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This lesson provided by:
Author: Kenneth Webb
System:Tuscaloosa City
School:Tuscaloosa City Board Of Education
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.

Content Standard(s):
MA2013(9-12) Algebra II with Trigonometry35. Find inverse functions. [F-BF4]
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 critera 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.  

Approximate Duration of the Lesson: 61 to 90 Minutes
Materials and Equipment:

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

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). 

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. 

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

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

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

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

 


Attachments:**Some files will display in a new window. Others will prompt you to download. RubricBursttheInverse.docx
Assessment Strategies:

EVALUATE

Use the rubric provided to assess student work individually and as a group. Students should justify all reasoning stated on 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.  Post 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.

Extension:

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!

Remediation:
Each area below is a direct link to general teaching strategies/classroom accommodations for students with identified learning and/or behavior problems such as: reading or math performance below grade level; test or classroom assignments/quizzes at a failing level; failure to complete assignments independently; difficulty with short-term memory, abstract concepts, staying on task, or following directions; poor peer interaction or temper tantrums, and other learning or behavior problems.

Presentation of Material Environment
Time Demands Materials
Attention Using Groups and Peers
Assisting the Reluctant Starter Dealing with Inappropriate Behavior

Be sure to check the student's IEP for specific accommodations.
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