Colorado Academic Standards Online

Colorado Essential Skills and Mathematical Practices:

1. Interpret expressions and their parts. (Entrepreneurial Skills: Inquiry/Analysis)
2. Make sense of variables, constants, constraints, and relationships in the context of a problem. (MP1)
3. Think abstractly about how terms in an expression can be rewritten or how terms can be combined and treated as a single object to be computed with. (MP2)
4. Discern a pattern or structure to see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, see \(5 - 3 \left(x - y \right)^2\) as \(5\) minus a positive number times a square and use that to realize that its value cannot be more than \(5\) for any real numbers \(x\) and \(y\). (MP7)

Inquiry Questions:

1. How could you show algebraically that the two expressions \(\left( n + 2 \right)^2 - 4\) and \(n^2 + 4n\) are equivalent? How could you show it visually, with a diagram or picture?

Coherence Connections:

1. This expectation represents major work of high school and includes a modeling (★) outcome.
2. In Grades 6 and 7, students use the properties of operations to generate equivalent expressions.
3. In high school, students continue to use properties of operations to rewrite expressions, gaining fluency and engaging in intentional manipulation of algebraic expressions, and strategically using different representations.
4. The separation of algebra and functions in the Standards is intended to specify the difference between the two, as mathematical concepts between expressions and equations on the one hand and functions on the other. Students often enter college-level mathematics courses conflating all three of these.

Colorado Essential Skills and Mathematical Practices:

1. Transform expressions to highlight properties and set up solution strategies. (Entrepreneurial Skills: Critical Thinking/Problem Solving)
2. Recognize the difference in the structure of linear, quadratic, and other equations and apply the appropriate strategies to solve. (MP7)
3. Notice, for example, the regularity in the way terms combine to make zero when expanding \(\left( x-1 \right) \left( x+1 \right)\), \(\left( x-1 \right) \left( x^2 + x + 1 \right)\), and \(\left( x - 1 \right) \left( x^3 + x^2 + x + 1 \right)\), and how it might lead to the general formula for the sum of a finite geometric series. (MP8)

Inquiry Questions:

1. What does the vertex form of a quadratic equation, \(y = a \left( x - h \right)^2 + k\), tell us about its graph that the standard form, \(y = ax^2 + bx + c\), does not?
2. What does the factored form of a quadratic equation, \(y = a \left( x - p \right) \left( x - q \right)\), tell us about the graph that the other two forms do not?

Coherence Connections:

1. This expectation represents major work of high school and includes modeling (★) and advanced (+) outcomes.
2. In middle school, students manipulate algebraic expressions to create equivalent expressions. In high school, students' manipulations become more strategic and advanced in response to increasingly complex expressions.
3. As students progress through high school, they should become increasingly proficient with mathematical actions such as ”doing and undoing”; for example, looking at expressions generated through the distributive property and identifying expressions that might have led to a given outcome. They do not use “FOIL” as a justification for the multiplication of two binomials, understanding that such mnemonics are not conceptually defensible and do not generalize.

Colorado Essential Skills and Mathematical Practices:

1. Students make hypotheses and draw conclusions about polynomials and operations on them. (Entrepreneurial Skills: Inquiry/Analysis)
2. Understand how types of numbers and operations form a closed system. (MP7)
3. See how operations on polynomials yield polynomials, much like how operations on integers yield integers. (MP8)

Inquiry Questions:

1. \(f(x) = x^2\) is a nonnegative polynomial because for all values of \(x\), \(f(x) \geq 0\). If you add two nonnegative polynomials together, do you always, sometimes, or never get another nonnegative polynomial? What if you multiply them?

Coherence Connections:

1. This expectation represents major work of high school.
2. In previous grades, students understand algebraic expressions as values in which one or more letters are used to stand for an unspecified or unknown number and use the properties of operations to write expressions in different but equivalent forms.
3. In high school, polynomials and rational expressions form a system in which they can be added, subtracted, multiplied, and divided. Polynomials are analogous to the integers; rational expressions are analogous to the rational numbers.

Colorado Essential Skills and Mathematical Practices:

1. Understand that the zeros of a polynomial that models a real-world context generally convey useful information about that context. (Professional Skills: Information Literacy)
2. Make sense of the relationship between zeros and factors of polynomials using graphs, tables, real-world contexts, and equations in factored forms. (MP1)
3. Reason with a factored quadratic, such as \(\left(x + 2\right)\left(x - 3\right) = 0\), abstractly as “something times zero is zero” and “zero times something is zero.” (MP2)
4. Look for the ways the structure of polynomial equations are different from linear equations and use appropriate methods, such as factoring, to reveal the polynomial’s zeros. (MP7)

Inquiry Questions:

1. What is the relationship between factoring a polynomial and finding its zeros?

Coherence Connections:

1. This expectation represents major work of high school.
2. In previous grades, students rewrite algebraic expressions in equivalent forms.
3. In high school, students construct polynomial functions with specified zeros. This is the first step in a progression that can lead, as an extension topic, to constructing polynomial functions whose graphs pass through any specified set of points in the plane.
4. A particularly important application of polynomial division is the case where a polynomial \(p(x)\) is divided by a linear factor of the form \(\left(x−a\right)\) for a real number \(a\). In this case, the remainder is the value \(p(a)\) of the polynomial at \(x=a\). This topic should not be reduced to “synthetic division,” which reduces the method to a matter of carrying numbers between registers while obscuring the reasoning that makes the result evident. It is important to regard the Remainder Theorem as a theorem, not a technique.
5. Experience with constructing polynomial functions satisfying given conditions is useful preparation not only for calculus (where students learn more about approximating functions), but for understanding the mathematics behind curve-fitting methods used in applications of statistics and computer graphics.

Colorado Essential Skills and Mathematical Practices:

1. Understand the connections between the coefficients in expansions of \(\left(x + y\right)^n\) and the values in Pascal’s Triangle. (Entrepreneurial Skills: Inquiry/Analysis)
2. Recognize patterns in the binomial coefficients as they appear in Pascal’s Triangle. (MP8)

Inquiry Questions:

1. Can you find a case (a specific value of \(x\) and \(y\)) for which the equation \(\left( x^2 + y^2 \right)^2 = \left ( x^2 - y^2 \right)^2 + \left( 2xy \right)^2\) does not generate a Pythagorean triple?

Coherence Connections:

1. This expectation represents advanced (+) work of high school.
2. Polynomials form a rich ground for mathematical explorations that reveal relationships in the system of integers.

Colorado Essential Skills and Mathematical Practices:

1. Reason with rational expressions like \(\frac{x^2 + 5x + 6}{x + 2}\) not as a sum divided by a sum, but as a yet-to-be-factored numerator where one of the factors, \(\left(x + 2\right)\), will make \(1\) when divided by the denominator. (MP2)
2. Determine when it is appropriate to use a computer algebra system or calculator instead of paper and pencil to rewrite rational expressions. (MP5)
3. Understand how types of numbers and operations form a closed system. (MP7)

Inquiry Questions:

1. How is dividing polynomials like and unlike long division with whole numbers?

Coherence Connections:

1. This expectation represents major work of high school and includes an advanced (+) outcome.
2. The analogy between polynomials and integers also applies to polynomial and integer division. Students should recognize the high school method of polynomial long division to find quotients and remainders of polynomials as similar to the method of integer long division first experienced in Grade 4.
3. In high school, polynomials and rational expressions form a system in which they can be added, subtracted, multiplied, and divided. Polynomials are analogous to the integers; rational expressions are analogous to the rational numbers.
4. Expressing rational expressions in different forms allows students to see different properties of the graph, such as horizontal asymptotes.

Colorado Essential Skills and Mathematical Practices:

1. Reason contextually, within the real-world context of the problem, and decontextually, about the mathematics, without regard to the context. (MP2)
2. Model and solve problems arising in everyday life, society, and the workplace. Interpret mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. (MP4)
3. Use pencil and paper, concrete models, a ruler, a calculator, a spreadsheet, a computer algebra system, and/or dynamic geometry software to make sense of and solve mathematical equations. (MP5)
4. Use the structure of an equation and a sequence of operations to rearrange the equation to isolate a variable by itself on one side of the equal sign. (MP7)

Inquiry Questions:

1. What are some similarities and differences in creating equations of different types?
2. What features of a real-world context might indicate that the equation that models it is quadratic instead of linear?

Coherence Connections:

1. This expectation represents major work of high school and includes modeling (★) outcomes.
2. In previous grades, students model real-world situations with mathematics. Modeling becomes a major objective in high school, including not only an increase in the complexity of the equations studied, but an upgrade of the student’s ability in every part of the modeling cycle.
3. The repertoire of functions that is acquired during high school allows students to create more complex equations, including equations arising from linear and quadratic expressions, and simple rational and exponential expressions. Students in high school start using parameters in their equations, to represent whole classes of equations or to represent situations where the equation is to be adjusted to fit data.

Colorado Essential Skills and Mathematical Practices:

1. Articulate steps of solving an equation using written communication skills. (Civic/Interpersonal Skills: Communication)
2. Describe a logical flow of mathematics, using stated assumptions, definitions, and previously established results in constructing arguments, and explain solving equations as a process of reasoning that demystifies “extraneous” solutions that can arise under certain solution procedures. (MP3)
3. Understand that solving equations is a process of reasoning where properties of operations can be used to change expressions on either side of the equation to equivalent expressions. (MP7)

Inquiry Questions:

1. What types of equations can have extraneous solutions? What types cannot? Why?
2. How are extraneous solutions generated?

Coherence Connections:

1. This expectation represents major work of high school.
2. In previous grades, students solve equations and inequalities.
3. In high school, students extend their skills with solving equations and inequalities to generalize about the solution methods themselves. They name assumptions, justify their steps, and view the process through the lens of proof rather than simple obtaining of a solution.
4. Students’ understanding of solving equations as a process of reasoning demystifies extraneous solutions that can arise under certain solution procedures. The reasoning begins from the assumption that \(x\) is a number that satisfies the equation and ends with a list of possibilities for \(x\). But not all the steps are necessarily reversible, and so it is not necessarily true that every number in the list satisfies the equation.

Colorado Essential Skills and Mathematical Practices:

1. Solve equations and draw conclusions from their solutions. (Entrepreneurial Skills: Inquiry/Analysis)
2. Strategically use calculators or computer technology, and recognize instances when the form of the equation doesn’t lend itself to using these tools. (MP5)
3. Analyze the structure of a quadratic equation to determine the most efficient solution strategy. (MP7)

Inquiry Questions:

1. How does the initial form of a quadratic equation cue us to an appropriate solution strategy?

Coherence Connections:

1. This expectation represents major work of high school.
2. With an understanding of solving equations as a reasoning process, students can organize the various methods for solving different types of equations into a coherent picture. For example, solving linear equations involves only steps that are reversible (adding a constant to both sides, multiplying both sides by a non-zero constant, transforming an expression on one side into an equivalent expression). Therefore, solving linear equations does not produce extraneous solutions. The process of completing the square also involves only this same list of steps, and so converts any quadratic equation into an equivalent equation of the form \(\left( x - p \right)^2 = q\) that has exactly the same solutions.
3. It is traditional for students to spend a lot of time on various techniques of solving quadratic equations, which are often presented as if they are completely unrelated (factoring, completing the square, the quadratic formula). In fact, the key step in completing the square involves factoring and the quadratic formula is nothing more than an encapsulation of the method of completing the square, expressing the actions repeated in solving a collection of quadratic equations with numerical coefficients with a single formula. Rather than long drills on techniques of dubious value, students with an understanding of the underlying reasoning behind all these methods are opportunistic in their application, choosing the method that best suits the situation at hand.

Colorado Essential Skills and Mathematical Practices:

1. Solve systems of equations by using graphs and algebraic methods. (Entrepreneurial Skills: Critical Thinking/Problem Solving)
2. Substitute expressions for variables when solving systems of equations, thinking of the expressions as single objects rather than a process that must be computed before substitution. (MP2)
3. Use a matrix to model a system of equations, which may itself be a model of a real-world situation. (MP4)

Inquiry Questions:

1. How is the solution to a system of equations related to the graph of the system? What if the system has no solution? What if the system has infinitely many solutions?
2. Two lines may intersect in zero, one, or infinitely many points. How many intersections may there be between a line and the graph of a quadratic equation?

Coherence Connections:

1. This expectation represents major work of high school and includes advanced (+) outcomes.
2. In previous grades, students solve systems of two linear equations graphically and by using substitution, and understand the concept of a solution of a system of equations.
3. In high school, students solve systems of equations using methods that include but are not limited to graphical, elimination/linear combination, substitution, and modeling. Students may use graphing calculators or other technology to model and find approximate solutions for systems of equations.

Colorado Essential Skills and Mathematical Practices:

1. Analyze and use information presented in equations and visually in graphs. (Entrepreneurial Skills: Literacy/Reading)
2. Make sense of correspondences between equations, verbal descriptions, tables, and graphs. (MP1)
3. Use graphing calculators and/or computer technology to reason about and solve systems of equations and inequalities. (MP5)
4. Specify units of measure, label axes to clarify the correspondence with quantities in a problem, calculate accurately and efficiently, and express numerical answers with a degree of precision appropriate for the problem context. (MP6)
5. Use the characteristics and structures of function families to understand and generalize about solutions to equations and inequalities. (MP7)

Inquiry Questions:

1. How is a solution to a system of inequalities different than the solution to a system of equations?
2. How are the types of functions in the system related to the number of solutions it might have? Can you give an example to explain your thinking?
3. How many different ways can you find to solve \(x^2 = \left( 2x - 9 \right)^2\)?

Coherence Connections:

1. This expectation represents major work of high school.
2. In Grade 8, students begin their study of systems of equations with systems of two linear equations. With a focus on graphical solutions, they build understanding of the concept of a system and its solution(s) or lack thereof. The concept is built upon in high school, extending to algebraic solution strategies as well as considering solutions of systems of non-linear equations and of inequalities.
3. In high school, students use algebraic solution methods that produce precise solutions and understand that these can be represented graphically or numerically. Students may use graphing calculators or other technology to generate tables of values, graph, and solve systems involving a variety of functions.

Colorado Essential Skills and Mathematical Practices:

1. Describe sequences as functions. (MP2)
2. Use accurate terms and symbols when describing functions and using function notation. (MP6)
3. Understand a function as a correspondence where each element of the domain is assigned to exactly one element of the range; this structure does not “turn inputs into outputs”; rather, it describes the relationship between elements in two sets. (MP7)

Inquiry Questions:

1. Besides the notation we use, what makes a function different from an equation?
2. Why is it important to know if an equation is a function?

Coherence Connections:

1. This expectation represents major work of high school.
2. In Grade 8, students define, evaluate and compare functions. Although students are expected to give examples of functions that are not linear functions, linear functions are the focus.
3. In high school, students deepen their understanding of the notion of function, expanding their repertoire to include quadratic and exponential functions.
4. In calculus, the concepts of function together with the rate of change are integral to reason about how variables operate together.

Colorado Essential Skills and Mathematical Practices:

1. Use functions and their graphs to model, interpret, and generalize about real-world situations. (MP4)
2. Graph functions and interpret key features of the graphs or use key features to construct a graph; use technology as a tool to visualize and understand how various functions behave in different representations. (MP5)
3. Make structural comparisons between linear, exponential, quadratic and higher order polynomial, rational, radical and trigonometric functions to describe commonalities, consistencies, and differences. (MP7)

Inquiry Questions:

1. In what ways does a real-world context influence the domain of the function that models it?
2. How are slope and rate of change related?

Coherence Connections:

1. This expectation represents major work of high school and includes modeling (★) outcomes.
2. In Grade 8, students understand the connections between proportional relationships, lines, and linear equations, and analyze and solve linear equations and pairs of simultaneous linear equations.
3. The rate of change of a linear function is equal to the slope of the line that is its graph. And because the slope of a line is constant, that is, between any two points it is the same, “the rate of change” has an unambiguous meaning for a linear function. In high school, the concept of slope is generalized to rates of change. Students understand that for linear functions, the rate of change is a constant, and for nonlinear functions, we refer to average rates of change over a given interval.

Colorado Essential Skills and Mathematical Practices:

1. Reason abstractly and understand the connections between the symbolic representation, the table of values, and the key features of the graph of a function. (MP2)
2. Use calculators or other graphing software to explore and analyze the graphs of complex and advanced functions. Use the understanding gained to sketch graphs by hand, when appropriate. (MP5)
3. Attend to important terms, definitions, and symbols when graphing, describing, and writing equivalent forms of functions. (MP6)

Inquiry Questions:

1. How can we rewrite a function to illustrate its key features? Give an example of a function written two different ways, one where one or more key features is evident, and another where they are not.
2. Which types of functions share underlying characteristics? How does this help us understand the function families?

Coherence Connections:

1. This expectation represents major work of high school and includes modeling (★) and advanced (+) outcomes.
2. In Grade 8, students understand the connections between proportional relationships, lines, and linear equations, graph linear equations, and analyze and solve linear equations and pairs of simultaneous linear equations.
3. In high school, students are able to recognize, construct, and apply attributes of exponential and quadratic functions, and also use the families of exponential and quadratic functions in a more general sense as a way to model and explain phenomena.
4. Across high school mathematics courses, students have opportunities to compare and contrast functions as they reason about the structure inherent in functions in general and the structure within specific families of functions. Considering functions with the same domains can be a useful classification for comparing and contrasting.

Colorado Essential Skills and Mathematical Practices:

1. Students apply their understanding of functions to real-world contexts. (MP4)

Inquiry Questions:

1. Why does a function require one output for every input?
2. How can the ideas of cause and effect be developed through the building of functions?

Coherence Connections:

1. This expectation represents major work of high school and includes modeling (★) and advanced (+) outcomes.
2. In previous grades, students understand how a function is defined and use equations to model relationships between two variables in context.
3. In high school, students build from the understanding of input and output to understanding dependence in mathematical relationships. Work with building functions is closely connected to expectations in the algebra domain, and provides opportunity to apply the modeling cycle (see Appendix).

Colorado Essential Skills and Mathematical Practices:

1. Use calculators or computer technology to create, describe, and analyze related functions. (Professional Skills: Use Information and Communication Technologies)
2. Create verbal and written explanations of the generalities they find across and between function families. (MP3)
3. Use accurate terms, definitions and mathematical symbols when building, describing, and manipulating functions. (MP6)
4. Extend the patterns of transformations of functions and make connections between function representations. (MP7)

Inquiry Questions:

1. What is meant by a “function family”?
2. Describe cases where the inverse of a function is only a function when the domain is restricted.

Coherence Connections:

1. This expectation is in addition to the major work of high school and includes advanced (+) outcomes.
2. In previous grades, students understand how a function is defined and describe how the slope and \(y\)-intercept of a linear function are evident on the graph of a linear equation.
3. Students develop a notion of naturally occurring families of functions that deserve particular attention. This can come from experimenting with the effect on the graph of simple algebraic transformations of the input and output variables. Quadratic and absolute value functions are good contexts for getting a sense of the effects of many of these transformations, but eventually, students need to understand these ideas abstractly and be able to talk about them for any function \(f\).

Colorado Essential Skills and Mathematical Practices:

1. Reason about and with situations that can be modeled by functions. In high school, focused study on multiple function types adds complexity to the reasoning required. (MP1)
2. Use linear, exponential, and logarithmic functions and their properties and graphs to model and reason about real-world situations. (MP4)
3. Distinguish between situations that can be modeled with linear functions and with exponential functions using understandings of rates of growth and factors of growth over equal intervals. (MP7)

Inquiry Questions:

1. In what ways are linear and exponential functions similar? In what ways are quadratic and exponential functions similar?
2. How can observing the connections between table, graph, and function notation help you better understand the function?

Coherence Connections:

1. This expectation represents major work of high school and includes modeling (★) outcomes.
2. In Grade 8, students understand that the ratio of the rise and run for any two distinct points on a line is the same and that this concept is referred to as slope or as rate of change.
3. To support the high school major work of deeply understanding functions, students’ work with linear and exponential functions as models of real-world phenomena lends itself to reasoning, analysis, comparison, and generalizations about linear and exponential functions.

Colorado Essential Skills and Mathematical Practices:

1. Make sense of a mathematical model of a real-world situation and describe and interpret its meaning both mathematically and contextually. (Entrepreneurial Skills: Critical Thinking/Problem Solving)
2. Both decontextualize—abstract a given situation and representing it symbolically and manipulate the representing symbols without necessarily attending to their referents—and contextualize—pause as needed during the manipulation process in order to probe into the referents for the symbols involved. (MP2)
3. Use mathematics to model, interpret, and reason about real-world contexts. (MP4)

Inquiry Questions:

1. What does the linear component, \(bx+c\), of a quadratic expression determine about the quadratic function?
2. How do the \(a\) and \(b\) values in the exponential function \(f(x)=ab^x\) compare to the \(a\) and \(b\) values in the linear function \(g(x)=a + bx\)?

Coherence Connections:

1. This expectation represents major work of high school and includes a modeling (★) outcome.
2. In Grade 8, students model linear relationships with functions with graphs and tables.
3. In high school, students describe rate of change between two quantities as well as initial values both within and apart from context. An understanding of how the interval remains the same in a linear situation as well as how the interval increases or decreases in a nonlinear situation is developed in high school. Students use recursive reasoning to analyze patterns and structures in tables in order to create functions which model the situation in context.

Colorado Essential Skills and Mathematical Practices:

1. Measure distance around a circle in units the length of the radius of the circle, or radians, and see how this measure stays the same for all equivalent angles, regardless of the circle's size. (MP7)

Inquiry Questions:

1. Trigonometric ratios are defined as ratios of one side of a right triangle to another. What are radians a ratio of?

Coherence Connections:

1. This expectation represents advanced (+) work of high school.
2. Trigonometry is a component of mathematics unique to high school where the functions standard and geometry standard overlap and support each other. In Grade 8, students understand and apply the Pythagorean Theorem.
3. In high school, students begin their study of trigonometry with right triangles. However, this limits the angles considered to those between \(0\) degrees and \(90\) degrees. Later, students expand the types of angles considered, and use the unit circle to make connections between the trigonometric ratios derived from right triangles and those of angles not representable by right triangles. Students learn, by similarity, that the radian measure of an angle can be defined as the quotient of arc length to radius. As a quotient of two lengths, therefore, radian measure is “dimensionless,” explaining why the “unit” is often omitted when measuring angles in radians.

Colorado Essential Skills and Mathematical Practices:

1. Students recognize a real-world situation as periodic and construct an appropriate trigonometric representation. (Entrepreneurial Skills: Critical Thinking/Problem Solving)
2. Make sense of periodic quantities and their relationships in problem situations, both within real-world contexts and with context removed. (MP2)
3. Apply trigonometric functions and graphs to model periodic situations arising in everyday life, society, and the workplace. (MP4)
4. Use the regularity inherent in periodic functions to gain a deeper understanding of their mathematical characteristics. (MP8)

Inquiry Questions:

1. How does an understanding of the unit circle support an understanding of periodic phenomena?
2. What are examples of phenomena that can be modeled using trigonometric functions?
3. How can you determine if a periodic phenomena should be represented with a sine function or a cosine function?

Coherence Connections:

1. This expectation is in addition to the major work of high school and includes modeling (★) and advanced (+) outcomes.
2. In previous grades, students calculate the area and circumference of circles.
3. In high school, students develop the ideas of periodic motion as simply being the graph of the movement around the circle. Transformations of trigonometric functions should be connected to the structures of transformations for other functions studied in high school.
4. In high school, students apply trigonometry to many different authentic contexts. Of all the subjects that students learn in geometry, trigonometry may have the greatest application in college and careers due in part to its ability to model real-world functions.

Colorado Essential Skills and Mathematical Practices:

1. Explain the relationship between algebra and trigonometry. (Civic/Interpersonal Skills: Communication)
2. Make sense of trigonometric quantities as expressions and use their relationships in problem situations. (MP2)
3. See trigonometric expressions as single objects or as being composed of several objects. (MP7)

Inquiry Questions:

1. How is the Pythagorean identity related to the Pythagorean Theorem?
2. How is the identity \(\sin^2(\theta) + \cos^2(\theta) = 1\) related to the equation of a circle centered at the origin?

Coherence Connections:

1. This expectation represents advanced (+) work of high school.
2. In Grade 8, students understand and apply the Pythagorean Theorem and its converse.
3. The Pythagorean Identity is a foundational trigonometric identity that must be understood through its components both in and out of context.