By Ryan Lyson, SEEC Math Specialist
Field properties are mathematical rules for addition and multiplication and should be explicitly taught using math vocabulary. For instance, many teachers teach x + 4x using the idea of like terms. While this does have value in the acquisition phase of learning, relying on a “trick” can have detrimental effects in the future. When I hear teachers say, “you can put a one in front of any variable and then just add the numbers,” my immediate reaction is to wonder why that is justified. Why does it work? What kind of trick am I learning? How many of these tricks do I need to memorize? Instead of teaching math tricks, we need to be teaching common language through math vocabulary to communicate the why. We know that one is the coefficient in front of the variable because of the multiplicative identity property which states that any number multiplied by one does not change its value. Using the term ‘multiplicative identity’ provides context for the why of this problem.
There is a reason for everything in mathematics, and most students and teachers at times use tricks as reminders of the procedures to utilize. However, it is important to remember that students in the acquisition phase of learning do not have the fluency and/or flexibility to make certain jumps that we may assume they should have. For example, when combining like terms students struggle with adding x + 4x and will often write 4x2. This is a clear misunderstanding of the multiplicative identity and closure of addition. It’s also a misunderstanding of distributive property factoring. See below for the two-column proof of x + 4x = 5x.
x + 4x | Given |
|---|---|
1x + 4x | Multiplicative Identity |
(1 + 4)x | Distributive Property Factoring |
5x | Closure of Addition |
Another major trick used to teach the multiplication of two binomials is called “The FOIL” Method (first, outer, inner, last). Instead of this trick, we should explicitly state that multiplying two binomials is the double distribution of multiplication over addition. Here is an example two-column proof demonstrating the properties used when solving (3x + 4)(2x – 5).
(3x + 4)(2x – 5) | Given |
(3x + 4)[2x + (-5)] | Definition of subtraction |
3x∙2x+3x∙(-5) + 4∙2x + 4∙(-5) | Distributive Property Expansion |
3∙(x∙2)∙x + 3∙[x∙(-5)] + (4∙2)x + 4∙(-5) | Associative Property of Multiplication |
3∙(2∙x)∙x + 3∙[(-5)∙x] + (4∙2)x + 4∙(-5) | Commutative Property of Multiplication |
(3∙2)∙(x∙x) + [3∙(-5)]∙x+ (4∙2)x + 4∙(-5) | Associative Property of Multiplication |
6x2+(-15x) + 8x + (-20) | Closure of Multiplication |
6x2+[(-15x) + 8x] + (-20) | Associative Property of Multiplication |
6x2+[(-15) + 8]x + (-20) | Distributive Property Factoring |
6x2+(-7x) + (-20) | Closure of Addition |
While the above proof seems a bit excessive, it’s a clear demonstration of each property used when multiplying two binomials. It’s also a good visual showing that these properties always occur in the same order when multiplying two binomials. Many curricular resources skip steps and create gaps of understanding with the problems. Associative and commutative properties are taught in early elementary school and this early learning should be used to help leverage understanding of the why behind the math. Ultimately, using math vocabulary to teach math concepts helps students build a deeper understanding of the subject while building a connection to the English language.