About the Mathematics: Multiplying Two Binomials – Elementary Strategies’ Vertical Connections to Algebra

James Brickwedde, Project for Elementary Mathematics

Multiplication strategies in third through fifth grades

have a direct vertical connection to what is necessary in 

Middle school grade algebra.

Overview

The strategies and mathematical number sense that students need to develop fluency with multiplication and division in grades three through five have a direct impact on the algebraic standards they encounter in middle school. Any focus on shortcuts and surface patterns tend to inhibit students’ progress in advancing in later grades. The language used around the mathematics, and the explicit  elevation of the properties of operations underlying multiplication and division strategies, is crucial in building a solid number sense to advance their algebraic understanding and to engage in upper level mathematics.

 

Mathematical notation For elementary students (and teachers) to engage in the mathematical discussion below, a flexible use of mathematical notation is helpful. The symbol ‘’ is most common for students to use when multiplying numbers. When moving into algebraic expressions, using letters other than ‘x’ is useful in the beginning as, for example, when discussing the commutative property of multiplication [a b = b a]. However, students can, if exposed, come to know that 3(2) or 3x also denote multiplication. The use of this notation makes following the links between elementary and middle school algebraic concepts more visible and more readily understood.

 

How is 54 38 related to multiplying two binomials?

The Standard Algorithm of Multiplication would direct students to operate in a vertical format and, using single-digitized language, start in the one’s place and multiply the sequence of single-digit combinations and if the digits are all placed in the proper location, the answer is determined. It saves space. Only knowing one’s single-digit combinations and procedural placement of those digits is required. The algebraic properties of operations are submerged and often, if ever, explored. (See Figure 1.) Partial products, on the other hand, while it initially takes up more space, allows students to use the commutative property of multiplication, work in values, and have a deeper number sense of the resulting product. Partial product strategy uses the distributive property of multiplication over addition and the explicit use of the place value factors of ten to determine the decomposed sub-elements of the solution. It is a direct link to the area model articulated in the 9th Century by the Islamic mathematician al-Khwarizmi.  al-Khwarizmi is the mathematician credited as the father of algebra. The Standard Algorithm for Multiplication and its ancient cousin Lattice Multiplication are simplified versions of the mathematically richer algebraic originals. While the digitized versions may demand a lower cognitive level of thinking, they obfuscate the underlying algebra.  

Building capacity around the number sense

Building the number sense and properties of operations for either the area model or partial products to determine the first, larger partial product itself involves developing another algebraic property of multiplication. To use the distributive property of multiplication over addition requires the student to comprehend decomposing numbers into their addends, as in 54 = 50 + 4 and 38 = 30 + 8. To multiply with understanding 50 x 30 requires comprehending that numbers can also be decomposed into factors. 50 = 5 x 10 and 30 = 3 x 10. So, when multiplying 5 x 10 x 3 x 10 the commutative property of multiplication allows one to reorder the terms to be 5 x 3 x 10 x 10 to make the calculations easier to work with. The associative property of multiplication is then used to recombine the factors to being 15 x 100, and finally, 1500 for the total partial product. The use of these properties are often intuitive as the actions occur rapidly in one’s head. Making the properties explicit allows a learner to be intentional in one’s actions. 

Taking the time, starting in third grade with combinations such as 5 x 30, and consistently focusing on explicitly using the factors of ten, builds the number sense and fluency behind quickly determining that the solution to 54 x 38 is at least above 1500. The insistence of starting with the one’s place in the standard algorithm limits students’ ability to estimate a reasonable answer as well as limit the use of a strategic algebraic property; the commutative property. And digitizing the language as one executes the standard algorithm further limits that ability to estimate a reasonable answer. 

Linkage of middle school algebra

Consider the following equation:

5a(3a) = 15a2

Now let a = 10

(5 x 10)(3 x 10) = 15(100) = 1500

There is a direct linkage between these mathematical equations. The former allows ‘a’ to be any number while the later is a specific case. If elementary students in grades three through five are well grounded in working explicitly with powers of ten, then the linkage to square numbers (a number times itself), understanding exponent notations, and generalizing a pattern to the nth case, the vertical flow of understanding between the grades is smoother and the mathematical ideas connected.

Now consider this.

(5x + a) (3x + b)

This is an example of what eighth graders need to comprehend when learning to multiply two binomials. Distributive property of multiplication over addition is used to determine the partial products. The classic ordered approach to multiplying the partial products would result in…

15x2 + 5xb + 3xa + ab

For a specific case, again let x = 10, but also let a = 4 and b = 8

(5 x 10 + 4) ( 3 x 10 + 8)

Use the order of operations to multiply first terms within the parentheses…

(50 + 4) (30 + 8)

Following the same classic order as with the above two binomials, that resolves to the partial products of…

1500 + 400 + 120 + 32

In vertical format, and represented in an expanded form, the same mathematical steps looks as follows:

Learning to multiply 54 x 38 should ground students in the core algebraic properties of operations that flow smoothly and vertically to the mathematical ideas of middle school standards. This is the core message for intermediate grade educators to grapple with. The shortcuts, surface patterns, and mnemonic devices used to “help” students more often than not freeze students in low levels of understanding that makes it difficult for them later on to rewire their brains in middle school. 

Does working explicitly with factors of ten seem slower than just ‘counting zeros’ or ignoring them altogether through the use of digitized language and a focus on procedural placement? On the surface, yes. But taking the time at the front end pays massive dividends in the long run. Building a house of bricks rather than houses or either straw or sticks sustains students in connecting mathematical ideas as they vertically progress through the grades. There should be clear linkage in how algorithms and strategies are taught in elementary school with the mathematical concepts at middle school. Viewing elementary arithmetic through the lens of algebra allows that linkage to develop.