TITLE: Simplifying Radicals
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.
Real World Setting: Education
Level I: Acquiring Data - Data students will acquire 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.
Student Involvement - The
students will complete the task in project groups of 4 students per group.
Assessment of Student Work - The
following forms of assessment will be used to determine progress and results:
Reporting Results - The assessment results will be reported as a letter grade.
Teaching for Understanding in Mathematics
Activity 1: The teacher reviews yesterday's lesson and assigns a problem that was not finished
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.
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.)
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.
Activity 8: The teacher highlights a good method for solving these problems.
Source: Adapted from:
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.
Cognitive Information: I will collect the following information by asking the questions to students in class and summarizing their verbal responses.
Attitude Information: I will collect information by asking the questions to students in class and summarizing their verbal responses.
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.
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.
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.