ALEX | Alabama Learning Exchange

Burst the Inverse!

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This lesson provided by:

Author:	Kenneth Webb
System:	Tuscaloosa City
School:	Tuscaloosa City Board Of Education
The event this resource created for:	CCRS

General Lesson Information

Lesson Plan ID:	33187
Title:	Burst the Inverse!
Overview/Annotation:	This lesson allows students to investigate functions and their inverses by hand and using graphing calculators. Students will also use equations and graphs. Students will work cooperatively to develop and justify ideas/conjectures about functions and inverses. This is a College- and Career-Ready Standards showcase lesson plan.

Associated Standards and Objectives

Content Standard(s):

Mathematics
MA2019 (2019)
Grade: 9-12
Algebra II with Statistics

12. Solve simple equations involving exponential, radical, logarithmic, and trigonometric functions using inverse functions.

Unpacked Content

Evidence Of Student Attainment:

Students:
Given a contextual situation,

Develop an appropriate model for the situation (exponential, radical, logarithmic, or trigonometric).
Rewrite the function as an equivalent inverse function.
Use algebraic properties to rearrange the function, to isolate the variable, and to find a solution (using technology as needed).

Teacher Vocabulary:

Radical equations
Exponential equations
Logarithmic equations
Trigonometric equations
Inverse functions

Knowledge:

Students know:

Techniques for rewriting algebraic expressions using properties of equality.
Methods for solving exponential, logarithmic, radical, and trigonometric equations.

Skills:

Students are able to:

Accurately use properties of inverse to rewrite and solve an exponential, logarithmic, radical, or trigonometric equation.
Use technology to approximate solutions to equation, if necessary.

Understanding:

Students understand that:

The inverse of exponential, logarithmic, radical, and trigonometric functions may be used to aid in the solution of problems.

Diverse Learning Needs:

Essential Skills:

Learning Objectives:
ALGII.12.1: Define function, function notation, linear, polynomial, rational, radical, absolute value, exponential, and logarithmic functions, and transitive property.
ALGII.12.2: Solve an equation of the form f(x) = c for a simple linear function f that has an inverse.
ALGII.12.3: Write an expression for the inverse of a simple linear function f of the form f(x) = c.
ALGII.12.4: Apply the properties of multiplicative inverses.
ALGII.12.5: Apply the properties of exponentials.
ALGII.12.6: Apply the substitution principle.
ALGII.12.7: Solve a multi-step equation.

Prior Knowledge Skills:

Evaluate a function for an output given the input.
Recall absolute value, radicals, exponents, and linear functions.
Recall how to substitute a value for a variable.
Solve an equation for a missing value.

Local/National Standards:

Standard 1: Make sense of problems and persevere in solving them

Standard 2: Reason abstractly and quantitatively

Standard 3: Construct viable arguments and critique the reasoning of others

Standard 4: Model with mathematics

Standard 5: Use appropriate tools strategically

Standard 6: Attend to precision

Standard 7: Look for and make use of structure

Standard 8: Look for and express regularity in repeated reasoning

Primary Learning Objective(s):

Students will discover and use the properties of functions and their inverses. Students will find inverse equations of linear functions given other linear function equations. Students will use slope criteria and formulas to solve problems and justify reasoning for developed discoveries.

Additional Learning Objective(s):

Students will work cooperatively to discover patterns in the graph table of values while problem solving. Students will verify conclusions by finding slope, developing equations and graphing both functions on the same axes. Students can manually or use graphing calculators to complete the tasks.

Preparation Information

Total Duration:	61 to 90 Minutes
Materials and Resources:	Poster Paper or Poster with grid lines, sticky notes of different types, straight edge, markers, loose leaf graph paper, student may use their own devices or teacher issued graphing calculators.
Technology Resources Needed:	Computer with Internet access, LCD Projector and Document camera to share student paper or calculator work (helpful but not necessary). Smartview with calculator can be used too.
Background/Preparation:	Teacher: Have loose leaf graph paper and chart paper available to be used for lesson. Have graphing calculator ready to be assigned. Students: Algebra students should be able to graph coordinates on Cartesian plane and find Slopes of lines. Students should be able to use slope intercept form to solve problems and have some knowledge of using the graphing calculators.

Procedures/Activities:

ENGAGE

As students enter the classroom, they should be given a group indicator that tells where to sit. Inform the students that their mission will be to determine the inverse of linear functions.

EXPLORE

1. Student groups should develop a table of values for each linear function assigned, reverse the coordinates, find the slope of this set of coordinates, develop equations for the coordinate sets, and graph both equations on the same axes. They should record these findings on their own paper (this information can be keyed into their graphing calculators).

2. The students should come back to the classroom and key their information into their graphing calculators (using the list key on the graphing calculator) to determine the slopes and equations. They should look for patterns of symmetry from graphs. The groups must agree on findings and be able to justify.

3. For students who are unable to find slopes and understand function inverses, provide necessary feedback to move them forward with their thinking. Refer to Slope (Rise over Run): https://www.youtube.com/watch?v=zTa0xTu9Yv4. (If your school system does not allow youtube access, you may download the video using http://www.keepvid.com/.)

Refer to:

https://www.khanacademy.org/math/algebra/algebra-functions/function_inverses/v/introduction-to-function-inverses

https://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/slope-and-intercepts/v/slope-of-a-line

http://www.coolmath.com/algebra/16-inverse-functions/05-how-to-find-the-inverse-of-a-function-01.htm

EXPLAIN

1. The teacher should be sure that students are using graphing tables and slope correctly to develop equations.

2. Groups will discuss properties discovered and begin justifying their ideas as well as creating equations and graphs to match their tables in the calculator.

3. Allow students the time to prepare attractive posters to share with the class to begin discussions about Inverse functions and tables, along with having the groups share their findings using the Smartview function of the calculator.

EVALUATE

Use the provided rubric to assess student work individually and as a group. Students should justify all reasoning stated on the poster. The Assessment Process is on-going. The teacher should visit each group and ask questions to make sure students apply the properties of functions and their inverses. Display the posters in the classroom and have students present conclusions. As students present posters and graphing calculator findings, they should compare posters for other information not listed on their group's poster.

Assessment

Assessment Strategies

Use the rubric provided to continuously assess student work individually and as a group. The teacher should visit each group and ask questions to make sure students apply the properties of functions and their inverses.

Acceleration:	Extensions can be to include understanding the process for algebraically finding inverses of functions and why the process works. Also, Does the process work for quadratic and higher functions? Justify the reasoning!
Intervention:	For extra practice or for students who were absent have them visit: http://www.mathsisfun.com/sets/function-inverse.html http://www.regentsprep.org/regents/math/algtrig/ATP8/inverselesson.htm

View the Special Education resources for instructional guidance in providing modifications and adaptations for students with significant cognitive disabilities who qualify for the Alabama Alternate Assessment.

ALEX Lesson Plan

Burst the Inverse!