# Cluster 1: Construct and compare linear, quadratic, and exponential models and solve problems. (Algebra 1 - Supporting Cluster) (Algebra 2 - Supporting Cluster)Archived

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

General Information
Number: MAFS.912.F-LE.1
Title: Construct and compare linear, quadratic, and exponential models and solve problems. (Algebra 1 - Supporting Cluster) (Algebra 2 - Supporting Cluster)
Type: Cluster
Subject: Mathematics - Archived
Domain-Subdomain: Functions: Linear, Quadratic, & Exponential Models

## Related Standards

This cluster includes the following benchmarks.

## Related Access Points

This cluster includes the following access points.

## Access Points

MAFS.912.F-LE.1.AP.1a
Select the appropriate graphical representation of a linear model based on real-world events.
MAFS.912.F-LE.1.AP.1b
In a linear situation using graphs or numbers, predict the change in rate based on a given change in one variable (e.g., If I have been adding sugar at a rate of 1T per cup of water, what happens to my rate if I switch to 2T of sugar for every cup of water?).
MAFS.912.F-LE.1.AP.3a
Compare graphs of linear, exponential, and quadratic growth graphed on the same coordinate plane.
MAFS.912.F-LE.1.AP.2a
Select the graph, the description of a relationship or two input-output pairs of linear functions.
MAFS.912.F-LE.1.AP.4a
Select the logarithm that models the function in the form MAFS.912.F-LE.1.AP.4b
Use technology to solve exponential models using logarithms with base 10 or e.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this topic.

## Formative Assessments

Students are given a linear and an exponential function, one represented verbally and the other by a table. Then students are asked to compare the rate of change in each in the context of the problem.

Type: Formative Assessment

Exponential Growth:

Students are given two functions, one represented verbally and the other by a table, and are asked to compare the rate of change in each in the context of the problem.

Type: Formative Assessment

Prove Linear:

Students are asked to prove that a linear function grows by equal differences over equal intervals.

Type: Formative Assessment

Prove Exponential:

Students are asked to prove that an exponential function grows by equal factors over equal intervals.

Type: Formative Assessment

Linear or Exponential?:

Students are given four verbal descriptions of functions and asked to identify each as either linear or exponential and to justify their choices.

Type: Formative Assessment

Students are asked to compare a quadratic and an exponential function in context.

Type: Formative Assessment

Compare Linear and Exponential Functions:

Students are asked to compare a linear function and an exponential function in context.

Type: Formative Assessment

What Is the Function Rule?:

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

Type: Formative Assessment

Writing an Exponential Function From a Table:

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

Type: Formative Assessment

Writing an Exponential Function From a Description:

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

Type: Formative Assessment

Writing an Exponential Function From Its Graph:

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

Type: Formative Assessment

Functions From Graphs:

Students are asked to write a function given its graph.

Type: Formative Assessment

Writing a Function From Ordered Pairs:

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

Type: Formative Assessment

The Cost of Water:

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

Type: Formative Assessment

## Lesson Plans

The Towers of Hanoi: Experiential Recursive Thinking:

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.

Type: Lesson Plan

You’re Pulling My Leg – or Candy!:

Students will watch a Perspectives Video to see how exponential growth is modeled in the real world. Students then explore how exponential growth and decay can model other real world problems. Students will also discover how manipulating the variables in an exponential equation changes the graph.

Type: Lesson Plan

Modeling: Having Kittens:

This lesson unit is intended to help you assess how well students are able to interpret a situation and represent the constraints and variables mathematically, select appropriate mathematical methods to use, make sensible estimates and assumptions, investigate an exponentially increasing sequence and communicate their reasoning clearly.

Type: Lesson Plan

Functions and Everyday Situations:

This lesson unit is intended to help you 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.

Type: Lesson Plan

Appreciation for Car Depreciation:

Students will use information from the internet or a car dealership's advertisement to identify a car and determine the future value of the car using different depreciation rates over different intervals of time. Students will graph their data to show exponential decay and compare to a linear decrease on the same graph.

Type: Lesson Plan

BIOSCOPES Summer Institute 2013 - Motion:

This lesson is the first in a sequence of grade 9-12 physical science lessons that are organized around the big ideas that frame motion, forces, and energy. It directly precedes resource # 52648 "BIOSCOPES Summer Institute 2013 - Forces." This lesson is designed along the lines of an iterative 5-E learning cycle and employs a predict, observe, and explain (POE) activity at the beginning of the "Engage" phase in order to elicit student prior knowledge. The POE is followed by a sequence of inquiry-based activities and class discussions that are geared toward leading the students systematically through the exploration of 1-dimensional motion concepts. Included in this resource is a summative assessment as well as a teacher guide for each activity.

Type: Lesson Plan

Falling for Gravity:

Students will investigate the motion of three objects of different masses undergoing free fall. Additionally, students will:

• Use spark timers to collect displacement and time data.
• Use this data to calculate the average velocity for the object during each interval.
• Graph this data on a velocity versus time graph, V-t. They find the slope of this graph to calculate acceleration.
• Calculate the falling object's acceleration from their data table and graph this data on an acceleration versus time graph, a-t.
• Use their Spark timer data paper, cut it into intervals, and paste these intervals into their displacement versus time graph.

Type: Lesson Plan

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.

Type: Lesson Plan

Piles of Paper:

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.

Type: Lesson Plan

Determining the Hubble Constant:

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.

Type: Lesson Plan

The Power of Exponentials, Big and Small - MIT Blossoms:

Exponential growth is keenly applicable to a variety of different fields ranging from cell growth in biology, nuclear chain reactions in physics to computational complexity in computer science. In this video-based lesson, through various examples and activities, we have tried to compare exponential growth to polynomial growth and to develop an insight about how quickly the number can grow or decay in exponentials. A basic knowledge of scientific notation, plotting graphs and finding intersection of two functions is assumed. It would be better if the students have done pre-calculus, though this is not a requirement. The lesson is about 20 minutes, interspersed with simple activities that can require up to half an hour.

The webpage for this video also includes tabs where additional resources and information can be found. These include a Teacher's Guide, a Powers of 2 table, links to other helpful lessons and resources, a transcript of the video, and even an option to download the video.

Type: Lesson Plan

## Lesson Study Resource Kit

The Motion of Objects:

This 9-12 Lesson study resource kit is designed to engage teachers of physical science and physics in the planning and design of an instructional unit and research lesson pertaining to the motion of objects. Included in this resource kit are unit plans, concept progressions, formative and summative assessments, complex informational texts, and etc. that align to academic standards for science, mathematics, and English language arts.

Type: Lesson Study Resource Kit

## Original Student Tutorials

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.

Type: Original Student 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.

Type: Original Student Tutorial

## Perspectives Video: Expert

Tree Rings Research to Inform Land Management Practices:

In this video, fire ecologist Monica Rother describes tree ring research and applications for land management.

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiasts

Slope and Deep Sea Sharks:

Shark researcher, Chip Cotton, discusses the use of regression lines, slope, and determining the strength of the models he uses in his research.

Type: Perspectives Video: Professional/Enthusiast

Linear Regression for Analysis of Sea Anemone Data:

Will Ryan describes how linear regression models contribute towards his research on sea anemones.

Type: Perspectives Video: Professional/Enthusiast

KROS Pacific Ocean Kayak Journey: Training, Simulation, and Modeling:

Complex problems require complex plans and training. Get in shape to get things done.

Related Resources:
KROS Pacific Ocean Kayak Journey: GPS Data Set[.XLSX]
KROS Pacific Ocean Kayak Journey: Path Visualization for Google Earth[.KML]

Type: Perspectives Video: Professional/Enthusiast

KROS Pacific Ocean Kayak Journey: Kites, Wind, and Speed:

Lofty ideas about kites helped power a kayak from California to Hawaii.

Related Resources:
KROS Pacific Ocean Kayak Journey: GPS Data Set[.XLSX]
KROS Pacific Ocean Kayak Journey: Path Visualization for Google Earth[.KML]

Type: Perspectives Video: Professional/Enthusiast

KROS Pacific Ocean Kayak Journey: Water Usage Rates:

A seafaring teacher filters all the good information you need to understand water purification rates for distance traveling.

Related Resources:
KROS Pacific Ocean Kayak Journey: GPS Data Set[.XLSX]
KROS Pacific Ocean Kayak Journey: Path Visualization for Google Earth[.KML]

Type: Perspectives Video: Professional/Enthusiast

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 1982-1988:

This problem solving task asks students to predict and model US population based on a chart of US population data from 1982 to 1988.

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.

Sandia Aerial Tram:

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

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.

Linear or exponential?:

This task gives a variation of real-life contexts which could be modeled by a linear or exponential function. The key distinguishing feature between the two is whether the change by equal factors over equal intervals (exponential functions), or by a constant increase per unit interval (linear functions). The task could either be used as an assessment problem on this distinction, or used as an introduction to the differences between these very important classes of functions.

Linear Functions:

This task requires students to use the fact that on the graph of the linear function h(x) = ax + b, the y-coordinate increases by a when x increases by one. Specific values for a and b were left out intentionally to encourage students to use the above fact as opposed to computing the point of intersection, (p,q), and then computing respective function values to answer the question.

Do two points always determine an exponential function?:

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.

Do two points always determine a linear function?:

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.

Comparing Exponentials:

This task gives students an opportunity to work with exponential functions in a real world context involving continuously compounded interest. They will study how the base of the exponential function impacts its growth rate and use logarithms to solve exponential equations.

Carbon 14 Dating, Variation 2:

This exploratory task requires the student to use properties of exponential functions in order to estimate how much Carbon 14 remains in a preserved plant after different amounts of time.

Carbon 14 Dating in Practice II:

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.

Carbon 14 Dating:

The task requires the student to use logarithms to solve an exponential equation in the realistic context of carbon dating, important in archaeology and geology, among other places. Students should be guided to recognize the use of the natural logarithm when the exponential function has the given base of e, as in this problem. Note that the purpose of this task is algebraic in nature -- closely related tasks exist which approach similar problems from numerical or graphical stances.

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.

Bacteria Populations:

This task provides a real world context for interpreting and solving exponential equations. There are two solutions provided for part (a). The first solution demonstrates how to deduce the conclusion by thinking in terms of the functions and their rates of change. The second approach illustrates a rigorous algebraic demonstration that the two populations can never be equal.

Population and Food Supply:

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).

Exponential growth versus linear growth I:

The purpose of this task it to have students discover how (and how quickly) an exponentially increasing quantity eventually surpasses a linearly increasing quantity. Students' intuitions will probably have them favoring Option A for much longer than is actually the case, especially if they are new to the phenomenon of exponential growth. Teachers might use this surprise as leverage to segue into a more involved task comparing linear and exponential growth.

Exponential Functions:

This task requires students to use the fact that the value of an exponential function f(x) = a · b^x increases by a multiplicative factor of b when x increases by one. It intentionally omits specific values for c and d in order to encourage students to use this fact instead of computing the point of intersection, (p,q), and then computing function values to answer the question.

Equal Factors over Equal Intervals:

This problem assumes that students are familiar with the notation x0 and ?x. However, the language "successive quotient" may be new.

Equal Differences over Equal Intervals 2:

This task assumes that students are familiar with the ?x and ?y notations. Students most likely developed this familiarity in their work with slope.

Equal Differences over Equal Intervals 1:

An important property of linear functions is that they grow by equal differences over equal intervals. In F.LE Equal Differences over Equal Intervals 1, students prove this for equal intervals of length one unit, and note that in this case the equal differences have the same value as the slope.

In the Billions and Linear Modeling:

This problem-solving task asks students to examine if linear modeling would be appropriate to describe and predict population growth from select years.

In the Billions and Exponential Modeling:

This problem-solving task provides students an opportunity to experiment with modeling real data by using population growth rates from the past two centuries.

Interesting Interest Rates:

This problem-solving task challenges students to write expressions and create a table to calculate how much money can be gained after investing at different banks with different interest rates.

Illegal Fish:

This problem-solving task asks students to describe exponential growth through a real-world problem involving the illegal introduction of fish into a lake.

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.

Identifying Functions:

This problem-solving emphasizes the expectation that students know linear functions grow by constant differences over equal intervals and exponential functions grow by constant factors over equal intervals.

Exponential growth versus polynomial growth:

This problem solving task shows that an exponential function takes larger values than a cubic polynomial function provided the input is sufficiently large. This resource also includes standards alignment commentary and annotated solutions.

Exponential growth versus linear growth II:

This task asks students to calculate exponential functions with a base larger than one.

## Professional Development

Mathematical Modeling: Insights into Algebra, Teaching for Learning:

This professional development resource provides a rich collection of information to help teachers engage students more effectively in mathematical modeling. It features videos of two complete lessons with commentary, background information on effective teaching, modeling, and lesson study, full lesson plans to teach both example lessons, examples of student work from the lessons, tips for effective teaching strategies, and list of helpful resources.

• In Lesson 1 students use mathematical models (tables and equations) to represent the relationship between the number of revolutions made by a "driver" and a "follower" (two connected gears in a system), and they will explain the significance of the radii of the gears in regard to this relationship.
• In Lesson 2 students mathematically model the growth of populations and use exponential functions to represent that growth.

Type: Professional Development

## Unit/Lesson Sequences

Sample Algebra 1 Curriculum Plan Using CMAP:

This sample Algebra 1 CMAP is a fully customizable resource and curriculum-planning tool that provides a framework for the Algebra 1 Course. The units and standards are customizable and the CMAP allows instructors to add lessons, worksheets, and other resources as needed. This CMAP also includes rows that automatically filter and display Math Formative Assessments System tasks, E-Learning Original Student Tutorials and Perspectives Videos that are aligned to the standards, available on CPALMS.

Learn more about the sample Algebra 1 CMAP, its features and customizability by watching the following video:

### Using this CMAP

To view an introduction on the CMAP tool, please .

To view the CMAP, click on the "Open Resource Page" button above; be sure you are logged in to your iCPALMS account.

To use this CMAP, click on the "Clone" button once the CMAP opens in the "Open Resource Page." Once the CMAP is cloned, you will be able to see it as a class inside your iCPALMS My Planner (CMAPs) app.

To access your My Planner App and the cloned CMAP, click on the iCPALMS tab in the top menu.

All CMAP tutorials can be found within the iCPALMS Planner App or at the following URL: http://www.cpalms.org/support/tutorials_and_informational_videos.aspx

Type: Unit/Lesson Sequence

Direct and Inverse Variation:

"Lesson 1 of two lessons teaches students about direct variation by allowing them to explore a simulated oil spill using toilet paper tissues (to represent land) and drops of vegetable oil (to simulate a volume of oil). Lesson 2 teaches students about inverse variation by exploring the relationship between the heights of a fixed amount of water poured into cylindrical containers of different sizes as compared to the area of the containers' bases." from Insights into Algebra 1 - Annenberg Foundation.

Type: Unit/Lesson Sequence

## Video/Audio/Animation

MIT BLOSSOMS - Flu Math Games:

This video lesson shows students that math can play a role in understanding how an infectious disease spreads and how it can be controlled. During this lesson, students will see and use both deterministic and probabilistic models and will learn by doing through role-playing exercises. There are no formal prerequisites, as students in any high school or even middle school math class could enjoy this learning video. But more advanced classes can go into the optional applied probability modeling that accompanies the module in a downloadable pdf file. The primary exercises between video segments of this lesson are class-intensive simulation games in which members of the class 'infect' each other under alternative math modeling assumptions about disease progression. Also there is an occasional class discussion and local discussion with nearby classmates.

Type: Video/Audio/Animation

## Virtual Manipulatives

Functions and Vertical Line Test:

This lesson is designed to introduce students to the vertical line test for functions as well as practice plotting points and drawing simple functions. The lesson provides links to discussions and activities related to the vertical line test and functions as well as suggested ways to integrate them into the lesson.

Type: Virtual Manipulative

Linear Function Machine:

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.

Type: Virtual Manipulative

Introduction to Functions:

This lesson is designed to introduce students to functions as rules and independent and dependent variables. The lesson provides links to discussions and activities that motivate the idea of a function as a machine as well as proper terminology when discussing functions. Finally, the lesson provides links to follow-up lessons designed for use in succession to the introduction of functions.

Type: Virtual Manipulative

Data Flyer:

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.

Type: Virtual Manipulative

Function Flyer:

In this online tool, students input a function to create a graph where the constants, coefficients, and exponents can be adjusted by slider bars. This tool allows students to explore graphs of functions and how adjusting the numbers in the function affect the graph. Using tabs at the top of the page you can also access 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.

Type: Virtual Manipulative

Equation Grapher:

This interactive simulation investigates graphing linear and quadratic equations. Users are given the ability to define and change the coefficients and constants in order to observe resulting changes in the graph(s).

Type: Virtual Manipulative

## Student Resources

Vetted resources students can use to learn the concepts and skills in this topic.

## Original Student Tutorials

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.

Type: Original Student 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.

Type: Original Student Tutorial

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 1982-1988:

This problem solving task asks students to predict and model US population based on a chart of US population data from 1982 to 1988.

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.

Sandia Aerial Tram:

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

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.

Linear or exponential?:

This task gives a variation of real-life contexts which could be modeled by a linear or exponential function. The key distinguishing feature between the two is whether the change by equal factors over equal intervals (exponential functions), or by a constant increase per unit interval (linear functions). The task could either be used as an assessment problem on this distinction, or used as an introduction to the differences between these very important classes of functions.

Linear Functions:

This task requires students to use the fact that on the graph of the linear function h(x) = ax + b, the y-coordinate increases by a when x increases by one. Specific values for a and b were left out intentionally to encourage students to use the above fact as opposed to computing the point of intersection, (p,q), and then computing respective function values to answer the question.

Do two points always determine an exponential function?:

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.

Do two points always determine a linear function?:

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.

Comparing Exponentials:

This task gives students an opportunity to work with exponential functions in a real world context involving continuously compounded interest. They will study how the base of the exponential function impacts its growth rate and use logarithms to solve exponential equations.

Carbon 14 Dating, Variation 2:

This exploratory task requires the student to use properties of exponential functions in order to estimate how much Carbon 14 remains in a preserved plant after different amounts of time.

Carbon 14 Dating in Practice II:

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.

Carbon 14 Dating:

The task requires the student to use logarithms to solve an exponential equation in the realistic context of carbon dating, important in archaeology and geology, among other places. Students should be guided to recognize the use of the natural logarithm when the exponential function has the given base of e, as in this problem. Note that the purpose of this task is algebraic in nature -- closely related tasks exist which approach similar problems from numerical or graphical stances.

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.

Bacteria Populations:

This task provides a real world context for interpreting and solving exponential equations. There are two solutions provided for part (a). The first solution demonstrates how to deduce the conclusion by thinking in terms of the functions and their rates of change. The second approach illustrates a rigorous algebraic demonstration that the two populations can never be equal.

Population and Food Supply:

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).

Exponential growth versus linear growth I:

The purpose of this task it to have students discover how (and how quickly) an exponentially increasing quantity eventually surpasses a linearly increasing quantity. Students' intuitions will probably have them favoring Option A for much longer than is actually the case, especially if they are new to the phenomenon of exponential growth. Teachers might use this surprise as leverage to segue into a more involved task comparing linear and exponential growth.

Exponential Functions:

This task requires students to use the fact that the value of an exponential function f(x) = a · b^x increases by a multiplicative factor of b when x increases by one. It intentionally omits specific values for c and d in order to encourage students to use this fact instead of computing the point of intersection, (p,q), and then computing function values to answer the question.

Equal Factors over Equal Intervals:

This problem assumes that students are familiar with the notation x0 and ?x. However, the language "successive quotient" may be new.

Equal Differences over Equal Intervals 2:

This task assumes that students are familiar with the ?x and ?y notations. Students most likely developed this familiarity in their work with slope.

Equal Differences over Equal Intervals 1:

An important property of linear functions is that they grow by equal differences over equal intervals. In F.LE Equal Differences over Equal Intervals 1, students prove this for equal intervals of length one unit, and note that in this case the equal differences have the same value as the slope.

In the Billions and Linear Modeling:

This problem-solving task asks students to examine if linear modeling would be appropriate to describe and predict population growth from select years.

In the Billions and Exponential Modeling:

This problem-solving task provides students an opportunity to experiment with modeling real data by using population growth rates from the past two centuries.

Interesting Interest Rates:

This problem-solving task challenges students to write expressions and create a table to calculate how much money can be gained after investing at different banks with different interest rates.

Illegal Fish:

This problem-solving task asks students to describe exponential growth through a real-world problem involving the illegal introduction of fish into a lake.

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.

Identifying Functions:

This problem-solving emphasizes the expectation that students know linear functions grow by constant differences over equal intervals and exponential functions grow by constant factors over equal intervals.

Exponential growth versus polynomial growth:

This problem solving task shows that an exponential function takes larger values than a cubic polynomial function provided the input is sufficiently large. This resource also includes standards alignment commentary and annotated solutions.

Exponential growth versus linear growth II:

This task asks students to calculate exponential functions with a base larger than one.

## Virtual Manipulatives

Linear Function Machine:

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.

Type: Virtual Manipulative

Data Flyer:

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.

Type: Virtual Manipulative

Function Flyer:

In this online tool, students input a function to create a graph where the constants, coefficients, and exponents can be adjusted by slider bars. This tool allows students to explore graphs of functions and how adjusting the numbers in the function affect the graph. Using tabs at the top of the page you can also access 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.

Type: Virtual Manipulative

Equation Grapher:

This interactive simulation investigates graphing linear and quadratic equations. Users are given the ability to define and change the coefficients and constants in order to observe resulting changes in the graph(s).

Type: Virtual Manipulative

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this topic.

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 1982-1988:

This problem solving task asks students to predict and model US population based on a chart of US population data from 1982 to 1988.

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.

Sandia Aerial Tram:

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

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.

Linear or exponential?:

This task gives a variation of real-life contexts which could be modeled by a linear or exponential function. The key distinguishing feature between the two is whether the change by equal factors over equal intervals (exponential functions), or by a constant increase per unit interval (linear functions). The task could either be used as an assessment problem on this distinction, or used as an introduction to the differences between these very important classes of functions.

Linear Functions:

This task requires students to use the fact that on the graph of the linear function h(x) = ax + b, the y-coordinate increases by a when x increases by one. Specific values for a and b were left out intentionally to encourage students to use the above fact as opposed to computing the point of intersection, (p,q), and then computing respective function values to answer the question.

Do two points always determine an exponential function?:

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.

Do two points always determine a linear function?:

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.

Comparing Exponentials:

This task gives students an opportunity to work with exponential functions in a real world context involving continuously compounded interest. They will study how the base of the exponential function impacts its growth rate and use logarithms to solve exponential equations.

Carbon 14 Dating, Variation 2:

This exploratory task requires the student to use properties of exponential functions in order to estimate how much Carbon 14 remains in a preserved plant after different amounts of time.

Carbon 14 Dating in Practice II:

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.

Carbon 14 Dating:

The task requires the student to use logarithms to solve an exponential equation in the realistic context of carbon dating, important in archaeology and geology, among other places. Students should be guided to recognize the use of the natural logarithm when the exponential function has the given base of e, as in this problem. Note that the purpose of this task is algebraic in nature -- closely related tasks exist which approach similar problems from numerical or graphical stances.

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.

Bacteria Populations:

This task provides a real world context for interpreting and solving exponential equations. There are two solutions provided for part (a). The first solution demonstrates how to deduce the conclusion by thinking in terms of the functions and their rates of change. The second approach illustrates a rigorous algebraic demonstration that the two populations can never be equal.

Population and Food Supply:

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).

Exponential growth versus linear growth I:

The purpose of this task it to have students discover how (and how quickly) an exponentially increasing quantity eventually surpasses a linearly increasing quantity. Students' intuitions will probably have them favoring Option A for much longer than is actually the case, especially if they are new to the phenomenon of exponential growth. Teachers might use this surprise as leverage to segue into a more involved task comparing linear and exponential growth.

Exponential Functions:

This task requires students to use the fact that the value of an exponential function f(x) = a · b^x increases by a multiplicative factor of b when x increases by one. It intentionally omits specific values for c and d in order to encourage students to use this fact instead of computing the point of intersection, (p,q), and then computing function values to answer the question.

Equal Factors over Equal Intervals:

This problem assumes that students are familiar with the notation x0 and ?x. However, the language "successive quotient" may be new.

Equal Differences over Equal Intervals 2:

This task assumes that students are familiar with the ?x and ?y notations. Students most likely developed this familiarity in their work with slope.

Equal Differences over Equal Intervals 1:

An important property of linear functions is that they grow by equal differences over equal intervals. In F.LE Equal Differences over Equal Intervals 1, students prove this for equal intervals of length one unit, and note that in this case the equal differences have the same value as the slope.

In the Billions and Linear Modeling:

This problem-solving task asks students to examine if linear modeling would be appropriate to describe and predict population growth from select years.

In the Billions and Exponential Modeling:

This problem-solving task provides students an opportunity to experiment with modeling real data by using population growth rates from the past two centuries.

Interesting Interest Rates:

This problem-solving task challenges students to write expressions and create a table to calculate how much money can be gained after investing at different banks with different interest rates.

Illegal Fish:

This problem-solving task asks students to describe exponential growth through a real-world problem involving the illegal introduction of fish into a lake.

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.

Identifying Functions:

This problem-solving emphasizes the expectation that students know linear functions grow by constant differences over equal intervals and exponential functions grow by constant factors over equal intervals.