| Code |
Description |
| MA.912.AR.5.1: | Solve one-variable exponential equations using the properties of exponents. |
| MA.912.AR.5.2: | Solve one-variable equations involving logarithms or exponential expressions. Interpret solutions as viable in terms of the context and identify any extraneous solutions. |
| MA.912.AR.5.3: | Given a mathematical or real-world context, classify an exponential function as representing growth or decay.Clarifications:
Clarification 1: Within the Algebra 1 course, exponential functions are limited to the forms  , where b is a whole number greater than 1 or a unit fraction, or  , where  .
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| MA.912.AR.5.4: | Write an exponential function to represent a relationship between two quantities from a graph, a written description or a table of values within a mathematical or real-world context. |
| MA.912.AR.5.5: | Given an expression or equation representing an exponential function, reveal the constant percent rate of change per unit interval using the properties of exponents. Interpret the constant percent rate of change in terms of a real-world context. |
| MA.912.AR.5.6: | Given a table, equation or written description of an exponential function, graph that function and determine its key features.Clarifications:
Clarification 1: Key features are limited to domain; range; intercepts; intervals where the function is increasing, decreasing, positive or negative; constant percent rate of change; end behavior and asymptotes.
Clarification 2: Instruction includes representing the domain and range with inequality notation, interval notation or set-builder notation.
Clarification 3: Within the Algebra 1 course, notations for domain and range are limited to inequality and set-builder.
Clarification 4: Within the Algebra 1 course, exponential functions are limited to the forms
 , where b is a whole number greater than 1 or a unit fraction or  , where  . |
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| MA.912.AR.5.7: | Solve and graph mathematical and real-world problems that are modeled with exponential functions. Interpret key features and determine constraints in terms of the context.Clarifications:
Clarification 1: Key features are limited to domain; range; intercepts; intervals where the function is increasing, decreasing, positive or negative; constant percent rate of change; end behavior and asymptotes. Clarification 2: Instruction includes representing the domain, range and constraints with inequality notation, interval notation or set-builder notation. Clarification 3: Instruction includes understanding that when the logarithm of the dependent variable is taken and graphed, the exponential function will be transformed into a linear function. Clarification 4: Within the Mathematics for Data and Financial Literacy course, problem types focus on money and business.
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| MA.912.AR.5.8: | Given a table, equation or written description of a logarithmic function, graph that function and determine its key features.Clarifications:
Clarification 1: Key features are limited to domain; range; intercepts; intervals where the function is increasing, decreasing, positive or negative; end behavior; and asymptotes.
Clarification 2: Instruction includes representing the domain and range with inequality notation, interval notation or set-builder notation.
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| MA.912.AR.5.9: | Solve and graph mathematical and real-world problems that are modeled with logarithmic functions. Interpret key features and determine constraints in terms of the context.Clarifications:
Clarification 1: Key features are limited to domain; range; intercepts; intervals where the function is increasing, decreasing, positive or negative; end behavior; and asymptotes.Clarification 2: Instruction includes representing the domain, range and constraints with inequality notation, interval notation or set-builder notation.
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This cluster includes the following access points.
| Access Point Number |
Access Point Title |
| MA.912.AR.5.AP.2: | Solve one-variable equations involving logarithms or exponential expressions. Identify any extraneous solutions. |
| MA.912.AR.5.AP.3: | Given a real-world context, identify an exponential function as representing growth or decay. |
| MA.912.AR.5.AP.4: | Select an exponential function to represent two quantities from a graph or a table of values. |
| MA.912.AR.5.AP.5: | Given an expression or equation representing an exponential function, reveal the constant percent rate of change per unit interval using the properties of exponents. |
| MA.912.AR.5.AP.6: | Given a table, equation or written description of an exponential function, select the graph that represents the function. |
| MA.912.AR.5.AP.7: | Given a mathematical and/or real-world problem that is modeled with exponential functions, solve the mathematical problem, or select the graph using key features (in terms of context) that represents this model. |
| MA.912.AR.5.AP.8: | Given an equation of a logarithmic function, select the graph of that function. |
| MA.912.AR.5.AP.9: | Given a mathematical and/or real-world problem that is modeled with logarithmic functions, solve the mathematical problem, or select the graph using key features (in terms of context) that represents this model. |
Vetted resources educators can use to teach the concepts and skills in this topic.
| Name |
Description |
| Exponential Functions Part 3: Decay: | Learn about exponential decay as you calculate the value of used cars by examining equations, graphs, and tables in this interactive tutorial. |
| Exponential Functions Part 2: Growth: | Learn about exponential growth in the context of interest earned as money is put in a savings account by examining equations, graphs, and tables in this interactive tutorial. |
| Exponential Functions Part 1: | Learn about exponential functions and how they are different from linear functions by examining real world situations, their graphs and their tables in this interactive tutorial. |
| Creating Exponential Functions: | Follow as we construct an exponential function from a graph, from a table of values, and from a description of a relationship in the real world in this interactive tutorial. |
| Name |
Description |
| You’re Pulling My Leg – or Candy!: | Students will explore how exponential growth and decay equations can model real-world problems. Students will also discover how manipulating the variables in an exponential equation changes the graph. Students will watch a Perspectives Video to see how exponential growth is modeled in the real world. |
| My Geometry Classroom: | Students will learn how to find the area and perimeter of multiple polygons in the coordinate plane using the composition and decomposition methods, applying the Distance Formula and Pythagorean Theorem. Students will complete a Geometry Classroom Floor Plan group activity. Students will do a short presentation to discuss their results which leads to the realization that polygons with the same perimeter can have different areas. Students will also complete an independent practice and submit an exit ticket at the end of the lesson. |
| The Copernicus' Travel: | This lesson uses Inverse Trigonometric Ratios to find acute angle measures in right triangles. Students will analyze the given information and determine the best method to use when solving right triangles. The choices reviewed are Trigonometric Ratios, The Pythagorean Theorem, and Special Right Triangles. |
| Which Function?: | This activity has students apply their knowledge to distinguish between numerical data that can be modeled in linear or exponential forms. Students will create mathematical models (graph, equation) that represent the data and compare these models in terms of the information they show and their limitations. Students will use the models to compute additional information to predict future outcomes and make conjectures based on these predictions. |
| Exponential Graphing Using Technology: | This lesson is teacher/student directed for discovering and translating exponential functions using a graphing app. The lesson focuses on the translations from a parent graph and how changing the coefficient, base and exponent values relate to the transformation. |
| Name |
Description |
| Algae Blooms: | In this example, students are asked to write a function describing the population growth of algae. It is implied that this is exponential growth. |
| What functions do two graph points determine?: | This problem solving task challenges students to find the linear, exponential and quadratic functions based on two points. |
| US Population 1790-1860: | This problem solving task asks students to solve five exponential and linear function problems based on a US population chart for the years 1790-1860. |
| Two Points Determine an Exponential Function II: | This problem solving tasks asks students to find the values of points on a graph. |
| Two Points Determine an Exponential Function I: | This problem solving task asks students to graph a function and find the values of points on a graph. |
| Snail Invasion: | The purpose of this task is to give students experience modeling a real-world example of exponential growth, in a context that provides a vivid illustration of the power of exponential growth, for example the cost of inaction for a year. There is an opportunity for further discussion based on part (c), since the ratio of costs from one year to the next is the same in each part. |
| Rumors: | This problem is an exponential function example that uses the real-world problem of how fast rumors spread. |
| Rising Gas Prices - Compounding and Inflation: | The purpose of this task is to give students an opportunity to explore various aspects of exponential models (e.g., distinguishing between constant absolute growth and constant relative growth, solving equations using logarithms, applying compound interest formulas) in the context of a real world problem with ties to developing financial literacy skills. In particular, students are introduced to the idea of inflation of prices of a single commodity, and are given a very brief introduction to the notion of the Consumer Price Index for measuring inflation of a body of goods. |
| Newton's Law of Cooling: | The coffee cooling experiment is a popular example of an exponential model with immediate appeal. The model is realistic and provides a good context for students to practice work with exponential equations. |
Vetted resources students can use to learn the concepts and skills in this topic.
| Title |
Description |
| Exponential Functions Part 3: Decay: | Learn about exponential decay as you calculate the value of used cars by examining equations, graphs, and tables in this interactive tutorial. |
| Exponential Functions Part 2: Growth: | Learn about exponential growth in the context of interest earned as money is put in a savings account by examining equations, graphs, and tables in this interactive tutorial. |
| Exponential Functions Part 1: | Learn about exponential functions and how they are different from linear functions by examining real world situations, their graphs and their tables in this interactive tutorial. |
| Creating Exponential Functions: | Follow as we construct an exponential function from a graph, from a table of values, and from a description of a relationship in the real world in this interactive tutorial. |
| Title |
Description |
| Algae Blooms: | In this example, students are asked to write a function describing the population growth of algae. It is implied that this is exponential growth. |
| What functions do two graph points determine?: | This problem solving task challenges students to find the linear, exponential and quadratic functions based on two points. |
| US Population 1790-1860: | This problem solving task asks students to solve five exponential and linear function problems based on a US population chart for the years 1790-1860. |
| Two Points Determine an Exponential Function II: | This problem solving tasks asks students to find the values of points on a graph. |
| Two Points Determine an Exponential Function I: | This problem solving task asks students to graph a function and find the values of points on a graph. |
| Snail Invasion: | The purpose of this task is to give students experience modeling a real-world example of exponential growth, in a context that provides a vivid illustration of the power of exponential growth, for example the cost of inaction for a year. There is an opportunity for further discussion based on part (c), since the ratio of costs from one year to the next is the same in each part. |
| Rumors: | This problem is an exponential function example that uses the real-world problem of how fast rumors spread. |
| Rising Gas Prices - Compounding and Inflation: | The purpose of this task is to give students an opportunity to explore various aspects of exponential models (e.g., distinguishing between constant absolute growth and constant relative growth, solving equations using logarithms, applying compound interest formulas) in the context of a real world problem with ties to developing financial literacy skills. In particular, students are introduced to the idea of inflation of prices of a single commodity, and are given a very brief introduction to the notion of the Consumer Price Index for measuring inflation of a body of goods. |
| Newton's Law of Cooling: | The coffee cooling experiment is a popular example of an exponential model with immediate appeal. The model is realistic and provides a good context for students to practice work with exponential equations. |
Vetted resources caregivers can use to help students learn the concepts and skills in this topic.
| Title |
Description |
| Algae Blooms: | In this example, students are asked to write a function describing the population growth of algae. It is implied that this is exponential growth. |
| What functions do two graph points determine?: | This problem solving task challenges students to find the linear, exponential and quadratic functions based on two points. |
| US Population 1790-1860: | This problem solving task asks students to solve five exponential and linear function problems based on a US population chart for the years 1790-1860. |
| Two Points Determine an Exponential Function II: | This problem solving tasks asks students to find the values of points on a graph. |
| Two Points Determine an Exponential Function I: | This problem solving task asks students to graph a function and find the values of points on a graph. |
| Snail Invasion: | The purpose of this task is to give students experience modeling a real-world example of exponential growth, in a context that provides a vivid illustration of the power of exponential growth, for example the cost of inaction for a year. There is an opportunity for further discussion based on part (c), since the ratio of costs from one year to the next is the same in each part. |
| Rumors: | This problem is an exponential function example that uses the real-world problem of how fast rumors spread. |
| Rising Gas Prices - Compounding and Inflation: | The purpose of this task is to give students an opportunity to explore various aspects of exponential models (e.g., distinguishing between constant absolute growth and constant relative growth, solving equations using logarithms, applying compound interest formulas) in the context of a real world problem with ties to developing financial literacy skills. In particular, students are introduced to the idea of inflation of prices of a single commodity, and are given a very brief introduction to the notion of the Consumer Price Index for measuring inflation of a body of goods. |
| Newton's Law of Cooling: | The coffee cooling experiment is a popular example of an exponential model with immediate appeal. The model is realistic and provides a good context for students to practice work with exponential equations. |
