Gizmos for Colorado - Mathematics: High School (2020 Academic Standards adopted 2018)

This correlation lists the recommended Gizmos for this state's curriculum standards. Click any Gizmo title below for more information.

1: : Number and Quantity

MA.HS.N-RN.A: : The Real Number System: Extend the properties of exponents to rational exponents.

MA.HS.N-RN.A.2: : Students can: Rewrite expressions involving radicals and rational exponents using the properties of exponents.

MA.HS.N-Q.A: : Quantities: Reason quantitatively and use units to solve problems.

MA.HS.N-Q.A.1: : Students can: Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

MA.HS.N-Q.A.3: : Students can: Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

MA.HS.N-CN.A: : The Complex Number System: Perform arithmetic operations with complex numbers.

MA.HS.N-CN.A.1: : Students can: Define complex number i such that i² = –1, and show that every complex number has the form a + bi where a and b are real numbers.

MA.HS.N-CN.A.2: : Students can: Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

MA.HS.N-CN.A.3: : Students can: Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

MA.HS.N-CN.B: : The Complex Number System: Represent complex numbers and their operations on the complex plane.

MA.HS.N-CN.B.4: : Students can: 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.

MA.HS.N-CN.B.5: : Students can: Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.

MA.HS.N-CN.B.6: : Students can: 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.

MA.HS.N-CN.C: : The Complex Number System: Use complex numbers in polynomial identities and equations.

MA.HS.N-CN.C.7: : Students can: Solve quadratic equations with real coefficients that have complex solutions.

MA.HS.N-CN.C.8: : Students can: Extend polynomial identities to the complex numbers.

MA.HS.N-VM.A: : Vector & Matrix Quantities: Represent and model with vector quantities.

MA.HS.N-VM.A.1: : Students can: 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 (e.g., v, |v|, ||v||, v).

MA.HS.N-VM.A.2: : Students can: Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

MA.HS.N-VM.A.3: : Students can: Solve problems involving velocity and other quantities that can be represented by vectors.

MA.HS.N-VM.B: : Vector & Matrix Quantities: Perform operations on vectors.

MA.HS.N-VM.B.4: : Students can: Add and subtract vectors.

MA.HS.N-VM.B.4.a: : 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.

MA.HS.N-VM.C: : Vector & Matrix Quantities: Perform operations on matrices and use matrices in applications.

MA.HS.N-VM.C.7: : Students can: Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

MA.HS.N-VM.C.8: : Students can: Add, subtract, and multiply matrices of appropriate dimensions.

MA.HS.N-VM.C.10: : Students can: 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. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

MA.HS.N-VM.C.12: : Students can: Work with 2 × 2 matrices as transformations of the plane and interpret the absolute value of the determinant in terms of area.

2: : Algebra and Functions

MA.HS.A-SSE.A: : Seeing Structure in Expressions: Interpret the structure of expressions.

MA.HS.A-SSE.A.1: : Students can: Interpret expressions that represent a quantity in terms of its context.

MA.HS.A-SSE.A.1.a: : Interpret parts of an expression, such as terms, factors, and coefficients.

MA.HS.A-SSE.A.1.b: : Interpret complicated expressions by viewing one or more of their parts as a single entity.

MA.HS.A-SSE.A.2: : Students can: Use the structure of an expression to identify ways to rewrite it.

MA.HS.A-SSE.B: : Seeing Structure in Expressions: Write expressions in equivalent forms to solve problems.

MA.HS.A-SSE.B.3: : Students can: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

MA.HS.A-SSE.B.3.a: : Factor a quadratic expression to reveal the zeros of the function it defines.

MA.HS.A-SSE.B.3.b: : Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

MA.HS.A-APR.A: : Arithmetic with Polynomials & Rational Expressions: Perform arithmetic operations on polynomials.

MA.HS.A-APR.A.1: : Students can: Explain that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

MA.HS.A-APR.B: : Arithmetic with Polynomials & Rational Expressions: Understand the relationship between zeros and factors of polynomials.

MA.HS.A-APR.B.2: : Students can: Know and apply the Remainder Theorem. For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x). (Students need not apply the Remainder Theorem to polynomials of degree greater than 4.)

MA.HS.A-APR.B.3: : Students can: Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

MA.HS.A-APR.C: : Arithmetic with Polynomials & Rational Expressions: Use polynomial identities to solve problems.

MA.HS.A-APR.C.5: : Students can: Know and apply the Binomial Theorem for the expansion of 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. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.)

MA.HS.A-CED.A: : Creating Equations: Create equations that describe numbers or relationships.

MA.HS.A-CED.A.1: : Students can: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

MA.HS.A-CED.A.2: : Students can: Create equations in two or more variables to represent relationships between quantities and graph equations on coordinate axes with labels and scales.

MA.HS.A-CED.A.3: : Students can: 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.

MA.HS.A-CED.A.4: : Students can: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

MA.HS.A-REI.A: : Reasoning with Equations & Inequalities: Understand solving equations as a process of reasoning and explain the reasoning.

MA.HS.A-REI.A.1: : Students can: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

MA.HS.A-REI.A.2: : Students can: Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

MA.HS.A-REI.B: : Reasoning with Equations & Inequalities: Solve equations and inequalities in one variable.

MA.HS.A-REI.B.3: : Students can: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

MA.HS.A-REI.B.4: : Students can: Solve quadratic equations in one variable.

MA.HS.A-REI.C: : Reasoning with Equations & Inequalities: Solve systems of equations.

MA.HS.A-REI.C.5: : Students can: Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

MA.HS.A-REI.C.6: : Students can: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

MA.HS.A-REI.C.8: : Students can: Represent a system of linear equations as a single matrix equation in a vector variable.

MA.HS.A-REI.C.9: : Students can: Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

MA.HS.A-REI.D: : Reasoning with Equations & Inequalities: Represent and solve equations and inequalities graphically.

MA.HS.A-REI.D.10: : Students can: Explain 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).

MA.HS.A-REI.D.11: : Students can: Explain why the x-coordinates of the points where the graphs of the equations y ϟ(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

MA.HS.A-REI.D.12: : Students can: Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

MA.HS.F-IF.A: : Interpreting Functions: Understand the concept of a function and use function notation.

MA.HS.F-IF.A.1: : Students can: Explain that a function is a correspondence from one set (called the domain) to another set (called the range) that assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

MA.HS.F-IF.A.2: : Students can: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

MA.HS.F-IF.A.3: : Students can: Demonstrate that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

MA.HS.F-IF.B: : Interpreting Functions: Interpret functions that arise in applications in terms of the context.

MA.HS.F-IF.B.4: : Students can: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

MA.HS.F-IF.B.5: : Students can: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

MA.HS.F-IF.B.6: : Students can: Calculate and interpret the average rate of change presented symbolically or as a table, of a function over a specified interval. Estimate the rate of change from a graph.

MA.HS.F-IF.C: : Interpreting Functions: Analyze functions using different representations.

MA.HS.F-IF.C.7: : Students can: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

MA.HS.F-IF.C.7.a: : Graph linear and quadratic functions and show intercepts, maxima, and minima.

MA.HS.F-IF.C.7.b: : Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

MA.HS.F-IF.C.7.c: : Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

MA.HS.F-IF.C.7.d: : Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

MA.HS.F-IF.C.7.e: : Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

MA.HS.F-IF.C.8: : Students can: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

MA.HS.F-IF.C.8.a: : Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

MA.HS.F-IF.C.8.b: : Use the properties of exponents to interpret expressions for exponential functions.

MA.HS.F-BF.A: : Building Functions: Build a function that models a relationship between two quantities.

MA.HS.F-BF.A.1: : Students can: Write a function that describes a relationship between two quantities.

MA.HS.F-BF.A.1.a: : Determine an explicit expression, a recursive process, or steps for calculation from a context.

MA.HS.F-BF.A.1.b: : Combine standard function types using arithmetic operations.

MA.HS.F-BF.A.2: : Students can: Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

MA.HS.F-BF.B: : Building Functions: Build new functions from existing functions.

MA.HS.F-BF.B.3: : Students can: Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k both positive and negative; find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

MA.HS.F-BF.B.4: : Students can: Find inverse functions.

MA.HS.F-BF.B.4.a: : Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.

MA.HS.F-BF.B.4.c: : Read values of an inverse function from a graph or table, given that the function has an inverse.

MA.HS.F-BF.B.5: : Students can: Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

MA.HS.F-LE.A: : Linear, Quadratic & Exponential Models: Construct and compare linear, quadratic, and exponential models and solve problems.

MA.HS.F-LE.A.1: : Students can: Distinguish between situations that can be modeled with linear functions and with exponential functions.

MA.HS.F-LE.A.1.a: : Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

MA.HS.F-LE.A.1.b: : Identify situations in which one quantity changes at a constant rate per unit interval relative to another.

MA.HS.F-LE.A.1.c: : Identify situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

MA.HS.F-LE.A.2: : Students can: 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).

MA.HS.F-LE.A.3: : Students can: Use graphs and tables to describe that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

MA.HS.F-LE.A.4: : Students can: For exponential models, express as a logarithm the solution to ab to the ct power = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

MA.HS.F-LE.B: : Linear, Quadratic, & Exponential Models: Interpret expressions for functions in terms of the situation they model.

MA.HS.F-LE.B.5: : Students can: Interpret the parameters in a linear or exponential function in terms of a context.

MA.HS.F-TF.A: : Trigonometric Functions: Extend the domain of trigonometric functions using the unit circle.

MA.HS.F-TF.A.1: : Students can: Use radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

MA.HS.F-TF.A.2: : Students can: Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

MA.HS.F-TF.A.3: : Students can: Use special triangles to determine geometrically the values to sine, cosine, tangent for pi/3, pi/4, and pi/6 and use the unit circle to express the values sine, cosine, and tangent for x, pi + x, and 2pi - x and in terms of their values for x where x is any real number.

MA.HS.F-TF.A.4: : Students can: Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

MA.HS.F-TF.B: : Trigonometric Functions: Model periodic phenomena with trigonometric functions.

MA.HS.F-TF.B.5: : Students can: Model periodic phenomena with trigonometric functions with specified amplitude, frequency, and midline.

MA.HS.F-TF.C: : Trigonometric Functions: Prove and apply trigonometric identities.

MA.HS.F-TF.C.8: : Students can: Prove the Pythagorean identity sin² (theta) + cos² (theta) = 1 and use it to find sin(theta), cos(theta), or tan(theta) given sin(theta), cos(theta), or tan(theta) and the quadrant of the angle.

MA.HS.F-TF.C.9: : Students can: Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

3: : Data, Statistics, and Probability

MA.HS.S-ID.A: : Interpreting Categorical & Quantitative Data: Summarize, represent, and interpret data on a single count or measurement variable.

MA.HS.S-ID.A.1: : Students can: Model data in context with plots on the real number line (dot plots, histograms, and box plots).

MA.HS.S-ID.A.2: : Students can: Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

MA.HS.S-ID.A.3: : Students can: Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

MA.HS.S-ID.A.4: : Students can: Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages and identify data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

MA.HS.S-ID.B: : Interpreting Categorical & Quantitative Data: Summarize, represent, and interpret data on two categorical and quantitative variables.

MA.HS.S-ID.B.6: : Students can: Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

MA.HS.S-ID.B.6.a: : Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.

MA.HS.S-ID.B.6.b: : Informally assess the fit of a function by plotting and analyzing residuals.

MA.HS.S-ID.B.6.c: : Fit a linear function for a scatter plot that suggests a linear association.

MA.HS.S-ID.B.7: : Students can: Distinguish between correlation and causation.

MA.HS.S-ID.C: : Interpreting Categorical & Quantitative Data: Interpret linear models.

MA.HS.S-ID.C.7: : Students can: Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

MA.HS.S-ID.C.8: : Students can: Using technology, compute and interpret the correlation coefficient of a linear fit.

MA.HS.S-IC.A: : Making Inferences & Justifying Conclusions: Understand and evaluate random processes underlying statistical experiments.

MA.HS.S-IC.A.1: : Students can: Describe statistics as a process for making inferences about population parameters based on a random sample from that population.

MA.HS.S-IC.A.2: : Students can: Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.

MA.HS.S-IC.B: : Making Inferences & Justifying Conclusions: Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

MA.HS.S-IC.B.3: : Students can: Identify the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

MA.HS.S-IC.B.4: : Students can: 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.

MA.HS.S-IC.B.5: : Students can: Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

MA.HS.S-CP.A: : Conditional Probability & the Rules of Probability: Understand independence and conditional probability and use them to interpret data.

MA.HS.S-CP.A.2: : Students can: Explain that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

MA.HS.S-CP.A.3: : Students can: Using the conditional probability of A given B as P(A and B)/P(B), interpret the independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

MA.HS.S-CP.A.5: : Students can: Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

MA.HS.S-CP.B: : Conditional Probability & the Rules of Probability: Use the rules of probability to compute probabilities of compound events in a uniform probability model.

MA.HS.S-CP.B.6: : Students can: Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.

MA.HS.S-CP.B.8: : Students can: Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A) P(B | A) = P(B) P(A | B), and interpret the answer in terms of the model.

MA.HS.S-CP.B.9: : Students can: Use permutations and combinations to compute probabilities of compound events and solve problems.

MA.HS.S-MD.A: : Using Probability to Make Decisions: Calculate expected values and use them to solve problems.

MA.HS.S-MD.A.1: : Students can: Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.

MA.HS.S-MD.A.2: : Students can: Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

MA.HS.S-MD.A.3: : Students can: Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.

MA.HS.S-MD.A.4: : Students can: Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.

MA.HS.S-MD.B: : Using Probability to Make Decisions: Use probability to evaluate outcomes of decisions.

MA.HS.S-MD.B.5: : Students can: Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

MA.HS.S-MD.B.5.a: : Find the expected payoff for a game of chance.

MA.HS.S-MD.B.5.b: : Evaluate and compare strategies on the basis of expected values.

MA.HS.S-MD.B.6: : Students can: Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

MA.HS.S-MD.B.7: : Students can: Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

4: : Geometry

MA.HS.G-CO.A: : Congruence: Experiment with transformations in the plane.

MA.HS.G-CO.A.1: : Students can: State precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

MA.HS.G-CO.A.2: : Students can: Represent transformations in the plane using e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

MA.HS.G-CO.A.3: : Students can: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

MA.HS.G-CO.A.4: : Students can: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

MA.HS.G-CO.A.5: : Students can: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using appropriate tools (e.g., graph paper, tracing paper, or geometry software). Specify a sequence of transformations that will carry a given figure onto another.

MA.HS.G-CO.B: : Congruence: Understand congruence in terms of rigid motions.

MA.HS.G-CO.B.6: : Students can: Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

MA.HS.G-CO.C: : Congruence: Prove geometric theorems.

MA.HS.G-CO.C.9: : Students can: Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

MA.HS.G-CO.C.10: : Students can: Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

MA.HS.G-CO.C.11: : Students can: Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

MA.HS.G-CO.D: : Congruence: Make geometric constructions.

MA.HS.G-CO.D.12: : Students can: Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

MA.HS.G-SRT.A: : Similarity, Right Triangles, and Trigonometry: Understand similarity in terms of similarity transformations.

MA.HS.G-SRT.A.1: : Students can: Verify experimentally the properties of dilations given by a center and a scale factor.

MA.HS.G-SRT.A.1.a: : Show that a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

MA.HS.G-SRT.A.1.b: : Show that the dilation of a line segment is longer or shorter in the ratio given by the scale factor.

MA.HS.G-SRT.A.2: : Students can: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

MA.HS.G-SRT.A.3: : Students can: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

MA.HS.G-SRT.B: : Similarity, Right Triangles, and Trigonometry: Prove theorems involving similarity.

MA.HS.G-SRT.B.4: : Students can: Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

MA.HS.G-SRT.B.5: : Students can: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

MA.HS.G-SRT.C: : Similarity, Right Triangles, and Trigonometry: Define trigonometric ratios and solve problems involving right triangles.

MA.HS.G-SRT.C.6: : Students can: Explain that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

MA.HS.G-SRT.C.7: : Students can: Explain and use the relationship between the sine and cosine of complementary angles.

MA.HS.G-SRT.C.8: : Students can: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

MA.HS.G-C.A: : Circles: Understand and apply theorems about circles.

MA.HS.G-C.A.2: : Students can: Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

MA.HS.G-C.A.3: : Students can: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

MA.HS.G-C.B: : Circles: Find arc lengths and areas of sectors of circles.

MA.HS.G-C.B.5: : Students can: Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

MA.HS.G-GPE.A: : Expressing Geometric Properties with Equations: Translate between the geometric description and the equation for a conic section.

MA.HS.G-GPE.A.1: : Students can: 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.

MA.HS.G-GPE.A.2: : Students can: Derive the equation of a parabola given a focus and directrix.

MA.HS.G-GPE.A.3: : Students can: Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

MA.HS.G-GPE.B: : Expressing Geometric Properties with Equations: Use coordinates to prove simple geometric theorems algebraically.

MA.HS.G-GPE.B.4: : Students can: Use coordinates to prove simple geometric theorems algebraically.

MA.HS.G-GPE.B.7: : Students can: Use coordinates and the distance formula to compute perimeters of polygons and areas of triangles and rectangles.

MA.HS.G-GMD.A: : Geometric Measurement and Dimension: Explain volume formulas and use them to solve problems.

MA.HS.G-GMD.A.1: : Students can: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

MA.HS.G-GMD.A.2: : Students can: Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

MA.HS.G-GMD.A.3: : Students can: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

MA.HS.G-GMD.B: : Geometric Measurement and Dimension: Visualize relationships between two-dimensional and three-dimensional objects.

MA.HS.G-GMD.B.4: : Students can: Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Correlation last revised: 6/6/2022

Colorado - Mathematics: High School

2020 Academic Standards | Adopted: 2018

1: : Number and Quantity

2: : Algebra and Functions

3: : Data, Statistics, and Probability

4: : Geometry