Gizmos for Colorado - Mathematics: High School (Academic Standards adopted 2010)

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

1: : Number Sense, Properties, and Operations

1.1: : Understand the structure and properties of our number system. At their most basic level numbers are abstract symbols that represent real-world quantities

1.1: : The complex number system includes real numbers and imaginary numbers

1.1.a: : Students can: Extend the properties of exponents to rational exponents.

1.1.a.i: : Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.

1.1.c: : Students can: Perform arithmetic operations with complex numbers.

1.1.c.i: : Define the 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.

1.1.c.ii: : Use the relation ??² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

1.1.d: : Students can: Use complex numbers in polynomial identities and equations.

1.1.d.i: : Solve quadratic equations with real coefficients that have complex solutions.

2: : Patterns, Functions, and Algebraic Structures

2.1: : Make sound predictions and generalizations based on patterns and relationships that arise from numbers, shapes, symbols, and data

2.1: : Functions model situations where one quantity determines another and can be represented algebraically, graphically, and using tables

2.1.a: : Students can: Formulate the concept of a function and use function notation.

2.1.a.i: : 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.

2.1.a.ii: : Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

2.1.a.iii: : Demonstrate that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

2.1.b: : Students can: Interpret functions that arise in applications in terms of the context

2.1.b.i: : For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.

2.1.b.ii: : Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

2.1.b.iii: : Calculate and interpret the average rate of change of a function over a specified interval. Estimate the rate of change from a graph.

2.1.c: : Students can: Analyze functions using different representations

2.1.c.i: : Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

2.1.c.ii: : Graph linear and quadratic functions and show intercepts, maxima, and minima.

2.1.c.iii: : Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

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

2.1.c.v: : Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

2.1.c.vi: : Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

2.1.c.vi.1: : 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.

2.1.c.vi.2: : Use the properties of exponents to interpret expressions for exponential functions.

2.1.c.vi.3: : Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

2.1.d: : Students can: Build a function that models a relationship between two quantities

2.1.d.i: : Write a function that describes a relationship between two quantities.

2.1.d.i.1: : Determine an explicit expression, a recursive process, or steps for calculation from a context.

2.1.d.i.2: : Combine standard function types using arithmetic operations.

2.1.d.ii: : Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

2.1.e: : Students can: Build new functions from existing functions

2.1.e.i: : Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k, and find the value of k given the graphs.

2.1.e.ii: : Experiment with cases and illustrate an explanation of the effects on the graph using technology.

2.1.e.iii: : Find inverse functions.

2.1.f: : Students can: Extend the domain of trigonometric functions using the unit circle

2.1.f.i: : Use radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

2.1.f.ii: : 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.

2.2: : Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present and defend solutions

2.2: : Quantitative relationships in the real world can be modeled and solved using functions

2.2.a: : Students can: Construct and compare linear, quadratic, and exponential models and solve problems

2.2.a.i: : Distinguish between situations that can be modeled with linear functions and with exponential functions.

2.2.a.i.1: : Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

2.2.a.i.2: : Identify situations in which one quantity changes at a constant rate per unit interval relative to another.

2.2.a.i.3: : Identify situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

2.2.a.ii: : Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs.

2.2.a.iii: : 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.

2.2.a.iv: : 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.

2.2.b: : Students can: Interpret expressions for functions in terms of the situation they model

2.2.b.i: : Interpret the parameters in a linear or exponential function in terms of a context.

2.2.c: : Students can: Model periodic phenomena with trigonometric functions.

2.2.c.i: : Choose the trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

2.3: : Understand that equivalence is a foundation of mathematics represented in numbers, shapes, measures, expressions, and equations

2.3: : Expressions can be represented in multiple, equivalent forms

2.3.a: : Students can: Interpret the structure of expressions

2.3.a.i: : Interpret expressions that represent a quantity in terms of its context.

2.3.a.i.1: : Interpret parts of an expression, such as terms, factors, and coefficients.

2.3.a.i.2: : Interpret complicated expressions by viewing one or more of their parts as a single entity.

2.3.a.ii: : Use the structure of an expression to identify ways to rewrite it.

2.3.b: : Students can: Write expressions in equivalent forms to solve problems

2.3.b.i: : Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

2.3.b.i.1: : Factor a quadratic expression to reveal the zeros of the function it defines.

2.3.b.i.2: : Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

2.3.b.i.3: : Use the properties of exponents to transform expressions for exponential functions.

2.3.c: : Students can: Perform arithmetic operations on polynomials

2.3.c.i: : 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.

2.3.d: : Students can: Understand the relationship between zeros and factors of polynomials

2.3.d.i: : State and apply the Remainder Theorem.

2.3.d.ii: : 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.

2.3.e: : Students can: Use polynomial identities to solve problems

2.3.e.i: : Prove polynomial identities and use them to describe numerical relationships.

2.4: : Are fluent with basic numerical and symbolic facts and algorithms, and are able to select and use appropriate (mental math, paper and pencil, and technology) methods based on an understanding of their efficiency, precision, and transparency

2.4: : Solutions to equations, inequalities and systems of equations are found using a variety of tools

2.4.a: : Students can: Create equations that describe numbers or relationships

2.4.a.i: : Create equations and inequalities in one variable and use them to solve problems.

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

2.4.a.iii: : Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.

2.4.a.iv: : Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

2.4.b: : Students can: Understand solving equations as a process of reasoning and explain the reasoning

2.4.b.i: : 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.

2.4.b.ii: : Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

2.4.c: : Students can: Solve equations and inequalities in one variable

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

2.4.c.ii: : Solve quadratic equations in one variable.

2.4.c.ii.1: : 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.

2.4.c.ii.2: : Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation.

2.4.c.ii.3: : Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

2.4.d: : Students can: Solve systems of equations

2.4.d.i: : 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.

2.4.d.ii: : Solve systems of linear equations exactly and approximately, focusing on pairs of linear equations in two variables.

2.4.e: : Students can: Represent and solve equations and inequalities graphically

2.4.e.i: : 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.

2.4.e.ii: : 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.

2.4.e.iii: : 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.

3: : Data Analysis, Statistics, and Probability

3.1: : Solve problems and make decisions that depend on understanding, explaining, and quantifying the variability in data

3.1: : Visual displays and summary statistics condense the information in data sets into usable knowledge

3.1.a: : Students can: Summarize, represent, and interpret data on a single count or measurement variable

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

3.1.a.ii: : 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.

3.1.a.iii: : Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

3.1.a.iv: : 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.

3.1.a.v: : Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

3.1.b: : Students can: Summarize, represent, and interpret data on two categorical and quantitative variables

3.1.b.i: : 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.

3.1.b.ii: : Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

3.1.b.ii.1: : 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.

3.1.b.ii.2: : Informally assess the fit of a function by plotting and analyzing residuals.

3.1.b.ii.3: : Fit a linear function for a scatter plot that suggests a linear association.

3.1.c: : Students can: Interpret linear models

3.1.c.i: : Interpret the slope and the intercept of a linear model in the context of the data.

3.1.c.ii: : Using technology, compute and interpret the correlation coefficient of a linear fit.

3.1.c.iii: : Distinguish between correlation and causation.

3.2: : Communicate effective logical arguments using mathematical justification and proof. Mathematical argumentation involves making and testing conjectures, drawing valid conclusions, and justifying thinking

3.2: : Statistical methods take variability into account supporting informed decisions making through quantitative studies designed to answer specific questions

3.2.a: : Students can: Understand and evaluate random processes underlying statistical experiments

3.2.a.i: : Describe statistics as a process for making inferences about population parameters based on a random sample from that population.

3.2.a.ii: : Decide if a specified model is consistent with results from a given data-generating process.

3.2.b: : Students can: Make inferences and justify conclusions from sample surveys, experiments, and observational studies

3.2.b.i: : Identify the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

3.2.b.ii: : Use data from a sample survey to estimate a population mean or proportion.

3.2.b.iii: : Develop a margin of error through the use of simulation models for random sampling.

3.2.b.iv: : Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

3.2.b.v: : Define and explain the meaning of significance, both statistical (using p-values) and practical (using effect size).

3.2.b.vi: : Evaluate reports based on data.

3.3: : Recognize and make sense of the many ways that variability, chance, and randomness appear in a variety of contexts

3.3: : Probability models outcomes for situations in which there is inherent randomness

3.3.a: : Students can: Understand independence and conditional probability and use them to interpret data

3.3.a.i: : Describe events as subsets of a sample space using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events.

3.3.a.ii: : 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.

3.3.a.iii: : 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.

3.3.a.iv: : Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.

3.3.a.v: : Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

3.3.b: : Students can: Use the rules of probability to compute probabilities of compound events in a uniform probability model

3.3.b.i: : 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.

4: : Shape, Dimension, and Geometric Relationships

4.1: : Apply transformation to numbers, shapes, functional representations, and data

4.1: : Objects in the plane can be transformed, and those transformations can be described and analyzed mathematically

4.1.a: : Students can: Experiment with transformations in the plane

4.1.a.i: : 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.

4.1.a.ii: : Represent transformations in the plane using appropriate tools.

4.1.a.iii: : Describe transformations as functions that take points in the plane as inputs and give other points as outputs.

4.1.a.iv: : Compare transformations that preserve distance and angle to those that do not.

4.1.a.v: : Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

4.1.a.vi: : Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

4.1.a.vii: : Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using appropriate tools.

4.1.a.viii: : Specify a sequence of transformations that will carry a given figure onto another.

4.1.b: : Students can: Understand congruence in terms of rigid motions

4.1.b.i: : Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure.

4.1.b.ii: : Given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

4.1.b.iv: : Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

4.1.c: : Students can: Prove geometric theorems

4.1.c.i: : Prove theorems about lines and angles.

4.1.c.ii: : Prove theorems about triangles.

4.1.c.iii: : Prove theorems about parallelograms.

4.1.d: : Students can: Make geometric constructions

4.1.d.i: : Make formal geometric constructions with a variety of tools and methods.

4.1.d.ii: : Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

4.2: : Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present and defend solutions

4.2: : Concepts of similarity are foundational to geometry and its applications

4.2.a: : Students can: Understand similarity in terms of similarity transformations

4.2.a.i: : Verify experimentally the properties of dilations given by a center and a scale factor:

4.2.a.i.1: : 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.

4.2.a.i.2: : Show that the dilation of a line segment is longer or shorter in the ratio given by the scale factor.

4.2.a.ii: : Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar.

4.2.a.iii: : 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.

4.2.a.iv: : Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

4.2.b: : Students can: Prove theorems involving similarity

4.2.b.i: : Prove theorems about triangles.

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

4.2.c: : Students can: Define trigonometric ratios and solve problems involving right triangles

4.2.c.i: : 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.

4.2.c.iii: : Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

4.2.d: : Students can: Prove and apply trigonometric identities

4.2.d.i: : Prove the Pythagorean identity sin²(theta) + cos²(theta) = 1.

4.2.d.ii: : Use the Pythagorean identity to find sin(theta), cos(theta), or tan(theta) given sin(theta), cos(theta), or tan(theta) and the quadrant of the angle.

4.2.e: : Students can: Understand and apply theorems about circles.

4.2.e.i: : Identify and describe relationships among inscribed angles, radii, and chords.

4.2.e.ii: : Construct the inscribed and circumscribed circles of a triangle.

4.2.e.iii: : Prove properties of angles for a quadrilateral inscribed in a circle.

4.2.f: : Students can: Find arc lengths and areas of sectors of circles.

4.2.f.i: : 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.

4.2.f.ii: : Derive the formula for the area of a sector.

4.3: : Make claims about relationships among numbers, shapes, symbols, and data and defend those claims by relying on the properties that are the structure of mathematics

4.3: : Objects in the plane can be described and analyzed algebraically

4.3.a: : Students can: Express Geometric Properties with Equations.

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

4.3.a.i.1: : Derive the equation of a circle of given center and radius using the Pythagorean Theorem.

4.3.a.i.2: : Complete the square to find the center and radius of a circle given by an equation.

4.3.a.i.3: : Derive the equation of a parabola given a focus and directrix.

4.3.a.ii: : Use coordinates to prove simple geometric theorems algebraically

4.3.a.ii.4: : Use coordinates and the distance formula to compute perimeters of polygons and areas of triangles and rectangles.

4.4: : Attributes of two- and three-dimensional objects are measurable and can be quantified

4.4.a: : Students can: Explain volume formulas and use them to solve problems

4.4.a.i: : Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.

4.4.a.ii: : Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Correlation last revised: 9/22/2020

Colorado - Mathematics: High School

Academic Standards | Adopted: 2010

1: : Number Sense, Properties, and Operations

2: : Patterns, Functions, and Algebraic Structures

3: : Data Analysis, Statistics, and Probability

4: : Shape, Dimension, and Geometric Relationships