Simplifying Square Roots

Present the warm-up problem (5 minutes).  The warm-up problem for this lesson (found in the presentation MTPGeometry_Lesson6_0_7.pptx) asks students to simplify a square root by factoring perfect squares out of the radicand.  This warm-up activity is intended as a formative assessment.

I begin every lesson by displaying a warm-up problem on the front board using an overhead projector.  This is a classroom routine, so students already know what to do.  Students get whiteboards, rags, and markers when they arrive in class, and arrange their desks in pinwheels.  Warm-up problems must be completed within the time limit (5 minutes).  Teams earn points only if every team member has the same answer (or equivalent answers) and all members show their work. 

I want students to work together while reviewing simple skills.  This warm-up routine has been fairly effective.  I often observe students coaching their peers so that the team can earn points.  While students do sometimes copy their team-mates, I know that at least they are copying the entire procedure as modeled by another student.

Complete administrative tasks (during the warm-up).  While students complete the warm-up problem, I take attendance and circulate around the classroom noting who does not have their homework or their materials. Before students begin the warm-up problem, they put their homework and one of their required materials (textbook, planner, pencil, or course notebook—a different item chosen at random each lesson) where I can see them.  Teams earn the maximum points each lesson only if every member of the team is punctual, prepared with both homework and materials, polite, and positive. 

I want to reinforce the expectation that students will come prepared in order to contribute to their team and to the class as a whole.  Students can be stubbornly resistant to both extrinsic rewards (points) and peer pressure, yet I regularly see evidence that students are at least getting the message.  Students chide each other for failing to come prepared, and when students who rarely complete homework do come prepared, they often proudly announce the fact to me and to their teammates. 

Homework Review (10  minutes).   The second side (problems #2-4) of handout Homework 6-9 was assigned to be completed at the beginning of this lesson.  This homework assignment is review for the unit test.

Following the warm-up, I display solutions to homework problems using the document camera and overhead projector.  I ask for and answer questions about the homework.  Having model solutions prepared saves time when students have questions.   I collect homework every other day and review one-half the assigned problems at the beginning of every lesson.   

 Motivate the lesson (1 minute). In a previous lesson, I told my students that they must learn to simplify square roots, because many beautiful relationships involving special right triangles and trigonometry will ‘pop out’ when they work with square root expressions, but will be very hard to see if they convert the roots to decimal approximations.  Also,  many textbooks—as well as standardized tests—will expect students to be able to convert between different forms of a square root expression.  In this lesson, I remarked that most students seemed to pick up the skill of simplifying square roots quickly when we first learned it.  Since then, however, many seem confused about it.  This lesson will give the students notes to study from as well as additional practice.

Review learning targets and the agenda for the lesson (2 minutes).   Display the agenda and learning goals for the lesson as you distribute materials for the activity.  Ask students to read them over and invite questions. 

Ask students to take notes as you review the geometric model and demonstrate methods of simplifying square roots (10 minutes).  Before class, print the guided notes (MTPGeometry_Notes_6-7.docx).  Make one copy for each student.  Distribute the guided notes.    For this lesson, complete only the front side of the guided notes (through Example 3).

I give all notes in this course in the form of guided notes, which I print on goldenrod paper.  I ask my students to keep all notes for the entire year, and inspect notebooks at the end of every unit.  I give the notes using the mimeo board, which allows me to save what I write on the board.  I try to make note-taking into a class discussion.  I ask students to call out the words that I write in the blanks if they think they know them.  I encourage them to ask questions.  Although students would love to write on the mimeo board if I asked for a volunteer to be the class scribe, I rarely do this as it increases the time required. 

Five years ago, no more than a quarter of my students would use their notes on tests or quizzes, and students occasionally would simply refuse to take notes in class.  Now, students regularly refer to their notes during tests, and many students tell me that they have saved all the notes I gave them in prior courses.  These differences are a reflection of a changing school culture, so I can’t point to my own practices as the cause.  Nevertheless, I believe that my use of guided notes has helped.  My current focus is to change the purpose of class notes: from ‘telling’ students how to solve a particular type of problem to asking them to summarize the knowledge they have constructed as a class. 

Have students practice (15 minutes).  The practice activity is intended to be completed using individual white boards.  The problem set (found in the presentation MTPGeometry_Lesson6_0_7.pptx)  is displayed using the document projector.  Problems are presented in pairs, allowing students to work at different speeds.  Students may choose to complete either the problem on the left or the problem on the right.  Ask students to write the number of the problem they completed.   Insist that students show their work, both to help you make corrections when necessary and so that they can explain their thinking to teammates who may need their help.  Students hold their whiteboards in the air when they are finished with a problem so that you can check it.  If students are working fast, they should complete both problems in each pair, but all students should concentrate on answering one problem of each pair correctly first. 

As students are working, circulate around the classroom.  Common problems to look for: 

• Students may forget that the square root of a number is represented by the length of one side of a square whose area is equal to the number.  This misunderstanding shows up when students are asked to write the square root in simplified form.  To ensure that students have the opportunity to recognize and correct this problem, ask students to write their answer to every problem by completing the second half of each equation. 
• Students my write both factors of a perfect square outside the radical.  This probably happens because students  forget the meaning of square root and feel that they need to include all the factors of the number in their answer.  Remind students that the square root of a number is represented by the length of one side of the square whose area is equal to the number.  They do not have to include the length of another side of the square in their answer, because all sides of a square are the same.
• Students who are confused may erase their work before you have time to come around and check it.  Ask those students to concentrate on working a single problem to completion and not to be concerned with how fast others are working.  Stay with a student if necessary, to give them support and ensure that they have success.