Mathematics

6.EE.B.5 Transcript

This is Common Core State Standards support video in mathematics. The standard is 6.EE.B.5. This standard reads: Understand solving an equation or inequality as a process of answering a question; which values from a specific set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

One fundamental idea here is that of solving an equation. The expectations of this standard regarding the solving of an equation should not be new to students due to previous exposure and practice in earlier grade levels. If you do a search of the standards using the term equation, it’s going to result in multiple hits in grade levels K through 5. For example, there’s standard 1.OA.D.8 that deals with equations determining the unknown number that would make that equation true. In third grade, there’s standard 3.OA.A.4, and again it deals with determining unknown numbers that would make the equation true. Back in second grade, you had standard 2.MD.B.5 that dealt with word problems, and again, students were expected to use drawings or equations with a symbol for the unknown number to represent the problem.

In third grade, a similar standard, 3.OA.D.8, deals with word problems, and again, using equations with a letter to stand for the unknown quantity. Standard 4.NF.B.3d deals with fractions and again using equations to represent a problem. Standard 4.MD.C.7 deals with angles, and there you also have equations involved. In the previous grade level, there’s standard 5.NF.B.6 where we’re dealing with multiplication of fractions, and again, students will be using visual fraction models or equations. Now in the same grade level, Grade 6, you have a standard that’s in this same cluster, 6.EE.B.7, that deals with solving real-world and mathematical problems by writing and solving equations in the form X + P = Q and PX = Q for P, Q, and X being non-negative rational numbers. Notice that we’re dealing with non-negatives here. So we’re not going to be dealing with any negative numbers.

Now, there’s a focus here on this idea of solving an equation being a process for answering a question and this question is: What values from a specific set, if any, make that equation or inequality true? What’s a little different here is this idea of the values being from a specific set. Typically, this is what we deal with. We just jump in, and we solve equations, and we’re looking for values that make the equation or inequality true. So for example, let’s say we had this basic equation: what plus 8 equals 12? In mathematics, it’s typically assumed that the set of numbers we have to choose from for the solution is the set of real numbers, and at lower grade levels, the set is usually the set of whole numbers. So after a while, teachers and students lose sight of the fact that there’s a given set of numbers that they can choose from for a solution to an equation or a problem. Again, we lose sight of that though, that we actually have a specific set to choose from.

Typically, okay, we’re solving this basic equation, and the solution set will be the set of whole numbers. But what’s different about this standard is now we need to deal with situations like this, where, let’s say the set of numbers that we can choose from for the solution is just these numbers, the whole numbers from 0 to 10. Then, of course, we know that the answer is 4. Let’s change the context. This time, the set of numbers from which we can choose is just these numbers—6, 7, 8, 9, and 10. Now, we figured out that the answer has to be 4, and that’s not one of the numbers in our set. So there’s no solution. So that seems a little strange. It’s something that students will need to get used to, but this was our context. We only had these five numbers to choose from and the correct solution, the 4, wasn’t one of them.

Let’s try a slightly different context. This time, the numbers that we have to choose from for our solution is the set of odd numbers. So there’s no solution. Again, this whole idea of our solution coming from a specific set is something new here that students will need to get accustomed to. This last part of the standard talks about substitution, using that to determine whether a given number in a set makes an equation or an inequality true. At this level this would be a typical equation, a simple addition equation like N + P = M. So let’s say the equation was N + 7 = 15, and the set of numbers that we have to choose from for our solution would be the whole numbers from 0 to 10. So the question here is what value for N will make this a true statement? So we use substitution, we plug in numbers, and we determine that 8 is the correct solution, and it is part of our set that we had to choose from. Let’s try a similar problem but subtraction, and let’s say the problem is N - 15 = 16. This time, we have a small set of numbers to choose from: 1, 8, 11, 24, 31, and 43. So our solution set is just these numbers, nothing else.

So the question would be, well, what value for N would make this true? Students might look at this and see that, well, we’re subtracting. There’s 15 and 16. Well, the difference between 16 and 15 is 1, so the answer’s going to be 1, and it is part of our set. Now, if we pay attention, you know the standard says to use substitution to determine if a number makes an equation true. If the students just jump in and try to do the calculations in their heads, well, they might make a mistake like what happens here. Apparently, this student did not follow the standard. They did not use substitution. They just figured, okay, the difference is 1. Now if we do use substitution and we plug in the 1, we’ll notice that, hey, this isn’t correct. That doesn’t come out to be 16. If we follow the process and we use substitution, we will discover that, whoa, 31 is the correct answer because 31 - 15 is 16.

Let’s try a multiplication type of equation, A times B equal C. Specifically, let’s try 7B = 56, and the set of numbers that we have to choose from are the whole numbers from 0 to 10 and the question is what value for B will make this a true statement? Use substitution to determine which one works, and we see that 8 works because 8 times 7 is 56. Let’s try a division example, and let’s make this problem 6 divided by 1/2 is equal to C. The numbers that we have to choose from are just these few here: 0, 1, 3, 6, 9, and 12. And of course, the question would be what value for C will make this a true statement? Well, some students will just jump in and figure, well, it’s got to be a 3. But wait a minute. That’s not correct. I thought 6 divided by 2 was 3, so that can’t be the right solution. Now we use substitution, and we come up with the correct choice. That is a 12. It was part of our given set for our solutions, and 12 did work, so that is the correct one.

So at this grade level, these would be the typical types of equations that students would be expected to solve, and again, the difference is that they would be given values from a specific set to choose from to see what solution would work and make that a true equation. Dealing with equations, and these are typical examples again of what we will use at this grade level, if the set is understood to be the set of whole numbers or the set of real numbers, that last statement about using substitution is not as applicable because subconsciously, that suggests that students check their answer for correctness. The tendency here will be to use substitution not for finding the solution, but to check for correctness. Students need to get in the habit of using substitution to actually find the solution.

Up to this point, we’ve been looking for values from a specific set to see what would make an equation true. But, what about inequalities? Grade 6 is the first grade level where the term inequality first appears in the Common Core Standards. Standard 6.NS.C.7a deals with inequalities. It states: Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. In this same cluster of standards, 6.EE.B.8 talks about writing inequalities in the form X is greater than C or X is less than C to represent a constraint or condition in a real-world or mathematical problem. So this informs us that this type of inequality is what we will deal with, something like X is greater than C or X is less than C, and specifically, maybe something like X is greater than 9 or X is less than 9.

But in looking ahead to the next grade level, seventh grade, there’s standard 7.EE.B.4b and the types of inequalities that students will be faced with there are something like this in the form PX + Q > R. So it stands to reason that we need to prepare students for that and go above just this simple idea of inequalities like X > C and maybe something like X + N > C. So rather than just X > 9, we need to do inequalities such as X + 3 > 8. Of course, we will have equations that model this and those first inequalities dealt with greater than. But of course, we also need to deal with the inequality less than. So we will need to use inequalities such as X + 3 < 8.

So let’s start off basic and have something like X < C, and let’s specifically go with X < 6, and the set of numbers from which we can choose our solution are the whole numbers 0 through 10. Now, for inequalities it is definitely the best strategy to follow the standard, to use substitution to determine whether a given number in a specified set would make the inequality true. It’s a lot more applicable for inequalities than it was for equations. So, following the standard, we plug in a 0. Yes, that works, 0 < 6. One works, 2 works, and we keep going, and we figured out that well, 0, 1, 2, 3, 4, and 5 work, but the rest of the numbers didn’t. Six is not less than 6, 7 isn’t less than 6, and so forth. So our solution set would be the numbers 0, 1, 2, 3, 4, and 5.

Let’s try an inequality that’s greater, and let’s go with 7 > C. Now there’s a little bit of a curve here because the variable isn’t on the left-hand side. It’s on the right-hand side, but students will have to get this type of experience. Okay, the set of numbers that we can choose from for our solution are the whole numbers 0 through 10. Students see, well, 7 is greater than. Okay, I need numbers bigger than 7, so they just write, okay, well that’s 8, 9, and 10. That’s our solution set. Now here’s the danger of not following the standard and not using substitution to determine our solution set. If we use substitution, we’ll see that, hey, wait a minute, 7 is greater than 8. Wait, no it’s not. Seven is not bigger than 8. Something is wrong here. So this is not the right solution set. Again, students might want to jump in and just quickly do these things in their heads, but they need to use substitution, especially with inequalities. So, following what the standard says, let’s use substitution. So we plug in a 0. Yes, that works, 1 works, 2 works, so we keep going and we see that the numbers 0 through 6 work whereas the numbers 7, 8, 9 and 10 didn’t. So our solution set is 0, 1, 2, 3, 4, 5, and 6; again, a good example here of the importance of using substitution to find your solution set.

Okay, let’s start off with our basic inequality, but let’s make this a little bit more complicated to prepare students for what they’ll face in seventh grade. Let’s go with X - Y > C, and specifically let’s go with X - 4 > 2, and again, the set of numbers that we can choose from for our solution are the whole numbers 0 through 10. Now let’s adhere to what the standard says. Let’s use substitution, so let’s plug in these numbers to see what works. By doing this, we see that 0 through 6 did not make that a correct statement whereas 7, 8, 9, and 10 did. So the solution set was 7, 8, 9, and 10; seven minus 4 is 3 and 3 is bigger than 2, and so forth.

Okay, starting with just the basic format of X < C, but let’s ratchet this up a little bit and let’s go with something like 2X < C, and specifically let’s try 2X < 9. The set of numbers that we can choose from for our solution set are the whole numbers 0 through 10. Follow the standard, and let’s use substitution. Let’s just plug in these numbers and see what works and what doesn’t. So we plug them in. Let’s see, 2 times 0 is 0, 0 < 9; 2 times 1 is 2, 2 < 9, and if we keep going, we’ll see that 0, 1, 2, 3, and 4 work whereas the rest of these whole numbers, 5 through 10, do not. For example, if I plug in a 5, 2 times 5 is 10. Ten is not less than 9, so our solution set is 0, 1, 2, 3, and 4. These values will make this inequality, 2X < 9, a true statement.

Starting with our basic format, X > C, we have something like X > 12, but this time the set of numbers that we have to choose from is the set of whole numbers. It’s not just a limited number of values to choose from like before. Well, let’s use substitution. What will work here if we have all the whole numbers to choose from? Well, we’ll see that well, 13 is bigger than 12, 14 is bigger than 12, and on down the line. So our solution set would be 13, 14, 15, 16, et cetera. It just keeps going. Let’s change the context a little bit, and the solution set can be chosen from the set of real numbers. However, this type of context is better addressed in standard 6.EE.B.8. That one talks about inequalities in the form X > C or X < C to represent a constraint or condition in a real-world or mathematical problem. Now, given the set of real numbers, we’re going to have in many cases infinitely many solutions, and standard 6.EE.B.5 really doesn’t address that. So we’re better off, again not doing these types of examples for standard 6.EE.B.5 because again, it applies better to standard 6.EE.B.8, the reason being that we’re talking about from a specific set. Also, standard 6.EE.B.8 deals with representing solutions of such inequalities on a number line diagram, and again, that’s not part of standard 6.EE.B.5.

Now what about inequalities such as X is greater than or equal to C or X is less than or equal to C? Even standard 6.EE.B.8 doesn’t address that because it only deals with less than or greater than, not greater than or equal to and less than or equal to. When we compare, say, X is greater than C to X is greater than or equal to C, the only difference is the inclusion or lack of inclusion of your starting point. But again, that deals more with your number line diagrams. So again, this question is better handled in the future with standard 6.EE.B.8 and not this particular standard 6.EE.B.5.

In looking at the standards for mathematical practice, looking at the first four, students doing some of the activities to address this standard would reason abstractly and quantitatively, and they would also construct viable arguments and critique the reasoning of others. Looking at the last four of these standards, students would be attending to precision, and they would be looking for and using structure.