Mathematics

K.CC.C.6 Transcript

This is common Core State Standards support video in mathematics. This is standard K.CC.C.6.

The standard reads: Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g. by using matching and counting strategies. Now first of all students must understand that we begin with one set or quantity, and we compare it to a second set or quantity. Students must be very aware that when comparing two quantities there's only three possibilities, either the first set or quantity is greater than the second, its less than the second, or its equal to it. Those are the only three possibilities.

We can lay the foundation for these the numeric comparisons by using analogies using standard English context, for example we can make comparisons using taller versus shorter, or older versus younger. For example, here Natalie is shorter than her father. We're making a comparison but we’re not interjecting any numbers in here. Another analogy would be something like this, where Pablo is older than his brother Henry. Students need to realize also, that comparisons have direction, for example here the tree on the left is shorter than the tree on the right, so we're going from left to right. But we could easily do the comparison in the opposite direction; we could say that the tree on the right is taller than the tree on the left. However, to lay the foundation for the numeric symbolism, where we are going to use greater than, or less than symbols the direction of the comparison should be from left to right, so that the symbols will match.

So let's start with an example where we have two quantities, two sets and they happen to be equal. Now one thing that we need to do is change the spatial orientation of the objects to focus on the idea that were comparing the quantities and nothing else. So we can do something like this, or maybe rearrange them something like this, to again emphasize that idea, it's quantity that are being compared, nothing else. We can also do something like this we could possibly change the color of the items to again focus on that idea that we’re only comparing the quantities

The standard says to use matching and counting strategies, so let’s look at matching. So in this example here, we match this one from the first set to the other one in the second. We match those together, and we match these third ones together. This works just fine as long as we have no small amounts. Now let’s refer to the counting strategy, so we count the quantity that we have on the left, we have a total of three. And on the right we count again, so on the right we also have a quantity of three. What's good about these kinds of activities is that we can address other standards simultaneously, so for example here, the standard K.CC.B.4c: Understand that the last number name said tells the number of objects counted. Also the number of objects is the same regardless of their arrangement for the order, which they recounted, and so we're doing both here. So we are addressing this other standard at the same time. There's this other standard K.CC.A.3 that reads: Write numbers from 0 to 20. Represent a number of objects with a written numeral. So here we can do the same thing, we can represent our quantities now with the symbolism, and this case with the numerals 3 and 3. There’s yet another standard that we can addressK.CC.C.7: Compare two numbers between 1 and 10 presented as numerals. So after we've done our comparisons with the actual physical objects, we can also write it down symbolically, and now we’re comparing them as numerals. So, orally or in written language we know that 3 is equal to 3. Now the question is will kindergarten students be expected to use the equal sign and the answer is yes. There's yet another standard we can address K.OA A.3 that states: Decompose numbers less than or equal to 10 into pairs in more than one way, e.g. by using objects or drawings, and record each decomposition by a drawing or equation. So there's the statement right there, so yes students would be expected to write this symbolically using the equal sign.

Now let’s take a scenario where it's either a greater than or less than comparison. Let's start off with these apples. So we can start off with a matching strategy, but what happens when you start getting into bigger numbers, students might get confused as to which one did I already matchup, which one goes with which one. So here’s another possibility, another suggestion, what we can do is this, rearrange them, and then take the top group and arrange them horizontally. Then we can easily match them up and not lose track of what goes with what. So we can do like so, match up the first one, match up the second, match up the third, match up the fourth. So at this point we can easily tell we've got you equal quantities there we have 4 and 4. I've got more on the top, so I know that the top group is going to have more than the bottom group. Now let’s look at the counting strategies. We can incorporate that, we can count the top group there's 6, we count the bottom group, there's 4 there. Now we line these up like this for a purpose, what we're really doing is laying the foundation at the kindergarten level for the number line. If you can visualize, or get something physical like a wooden dowel, or something that represents a line. Again notice what we're doing we're laying the foundation again for the idea of the number line. Students can see that our larger number of six is further to the right in our smaller number of 4. Students will see that as a number is further and further to the right, its larger, and if a numbers is further to the left it’s smaller. We can also address the decomposition idea back in that standard K.OA.A.3. Here we can decompose the 6 on top to a 4, and then a 2, and of course that's our whole quantity of 6.

So let's look at our original up situation, we’ve compared the numbers, we know we have 6 on the left, and 4 on the right, and we can also write the numbers down, and represent them symbolically, so we can address standard K.CC.C.7. We know the 6 is greater than 4. Now the question, should we expect in kindergarten students to use greater than or less and symbols. And the answer is no, it’s not until 1st grade that we have that expectation. Standard 1.NBT.B.3 states: Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results and comparisons with the symbols >, =, and <. So the expectation doesn't appear until first-grade, now it's up to you as the teacher to decide whether or not your kindergarten students can handle this symbolism using the greater than, or less and sign. But if you want to attempt this, if you want to see if they can handle it, here's what you can do. So in this situation we think that this is the greater than sign. Now teachers have developed all kinds of little tricks and acronyms and so forth; alligators eating stuffed things like that. Here's a nice simple little way for students remember which symbol is which, and again it doesn't involve any memorization or anything. Here's all that you need the students to do, have them draw a vertical bar thatconnects the top and bottom segments of your sign, and then do the same thing further to the right in the interior. Now notice that you have a longer bar here and a shorter vertical bar there, and there's a direct connection to the actual quantities involved. So you have the longer bar here so we should have the larger number on the left, and then over here you have the smaller of the two vertical bars, so your smaller numbers correspondingly should go over here on the right.

So let’s reverse the quantities and put the 4 on the left and the 6 on the right. We know the 4 is less than 6, and we think this is the correct sign for less than. Well let's check again, draw your vertical bar there, draw your second vertical bar here. And the smaller bar is to the left so that should be where our smaller number goes. The larger bar is to the right, so that should be where our larger numbers is. Now we have a high degree of certainty that we have the right symbol here for less than. Let's take another scenario and let’s look at this less than context. And let’s use our county strategies. So on the left we count 'em up there are 6, on the right we count them up there are 7 of them. So we know that 6 is less than 7. And again if you want to see if your students can handle it at this level, we think that this is the correct symbol for less than. We do our little vertical bar strategy and yes it checks out we have the smaller number on the left and the larger number on the right.

Now let’s combine our matching and counting strategies. Let’s rearrange these horizontally like we did before. Let's count the top group there are 6. Let’s count the bottom group there are 7. And again the nice thing that you’re doing here is that you're laying the foundation for what happens on the number line. We know the 6 is less than 7, and we think this is a correct symbol, then again double check. Draw vertical bars to make sure that the size of the bar corresponds to the size of the number. And so we're done, and we have again a high degree of certainty that we have this stated correctly.

Couple of other items, teachers should use a structural language that matches the language of the standards. Now that doesn't always apply, because sometimes the language of the standard is more for the teacher, not for the student. But in this case we should state the comparisons using greater than, less than, or equal to. The reason being that expressions such as larger than or smaller then can be misleading because they can refer to size rather than just quantity and that might cause confusion with students. So again a in this scenario stick with the language of the standards, use greater than, less than, or equal to, instead of substituting other phrases such as larger than or smaller than. As you saw from these activities using manipulatives you can address several other standards simultaneously, you can address a lot more than just standard K.CC.C.6.