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Once students find the correct sum, the calculations are done. They do not need to try the remaining possible factors, since a quadratic equation has a maximum of two roots. The Tic Tac Toe method focuses attention on the middle term, which allows the students to notice how the middle term is split. I found this method very helpful because students already know the Tic Tac Toe game and I found many students remem ber this method better and are able to implement it more quickly than the other methods I shared. This method is explained using the Free Algebra Tiles simulator, which is available from the CPM Educational Program. The Geometric Model uses algebra tiles to model the trinomial. Different sizes of algebra tiles represent different terms in the tri nomial, which are shown by areas. We will use the x 2 - tiles, x - tiles and the unit 1 - tiles. The solution is found by physically rearranging the various sized tiles into a rectangle that has no gaps or leftover tiles. Sometimes, in order to make a rectangle, we may need to add a few equal pairs of positive and negative tiles that sum to zero, which is sometimes called “ zero pairs. ” Again, we will use the trinomi al, 6 x 2 + 5 x - 4, for our example. Step 1: Let us use the blue color tiles to represent positive values and red color tiles to represent neg ative values. We use 6, x - by - x sized square tiles to represent 6 x 2 , 5 x - by - 1 rectangular tiles to represent 5 x , and 4 red color unit squares tiles to represent - 4 (see Figure 1). Method 4: Factoring by a Geometric Model Us ing Algebra Tiles

als with positive constants, we normally do not need to use zero pairs to create a rectangular shape with our tiles. These questions are much simpler than polynomials with negative constants and may be your first choice when introducing factoring tri nomials to your students using this method. Step 3: When the tiles are placed into a rectangular shape, the length and width of the sides show the factor solutions for the trinomial. For our example, the rectangle has a width of (3 x + 4) and length of (2 x – 1), which are the factors for the trinomial 6 x 2 + 5 x – 4, when we multiply the binomials.

Mathematical explanation: since students know the area of a rectangle is length times width, they can easily grasp that a polynomial can be represented by an area model with the factors representing the length and width. We use algebra tiles to model polynomials and shows the relationship between geometrical area and algebraic factoring. Algebra tiles are mathematical tools that allow students to better understand algebraic ways of thinking, spe cifically for factoring trinomials. This method, factor by Polynomial Graphing, uses a calculator or another type of software. The stu dent traces the graph of the polynomial until if crosses the x - axis, to find its roots. If the roots are decimals, we can convert them into fractions. We can remove the fractions by multiplying the root equations by the denominators of the roots. For ex ample, x = 2.33 becomes x = 2 - 1/3, or 3 x = 7. We work backwards to change this root into one of the binomials by having the equation equal to 0, or 3 x – 7 = 0. We do the same thing with the second bi nomial factor, (x— 1/2). Method 5: Factoring by Polynomial Graphing

Figure 1

Step 2: Next, rearrange the tiles, like a puzzle, to form a rectangle shape. After a few trials, we deter mine we cannot arrange the tiles to create a rectan gle shape with the current tiles available. Since adding zeros to the polynomial will not change its value, we can add equal values of positive and neg ative pairs of tiles that will sum to zero, our zero pairs. We see that we are 6 x - tiles away from mak ing a rectangle shape, which suggests we need to add 3 blue x - tiles and 3 red x - tiles. For polynomi

Virginia Mathematics Teacher vol. 46, no. 2

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