polynomials form a system closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Know and apply the Remainder Theorem.

Identify zeros of polynomials by factoring.

When suitable factorizations are available, use the zeros to construct a rough graph of the related function.

When given a graph, use the zeros to construct a possible factorization of a polynomial.

(+)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 simple rational expressions in different forms; using inspection, synthetic division, long division, box method or, for the more complicated examples, a computer algebra system.

(+)Discover 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.

Create equations and inequalities in one variable arising from situations in which linear, quadratic, and exponential functions are appropriate and use them to solve problems.

Create equations and inequalities in one variable and use them to solve problems.

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

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.

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.

Rewrite formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

Rewrite formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

Explain each step in solving an 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 linear, quadratic, polynomial, and rational inequalities in two variables algebraically and graphically.

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 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, including but not limited to 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, quadratic and exponential.

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, including but not limited to 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 a linear inequality (strict or inclusive) in two variables; graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Solve rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Rational functions are limited to those whose numerators are of degree at most 1 and denominators of degree at most 2. Radical functions are limited to square roots or cube roots of at most quadratic polynomials.

Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

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 completing the square.

Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation.

Solve quadratic equations in one variable.

Select, justify and apply appropriate methods to solve quadratic equations in one variable. Recognize complex solutions and write them as a +/- bi for real numbers a and b.

Understand the principles of the elimination method.

Solve systems of linear equations exactly and approximately by graphing, 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 a system of linear equations as a single matrix equation in a vector variable.

(+)Use matrices to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

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 in context.

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 in context.

Recognize and use the structure of an expression to identify ways to rewrite it.

Recognize and use the structure of an expression to identify ways to rewrite it.

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 write equivalent expressions for exponential functions.

(+)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.

(+)Use summation notation to describe the sums in a series.

(+)Write a function that describes a relationship between two quantities.

(+)Compose functions in context.

Write a function (linear, quadratic, and exponential) that describes a relationship between two quantities.

Determine an explicit expression, a recursive process, or steps for calculation from a context.

Determine an explicit expression from a graph.

Combine standard function types using arithmetic operations.

Write a function that describes a relationship between two quantities.

Determine an explicit expression, a recursive process, or steps for calculation from a context.

Determine an explicit expression from a graph.

Combine standard function types using arithmetic operations.

Compose functions.

Write arithmetic and geometric sequences both recursively and with an explicit formula and use them to model situations.

Identify the effect on the graph of f(x) (linear, exponential, quadratic) replaced with 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 contrasting cases and illustrate an explanation of the effects on the graph using technology.

Identify the effect on the graph of f(x) replaced with 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 contrasting cases and illustrate an explanation of the effects on the graph using technology.

(+)Find inverse functions.

(+)Verify by composition that one function is the inverse of another.

(+)Read values of an inverse function from a graph or a table, given that the function has an inverse.

(+)Produce an invertible function from a non-invertible function by restricting the domain.

Find inverse functions.

Solve an equation for the independent variable of a function f that has an inverse function and write an expression for the inverse.

Verify by composition that one function is the inverse of another.

Read values of an inverse function from a graph or a table, given that the function has an inverse.

(+)Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

(+)Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

(+)Use reciprocal properties to develop definitions for cotangent, cosecant, and secant.

Understand that a function maps each element of the domain to 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 off is the graph of the equation y = f(x).

Use function notation, evaluate functions, 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 functions, including linear, quadratic, and exponential, that model 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.

For functions that model 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 (including even, odd, or neither); end behavior; and periodicity.

Relate the domain of a function to its graph and find an appropriate domain in the context of the problem.

Relate the domain of a function to its graph and find an appropriate domain in the context of the problem.

Calculate and interpret the average rate of change of a function, both symbolically and from a table over a specified interval. Estimate the rate of change from a graph.

(+)Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

(+)Graph rational functions, identify zeros and vertical, horizontal, and slant asymptotes, and determine end behavior.

(+)Graph exponential and logarithmic functions, showing relationships, intercepts and end behavior.

(+)Graph all trigonometric functions, showing key features and applying transformations.

Graph parent functions and their transformations expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

Graph linear, exponential, and quadratic functions and show intercepts, maxima, and minima.

Graph parent functions and their transformations expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available and showing end behavior.

Graph logarithmic functions, showing intercepts and end behavior.

Graph trigonometric functions (sine and cosine), 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 graphing, 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.

Interpret expressions for exponential growth and decay.

Compare properties of two functions (linear, quadratic and exponential) each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

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 input-output pairs (include reading these from a table).

Recognize, using graphs and tables, that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically.

For exponential models, 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.

Interpret the parameters in a linear or exponential function in terms of a context.

Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

(+)Use fundamental trigonometric identities.

(+)Verify trigonometric identities

(+)Evaluate trigonometric functions

(+)Write equivalent trigonometric expressions

(+)Solve trigonometric equations.

Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions (sine and cosine) to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

(+)Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π-x, π+x, and 2π–x in terms of their values for x, where x is any real number.

(+)Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Choose trigonometric functions (sine and cosine) to model periodic phenomena with specified amplitude, frequency, and midline.

(+)Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

(+)Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.

Prove the Pythagorean identity sin²(A) + cos²(A) = 1 and use it to calculate trigonometric ratios.

(+)Prove the addition and subtraction, half-angle, and double-angle formulas for sine, cosine, and tangent and use them to solve problems.

Prove that all circles are similar.

Identify and describe relationships among central angles, inscribed angles, circumscribed angles, radii, and chords.

Construct, using a compass and straight edge, the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

Construct a tangent line from a point outside a given circle to the circle.

Derive using similarity the length of the arc intercepted by an angle is proportional to the radius.

Define the radian measure of the angle as the constant of proportionality;

Derive and apply the formula for the area of a sector.

State and apply precise definitions of angle, circle, perpendicular, parallel, ray, line segment, and distance based on the undefined notions of point, line, and plane.

Prove congruence theorems about triangles. Theorems must include but not limited to: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the mid segment 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 must include but not limited to: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Perform geometric constructions with a compass and straightedge. including copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines/segments, 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.

Represent transformations in the plane. (e.g., using transparencies and/or 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 dilation).

Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and/or reflections that map the figure 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, (e.g., using graph paper, tracing paper, or geometry software). Specify a sequence of transformations that will map a given figure onto another.

Use geometric descriptions of rigid motions to transform figures.

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 theorems about lines and angles. Theorems must include but not limited to: vertical angles are congruent; when a transversal intersects parallel lines, alternate interior angles are congruent and same side interior angles are supplementary (using corresponding angles postulate); points on a perpendicular bisector of a line segment are equidistant from the segment's endpoints.

Give an informal argument for the formulas for the volume of a cylinder, pyramid, sphere, and cone. Use dissection arguments, and informal limit arguments.

Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures

Know and apply volume and surface area formulas for cylinders, pyramids, cones, and spheres for composite figures to solve problems.

Identify two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

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.

Use coordinates to prove geometric relationships algebraically. For example, determine whether a figure defined by four given points in the coordinate plane is a rectangle; determine whether the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

Define and use the slope criteria for parallel and perpendicular lines. (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

Find the point on a directed line segment between two given points that partitions the segment in a given ratio. e.g. Determine the point(s) that divide the segment with endpoints of (-4, 7) and (6, 3) into the ratio 2:3

Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

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

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

Apply geometric concepts 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).

Verify experimentally and apply the properties of dilations as determined by a center and a scale factor.

Determine whether figures are similar, using the definition of similarity and using similarity transformations.

Use the properties of similarity transformations to establish similarity theorems. Theorems must include AA, SAS, and SSS.

Prove theorems about triangles involving similarity. Theorems must include but not limited to: a line parallel to one side of a triangle divides the other two proportionally, and its converse; 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, using similarity, that side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios (sine, cosine, and tangent) 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.

(+)Analyze conic sections using equations and graphs.

(+)Given a quadratic equation of the form Ax² +Bxy+Cy² + Dx + Ey + F =0 (where B = 0), determine whether the graph is a circle, parabola, ellipse, or hyperbola

(+)Use the process of completing the square to put the equation in standard form

(+)When given a graph, be able to write the equation of the conic section, and vice versa.

(+)Use permutations and combinations to compute probabilities of compound events and solve problems.

(+)Prove the Laws of Sines and Cosines and use them to solve problems involving right and non-right triangles.

(+)Derive the formula A = ½ ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side and use the formula to solve problems.

Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Model with mathematics.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.

Know there is a complex number i such that i² = –1, and every complex number has the form a + bi where a and b are real numbers.

Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

(+)Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

(+)Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.

(+)Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.

(+)Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.

Solve quadratic equations with real coefficients that have complex solutions.

(+)Extend polynomial identities to the complex numbers. For example, rewrite x² + 4 as (x + 2i)(x - 2i).

(+)Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

Use unit analysis to understand and guide the process of solving multi-step problems; choose and interpret units consistently in formulas; and choose and interpret the scale and origin in graphs and data displays.

Define appropriate quantities for the purpose of descriptive modeling.

Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Explain how the definition of rational exponents follows from extending the properties of integer exponents, allowing for a notation for radicals in terms of rational exponents.

Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Explain why the sum or product of two rational numbers is rational; the sum of a rational and an irrational number is irrational; and the product of a nonzero rational and an irrational number is irrational.

(+)Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes

(+)Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. Discover that the determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

(+)Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.

(+)Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

(+)Write a vector in component form.

(+)Solve problems involving velocity and other quantities that can be represented by vectors.

(+)Add and subtract vectors.

(+)Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.

(+)Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

(+)Understand vector subtraction v – w as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.

(+)Multiply a vector by a scalar.

(+)Represent scalar multiplication graphically by scaling vectors and/or reversing their direction; perform scalar multiplication component-wise.

(+)Compute the magnitude of a scalar multiple cv. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).

(+)Use matrices to represent and manipulate data.

(+)Multiply matrices by scalars to produce new matrices.

(+)Add, subtract, and multiply matrices of appropriate dimensions.

(+)Understand that, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

(+)Determine if a function is continuous at a point. Find the types of discontinuities of a function and relate them to finding limits of a function. Use the concept of limits to describe discontinuity and end-behavior of the function.

(+)Demonstrate knowledge of both the definition and graphical interpretation of limits of values of functions and sequences. Verify and estimate limits using graphs, tables, and technology.

(+)Evaluate limits of functions and apply properties of limits, including one-sided limits and limits at infinity using algebra.

(+)Define polar coordinates and the relationship between polar coordinates and Cartesian coordinates with and without the use of technology.

(+)Use polar equations to model and solve problems using graphs and algebraic properties.

(+)Given equations for a parametric function, plot the graph and make conclusions about the geometric figure that result.

(+)Convert between a pair of parametric equations and an equation in x and y. Model and solve problems using parametric equations.

(+)Define arithmetic and geometric sequences and series. Model and solve word problems involving applications of sequences and series, interpret the solutions and determine whether the solutions are reasonable.

Describe events as subsets of a sample space or as unions, intersections, or complements of other events.

Determine whether two events A and B are independent.

Determine conditional probabilities and interpret independence by analyzing conditional probability.

Construct and interpret two-way frequency tables of data. Use the two-way 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 situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.

Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the result.

Apply the Addition Rule, P(A or B), and interpret the result.

Apply the general Multiplication Rule, P(A and B), and interpret the result.

Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

Determine whether a specified model is consistent with results from a given data-generating process.

Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

Use data from a randomized experiment to compare two treatment groups; use simulations to decide if differences between parameters are significant.

Evaluate reports based on data.

Represent data with plots on the real number line (dot plots, histograms, and box plots).

Use statistics appropriate to the shape and context 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).

Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

Summarize categorical data for two categories in two-way 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.

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.

Determine the function (linear, quadratic, or exponential model) that best fits a set of data and use that function fitted to data to solve problems within context.

Informally and using technology assess the fit of a function by plotting and analyzing residuals.

Fit a linear function for a scatter plot that suggests a linear association.

(+)Assign a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.

(+)Calculate the expected value of a random variable; understand that it is the mean of the probability distribution.

(+)Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; calculate the expected value.

(+)Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; calculate the expected value.

(+)Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding calculating the expected values.

(+)Calculate the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.

(+)Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.

(+)Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

(+)Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

Patterns and Sums

Sequences & Series

Properties, Equations and Graphs

Conic Sections

Matrices and Systems

Vector Operations and Applications

Vectors & Matrices

Coordinates and Equations

Polar & Parametric

Unit Circle and Applications

Graphs and Key Features

Trigonometric Functions and Equations

Trigonometry

Operations and Graphing

Functions & Relationships

Identify and describe the properties of conic sections (parabolas, ellipses, circles, and hyperbolas).

Write equations and graph conic sections given specific properties, features, or transformations.

Add, subtract, multiply, and divide polynomial and rational functions.

Use the properties of exponents to find equivalent expressions and to solve equations.

Use properties of logarithms to simplify and evaluate logarithmic expressions, and to solve equations.

Find the composition of two functions and determine the domain of the composite function

Determine if a function has an inverse and find the inverse of a function algebraically and graphically, restricting the domain if necessary.

Analyze functions and identify key features including domain, range, intercepts, symmetry, asymptotes, and end behavior.

Apply transformations (translations, reflections, and dilations) to functions and graphs of polynomial, rational, exponential, and logarithmic functions.

Model data sets with exponential and logarithmic regression.

Choose an appropriate regression model and use it to make predictions.

Represent points and equations in both rectangular and polar coordinate systems and convert between them.

Represent and graph parametric equations and eliminate the parameter to convert to rectangular form.

Represent complex numbers in rectangular and polar form and multiply, divide, and evaluate powers (De Moivre's Theorem).

Find the modulus of a complex number, and the distance between two complex numbers

Define and use arithmetic and geometric sequences and series to model real-world situations.

Find partial sums of arithmetic and geometric series and understand the concept of limits as they relate to infinite series.

Know and apply the Binomial Theorem.

Use trigonometric functions to model and solve real-world problems

Develop the Pythagorean and Quotient Identities, and use them to simplify expressions and solve equations

Use the addition and subtraction, half-angle, and double-angle formulas for sine, cosine, and tangent to simplify expressions and solve equations.

Apply the Law of Sines (including the ambiguous case) and the Law of Cosines to find unknown sides and angles in any triangle.

Explain how changes in amplitude, period, phase shift, and midline affect the graph of sine, cosine, and tangent functions, using transformations and real-world contexts to support understanding.

Explain how changes in amplitude, period, phase shift, and midline affect the graph of secant, cosecant, and cotangent functions, using transformations.

Explain the relationship between the radian measure of an angle and the length of the arc it subtends on a unit circle.

Explain how the unit circle can be used to model sine, cosine, tangent, secant, cosecant, and cotangent for all real numbers.

Calculate arc length, area of a sector, linear speed, and angular speed in real-world contexts.

Represent vectors in component form and calculate magnitude and direction.

Perform vector addition, subtraction, and scalar multiplication both visually and component-wise.

Find the dot product of, and angle between, two vectors.

Use matrices to represent and solve systems of equations, in two and three variables, with and without technology

Perform matrix multiplication and use matrices to represent transformations in the plane.

Find the determinant and inverse of a matrix, with and without technology.