## Perfect Squares and Factoring

### Johnny WolfeSanta Rosa District Schools

#### Description

Numbers such as 1, 4, 9, and 16 are called perfect squares. Products of the form (a + b)^2 and (a – b)^2 are also called perfect squares, and these expansions are called perfect square trinomials.

#### Objectives

Understands and explains the effects of addition, subtraction, multiplication and division on real numbers, including square roots, exponents, and appropriate inverse relationships.

Describes, analyzes and generalizes relationships, patterns, and functions using words, symbols, variables, tables and graphs.

#### Materials

-Overhead transparencies (if using overhead) for PERFECT SQUARES AND FACTORING (See attached file)
-PERFECT SQUARES AND FACTORING Examples (See attached file)
-PERFECT SQUARES AND FACTORING Worksheet (See attached file)
-PERFECT SQUARES AND FACTORING CHECKLIST (See attached file)

#### Preparations

1. Prepare transparencies (if teacher uses overhead for examples) for PERFECT SQUARES AND FACTORING EXAMPLES. (See attached file.)

2. Have marking pens (for overhead).

3. Have PERFECT SQUARES AND FACTORING EXAMPLES prepared and ready to demonstrate to students. (See attached file.)

4. Have enough copies of PERFECT SQUARES AND FACTORING WORKSHEET for each student. (See attached file.)

5. Have enough copies of PERFECT SQUARES AND FACTORING CHECKLIST for each student. (See attached file.)

#### Procedures

Prior Knowledge: Students should be familiar with basic operation skills, such as addition, subtraction, multiplication, division, exponents, fractions, decimals, area, distributive property, and multiplying binomials.

1. Ask the students what is meant by “identical.” Get their responses and then explain that when we have two factors that are “identical,” we call them “perfect squares.” The term “perfect” and “identical” are synonyms for each other in this case.

2. Ask students to find the product for these problems:
a. (x + 3)
2
b. (x – 3)
2
c. (2x + 4)
2

3. Go through the expansion steps. (See 2a, 2b, and 2c on attached file: PERFECT SQUARES AND FACTORING EXAMPLES.) Encourage students to look for patterns.

4. Go over the “Perfect Square” model. (See #3 on attached file: PERFECT SQUARES AND FACTORING EXAMPLES.)

5. Have students find the product for (y + 8)
2. (See #4 on attached file: PERFECT SQUARES AND FACTORING EXAMPLES.) Then use the “perfect square” model to factor y
2 + 16y + 54.

6. Have students find the product for (2x – 5y)
2. (See #4 on attached file: PERFECT SQUARES AND FACTORING EXAMPLES.) Then use the “perfect square” model to factor 4x
2 – 20xy + 25y
2.

7. Caution students against confusing the “difference of squares” with the “square of a difference.” (See #5 on attached file: PERFECT SQUARES AND FACTORING EXAMPLES.)

8. Assist students in recognizing whether or not a trinomial can be factored using the “perfect square” model. (See #6 on attached file: PERFECT SQUARES AND FACTORING EXAMPLES.)

9. Ask students to do the following: Determine whether x
2 + 22x + 121 is a perfect square. If it is, factor it. Assist students in understanding the “3-step” model for determining if a polynomial is a “perfect square.” (See #7 and #8 on attached file: PERFECT SQUARES AND FACTORING EXAMPLES.)

10. Point out that before factoring a polynomial its terms should be arranged so that the powers of “x” are in descending or ascending order. For example, 8x + x
2+ 16 should be written as x
2 + 8x + 16 or 16 + 8x + x
2. (See #9 on attached file: PERFECT SQUARES AND FACTORING EXAMPLES.)

11. Ask students to do the following: Determine whether 16a
2 + 72a + 81 is a perfect square. If it is, factor it. Assist students in understanding the “3-step” model for determining if a polynomial is a “perfect square.” (See #10 on attached file: PERFECT SQUARES AND FACTORING EXAMPLES.)

12. Ask students to do the following: Determine whether 15 + 4a
2 – 20a is a perfect square. If it is, factor it. Remind students that the terms must be in ascending or descending order! Assist students in understanding the “3-step” model for determining if a polynomial is a “perfect square.” (See #11 on attached file: PERFECT SQUARES AND FACTORING EXAMPLES.)

13. Ask students to do the following: Determine whether 16x
2 - 26x + 49 is a perfect square. If it is, factor it. Assist students in understanding the “3-step” model for determining if a polynomial is a “perfect square.” (See #12 on attached file: PERFECT SQUARES AND FACTORING EXAMPLES.)

14. Ask students to do the following: Determine whether 9x
2 –12xy + 4y
2 is a perfect square. If it is, factor it. Assist students in understanding the “3-step” model for determining if a polynomial is a “perfect square.” (See #13 on attached file: PERFECT SQUARES AND FACTORING EXAMPLES.)

15. Distribute the PERFECT SQUARES AND FACTORING WORKSHEET. (See attached file.)

16. Distribute the PERFECT SQUARES AND FACTORING CHECKLIST. (See attached file.) Describe what constitutes an “A,” “B,” “C,” “D,” and an “F” in the CHECKLIST.

17. The students will write their responses on the worksheet.

18. The teacher will move from student to student observing the students work and lending assistance.

#### Assessments

The student worksheet will be collected and scored according to the PERFECT SQUARES AND FACTORING CHECKLIST. (See attached file.)

#### Extensions

Have a student draw a square on the board. The student will place a trinomial in the center of the square to represent its area. The other students are to determine if this can be the area of a square. If it is, then determine its sides.