From Arithmetic to Algebra: Variables, Word Problems, Fractions and the Rules

Rationale

The focus for this unit will be on giving my students a clear understanding of how to manipulate compound expressions and reduce them in order to create an equivalent expression in standard form. My goal is to avoid from the beginning misunderstandings that students may have about the meaning of expressions and their relationships. Teaching expressions to students in the ninth grade has been a challenge because they did not have the foundation they needed to steer clear of misconceptions while simplifying expressions. I have often seen students turn a two-term algebraic expression into either a one term algebraic expression or a one term numeric “answer”. While teaching eighth graders expressions, we faced the same issues. The only way to stop this from continuing in high school and to enable students to begin building on their math skills for college is to catch these mistakes when they are learning pre-algebra.

The main mathematical theme of this unit is that any first order expression in some variable, say x, can be simplified using the rules of arithmetic, to an equivalent expression of the form ax+b, known as standard form, where a and b are constants (meaning, known numbers in any particular problem). But in general, it cannot be made any simpler. Indeed, two expressions ax+b and a’x+b’ are equivalent only when a = a’ and b = b’. The condition for equivalence is: two expressions in the variable x are equivalent if, whenever a particular number is substituted for x, the two expressions evaluate to the same number. So if ax+b and a’x+b’ are equivalent, then setting x = 0 would give the condition b = b’. Then setting x = 1 gives the condition a+b = a’+b’. Since we already know that b = b’, we can subtract these equal quantities from both sides of the equation to conclude that a=a’ also.

One of the main issues that students have while working with expressions is that they try to simplify algebraic expressions down to one term even when it is not possible. In 1998, a study was conducted of four seventh grade math teachers.¹ The strategies the teachers used when teaching how to simplify algebraic expressions, and how aware the teachers were of the common mistakes their students tended to make were observed, recorded, and investigated. In four separate teaching situations, it was observed that there are “different sources to students’ tendency to conjoin open expressions”.² When students conjoin open expressions, an expression that includes combinations of numerical terms, algebraic terms and operations, they are calculating for one numerical or algebraic term. This being said, if they simplify an expression and end up with the open expression 3x + 5, they do not believe this is the answer because there remains an operation and it looks as if it is not finished. They do not understand that this expression does not and cannot have a specific value unless numbers are assigned for x. What they tend to do is add the two terms as if the 5 also has the variable, x, to get 3x + 5x and say it is the same as 8x. Another way students conjoin these open expressions is by removing the variable from the term 3x to get 3 and adding 3 + 5 to get 8. To test if this is correct, students should substitute various values for x in order to see if they get the same answer from the two expressions. It should be made clear that the role of the variable is to represent an unknown number and when in an algebraic expression, it allows us to identify equivalent expressions. For two expressions, if substituting in the same value for the variable in both always results in the same numerical value, then the expressions are equivalent. The topic of the role of variables in expressions is covered more deeply in the unit of Jeffrey Rossiter.

One of the reasons students want to conjoin, or oversimplify, open expressions is because “students face cognitive difficulty in accepting lack of closure”.³ When students start learning math, they are taught that when they see the addition sign between two or more terms, they must add the terms together in order to get an answer. This might not always be the case. As they get to higher levels of math, it is crucial for them to be able to identify not only what the addition symbol means, but all the symbols of the operations, and especially the equal sign, which has been shown to be often misinterpreted.⁴ When they see an equal sign, they are accustomed to computing for one specific value, not finding equivalence through other forms of expressions. This could probably be avoided, by using the equal sign more flexibly in elementary school. Students should be able to notice the difference between the case of 5 + 2 and 5x + 2. Being that in elementary school when they saw 5 + 2 it meant that they had 7, when they see 5x + 2 in middle or high school they feel as though that open expression is incomplete. To discourage this misunderstanding, they must get used to the idea of using substitution to check their work in order to see if they are correct. By substituting values for the variable, they will be able to see if the expressions are equivalent. I have concluded that I must not skip these crucial steps in our teaching and assume that students understand or have been taught the material as clearly as they should have been, especially in math, because it is a new language that can get lost in translation. I will try to listen to them carefully and when they make these mistakes of “finishing expressions,” focus on understanding their thinking in order to fix it. I am afraid that if I do not, they will go back to what is comfortable to them and continue making these errors.

In schools, teachers teach “syntactics (the study of rules governing the behavior of systems, without referring to meaning)” rather than “semantics (the study of meaning)”.⁵ If students do not learn the meaning behind the symbols, they will not be able to understand why and how the operations are able to do what they do, which causes them to not understand why they cannot “finish expressions” the way their minds want them to. I will aim to provide a combination of both syntax and semantics when teaching mathematical concepts to our students, whether they are simple or more complex. If they have the background knowledge they need when solving problems in math, as they reach higher levels, if they have forgotten the “rules” of certain topics, they will still have the knowledge to be able to get back to that “rule” in order to solve the problem. This split focus between semantics and syntax will be most helpful while covering the rules of arithmetic, which students know as the Properties of Operations. These Properties should not simply be memorized by students as a list of rules when first learning them. They should be clear on what each of these rules mean in order to understand how and why they can manipulate numbers.

As students get to more complex expressions, they tend to struggle with the simplification process. In textbooks, students are always advised to combine like terms when they simplify. However, it is never made clear that “combining like terms” is an application of the Distributive Rule. This is where they manipulate an expression, for example 12x+25-3x-3, to have the “like terms,” which are the terms that have the same variable raised to the same power, next to each other, 12x-3x+25-3 to be able to add or subtract them easier and get an expression in standard form, 9x+22. The confusion with a problem like this is usually related to the subtraction symbol. When students manipulate the expression, they do not realize that the subtraction symbol should be interpreted to mean that they are adding the additive inverse of the indicated term. The result of this is that they often write the expression as 12x+3x-25-3, keeping the signs where they were while permuting the symbolic terms, instead of moving the signs as integral parts of the whole symbolic term. Another way that students struggle working with simplification is when an expression involves variables raised to different powers. Although my unit focuses on linear expressions, it is crucial that I touch upon the fact that a variable raised to the power of one, x, is not the same as a variable raised to the power of two, x²,  because when we substitute a number greater than 1 for x, and then substitute the same number in x², we will get different answers, and see that they are not equivalent. For example, if x = 2, then x = 2, but x² = 2² = 2×2= 4.