This is Common Core State Standards support video in mathematics. The standard is 6.EE.C.9. This standard states: Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.
Let’s first look at what standards that 6.EE.C.9 might be connected to. There is a standard in the same grade level, 6.EE.B.7 that talks about real-world and mathematical problems and writing and solving equations. Now, notice that these are in very simplistic form. It would only take one operation, one step, for the solution. This standard, 6.EE.C.9 sets the stage, sets the foundation, for a seventh-grade standard, in this same area. Standard 7.EE.B.4a deals with solving word problems leading to equations of the form px + q = r and p(x + q) = r. Notice that this is a little bit of a step up in terms of the complexity of the equation because these, it appears, would be those that it would take two or more steps to solve.
Let’s focus on this idea of two quantities in a real-world problem that change in relationship to one another. This standard in sixth grade sets the foundation for this eighth-grade standard, 8.EE.B.5, that talks about proportional relationships and interpreting the unit rate as the slope of the graph. Now, notice that our sixth-grade standard deals with two quantities that change in relationship to one another and that is slope. Now, whether or not we call that slope is pretty much up to the district or the school. But, still, it’s the same idea. It’s just a matter of whether or not we apply the mathematical terminology to it.
This standard talks about the dependent and independent variables. Now, at this point students would have little, if any, experience with those terms. So, it’s important that students get the practice and experience on how to differentiate between those two types of variables. Let’s look at a couple of examples. Let’s say the two quantities involved are total cost and quantity. Upon examination, it seems that the total cost would depend on the quantity of items that are bought. So, the total cost would be your dependent variable because again, the cost depends on the quantity, and the quantity would be the independent variable.
Now, remember with your ordered pairs that your independent variable, typically your x, goes first, and the y value is your dependent variable; and that’s the second number in your ordered pair. Be very careful to make sure that you place the right one in the right location. So, in this case, the independent variable will be our quantity, and the dependent variable is our total cost. So, when we establish our ordered pair for our situation here, we would put the quantity first and the total cost second.
Let’s look at a second example. Lets say we’re comparing time and distance traveled. In this situation, the distance that you travel depends on the amount of time that you are traveling. So, your distance traveled would be your dependent variable, and the time involved would be your independent variable. When we put this as ordered pairs, your time would go first because that’s your independent variable. That’s your x value, and the distance traveled is your dependent variable. That would be your y value. Notice that another part of the standard deals with analyzing the relationship between the dependent and independent variables using graphs and tables, and students also have an expectation of writing an equation to express the relationship between the two variables. So, one of the best ways to do this is to just look at some contexts and give students some practice and the experience in doing it.
Let’s look at this one. Your class is raising funds by selling candy bars at $2 each. Well, we need to analyze the relationship. First of all though, you have to figure out, well, which is your independent and which is your dependent variable? Well, in this case, how much money you raise is going to depend on how many candy bars that you sell. So, the independent variable versus the dependent variable, in this case, the money raised is going to depend on how many candy bars we sell. So, here’s the setup. Your candy bars is your independent variable, and money raised is your dependent variable.
Now, let’s use a table for this. Money raised, dependent variable, that’s your y. Quantity sold, that’s our independent variable, which would be our x value. Now, let’s just try some numbers. If we didn’t sell anything, well, we didn’t raise any money. If we sold one bar, that would raise $2. Two bars would raise $4. Three would result with $6, and so forth. Now we need to work on writing an equation. Well, let’s see. If our unknown x is how many candy bars that we sell, this pattern seems to be that it’s doubling, and of course, that’s from the $2 each. So, it stands to reason that to get the y value, the amount raised, we simply take the number of candy bars and multiply by two. So, if we let x be the candy bars and y is the amount raised, then our equation should be y = 2x.
Let’s try a second context. A truck driver averages 60 miles per hour on a trip. Develop a data table and equation that compare time and distance traveled. The first decision—which is the dependent and which is the independent variable? Well, here, the distance is going to depend on the amount of hours driven. So, the independent variable versus the dependent variable: our dependent is the distance traveled, and our independent is the number of hours traveled. Let’s set up a table and let’s figure out some numbers. Again, obviously you always have to consider your x value of 0. So, if we travel 0 hours, we travel 0 distance. One hour at 60 miles an hour will be 60 miles, 2 hours—120, 3 hours—180 miles traveled.
Now let’s see if we can figure out an equation. If we had so many hours, x number of hours, well, we simply multiply by 60 to get the miles driven. So, that would be 60 times x, with x being the number of hours driven. So, in looking at this, if we let x be the number of hours driven, and y is the distance traveled, then we simply take our hours driven and multiply by 60. So, our equation should be y = 60x. Notice that it’s always a good habit to always have students write down exactly what each of the variables represents. Again, in this case, x is the hours driven; y is the distance traveled. Now, there’s nothing that says that you have to use x and y, so it might be a good idea to maybe for the hours driven, use h for hours, and for distance, used d for distance. Just be sure that students understand which one is the independent variable and which one is the dependent variable, in those cases where it might be better to use something other than x and y as your variables.
The standard also talks about using graphs, so let’s focus on that. Let’s look at this context. Job A pays $9 an hour and Job B pays $15 per hour. Develop an equation for each, and compare them graphically. Okay, first of all, the decision—dependent versus independent variable—in this case, the total pay would depend on the number of hours that you work. So, our independent variable and our dependent variable, total pay is dependent on the hours worked. Now let’s develop tables for each job. So, let’s see, 0 hours, 0 pay for each. For Job A, if we work 1 hour, that would be $9. For Job B, 1 hour would result in $15 for our pay. Two hours, $18 for Job A and $30 for Job B, and then our values for 3 hours would be $27 for Job A and $45 for Job B.
What if our number of hours was unknown, let’s say x? Well, let’s see. For Job A, we’re simply multiplying by $9 an hour, and for Job B we’d have to multiply the number of hours by 15 because we’re earning $15 an hour. Now it’s just a matter of putting this into equation form. Again, always be sure to write down what your variables represent—x is the number of hours worked, y is the total pay. That would make the equation for Job A y = 9x, and for Job B, our equation would be y = 15x.
Now, let’s address the idea of using graphs. Lets go ahead and focus also on the tables that we developed. That would really help us. We’ve already done some work to get some values. One of the big decisions when you’re doing your labeling is to decide, well, what numbers should I use for my x-axis and for my y-axis? Let’s give ourselves some room and label the x-axis this way using whole numbers. We’re going to have larger numbers for our monetary values, for our y. So, we need to maybe go by twos, so let’s do that. Now, let’s graph Job A. So, we have the point (0, 0), 0 hours worked, 0 money earned. Then for 1 hour, that would be $9; 2 hours, $18 for our y-value; 3 hours would be a $27 y-value. It looks like we have a linear pattern, so we connect those; and we have our ray this way.
Now let’s graph Job B. We also have (0,0) as our starting point; 1 hour, $15 for the y value, 2 hours would give us $30; working 3 hours would give us $45. Again, we see the linear pattern, and we complete that. Notice that the relationship would be still the same. It might look different. Like here, we went ahead and changed the spacing, how we numbered the x values on our x-axis. So, it changed the look of the graph, but it’s still the same relationship. An important thing to notice here, and it’s important that students recognize this. Notice that the greater the rate of change, your slope, the steeper the incline of the linear graph is going to be. Like in this case, Job B, our slope was 15/1. Again, for every 1 hour we earn $15, and for Job A, our ratio, our slope (whatever you want to call it at this stage, in sixth grade). For every 1 hour that we work, there’s a change of $9 in the amount of money.
In looking at our standards for mathematical practice, by doing the activities involved with this standard, students would make sense of problems and persevere in solving them. We saw that. Students would reason abstractly and quantitatively. If you have them work in teams or groups, it would make it very easy for them to construct viable arguments and critique the reasoning of others, and they would be modeling the mathematics. Then when we look at the other four standards for mathematical practice, by doing the activities, we’d be doing all four of these. Students would use appropriate tools strategically. They will attend to precision and look for and make use of structure. Students will also look for and express regularity in repeated reasoning.