This three-part series started with using the graphical representation of “multiplying” two lines to create a new function and parabolic graph as a way to lay the foundation for factoring quadratic equations. Now let’s look at using algebra tiles for helping students “see” how multiplying linear factors creates a quadratic function and how we can use this as another representation to build on for getting students ready for understanding factoring.

Let’s return to the two lines that we were multiplying in my first post in this series: f₁(x) = x + 1 and f₂(x) = x – 2. Before we multiply them, however, I suggest a trip back to elementary school, where multiplication is introduced. Kieran (1992) has suggested that teachers spend time connecting algebra to arithmetic before moving on to the structural ideas of algebra, and the Common Core State Standards for Mathematics suggest making explicit connections between multiplication and finding areas (e.g., CCSS.MATH.CONTENT.4.NBT.B.5; CCSSI 2010). Multiplication can be represented by an array, and we can connect multiplying the linear factors of quadratic function to multiplying two-digit numbers, such as 29 × 14 = (20 + 9)(10 + 4), using partial products in the area model and the “box method” (a method for organizing the numbers in boxes to aid in multiplication).

The figure below shows a representation for multiplying whole numbers: 29 × 14 = 406:

Note the shape that is created when two linear measurements (lengths) are multiplied: a rectangle.

Students can set up the multiplication of linear factors exactly as we did with whole numbers. Algebra tiles can show how x times x produces x², using x as the length to create a square with an area of x² (check out Advocating for Algebra Tiles blog post for more introduction to using algebra tiles). Students can work backward from the tiles to figure out the factors by representing the terms of the product with the tiles, first making sure to arrange them in the proper shape (the rectangle). They can look for and make use of the structure of multiplication (Standards for Mathematical Practice 7, CCSSI 2010) to help them factor quadratic equations.

Physical or virtual algebra tiles, such as those from Glencoe/McGraw-Hill or Michigan Virtual University , can be used:

Having students see the connections between the different representations is key: When they are able to move back and forth between the different representations fluently, they have developed a deeper understanding of the mathematics. What connections can we make between the graphical, algebraic, and physical representations of quadratic functions and their linear factors? Ask your students and find out what they have to say!

References

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. http://www.corestandards.org/wp-content/uploads/Math_Standards.pdf

Kieran, Carolyn. 1992. “The Learning and Teaching of School Algebra.” In Handbook of Research on Mathematics Teaching and Learning, edited by D. A. Grouws, pp. 390–419. Reston, VA: National Council of Teachers of Mathematics.

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.