Standard #: MAFS.912.F-LE.1.2 (Archived Standard)

Students are asked to write function rules for sequences given tables of values.

Students are asked to write an exponential function represented by a table of values.

Students are asked to write an exponential function from a written description of an exponential relationship.

Students are asked to write an exponential function given its graph.

Students are asked to write a function given its graph.

Students are given a table of values and are asked to write a linear function.

Students are asked to write a function to model the relationship between two variables described in a real-world context.

This lesson is about the Towers of Hanoi problem, a classic famous problem involving recursive thinking to reduce what appears to be a very large and difficult problem into a series of simpler ones.  The learning objective is for students to begin to understand recursive logic and thinking, relevant to computer scientists, mathematicians and engineers.   The lesson is experiential, in that each student will be working with her/his own Towers of Hanoi manipulative, inexpensively obtained.  There is no formal prerequisite, although some familiarity with set theory and functions is helpful.  The last three sections of the lesson involve some more formal concepts with recursive equations and proof by induction, so the students who work on those sections should probably be level 11 or 12 in a K-12 educational system.  The lesson has a Stop Point for 50-minute classes, followed by three more segments that may require a half to full additional class time.  So the teacher may use only those segments up to the Stop Point, or if two class sessions are to be devoted to the lesson, the entire set of segments.  Supplies are modest, and may be a set of coins or some washers from a hardware store to assemble small piles of disks in front of each student, each set of disks representing a Towers of Hanoi manipulative.  Or the students may assemble before the class a more complete Towers of Hanoi at home, as demonstrated in the video.  The classroom activities involve attempting to solve with hand and mind the Towers of Hanoi problem and discussing with fellow students patterns in the process and strategies for solution.

Piles of Paper is a student activity that demonstrates linear and exponential growth using heights of flat and folded paper. Data tables are created and then algebraic models are developed. Real world types of linear and exponential growth are also introduced.

Students will graph distance/velocity data of real galaxies to arrive at their own value of the Hubble constant (H). Once they have calculated their own value of H, they will use it to determine distances to real galaxies with known recessional velocities.

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.

In this example, students are asked to write a function describing the population growth of algae. It is implied that this is exponential growth.

This problem solving task challenges students to find the linear, exponential and quadratic functions based on two points.

This problem solving tasks asks students to find the values of points on a graph.

This problem solving task asks students to graph a function and find the values of points on a graph.

The task provides an opportunity for students to engage in detailed analysis of the rate of change of the elevation.

This problem is an exponential function example that uses the real-world problem of how fast rumors spread.

This problem complements the problem "Do two points always determine a linear function?'' There are two constraints on a pair of points R1 and R2 if there is an exponential function f(x) = ae^bx whose graph contains R1 and R2.

This problem complements the problem "Do two points always determine a linear function?'' There are two constraints on a pair of points R1 and R2 if there is an exponential function f(x) = ae^bx whose graph contains R1 and R2. First, the y-coordinates of R1 and R2 cannot have different signs, that is it cannot be that one is positive while the other is negative. This is because the function g(x) = ex takes only positive values. Consequently, f(x) = ae^bx cannot take both positive and negative values. Furthermore, the only way aebx can be zero is if a = 0 and then the function is linear rather than exponential. As long as the y-coordinates of R1 and R2 are non-zero and have the same sign, there is a unique exponential function f(x) = ae^bx whose graph contains R1 and R2.

This problem introduces the method used by scientists to date certain organic material. It is based not on the amount of the Carbon 14 isotope remaining in the sample but rather on the ratio of Carbon 14 to Carbon 12. This ratio decreases, hypothetically, at a constant exponential rate as soon as the organic material has ceased to absorb Carbon 14, that is, as soon as it dies.

This task involves a fairly straightforward decaying exponential. Filling out the table and developing the general formula is complicated only by the need to work with a fraction that requires decisions about rounding and precision.

This task asks students to analyze a set of data from a physical context, choose a model that reasonably fits the data, and use the model to answer questions about the physical context. This variant of the task is not scaffolded; for a more scaffolded version, see F-LE Basketball Bounces, Assessment Variation 1.

Students are asked to select the best model for a given context and use the model to make predictions.

The purpose of this task is to assess aspects of students' modeling skills in the context of standards:

F-LE.1.1.c Distinguish between situations that can be modeled with linear functions and with exponential functions; Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

F-LE.1.2 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).

This task asks students to analyze a set of data from a physical context, choose a model that reasonably fits the data, and use the model to answer questions about the physical context. This variant of the task is scaffolded; for a less scaffolded version, see F-LE Basketball Bounces, Assessment Variation 2.

In this task students use verbal descriptions to construct and compare linear and exponential functions and to find where the two functions intersect (F-LE.2, F-LE.3, A-REI.11).

This task asks students to write equations to predict how much money will be in a savings account at the end of each year, based on different factors like interest rates.

In this activity, students plug values into the independent variable to see what the output is for that function. Then based on that information, they have to determine the coefficient (slope) and constant(y-intercept) for the linear function. This activity allows students to explore linear functions and what input values are useful in determining the linear function rule. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the Java applet.

Using this virtual manipulative, students are able to graph a function and a set of ordered pairs on the same coordinate plane. The constants, coefficients, and exponents can be adjusted using slider bars, so the student can explore the affect on the graph as the function parameters are changed. Students can also examine the deviation of the data from the function. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

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.

In this example, students are asked to write a function describing the population growth of algae. It is implied that this is exponential growth.

This problem solving task challenges students to find the linear, exponential and quadratic functions based on two points.

This problem solving tasks asks students to find the values of points on a graph.

This problem solving task asks students to graph a function and find the values of points on a graph.

The task provides an opportunity for students to engage in detailed analysis of the rate of change of the elevation.

This problem is an exponential function example that uses the real-world problem of how fast rumors spread.

This problem complements the problem "Do two points always determine a linear function?'' There are two constraints on a pair of points R1 and R2 if there is an exponential function f(x) = ae^bx whose graph contains R1 and R2.

This problem complements the problem "Do two points always determine a linear function?'' There are two constraints on a pair of points R1 and R2 if there is an exponential function f(x) = ae^bx whose graph contains R1 and R2. First, the y-coordinates of R1 and R2 cannot have different signs, that is it cannot be that one is positive while the other is negative. This is because the function g(x) = ex takes only positive values. Consequently, f(x) = ae^bx cannot take both positive and negative values. Furthermore, the only way aebx can be zero is if a = 0 and then the function is linear rather than exponential. As long as the y-coordinates of R1 and R2 are non-zero and have the same sign, there is a unique exponential function f(x) = ae^bx whose graph contains R1 and R2.

This problem introduces the method used by scientists to date certain organic material. It is based not on the amount of the Carbon 14 isotope remaining in the sample but rather on the ratio of Carbon 14 to Carbon 12. This ratio decreases, hypothetically, at a constant exponential rate as soon as the organic material has ceased to absorb Carbon 14, that is, as soon as it dies.

This task involves a fairly straightforward decaying exponential. Filling out the table and developing the general formula is complicated only by the need to work with a fraction that requires decisions about rounding and precision.

In this task students use verbal descriptions to construct and compare linear and exponential functions and to find where the two functions intersect (F-LE.2, F-LE.3, A-REI.11).

This task asks students to write equations to predict how much money will be in a savings account at the end of each year, based on different factors like interest rates.

In this activity, students plug values into the independent variable to see what the output is for that function. Then based on that information, they have to determine the coefficient (slope) and constant(y-intercept) for the linear function. This activity allows students to explore linear functions and what input values are useful in determining the linear function rule. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the Java applet.

Using this virtual manipulative, students are able to graph a function and a set of ordered pairs on the same coordinate plane. The constants, coefficients, and exponents can be adjusted using slider bars, so the student can explore the affect on the graph as the function parameters are changed. Students can also examine the deviation of the data from the function. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

In this example, students are asked to write a function describing the population growth of algae. It is implied that this is exponential growth.

This problem solving task challenges students to find the linear, exponential and quadratic functions based on two points.

This problem solving tasks asks students to find the values of points on a graph.

This problem solving task asks students to graph a function and find the values of points on a graph.

The task provides an opportunity for students to engage in detailed analysis of the rate of change of the elevation.

This problem is an exponential function example that uses the real-world problem of how fast rumors spread.

This problem complements the problem "Do two points always determine a linear function?'' There are two constraints on a pair of points R1 and R2 if there is an exponential function f(x) = ae^bx whose graph contains R1 and R2.

This problem complements the problem "Do two points always determine a linear function?'' There are two constraints on a pair of points R1 and R2 if there is an exponential function f(x) = ae^bx whose graph contains R1 and R2. First, the y-coordinates of R1 and R2 cannot have different signs, that is it cannot be that one is positive while the other is negative. This is because the function g(x) = ex takes only positive values. Consequently, f(x) = ae^bx cannot take both positive and negative values. Furthermore, the only way aebx can be zero is if a = 0 and then the function is linear rather than exponential. As long as the y-coordinates of R1 and R2 are non-zero and have the same sign, there is a unique exponential function f(x) = ae^bx whose graph contains R1 and R2.

This problem introduces the method used by scientists to date certain organic material. It is based not on the amount of the Carbon 14 isotope remaining in the sample but rather on the ratio of Carbon 14 to Carbon 12. This ratio decreases, hypothetically, at a constant exponential rate as soon as the organic material has ceased to absorb Carbon 14, that is, as soon as it dies.

This task involves a fairly straightforward decaying exponential. Filling out the table and developing the general formula is complicated only by the need to work with a fraction that requires decisions about rounding and precision.

In this task students use verbal descriptions to construct and compare linear and exponential functions and to find where the two functions intersect (F-LE.2, F-LE.3, A-REI.11).

This task asks students to write equations to predict how much money will be in a savings account at the end of each year, based on different factors like interest rates.