TITLE: Rational Exponents and Radicals
CONTENT AREA AND GRADE: Algebra - Grade 11
TARGET TEACHING DATE: March 19th, 20th, 21st, 22nd
SCHOOL: John F. Kennedy High School



STANDARD 4.3 PATTERNS AND ALGEBRA: All students will represent and analyze relationships among variable quantities and solve problems involving patterns, functions, and algebraic concepts and processes.

Strand D. Procedures: Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:

1. Evaluate and simplify expressions containing Rational Exponents. 

3. Judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology.

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  1. The students will complete the task individually, using properties of rational exponents and as a group to evaluate and simplify expressions involving nth roots and rational exponents.

  2. The student will construct a bulletin board of several examples of powers, nth roots and Rational Exponents by simplifying, combining Roots and Radicals. 

  3. The students will extend  the knowledge to simplify variable expressions and to solve real-life problems such as evaluating the surface area of a human, given the mass etc...

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Real world settings: You are a Financial Planner.  You are faced with finding inflation rate for homes.  To calculate the inflation of homes that increases from p1 to p2 over a period of n years, the annual rate of inflation (expressed as a decimal) can be modeled by i = (p2/p1)^(1/n) -1 .  This is problem # 64 on page 405 on the class work.  We will thoroughly discuss this when we cover this problem on the class work.  As a follow up, problem #63 on page 442 on the homework is similar to this was done.

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Level 1:  Acquiring Data - Data students will acquire in this standards-based task.


  • Rational Numbers
  • Squares- Square roots
  • cubes - cube roots
  • nth roots
  • Simplification of Radicals
  • Properties of Rational  Exponents

Level 2:  Visualizing Information - Data from Level 1 are visualized as information in this standards-based task.

Level 3: Applying Knowledge - Visualized information from level 2 that is applied knowledge in this task.

Solving real-life problems.

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Task Approach - The students will complete the task individually and working in cooperative groups.

Task Delivery - The task will be organized and delivered in the student class work during class time.

Use of Resources - The students will provide classroom materials such as pencils, paper, notebooks.

Customer for  Student Work -  The students will present their work as evidence of task completion to the teacher.

Assessment of Student work - The following people will be involved in assessing student work generated to complete the task: The student's teacher and peers.

The following forms of assessment will be used to determine progress and results:

A performance assessment

Reporting results - The assessment results will be reported as letter grade.

Timeline -  The estimated time to plan, teach, and score this task is  four class periods(41 minutes each).

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Activity 1.  The teacher reviews the previous lesson discussed earlier and assigns a problem to reinforce the lesson.

Developmental Activities: ( estimated time- 3 minutes each)

  1. Ask the students to define a square and give an example.
  2. Ask the students to define cube root,4th root, nth root etc..and give examples of those.
  3. Review properties of Rational Exponents and use them to do sample problems.
  4. Do practice problems using properties of exponents.
  5. Check the homework on previous day's lesson.

Activity 2. Guided Practice - (estimated time 3 minutes each)

  1. The teacher connects the different parts of the lesson.
  2. Check the ability to rewrite the terms using radical and rational exponents.
  3. The teacher presents a compound radical or cube root or 4th root that are squares,cubes or 4th powers in the radicand respectively.
  4. The teacher solves equations with higher powers than two and gives the students practice problems.

Activity 3.

The teacher presents the task for the day and asks  the students to work on it individually and in groups.

(optional: Do some  on the board)

(Estimated time 10 minutes)

Activity 4.

The teacher suggests the work from the test book (Algebra 2 by Larson, Boswell, Kanold, Stiff) - page 411 (22 - 66) even numbers as home work (Estimated time for this is 20 minutes)

Materials needed for this task: Text book, Note book, pen/pencil. 

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Student Performance One: (class work: page 411 problems 5 - 16, and 50 - 55)

1. Assessment Benchmarking Example: Home work Page 411 - 412 (22 -66 )even numbers
2. Real World Benchmarking Example: Page 405 ,problem #64

[You are a Financial Planner.  You are faced with finding inflation rate for homes.  To calculate the inflation of homes that increases from p1 to p2 over a period of n years, the annual rate of inflation i(expressed as a decimal) can be modeled by i = (p2/p1)^(1/n) -1.]

NOTES:  Many examples were taken from the Text Book Algebra 2 by Larson R., BOSWELL L., Kanold D.and Stiff L.  Published by Mc Dougal Littell. 

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New Jersey High School Mathematics Rubric: Extended Constructed Response
Level 3

The response indicates application of a reasonable strategy that may or may not lead to a correct solution. The representations are essentially correct. The explanation and/or justification is generally well developed, feasible, and supports the solution. The response demonstrates a clear understanding and analysis of the problem.

Level 2

The response indicates an incomplete application of a reasonable strategy that may or may not lead to a correct solution. The representations are fundamentally correct. The explanation and/or justification supports the solution and is plausible, although it may not be well developed or complete. The response demonstrates a conceptual understanding and analysis of the problem.

Level 1

The response indicates little or no application of a reasonable strategy. It may or may not have the correct answer. The representations are incomplete or missing. The explanation and/or justification reveals serious flaws in reasoning. The explanation and/or justification may be incomplete or missing. The response demonstrates a minimal understanding and analysis of the problem.

Level 0 The response is completely incorrect or irrelevant. There may be no response, or the response may state, "I don't know."
Notes: Explanation refers to the student using the language of mathematics to communicate how the student arrived at the solution.

Justification refers to the student using mathematical principles to support the reasoning used to solve the problem or to demonstrate that the solution is correct. This could include the appropriate definitions.

Essentially correct representations may contain a few minor errors .

Fundamentally correct representations may contain several minor errors .

reversed axes, or scales that are not uniform.
Source: http://www.mdk12.org/mspp/high_school/structure/algebra/index.html

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Attitude Information: I will collect the following information (Items to keep in mind for students.)

  1. Do you feel that you are good in (mathematics, Algebra)?
  2. Did you find this task to be difficult?
  3. Did you see the usefulness of what you were asked to do in real life?(such as finding the surface area of a mammal,if you know the weight,Financial Planning etc...)
  4. Did you enjoy the task?
  5. What would you tell your friends about this task?

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Analysis:   I will ask the students to explain the skills they have used in solving problems involving rational exponents. I will examine the data in the chart to look for trends, contributing factors, and implications of student performance over a series of assessments of the same learning standard.

  • Trends: The class work grades, home work grades and quiz grades from this lesson using BCR Rubric.
  • Contributing factors: Regular attendance on time made a big difference.
  • Implications for student performance: I will check the number of students who performed at or above proficient level.

Reflect: As I relate my students' results with my lesson activities, I noticed that...

  • Students,  who were present every day had the most influence on student achievement because they  were there to listen 
  • (Some focussing activity at the beginning of the class) has the most promise for becoming a best practice in my classroom because students were really focussed for the lesson.

Summarize: The homework assignment and focussing activity were valuable to the lesson.  This assignment would not be successful if students were not aggressively encouraged to complete the work assigned.

Three- fifths of the students reached the proficient level on the scoring tool. This was not an encouraging result. Many of the students who did not reach the proficient level were on the right track. The trend in difficulty lied in establishing effective supporting details. Often students made silly mistakes in the proficiency they have already acquired.  I was surprised that these activities dramatically helped clear up many of the students' organizational problems. I will probably focus more on non-supportive details the next time I attempt this lesson. I noticed that the attendance and tardiness is a main factor for their proficiency of the lesson.

As soon as I achieve an instructional improvement and make it a routine process, it becomes a standard to be challenged with new plans for additional improvement.
The task took four-41 minutes to complete, this is longer than the normal instruction for a topic.  As I mentioned  earlier less than 60% of the students scored proficient. I believe the students needed more guided practice to achieve the goal.  

The students took a test on March 26th, the blue prints created by the students lacked the details. Only 60 % proficient.

Action Plan: I will complete the following Task Builder Figure 8 Strategy Action Plan to prepare for my next standards-based task. I will write a summary paragraph about my findings, contributing factors, and implications for improvement.

1. Plan - My next standards-based task will focus on:

  • Title: operations with power functions and function operations.
  • Content Area:Mathematics, Algebra 2
  • Learning Standard(s):4.3 ,D,2.
  • I will prepare the students with more detailed examples and guided practice in future

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