This is Common Core State Standards Support Video for Mathematics. The standard is K.OA.A.3. The standard reads: Decompose numbers less than or equal to 10 into pairs in more than one way, for example, by using objects or drawings, and record each decomposition by a drawing or equation; for example, 5 is equal to 2 plus 3 and 5 is equal to 4 plus 1. If you look at this standard and make connections to others, there is another kindergarten standard right before this one, K.OA.A.2, that connects directly to it and this one states: Solve addition and subtraction word problems and add and subtract within 10, and again, example; by using objects or drawings to represent the problem. Also, in kindergarten, there’s K.OA.A.5 which states: Fluently and subtract within 5.
Let’s look at this idea—record each decomposition by a drawing or equation. With this standard, we’re laying the foundation for standard 1.OA.D.7 in the next grade, and this one talks about understanding the meaning of the equal sign and determining if equations involving addition and subtraction are true or false. Okay, let’s look at the number 6 and look at this term, pairs. Now pairs typically, for most people, mean sets of two. So, if we were to take six, and let’s take some physical objects; I just got some cardinals, and do what it says, break it up into pairs. So this would be 2 plus 2 plus 2. If this would have been 10, it would have been 2 plus 2 plus 2 plus 2 plus 2, and it doesn’t seem like this was the intent of the standard. So really, what we need to do as far as interpreting pairs, we need to really see that as two groups. So we need to see this as decompose numbers less than or equal to 10 into two groups in more than one way. So for example, we could take the six and break it up into two groups, say four and two.
Now let’s focus on the idea of decompose. Let’s look at the number 6, a quantity of six. So we have six objects here, six basketballs, and if we were to rearrange them, we’d set it up to where it’s 4 plus 2, which of course, is still equal to 6. And if we represent that symbolically, we have 4 plus 2. The idea of conservation of number, that’s a very critical fundamental concept. Conservation of number simply says that if you have a quantity, if you rearrange it, it doesn’t change the amount. It still stays the same. So we have our six split up into two groups of four and two, but if I did this, if I moved one of them, that’s still a quantity of six. If I move another, that’s still six. Move another, I still have six; one more, still have a six; move another one, still six; move the last one, still six. So even though again, I was rearranging the groupings and everything, I never really changed the amount. It stayed at a constant quantity of six.
Let’s take five and decompose it. Let’s say we decompose it to three and two. If we pay attention to the last part of the standard, we’re going to record this with an equation. And if we’re going to do that as an equation, we do 3 plus 2 equals to 5. But we have a little bit of a problem with the physical representation. We have a 3 and we have a 2 physically represented, but where’s the 5? So when we start off with a quantity and we decompose it, but we have it like this, students might lose the connection. Now if we do it this way, the physical representation connects directly and relates to the equation that we had set up for it. We have our physical representation of three and two, and it is in fact, equal to five. The idea of conservation of number needs to be instilled in students, and this in turn transitioned to the idea of equality, because of course, equality connects directly to this idea of an equation.
So let’s take the quantity of four, but rather than do what we typically would do, why don’t we start off with two sets of four? Now of course, these are equal because 4 is equal to itself; 4 is equal to 4. What we can do is, for example, on the left-hand side, let’s decompose the 4 to 3 and 1. And if we’re going to represent this with an equation, we need our plus sign. On the right-hand side, we have our original quantity of 4, and on the left, we decompose it to 3 plus 1. So now we have our equation symbolically, 3 plus 1 is equal to 4. But then we also have the exact same representation with our physical objects right above it so students can really make the connection. And again, we’re reinforcing this idea of conservation of number, because we rearranged it but they can see that it still remained the same as the original 4.
Let’s do another example. We typically always change what’s on the left-hand side and leave the right-hand side alone. But this gives us the advantage of the flexibility of decomposing either side. So let’s decompose the right to 1 plus 3. And now let’s work on our equation. On the left-hand side, we have our original quantity of 4. And on the right-hand side, we decompose it to 1 plus 3. So, again, the symbolic representation and the physical representation right next to each other so the students can really see what’s happening. On the right, we decompose the 4 to 2 plus 2. So we put our plus sign. On the left, we have our original 4, and on the right-hand side, we have 2 plus 2. Now what we can do is, hey, decompose the left-hand side, let’s say to 1 plus 3. So now we have a little bit more of a complicated equation, but it’s still equal quantities, because 4 is equal to 4. So the students can readily see that 1 plus 3 would be equivalent to 2 plus 2.
Let’s try this again, and let’s do this. On the left-hand side I have 1 plus 3. On the right-hand side I have 3 plus 1. Notice that this lays the foundation for the idea of the commutative property where on the left-hand side I have 1 plus 3. On the right-hand side, I have 3 plus 1. It’s actually the same groupings. I just have them in a different order. But again, it’s a nice simple way to teach another fundamental property simultaneously. But you don’t even have to call it the commutative property. Students can kind of figure this out by themselves, and we worry about naming it later.
So to summarize, you might want to try decomposing numbers, but start off with two of the same set. That way you can put your equal sign and then manipulate one side or the other. So for example, I can start off here, and I can manipulate the left-hand side and maybe regroup this to 4 plus 3 is equal to 7. Or I can also manipulate the right-hand side and maybe regroup the 7 to 2 plus 5. Let’s look at the standards for mathematical practice.
If students do these activities to address this standard, they will reason abstractly and quantitatively. They will construct viable arguments and critique the reasoning of others. We did model with mathematics. If we look at 5, 6, 7, and 8, students would use tools strategically if we do this with drawings or with manipulatives. And we also would be looking for and making use of structure and looking for and expressing regularity in repeated reasoning.