TITLE:  Simplifying Radicals
LESSON DEVELOPER: Chuck Coronato
GRADE AND CONTENT AREA: 10th grade
TARGET TEACHING DATE: February 2nd, 5th, 6th, 7th,
SCHOOL: John F. Kennedy High School

STANDARDS:

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 10, students will:

  1. Evaluate and simplify expressions.
    • Simplifying Radicals

Back to top

PERFORMANCES:
  1. The students will construct a step by step blueprint for simplifying radicals.
  2. The students will submit items to be used in constructing a bulletin board of several examples of radicals becoming simplified.

Back to top

SETTING:

Real World Setting: Education

You are a math teacher. You are faced with teaching an algebra class. You must teach your students how to simplify radicals. Once you have completed your lesson on how to make a blueprint to simplify radicals, you will have your students construct a bulletin board with examples of simplifying radicals.

Back to top

SMARTSKILLS:

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

Numbers:  Irrational numbers
Concepts:  Simplifying radicals
Processes: Expressing radicands as factors containing squares or cubes and rationalizing denominators.

Level II:  Visualizing Information - Data from Level I that are visualized as information in this standards-based task.

The radical symbol is to be recognized visually.

Arranging:  Students will arrange their presentations in a manner which shows all intermediate steps in simplifying a radical.

Level III: Applying Knowledge - Visualized information from Level II that is applied knowledge in this standards-based task:

Making decisions: Students will decide which types of examples are appropriate for the lesson being presented and for the age group and skill level of students that the lesson is targeting. Creating their own examples indicates an extension of the lesson.

Back to top

PREFERENCES:

Student Involvement - The students will complete the task in project groups of 4 students per group.

Instruction - Activities will be organized and delivered by a teacher facilitating a set of hands-on activities.

Special Education Accommodations - Students with special needs will require calculators.

Use of Resources - The school will provide classroom time to complete the task, and the students will provide classroom materials such as pencils, paper, notebooks, and homework time.

Customer for Student Work - The student will present their work as evidence of task completion to teachers and their peers.

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.

Assessment of Student Work - The following forms of assessment will be used to determine progress and results:

A Performance assessment:  A unit test which includes 10 questions requiring simplification of radicals of the type that students are created for their bulletin boards.

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

Timeline - The estimated time needed to plan, teach, and score this task is three to four class periods.

Back to top

ACTIVITIES:

Teaching for Understanding in Mathematics

Activity 1: The teacher reviews yesterday's lesson and assigns a problem that was not finished

  • Step 1: Ask the students to define a square root and give an example.
  • Step 2: Ask the students to define cube root and give and example.
  • Step 3: Check the homework on yesterday's lesson.

Materials:  Notebooks
Student product or performance: Students produce their homework and give examples of square and cube roots.
Links or connections between different parts of the lesson:  The ability to find square and cube roots will be needed to simplify radicals.
Scoring:  Teacher records if homework was completed.

Activity 2:   The teacher presents an unsimplified radical which has squares that are factors of the radicand.

Example:  Simplify the square root of 20.

Show students how to break the radical into separate radicals with one radicand containing a square.

Example:  the square root of 20 becomes the square root of 4 times the square root of 5.

Find the square root of the square radicand and remove the radical.  The answer will be 2 times the square root of 5.

The teacher shows an unsimplified radical which has a 3 for its index, and shows how factors of the radicand which are cubes must be identified. The rest of the problem is reduced to the above procedure.

Example:  simplify the cube root of 40.

Activity 3:  Teacher shows example of a radical in rational form, and how to rewrite the radical as two separate radicals for numerator and denominator.  The process known as rationalizing the denominator is demonstrated.

Example:  The square root of the fraction 4 over 9.

Example: The square root of the fraction 2 over 5.

Activity 4:  Teacher shows radical with index of 3 with the radicand in rational form.  Repeat the process in Activity 3, but emphasize the need to use the square of the radicand to rationalize the denominator.

Example:  simplify the cube root of the fraction 1 over 4.

Activity 5:  Teacher shows an example of variables under the radical.  Show how to break radicand into factors that are squares or cubes as needed and continue as shown in activity #1.

Example:  simplify the square root of x to the 5th power.

Note:  Many examples can be taken from the Text Book: Algebra and Trigonometry structure and method Book 2 published by McDougal Littell.  Simplifying radicals is in section 6-2 of this text and there are many fine examples on pages 267 and 268.

Activity 6: The teacher presents the task for the day and asks the students to work on it independently (Task is to invent a problem for classmates to solve.)
Note: It is typical to present the task for the day and allow students to solve it in their own way. Often, the task can be solved using a method the students have learned recently.
(Estimated time: 10 minutes)

  • Step 1: Create radicals which need to be simplified with an index of 2.
  • Step 2:  Create radicals which need to be simplified with an index of 3.
  • Step 3:< Each group adds their 2 radicals to a sheet passed around the room.

Activity 7: The teacher suggests that students continue their work in small groups. Leaders of groups share their problems with the teacher, who makes them public, e.g., writes them on the board. Students copy the problems and begin working on them.
Note: It is unusual for students to work this long without a class discussion. Also, it is typical for students to struggle with the task before the teacher intervenes.
(Estimated time: 20 minutes)

  • Step 1: Clarify and troubleshoot problems that groups are having.

Activity 8: The teacher highlights a good method for solving these problems.

Source: Adapted from:

  1. The 1996 Third International Mathematics and Science Study (TIMSS)
  2. Stigler, J. & Hiebert, J. (1999) The teaching gap: Best ideas from the world's teachers for improving education in the classroom. New York: The Free Press.

Back to top

BENCHMARKING:

Student Performance: Students can create their own examples of 5 types of simplifying radical questions.  The 5 types should mirror the examples used in the teaching activity 1 through 5.

Back to top

SCORING:

Holistic Rubric: Completing a Task

Distinguished
  • The student completes all important components of the task and communicates ideas clearly.
  • The student demonstrates in-depth understanding of the relevant concepts and/or process.
  • Where appropriate, the student offers insightful interpretations or extensions (generalizations, applications, and analogies).

Proficient
  • The student completes most important components of the task and communicates clearly.
  • The student demonstrates understanding of major concepts even
    though she/he overlooks or misunderstands some less important ideas or details.

Apprentice
  • The student completes some important components of the task and communicates those clearly.
  • The student demonstrates that there are gaps in his/her conceptual understanding.

Novice
  • The student shows minimal understanding.
  • The student is unable to generate strategy; answers may display only recall effect, lack clear communication and/or be totally incorrect or irrelevant.

Source: Kentucky Department of Education

Back to top

METACOGNITION:

Cognitive Information: I will collect the following information by asking the questions to students in class and summarizing their verbal responses.

  1. Describe what skills you needed to complete this task.
  2. Explain how you solved the goal, problem, or issue in this task.

Attitude Information: I will collect information by asking the questions to students in class and summarizing their verbal responses.

  1. Do you feel that you are good in simplifying radicals?
  2. Did you find this task to be difficult?
  3. Did you enjoy the task?

Back to top

RESULTS:

Reflect: I noticed that the students generated some very good and appropriate questions for the bulletin board.  According to my selected scoring rubric, I graded the projects all as Proficient.  I consider that a good result for the task considering this was the first time that the students were asked to complete this type of task in a structured way.  Almost all of the examples generated would be in the Distinguished category, but the student generated blueprints lacked some detail and failed to accommodate as many strategies as were included in the examples. I'll address this in the next reflection question.  

I also noticed that although the students were confident in their abilities based on the metacognition questions, their scores on the unit test were quite low in the subject area.  The class average for the 10 questions requiring simplifying radicals was only 68, even though the 10 questions were of the type for which students created their own examples. This is much lower than the usual  class average in this topic based on results from previous years.

I also noticed that this task took the students four class periods (41 minutes each) to complete. This is twice the amount of time that is usually used to present the topic of simplifying radicals, so I was very disappointed in the testing results.  Twice as much time on task with lower test results indicates an inefficient method of teaching the lesson.  

What worked and what didn't work:

The students demonstrated enthusiasm for the task.  They generated many good questions.

What didn't work initially was the tendency for students to try and create the hardest examples to simplify.  They needed to be reminded of the Real World Setting that we were using; they are math teachers trying to teach the topic.  Therefore, the task wasn't to create the most difficult examples, but to create examples that would be useful to teach someone who didn't already know how to simplify radicals.  This reminder helped.  

As mentioned earlier, the Blueprints created by the students lacked detail and failed to accommodate all situations.  I believe that the students need to have a few examples of blueprints from other topics given to them before this task.  This may provide the needed guidance.

Action Plan: I will complete the following TaskBuilder Figure 8 Strategy Action Plan to prepare for my next standards-based task.

1. Plan - My next standards-based task will focus on:
Title: Creating quadratic equations to solve using the quadratic formula
Content Area: Mathematics, Algebra II,  Quadratics
Learning Standard(s): 4.3, D, 2
Intent: (Instruction or Assessment)

Students create quadratic equations which have rational solutions, irrational solutions, and complex number solutions.  Students create the data base to be used for their test in this topic.

Back to top