By Barbara A. Swartz, posted February 1, 2016 —

Learning algebra poses unique challenges to students: It requires them to reason abstractly, learn a “new” language of mathematical symbols and vocabulary, and understand mathematical structures such as equations, functions, and equality (Rakes et al. 2010). In my experience, factoring has been one of these particularly difficult topics. When I was a beginning teacher, following the textbook’s lead only seemed to further confuse and even frustrate my students; to their credit, they weren’t satisfied with simply memorizing factoring procedures without developing any understanding of what they were doing and why.

Here and in the next two MT blogs, I outline a few activities to help algebra learners:

Part 1: Understanding how linear factors make up the quadratic function

Part 2: Using a context to provide the rationale for why we want to solve for the zeros of the quadratic function

Part 3: Building on students’ prior knowledge about multiplication and making connections between representations for multiplying numbers and multiplying binomials

In all these posts, I want to encourage teachers to help students look for and make connections across the algebraic, tabular, and graphical representations of the linear factors and resulting quadratic equations.

The first step is to set the stage for students long before we expect them to perform factoring procedures. Before presenting factoring quadratic functions or other higher-order polynomials, many algebra materials and current curricular standards start with creating such functions by multiplying monomials, binomials, and other polynomials (CCSS.MATH.CONTENT.HSA.APR.A.1, CCSSI 2010).

It is all too easy when teaching this topic to jump right into the symbolic manipulation using only the algebraic representations of these functions. Far too many students are simply told to “FOIL” two binomials without developing an understanding of why one linear function multiplied by another yields a quadratic function—for example, (x + 1)(x – 2) = x² – x – 2—or how two lines can be combined to create a parabola.

Let’s look at the graphical representation of “multiplying” these lines (meaning multiplying the y-values of points on the line) without the algebraic or symbolic representations (adapted from Garofalo and Trinter 2012; for more, see Ceyanes, Lockwood, and Gill 2014).

First, let’s graph both functions on the same plane and see what “important points” we can begin to make sense of. Note that on the product function, there will be two places (the x-intercepts of each line) where the y-value will be zero because zero times anything is zero.

Now, take a closer look. Multiplying the lines in the region x < –1 will give positive y-values for the new function (the negative y-values of the red line times the negative y-values of the blue line). Between x = –1 and x = 2, the y-values for the new function will be negative (resulting from the positive y-values from the red line and the negative y-values from the blue). Last, the positive y-values of the red line times the positive y-values of the blue line will yield positive y-values for the new function. (See the figure below; the graphs for the figures were created using Desmos.com.)

We can now start to see the shape of this “new” function and can even make a table of values.

First, calculate and plot the points of f₃(x) = f₁(x)f₂(x) and then connect the dots to reveal the graph of the new function. Similarly, we can graph the product using Desmos® or other software.

By first graphing the two lines and then multiplying values, instead of multiplying the two linear expressions and graphing the resulting quadratic function, teachers can help students connect how the linear factors make the new quadratic function and parabolic shape. This activity lays the groundwork for solving quadratic equations and helps students see where the linear factors come. For an extension activity or an introduction into the zeros of higher-degree polynomials, try using three or more linear factors!

REFERENCES

Ceyanes, Ben, Pamela Lockwood, and Kristina Gill. 2014. “Three Lessons on Parabolas—What, Where, Why.” Mathematics Teacher 108 (5): 368–75.

Garofalo, Joe, and Christine Trinter. 2012. “Tasks That Make Connections through Representations.” Mathematics Teacher 106 (4): 302–7.

Common Core State Standards Initiative (CCSSI). 2010. Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

Rakes, Christopher R., J. C. Valentine, Maggie B. McGatha, and Robert N. Ronau. 2010. “Methods of Instructional Improvement in Algebra: A Systematic Review and Meta-Analysis.” Review of Educational Research 80 (3): 372–400.

BARBARA A. SWARTZ, bswartz@mcdaniel.edu, is an assistant professor of mathematics education at McDaniel College in Westminster, Maryland. She is interested in mathematics teacher education and teaches mathematics courses for prospective elementary and secondary school mathematics methods.