This is Common Core State Standards support video in mathematics. The standard is 6.EE.B.8. This standard reads: Write an inequality of the form X > C or X < C to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form X > C or X < C have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
Grade 6 is the first grade level where the term inequality appears in the Common Core State Standards in math. Standard 6.NS.C.7a states: Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. There’s a standard in this same cluster, 6.EE.B.5 that talks about understanding and solving an equation or inequality as a process of answering the question, and that question would be well, what values from a specific set, if any, make the equation or inequality true? Looking forward to seventh grade, there’s standard 7.EE.B.4b that deals with solving word problems leading to inequalities of the form PX + Q > R or PX + Q < R, and again, this deals with inequalities.
Since the idea of inequality is fairly new to students, it might be a good idea to make absolutely sure that students understand what an inequality is. The key knowledge for students is that they realize that there are only three possibilities when comparing two numbers. So, these three possibilities would be either the two numbers are equal or they’re not equal to each other, and if they’re not equal to each other, then a given number is either going to be smaller than the other one or it’s going to be bigger than the other one. So there are our three possibilities. If I’m comparing one number to another one, it’s either going to be equal to, smaller, or greater than the other one. Also relatively new to students would be this idea that with an inequality, you can have infinitely many solutions. They’re used to equations where typically you just have one solution. So, let’s say we have C > 5. Well, 6 is greater than 5, so is 7, so is 8, so is 9, so is 10, and so forth. So initially, we might be tempted to do this, that our solution set is 6, 7, 8, 9, 10, and so forth.
But wait, there’s another standard in this same grade level, 6.NS.C.6c, that states: Find and position integers and other rational numbers on a horizontal or vertical number line diagram. So we’re not just dealing with whole numbers here. Well, what about 5 1/2? That’s bigger than 5. What about 7.2? What about 9.671? That’s bigger than 5. So now, realizing that, hey, there’s these other solutions, I pop them into my solution set. But wait, what about 5 1/5, or 5 2/3, or 7.1, or 7.8, or 9.675? At this point, it’s starting to look like, hey, there’s a whole lot of solutions to this, and one thing that should be apparent is that there’s no way that you can state all the solutions in set notation. It would be physically impossible to write down every single possible number that’s bigger than 5.
So here’s where this last part of the standard comes into play, this idea of representing solutions of inequalities on number line diagrams. So stating our solution as a set in set notation just won’t work in these cases, so we can’t do that. We have to go with the number line diagram. So let’s say okay, we have 5, 6, 7, 8, 9, 10, 13, and let’s not worry about the 5 for now. We had all these other numbers like 5 1/2, 7.2, 9.671, but that’s still not enough. We know that all the numbers from 5 to 6 would work, and from 6 to 7, from 7 to 8, and so forth. It includes all of those numbers, so that’s the reason for having the line because that indicates inclusion of all of those possibilities.
Now what we need to do is resolve the issues or the questions at the end of these segments. Well, let’s see, our number needs to be bigger than 5. Now that indicates that the 5 is not included, so how do we represent that on a number line? Well, when we have a circle, a little circle with the interior included, that indicates that it includes that number. If it’s just the circle, that indicates that particular number is excluded, it’s not part of the solution set. So here we have C > 5, so the 5 is not included. So we have to put an open circle to indicate that 5 is not part of the solution set. Now, what about at the other end? What about numbers past 13? Those are bigger than 5. So yes, we have to keep going and the way that we do that is we simply convert this to a ray to indicate that numbers keep going infinitely in that direction. Number lines can take different appearances. Some might decide that well, we don’t need those dots any more, so let’s get rid of them. In fact, typically on a number line, you use small hash marks to indicate the location of various numbers on the number line. So this is one possible format or look to the solution set for this number line. But a lot of times this will suffice, because mathematically, that’s correct. I have all the numbers that are greater than 5 but not including the 5 indicated with this ray.
Let’s take another example. We have some number D is less than 12, and we’re going to do our number line diagram. So we know we have to start at 12. It’s less than, so we know it doesn’t include it, so it’s open circle and less than, so it should be everything to the left. And of course, this is a habit that students need to get into. They need to test the result with these inequalities. Well, let’s see, a 10; that lies in that part of the number line, and that is less than 12. So, it works, so that is the correct solution.
Let’s try another example. Let’s try 9 < C. Well, it doesn’t include the 9, so it’s an open circle, and we have a less than symbol, so I guess it’s everything to the left. Oh, but wait a minute. Test your result! Well, let’s see. Let’s try a 7. Wait a minute, 9 is not less than 7. That doesn’t work. Ah, so that’s not correct. The trick here was that the variable was on the right-hand side of the inequality, which throws things off. So let’s try the other side, a 10 for example. Hey, 10 works. Nine is less than 10, and 11 works. Nine is less than 11. Ah, so it looks like it’s all the numbers this way. So our solution set is all the numbers larger than 9. Again, be very careful, especially in situations like this where the variable in an inequality is to the right of the inequality sign.
What about X is greater than or equal to C or X is less than or equal to C? The symbols greater than or equal to and less than or equal to do not appear anywhere in the standards. But it needs to be addressed because students will see this symbolism, either on a test or textbooks or somewhere. X is greater than or equal to C is actually a double statement. It means that X is either bigger than C or it’s equal to C. If we use some numbers, let’s say we have X is greater than or equal to 4. That means that our unknown is either bigger than 4 or it is exactly 4. If we do this as number line diagrams, one statement says that our number is bigger than 4, and the other statement says or it’s equal to 4. It can be one or the other, so it includes numbers in both statements. So we need to put them together and here’s the result. So it would be everything larger than 4 and it does include the 4 because it can be equal to 4. Okay, we did this example. We have numbers smaller than 5, and we have our number line diagram, and it is an open circle at the 5 because it doesn’t include it. Then D is less than or equal to 5—well, the only difference is that it does include the 5. So the difference between the two number line diagrams is simply whether or not the 5 is shaded because that indicates if it includes it or not.
If we look at the standard just prior to this one, 6.EE.B.7, it deals with equations in the form X + P = Q and PX = Q. So they’re very simplistic equations that just involve one operation. Looking ahead to Grade 7, students will have to deal with inequalities in the form PX + Q > R or PX + Q < R. Notice that the complexity level goes up a notch because these inequalities have more than one operation involved. So we have to address this; we have to get students ready for that complexity in Grade 7 by doing some things here in Grade 6. So according to this standard 6.EE.B.7, we have equations that have strictly one operation involved, for example, just adding or multiplying. But here’s something that’s very important, non-negative rational numbers. It’s not that big of a factor with equations, but it is critical with inequalities, in particular when you’re dealing with multiplication or division. As you know, when you have an inequality, if you multiply or divide by a negative, that involves switching the inequality sign around, and that’s a lot more complexity. But that is not an expectation here at this grade level in Grade 6. So again P, Q, and X are non-negative. They’re positive numbers, which makes a huge difference when dealing with inequalities.
Let’s say we have D + 5 < 12. Now the process here is very similar to what you would do for an equation. We want to isolate the variable, so we need to subtract 5 from both sides, so we would have D + 0 < 7 or just D < 7. Then the number line diagram would look like this. We know that it does not include the 7, and it’s all the numbers to the left of 7 because it’s all the ones that are less than than number 7. Let’s try this example, 3D > 12. We have to isolate the D and get that by itself. Some might look at this as having to multiply each side by 1/3. Others might see it as dividing both sides by 3, but either way we’re going to arrive at the same place. So we would have D > 4. We’ve isolated our variable, and the number line diagram would look like this where it doesn’t include the 4, and it’s everything larger than 4.
Now let’s address this critical idea of representing a constraint or condition in a real-world or mathematical problem. This is something relatively new. Let’s look at this example. Manny has a sales job and gets paid on commission. This week, he wants to earn more than $700, so we know that the number line diagram would look like this. It’s more than $700, so it doesn’t include 700, and it’s everything to the right because that’s everything that’s larger than 700. We have to pay attention to the exact wording on these statements. What if it said that he wants to earn at least $700? Students might look at this and focus on this idea of least and might think that well, is it everything to the left of 700, or is it everything to the right of 700? Well, simple test—try numbers on each side, and see what works. Let’s see, 600. No, that’s not at least 700, so no, that doesn’t work. How about 800? Yes, that’s at least 700. That works, so we know that it’s everything to the right of the 700. But now we have a question of, okay, wait a minute, does it include the 700? Well, at least means that much or more, so it does include the 700. So we need to have a darkened-in endpoint to indicate that 700 is part of the solution.
Let’s look at another example. Safety rules limit the average workweek for truck drivers to 70 hours. So the interpretation here would be that a truck driver can drive no more than 70 hours per week, or maybe a slightly different interpretation would be that a truck driver can drive 70 hours or less per week. So the number line diagram should look like this: 70 hours or less, and since it’s 70 hours or less, it does include the 70. That’s another big decision, so we have to have a filled-in endpoint.
Let’s go back and look at that first example about Manny and the sales job. Here’s our number line diagram, and mathematically, that would be X is greater than or equal to 700. Although mathematically correct, is that really realistic? For example, could he really make like $20,000,000 in that week on commission? So it seems a little bit unrealistic for him to be able to make $20,000,000. Let’s look at the second example with the truck drivers. Here’s our solution, our number line and numerically, mathematically, that would be X is less than or equal to 70. Now, based on our solution, negative 23 hours could be part of that, but you can’t drive negative 23 hours. Realistically, zero is as small as we can get here. A truck driver might take the whole week off, so he drove 0 hours. But that’s it, we can’t have any negative numbers.
So really, for this situation, there are two conditions that have to be met. The number of hours that a truck driver can go has to be 70 or less, so that’s the first condition. But since we can’t drive negative hours, the number of hours has to be 0 or more. So if we look at satisfying both conditions, here’s where we are. It has to be everything from 0 to 70, and so that’s the realistic solution here. Okay, now, these two conditions: the number of hours has to be greater than or equal to 0, but they also have to be less than or equal to 70. So we know that the number of hours is going to fall somewhere in this interval, and in plain English, we know how to state it, that the number of hours, the H, is a number from 0 to 70.
But we have a problem. How do we represent this mathematically? In the standard, they only talk about X > C or X < C. There’s nothing to handle something like this that actually has two different conditions that have to be met. The standards don’t address this. How do you express this? Well, let’s see. The first condition is the number of hours has to be 70 or less. The second condition is that the number of hours has to be zero or more. In order to express this mathematically, here’s a problem. The inequality signs are opposites. On the first one, it’s greater than or equal to, but on the second condition, it’s less than or equal to, and we can’t have that. We’ve got to have them both going the same direction. Typically, we like to use less than, so let’s figure out how we can make both of these into less than statements.
Well, no problem on the right-hand side; we already have H is less than or equal to 70. It’s the left hand where we have to do something. Well, let’s look at this numerically. We can say 15 > 0, but another way to say that same exact thing is that 0 < 15. So we can take that statement, H is greater than or equal to 0 and state it differently. We can say 0 is less than or equal to H, and now, by doing this we have both our inequalities going the same way. They’re both less than or equal to. So now we know where the number of hours has to fall. How do we express this? Well, here’s what we do. We can actually do this visually where what we have to do is combine these two into one statement. Well, let’s do that. We’ll just move this over, and there it is. This is how you express this type of situation: 0 is less than or equal to H is less than or equal to 70.
Now that doesn’t sound too good mathematically. It kind of gets jumbled up. In plain English, the way to interpret the symbolism is that H is a number from 0 to 70. Now what if it didn’t include the 0 and the 70? Well, all you have to do is get rid of the equal part of it. So you’d have 0 < H < 70, but that gets a little bit lost in the translation. It’s better to look at this and read that as H being a number between 0 and 70, and the word between indicates that it doesn’t include either of the endpoints. It doesn’t include the 0 or the 70. Since this is not addressed anywhere in the standards, it’s going to be a school or a district decision as to when this representation should be taught. Again, it’s not in the standards, but this needs to be taught at some point. But again, it’s going to be up to the district or the school to make that decision as to when it’s most appropriate.
In looking at the standards for mathematical practice, doing the activities that we did in this standard, the first four, we would be doing all of that. Students would have to make sense of problems and persevere in solving them. They have to reason abstractly and quantitatively. They have to construct viable arguments and critique the reasoning of others, and we did model the mathematics. Looking at the other four standards, students would be attending to precision and they would look for and express regularity in repeated reasoning.