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In this section, we feature teacher reflections related to mathematics topics challenging to teach. These reflections include some common student difficulties and errors as well as ways of addressing these challenges. If you have such a topic you wish to share with other practitioners in the field, please respond to the Call for Manuscripts on page 22. Teaching Dilemmas

A Few Interesting Methods to Teach Factoring of Trinomials

Zhenqiang Li

factor out that greatest common factor. Next, we examine the remaining trinomial in the form, ax 2 + bx + c . We locate the middle term and split the middle term in the following way: ax 2 + bx + c = ax 2 + mx + nx + c = ( dx + e ) ( fx + g ). We have m + n = b , and mn = ac . We use factoring by grouping to find the factors (Ratti, McWaters, & Skrzypek, 2019). When you find m + n = b is a true statement, you do not need to keep trying all of the possible factors of ac, since we know a quadratic equation has a maximum of two real roots of the form, ( dx + e ) ( fx + g ). Notice that a = df ; c = eg ; b = gd + ef , in which a, b, c, d, e, f, g are all constants. An example of Method I is given using a cubic pol ynomial after the greatest common factor among all of the terms is factored out, 12 x 3 + 10 x 2 – 8 x . The greatest common factor factored out is 2x, 12 x 3 + 10 x 2 – 8 x = 2 x (6 x 2 + 5 x – 4), which most students can do. The remaining section shows how Method I is used to factor the remaining expres sion, 6 x 2 + 5 x – 4. Basically, this method takes the remaining expres sion that is in the standard form, ( ax ) 2 + bx + c . We make a list in a table of all possible factors for the product of ac and use those factors to calculate their sums to locate the one that is equal to b. Next, we split b into two factor groups to create the ex pression for each factor group. The details for find ing the factors for 6 x 2 + 5 x - 4 using this method are given below. Step 1: Write your polynomial in standard form, and identify the coefficients a, b, c . For this example, a = 6, b = 5 and c = - 4. Step 2: Make a table of the two factors with the product value for a × c . For example, ac = 6( - 4) = - 24.

Motivation and Approach:

The purpose of this paper is to (i) create a frame work for students who have difficulties factoring trinomials; (ii) share some interesting methods with educators to teach the content; and (iii) describe some teaching activities that are instructionally powerful and motivates students. According to re search by Hülya Burhanzade and Nilgün Aygör (2015), students lack a solid foundation to factor trinomials. Many exams administered in the middle and high school classrooms are multiple choice and students are proficient using strategies to find the correct answer without factoring. The step - by - step approach used in this paper helps students master factoring Likewise, in my teaching experience, I found some students do not understand how factor ing connects to their prior math knowledge, specifi cally, the distributive property. Therefore, they are unable to transfer the knowledge they previously learned in elementary school to this new topic. Most textbooks provide only one method of factor ing. In this article, five additional ways to teach students how to factor trinomials in the standard form, ax 2 + bx + c, are offered. This allows stu dents to understand how factoring connects to other areas in math. The multiple ways to solve the same type of problems are presented here to addresses students ’ needs and give instructors the option of exposing students to a variety of methods, depend ing upon their students ’ prior knowledge and pref erence.

Method 1: Factoring by Grouping

For any factoring problem, the first step is to al ways check to determine if there are any common factors among all of the terms. If there is, then we

Virginia Mathematics Teacher vol. 46, no. 2

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