<!-- Array
(
    [Mathematics] => Array
        (
            [Eighth Grade] => Array
                (
                    [1. Number and Quantity] => Array
                        (
                            [0] => 530
                        )

                    [2. Algebra and Functions] => Array
                        (
                            [0] => 645
                            [1] => 646
                            [2] => 647
                            [3] => 648
                            [4] => 649
                        )

                    [3. Data, Statistics, and Probability] => Array
                        (
                            [0] => 655
                        )

                    [4. Geometry] => Array
                        (
                            [0] => 723
                            [1] => 725
                            [2] => 724
                        )

                )

        )

)
Array
(
    [530] => Array
        (
            [8.NS.A. The Number System: Know that there are numbers that are not rational, and approximate them by rational numbers.] => Array
                (
                    [Students Can] => Array
                        (
                            [0] => Demonstrate informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. Define irrational numbers as numbers that are not rational. (CCSS: 8.NS.A.1)
                            [1] => Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., \(\pi^2\)). For example, by truncating the decimal expansion of \(\sqrt{2}\), show that \(\sqrt{2}\) is between \(1\) and \(2\), then between \(1.4\) and \(1.5\), and explain how to continue on to get better approximations. (CCSS: 8.NS.A.2)
                        )

                    [Colorado Essential Skills and Mathematical Practices] => Array
                        (
                            [0] => Investigate rational and irrational numbers and their relative approximate positions on a number line. (Entrepreneurial Skills: Inquiry/Analysis)
                            [1] => Use technology to look for repetition in decimal expansions and use number lines to order and compare irrational numbers relative to rational numbers. (MP5)
                            [2] => Apply understanding of rational and irrational numbers to describe and work within the structure of the real number system effectively and efficiently. (MP7)
                            [3] => Recognize repetition in decimal expansions of rational numbers and recognize when a decimal expansion cannot be represented by a rational number. (MP8)
                        )

                    [Inquiry Questions] => Array
                        (
                            [0] => How many irrational numbers exist?
                            [1] => Why is there no real number closest to zero?
                            [2] => Can you accurately plot an irrational number on the number line? How do you know?
                        )

                    [Coherence Connections] => Array
                        (
                            [0] => This expectation supports the major work of the grade.
                            [1] => In Grade 7, students apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
                            [2] => In Grade 8, this expectation supports working with radicals and integer exponents. This concludes the introduction of all numbers that comprise the real number system.
                            [3] => In high school, students use properties of rational and irrational numbers, work with rational exponents, and extend their understanding of number systems to include complex numbers.
                        )

                )

        )

    [645] => Array
        (
            [8.EE.A. Expressions & Equations: Work with radicals and integer exponents.] => Array
                (
                    [Students Can] => Array
                        (
                            [0] => Know and apply the properties of integer exponents to generate equivalent numerical expressions. <i>For example, \(3^2 × 3^{-5} = 3^{-3} = \frac{1}{3^3} = \frac{1}{27}\).</i> (CCSS: 8.EE.A.1)
                            [1] => Use square root and cube root symbols to represent solutions to equations of the form \(x^2 = p\) and \(x^3 = p\), where \(p\) is a positive rational number. Evaluate square roots of small perfect squares (up to \(100\)) and cube roots of small perfect cubes (up to \(64\)). Know that \(\sqrt{2}\) is irrational. (CCSS: 8.EE.A.2)
                            [2] => Use numbers expressed in the form of a single digit times an integer power of \(10\) to estimate very large or very small quantities, and to express how many times as much one is than the other. <i>For example, estimate the population of the United States as \(3\) times \(10^8\) and the population of the world as \(7\) times \(10^9\), and determine that the world population is more than \(20\) times larger.</i> (CCSS: 8.EE.A.3)
                            [3] => Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. (CCSS: 8.EE.A.4)
                        )

                    [Colorado Essential Skills and Mathematical Practices] => Array
                        (
                            [0] => Reason about unusually large or small quantities. (Entrepreneurial Skills: Inquiry/Analysis)
                            [1] => Use calculators or computers to compute with and approximate radicals and roots, and understand how such tools represent scientific notation. (MP5)
                            [2] => Explore the structure of numerical expressions with integer exponents to generate equivalent expressions. (MP7)
                            [3] => Look for how positive integer exponents are equivalent to repeated multiplication by the base and how negative integer exponents are equivalent to repeated division by the base. (MP8)
                        )

                    [Inquiry Questions] => Array
                        (
                            [0] => How is performing operations on numbers in scientific notation similar to or different from performing operations on numbers in standard notation?
                            [1] => Why does a positive number raised to a negative exponent not equal a negative number?
                        )

                    [Coherence Connections] => Array
                        (
                            [0] => This expectation represents major work of the grade.
                            [1] => In Grade 5, students use whole-number exponents to denote powers of ten. In Grades 6 and 7, they work with algebraic and numerical expressions containing whole-number exponents.
                            [2] => In Grade 8, this expectation connects with knowing that there are numbers that are not rational and approximating them by rational numbers, understanding and applying the Pythagorean Theorem, and solving real-world and mathematical problems involving volume of cylinders, cones, and spheres. In high school, students extend work with exponents to rational exponents.
                        )

                )

        )

    [646] => Array
        (
            [8.EE.B. Expressions & Equations: Understand the connections between proportional relationships, lines, and linear equations.] => Array
                (
                    [Students Can] => Array
                        (
                            [0] => Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. <i>For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.</i> (CCSS: 8.EE.B.5)
                            [1] => Use similar triangles to explain why the slope \(m\) is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation \(y = mx\) for a line through the origin and the equation \(y = mx + b\) for a line intercepting the vertical axis at \(b\). (CCSS: 8.EE.B.6)
                        )

                    [Colorado Essential Skills and Mathematical Practices] => Array
                        (
                            [0] => Make connections between representations of linear growth. (Entrepreneurial Skills: Inquiry/Analysis)
                            [1] => Record information about constant rates of change in graphs and equations. (Entrepreneurial Skills: Literacy/Writing)
                            [2] => Make sense of and compare proportional relationships represented in different forms. (MP1)
                            [3] => Compare, contrast, and make claims with proportional relationships based on properties of equations, tables, and/or graphs. (MP3)
                            [4] => Explore the structure of proportional relationships expressed as equations or graphs for methods of comparison. (MP7)
                        )

                    [Inquiry Questions] => Array
                        (
                            [0] => How is the unit rate of a proportional relationship related to the slope of its graphical representation?
                            [1] => Why are similar triangles effective for describing slope geometrically?
                        )

                    [Coherence Connections] => Array
                        (
                            [0] => This expectation represents major work of the grade.
                            [1] => In Grade 7, students recognize and represent proportional relationships, calculate the constant of proportionality, and graph proportional relationships on the coordinate plane, recognizing that they always pass through the origin.
                            [2] => In Grade 8, this expectation connects with analyzing and solving linear equations and pairs of simultaneous linear equations, with defining, evaluating, and comparing functions, and with understanding congruence and similarity using physical models, transparencies, or geometry software.
                            [3] => In high school, students compare multiple representations of inverse proportional relationships.
                        )

                )

        )

    [647] => Array
        (
            [8.EE.C. Expressions & Equations: Analyze and solve linear equations and pairs of simultaneous linear equations.] => Array
                (
                    [Students Can] => Array
                        (
                            [0] => Solve linear equations in one variable. (CCSS: 8.EE.C.7)
<ol style="list-style-type: lower-alpha">
<li>Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form \(x = a\), \(a = a\), or \(a = b\) results (where \(a\) and \(b\) are different numbers). (CCSS: 8.EE.C.7.a)</li>
<li>Solve linear equations with rational number coefficients, including equations with variables on both sides and whose solutions require expanding expressions using the distributive property and collecting like terms. (CCSS: 8.EE.C.7.b)</li>
</ol>
                            [1] => Analyze and solve pairs of simultaneous linear equations. (CCSS: 8.EE.C.8)
<ol style="list-style-type: lower-alpha">
<li>Explain that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. (CCSS: 8.EE.C.8.a)</li>
<li>Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. <i>For example, \(3x + 2y = 5\) and \(3x + 2y = 6\) have no solution because \(3x + 2y\) cannot simultaneously be \(5\) and \(6\).</i> (CCSS: 8.EE.C.8.b)</li>
<li>Solve real-world and mathematical problems leading to two linear equations in two variables. <i>For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.</i> (CCSS: 8.EE.C.8.c)</li>
</ol>
                        )

                    [Colorado Essential Skills and Mathematical Practices] => Array
                        (
                            [0] => Solve problems involving linear equations and systems of linear equations. (Entrepreneurial Skills: Critical Thinking/Problem Solving)
                            [1] => Solve problems that require a system of linear equations in two variables. (MP1)
                            [2] => Model real-world problems with linear equations and systems of linear equations, with variables defined in their real-world context. (MP4)
                            [3] => Solve equations and systems of equations and express solutions with accuracy that makes sense in the real-world context modeled by the equations. (MP6)
                            [4] => Recognize the structure of equations and of systems of equations that produce one, infinitely many, or no solution. (MP7)
                        )

                    [Inquiry Questions] => Array
                        (
                            [0] => What is meant by a “solution” to a linear equation? What is meant by a “solution” to a system of two linear equations? How are these concepts related?
                            [1] => How is it possible for an equation to have more than one solution? How is it possible for an equation to have no solution?
                            [2] => Why can’t a system of linear equations have a solution set other than one, zero, or infinitely many solutions?
                            [3] => What connections exist between the graphical solution and the algebraic solution of a system of linear equations?
                        )

                    [Coherence Connections] => Array
                        (
                            [0] => This expectation represents major work of the grade.
                            [1] => In previous grades, students reason about and solve one-step and two-step, one-variable equations and inequalities, use properties of operations to generate equivalent expressions, and solve real-world and mathematical problems using numerical and algebraic expressions and equations.
                            [2] => In Grade 8, this expectation connects with understanding the connections between proportional relationships, lines, and linear equations and with investigating patterns of association in bivariate data.
                            [3] => In high school, students abstract and generalize about linear functions and how they compare and contrast to nonlinear functions. Students also reason about and solve systems of equations that include one or more nonlinear equations.
                        )

                )

        )

    [648] => Array
        (
            [8.F.A. Functions: Define, evaluate, and compare functions.] => Array
                (
                    [Students Can] => Array
                        (
                            [0] => Define a function as a rule that assigns to each input exactly one output. Show that the graph of a function is the set of ordered pairs consisting of an input and the corresponding output. (Function notation is not required for Grade 8.) (CCSS: 8.F.A.1)
                            [1] => Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). <i>For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.</i> (CCSS: 8.F.A.2)
                            [2] => Interpret the equation \(y = mx + b\) as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. <i>For example, the function \(A = s^2\) giving the area of a square as a function of its side length is not linear because its graph contains the points \(\left(1,1\right)\), \(\left(2,4\right)\) and \(\left(3,9\right)\), which are not on a straight line.</i> (CCSS: 8.F.A.3)
                        )

                    [Colorado Essential Skills and Mathematical Practices] => Array
                        (
                            [0] => Make connections between the information gathered through tables, equations, graphs, and verbal descriptions of functions. (Entrepreneurial Skills: Inquiry/Analysis)
                            [1] => Define variables as quantities and interpret ordered pairs from a functional relationship with respect to those variables. (MP2)
                            [2] => With and without technology, analyze and describe functions that are not linear with the use of equations, graphs, and tables. (MP5)
                            [3] => See a function as a rule that assigns each input to exactly one output; this structure does not “turn inputs into outputs”; rather, it describes the relationship between items in two sets. (MP7)
                            [4] => Recognize patterns of linear growth in different representations of linear functions. (MP8)
                        )

                    [Inquiry Questions] => Array
                        (
                            [0] => Why is it important to know if a mathematical relationship is a function?
                            [1] => How can you determine if a function is linear or nonlinear?
                        )

                    [Coherence Connections] => Array
                        (
                            [0] => This expectation represents major work of the grade.
                            [1] => In Grade 7, students analyze proportional relationships and use them to solve real-world and mathematical problems.
                            [2] => In Grade 8, this expectation connects with understanding the connections between proportional relationships, lines, and linear equations and with using functions to model relationships between quantities.
                            [3] => In high school, students use function notation, analyze functions using different representations, build new functions from existing functions, and extend from linear functions to quadratic, exponential, and other more advanced functions.
                        )

                )

        )

    [649] => Array
        (
            [8.F.B. Functions: Use functions to model relationships between quantities.] => Array
                (
                    [Students Can] => Array
                        (
                            [0] => Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two \(\left(x, y\right)\) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. (CCSS: 8.F.B.4)
                            [1] => Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. (CCSS: 8.F.B.5)
                        )

                    [Colorado Essential Skills and Mathematical Practices] => Array
                        (
                            [0] => Describe in writing the qualitative features of linear or nonlinear functions. (Entrepreneurial Skills: Literacy/Writing)
                            [1] => Model real-world situations with linear functions. (MP4)
                            [2] => Explore properties of linear functions and how those properties appear in the structure of linear equations in slope-intercept form. (MP7)
                            [3] => Use strategies to calculate the rate of change in a linear function (slope) and use properties of linear functions to create equations. (MP8)
                        )

                    [Inquiry Questions] => Array
                        (
                            [0] => What is the minimum information needed to write a linear function for a relationship between two quantities?
                            [1] => What are some quantitative and qualitative features of graphs of functions?
                        )

                    [Coherence Connections] => Array
                        (
                            [0] => This expectation represents major work of the grade.
                            [1] => In Grade 7, students analyze proportional relationships and use them to solve real-world and mathematical problems.
                            [2] => In Grade 8, this expectation connects with defining, evaluating, and comparing functions and with investigating patterns of association in bivariate data.
                            [3] => In high school, students use function notation, analyze functions using different representations, build new functions from existing functions, and extend from linear functions to quadratic, exponential, and other more advanced functions.
                        )

                )

        )

    [655] => Array
        (
            [8.SP.A. Statistics & Probability: Investigate patterns of association in bivariate data.] => Array
                (
                    [Students Can] => Array
                        (
                            [0] => Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. (CCSS: 8.SP.A.1)
                            [1] => Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. (CCSS: 8.SP.A.2)
                            [2] => Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. <i>For example, in a linear model for a biology experiment, interpret a slope of \(1.5\) cm/hr as meaning that an additional hour of sunlight each day is associated with an additional \(1.5\) cm in mature plant height.</i> (CCSS: 8.SP.A.3)
                            [3] => Explain that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. <i>For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?</i> (CCSS: 8.SP.A.4)
                        )

                    [Colorado Essential Skills and Mathematical Practices] => Array
                        (
                            [0] => Recognize and describe patterns in bivariate data. (Entrepreneurial Skills: Inquiry/Analysis)
                            [1] => Interpret the contextual meaning of slope and \(y\)-intercept, where applicable in a linear model fit to bivariate data. (MP2)
                            [2] => Build statistical models to explore, describe, and generalize the relationship between two variables. (MP4)
                            [3] => Use scatter plots and two-way tables to describe possible associations in bivariate data. (MP7)
                        )

                    [Inquiry Questions] => Array
                        (
                            [0] => In what ways is a scatter plot useful in describing and interpreting the relationship between two quantities?
                            [1] => Why would we create a linear model for a set of bivariate data?
                            [2] => How do you know when a credible prediction can be made from a linear model of bivariate data?
                            [3] => What does a pattern of association look like for categorical data?
                        )

                    [Coherence Connections] => Array
                        (
                            [0] => This expectation supports the major work of the grade.
                            [1] => In previous grades, students apply and extend previous understandings of numbers to the system of rational numbers.
                            [2] => In Grade 8, this expectation supports using functions to model relationships between quantities.
                            [3] => In high school, students summarize, represent, and interpret data on two categorical and quantitative variables, interpret linear models, and understand independence and conditional probability.
                        )

                )

        )

    [723] => Array
        (
            [8.G.A. Geometry: Understand congruence and similarity using physical models, transparencies, or geometry software.] => Array
                (
                    [Students Can] => Array
                        (
                            [0] => Verify experimentally the properties of rotations, reflections, and translations: (CCSS: 8.G.A.1)
<ol style="list-style-type: lower-alpha">
<li>Lines are taken to lines, and line segments to line segments of the same length. (CCSS: 8.G.A.1.a)</li>
<li>Angles are taken to angles of the same measure. (CCSS: 8.G.A.1.b)</li>
<li>Parallel lines are taken to parallel lines. (CCSS: 8.G.A.1.c)</li>
</ol>
                            [1] => Demonstrate that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. (CCSS: 8.G.A.2)
                            [2] => Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. (CCSS: 8.G.A.3)
                            [3] => Demonstrate that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. (CCSS: 8.G.A.4)
                            [4] => Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. <i>For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.</i> (CCSS: 8.G.A.5)
                        )

                    [Colorado Essential Skills and Mathematical Practices] => Array
                        (
                            [0] => Think about how rotations, reflections, and translations of a geometric figure preserve congruence as similar to how properties of operations such as the associative, commutative, and distributive properties preserve equivalence of arithmetic and algebraic expressions. (Entrepreneurial Skills: Critical Thinking/Problem Solving and Inquiry/Analysis)
                            [1] => Explain a sequence of transformations that results in a congruent or similar triangle. (MP3)
                            [2] => Use physical models, transparencies, geometric software, or other appropriate tools to explore the relationships between transformations and congruence and similarity. (MP5)
                            [3] => Use the structure of the coordinate system to describe the locations of figures obtained with rotations, reflections, and translations. (MP7)
                            [4] => Reason that since any one rotation, reflection, or translation of a figure preserves congruence, then any sequence of those transformations must also preserve congruence. (MP8)
                        )

                    [Inquiry Questions] => Array
                        (
                            [0] => How are properties of rotations, reflections, translations, and dilations connected to congruence?
                            [1] => How are properties of rotations, reflections, translations, and dilations connected to similarity?
                            [2] => Why are angle measures significant regarding the similarity of two figures?
                        )

                    [Coherence Connections] => Array
                        (
                            [0] => This expectation represents major work of the grade.
                            [1] => In previous grades, students solve problems involving angle measure, area, surface area, and volume, and draw, construct, and also describe geometrical figures and the relationships between them.
                            [2] => In Grade 8, this expectation connects with understanding the connections between proportional relationships, lines, and linear equations.
                            [3] => In high school, students extend their work with transformations, apply the concepts of transformations to prove geometric theorems, and use similarity to define trigonometric functions.
                        )

                )

        )

    [724] => Array
        (
            [8.G.C. Geometry: Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.] => Array
                (
                    [Students Can] => Array
                        (
                            [0] => State the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. (CCSS: 8.G.C.9)
                        )

                    [Colorado Essential Skills and Mathematical Practices] => Array
                        (
                            [0] => Efficiently solve problems using established volume formulas. (Professional Skills: Task/Time Management)
                            [1] => Describe how the formulas for volumes of cones, cylinders, and spheres relate to one another and to the volume formulas for solids with rectangular bases. (MP3)
                            [2] => Use appropriate precision when solving problems involving measurements and volume formulas that describe real-world shapes. (MP6)
                        )

                    [Inquiry Questions] => Array
                        (
                            [0] => How are the formulas of cones, cylinders, and spheres similar to each other?
                            [1] => How are the formulas of cones, cylinder, and spheres connected to the formulas for pyramids, prisms, and cubes?
                        )

                    [Coherence Connections] => Array
                        (
                            [0] => This expectation is in addition to the major work of the grade.
                            [1] => In Grade 7, students solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
                            [2] => In Grade 8, this expectation connects with radicals and integer exponents.
                            [3] => In high school, students apply geometric concepts in mathematical modeling situations and to solve design problems.
                        )

                )

        )

    [725] => Array
        (
            [8.G.B. Geometry: Understand and apply the Pythagorean Theorem.] => Array
                (
                    [Students Can] => Array
                        (
                            [0] => Explain a proof of the Pythagorean Theorem and its converse. (CCSS: 8.G.B.6)
                            [1] => Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. (CCSS: 8.G.B.7)
                            [2] => Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. (CCSS: 8.G.B.8)
                        )

                    [Colorado Essential Skills and Mathematical Practices] => Array
                        (
                            [0] => Think of the Pythagorean Theorem as not just a formula, but a formula that only holds true under certain conditions. (Entrepreneurial Skills: Inquiry/Analysis)
                            [1] => Construct a viable argument about why a proof of the Pythagorean Theorem is valid. (MP3)
                            [2] => Test to see if a triangle is a right triangle by applying the Pythagorean Theorem. (MP7)
                            [3] => Use patterns to recognize and generate Pythagorean triples. (MP8)
                        )

                    [Inquiry Questions] => Array
                        (
                            [0] => What is the relationship between the Pythagorean Theorem and its converse? In what ways is each useful?
                            [1] => Is it always possible to use the Pythagorean Theorem to find the distance between points on the coordinate plane? How do you know?
                        )

                    [Coherence Connections] => Array
                        (
                            [0] => This expectation represents major work of the grade.
                            [1] => In Grades 6 and 7, students solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
                            [2] => In Grade 8, this expectation connects with radicals and integer exponents, square roots, and solving simple equations in the form \(x^2 = p\).
                            [3] => In high school, students (a) prove and apply trigonometric identities, (b) prove theorems involving similarity, (c) define trigonometric ratios and solve problems involving right triangles, (d) translate between the geometric description and the equation for a conic section, and (e) use coordinates to prove simple geometric theorems algebraically.
                        )

                )

        )

)
Array
(
    [530] => Array
        (
            [0] => MP5. Use appropriate tools strategically.
            [1] => MP7. Look for and make use of structure.
            [2] => MP8. Look for and express regularity in repeated reasoning.
        )

    [645] => Array
        (
            [0] => MP5. Use appropriate tools strategically.
            [1] => MP7. Look for and make use of structure.
            [2] => MP8. Look for and express regularity in repeated reasoning.
        )

    [646] => Array
        (
            [0] => MP1. Make sense of problems and persevere in solving them.
            [1] => MP3. Construct viable arguments and critique the reasoning of others.
            [2] => MP7. Look for and make use of structure.
        )

    [647] => Array
        (
            [0] => MP1. Make sense of problems and persevere in solving them.
            [1] => MP4. Model with mathematics.
            [2] => MP6. Attend to precision.
            [3] => MP7. Look for and make use of structure.
        )

    [648] => Array
        (
            [0] => MP2. Reason abstractly and quantitatively.
            [1] => MP5. Use appropriate tools strategically.
            [2] => MP7. Look for and make use of structure.
            [3] => MP8. Look for and express regularity in repeated reasoning.
        )

    [649] => Array
        (
            [0] => MP4. Model with mathematics.
            [1] => MP7. Look for and make use of structure.
            [2] => MP8. Look for and express regularity in repeated reasoning.
        )

    [655] => Array
        (
            [0] => MP2. Reason abstractly and quantitatively.
            [1] => MP4. Model with mathematics.
            [2] => MP7. Look for and make use of structure.
        )

    [723] => Array
        (
            [0] => MP3. Construct viable arguments and critique the reasoning of others.
            [1] => MP5. Use appropriate tools strategically.
            [2] => MP7. Look for and make use of structure.
            [3] => MP8. Look for and express regularity in repeated reasoning.
        )

    [724] => Array
        (
            [0] => MP3. Construct viable arguments and critique the reasoning of others.
            [1] => MP6. Attend to precision.
        )

    [725] => Array
        (
            [0] => MP3. Construct viable arguments and critique the reasoning of others.
            [1] => MP7. Look for and make use of structure.
            [2] => MP8. Look for and express regularity in repeated reasoning.
        )

)
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<h1>2020 Colorado Academic Standards Online</h1>

	<p>Use the options below to create customized views of the 2020 Colorado Academic Standards. For all standards resources, see the <a href="/standardsandinstruction">Office of Standards and Instructional Support</a>.</p>

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				<span><em>Content Area:</em> Mathematics
					// <em>Grade Level:</em> Eighth Grade
					// <em>Standard Category:</em> All Standards Categories</span>
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<div class="row">
	<div class="col">
								<div class="caBlock Mathematics">
			<div class="row ">
				<div class="col">
					<div class="standardhead">
						<div class="row">
							<div class="col-3"><p><strong>Mathematics</strong></p></div>
							<div class="col-9"><p style="text-align: right;"><strong>Eighth Grade, Standard 1. Number and Quantity</strong></p></div>
						</div>
					</div>
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				<div class="col">
					<p style="text-align: right;">


	<a href="/comath" target="_blank">Learn More About Mathematics in Colorado</a>


						| <a href="/standardsandinstruction/essentialskills" target="_blank">Read the Colorado Essential Skills</a>
						| <a href="#" onclick="$(this).closest('.caBlock').find('.gleDetail').removeClass( 'notvisible' );$(this).closest('.caBlock').find('.expanderDown').addClass('notvisible');return false;">Expand Set of GLEs Below</a>
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					<div class="gle">


						<p class="float-right">
							<a href="#" onclick="$(this).closest('.col').find('.gleDetail').toggleClass('notvisible');return false;">
								<i class="material-icons gleDetail expanderDown">keyboard_arrow_down</i>
								<i class="material-icons gleDetail expanderUp notvisible">keyboard_arrow_up</i>
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						</p>
						<div class="gleDetail notvisible">
							<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsPGC"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a> Prepared Graduates:</p>
							<ul style="list-style-type:none;">
														<li>MP5. Use appropriate tools strategically.</li>
														<li>MP7. Look for and make use of structure.</li>
														<li>MP8. Look for and express regularity in repeated reasoning.</li>
													</ul>
						</div>

						<p class="gleDetail notvisible"><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsGLE"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a> 


Grade Level Expectation:


</p>
						<p style="padding-left:40px;">8.NS.A. The Number System: Know that there are numbers that are not rational, and approximate them by rational numbers.</p>
						<div class="gleDetail notvisible">
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								<div class="col eocol leftcol">
									<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsEO"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a> 

Evidence Outcomes:


									</p>

																						<div class="eoinfo" style="display: block;">

			<p><em>Students Can:</em></p>
	
										<ol type="a">

																			<li>Demonstrate informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. Define irrational numbers as numbers that are not rational. (CCSS: 8.NS.A.1)</li>
																			<li>Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., \(\pi^2\)). For example, by truncating the decimal expansion of \(\sqrt{2}\), show that \(\sqrt{2}\) is between \(1\) and \(2\), then between \(1.4\) and \(1.5\), and explain how to continue on to get better approximations. (CCSS: 8.NS.A.2)</li>
								
										</ol>
									</div>
																																																				
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								<div class="col eocol rightcol">
									<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsConnections"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a> 


Academic Contexts and Connections:


									</p>

																																			<div class="eoinfo" style="display: block;">
										<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsColoradoEssentialSkillsandMathematicalPractices"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a><em> 


Colorado Essential Skills and Mathematical Practices:

							
									</em></p>

										<ol type="1">

																			<li>Investigate rational and irrational numbers and their relative approximate positions on a number line. (Entrepreneurial Skills: Inquiry/Analysis)</li>
																			<li>Use technology to look for repetition in decimal expansions and use number lines to order and compare irrational numbers relative to rational numbers. (MP5)</li>
																			<li>Apply understanding of rational and irrational numbers to describe and work within the structure of the real number system effectively and efficiently. (MP7)</li>
																			<li>Recognize repetition in decimal expansions of rational numbers and recognize when a decimal expansion cannot be represented by a rational number. (MP8)</li>
								
										</ol>
									</div>
																													<div class="eoinfo" style="display: block;">
										<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsInquiryQuestions"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a><em> 


Inquiry Questions:

							
									</em></p>

										<ol type="1">

																			<li>How many irrational numbers exist?</li>
																			<li>Why is there no real number closest to zero?</li>
																			<li>Can you accurately plot an irrational number on the number line? How do you know?</li>
								
										</ol>
									</div>
																													<div class="eoinfo" style="display: block;">
										<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsCoherenceConnections"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a><em> 


Coherence Connections:

							
									</em></p>

										<ol type="1">

																			<li>This expectation supports the major work of the grade.</li>
																			<li>In Grade 7, students apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.</li>
																			<li>In Grade 8, this expectation supports working with radicals and integer exponents. This concludes the introduction of all numbers that comprise the real number system.</li>
																			<li>In high school, students use properties of rational and irrational numbers, work with rational exponents, and extend their understanding of number systems to include complex numbers.</li>
								
										</ol>
									</div>
													
								</div>
							</div>

						</div>

					</div>
				</div>

			</div>

					
		</div>

					<div class="caBlock Mathematics">
			<div class="row ">
				<div class="col">
					<div class="standardhead">
						<div class="row">
							<div class="col-3"><p><strong>Mathematics</strong></p></div>
							<div class="col-9"><p style="text-align: right;"><strong>Eighth Grade, Standard 2. Algebra and Functions</strong></p></div>
						</div>
					</div>
				</div>
			</div>
			<div class="row">
				<div class="col">
					<p style="text-align: right;">


	<a href="/comath" target="_blank">Learn More About Mathematics in Colorado</a>


						| <a href="/standardsandinstruction/essentialskills" target="_blank">Read the Colorado Essential Skills</a>
						| <a href="#" onclick="$(this).closest('.caBlock').find('.gleDetail').removeClass( 'notvisible' );$(this).closest('.caBlock').find('.expanderDown').addClass('notvisible');return false;">Expand Set of GLEs Below</a>
					</p>
				</div>
			</div>
												<div class="row">
				<div class="col">

					<div class="gle">


						<p class="float-right">
							<a href="#" onclick="$(this).closest('.col').find('.gleDetail').toggleClass('notvisible');return false;">
								<i class="material-icons gleDetail expanderDown">keyboard_arrow_down</i>
								<i class="material-icons gleDetail expanderUp notvisible">keyboard_arrow_up</i>
							</a>
						</p>
						<div class="gleDetail notvisible">
							<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsPGC"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a> Prepared Graduates:</p>
							<ul style="list-style-type:none;">
														<li>MP5. Use appropriate tools strategically.</li>
														<li>MP7. Look for and make use of structure.</li>
														<li>MP8. Look for and express regularity in repeated reasoning.</li>
													</ul>
						</div>

						<p class="gleDetail notvisible"><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsGLE"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a> 


Grade Level Expectation:


</p>
						<p style="padding-left:40px;">8.EE.A. Expressions &amp; Equations: Work with radicals and integer exponents.</p>
						<div class="gleDetail notvisible">
							<div class="row eorow">
								<div class="col eocol leftcol">
									<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsEO"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a> 

Evidence Outcomes:


									</p>

																						<div class="eoinfo" style="display: block;">

			<p><em>Students Can:</em></p>
	
										<ol type="a">

																			<li>Know and apply the properties of integer exponents to generate equivalent numerical expressions. <i>For example, \(3^2 × 3^{-5} = 3^{-3} = \frac{1}{3^3} = \frac{1}{27}\).</i> (CCSS: 8.EE.A.1)</li>
																			<li>Use square root and cube root symbols to represent solutions to equations of the form \(x^2 = p\) and \(x^3 = p\), where \(p\) is a positive rational number. Evaluate square roots of small perfect squares (up to \(100\)) and cube roots of small perfect cubes (up to \(64\)). Know that \(\sqrt{2}\) is irrational. (CCSS: 8.EE.A.2)</li>
																			<li>Use numbers expressed in the form of a single digit times an integer power of \(10\) to estimate very large or very small quantities, and to express how many times as much one is than the other. <i>For example, estimate the population of the United States as \(3\) times \(10^8\) and the population of the world as \(7\) times \(10^9\), and determine that the world population is more than \(20\) times larger.</i> (CCSS: 8.EE.A.3)</li>
																			<li>Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. (CCSS: 8.EE.A.4)</li>
								
										</ol>
									</div>
																																																				
								</div>
								<div class="col eocol rightcol">
									<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsConnections"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a> 


Academic Contexts and Connections:


									</p>

																																			<div class="eoinfo" style="display: block;">
										<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsColoradoEssentialSkillsandMathematicalPractices"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a><em> 


Colorado Essential Skills and Mathematical Practices:

							
									</em></p>

										<ol type="1">

																			<li>Reason about unusually large or small quantities. (Entrepreneurial Skills: Inquiry/Analysis)</li>
																			<li>Use calculators or computers to compute with and approximate radicals and roots, and understand how such tools represent scientific notation. (MP5)</li>
																			<li>Explore the structure of numerical expressions with integer exponents to generate equivalent expressions. (MP7)</li>
																			<li>Look for how positive integer exponents are equivalent to repeated multiplication by the base and how negative integer exponents are equivalent to repeated division by the base. (MP8)</li>
								
										</ol>
									</div>
																													<div class="eoinfo" style="display: block;">
										<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsInquiryQuestions"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a><em> 


Inquiry Questions:

							
									</em></p>

										<ol type="1">

																			<li>How is performing operations on numbers in scientific notation similar to or different from performing operations on numbers in standard notation?</li>
																			<li>Why does a positive number raised to a negative exponent not equal a negative number?</li>
								
										</ol>
									</div>
																													<div class="eoinfo" style="display: block;">
										<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsCoherenceConnections"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a><em> 


Coherence Connections:

							
									</em></p>

										<ol type="1">

																			<li>This expectation represents major work of the grade.</li>
																			<li>In Grade 5, students use whole-number exponents to denote powers of ten. In Grades 6 and 7, they work with algebraic and numerical expressions containing whole-number exponents.</li>
																			<li>In Grade 8, this expectation connects with knowing that there are numbers that are not rational and approximating them by rational numbers, understanding and applying the Pythagorean Theorem, and solving real-world and mathematical problems involving volume of cylinders, cones, and spheres. In high school, students extend work with exponents to rational exponents.</li>
								
										</ol>
									</div>
													
								</div>
							</div>

						</div>

					</div>
				</div>

			</div>

					
												<div class="row">
				<div class="col">

					<div class="gle">


						<p class="float-right">
							<a href="#" onclick="$(this).closest('.col').find('.gleDetail').toggleClass('notvisible');return false;">
								<i class="material-icons gleDetail expanderDown">keyboard_arrow_down</i>
								<i class="material-icons gleDetail expanderUp notvisible">keyboard_arrow_up</i>
							</a>
						</p>
						<div class="gleDetail notvisible">
							<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsPGC"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a> Prepared Graduates:</p>
							<ul style="list-style-type:none;">
														<li>MP1. Make sense of problems and persevere in solving them.</li>
														<li>MP3. Construct viable arguments and critique the reasoning of others.</li>
														<li>MP7. Look for and make use of structure.</li>
													</ul>
						</div>

						<p class="gleDetail notvisible"><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsGLE"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a> 


Grade Level Expectation:


</p>
						<p style="padding-left:40px;">8.EE.B. Expressions &amp; Equations: Understand the connections between proportional relationships, lines, and linear equations.</p>
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Evidence Outcomes:


									</p>

																						<div class="eoinfo" style="display: block;">

			<p><em>Students Can:</em></p>
	
										<ol type="a">

																			<li>Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. <i>For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.</i> (CCSS: 8.EE.B.5)</li>
																			<li>Use similar triangles to explain why the slope \(m\) is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation \(y = mx\) for a line through the origin and the equation \(y = mx + b\) for a line intercepting the vertical axis at \(b\). (CCSS: 8.EE.B.6)</li>
								
										</ol>
									</div>
																																																				
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Academic Contexts and Connections:


									</p>

																																			<div class="eoinfo" style="display: block;">
										<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsColoradoEssentialSkillsandMathematicalPractices"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a><em> 


Colorado Essential Skills and Mathematical Practices:

							
									</em></p>

										<ol type="1">

																			<li>Make connections between representations of linear growth. (Entrepreneurial Skills: Inquiry/Analysis)</li>
																			<li>Record information about constant rates of change in graphs and equations. (Entrepreneurial Skills: Literacy/Writing)</li>
																			<li>Make sense of and compare proportional relationships represented in different forms. (MP1)</li>
																			<li>Compare, contrast, and make claims with proportional relationships based on properties of equations, tables, and/or graphs. (MP3)</li>
																			<li>Explore the structure of proportional relationships expressed as equations or graphs for methods of comparison. (MP7)</li>
								
										</ol>
									</div>
																													<div class="eoinfo" style="display: block;">
										<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsInquiryQuestions"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a><em> 


Inquiry Questions:

							
									</em></p>

										<ol type="1">

																			<li>How is the unit rate of a proportional relationship related to the slope of its graphical representation?</li>
																			<li>Why are similar triangles effective for describing slope geometrically?</li>
								
										</ol>
									</div>
																													<div class="eoinfo" style="display: block;">
										<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsCoherenceConnections"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a><em> 


Coherence Connections:

							
									</em></p>

										<ol type="1">

																			<li>This expectation represents major work of the grade.</li>
																			<li>In Grade 7, students recognize and represent proportional relationships, calculate the constant of proportionality, and graph proportional relationships on the coordinate plane, recognizing that they always pass through the origin.</li>
																			<li>In Grade 8, this expectation connects with analyzing and solving linear equations and pairs of simultaneous linear equations, with defining, evaluating, and comparing functions, and with understanding congruence and similarity using physical models, transparencies, or geometry software.</li>
																			<li>In high school, students compare multiple representations of inverse proportional relationships.</li>
								
										</ol>
									</div>
													
								</div>
							</div>

						</div>

					</div>
				</div>

			</div>

					
												<div class="row">
				<div class="col">

					<div class="gle">


						<p class="float-right">
							<a href="#" onclick="$(this).closest('.col').find('.gleDetail').toggleClass('notvisible');return false;">
								<i class="material-icons gleDetail expanderDown">keyboard_arrow_down</i>
								<i class="material-icons gleDetail expanderUp notvisible">keyboard_arrow_up</i>
							</a>
						</p>
						<div class="gleDetail notvisible">
							<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsPGC"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a> Prepared Graduates:</p>
							<ul style="list-style-type:none;">
														<li>MP1. Make sense of problems and persevere in solving them.</li>
														<li>MP4. Model with mathematics.</li>
														<li>MP6. Attend to precision.</li>
														<li>MP7. Look for and make use of structure.</li>
													</ul>
						</div>

						<p class="gleDetail notvisible"><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsGLE"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a> 


Grade Level Expectation:


</p>
						<p style="padding-left:40px;">8.EE.C. Expressions &amp; Equations: Analyze and solve linear equations and pairs of simultaneous linear equations.</p>
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Evidence Outcomes:


									</p>

																						<div class="eoinfo" style="display: block;">

			<p><em>Students Can:</em></p>
	
										<ol type="a">

																			<li>Solve linear equations in one variable. (CCSS: 8.EE.C.7)
<ol style="list-style-type: lower-alpha">
<li>Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form \(x = a\), \(a = a\), or \(a = b\) results (where \(a\) and \(b\) are different numbers). (CCSS: 8.EE.C.7.a)</li>
<li>Solve linear equations with rational number coefficients, including equations with variables on both sides and whose solutions require expanding expressions using the distributive property and collecting like terms. (CCSS: 8.EE.C.7.b)</li>
</ol></li>
																			<li>Analyze and solve pairs of simultaneous linear equations. (CCSS: 8.EE.C.8)
<ol style="list-style-type: lower-alpha">
<li>Explain that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. (CCSS: 8.EE.C.8.a)</li>
<li>Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. <i>For example, \(3x + 2y = 5\) and \(3x + 2y = 6\) have no solution because \(3x + 2y\) cannot simultaneously be \(5\) and \(6\).</i> (CCSS: 8.EE.C.8.b)</li>
<li>Solve real-world and mathematical problems leading to two linear equations in two variables. <i>For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.</i> (CCSS: 8.EE.C.8.c)</li>
</ol></li>
								
										</ol>
									</div>
																																																				
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Academic Contexts and Connections:


									</p>

																																			<div class="eoinfo" style="display: block;">
										<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsColoradoEssentialSkillsandMathematicalPractices"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a><em> 


Colorado Essential Skills and Mathematical Practices:

							
									</em></p>

										<ol type="1">

																			<li>Solve problems involving linear equations and systems of linear equations. (Entrepreneurial Skills: Critical Thinking/Problem Solving)</li>
																			<li>Solve problems that require a system of linear equations in two variables. (MP1)</li>
																			<li>Model real-world problems with linear equations and systems of linear equations, with variables defined in their real-world context. (MP4)</li>
																			<li>Solve equations and systems of equations and express solutions with accuracy that makes sense in the real-world context modeled by the equations. (MP6)</li>
																			<li>Recognize the structure of equations and of systems of equations that produce one, infinitely many, or no solution. (MP7)</li>
								
										</ol>
									</div>
																													<div class="eoinfo" style="display: block;">
										<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsInquiryQuestions"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a><em> 


Inquiry Questions:

							
									</em></p>

										<ol type="1">

																			<li>What is meant by a “solution” to a linear equation? What is meant by a “solution” to a system of two linear equations? How are these concepts related?</li>
																			<li>How is it possible for an equation to have more than one solution? How is it possible for an equation to have no solution?</li>
																			<li>Why can’t a system of linear equations have a solution set other than one, zero, or infinitely many solutions?</li>
																			<li>What connections exist between the graphical solution and the algebraic solution of a system of linear equations?</li>
								
										</ol>
									</div>
																													<div class="eoinfo" style="display: block;">
										<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsCoherenceConnections"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a><em> 


Coherence Connections:

							
									</em></p>

										<ol type="1">

																			<li>This expectation represents major work of the grade.</li>
																			<li>In previous grades, students reason about and solve one-step and two-step, one-variable equations and inequalities, use properties of operations to generate equivalent expressions, and solve real-world and mathematical problems using numerical and algebraic expressions and equations.</li>
																			<li>In Grade 8, this expectation connects with understanding the connections between proportional relationships, lines, and linear equations and with investigating patterns of association in bivariate data.</li>
																			<li>In high school, students abstract and generalize about linear functions and how they compare and contrast to nonlinear functions. Students also reason about and solve systems of equations that include one or more nonlinear equations.</li>
								
										</ol>
									</div>
													
								</div>
							</div>

						</div>

					</div>
				</div>

			</div>

					
												<div class="row">
				<div class="col">

					<div class="gle">


						<p class="float-right">
							<a href="#" onclick="$(this).closest('.col').find('.gleDetail').toggleClass('notvisible');return false;">
								<i class="material-icons gleDetail expanderDown">keyboard_arrow_down</i>
								<i class="material-icons gleDetail expanderUp notvisible">keyboard_arrow_up</i>
							</a>
						</p>
						<div class="gleDetail notvisible">
							<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsPGC"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a> Prepared Graduates:</p>
							<ul style="list-style-type:none;">
														<li>MP2. Reason abstractly and quantitatively.</li>
														<li>MP5. Use appropriate tools strategically.</li>
														<li>MP7. Look for and make use of structure.</li>
														<li>MP8. Look for and express regularity in repeated reasoning.</li>
													</ul>
						</div>

						<p class="gleDetail notvisible"><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsGLE"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a> 


Grade Level Expectation:


</p>
						<p style="padding-left:40px;">8.F.A. Functions: Define, evaluate, and compare functions.</p>
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Evidence Outcomes:


									</p>

																						<div class="eoinfo" style="display: block;">

			<p><em>Students Can:</em></p>
	
										<ol type="a">

																			<li>Define a function as a rule that assigns to each input exactly one output. Show that the graph of a function is the set of ordered pairs consisting of an input and the corresponding output. (Function notation is not required for Grade 8.) (CCSS: 8.F.A.1)</li>
																			<li>Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). <i>For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.</i> (CCSS: 8.F.A.2)</li>
																			<li>Interpret the equation \(y = mx + b\) as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. <i>For example, the function \(A = s^2\) giving the area of a square as a function of its side length is not linear because its graph contains the points \(\left(1,1\right)\), \(\left(2,4\right)\) and \(\left(3,9\right)\), which are not on a straight line.</i> (CCSS: 8.F.A.3)</li>
								
										</ol>
									</div>
																																																				
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Academic Contexts and Connections:


									</p>

																																			<div class="eoinfo" style="display: block;">
										<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsColoradoEssentialSkillsandMathematicalPractices"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a><em> 


Colorado Essential Skills and Mathematical Practices:

							
									</em></p>

										<ol type="1">

																			<li>Make connections between the information gathered through tables, equations, graphs, and verbal descriptions of functions. (Entrepreneurial Skills: Inquiry/Analysis)</li>
																			<li>Define variables as quantities and interpret ordered pairs from a functional relationship with respect to those variables. (MP2)</li>
																			<li>With and without technology, analyze and describe functions that are not linear with the use of equations, graphs, and tables. (MP5)</li>
																			<li>See a function as a rule that assigns each input to exactly one output; this structure does not “turn inputs into outputs”; rather, it describes the relationship between items in two sets. (MP7)</li>
																			<li>Recognize patterns of linear growth in different representations of linear functions. (MP8)</li>
								
										</ol>
									</div>
																													<div class="eoinfo" style="display: block;">
										<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsInquiryQuestions"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a><em> 


Inquiry Questions:

							
									</em></p>

										<ol type="1">

																			<li>Why is it important to know if a mathematical relationship is a function?</li>
																			<li>How can you determine if a function is linear or nonlinear?</li>
								
										</ol>
									</div>
																													<div class="eoinfo" style="display: block;">
										<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsCoherenceConnections"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a><em> 


Coherence Connections:

							
									</em></p>

										<ol type="1">

																			<li>This expectation represents major work of the grade.</li>
																			<li>In Grade 7, students analyze proportional relationships and use them to solve real-world and mathematical problems.</li>
																			<li>In Grade 8, this expectation connects with understanding the connections between proportional relationships, lines, and linear equations and with using functions to model relationships between quantities.</li>
																			<li>In high school, students use function notation, analyze functions using different representations, build new functions from existing functions, and extend from linear functions to quadratic, exponential, and other more advanced functions.</li>
								
										</ol>
									</div>
													
								</div>
							</div>

						</div>

					</div>
				</div>

			</div>

					
												<div class="row">
				<div class="col">

					<div class="gle">


						<p class="float-right">
							<a href="#" onclick="$(this).closest('.col').find('.gleDetail').toggleClass('notvisible');return false;">
								<i class="material-icons gleDetail expanderDown">keyboard_arrow_down</i>
								<i class="material-icons gleDetail expanderUp notvisible">keyboard_arrow_up</i>
							</a>
						</p>
						<div class="gleDetail notvisible">
							<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsPGC"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a> Prepared Graduates:</p>
							<ul style="list-style-type:none;">
														<li>MP4. Model with mathematics.</li>
														<li>MP7. Look for and make use of structure.</li>
														<li>MP8. Look for and express regularity in repeated reasoning.</li>
													</ul>
						</div>

						<p class="gleDetail notvisible"><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsGLE"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a> 


Grade Level Expectation:


</p>
						<p style="padding-left:40px;">8.F.B. Functions: Use functions to model relationships between quantities.</p>
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								<div class="col eocol leftcol">
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Evidence Outcomes:


									</p>

																						<div class="eoinfo" style="display: block;">

			<p><em>Students Can:</em></p>
	
										<ol type="a">

																			<li>Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two \(\left(x, y\right)\) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. (CCSS: 8.F.B.4)</li>
																			<li>Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. (CCSS: 8.F.B.5)</li>
								
										</ol>
									</div>
																																																				
								</div>
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									<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsConnections"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a> 


Academic Contexts and Connections:


									</p>

																																			<div class="eoinfo" style="display: block;">
										<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsColoradoEssentialSkillsandMathematicalPractices"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a><em> 


Colorado Essential Skills and Mathematical Practices:

							
									</em></p>

										<ol type="1">

																			<li>Describe in writing the qualitative features of linear or nonlinear functions. (Entrepreneurial Skills: Literacy/Writing)</li>
																			<li>Model real-world situations with linear functions. (MP4)</li>
																			<li>Explore properties of linear functions and how those properties appear in the structure of linear equations in slope-intercept form. (MP7)</li>
																			<li>Use strategies to calculate the rate of change in a linear function (slope) and use properties of linear functions to create equations. (MP8)</li>
								
										</ol>
									</div>
																													<div class="eoinfo" style="display: block;">
										<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsInquiryQuestions"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a><em> 


Inquiry Questions:

							
									</em></p>

										<ol type="1">

																			<li>What is the minimum information needed to write a linear function for a relationship between two quantities?</li>
																			<li>What are some quantitative and qualitative features of graphs of functions?</li>
								
										</ol>
									</div>
																													<div class="eoinfo" style="display: block;">
										<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsCoherenceConnections"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a><em> 


Coherence Connections:

							
									</em></p>

										<ol type="1">

																			<li>This expectation represents major work of the grade.</li>
																			<li>In Grade 7, students analyze proportional relationships and use them to solve real-world and mathematical problems.</li>
																			<li>In Grade 8, this expectation connects with defining, evaluating, and comparing functions and with investigating patterns of association in bivariate data.</li>
																			<li>In high school, students use function notation, analyze functions using different representations, build new functions from existing functions, and extend from linear functions to quadratic, exponential, and other more advanced functions.</li>
								
										</ol>
									</div>
													
								</div>
							</div>

						</div>

					</div>
				</div>

			</div>

					
		</div>

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							<a href="#" onclick="$(this).closest('.col').find('.gleDetail').toggleClass('notvisible');return false;">
								<i class="material-icons gleDetail expanderDown">keyboard_arrow_down</i>
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							<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsPGC"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a> Prepared Graduates:</p>
							<ul style="list-style-type:none;">
														<li>MP2. Reason abstractly and quantitatively.</li>
														<li>MP4. Model with mathematics.</li>
														<li>MP7. Look for and make use of structure.</li>
													</ul>
						</div>

						<p class="gleDetail notvisible"><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsGLE"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a> 


Grade Level Expectation:


</p>
						<p style="padding-left:40px;">8.SP.A. Statistics &amp; Probability: Investigate patterns of association in bivariate data.</p>
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Evidence Outcomes:


									</p>

																						<div class="eoinfo" style="display: block;">

			<p><em>Students Can:</em></p>
	
										<ol type="a">

																			<li>Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. (CCSS: 8.SP.A.1)</li>
																			<li>Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. (CCSS: 8.SP.A.2)</li>
																			<li>Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. <i>For example, in a linear model for a biology experiment, interpret a slope of \(1.5\) cm/hr as meaning that an additional hour of sunlight each day is associated with an additional \(1.5\) cm in mature plant height.</i> (CCSS: 8.SP.A.3)</li>
																			<li>Explain that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. <i>For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?</i> (CCSS: 8.SP.A.4)</li>
								
										</ol>
									</div>
																																																				
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Academic Contexts and Connections:


									</p>

																																			<div class="eoinfo" style="display: block;">
										<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsColoradoEssentialSkillsandMathematicalPractices"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a><em> 


Colorado Essential Skills and Mathematical Practices:

							
									</em></p>

										<ol type="1">

																			<li>Recognize and describe patterns in bivariate data. (Entrepreneurial Skills: Inquiry/Analysis)</li>
																			<li>Interpret the contextual meaning of slope and \(y\)-intercept, where applicable in a linear model fit to bivariate data. (MP2)</li>
																			<li>Build statistical models to explore, describe, and generalize the relationship between two variables. (MP4)</li>
																			<li>Use scatter plots and two-way tables to describe possible associations in bivariate data. (MP7)</li>
								
										</ol>
									</div>
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										<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsInquiryQuestions"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a><em> 


Inquiry Questions:

							
									</em></p>

										<ol type="1">

																			<li>In what ways is a scatter plot useful in describing and interpreting the relationship between two quantities?</li>
																			<li>Why would we create a linear model for a set of bivariate data?</li>
																			<li>How do you know when a credible prediction can be made from a linear model of bivariate data?</li>
																			<li>What does a pattern of association look like for categorical data?</li>
								
										</ol>
									</div>
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										<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsCoherenceConnections"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a><em> 


Coherence Connections:

							
									</em></p>

										<ol type="1">

																			<li>This expectation supports the major work of the grade.</li>
																			<li>In previous grades, students apply and extend previous understandings of numbers to the system of rational numbers.</li>
																			<li>In Grade 8, this expectation supports using functions to model relationships between quantities.</li>
																			<li>In high school, students summarize, represent, and interpret data on two categorical and quantitative variables, interpret linear models, and understand independence and conditional probability.</li>
								
										</ol>
									</div>
													
								</div>
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							<div class="col-3"><p><strong>Mathematics</strong></p></div>
							<div class="col-9"><p style="text-align: right;"><strong>Eighth Grade, Standard 4. Geometry</strong></p></div>
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	<a href="/comath" target="_blank">Learn More About Mathematics in Colorado</a>


						| <a href="/standardsandinstruction/essentialskills" target="_blank">Read the Colorado Essential Skills</a>
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							<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsPGC"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a> Prepared Graduates:</p>
							<ul style="list-style-type:none;">
														<li>MP3. Construct viable arguments and critique the reasoning of others.</li>
														<li>MP5. Use appropriate tools strategically.</li>
														<li>MP7. Look for and make use of structure.</li>
														<li>MP8. Look for and express regularity in repeated reasoning.</li>
													</ul>
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						<p class="gleDetail notvisible"><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsGLE"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a> 


Grade Level Expectation:


</p>
						<p style="padding-left:40px;">8.G.A. Geometry: Understand congruence and similarity using physical models, transparencies, or geometry software.</p>
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Evidence Outcomes:


									</p>

																						<div class="eoinfo" style="display: block;">

			<p><em>Students Can:</em></p>
	
										<ol type="a">

																			<li>Verify experimentally the properties of rotations, reflections, and translations: (CCSS: 8.G.A.1)
<ol style="list-style-type: lower-alpha">
<li>Lines are taken to lines, and line segments to line segments of the same length. (CCSS: 8.G.A.1.a)</li>
<li>Angles are taken to angles of the same measure. (CCSS: 8.G.A.1.b)</li>
<li>Parallel lines are taken to parallel lines. (CCSS: 8.G.A.1.c)</li>
</ol></li>
																			<li>Demonstrate that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. (CCSS: 8.G.A.2)</li>
																			<li>Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. (CCSS: 8.G.A.3)</li>
																			<li>Demonstrate that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. (CCSS: 8.G.A.4)</li>
																			<li>Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. <i>For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.</i> (CCSS: 8.G.A.5)</li>
								
										</ol>
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Academic Contexts and Connections:


									</p>

																																			<div class="eoinfo" style="display: block;">
										<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsColoradoEssentialSkillsandMathematicalPractices"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a><em> 


Colorado Essential Skills and Mathematical Practices:

							
									</em></p>

										<ol type="1">

																			<li>Think about how rotations, reflections, and translations of a geometric figure preserve congruence as similar to how properties of operations such as the associative, commutative, and distributive properties preserve equivalence of arithmetic and algebraic expressions. (Entrepreneurial Skills: Critical Thinking/Problem Solving and Inquiry/Analysis)</li>
																			<li>Explain a sequence of transformations that results in a congruent or similar triangle. (MP3)</li>
																			<li>Use physical models, transparencies, geometric software, or other appropriate tools to explore the relationships between transformations and congruence and similarity. (MP5)</li>
																			<li>Use the structure of the coordinate system to describe the locations of figures obtained with rotations, reflections, and translations. (MP7)</li>
																			<li>Reason that since any one rotation, reflection, or translation of a figure preserves congruence, then any sequence of those transformations must also preserve congruence. (MP8)</li>
								
										</ol>
									</div>
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										<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsInquiryQuestions"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a><em> 


Inquiry Questions:

							
									</em></p>

										<ol type="1">

																			<li>How are properties of rotations, reflections, translations, and dilations connected to congruence?</li>
																			<li>How are properties of rotations, reflections, translations, and dilations connected to similarity?</li>
																			<li>Why are angle measures significant regarding the similarity of two figures?</li>
								
										</ol>
									</div>
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Coherence Connections:

							
									</em></p>

										<ol type="1">

																			<li>This expectation represents major work of the grade.</li>
																			<li>In previous grades, students solve problems involving angle measure, area, surface area, and volume, and draw, construct, and also describe geometrical figures and the relationships between them.</li>
																			<li>In Grade 8, this expectation connects with understanding the connections between proportional relationships, lines, and linear equations.</li>
																			<li>In high school, students extend their work with transformations, apply the concepts of transformations to prove geometric theorems, and use similarity to define trigonometric functions.</li>
								
										</ol>
									</div>
													
								</div>
							</div>

						</div>

					</div>
				</div>

			</div>

					
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				<div class="col">

					<div class="gle">


						<p class="float-right">
							<a href="#" onclick="$(this).closest('.col').find('.gleDetail').toggleClass('notvisible');return false;">
								<i class="material-icons gleDetail expanderDown">keyboard_arrow_down</i>
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							<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsPGC"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a> Prepared Graduates:</p>
							<ul style="list-style-type:none;">
														<li>MP3. Construct viable arguments and critique the reasoning of others.</li>
														<li>MP7. Look for and make use of structure.</li>
														<li>MP8. Look for and express regularity in repeated reasoning.</li>
													</ul>
						</div>

						<p class="gleDetail notvisible"><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsGLE"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a> 


Grade Level Expectation:


</p>
						<p style="padding-left:40px;">8.G.B. Geometry: Understand and apply the Pythagorean Theorem.</p>
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Evidence Outcomes:


									</p>

																						<div class="eoinfo" style="display: block;">

			<p><em>Students Can:</em></p>
	
										<ol type="a">

																			<li>Explain a proof of the Pythagorean Theorem and its converse. (CCSS: 8.G.B.6)</li>
																			<li>Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. (CCSS: 8.G.B.7)</li>
																			<li>Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. (CCSS: 8.G.B.8)</li>
								
										</ol>
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Academic Contexts and Connections:


									</p>

																																			<div class="eoinfo" style="display: block;">
										<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsColoradoEssentialSkillsandMathematicalPractices"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a><em> 


Colorado Essential Skills and Mathematical Practices:

							
									</em></p>

										<ol type="1">

																			<li>Think of the Pythagorean Theorem as not just a formula, but a formula that only holds true under certain conditions. (Entrepreneurial Skills: Inquiry/Analysis)</li>
																			<li>Construct a viable argument about why a proof of the Pythagorean Theorem is valid. (MP3)</li>
																			<li>Test to see if a triangle is a right triangle by applying the Pythagorean Theorem. (MP7)</li>
																			<li>Use patterns to recognize and generate Pythagorean triples. (MP8)</li>
								
										</ol>
									</div>
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Inquiry Questions:

							
									</em></p>

										<ol type="1">

																			<li>What is the relationship between the Pythagorean Theorem and its converse? In what ways is each useful?</li>
																			<li>Is it always possible to use the Pythagorean Theorem to find the distance between points on the coordinate plane? How do you know?</li>
								
										</ol>
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Coherence Connections:

							
									</em></p>

										<ol type="1">

																			<li>This expectation represents major work of the grade.</li>
																			<li>In Grades 6 and 7, students solve real-life and mathematical problems involving angle measure, area, surface area, and volume.</li>
																			<li>In Grade 8, this expectation connects with radicals and integer exponents, square roots, and solving simple equations in the form \(x^2 = p\).</li>
																			<li>In high school, students (a) prove and apply trigonometric identities, (b) prove theorems involving similarity, (c) define trigonometric ratios and solve problems involving right triangles, (d) translate between the geometric description and the equation for a conic section, and (e) use coordinates to prove simple geometric theorems algebraically.</li>
								
										</ol>
									</div>
													
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							<a href="#" onclick="$(this).closest('.col').find('.gleDetail').toggleClass('notvisible');return false;">
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							<ul style="list-style-type:none;">
														<li>MP3. Construct viable arguments and critique the reasoning of others.</li>
														<li>MP6. Attend to precision.</li>
													</ul>
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Grade Level Expectation:


</p>
						<p style="padding-left:40px;">8.G.C. Geometry: Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.</p>
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Evidence Outcomes:


									</p>

																						<div class="eoinfo" style="display: block;">

			<p><em>Students Can:</em></p>
	
										<ol type="a">

																			<li>State the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. (CCSS: 8.G.C.9)</li>
								
										</ol>
									</div>
																																																				
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Academic Contexts and Connections:


									</p>

																																			<div class="eoinfo" style="display: block;">
										<p><a style="cursor: pointer" data-toggle="modal" data-target="#myModalDefinitionsColoradoEssentialSkillsandMathematicalPractices"><img src="/apps/standards/images/icon-info.png" alt="More information icon"></a><em> 


Colorado Essential Skills and Mathematical Practices:

							
									</em></p>

										<ol type="1">

																			<li>Efficiently solve problems using established volume formulas. (Professional Skills: Task/Time Management)</li>
																			<li>Describe how the formulas for volumes of cones, cylinders, and spheres relate to one another and to the volume formulas for solids with rectangular bases. (MP3)</li>
																			<li>Use appropriate precision when solving problems involving measurements and volume formulas that describe real-world shapes. (MP6)</li>
								
										</ol>
									</div>
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Inquiry Questions:

							
									</em></p>

										<ol type="1">

																			<li>How are the formulas of cones, cylinders, and spheres similar to each other?</li>
																			<li>How are the formulas of cones, cylinder, and spheres connected to the formulas for pyramids, prisms, and cubes?</li>
								
										</ol>
									</div>
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Coherence Connections:

							
									</em></p>

										<ol type="1">

																			<li>This expectation is in addition to the major work of the grade.</li>
																			<li>In Grade 7, students solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.</li>
																			<li>In Grade 8, this expectation connects with radicals and integer exponents.</li>
																			<li>In high school, students apply geometric concepts in mathematical modeling situations and to solve design problems.</li>
								
										</ol>
									</div>
													
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	      			<p><strong>Prepared Graduates:</strong></p>
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	      					<p>The P-12 concepts and skills that all students who complete the Colorado education system must master to ensure their success in a postsecondary and workforce setting.</p>
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	      					<p><em>What do students need to know?</em></p>
	      					<p><strong>Grade Level Expectation: High Schools</strong>:<br>
	      					The articulation of the concepts and skills of a standard that indicates a student is making progress toward being a prepared graduate.</p>
	      					<p><strong>Grade Level Expectations</strong>:<br>
	      					The articulation, at each grade level, the concepts and skills of a standard that indicates a student is making progress toward being ready for high school.</p>

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	      			<p><strong>Evidence Outcomes:</strong></p>
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	      					<p><em>How do we know that a student can do it?</em></p>
	      					<p>The indication that a student is meeting an expectation at the mastery level.</p>
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				<p><strong>Academic Context and Connections:</strong></p>
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					<p>Academic context and connections are the subject-specific elements needed to create context for learning. This right side section highlights essential skills, practices and other important connections necessary for students to understand, apply and transfer the knowledge and skills within the Grade Level Expectation.</p>
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	      			<p><strong>Inquiry Questions:</strong></p>
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	      					<p>Sample questions intended to promote deeper thinking, reflection and refined understandings precisely related to the grade level expectation.</p>
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	      			<p><strong>Relevance and Application:</strong></p>
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	      					<p>Examples of how the grade level expectation is applied at home, on the job or in a real-world, relevant context.</p>
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				<p><strong>Nature of the Discipline:</strong></p>
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					<p>The characteristics and viewpoint one keeps as a result of mastering the grade level expectation.</p>
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				<p><strong>Supportive Teaching Practices/Adults May</strong></p>
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					<p>Embed standards throughout daily routines, within and across the environment and activities to ensure developmentally appropriate learning opportunities.</p>
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				<p><strong>Colorado Essential Skills</strong></p>
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					<p>The critical skills needed to prepare students to successfully enter the workforce or educational opportunities beyond high school embedded within statute (C.R.S. 22-7-1005) and identified by the Colorado Workforce Development Committee.</p>
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				<p><strong>Colorado Essential Skills and Mathematical Practices</strong></p>
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					<p>This section describes ways students engage with the mathematical content using mathematical practices and essential skills needed to prepare students to successfully enter the workforce or educational opportunities beyond high school pursuant to Colorado Revised Statute 22-7-1005 and identified by the Colorado Workforce Development Committee.</p>
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				<p><strong>Inquiry Questions</strong></p>
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					<p>The sample questions that are intended to promote deeper thinking, reflection and refined understandings precisely related to the grade level expectation.</p>
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				<p><strong>Coherence Connections</strong></p>
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					<p>This section describes how the content described by the Grade Level Expectation and Evidence Outcomes builds from content learned in prior grades, connects to content in the same grade, and supports student learning in later grades.</p>
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				<p><strong>Elaboration on the GLE</strong></p>
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					<p>This section provides greater context for the Grade Level Expectation (GLE) through a description of the understanding about the core ideas that should be developed by students.</p>
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				<p><strong>Colorado Essential Skills and Science and Engineering Practices</strong></p>
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					<p>Skills and major practices that scientists employ as they investigate and build models and theories about the world.  These terms are used to emphasize that engaging in scientific investigation requires not only skill but also knowledge that is specific to each practice. </p>
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				<p><strong>Cross Cutting Concepts</strong></p>
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					<p>The crosscutting concepts have application across all domains of science. As such, they provide one way of linking across the domains through core ideas.</p>
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				<p><strong>Computer Science Practices</strong></p>
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					<p>This section highlights computer science practices connected to the GLE which describe the behaviors and ways of thinking that computationally literate students use to fully engage in today’s data-rich and interconnected world.  The practices naturally integrate with one another and contain language that intentionally overlaps to illuminate the connections among them.</p>
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				<p><strong>Essential Questions</strong></p>
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					<p>These "big picture" questions ask students to more deeply explore the concepts and skills expressed in the GLE.</p>
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				<p><strong>Essential Reasoning Skills</strong></p>
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					<p>These skills develop critical thinking, building awareness to multiple perspectives, and engage students in "thinking about their thinking" and to consider their own attitudes, beliefs, and biases on issues.</p>
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				<p><strong>Colorado Essential Skills and Real-World Application</strong></p>
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					<p>The critical skills needed to prepare students to successfully enter the workforce or educational opportunities beyond high school embedded within statute (C.R.S. 22-7-1005) and identified by the Colorado Workforce Development Committee. Connections to how these skills relate to lifelong learning have been provided.</p>
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				<p><strong>Expand and Connect</strong></p>
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					<p>Ideas that can be used to expand student thinking around the concepts, connect to other musical concepts, or connect to other content areas outside of music. </p>
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				<p><strong>Colorado Essential Skills and Meaning Making</strong></p>
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					<p>Ways in which students demonstrate the ability to form, grapple with, and convey concepts and ideas through visual art and design with real-world application.</p>
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				<p><strong>Learning Experience and Transfer</strong></p>
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					<p>Ideas that can be used to expand student thinking, encourage conceptual curiosity, and connect multiple disciplines and literacies.</p>
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				<p><strong>Nature and Skills of History</strong></p>
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					<p>The characteristics and viewpoint one keeps as a result of mastering the grade level expectation.</p>
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				<p><strong>Nature and Skills of Geography</strong></p>
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					<p>The characteristics and viewpoint one keeps as a result of mastering the grade level expectation.</p>
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				<p><strong>Nature and Skills of Economics</strong></p>
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					<p>The characteristics and viewpoint one keeps as a result of mastering the grade level expectation.</p>
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				<p><strong>Nature and Skills of Civics</strong></p>
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	      					<p>The characteristics and viewpoint one keeps as a result of mastering the grade level expectation.</p>
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				<p><strong>Disciplinary, Information, & Media Literacy</strong></p>
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					<p>The disciplinary, information, and media literacy skills necessary to demonstrate mastery of the Evidence Outcomes. </p>
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				<p><strong>Nature of Dance</strong></p>
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					<p>The characteristics and viewpoint one keeps as a  result of mastering the grade level expectation</p>
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				<p><strong>Health Skills</strong></p>
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					<p>This section connects and focuses on the key health specific skills connected to this grade level expectation.</p>
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				<p><strong>Components of a Physically Literate Individual</strong></p>
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					<p>Connects the GLE to physical literacy and how it supports students ability to move with competence and confidence in a wide variety of physical activities in multiple environments that benefit the healthy development of the whole person. </p>
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        <p style="text-align:center;font-size:.90em;margin-top:50px;"><strong>Need Help?</strong> Submit questions or requests for assistance to <a href="mailto:bruno_j@cde.state.co.us">bruno_j@cde.state.co.us</a></p>
      </div>
    </div>
    <script src="https://code.jquery.com/jquery-3.3.1.slim.min.js" integrity="sha384-q8i/X+965DzO0rT7abK41JStQIAqVgRVzpbzo5smXKp4YfRvH+8abtTE1Pi6jizo" crossorigin="anonymous"></script>
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    <script src="/apps/standards/js/bootstrap.min.js"></script>
    <script src="/apps/standards/js/scripts.js"></script>
  

<script type="text/javascript">
	
	var selectedClass;
	var currentFilter = [
    "4,10,0"		];
	var gradeSelect = $("#srchFltrGl").html();
	var categorySelect = $("#srchFltrSt").html();


	function updateSearch() {

		selectedClass = $("#srchFltrCa option:selected").data('caclass');

		$("#srchFltrGl option").each(function( index, value ) {
			if (!$(value).hasClass( "notactive" )) {
				$(value).addClass( "notactive" ); 
			}
		});

		$("#srchFltrSt option").each(function( index, value ) {
			if (!$(value).hasClass( "notactive" )) {
				$(value).addClass( "notactive" ); 
			}
		});

		$("#srchFltrGl option.filterDefault, #srchFltrSt option.filterDefault").removeClass( "notactive" );
		
		if ($("#srchFltrCa").val() != "0") {

			$("#srchFltrGl option." + selectedClass ).removeClass( "notactive" );

			$("#srchFltrGl option.notactive").remove();
			// $("#srchFltr01").removeClass( "notactive" ); 

			$("#srchFltrGl").prop("disabled", false);

			if ($("#srchFltrGl").val() != "0") {

				$("#srchFltrSt option." + selectedClass ).removeClass( "notactive" );
				$("#srchFltrSt option.notactive").remove();

				// $("#srchFltr02").removeClass( "notactive" ); 
				// $("#srchFltrSt").removeAttr('disabled');
				// $("#addSearchFilter").removeAttr('disabled');

				$("#srchFltrSt, #addSearchFilter").prop("disabled", false);
				// $("").prop("disabled", false);
				
			} else {

				// $("#srchFltrSt").attr('disabled');
				// $("#addSearchFilter").attr('disabled');

				$("#srchFltrSt, #addSearchFilter").prop("disabled", true);
				// $("").prop("disabled", true);

			}

		} else {

			$("#srchFltrGl, #srchFltrSt, #addSearchFilter").prop("disabled", true);
			// $("").prop("disabled", true);

			// if (!$( "#srchFltr01" ).hasClass( "notactive" )) {
			// 	$("#srchFltr01").addClass( "notactive" ); 
			// }

			// if (!$( "#srchFltr02" ).hasClass( "notactive" )) {
			// 	$("#srchFltr02").addClass( "notactive" ); 
			// }

			// $("#srchFltrGl option, #srchFltrSt option").removeClass( "notactive" );
			// $("#srchFltrGl option, #srchFltrSt option").addClass( "notactive" );

		}

	}

	function resetGradeSelect() {
		$("#srchFltrGl").html(gradeSelect);
	}

	function resetCategorySelect() {
		$("#srchFltrSt").html(categorySelect);
	}

	function cleanSearch() {
		$("#srchFltrGl option").addClass( "hidden" );
		$("#srchFltrSt option").addClass( "hidden" );
		if ($("#srchFltrCa").val() != "0") {
			selectedClass = $("#srchFltrCa option:selected").data('caclass');
			$(".srchFltr01").removeClass( "hidden" );
			$("#srchFltrGl ." + selectedClass + ", #srchFltrGl .default").removeClass( "hidden" );
			if ($("#srchFltrGl").val() != "0") {
				$(".srchFltr02").removeClass( "hidden" );
				$("#srchFltrSt ." + selectedClass + ", #srchFltrSt .default").removeClass( "hidden" );
			}
		}
	}

	function updateSearch01() {
		if ($("#srchFltrCa").val() != "0") {
			selectedClass = $("#srchFltrCa option:selected").data('caclass');
			$(".srchFltr02").addClass( "hidden" );
			$("#srchFltrGl").val("0");
			$("#srchFltrSt").val("0");
			// $(".srchFltr01").toggleClass( "hidden" );
			// $( "#srchFltrGl" ).each(function( index ) {
			//   console.log( index + ": " + $( this ).text() );
			// });
			$("#srchFltrGl option").removeClass( "hidden" );
			$("#srchFltrGl option").addClass( "hidden" );
			$("#srchFltrGl ." + selectedClass + ", #srchFltrGl .default").removeClass( "hidden" );
			$(".srchFltr01").removeClass( "hidden" );
		} else {
			$(".srchFltr01").addClass( "hidden" );
			$(".srchFltr02").addClass( "hidden" );
			$("#srchFltrGl").val("0");
			$("#srchFltrSt").val("0");
		}
	}

	function updateSearch02() {
		if ($("#srchFltrGl").val() != "0") {
			// $(".srchFltr01").toggleClass( "hidden" );
			// $( "#srchFltrGl" ).each(function( index ) {
			//   console.log( index + ": " + $( this ).text() );
			// });
			$("#srchFltrSt option").removeClass( "hidden" );
			$("#srchFltrSt option").addClass( "hidden" );
			$("#srchFltrSt ." + selectedClass + ", #srchFltrSt .default").removeClass( "hidden" );
			$(".srchFltr02").removeClass( "hidden" );
		} else {
			$(".srchFltr02").addClass( "hidden" );
			$("#srchFltrSt").val("0");
		}
	}


	function updateSearchForm() {
		var searchItem01a = $("#srchFltrCa").val();
		var searchItem01b = $("#srchFltrGl").val();
		var searchItem01c = $("#srchFltrSt").val();

		// var searchItem02a = $("#srchFltrCa").val();
		// var searchItem02b = $("#srchFltrGl").val();
		// var searchItem02c = $("#srchFltrSt").val();

		// var searchItem03a = $("#srchFltrCa").val();
		// var searchItem03b = $("#srchFltrGl").val();
		// var searchItem03c = $("#srchFltrSt").val();

		var searchVar1 = searchItem01a + ',' + searchItem01b + ',' + searchItem01c;
		var searchVar2 = "0" // searchItem02a + ',' + searchItem02b + ',' + searchItem02c;
		var searchVar3 = "0" // searchItem03a + ',' + searchItem03b + ',' + searchItem03c;

		$("#searchForm").attr("action", "/apps/standards/" + searchVar1 + "/" + searchVar2 + "/" + searchVar3 + "/");

		if (searchItem01b != "0") {
			$("#searchForm").submit();
		}

	}

	function displayFilters() {
		var thefilter01a = $("#srchFltrCa option[value='4']").text();
		$("#fltr0a").text(thefilter01a);
		var thefilter01b = $("#srchFltrGl option[value='10']").text();
		$("#fltr0b").text(thefilter01b);
		var thefilter01c = $("#srchFltrSt option[value='0']").text();
		$("#fltr0c").text(thefilter01c);
	}

	function buildCurrentFilter() {
	    		currentFilter.push("4,10,0");
			}

	function buildSearchSet() {
		if (($("#srchFltrCa option:selected").val() != "0") && ($("#srchFltrGl option:selected").val() != "0")) {
			currentFilter.push($("#srchFltrCa option:selected").val() + "," + $("#srchFltrGl option:selected").val() + "," + $("#srchFltrSt option:selected").val());
		}
		
		$("#searchForm").attr("action", "/apps/standards/" + currentFilter.join("/"));
		return true;
	}

	function removeFilter(theFilter) {
		currentFilter.splice(theFilter, 1);
		// currentFilter[theFilter] = 0;
		$("#searchForm").attr("action", "/apps/standards/" + currentFilter.join("/"));
		$("#searchForm").submit();
	}


	$(function() {
		// $("#srchFltrCa").val("4");
		// $("#srchFltrGl").val("10");
		// $("#srchFltrSt").val("0");
		// cleanSearch();
		// displayFilters();
		// buildCurrentFilter();
	});

</script>

  </body>
</html>