Introduction to Radicals

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SWBAT simplify radical expressions that are perfect squares and non-perfect squares.

Big Idea

The point of this lesson is for students to understand the difference between exact and approximate solutions.

Warm up

10 minutes

The objective of this lesson is for students to have a clear understanding of the difference between an exact solution and an approximate solution. The purpose of this Warm up is to check my students' prior knowledge of simplifying radicals.  

Even though students have previously learned simplifying radicals, my goal in this lesson is for students to develop a deeper understanding of radicals.  Having a deeper understanding of radicals will help students be able to simplify and solve problems involving quadratics in the next unit.  I use Problem 4 of the Warm Up to introduce two different methods for simplifying radicals.  I intend for the Warm Up to take about 10 minutes for the students to complete and for me to review with the class.  I review the Warm Up in the video below.


Guided Practice

10 minutes

After the Warm Up is completed, I hand students a piece of Dot Paper. I work with students in a Guided Practice situation to review some concepts of Perfect Squares.  I have students select three whole number side lengths for three squares, and draw the three squares on the dot paper.  I use the activity for students to discover the relationship between the side length of a square and its area.  I want students to use the geometric representation of a square to easily see that:

  1. The side squared is equal to a perfect square if the side is a whole number.
  2. The square root of the area is equal to length of the side of the square.

I expect my students to know the perfect squares through 15 by memory. So, after the exploration of the perfect squares activity, I have students do a quick self-check of their recall of these number facts.  I quiz the class briefly on the perfect squares from one to fifteen.  Once students complete the quiz, I provide the answers. My goal is to help students recognize which perfect squares they need to learn.  I assign this as homework: each student needs to memorize the squares of the numbers up to 15.  


Independent Practice

25 minutes

I work the examples with the students from the Simplifying Radicals worksheet provided in the resources.

Worksheet Source: (accessed online July 21 2014)  

I chose this practice because I want students to have a short practice with a variety of problems.  I plan to work a few example problems with the students, showing both simplifying the radicals by prime factorization and by using perfect squares again.  

I will model Problem 11 in the practice before handing them the worksheet.  I review the examples in the video below.  



I allow my students to use any method of their choice. Students must show work and be able to explain the method that they used.

I modify this assignment by asking students to provide an approximate as well as an exact answer. Again, like in today's Warm Up, I remind students that exact answers are in radical form because they are not rounded.  Decimals are approximate because they are rounded answers.

I allow students about 25 minutes to complete the independent practice.  As students are working, I walk around to monitor student work and assist students one on one or in small groups.  At the end of the 25 minutes I provide the students the answers.  The check is to provide students with immediate feedback.  I review any problems of difficulty, or that they need explained.

Exit Slip

5 minutes

I provide each student with an Exit Slip with about five minutes left in class.  I use the Exit Slip as a formative assessment to check for student understanding on the difference between providing an exact answer and an approximate answer.  I take up the Exit Slips as students finish.  Students that do not complete the Exit Slip may take it home as homework.  However, I try to take up Exit Slips before class is over to ensure independent work.

The highest percentage of mistakes on the Exit Slip was confusing which answer is approximate and which answer is exact. I plan the Exit Slip so that students find the exact answer in simplest radical form by taking the square root of the area of the square.  I purposely plan the second problem so that students could approximate  their answers with decimals instead of multiplying, adding or subtracting radicals which I teach later in this unit.

When discussing the Exit Slip with students, I want students to realize how to estimate radicals using mental math.  The exact answer for the side of the square is 2 square root of 33.  I model estimating radicals often so that students form an idea of what the radical form represents even if they are going to use a calculator.  For example, square root of 33 is between the two perfect squares of 25 and 36, so it is a number between 5 and 6 that is being multiplied by 2.