This is Common Core State Standards support video in mathematics. The standard is 1.OA.D.7. This standard states: Understand the meaning of the equal sign and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false: 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2?
Let’s look at related or connected standards to 1.OA.D.7. Back in kindergarten, there was standard K.OA.A.3, which states: decompose numbers less than or equal to 10 into pairs in more than one way. At this same grade level, there’s standard 1.OA.C.6, which talks about adding and subtracting within 20, and demonstrating fluency for addition and subtraction within 10. Also in first grade, there’s standard 1.OA.D.8, which states: determine the unknown number in an addition or subtraction equation relating three whole numbers. So, as you can tell, this one is related because it deals with adding and subtracting and equations.
At the next grade level, there’s standard 2.OA.B.2, which is about fluently adding and subtracting within 20 using mental strategies. Let’s focus on this initial idea of the standard that deals with understanding the meaning of the equal sign. There’s a tendency in our system to do something like this—we always put the actions that are needed to be done on the left, and we always put the answer on the right. As a result of that, the equal sign can easily become a symbol that just says, “do something.”
For example, if we were to give this problem to students, here’s a possible tendency. They might do something like this where they’ll just say, “well, 4 + 1 is 5.” They fill that into the answer box, and they think that they are done and that it is correct; when in fact, of course, they totally forgot about that + 2, and consequently this is a false statement. In fact, what some students might do is this. They might even add another equal sign and say “well, 5 + 2 = 7.” So, that’s a good indication that there’s some confusion here and students have, in fact, lost the meaning of the equal sign. To embed the meaning of the equal sign, be sure to vary the representations as done in the examples provided in the standard.
We need to vary where the action is needed and where the answer is placed, those types of things. So, something like this, 7 = 8 - 1, which is one of the examples in this standard—notice that this might have started off with 8 - 1 on the right-hand side, and we would have our unknown, which turns out to be 7 on the left-hand side. We might adjust that same example a little bit and put something like 7 = some unknown - 1.
At this grade level, it is definitely a good idea to use some physical representations, some manipulatives to help with the understanding. So, let’s say we start off typically with something like this to represent five. Maybe we can regroup it and do it something like this. Then symbolically we put that representation as 2 + 3, which, of course, connects back to our original quantity of five, and then we finish out the abstract representation with our equation to be 2 + 3 = 5. So, again, if we would have started like this, and we do this, we have 2 + 3.
But, there’s a little bit of a problem here, because students might lose the connection; and that connection is the original five that we started off with. Starting off with the five physical objects, these circles, we split it up to 2 + 3. Put our equation 2 + 3. That’s equal to five. But, again, there’s nothing with the physical objects. There’s just empty space out there. So, my physical objects, my two circles plus three more circles, well there’s no connection back to the original five that we began with. So, we need to do something like this where we have a second group of five to cement the idea with students that these are equivalent.
Perhaps a good adaptation would be to always do this—start off with two sets of the same thing. Initially, here we know these are equal because 5 = 5. Just like back in our standard, we have the sample 6 = 6. Let’s say on the left side we regroup this to two and three. So, now we adapt our representation to be 2 + 3, and of course, that’s equal to five, and so this is a true statement. Students can really understand that with the physical objects because nothing was taken away. Nothing extra was thrown in. We still have the original five on both sides of the equal sign. Again, it’s really easy for the students to make the connection and to realize that this is still equal. What we have on the left is equivalent to what we have on the right because we still have the same quantities of five on each side that we started off with.
Let’s say we start off with this physical representation, but again, start off with two sets of the same items. Students would count what we have on the left and what we have on the right. They realize that, hey, I have seven on both sides, and again, that’s similar to the example that was given in the standard of 6 = 6, except in this case, we have 7 = 7, which of course is true. Let’s say we have this example. Students do the counting and realize that I have seven on the left, but I have eight on the right-hand side of the equal sign; so that is not a true statement. These are not equal, so that’s false.
A good extension of something like this when you get a false statement, would be to ask, well, how can I make this into a true statement? So, in this case, some students might do this. They might decide, okay, on the right-hand side I can take one away, which in fact, is a representation of one of the examples back on this standard, 7 = 8 - 1. So, that’s what I have. I have 7 = 8 - 1, because on the right-hand side I took one away, and so now it is a true statement because I do, in fact, have seven on both sides. Other students might decide to work with what’s on the left-hand side, and maybe they’ll do something like this where they added one. So, now these are equal because I have 7 + 1 on the left, which is the same amount as eight on the right-hand side. So, that is a true statement.
Let’s start off with these two sets. Students count them, and they realize that this is a true statement because we have seven on each side, so 7 = 7. By starting off like this, students can manipulate one side or the other. For example, on the left-hand side, maybe we can split this up to 5 + 2, and we know this is a true statement because again, nothing extra was thrown in. Nothing was taken away. But, we can also manipulate the right-hand side, except this time we have 2 + 5. Notice that we can actually lay the foundation for the commutative property here because on the left we have 5 + 2. On the right, we have 2 + 5. It’s actually the same groupings. We just have them in different order. This is a true statement, and in fact, it’s one of the examples given in the standard. But, it’s a good idea to do a physical representation and don’t depend just on the abstract representation with the writing and with the equal signs.
Let’s take that last example that’s given in the standard, 4 + 1 = 5 + 2, but let’s take it a step further. Let’s actually do some physical representation so that students can see really what happens with each side of the equal sign. On the left-hand side, we’ve represented 4 + 1. On the right-hand side, we’ve represented 5 + 2. Students will realize, well, I’ve got five on the left, and I’ve got seven on the right, so, that is a false statement. Those are not equivalent. Taking this a step further, let’s rearrange the quantities on each side of the equal sign. Students can now easily tell because of the orientation that I do, in fact, have more on the right than I have on the left-hand side of the equal sign. Students know that this is a false statement.
Again, a good extension of this would be to take a false statement and ask students, “how can I make this a true statement?” If we work with our physical representations, some students might do something like this. On the right-hand side, they could take away one. Take away another one, and now we do have equivalent amounts on each side of the equal sign, except on the right-hand side we have 3 + 2 now instead of 5 + 2. But, we know that these are equal because they’re both equal to five. Students think differently, so some students might work on the left-hand side and do something like this where they will add one and add another one. So, they’ve changed the 4 + 1 to 4 + 3, which of course is 7. So, now this is a true statement because we have seven on both sides of the equal sign, 7 = 7.
Hopefully you’ve seen the advantage of starting off with two sets, two physical representations instead of just one. In this way, initially they can determine if you do, in fact, have the same quantity on each side. In this case I do because 4 = 4. I’ve got four on each side. This gives you the advantage of having physical representations on both sides of the equal sign, and you can manipulate one side or the other. In this case I’ve manipulated the left—changed that to 3 + 1, which I know is equal to four. I can also at the same time manipulate the right-hand side. I’ve changed that to 2 + 2. This is a true statement. But again, I made this nice and easy because you’ve got the quantities already that you can manipulate on one side of the equal sign or the other.
Let’s do another example. We’re starting off with the same quantities, 4 = 4. On the left-hand side, I’ve made that 3 + 1. On the right-hand side though, I’ve made it 1 + 3. Just like before, a strength here is that you can actually lay the foundation for the commutative property without ever even using that terminology. Here, 3 + 1 = 1 + 3; same groupings, you just put them in a different order.
Let’s look at our standards for mathematical practice. Looking at the first four, by doing the activities that we used as examples in this standard, they would reason abstractly and quantitatively. Students would also construct viable arguments and critique the reasoning of others. Looking at the other four standards of mathematical practice, students would attend to precision, and they would look for any express regularity in repeated reasoning.