Mathematics

3.OA.7, Part 1, Transcript

This is Common Core State Standards Support Video in Mathematics. The standard is 3.OA.7. This standard states: Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (for example, knowing that 8 times 5 is 40, one would know that 40 divided by 5 is 8) or use properties of operations. By the end of Grade 3, know from memory all products of 2 one-digit numbers.

So the focus of this standard is to fluently multiply and divide within 100. The goal is that by the end of Grade 3, students should know from memory all products of 2 one-digit numbers. Now, this automatic recall is vital for procedural fluency. If they don’t have these multiplication facts down, committed to memory, it’s really going to bog them down in mathematics in the future...so very, very important.

Now, realistically, for whatever reason, some students really struggle with this, and as teachers, there is no magic bullet. Now, memorization is no guarantee of understanding. Students can memorize stuff, but they still may not have a clue what they’re doing or what it means. Instead, it’s really important that they understand because this facilitates the memorization. It will make it a lot easier if they really understand the concepts. So what we need to do is focus on understanding multiplication conceptually, and this, coupled with properties of operations, will empower students and give them alternative pathways to the desired goal.

Okay, so this idea of understanding multiplication conceptually...well, first of all, as far as a definition, what is multiplication? Now, a lot of us were taught that multiplication is repeated addition, but if that’s our definition, that falls way short. It really is insufficient. If we go to the Common Core document, and we can find it at this website, on page 21 there is a summary of multiplication and it focuses on this idea of equal- sized groups. So, a much better definition of multiplication is that it is repeated addition, but it’s based on the use of equal-sized groups. And this is important because by defining it that way, it enables us to make some connections, especially between multiplication and division because defining it as just repeated addition leaves us this big river of whatever in the middle, and there’s no way to cross and connect the two. The equal-sized groups are important because it serves as the bridge that connects these two ideas of multiplication and division.

So along that line, making these connections, well, how are multiplication and division connected? And a related question would be how are multiplication and division the same, which is really an easier question than how are multiplication and division different. Okay, now at this level, the typical context would be one where you would have a certain number of groups of a constant equal size that would constitute a total quantity.

Continuing on, again, we have a certain number of groups of a constant equal size that constitute a total quantity. Now let’s say we have eight groups of seven each. That would be our total of 56. Now, at this stage, this would be considered a multiplication problem: 8 times 7 equals what? And then, these next two situations in the blue font would be considered division contexts. In this case, the first one, we know the total is 56. We know that we want the size of the sets to be seven. We’re looking for the number of groups that that would constitute. Over here we have a total of 56, and we know we have a certain number of groups, in this case eight. So we want to know what would the size of the groups need to be.

Now, notice that in all of these scenarios, whether you are thinking of it as multiplication or if you’re thinking of it as division, notice that we still had the same three things involved. We had so many groups of an equal constant size and a total quantity. Now, notice that for multiplication, we were looking for the total. And for division, we knew the total and we were either looking for the number of groups or the size of the groups. So, multiplication and division are very similar because in contexts at this level, it involves the same things. It involves a certain number of groups of a constant equal size that constitute a total quantity.

So really, the only difference between multiplication and division is what you know and what you don’t know. So in the case multiplication, you know the size of the groups and the number of groups. You’re looking for the total. And in division, you know the total and you’re looking again, for either the size of the groups or the number of groups. So again, what makes multiplication and division the same is that it involves the same three components, and the only thing that makes them different is what you know and what you’re looking for.

Now let’s look at this idea of representation: 8 times 7 being 56. Now, is eight the number of groups and seven is the size of the group, or is it reversed? Is the eight the size of the group and the seven the number of groups? If you do research, there’s no national consensus. There’s no definite answer out there that says one way or the other is correct. Now, standard 3.OA.1 does speak to the interpretation of products of whole numbers and states that something like 8 times 7 should be interpreted as eight groups of seven. So that’s the way that we’re going to approach it, where the first number tells us how many groups, and the second number will tell us the size of the group. Now that interpretation parallels standard English usage, and it leads to the flexible perspective of 8 times 7 as 8 sevens.

Let’s go into this a little bit deeper. In standard English, we might have something like eight bears where the rule is, okay, we have the adjective here followed by the noun. Now, interpreting the 8 times 7 as eight groups of seven each parallels this so that we eventually will see 8 times 7 as 8 sevens. So again, it makes sense, and this will also help algebraically down the road because, when students hit something like this, then they’ll also interpret that not just as 8 times y, but really as 8 ys.

Now, why memorize? Students need some motivation. Well, let’s say you gave them a bunch of blocks like this. It’s going to take the students some time to count by ones to get to the total of 21. Counting by ones can be a lot of trouble and a pretty slow process. But, let’s say they organize those blocks into three groups of seven each, which would be 21. So rather than count by ones, here they’re counting by sevens, and knowing the multiplication fact 3 times 7 is 21 will really help them out. Other students might think a little bit differently and come up with seven groups of three, which is still going to be 21. Now, when your students compare what they did, they’ll see this—that, hey, three groups of seven and seven groups of three are the same because in each case we got 21. Three groups of seven is 21. Seven groups of three is also 21.

Now, 21– that’s not a whole bunch, but let’s say you gave them 72 blocks. They’ll start to get the message that man, if they have to take the 72 and count them one by one, that’s a lot of trouble. But, if they take them and organize them into say, eight groups of nine, then they’ll automatically know, hey, that’s 72. Or, some other students might do nine groups of eight, which of course, is still 72—so again, motivation there because it will save them a lot of trouble.

3.OA.7, Part 2, Transcript

In essence, this standard really isn’t all that complicated, but we know that students have trouble with this. We need to concentrate on the strategies that will help our students memorize these multiplication facts. So let’s focus on the properties of operations. Okay this is what we typically call the times table, but what we ought to do is think of this as the grouping table, because that in essence at this grade level, that’s what students will be doing. They’ll be working with groups or sets of things. Now the way that this table is organized, we have the size of the groups horizontally, and the number of groups is represented vertically. And typically, what we do when kids are learning the different multiplication facts is to keep the size of the groups constant. So they’ll learn for example all their, you know, all their sets of one. All the combinations as far as the sets of two sets of three and so forth this idea of commutativity for example, three times eight and eight times three, both being 24.

Well notice that if students know one they’ll pretty much already know the other. The only difference is how they’re grouped—three groups of eight instead of eight groups of three. If we use the commutative property, how does that help with the memorization? Let’s look at this grouping table from this perspective. Now initially, we would have started okay. The very simplistic idea of sets of one so, for example, four sets of one is four, five sets of one is five, etc. Then, we work over to our sets of two. For example, three groups of two is six; five groups of two is 10 etc. If we look at this set of cells diagonally, in this pinkish sort of color, those are the perfect squares. Two squared is four; three squared is nine etc. Now look at what’s happening here, our columns are getting shorter and shorter, with this turquoise color, so this is what’s happening. Let’s say for example, we have six sets of two being 12. Well notice that over here, I have another 12, and this one is two sets of six. In fact, if you look at this column, all of these here also appear over there. So basically, what you have is the same factor combinations, except they’re in reverse order as far as the size of the groups and the number of groups.

So in essence, when students are learning their sets of two, they’re also learning all of these combinations over here, and that’s what happens as we go to the right—this column here. When they learn these, they’re already learning these by using the commutative property. And notice by the time that we get here, the only new fact that they had to learn is that nine times eight was 72. So using the commutative property here, you could almost make a deal with your students, say hey, I’ll be happy if you memorize half of the table. Because by doing that, they’ll actually be memorizing the whole thing. Now there’s going to be some combinations for whatever reason that are problematic for students. They just have trouble memorizing some of them. Now one strategy that we can use is the distributive property. We need to really use this, because the distributive property focuses on composing and decomposing the operations into known facts. So use the distributive property. It can really be your friend. So how can we use the distributive property? The best way to see this is to do some examples.

Okay six times seven, that’s one combination that’s difficult for students to memorize. Now notice here, I have the visual model, and vertically what we have is our sets of seven, because if you count vertically ... we have seven. So here, we have six groups of seven. Now here’s how we can use the distributive property. Let’s do this. We’ll just move this over; so what we are really doing here is we’re actually manipulating the groups instead of having six groups. I changed this to three groups here, so this is what we have. We have three sets of seven here and three sets of seven here. Okay, now most students should have this multiplication fact down—three times seven is 21. So by knowing that, students simply have to take the 21 and the 21, combine them to be 42. Now this is a little bit of a detour. It’s a little bit more work, but the students will get there. Students don't all think the same way. So let’s take another example. Some students might do this instead. So what happened here? Well in essence, what the students have done instead of manipulating the number of groups, they’re manipulating the size of the group. Because notice, okay, I have sets of seven vertically. Well look what happened. Now we have converted that to groups of five and groups of two. So this is what happened. We took the seven and we made it five plus two; but again, that’s six of them. So in essence, what we did is we took the size of the groups, and we changed it to sets of five and sets of two. And we’re going to have six sets of each. So here we have six sets of five, and most students should have this committed to memory. And over here, we have six sets of two, and kids will have that committed to memory to being 12. Combine those together, and we get 42. Again a little bit of a detour, but we’ll get there.

Let’s take another one that’s a little bit of trouble for some students—nine times eight. Thinking along the same line as that first example, here’s what some student might do— here is something like this. And again what we’re doing is we’re manipulating the number of groups. We’re manipulating the nine. So what we’re doing is we’re splitting the nine up to five plus four. And of course, we’re going to have eight of each. So again, so what we did is we split the nine up to five plus four, so that instead of having just nine groups of eight, we’ve got five sets of eight here, and we have four sets of eight over here. And again, committed to memory, five times eight being 40 and hope the kids have four times eight being 32 committed to memory. And combined, that’s 72. A little bit more trouble, and it took us a little longer, but again, the students got there; and that beats not getting there at all. Now some students might think this way. It might be a little bit unorthodox, but not really. Some students might look at the nine and start thinking, hey, this is pretty close to 10. In fact, 10 minus one is nine. So using that thinking, let’s do this.

Okay, instead of having nine groups of eight, what we’re really starting off with is 10 groups of eight. So we have 10 groups of eight to begin with. Now what we’re doing is we’re converting the nine to 10 minus one. Okay, so in essence, okay we’re going to have 10 groups of eight to start off with, which we know is 80. But we need nine groups of eight, so what we’re going to have to do is take one of these sets of eight and knock it out. So we need one group of eight, and we’re going to subtract that. And of course, one times eight is eight; so now, eighty minus eight is 72. So again, a little bit different approach, little bit unorthodox; but that’s okay. Now this idea of memorization and using the distributive property as an alternative strategy. Now this might be viewed initially as a crutch, but if teachers really make sure that the students keep repeating what they’re doing, this repeated practice will eventually lead to proficient recall.

So for example, on that six times seven example, if they repeatedly do that and split it up to three times seven and three times seven and keep doing, you know, that 21 plus 21, sooner or later they’re going to get there to, hey, six times seven is 42. The distributive property is not stressed at this level, but the last sentence in standard 3.O.5 does state knowing that eight times five equals 40 and eight times two equals 16 one can find eight times seven as eight times parenthesis five plus two being equal to eight times five plus eight times two, which would be 40 plus 16. And that is 56. So this is the distributive property, and this is very similar to some of the examples that we used.

Again the distributive property is not stressed at this grade level, but the distributive property at this level can really be viewed as a form of composing and decomposing, and that...and this idea of composing and decomposing is stressed at the elementary level in the standards. Now what is this distributive property being a form of composing and decomposing?

Well let’s say we have a quantity of six. These six little blocks. Now I can take that six, those six blocks, and regroup them to where let’s say I’ve got three here and three there. So that’s still six. Now how does that connect back to the distributive property? Remember the example of six times seven. Okay here’s what's happening. All right, we broke this six down to three and three. Well the same thing happened here; we broke this down to three and three, but here’s the distinction over here. We’re dealing strictly with single items. We took the six single items, and we broke them down into three single items here, and three single items here, as you can see, with the manipulatives. Well the parallel is that with the distributive property instead of working with single items, we’re working with sets. So over here, instead of regrouping single items, we’re regrouping sets of seven. So that’s the parallel. We are composing and decomposing with the distributive property, but we’re doing it with sets not single items.

The focus of the standard, fluently multiply and divide within a hundred. It’s important to note that the more simplistic factor combinations are just as important. So don’t overlook them; don’t neglect them. So for example, you know five times one is five, and one times five is five—kinda simple—but here’s why this is important. This is your multiplicative identity; this basic idea that anything times one is not going to change anything. I’m still gonna to get the same answer. But the reason this is so important is because the multiplicative identity is in a lot of equations. That’s part of what happens, you need to convert things toward something. Times one it’s the linchpin for a lot of the things that we do in algebra, so it is vital. So again, do not overlook, you know, the basic combinations, especially those where you’re multiplying by one. And what also happens is that these more simplistic factor combinations can also help with some of the little nuances as far as the symbolism.

So for example, you know five times one and one times five. There is a difference. You know five times one is five ones, which would be like five pennies. And then one times five, that’s just one set of five, which would be like one nickel. They’re both five cents, but they don’t look the same. If we look at the standard more closely, what we haven’t done too much of is use the relationship between multiplication and division. And yes, the focus has been on multiplication facts. It looks like we’ve neglected division totally, but we know, for example, that if a student knows eight times seven is 56 they’re automatically going to know this. Okay if they know eight times seven is 56, they’re going to know that 56 divided by eight is seven and that 56 divided by seven is eight. So they’re so connected; I mean, there’s no distinction.

So here’s the key, I mean, they have to know the multiplication fact, and if they know that, they automatically know the division also. So in summary, again, there’s no magic bullet. It’s going to take practice and persistence to get the students to know their multiplication facts. You know the products of two one-digit factors from memory by the end of third grade. We spoke about focusing on multiplication as a concept not a procedure. If students understand multiplication is a concept, it’s really going to facilitate the memorization—make it a lot easier. And we do that by first of all emphasizing this idea of equal-sized groups. Very important because that automatically is a bridge to connect to division. It’s also very important; we talked about understanding the symbolism, you know, what does something like three times five really mean?

We did a couple of things on the utility of rapid recall, that it really is a lot less trouble; it really helps them in the long run. And of course, it’s important to make sure to connect the abstract and the concrete with visual models, and so forth. And then, we also looked at using the power of the commutative and distributive properties to lessen the burden of memorization and, in that case, the commutative property was the linchpin. And in the second, if we use the properties, and in particular the distributive property, that really will give students alternatives as you saw with the examples that we did like taking six times seven and breaking it down to three times seven and three times seven.