Two-step equations are in the form ax + b = c. Although they seem straightforward, students will do things like divide the ax term by the coefficient without first moving the b term to the other side. This isn’t to say that one cannot divide first, but that requires dividing all the terms by the coefficient, which students 1) often forget to do or 2) have difficulty dealing with the fractional terms that may result.
Warm-Ups. Problems should include translating English into algebraic expressions, including some that translate into equations.
1. 3m = -13 Answer: m= -13/3
2. (5/6)b= 15 Answer: b = 15 (6/5) = 18
3. Write as an algebraic equation. Three times a number is twenty-four. Answer: 3x=24
4. Solve the equation in problem 3. Answer: x = 8
5. Write as an algebraic expression. Five more than two times a number is fifteen.
Answer: 5 +2x = 15
Students may get stuck on problems 3 and 5, but usually need only be reminded that “is” translates to “equals”. They are now writing full equations, rather than algebraic expressions.
Also, for problems in which students must multiply a fraction, they often forget to do cancellation before multiplying, or simplifying (reducing) afterward. This tends to happen on quizzes and tests as well. In an effort to get them in that habit, I would give them an extra credit point on quizzes and tests for each fraction that was expressed in lowest terms. A disappointing few would respond to this, but I would tell myself that a few is better than none.
Addition and Subtraction Step First. Students by now should be comfortable solving one step equations like y + 5 = 15. They know they need to “undo” the addition on the left hand side by adding -5 to both sides of the equation to get y= 10. For equations like 2y= 10, they know to undo the multiplication step by dividing both sides by 2 (or multiplying by the reciprocal of 2 if we want to be more formal about it).
For two step equations we do the same thing, but one at a time. I use a warm-up problem, in this case, Problem 5: 2x + 5 = 15. I tell students “The name of the game is to get the variable term on one side, and numbers on the other. We do this in two steps; we move the number over first, and then we take care of the variable.”
I write the two steps on the board as I do the problem with the students; as the students write the steps in their notebooks and solve the problem along with me:
Step 1: Undo the Addition/Subtraction step.
Working this step I write:
2x + 5 = 15
-5 -5
2x = 10
I explain that we undo the addition step by adding the opposite of 5 (which is -5) to each side.
Step 2: Undo the Multiplication/Division step. Undo multiplication by dividing each side by 2
I now write:
2x /2= 10/2
x = 5
Examples. While the worked example is still fresh in their heads, and the steps written on the board and in their notebooks, I have them try some more.
3x + 5 = 2 Answer: 3x = -3, x = -1
t/5 +20 = 22 Answer: t/5 = 2, t = 10
-2m – 7 = -27 Answer: -2m = -20; m = 10
5 – 2m = 18 Answer: -2m =13; m = -13/2 or -6 ½
What Goes Wrong. Students do not like to show their work, and when forced to do so, will sometimes do it in messy fashion. One solution to this is to write the steps horizontally, so that the first example would be written as follows:
3x + 5 -5 = 2 -5
3x = -3
3x/3 = -3/3
x = -1
Note that the second and third steps are usually combined; that is, after students write 3x = 3, they will then draw the fraction bar under each of the members of each side and proceed with the division step to solve the equation.
Also, problems in which equations are written in a different order, like 2 = 5 + 3x, might confuse students who are used to doing problems in a particular order. It is worthwhile to use examples that mix things up so they get used to doing the steps regardless of how the equation is written.
I emphasize that two-step equations are really two problems in one, and that doing the addition/subtraction step and the multiplication/division step is what they’ve been doing for one-step equations.
Nevertheless, students will get confused. Sometimes they forget to add or subtract the number on both sides, and only do it on one side. But the far more common mistake is to do the division step first, but without dividing each term by the coefficient.
I explain that we could divide first but it can be more complicated as shown:
With 3x +5 = 2 each term is divided by 3. The result is x + 5/3 = 2/3. Now -5/3 is added to each side: x = 2/3 – 5/3=-3/3 = -1. I tell the class I have no objection to them doing it this way, but if they choose this way, they must do it correctly. It is easy to make a mistake and not divide each term by the coefficient. I then take a poll to see which method students prefer. It’s generally the add/subtract operation first which I recommend they do because it’s easier to do and doesn’t lend itself to errors as might the “divide first” method.
Equations with Decimals. I will put an equation on the board with decimals in it, and ask how I should proceed: 0.5x + 0.15 = 0.30
Since we have had some experience with decimals in the last lesson, there may be some mixed responses. Some may keep the numbers in decimal form with the first step resulting in 0.5x = 0.15. The second step is then x = 0.15/0.5 = 0.3. Following what we did in the previous lesson, I will ask how we can get rid of the decimals so we have whole numbers. Since the greatest number of decimal places is two, all terms would be multiplied by 100, resulting in 50x+15 = 30. Solving, we get x =15/50 which equals 0.3.
The lesson may stop here with an assignment of homework problems that incorporate both one and two step equations. Some of the problems should be presented in different formats, like -20 = 10 + 2x so that students develop flexibility in applying the applicable steps.
It has been my experience that homework assignments sometimes need to be developed to include appropriate problems that are scaffolded and increase in difficulty. This may mean cherry picking problems from the textbook, or creating your own worksheets. The more difficult problems may have answers that are fractions or decimals.
Cognitive Overload Warning. I recommend that problems with fractions, such as (2/3)x +5/6 = 7/8 be addressed in the next lesson so that students do not feel overwhelmed with new information.
I like to tell them they’re undoing pemdas which is why they undo the adding and subtracting first. I also like to tell the that they’re undoing the operations and not the sign. I tend to have kids that want to negate the coefficient when dividing. This is hard since the simplifying part requires integer arithmetic and they have to think of the signs as positive and negative not adding and subtracting. It’s hard for them to compartmentalize as two different concepts in one problem. -Patty H