The Colorado Department of Education

Offices | Staff Contacts | Colorado.gov

New Colorado P-12 Academic Standards

Current Display Filter: Mathematics - All - by Specific Prepared Graduate Competency - (Remove PGC Filter)

Content Area: Mathematics
Grade Level Expectations: High School
Standard: 2. Patterns, Functions, and Algebraic Structures

Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter)

Concepts and skills students master:

1. Functions model situations where one quantity determines another and can be represented algebraically, graphically, and using tables
Evidence Outcomes 21st Century Skill and Readiness Competencies

Students Can:

  1. Formulate the concept of a function and use function notation. (CCSS: F-IF)
    • Explain that a function is a correspondence from one set (called the domain) to another set (called the range) that assigns to each element of the domain exactly one element of the range.1 (CCSS: F-IF.1)
    • Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. (CCSS: F-IF.2)
    • Demonstrate that sequences are functions,2 sometimes defined recursively, whose domain is a subset of the integers. (CCSS: F-IF.3)
  2. Interpret functions that arise in applications in terms of the context. (CCSS: F-IF)
    • For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features3 given a verbal description of the relationship. * (CCSS: F-IF.4)
    • Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.4 * (CCSS: F-IF.5)
    • Calculate and interpret the average rate of change5 of a function over a specified interval. Estimate the rate of change from a graph.* (CCSS: F-IF.6)
  3. Analyze functions using different representations. (CCSS: F-IF)
    • Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. * (CCSS: F-IF.7)
    • Graph linear and quadratic functions and show intercepts, maxima, and minima. (CCSS: F-IF.7a)
    • Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. (CCSS: F-IF.7b)
    • Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. (CCSS: F-IF.7c)
    • Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. (CCSS: F-IF.7e)
    • Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. (CCSS: F-IF.8)
      • Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. (CCSS: F-IF.8a)
      • Use the properties of exponents to interpret expressions for exponential functions. (CCSS: F-IF.8b)
      • Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (CCSS: F-IF.9)
  4. Build a function that models a relationship between two quantities. (CCSS: F-BF)
    • Write a function that describes a relationship between two quantities.* (CCSS: F-BF.1)
      • Determine an explicit expression, a recursive process, or steps for calculation from a context. (CCSS: F-BF.1a)
      • Combine standard function types using arithmetic operations.8 (CCSS: F-BF.1b)
    • Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. (CCSS: F-BF.2)
  5. Build new functions from existing functions. (CCSS: F-BF)
    • Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k, 9 and find the value of k given the graphs.10 (CCSS: F-BF.3)
    • Experiment with cases and illustrate an explanation of the effects on the graph using technology.
    • Find inverse functions.11 (CCSS: F-BF.4)
  6. Extend the domain of trigonometric functions using the unit circle. (CCSS: F-TF)
    • Use radian measure of an angle as the length of the arc on the unit circle subtended by the angle. (CCSS: F-TF.1)
    • Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. (CCSS: F-TF.2)

Inquiry Questions:

  1. Why are relations and functions represented in multiple ways?
  2. How can a table, graph, and function notation be used to explain how one function family is different from and/or similar to another?
  3. What is an inverse?
  4. How is “inverse function” most likely related to addition and subtraction being inverse operations and to multiplication and division being inverse operations?
  5. How are patterns and functions similar and different?
  6. How could you visualize a function with four variables, such as \(x^2 + y^2 +z^2 +w^2 =1\)?
  7. Why couldn’t people build skyscrapers without using functions?
  8. How do symbolic transformations affect an equation, inequality, or expression?

Relevance & Application:

  1. Knowledge of how to interpret rate of change of a function allows investigation of rate of return and time on the value of investments. (PFL)
  2. Comprehension of rate of change of a function is important preparation for the study of calculus.
  3. The ability to analyze a function for the intercepts, asymptotes, domain, range, and local and global behavior provides insights into the situations modeled by the function. For example, epidemiologists could compare the rate of flu infection among people who received flu shots to the rate of flu infection among people who did not receive a flu shot to gain insight into the effectiveness of the flu shot.
  4. The exploration of multiple representations of functions develops a deeper understanding of the relationship between the variables in the function.
  5. The understanding of the relationship between variables in a function allows people to use functions to model relationships in the real world such as compound interest, population growth and decay, projectile motion, or payment plans.
  6. Comprehension of slope, intercepts, and common forms of linear equations allows easy retrieval of information from linear models such as rate of growth or decrease, an initial charge for services, speed of an object, or the beginning balance of an account.
  7. Understanding sequences is important preparation for calculus. Sequences can be used to represent functions including \(e^x\), \(e^{x^2}\), sin x, and cos x.

Nature Of:

  1. Mathematicians use multiple representations of functions to explore the properties of functions and the properties of families of functions.
  2. Mathematicians model with mathematics. (MP)
  3. Mathematicians use appropriate tools strategically. (MP)
  4. Mathematicians look for and make use of structure. (MP)

1 If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). (CCSS: F-IF.1)

2 For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n \(\geq\) 1. (CCSS: F-IF.3)

3 Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. (CCSS: F-IF.4)

4 For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. (CCSS: F-IF.5)

5 presented symbolically or as a table. (CCSS: F-IF.6)

6 For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10,. (CCSS: F-IF.8b)

7 For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. (CCSS: F-IF.9)

8 For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. (CCSS: F-BF.1b)

9 both positive and negative. (CCSS: F-BF.3)

10 Include recognizing even and odd functions from their graphs and algebraic expressions for them. (CCSS: F-BF.3)

11 Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.
For example, f(x) =2 \(x^3\) or f(x) = (x+1)/(x–1) for x \(\neq\) 1. (CCSS: F-BF.4a)

Content Area: Mathematics
Grade Level Expectations: Fifth Grade
Standard: 2. Patterns, Functions, and Algebraic Structures

Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter)

Concepts and skills students master:

1. Number patterns are based on operations and relationships
Evidence Outcomes 21st Century Skill and Readiness Competencies

Students Can:

  1. Generate two numerical patterns using given rules. (CCSS: 5.OA.3)
  2. Identify apparent relationships between corresponding terms. (CCSS: 5.OA.3)
  3. Form ordered pairs consisting of corresponding terms from the two patterns, and graphs the ordered pairs on a coordinate plane.1 (CCSS: 5.OA.3)
  4. Explain informally relationships between corresponding terms in the patterns. (CCSS: 5.OA.3)
  5. Use patterns to solve problems including those involving saving and checking accounts.2 (PFL)
  6. Explain, extend, and use patterns and relationships in solving problems, including those involving saving and checking accounts such as understanding that spending more means saving less (PFL)

Inquiry Questions:

  1. How do you know when there is a pattern?
  2. How are patterns useful?

Relevance & Application:

  1. The use of a pattern of elapsed time helps to set up a schedule. For example, classes are each 50 minutes with 5 minutes between each class.
  2. The ability to use patterns allows problem-solving. For example, a rancher needs to know how many shoes to buy for his horses, or a grocer needs to know how many cans will fit on a set of shelves.

Nature Of:

  1. Mathematicians use creativity, invention, and ingenuity to understand and create patterns.
  2. The search for patterns can produce rewarding shortcuts and mathematical insights.
  3. Mathematicians construct viable arguments and critique the reasoning of others. (MP)
  4. Mathematicians model with mathematics. (MP)
  5. Mathematicians look for and express regularity in repeated reasoning. (MP)

1 For example, given the rule "add 3" and the starting number 0, and given the rule "add 6" and the starting number 0, generate terms and the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. (CCSS: 5.OA.3)

2 such as the pattern created when saving $10 a month

Content Area: Mathematics
Grade Level Expectations: Fourth Grade
Standard: 2. Patterns, Functions, and Algebraic Structures

Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter)

Concepts and skills students master:

1. Number patterns and relationships can be represented by symbols
Evidence Outcomes 21st Century Skill and Readiness Competencies

Students Can:

  1. Generate and analyze patterns and identify apparent features of the pattern that were not explicit in the rule itself.1 (CCSS: 4.OA.5)
    • Use number relationships to find the missing number in a sequence
    • Use a symbol to represent and find an unknown quantity in a problem situation
    • Complete input/output tables
    • Find the unknown in simple equations
  2. Apply concepts of squares, primes, composites, factors, and multiples to solve problems
    • Find all factor pairs for a whole number in the range 1–100. (CCSS: 4.OA.4)
    • Recognize that a whole number is a multiple of each of its factors. (CCSS: 4.OA.4)
    • Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. (CCSS: 4.OA.4)
    • Determine whether a given whole number in the range 1–100 is prime or composite. (CCSS: 4.OA.4)

Inquiry Questions:

  1. What characteristics can be used to classify numbers into different groups?
  2. How can we predict the next element in a pattern?
  3. Why do we use symbols to represent missing numbers?
  4. Why is finding an unknown quantity important?

Relevance & Application:

  1. Use of an input/output table helps to make predictions in everyday contexts such as the number of beads needed to make multiple bracelets or number of inches of expected growth.
  2. Symbols help to represent situations from everyday life with simple equations such as finding how much additional money is needed to buy a skateboard, determining the number of players missing from a soccer team, or calculating the number of students absent from school.
  3. Comprehension of the relationships between primes, composites, multiples, and factors develop number sense. The relationships are used to simplify computations with large numbers, algebraic expressions, and division problems, and to find common denominators.

Nature Of:

  1. Mathematics involves pattern seeking.
  2. Mathematicians use patterns to simplify calculations.
  3. Mathematicians model with mathematics. (MP)

1 For example, given the rule "Add 3" and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way. (CCSS: 4.OA.5)

Content Area: Mathematics
Grade Level Expectations: Preschool
Standard: 4. Shape, Dimension, and Geometric Relationships

Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter)

Concepts and skills students master:

1. Shapes can be observed in the world and described in relation to one another
Evidence Outcomes 21st Century Skill and Readiness Competencies

Students Can:

  1. Match, sort, group and name basic shapes found in the natural environment
  2. Sort similar groups of objects into simple categories based on attributes
  3. Use words to describe attributes of objects
  4. Follow directions to arrange, order, or position objects

Inquiry Questions:

  1. How do we describe where something is?
  2. Where do you see shapes around you?
  3. How can we arrange these shapes?
  4. Why do we put things in a group?
  5. What is the same about these objects and what is different?
  6. What are the ways to sort objects?

Relevance & Application:

  1. Shapes and position help students describe and understand the environment such as in cleaning up, or organizing and arranging their space.
  2. Comprehension of order and position helps students learn to follow directions.
  3. Technology games can be used to arrange and position objects.
  4. Sorting and grouping allows people to organize their world. For example, we set up time for clean up, and play.

Nature Of:

  1. Geometry affords the predisposition to explore and experiment.
  2. Mathematicians organize objects in different ways to learn about the objects and a group of objects.
  3. Mathematicians attend to precision. (MP)
  4. Mathematicians look for and make use of structure. (MP)