Cognition or Computation? Factoring in Practice

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1 Cognition or Computation? Factoring in Practice Keith Hubbard (1), Lesa Beverly (1), Leah Handrick (1), & Meagan Habluetzel (2) Picture this: Kayla is halfway through her first semester of teaching algebra at Blocker High. Her mentor teacher, Mr. Hiland, is fairly supportive, but always seems to suggest methods that she's never heard of before and that often feel to Kayla like magic tricks. When discussing order of operations, he used the phrase Pinch Each Midget During Any Story. For factoring, he likes the Hilander Hiyah method. These aren't exactly advocated in the standardized curriculum Kayla s district uses, but the students do seem to remember the catchy names. Kayla misses her college community so she makes a point of making it to her first college reunion and finds herself sitting next to one of her college math professors. Kayla can't resist asking about the Hilander Hiyah. Is it really true? Should I be using this method? After about 10 minutes, she and her professor confirm that it is true ONLY if the quadratic is simplified (has no common integer factors). But they go on to discuss shortcomings of this method. How come no one warned Kayla about these memory tricks? What should she be looking for in methods she keeps and tricks she discards? We've worked to compile a list of methods (certainly NOT exhaustive) with some thoughts about what makes methods beneficial or detrimental to students. Approach 1: Hilander Hiyah What would you say if a colleague of yours outlined the following approach to factoring? Given a factoring problem: f(x) = 4x 2 + x Remove the lead coefficient and multiply the constant by the lead coefficient. x 2 + x Factor this problem as you normally would. (x + 8)(x 5). The 8 is evenly divisible by the 4, so divide. The 5 is not divisible by the 4, so place the 4 in front of the x. (x + 2)(4x 5) This trick has a number of troubling shortcomings. First, notice that none of the expressions are equal to the expressions above or below it. Following this strategy ignores the importance of retaining equality when performing algebraic manipulations. Failing to conceptualize equality correctly appears to be a common but addressable problem in American classrooms (Li, et. al. 2008). Second, the strategies used by this method should never be used on an arbitrary mathematical equation. Imagine the issues students would have if they converted f(x)=4x 10 to x 40 believing both expressions to contain the same information. Third, this strategy downplays the significance of the distributive property. Fourth, this method does not generalize well to higher order polynomials. And finally, for some problem types, this strategy does not even work. Consider f(x) = 4x 2 + 4x. Following the steps above, we d get x 2 + 4x 12 (x + 6)(x 2) (4x + 6)(4x 2) One could argue that eliminating common factors would then yield the right answer: (2x + )(2x 1). But this adds yet another step, which is not mathematically tractable in its own right, and supplants the right conception of the distributive property. If one really likes this approach, a way that it might be justified is using substitution. First, multiply the entire equation by a clever one - the lead coefficient divided by itself. In our first example, we would then have f(x)= (4x 2 + x 10)= ((4x) 2 + (4x) 40). If we then set y=4x, the expression above (1) Stephen F. Austin State University (2) Mustang High School, Mustang, Oklahoma

2 begins to make mathematical sense as the expression inside the parentheses becomes: y 2 + y 40 =(y + 8)(y 5)=(4x + 8)(4x 5). We need to finish by factoring out 4 and simplifying: f(x)= ((4x) 2 + (4x) 40)= (4(x + 2)(4x 5))=(x + 2)(4x - 5). This substitution approach will work on the second problem as well (but is a bit cumbersome). It appears that the Hilander Hiyah is often known as kick the coefficient. Approach 2: A Way to Organize Coefficients This method was developed (certainly not for the first time) by an undergraduate student at our university for the purpose of helping high school students she was tutoring. Given a factoring problem: f(x)= x 2 + 5x Write the lead and constant coefficients below the problem Write the linear coefficient as a sum so that the first x2 2 + summand is the same multiple of the lead + 2 coefficient as the constant is of the second summand Split the linear term. f(x)= x 2 + 2x + x Use distributivity to remove common factors. = x(x + 2) + (x + 2) 5. Use distributivity again to complete the factorization. = (x + )(x + 2) This method emphasizes distributivity and easily generalizes to nontrivial lead coefficients. In fact, it reinforces to the student that, although no coefficient appears before the x 2, the coefficient is indeed one. When nontrivial lead coefficients are used, we might first consider sums that start with a multiple of the lead coefficient: f(x) = 4x 2 + x Then we have f(x)= 4x 2 + 8x 5x 10= =(x + 2)(4x 5) as before. (Writing a small (x2) on the lines emphasizes the relationship, but is not necessary.) One limitation of this method is that it does not work for f(x)=4x 2 + 4x unless non-integer multiples are used. f(x) = 4x 2 + 4x Then we have f(x)=4x 2 2x + 6x =2x(2x 1) + (2x 1) =(2x )(2x 1). This method generalizes beautifully however: f(x) = x + 5x 2 + 8x x2 2

3 We can then write f(x)=x + 2x 2 + x 2 + 6x + 2x + 4 =x 2 (x + 2) + x(x + 2) + 2(x + 2) =(x 2 + x + 2)(x + 2). We could complete the factorization using the same method: x 2 + x So x 2 + x + 2=x(x + 1) + 2(x + 1)=(x + 2)(x + 1), and f(x)=(x + 1)(x + 2)(x + 2) You may have already noticed that what has been developed is shorthand for synthetic division. In the last problem mentioned, we simply divided f(x) by (x + 2). In standard (American form) synthetic division, we would write: An interesting aspect of this modification is that it also finds the factor by which one should divide. Notice that the discovery of all chains through the factors amounts to finding all integer factors. Observe: f(x) = x 2 5x Since the common ratios are 2 and -7, the factorization is (x + 2)(x 7). Although it might take a little more work, we could use the same approach for cubics. f(x) = x + 5x 2 2x Since three common ratios appear (,4,-2) for a degree monic polynomial, the factorization must be precisely f(x)=(x + )(x + 4)(x 2). This method s weak point might be that there is no exact way to determine when one is finished checking for possible integer factors, but its links to synthetic (and, by extension, polynomial) division have merit. Approach : The AC Method Nearly all methods are based upon the concept of separating the linear term into two terms, then factoring repeatedly. The AC Method makes this explicit with the one twist that the lead and constant coefficients are multiplied to help determine how to decompose the linear term. Given a quadratic polynomial to factor: f(x)= 4x 2 + x Multiply the lead and constant coefficients. 4(-10)=

4 Find a factorization of this number that sums 40(-1)? 40-1=9 No. to the linear coefficient. 20(-2)? 20-2=18 No. 10(-4)? 10-4=6 No. 8(-5)? 8-5= Yes. 2. Decompose the linear term. f(x)=4x 2 + 8x 5x 10. Use left distributivity to factor twice. f(x)=4x(x + 2) 5(x + 2) 4. Use right distributivity to factor again. f(x)=(4x 5) (x + 2) Notice that every line in this progression preserves equality and that factoring is used explicitly three times. One drawback of this method, however, is that it is not particularly clear why one multiplies the quadratic and constant coefficients. A geometric depiction of why this works may be found in an article by Hubbard, Beverly, Handrick and Habluetzel (201). Approach 4: Using Manipulatives The manipulative route is less an independent method than a technique for heightening geometric meaning associated to other methods. Here are the general steps: 1. Interpret x 2 terms as x-by-x squares, x terms as x- by-1 bars, and constant term as 1-by-1 squares. (Use a different color for negative areas, but don t start with these.) 2. Group the x-by-x squares into a rectangle in the upper left.. Group the 1-by-1 squares into a rectangle in the lower right. 4. Complete a large rectangle by filling in the upper right and lower left corners with x- by-1 bars. The two pictures included demonstrate the application of the method to the quadratic f(x)=4x 2 + 4x. In their book The X s and Why s of Algebra, Collins and Dacey recommend the use of manipulatives for factoring, writing Research indicates that student learning occurs in three phases, from the concrete to the pictorial to the abstract (2011). This method allows students to start with the concrete and build to the abstract. Use of manipulatives for more complicated problems such as negative areas also pushes students geometric reasoning, abstraction and their connections between geometry and algebra. Approach 5: The Box Method 4

5 The Box Method is a factoring method that involves drawing a diagram like the Punnett square used in biology and builds on the Box Method for multiplying multi-digit numbers or polynomials employing the same box. It is primarily an organizational tool. Here are the steps: f(x)= 8x 2 + 6x Factor out any common factors to get a simplified f(x)= 2(4x 2 + x 10) quadratic: ax 2 + bx + c. (This was optional in the last three approaches.) 2. Draw a two-by-two box. Write ax 2 in the upperleft square and c in the lower-right square. x 2 4x x 40x -2x 20x -4x 10x -5x + 8x = x 4x 2 8x -5x -10. Find the factors of (ax 2 )(c) whose sum is equal to bx, and place the factors into the remaining two squares. (Order is irrelevant.) x 2 4. Find the greatest common factor of each column and row. Place this number above (in front of) the respective column (row). 4x 4x 2 8x -5-5x Write the expressions above and beside the box as linear factors. f(x)= 2(4x 5)(x + 2) Mathematically, this method is no different than the method described in Approach. Perhaps, if taught after the geometric arguments in Approach 4, it could effectively remind students of the underlying geometry without recreating a unique diagram for each problem. This method also has the benefit that it could extend the common Box Method for multiplication. One point of caution seems relevant. If the connection with distributivity or writing equivalent expressions is obscured, students miss out on two bedrock principles of mathematics far more ubiquitous than factoring a quadratic. Approach 6: The Mustang Method The Mustang Method is a quadratic factoring method whose name is derived from the mnemonic: My Father Drives A Red Mustang. These letters stand for: M = Multiply F = Factor D = Divide by A = R = Reduce M = Move and 5

6 This mnemonic assumes that the student begins with a simplified quadratic ax 2 +bx+c (which is potentially problematic if the student depends too heavily on the mnemonic). 4x 2 + x Multiply a by c Factor ac into terms whose sum is b =. Make both factors the constants in linear terms ( x +_ )( x + _ ). (x 5) (x + 8) 4. Divide the constant terms by a. (x ) (x + ) 5. Reduce the fractions. (x ) (x + 2) 6. Move any denominators to the front of that term. (4x 5) (x + 2) The Mustang Method is essentially the same approach as Approach 1. It ignores equality of expressions, distributivity, and makes generalization problematic. Notice that this method also teaches students that fractions never appear in a factorization. Thus, many educators favor the term simplify for fractions rather than reduce, since reduce carries the connotation of getting smaller, not remaining the same size. When everyday usage of a word differs from mathematical use of that same term, students face an added language barrier to comprehending the mathematics (Adams, Thangata, and King 2005). In the case of a simplified quadratic with rational roots, however, this algorithm yields an answer that is correct. Because students often think the goal of factoring is to find an answer in the fastest way possible, this method appears effective and appealing. Teachers must be aware of the bigger goal. Approach 7: Rooftop Method The Rooftop Method is a factoring method most similar to the Mustang Method. This method gets its name from the fact that the first step is illustrated by drawing a roof over the trinomial ax 2 + bx + c connecting a with c Multiply. 4x 2 + x Factor ac into terms whose sum is b =. Make both factors the constants in (4x +_ )( 4x + _ ). (4x 5)(4x + 8) 4. Divide each linear term by its greatest common factor. (4x 5)( x + 2) Although the middle terms differ from the Mustang Method, the Rooftop Method has the exact same flaws. Approach 8: Rescale the Roots The following method bears a strong similarity to the first method we examined. Given f(x)=ax 2 +bx+c f(x)=4x 2 + x Multiply the entire formula by a. 16x x Multiply the quadratic term by a -2, the linear term by x 2 + x 40 a -1, and the constant term by a 0.. Factor this problem as you normally would. (x + 8)(x 5) 4. Take the zeros of this polynomial and divide by a. x=-8/4, x=5/4 5. Place the lead coefficient in front of linear terms 4(x-2)(x-5/4) corresponding to the zeros in Step 4. 6

7 From an algebraic perspective Rescale the Roots has the same shortcomings as Approach 1. Steps 1 and 2 are simply combined in Approach 1: failure to preserve equality, obscuring distributivity, etc. However, rescaling the roots has a link to the graph of the equation which is worth examining. Notice graphically Step 1 is a vertical rescaling by 4, and Step 2 is a horizontal rescaling by 4. Since the vertical rescaling does not move roots, and horizontal rescaling scales the roots by 4, one need only divide the roots by 4 as in Step 4 to recover the original roots. Step 5 is geometrically a reconstruction of a polynomial from its roots and lead coefficient. If time is taken to link each step to a non-equality preserving manipulation of the graph of the equation, this method does indeed carry mathematics which can be used in other settings. However, teaching the algebraic steps alone would not comport with the rest of mathematics. If you are teaching college bound students methods such as Rescale the Roots, a mathematically sound yet non-standard method, please consider this. If your students forget even part of one step when in a university course, no one will be able to help them correct their mistake. They re previous learning will be perceived as rubbish and they will have to relearn a boring standard technique like using the distributivity property to factor. Approach 9: The Tic-Tac-Toe Method The Tic-Tac-Toe Method uses a tic-tac-toe grid as an organizational tool to factor simplified quadratics. For f(x)=ax 2 + bx + c: f(x)= 4x 2 + x Draw a tic-tac-toe grid with ax 2, c, and (ax 2 )(c) in the top row. 2. Find the factors of (ax 2 )(c) that sum to bx. Place these in the remaining two boxes of the last column. 4x 2 10 x 2 8x 5x 8x 5x = x. Find the greatest common factor between the last entry of each row and the first entry in each column. 4. Circle the diagonals of the bottom-left 4 squares. 5. Write the circled terms as factors of the polynomial. f(x)= (4x 5)(x + 2) The studious reader likely recognizes this organizational structure as a convolution of the Box Method, working from the outside of the box in, instead of vice versa. 7

8 x 2 4x 4x 2 8x 5 5x 10 4x 2 10 x 2 4x 2 8x x 5 5x While this method is an organizational tool, the steps can be confusing. Also the connection to the geometric structure is lost. The Tic-Tac-Toe Method also appears to obscure the use of the distributive property in finding the factors. Approach 9: The Asterisk Method The Asterisk Method uses a large asterisk as an organizational tool to factor simplied quadratics. Consider a factoring problem: f(x)= 4x 2 + x Draw an asterisk 2. Place the product of the lead and constant coefficients 4 4 above the asterisk, the linear coefficient below, and the lead coefficient in the upper left and upper right In the lower left and lower right, place two numbers that -5 8 multiply to the value at the top and add to the value at the bottom Treating the right and left pairs of numbers as fractions, simplify them if possible. 5. Each fraction represents one linear factor in the sense that the numerator is the linear coefficient and the denominator is the constant. f(x)= (4x 5)(x + 2) Approach 11: The Diamond Method The diamond method provides another framework for to factoring quadratics that may have a leading coefficient other than 1. Note, however, that finding the factors listed in step 2 essentially depends on a technique such as that in Approaches 2 or. Consider a factoring problem: f(x)= 4x 2 + x Draw a diamond 2. Place the product of the lead and constant coefficients above the diamond and the linear coefficient below.. At the left and right corners of the diamond, place two numbers that multiply to the value at the top and add to the value at the bottom. 4. Make the left and right values the denominators of fractions with the linear term as the numerator. 5. If possible, simplify these fractions. 6. Each fraction represents one factor; x 4x x 2x -5 2

9 i.e. numerator value + denominator value. f(x)= (4x 5)(x + 2) Although this is just a framework for holding numbers, the Diamond Method has a number of troubling properties. It seems to associate fractions with linear terms, which is a problematic habit at best. Second, the approach fails for quadratic functions that are not simplified. Finally, as with so many methods, it obscures the underlying factoring and distributivity at work. Conclusion Different factoring methods are often intriguing. It is a non-trivial task at times to establish whether a new approach is true in all cases, has additional benefits, or is really just a remixing of a more familiar method. If you know of other factoring methods in common use, be they good, bad or exotic, we would love to hear about them. We would encourage all our readers, however, to forgo the ah-ha that comes from producing a magical method in favor of methods the implement sound mathematical practice. Remember, the ultimate goal is not Can you solve this problem? but Can you apply the mathematics you ve learned to whatever problem comes your way? References Collins, A. & Dacey, L. (2011). The x s and why s of algebra: Key ideas and common misconceptions. Portland, ME: Stenhouse Publishers. Hubbard, Keith, Lesa Beverly, Leah Handrick and Megan Habluetzel Visual Factoring: Beyond Symbolic Success. To appear. King, Cindy, Thomasenia L. Adams, & Fiona Thangata " Weigh to Go!" Mathematics Teaching in the Middle School 10, no. 9: Li, Xiaobao, Meixia Ding, Mary Margaret Capraro & Robert M. Capraro Sources of differences in children s understandings of mathematical equality: Comparative analysis of teacher guides and student texts in China and the United States. Cognition and Instruction, 26: