## Description

- Overview:
- This lesson unit is intended to help teachers assess how well students are able to: articulate verbally the relationships between variables arising in everyday contexts; translate between everyday situations and sketch graphs of relationships between variables; interpret algebraic functions in terms of the contexts in which they arise; and reflect on the domains of everyday functions and in particular whether they should be discrete or continuous.

- Level:
- Lower Primary, Upper Primary, Middle School, High School
- Grades:
- Kindergarten, Grade 1, Grade 2, Grade 3, Grade 4, Grade 5, Grade 6, Grade 7, Grade 8, Grade 9, Grade 10, Grade 11, Grade 12
- Material Type:
- Assessment, Lesson Plan
- Provider:
- Shell Center for Mathematical Education, U.C. Berkeley
- Provider Set:
- Mathematics Assessment Project (MAP)
- Date Added:
- 04/26/2013

- License:
- Creative Commons Attribution Non-Commercial No Derivatives
- Media Format:
- Downloadable docs, Text/HTML

## Standards

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# Common Core State Standards Math

Grades 9-12,Functions: Linear, Quadratic, and Exponential ModelsCluster: Construct and compare linear, quadratic, and exponential models and solve problems

Standard: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Linear, Quadratic, and Exponential ModelsCluster: Construct and compare linear, quadratic, and exponential models and solve problems

Standard: Construct and compare linear, quadratic, and exponential models and solve problems. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Linear, Quadratic, and Exponential ModelsCluster: Construct and compare linear, quadratic, and exponential models and solve problems

Standard: For exponential models, express as a logarithm the solution to ab^(ct) = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Linear, Quadratic, and Exponential ModelsCluster: Construct and compare linear, quadratic, and exponential models and solve problems

Standard: Distinguish between situations that can be modeled with linear functions and with exponential functions.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Understand the concept of a function and use function notation.

Standard: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1 (n is greater than or equal to 1).

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Analyze functions using different representations

Standard: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Analyze functions using different representations

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

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Analyze functions using different representations

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

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their 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.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Interpret functions that arise in applications in terms of the context

Standard: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Analyze functions using different representations

Standard: 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.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Analyze functions using different representations

Standard: Use the properties of exponents to interpret expressions for exponential functions. 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), and classify them as representing exponential growth and decay.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Interpret functions that arise in applications in terms of the context

Standard: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. 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.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Interpret functions that arise in applications in terms of the context

Standard: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Linear, Quadratic, and Exponential ModelsCluster: Construct and compare linear, quadratic, and exponential models and solve problems

Standard: Prove that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Linear, Quadratic, and Exponential ModelsCluster: Construct and compare linear, quadratic, and exponential models and solve problems

Standard: Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Linear, Quadratic, and Exponential ModelsCluster: Construct and compare linear, quadratic, and exponential models and solve problems

Standard: Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.*

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Reason abstractly and quantitatively. Mathematically proficient students make sense of the quantities and their relationships in problem situations. Students bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Linear, Quadratic, and Exponential ModelsCluster: Interpret expressions for functions in terms of the situation they model

Standard: Interpret the parameters in a linear or exponential function in terms of a context.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Understand the concept of a function and use function notation.

Standard: 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. 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).

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Understand the concept of a function and use function notation.

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

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Analyze functions using different representations

Standard: Graph linear and quadratic functions and show intercepts, maxima, and minima.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Analyze functions using different representations

Standard: Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Analyze functions using different representations

Standard: Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Analyze functions using different representations

Standard: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Analyze functions using different representations

Standard: (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Linear, Quadratic, and Exponential Models

Standard: Construct and compare linear, quadratic, and exponential models and solve problems

Indicator: Construct and compare linear, quadratic, and exponential models and solve problems. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Linear, Quadratic, and Exponential Models

Standard: Construct and compare linear, quadratic, and exponential models and solve problems

Indicator: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Linear, Quadratic, and Exponential Models

Standard: Construct and compare linear, quadratic, and exponential models and solve problems

Indicator: Distinguish between situations that can be modeled with linear functions and with exponential functions.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Understand the concept of a function and use function notation.

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

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Understand the concept of a function and use function notation.

Indicator: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ‰ä´ 1 (n is greater than or equal to 1).

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Linear, Quadratic, and Exponential Models

Standard: Construct and compare linear, quadratic, and exponential models and solve problems

Indicator: For exponential models, express as a logarithm the solution to ab^(ct) = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Understand the concept of a function and use function notation.

Indicator: 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. 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).

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Analyze functions using different representations

Indicator: (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Analyze functions using different representations

Indicator: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Analyze functions using different representations

Indicator: Graph linear and quadratic functions and show intercepts, maxima, and minima.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Analyze functions using different representations

Indicator: Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Analyze functions using different representations

Indicator: Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Linear, Quadratic, and Exponential Models

Standard: Interpret expressions for functions in terms of the situation they model

Indicator: Interpret the parameters in a linear or exponential function in terms of a context.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Linear, Quadratic, and Exponential Models

Standard: Construct and compare linear, quadratic, and exponential models and solve problems

Indicator: Prove that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Linear, Quadratic, and Exponential Models

Standard: Construct and compare linear, quadratic, and exponential models and solve problems

Indicator: Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Linear, Quadratic, and Exponential Models

Standard: Construct and compare linear, quadratic, and exponential models and solve problems

Indicator: Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Analyze functions using different representations

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

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Analyze functions using different representations

Indicator: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Analyze functions using different representations

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

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Interpret functions that arise in applications in terms of the context

Indicator: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. 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.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Interpret functions that arise in applications in terms of the context

Indicator: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Interpret functions that arise in applications in terms of the context

Indicator: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Analyze functions using different representations

Indicator: 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.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Analyze functions using different representations

Indicator: Use the properties of exponents to interpret expressions for exponential functions. 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), and classify them as representing exponential growth and decay.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their 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.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Reason abstractly and quantitatively. Mathematically proficient students make sense of the quantities and their relationships in problem situations. Students bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize"Óto abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents"Óand the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?"ť They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Degree of Alignment: Not Rated (0 users)

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# Tags (9)

- Mathematics
- Algebra and Calculus
- Algebraic Functions
- CCSS
- Common Core Math
- Common Core PD
- Functions
- ODE Learning
- Real World Math

## Comments