OREGON MATH STANDARDS (2021): [HS.AFN]
Overview
The intent of clarifying statements is to provide additional guidance for educators to communicate the intent of the standard to support the future development of curricular resources and assessments aligned to the 2021 math standards.
Clarifying statements can be in the form of succinct sentences or paragraphs that attend to one of four types of clarifications: (1) Student Experiences; (2) Examples; (3) Boundaries; and (4) Connection to Math Practices.
2021 Oregon Math Guidance: HS.AFN.A.1
Cluster: HS.AFN.A - Describe functions by using both symbolic and graphical representations.
STANDARD: HS.AFN.A.1
Standards Statement (2021):
Understand a function as a rule that assigns a unique output for every input and that functions model situations where one quantity determines another.
Connections:
Preceding Pathway Content (2021) | Subsequent Pathway Content (2021) | Cross Domain Connections (2021) | Common Core (CCSS) (2010) |
8.AFN.A.1 | HS.AFN.A.2, HS.AFN.B.4, HS.AFN.B.5, HS.AFN.C.6, HS.AFN.C.7 | N/A | HSF.IF.A.1 HS.AFN.A Crosswalk |
Standards Guidance:
Clarification
- Functions are often represented by tables, expressions or graphs. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range.
- Modeling examples should include both contexts where only one quantity can be considered the independent variable as well as contexts where both quantities could.
Boundaries
- Standard included expectation students connect the concept of a function to use of notation where if 𝑓 is a function and 𝑥 is an element of its domain, then (𝑥) denotes the output of f corresponding to the input 𝑥. The graph of 𝑓 is the graph of the equation 𝑦=(𝑥).
- Concept of a function introduced in grade 8, but formal use of function notation is not an expectation until high school.
Examples
- Illustrative Mathematics:
2021 Oregon Math Guidance: HS.AFN.A.2
Cluster: HS.AFN.A - Describe functions by using both symbolic and graphical representations.
STANDARD: HS.AFN.A.2
Standards Statement (2021):
Use function notation and interpret statements that use function notation in terms of the context and the relationship it describes.
Connections:
Preceding Pathway Content (2021) | Subsequent Pathway Content (2021) | Cross Domain Connections (2021) | Common Core (CCSS) (2010) |
HS.AFN.A.1 | HS.AFN.D.9 | HS.GM.A.1 | HSF.IF.A.2 HSF.BF.A.1 HS.AFN.A Crosswalk |
Standards Guidance:
Clarifications
- Student should develop a deep understanding of function notation to build, evaluate, and interpret linear functions; this understanding will be applied to other functions studied hereafter.
- Students should be able to interpret the domain when given a function expressed numerically, algebraically, and graphically.
- Students should apply their understanding of function notation from their work with linear functions to build, evaluate, and interpret quadratic functions using function notation.
- Students should apply their understanding of function notation from their work with non-linear functions when needed to build, evaluate, and interpret functions in authentic contexts.
Progressions
- MP4: mathematical modeling
Examples
- Illustrative Mathematics:
- Cell phones
- The Random Walk [Version 1] [Version 2] [Version 3]
- Using Function Notation [Verson 1] [Version 2]
- Yam in the Oven
- 1,000 is half of 2,000
- A Sum of Functions
- Kimi and Jordan
- Lake Algae
- Skeleton Tower
- Summer Intern
2021 Oregon Math Guidance: HS.AFN.A.3
Cluster: HS.AFN.A - Describe functions by using both symbolic and graphical representations.
STANDARD: HS.AFN.A.3
Standards Statement (2021):
Calculate and interpret the average rate of change of a function over a specified interval.
Connections:
Preceding Pathway Content (2021) | Subsequent Pathway Content (2021) | Cross Domain Connections (2021) | Common Core (CCSS) (2010) |
8.AFN.B.4 | HS.AFN.C.6 | HS.DR.D.11 | HSF.IF.B.6 HS.AFN.A Crosswalk |
Standards Guidance:
Clarifications
- Students should be given opportunities to estimate the rate of change from a graph.
- Students should be able to show that linear functions grow by equal differences over equal intervals and recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
- Students should be able to compare this behavior to that of the average rate of change of quadratic functions. This can be shown by algebraic proof, with a table showing differences, or by calculating average rates of change over equal intervals.
Boundaries
- Work with functions presented as graphs, tables or symbolically.
- Students should choose specified intervals for analysis of functions with substantially varying rates of change.
- Interpreting also includes estimates of the rate of change from a graph.
- MP6: precision
- MP7: structural thinking
Teaching Strategies
- Functions can be presented symbolically, as a graph, or as a table.
Examples
- Illustrative Mathematics:
2021 Oregon Math Guidance: HS.AFN.B.4
Cluster: HS.AFN.B - Compare and relate functions using common attributes.
STANDARD: HS.AFN.B.4
Standards Statement (2021):
Compare properties of two functions using multiple representations. Distinguish functions as members of the same family using common attributes.
Connections:
Preceding Pathway Content (2021) | Subsequent Pathway Content (2021) | Cross Domain Connections (2021) | Common Core (CCSS) (2010) |
8.AFN.A.2, HS.AFN.A.1 | HS.AFN.D.10 | N/A | HSF.IF.C.9 HS.AFN.B Crosswalk |
Standards Guidance:
Clarifications
- Students should be able to compare key characteristics of exponential functions with the key characteristics of linear and quadratic function.
- Students should be able to observe using graphs and tables that a quantity is increasing .
Boundaries
- Functions can be represented algebraically, graphically, numerically in tables, or by verbal descriptions.
Examples
- Given a graph of one function and an algebraic expression for another, determine which has the larger y-intercept.
- Given a graph of one quadratic function and an algebraic equation for another, students should be able to determine which has the larger maximum.
- Given a graph of one function and an algebraic equation for another, students should be able to determine which has the larger y-intercept.
- Illustrative Mathematics:
2021 Oregon Math Guidance: HS.AFN.B.5
Cluster: HS.AFN.B - Compare and relate functions using common attributes.
STANDARD: HS.AFN.B.5
Standards Statement (2021):
Relate the domain of a function to its graph and to its context.
Connections:
Preceding Pathway Content (2021) | Subsequent Pathway Content (2021) | Cross Domain Connections (2021) | Common Core (CCSS) (2010) |
HS.AFN.A.1 | N/A | N/A | HSF.IF.B.5 HS.AFN.B Crosswalk |
Standards Guidance:
Boundaries
- Contexts can demand discrete vs. continuous and domain restrictions.
- MP4: mathematical model
- MP6: precision
Terminology
- Use symbolic notation to represent the domain and range of a linear function, considering the specific context.
- (-∞,∞)
- [3, ∞)
- D: {x| xϵƦ}
- D: {x| x > 0}
- D: {x| x = 1,2,3,4,5,…}
- R: {y| y = 10,20,30,…}
Examples
- If the function h(n) gives the number of hours it takes a person to assemble n engines in a factory, then the set of positive integers would be an appropriate domain for the function.
- If the function h(t) gives the path of a projectile over time, t, then the set of non- negative real numbers would be an appropriate domain for the function because time does not include negative values.
- A bird is building a nest in a tree 36 feet above the ground. The bird drops a stick from the nest. The function f(x) = -16x2 + 36 describes the height of the stick in feet after x seconds. Graph this function. Identify the domain and range of this function. (A student should be able to determine that the appropriate values for the domain and range of this graph are 0 ≤ x ≤ 1.5 and 0 ≤ y ≤ 36, respectively.)
- Illustrative Mathematics:
2021 Oregon Math Guidance: HS.AFN.C.6
Cluster: HS.AFN.C - Represent functions graphically and interpret key features in terms of the equivalent symbolic representation.
STANDARD: HS.AFN.C.6
Standards Statement (2021):
Interpret key features of functions, from multiple representations, and conversely predict features of functions from knowledge of context.
Connections:
Preceding Pathway Content (2021) | Subsequent Pathway Content (2021) | Cross Domain Connections (2021) | Common Core (CCSS) (2010) |
HS.AFN.A.1, HS.AFN.A.3 | HS.AFN.D.10 | N/A | HSF.IF.B.4 HS.AFN.C Crosswalk |
Standards Guidance:
Clarifications
- Students should be able to express characteristics in interval and set notation with linear functions.
- Students should be able to interpret the key characteristics of the graph in a contextual situation.
Boundaries
- Key features include: domain, range, discrete, continuous, intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums.
- Representations include: graphs, tables, spreadsheet representations, as well as symbolic.
Teaching Strategies
- Students should be able to use graphs created by hand and with technology, verbal descriptions, tables, and function notation when analyzing linear functions in context.
- Students should be given opportunities to use interactive graphing technologies to explore and analyze key characteristics of linear functions, including domain, range, intercepts, intervals where the function is increasing or decreasing, positive or negative, maximums and minimums over a specified interval, and end behavior.
Examples
- Illustrative Mathematics:
2021 Oregon Math Guidance: HS.AFN.C.7
Cluster: HS.AFN.C - Represent functions graphically and interpret key features in terms of the equivalent symbolic representation.
STANDARD: HS.AFN.C.7
Standards Statement (2021):
Graph functions using technology to show key features.
Connections:
Preceding Pathway Content (2021) | Subsequent Pathway Content (2021) | Cross Domain Connections (2021) | Common Core (CCSS) (2010) |
HS.AFN.A.1 | HS.AFN.D.9 | N/A | HSF.IF.C.7 HS.AFN.C Crosswalk |
Standards Guidance:
Clarifications
- Students should be able to sketch a graph showing key features including domain, range, and intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; asymptotes; end behavior.
- Key characteristics of the quadratic functions should be expressed in interval and set- builder notation using inequalities.
Boundaries
- Key features include: specific values when context demands; domain and range; discrete or continuous; intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maxima and minima.
- Use technology to graph functions expressed symbolically or in tables, with intentional choices of window and scale. In some simple cases, graphing functions could by hand or for approximations.
- Graph linear and quadratic functions and show intercepts, maxima, and minima.★
- Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.★
- Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.★
- Graph exponential and logarithmic functions, showing intercepts and end behavior.★
Teaching Strategies
- Students should be able to use verbal descriptions, tables, and graphs created using interactive technology tools.
Examples
- 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.
- The function can be presented symbolically, as a graph, or as a table.
- Students should be able to estimate the rate of change from a graph.
- Illustrative Mathematics:
2021 Oregon Math Guidance: HS.AFN.D.8
Cluster: HS.AFN.D - Model a wide variety of authentic situations using functions through the process of making and changing assumptions, assigning variables, and finding solutions to contextual problems.
STANDARD: HS.AFN.D.8
Standards Statement (2021):
Model situations involving arithmetic patterns. Use a variety of representations such as pictures, graphs, or an explicit formula to describe the pattern.
Connections:
Preceding Pathway Content (2021) | Subsequent Pathway Content (2021) | Cross Domain Connections (2021) | Common Core (CCSS) (2010) |
8.AFN.B.4 | HS.AFN.D.10 | N/A | HSF.BF.A.2 HS.AFN.D Crosswalk |
Standards Guidance:
Clarifications
- Students should be able to:
- make connections between linear functions and arithmetic sequences presented in contextual situations.
- build and interpret arithmetic sequences as functions presented graphically and algebraically.
- Sequences can be defined explicitly.
- The focus of this learning objective is on building and interpreting arithmetic sequences.
Examples
- By graphing or calculating terms, students should be able to show how the arithmetic sequence in explicit form a1=7, an = 2n-1 + 7; and the function f(x) = 2x + 5 (when x is a natural number) define the same sequence.
- MP2: quantitative and abstract reasoning
- MP4: mathematical modeling
- Illustrative Mathematics:
2021 Oregon Math Guidance: HS.AFN.D.9
Cluster: HS.AFN.D - Model a wide variety of authentic situations using functions through the process of making and changing assumptions, assigning variables, and finding solutions to contextual problems.
STANDARD: HS.AFN.D.9
Standards Statement (2021):
Identify and interpret the effect on the graph of a function when the equation has been transformed.
Connections:
Preceding Pathway Content (2021) | Subsequent Pathway Content (2021) | Cross Domain Connections (2021) | Common Core (CCSS) (2010) |
HS.AFN.A.2, HS.AFN.C.7 | N/A | HS.GM.A.1, HS.GM.A.2, HS.GM.A.4 | HSF.BF.B.3 HS.AFN.D Crosswalk |
Standards Guidance:
Teaching Strategies
- Students should be given opportunities to experiment with cases and illustrate an explanation of the effects on the graph using technology.
Boundaries
- Transformations include translations (f(x)+k, and f(x-h)), reflections (e.g. -f(x) and f(-x), and dilations (e.g. a*f(x)). Interpretations include accounting for different choices of variables, such as initial values or units.
- Full proficiency with linear functions and developing proficiency with exponential functions is expected. Technology provides opportunities for exploration with non-linear functions.
- MP4: mathematical modeling
- MP5: using graphing technology
Examples
- Illustrative Mathematics:
2021 Oregon Math Guidance: HS.AFN.D.10
Cluster: HS.AFN.D - Model a wide variety of authentic situations using functions through the process of making and changing assumptions, assigning variables, and finding solutions to contextual problems.
STANDARD: HS.AFN.D.10
Standards Statement (2021):
Explain why a situation can be modeled with a linear function, an exponential function, or neither. In a given model, explain the meaning of coefficients and features of functions used, such as slope for a linear model.
Connections:
Preceding Pathway Content (2021) | Subsequent Pathway Content (2021) | Cross Domain Connections (2021) | Common Core (CCSS) (2010) |
8.AFN.A.3, 8.AFN.B.4, 8.AFN.B.5, HS.AFN.B.4, HS.AFN.C.6, HS.AFN.D.8 | N/A | N/A | HSF.LE.A.1 HS.AFN.D Crosswalk |
Standards Guidance:
Clarifications
- Students should be provided with opportunities to learn mathematics in the context of real-life problems.
- Contextual, mathematical problems are mathematical problems presented in context where the context makes sense, realistically and mathematically, and allows for students to make decisions about how to solve the problem (model with mathematics).
Terminology
- Linear functions grow by equal differences over equal intervals.
- Exponential functions grow by equal factors over equal intervals.
Boundaries
- Identify situations in which one quantity changes at a constant rate per unit interval relative to another.
- Identify situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
Teaching Strategies
- Students should be able to fluently navigate between mathematical representations that are presented numerically, algebraically, and graphically.
- For graphical representations, students should be given opportunities to analyze graphs using interactive graphing technologies.
Progressions
- Students should be able to use the content learned in this course to create a mathematical model to explain real-life phenomena.
- MP4: Mathematical Modeling
Examples
- Illustrative Mathematics: