 # Common Core High School Functions Teaching and Learning Resources

## Domain: LE: Linear, Quadratic, and Exponential Models • 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 ### 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:

Interactive Mathematics: Exponential and Logarithmic Functions delves into definitions, graphs, logarithm laws, logs in base 10, natural logs, exponential and log equations, also includes applications and examples.  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.

Imagine Learning Classroom (formerly LearnZillion) subscription needed:

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.

PhET Interactive Simulations: Graphing Lines. Aligns with LE-A.1.a.

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:

YouTube.com: Introduction to Linear Functions video illustrates the concept and how linear functions represent a constant rate of change.

Multiple Choice:

Khan Academy: Practice questions with videos.

Constructed-response:

Khan Academy: HSF-F-LE includes HSF-LE.A.1 through HSF-LE.A.4 Common Core aligned problems.

Also see the following practice questions with videos.

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.

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:

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:

### 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:

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.

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.

PhET Interactive Simulations: Projectile Motion. Aligns with 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:

Constructed-response:

Khan Academy: HSF-F-LE includes HSF-LE.B.5 Common Core aligned problems.

Also see the practice questions with videos in Construct exponential models: Aligns with LE-A.1 and LE-B.5.

Illustrative Mathematics: Functions:

Mathematics Assessment Project: Standards: High School: Functions:

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: 