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Common Core Math

High School Functions Teaching and Learning Resources

Domain: LE: Linear, Quadratic, and Exponential Models

Common Core Math

Select the cluster for resources on this page:

  • LE-A: Construct and compare linear and exponential models and solve problems

  • LE-B: Interpret expressions for functions in terms of the situation they model

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LE-A: Construct and compare linear and exponential models and solve problems

Standards:

  • LE-A.1. Distinguish between situations that can be modeled with linear functions and with exponential functions.
    • LE-A.1.a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
    • LE-A.1.b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
    • LE-A.1.c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
  • LE-A.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).
  • LE-A.3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
  • LE-A.4. For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

 

Technology-enhanced investigations:

Mathwords.com: Math Dictionary: Key vocabulary for this domain.  Use with LE-A, LE-B:

arithmetic sequence exponential growth interval model
exponential decay function linear equation parameter
exponential function/exponential model geometric sequence logarithm quadratic equation

 

CK-12: Algebra: Concepts, videos, and practice problems for learners to complete:

NCTM Illuminations: Shedding Light on a Subject: This unit has four lessons in which learners explore "the development of a mathematical model for the decay of light passing through water.  The goal of this investigation is a rich exploration of exponential models in context." Movie clips and simulations are included, along with virtual manipulatives (interactive grapher for data and line of best fit), which help learners to identify a function that models their data. Use with LE-A.

Ohio Resource Center on YouTube: Tutorials for High School Mathematics: Logarithms. Definition of and computation with both common logarithms and base e natural logarithms.

LearnZillion:

MIT BLOSSOMS: Video lessons with additional teacher and learner resources.  Descriptions are from the video summaries.

  • 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."  Additional simulations are included.  Aligns with Algebra standards SSE-B.3.c and REI-A.1; Function standards IF-C.8.b, BF-B.4.a, and LE-A-1.(a, c); and Statistics and Probability standards ID-B.6.a, IC-A.1, IC-B.4, CP-A.2, and MD-A.1.
  • Free Fall: "This video lesson is an example of "teaching for understanding" in lieu of providing students with formulas for determining the height of a dropped (or projected) object at any time during its fall. The concept presented here of creating a chart to organize and analyze data collected in a simple experiment is broadly useful."  Aligns with Function standards IF-B.6, IF-C.7.a, LE-A.1.a, and Algebra standard CED-A.2.
  • The Power of Exponentials, Big and Small: "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 lesson, through various examples and activities, [the presenters] 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."  Aligns with Algebra standards SSE-B.3.c and CED-A.2 and Function standards IF-A.3, IF-C.7.(a, c, e), IF-C.8.b, and LE-A.1.(a, c).
  • 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."  Aligns with Algebra standard SSE-B.3.c, and Function standards IF-C.8.b and LE-A.1.a.
  • Tragedy of the Commons: "This lesson introduces students to the “Tragedy of the Commons,” an extended metaphor for problems of shared environmental or man-made resources that are overused and eventually depleted. In this metaphor, shared resources are compared to a common grazing pasture, or “commons,” on which any dairy farmer can graze as many cows as he/she wishes. If too many cows are added to the commons, they will overeat the grass in the pasture and the shared resource will become depleted – a disadvantage to everyone. In this lesson, students will be inspired to think about possible solutions to this problem."  Aligns with Algebra standard SSE-B.3.c and Function standards IF-B.4, IF-C.8.b, LE-A.1.a, LE-A.3, and LE-B.5.

Zweigmedia.com: Tutorials and examples using fill-in or multiple choice to test understanding.  Learners can also choose to do game versions of some topics.  Use with LE-A.1 and LE-A.2:

  • Functions and Models: Part A: Cost, revenue, and profit models; Part B: Demand, supply, and time-change models
  • Linear functions and models: In three parts, learners get the basics of slope and intercepts; finding the equation of a line; then applications of linear models.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives.  Note: Within the Wolfram Demonstration Project are 9 manipulatives addressing LE-A.1, 19 manipulatives for LE-A.2, 1 manipulative for LE-A.3, and 3 manipulatives for LE-A.4.  Among those:

McDougal Littell ClassZone, Algebra 2, 2011: Select resources in Ch. 7 Exponential and Logarithmic Functions for interactive practice: Animations (Virtual manipulatives), Games/Vocabulary Flashcards, and PowerPoint lesson examples, and interactive problems within the eWorkbook. See multiple choice below for related sections to use within the eWorkbook.  Use with LE-A.

Multiple Choice:

Glencoe Online Learning Center, Algebra 2 (2010) self-check quizzes:

Khan Academy: Practice questions with videos.

McDougal Littell ClassZone, Algebra 2, 2011:

OpenEd:

Constructed-response:

Khan Academy: Practice questions with videos.

Performance tasks:

Illustrative Mathematics: Functions:

Inside Mathematics: MARS Task: Functions (pdf) or see all MARS Tasks: Functions & Relations: Course 1 Algebra: Scroll down the page to select Functions: Learners work with graphs of linear and non-linear functions.  This task aligns with function standards IF-C.7, IF-C.9 and LE-A.3.

Mathematics Assessment Project: Standards: High School: Functions: http://map.mathshell.org/stds.php?standardid=1448

Mathematics Common Core Toolbox: PARCC Prototype Project:  Under High School Tasks, select:

  • Cellular growth: "Students write geometric sequences both recursively and with an explicit formula and use the two representations to model situations. In this task, students consider the use of one or more representations to solve a real-world problem and choose the type of sequence to represent a situation."  This four-part task aligns with function standards BF-A.2 and LE-A.2 and mathematical practice standards MP.2 and MP.4.
  • Golf balls in water: "Students model approximately linear data with a linear function. To do so, they must use contextual information and construct representations of linear relationships as well as interpret parameters of the linear models in the context. It makes use of common sense understandings of volume as it relates to changing the dimensions of a glass jar; thus, the mathematics is presented in a context that is novel, yet has multiple entry points for students." This three-part task aligns with function standards BF-A.1, LE-A.2, LE-B.5, and mathematical practice standards MP.1, MP.2 and MP.4.
  • Rabbit populations:  "Students analyze the situation (rabbit population growth) and create a mathematical model that represents it. Students are then presented with a second rabbit population that is slightly different and asked to compare the models for the two populations. This requires students to identify which function parameters and key features of the graph will change based on the adjustment in context. Students then use the two models together to answer mathematical questions." This three-part task aligns with function standards BF-A.1, LE-A.2, LE-B.5, and mathematical practice standard MP.4.

Mathematics Vision Project, Secondary 1 Student Edition: Module 3: Arithmetic and Geometric Sequences: This module contains 10 classroom tasks.  Function standards addressed in Module 2 include LE-A.1, LE-A.2, LE-B.5, and BF-A.1.

  • Task 1: Growing Dots; Task 2: Growing, Growing Dots; Task 3: Scott’s Workout; Task 4: Don’t Break the Chain; Task 5: Something to Chew On; Task 6: Chew On This; and Task 7: What Comes Next? What Comes Later? align to both standards LE-A.1 and LE-A.2. 
  • Task 8: What Does It Mean?; and Task 9: Geometric Meanies; Task 10: I Know … What Do You Know? align to standard LE-A.2.

Mathematics Vision Project, Secondary 1 Student Edition: Module 4: Linear and Exponential Functions: This module contains 10 classroom tasks.  Function standards addressed in Module 3 include IF-A.3, IF-C.7, LE-A.1, LE-A.2, LE-A.3, LE-B.5, and BF-A.1, BF-A.2.  Five of those include function standards LE-A.1, LE-A.2, LE-A.3.

  • Task 2: Sorting Out The Change; and Task 3: Where’s My Change? align to standards LE-A.1 and LE-A.2.
  • Task 5: Getting Down to Business aligns to standards LE-A.2 and LE-A.3.
  • Task 4: Linear, Exponential or Neither aligns to standard LE-A.3.
  • Task 6: Growing, Growing, Gone align to standards LE-A.1, LE-A.2, and LE-A.3.

Mathematics Vision Project, Secondary 3 Student Edition:

 

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LE-B: Interpret expressions for functions in terms of the situation they model

Standards:

  • LE-B.5. Interpret the parameters in a linear or exponential function in terms of a context.

Technology-enhanced investigations:

NCTM Illuminations: Function Matching: Students see the graph and determine parameters.

A. Dendane: Analyze Math: Exponential Functions: HTML5 applet to explore the properties of the graphs of exponential functions.  Learners change parameters a, b, c, d, and B in equations of the form f(x)= a(Bb(x+c))+ d.

Ron Blond:  Exponential Functions: Use this applet to examine the affect on a graph by changing the parameters a and b in exponential functions of the form y = a(bx).

MIT BLOSSOMS: Video lesson with additional teacher and learner resources.  Description is from the video summary.  Tragedy of the Commons: "This lesson introduces students to the “Tragedy of the Commons,” an extended metaphor for problems of shared environmental or man-made resources that are overused and eventually depleted. In this metaphor, shared resources are compared to a common grazing pasture, or “commons,” on which any dairy farmer can graze as many cows as he/she wishes. If too many cows are added to the commons, they will overeat the grass in the pasture and the shared resource will become depleted – a disadvantage to everyone. In this lesson, students will be inspired to think about possible solutions to this problem."  Aligns with Algebra standard SSE-B.3.c and Function standards IF-B.4, IF-C.8.b, LE-A.1.a, LE-A.3, and LE-B.5.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulative: Final number of bacteria cells.  Aligns with LE-B.5.

Multiple-choice:

OpenEd: Parts of an exponential or linear function: 5 questions that combine multiple choice and free response.  Includes related review resources.  Aligns with LE-B.5.

Constructed-response:

Khan Academy: Practice questions with videos. Modeling with exponential functions: Aligns with LE-A.1 and LE-B.5.

OpenEd: Interpreting parameters: 5 free response questions.  Includes related review resources.  Aligns with LE-B.5.

Performance tasks:

Illustrative Mathematics: Functions:

Mathematics Assessment Project: Standards: High School: Functions: http://map.mathshell.org/stds.php?standardid=1448

Mathematics Common Core Toolbox: PARCC Prototype Project:  Under High School Tasks, select:

  • Golf balls in water: "Students model approximately linear data with a linear function. To do so, they must use contextual information and construct representations of linear relationships as well as interpret parameters of the linear models in the context. It makes use of common sense understandings of volume as it relates to changing the dimensions of a glass jar; thus, the mathematics is presented in a context that is novel, yet has multiple entry points for students." This three-part task aligns with function standards BF-A.1, LE-A.2, LE-B.5, and mathematical practice standards MP.1, MP.2 and MP.4.
  • Rabbit populations:  "Students analyze the situation (rabbit population growth) and create a mathematical model that represents it. Students are then presented with a second rabbit population that is slightly different and asked to compare the models for the two populations. This requires students to identify which function parameters and key features of the graph will change based on the adjustment in context. Students then use the two models together to answer mathematical questions." This three-part task aligns with function standards BF-A.1, LE-A.2, LE-B.5, and mathematical practice standard MP.4.

Mathematics Vision Project, Secondary 1 Student Edition:

  • Module 3: Arithmetic and Geometric Sequences: This module contains 10 classroom tasks.  Six of those tasks include function standard LE-B.5.  Function standards addressed in Module 2 include LE-A.1, LE-A.2, LE-B.5, and BF-A.1.
  • Module 4: Linear and Exponential Functions: This module contains 10 classroom tasks.  Function standards addressed in Module 3 include IF-A.3, IF-C.7, LE-A.1, LE-A.2, LE-A.3, LE-B.5, and BF-A.1, BF-A.2. Task 4: Linear, Exponential, or Neither; Task 5: Getting Down to Business; Task 7: Making My Point; Task 9:  Up a Little, Down a Little align to function standard LE-B.5.

Mathematics Vision Project, Secondary 3 Student Edition:

 

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