Two Step Equations
Lesson 8
The warm up for lesson 8 is one step equations. We are going to review one step equations because they will be helpful in learning two step equations today. We are going to begin our lesson with a SOLVE problem. We are gong to “S” the problem. “One year of cable cost $685, including the $85 installation fee. What is the monthly cost?” In “S” we study the problem. Our first step is to underline the question. What is the monthly cost? Our second step is to answer the question, what is this problem asking me to find? This problem is asking me to find the cost of cable per month.
We are going to SOLVE our equation 2C plus 6 equals 8. WE will represent our 2C with 2 cups, we will represent our plus 6 with 6 yellows, the left hand side of the equation is plus 8 and we will represent our 8 with 8 yellows. In order to isolate our variables or isolate our cups, we must remove the 6 yellow’s from the left hand side. We can remove them by taking them away, to keep our equation balanced if we remove 6 yellows from the left hand side, we have to remove 6 yellow’s from the right hand side. This leaves us with 2 cups equal to 2 yellows. From here we want to know what the value is for one cup. We must split our integer chips; divide them evenly into each cup, seeing that the value of just one cup would be just one yellow chip. The value of C in the equation, 2C plus 6 equals 8 is 1. C equals 1. Now let’s check out answer, we are going to put the original equations of 2C or 2 cups plus 6, which is represented with 6 yellows, equal to 8 yellows. The value of our cup is one yellow, and when we replace out cups we see that the scale is balanced, 8 equals 8 so our answer is correct.
In order to model 2C plus 6 equals 8 pictorially, we are going to represent our 2C with 2 C’s. We will represent our plus 6 with 6 yellow and we will represent our 8 with 8 Y’s. Because we want to isolate our C we want to remove, or take away the 6 yellow’s on the left hand side, but if you take away 6 yellow’s on the left you must also take away 6 yellows on the right. This leaves us with 2 C equal to 2 Yellows. So when we split them up so that we see the value of one C, one C is equal to one yellow, or C equals 1.
If we look at the problem abstractly, we see that we want to isolate our 2C first. In order to isolate the 2C we have a plus 6 we are going to subtract 6 from both sides, leaving us with 2C equal to 2. To isolate the C we will divide both sides by 2, because the opposite of multiply by 2 is divide by 2, C equals 2 divided by 2 which is 1. And we can check our answer by rewriting our original equations 2 times 1, which is the value of C, equals 2 plus 6and 2 plus 6 equals 8, so our answer checks.
We are going to model the problem 4C minus 4 equals negative 12. We cannot model a subtraction problem…so we must change the 4C minus 4 to 4C plus negative 4. we can represent our 4C with 4 cups, we can represent our plus negative 4 with 4 reds, and we can represent our negative 12 with 12 reds. Our goal is to isolate the cups, in order to isolate the cups we must remove the 4 reds from the left hand side of the equation. If we remove 4 reds from the left hand side of the equation we must also remove 4 reds from the right hand side of the equation. This leaves us with 4C equal to 8 reds, or negative 8. We will then determine what the value is for each cup, by dividing or splitting the chips evenly, the value of one cup is 2 reds, or negative 2. So C equals negative 2. If we want to check our answer we will setup our original problem, of 4C plus negative 4 equals negative 12. C equals negative 2, so we will replace each cup with 2 reds, which has a value of negative 2. And we can see that our answer is correct because negative 12 equals negative 12.
If we want to model the problem pictorially, we will represent our 4C with 4 C’s. We will represent our negative 4 with 4 reds and that is equal to 12 reds. When we take away the 4 reds on the left hand side, we also have to take away 4 reds on the right hand side, leaving us with one cup equal to 2 reds. So our answer is C equal to negative 2.
If we look at the problem abstractly, we see that we have 4C plus negative 4 equals negative 12. We took away or subtracted negative 4 from both sides, leaving us with 4C equal to negative 8. We divide both sides by 4 in order to isolate the C, and C equals negative 2. To check our answer we rewrite our original equation 4C plus negative 4 equals negative 12. We plug in negative 2 for C. 4 times negative 2 is negative 8, and negative 8 plus negative 4 is negative 12, so negative 12 equals negative 12 our answer is correct.
To model the problem negative 7 plus 2C equals 5, we will first represent our negative 7 with 7 reds, plus 2 C equals 5. Your students may seem confused by having negative 7 plus 2C instead of 2C plus negative 7 but there is no different. We still want to isolate the 2 cups. And if we have negative 7 or 7 reds, we want to take away or add a positive 7 to both sides. We have been subtracting the negative 7 but we could also add 7 yellows to each side because that will create zero pairs on the left hand side leaving us with 2C. If I add 7 yellows on the left hand side, I must also add 7 yellows on the right hand side. We will remove our zero pairs one at the time in order to cancel the negative 7. We now have 2C equal to 8, 9, 10, 11, 12. If we want to separate and fine out the value, of one cup we will move the chips one at a time, so that we can see that the value of one cup is equal to positive 6. So C equals 6. To check our answer we will go back to the original equation of negative 7 plus 2C equals 5. We must replace each cup with 6 yellows, because the value of C is 6. Now we must remove all zero pairs from the left hand side. Leaving us with the correct answer of 5 equal to 5. To represent this pictorially, we will represent our negative 7 plus 2C equals to 5. We added 7 yellows in order to create zero pairs and isolate our 2 cups. If we add 7 yellows to the left hand side, we also have to add 7 yellows to the right hand side. From here we see that we have two equal groups, so the value of one C is 1, 2, 3, 4, 5, 6 yellows, or positive 6. To isolate the 2C we have a negative 7. We added 7 to both sides, negative 7 plus 7 cancels leaving us with 2C equal to 12. The 2 is multiplied by the C, so the opposite of multiply by 2 is to divide both sides by 2, and C equals 6. We can check our answer by rewriting the original equation and plugging in a 6 for C, negative 7 plus 12 equals 5 so 5 equals 5 and our answer is correct.
To complete our SOLVE problem from the beginning of the lesson we have already underlined our question, what is the monthly cost. In studying this problem we know that this problem is asking me to find the cost of cable per month. In “O” we are going to organize our facts. We will first identify each fact, One year of a local cable company cost $685, including an $85 instillation fee. Our first fact is the cost of one year of local cable is $685, it is important. Our second fact is that it includes the $85 installation fee, this fact is also necessary. In “L” we are going to line up a plan. We choose an operation or operations. And we write in words what your plan of action will be. Our operations in order to find the monthly cost will be subtraction and division. When we write in words what our plan of action will be…We will setup a two step equation, multiply the variable by the number of months in a year, add the instillation fee to equal the total cost. To solve, subtract the installation fee from both sides of the equation, then divide by the number of months. In “V” verify your plan with action, our first step would be to estimate. If we estimate, we can subtract the$85 from $685 leaving a cost of $600 per month. And we know there are 12 months in a year, and 600 divided by 12 is about $50. Some students may not be able to get that close of and estimate. But they should be able to tell you that the answer is going to be less than a $100. To carry out our plan we will write an equation 12M plus 85 equals 685. We subtract 85 from both sides, leaving us with 12M equal to 600, when we divide both sides by 12, M is equal to 50. In “E” we are going to examine our results. The first step in “E” is to ask does your answer make sense? If we go back to our question of what is the monthly cost, $50 is a normal monthly cost for cable. So our answer makes sense. Is our answer reasonable? Our estimate was about 50 so yes our answer is reasonable. Is your answer accurate? You can have your student check the answer by rewriting the equation, and plugging in 50 for the monthly cost, 12 times 50 is 600, and 600 plus 85 is 685. So yes our answer checks. To write our answer as a complete sentence…the monthly cost for cable is $50.
To close your lesson you will review the essential questions. Number 1, why do we use variable in equations? We use variables to take the place of unknown values. Number 2, which operations do we undo first in two step equations? We always undo addition or subtraction first in two step equations. Number 3, what is the inverse operation for multiplication? Division is the inverse for multiplication. And number 4, what is the goal in solving equations? The goal in solving equations is to isolate the variable.