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Rational Functions

Unit Plan

Rational Functions

Objectives

In this unit, students perform basic arithmetic and algebraic tasks while working with rational functions. Students will:

  1. use basic operations (add, subtract, multiply, and divide) with rational expressions.
  2. solve rational equations.
  3. simplify rational expressions involving complex fractions.

Essential Questions

  1. How can we extend arithmetic properties and processes to algebraic expressions and processes and how can we use these properties and processes to solve problems?

Related Unit and Lesson Plans

Related Materials & Resources

The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.

  1. Video–Simplifying Rational Expressions https://www.youtube.com/watch?v=B4bVlDgHF5I
  2. Video–Multiplying Rational Expressions https://www.youtube.com/watch?v=-aZ0a7aMiPE&feature=channel
  3. Video–Dividing Rational Expressions https://www.youtube.com/watch?v=Fdie6CqInyg&NR=1
  4. Video–Adding Rational Expressions https://www.youtube.com/watch?v=omv7Di2o8-Y&feature=channel
  5. Video–Subtracting Rational Expressions https://www.youtube.com/watch?v=gv8h9Ivs__Y&feature=channel
  6. http://www.beaconlearningcenter.com/documents/1551_01.pdf - document contains more step-by-step examples, practice problems, and scoring rubrics
  7. https://www.youtube.com/watch?v=QrsITlIiwUY- video with 3 more examples & explanations
  8. https://www.youtube.com/watch?v=01UAMu531og video showing process of simplifying multiple types of complex fractions, from basic to more complicated
  9. https://www.youtube.com/watch?v=q4aMMtSvsvI&feature=related video shows a slightly different method of solving a complex fraction involving addition in the numerator (LCD-Multiply-Divide Method)

Formative Assessment

  • View

    Multiple Choice Items

    1. Find the sum of and 01.PNG

    A

     

    01a.PNG

    B

     

    01b.PNG

    C

     

    01c.PNG

    D

     

    01d.PNG

    2. What is the common denominator needed to solve the following problem?

    02.PNG

    A

     

    x² + 5x + 6

    B

     

    x(x + 2)(x + 3)

    C

     

    x + 3

    D

     

    2x² + 8x + 6

    3. Find the area of a rectangle with length 03.PNG .

    A

     

    03a.PNG

    B

     

    03b.PNG

    C

     

    03c.PNG

    D

     

    03d.PNG

    4. Solve: 04.PNG

    A

    3

    B

    9

    C

    -9

    D

    -3

    5. Solve: 05.PNG

    A

    3

    B

    -4

    C

    -3, 4

    D

    -4, 3

    6. Solve: 06.PNG

    A

    -15

    B

    -35

    C

    15

    D

    5

    7. Simplify: 07.PNG

    A

     

    5ab

    B

     

    07b.PNG

    C

     

    07c.PNG

    D

     

    07d.PNG

    8. Simplify: 08.PNG

    A

     

    x – 1

    B

     

    08b.PNG

    C

     

    2(x + 1)

    D

     

    08d.PNG

    9. Simplify: 09.PNG

    A

     

    09a.PNG

    B

     

    y – 2

    C

     

    2y + 4

    D

     

    y + 2

    Multiple-Choice Answer Key

    1. C

    2. B

    3. A

    4. B

    5. D

    6. C

    7. B

    8. D

    9. D

     

     

    Short-Answer Items:

    10.       Divide the following; show all work.

                10.PNG

     

     

     

     

     

     

     

     

    11.       The sum of a number and 15 times its reciprocal is 8. Find the number(s).

     

     

     


     

    12.       Simplify: 12.PNG


     

    Short-Answer Key and Scoring Rubrics:

    10. 10key.PNG

    Points

    Description

    2

    ·      The student correctly factors all expressions.

    ·      The student correctly divides by the appropriate factors.

    ·      The student displays understanding of method of dividing rationals; remembers to use the reciprocal and multiply.

    1

    ·      The student displays understanding of method of dividing rationals; remembers to use the reciprocal and multiply.

    ·      The student’s work has some errors in factoring of expressions.

    ·      The student demonstrates knowledge of when to divide by common factors, but factoring errors leads to incorrect answer.

    0

    ·      The student does not demonstrate an understanding of a method for dividing rationals.

    ·      The student does not demonstrate an understanding that factoring expressions is necessary for correctly completing the problem.

    ·      The student does not complete the problem.

     


     

    11. n = 5 and n = 3

     

    Points

    Description

    3

    The student sets up the equation correctly.

    The student uses appropriate methods to solve the problem.

    The student reaches the correct conclusion.

    The student checks the solution.

    2

    ·         The student’s work has a small error in setting up the equation but appropriate techniques are used to solve the problem. The student comes to the correct conclusion for equation used.

    ·         The student uses the correct equation but has a small error in solving techniques.

    1

    ·         The student demonstrates a moderate understanding of required steps but has many arithmetic errors and gaps in understanding.

    0

    ·         The student demonstrates no understanding of processes required to solve problem.

     


     

    12. ¼

    Points

    Description

    3

    The student factors correctly.

    The student performs the use the reciprocal and multiply step correctly.

    The student divides by common factors correctly to get the correct answer.

    2

    The student’s work has a slight factoring error.

    The student performs the reciprocal and multiply step correctly with factors.

    The student divides by common factors appropriately.

    1

    The student understands the reciprocal and multiply step.

    The student’s work has many errors in factoring and dividing processes.

    0

    The student does not factor the polynomials.

    The student divides inappropriately.

    The student demonstrates no understanding of correct simplification processes.

     


     

    Performance Assessment:

    Apply the concepts used during this unit to solve the following problems. Show all work for full credit.

    a.       What would the common denominator be between two fractions if one has a denominator of x + 5 and the other has denominator x - 3?

    b.      Use the concepts from part a to find the sum of pa-b.PNG

    c.       Simplify the following: pa-c.PNG

    d.      Using your solutions to part b and part c, simplify the following complex fraction:

    pa-d.PNG

    e.       Would -5 be a reasonable solution for the equation: pa-e.PNG Why or why not?

    f.       Find the exact solution(s) to the problem in part e.

    Performance Assessment Scoring Rubric:

    Points

    Description

    4

    ·         The student shows all work.

    ·         The student answers all parts correctly.

    ·         The student interprets all answers appropriately.

    ·         The student’s work has no errors.

    3

    ·         The student shows understanding of concepts and procedures but has small error(s) in arithmetic that prevents him/her from getting all answers 100% correct.

    ·         The student demonstrates accurate interpretation of questions and procedures required to solve problems.

    2

    ·         The student’s work has multiple arithmetic errors.

    ·         The student demonstrates understanding of some parts but not all.

    ·         The student demonstrates a partial understanding of questions and procedures required to solve problems.

    1

    ·         The student’s work and arithmetic have errors or are not shown.

    ·         The student shows little work.

    ·         The student demonstrates minimal understanding of overall procedures.

    0

    ·         The student’s work is missing or a large majority is completely illogical.

    ·         The student demonstrates no understanding of questions or topics.

    Solutions:

    a.   (x + 5)(x – 3)  b. pa-bkey.PNG  c. pa-ckey.PNG

    d.   pa-dkey.PNG

    e.       No, −5 creates a zero denominator, which is undefined for a fraction.

    f.       31

DRAFT 11/05/2010
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