This is Common Core State Standards Support Video in Mathematics: the standard is 3.OA.9. This standard reads: Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations.
Since the standard specifically mentions the addition and multiplication tables, let’s go ahead and start with that. Let’s look at the addition table first. One thing that students might notice is that across these diagonals, the numbers are the same. So for example, for the fives, if we read from left to right, then this 5 comes from 1 plus 4. This 5 comes from 2 plus 3. This 5 comes from 3 plus 2, and this last 5 comes from 4 plus 1. If we were to take this situation, these different combinations, and use some manipulatives—if we were to take one from this group and move it over to here, we have changed the 1 plus 4 to 2 plus 3.
Now mathematically, something else happened here, and it’s a little bit abstract. What happened here; we started off with 4. We took 1 away from this group over here and moved it over here. So in essence, we added one to this side. Now notice that when we subtract 1 and add 1, the net result is 0. Mathematically this is our additive inverse, that anything plus its opposite is zero. So there’s a net change of zero, so our sum is still 5. But now we have 2 plus 3 instead of 1 plus 4.
Next, if we look at the diagonal with the nines, we have pretty much the same pattern. We start off with 1 plus 8 and then 2 plus 7 and so forth. And if we do the same thing, what happens again is that if I were to take 1 from this group and move it over here, I’ve changed the 1 plus 8 to 2 plus 7. Now when we move one from one group to the other, we did basically the same thing as we did before. We subtracted one from the right-side group, and we added it to the left group, and so our result is 2 plus 7 instead of 1 plus 8.
Let’s look at the other diagonals. The pattern here seems to be that each one down the diagonal is two larger than the previous one. So let’s see, the 5 comes from 4 plus 1, the 7 comes from 5 plus 2, the 9 comes from 6 plus 3, and so forth. Now let’s see what happened here. To get from 4 to 5, we had to add 1, and to go from the 1 to the 2, we had to add 1. Now notice that the increase in the sum was 2, which is exactly what we have. Notice that what we’re doing, we’re establishing the foundation for the basic idea that what you do to one side of the equation you add to the other, and that’s exactly what we did here. We added 2 to the right-hand side, and we added 2 to the left-hand side, but we did it in increments of 1 and 1. And the process will repeat itself for the next pairings where again, to get from the 5 to the 6, we have to add 1, and to get from the 2 to the 3, you had to add 1, and that resulted in a net change of 2.
Let’s look at the addition table one more time. Now let’s look diagonally and let’s say this pairing and that pairing, so that is 6 plus 6, and the other one is 5 plus 7. We can use that same basic idea—to get from the 6 to the 5, we had to subtract 1, and to get from the 6 to the 7, we had to add 1. And notice again, that’s a net change of zero. And that’s why the sum in both cases is still 12. What about other pairings? What about this one here, 6 and 10 and 8 and 8? Same basic pattern, they’re both 16, and again the change would result from adding 2 to the 6 and subtracting 2 from the 10 to get the 8. The plus 2 and the minus 2 (subtracting 2) is a net change of zero resulting in the same sum of 16.
Let’s look at our table one more time. What about this pairing here and that pairing? We have 5 plus 8 and 6 plus 7. Well let’s see, they’re both 13. Same basic idea...we would have to add 1 to the 5 to get the 6. We would have to subtract 1 from the 8 to get to 7, and again, we have a plus one and a minus one, net change of zero. So we still have the same sum of 13.
A pattern that students might notice is that some of these combinations are repeated in several places in the addition table. What about other patterns? What about what happens when we add even and odd numbers? Even plus even, what happens there? If we add an even plus an odd number, what happens there, or an odd plus and odd? Now of course, the first thing that we need to make sure here is that students understand the difference between even and odd numbers. They need to know the definitions.
Now students at this level are probably going to focus on examples to prove why a pattern works. Now mathematically, that’s not really a proof, but at this level that’s okay. But they still need to understand that, again that later on in pure mathematics that just providing examples is not really a proof. But for now, it’s okay.
So let’s take even plus even. Hah, it looks like the sums are coming out even. Now, any even number can be broken down to a sum of twos because any even number is divisible by two. So we take this example here; we can break the 4 to 2 plus 2. We can take the 10 and break it down to the sum of these 5 twos, and when we add them all together, we have a bunch of twos that come out to be 14. If we take a more generic case, the first even number would be a bunch of twos. The second even number would be another bunch of twos, and the end result is a whole bunch of twos added together which should still be an even number.
What about even plus odd? Hmmm, the pattern seems to be that the sums are coming out odd. If we take one example and break it down to see why, well, let’s see, the even number 4 breaks down to 2 plus 2. The odd number 7 breaks down to 2 plus 2 plus 2, but we have a plus 1 at the end. When we combine them all together we get 11. So again, we have that plus 1 at the end. So it looks like the answer’s again going to come out odd.
Let’s take the generic case. Our even number would be a bunch of twos. Our odd number would be a bunch of twos, plus 1 at the end, and we combine them all together. We have all the twos from the even number, all of the twos from the odd number plus one at the end which is still going to give us an odd number for the sum.
What about an odd number plus another odd number? The pattern seems to be even. Let’s jump straight to the generic case, the general case. My first odd number would be so many twos plus a 1. My second odd number would be so many twos plus 1 at the end. Now notice that we can take each of those ones and combine them together to be another 2. So now we have the sum of nothing but twos which should result in an even number. So that’s our pattern there. An odd number plus an odd number will give us an even result.
So now let’s look at multiplication, and let’s look at the table. What if we look diagonally, 3 and 8, and 4 and 6? We have 24 for both. Now look at this question. Why do 6 times 4 and 8 times 3 result in the same product? Now that’s not really the same question as why are 6 times 4 and 8 times 3 equal? We have already answered that one. They’re equal because they both came out to be 24.
Let’s look at it this way. The 3 times 8...we can break this down. We can’t bring down the 3 any more, but we can break down the 8 to 4 times 2. And then we can take the 4 and break it down to 2 times 2. So here’s all our factors for 3 times 8. We can do the same thing with the 4 times 6. We can break the 4 down to 2 times 2, and we can break the 6 down to 3 times 2. Now notice what happened here. In both cases, we had a 3 for a factor, and then we can also match up all the different twos. So again, notice now that we have the same factors in both cases. We have a 3 and we have 3 twos for both of them. So we’re really multiplying the same factors in both cases.
If we look at another possibility like 6 times 12 and 9 times 8, if we break down the 6 times 12, this is what we get. If we break down the 9 times 8, this is what we get. And again, notice the pattern. We can match them all up, so in essence, we have the same factors for both cases, and that explains why we get the same result.
What about other possibilities? What about this way? Let’s see, we have 2 times 20. That’s 40. And 5 times 8, that’s 40, that works. And what about this? Six times 40 is 240, 16 times 15, if we multiply it out, that will come out to be 240. So this can be a lot of fun...students looking for all these different possibilities where they can get equal products. And this will give them a lot of practice with basic multiplication skills. Now if go back to that question, why do 6 times 4 and 8 times 3 result in the same product? Well again, the key is that when you break it down to all of the factors and then the same thing here, we have the same factors for both cases. We’re multiplying the same things.
Students can use the commutative and associative properties of multiplication to derive all of the factor combinations for any given product. So let’s take 24 for example. If we break it all down, notice what we can do here. The 2 times 2 times 2 would be 8 times 3. And then here, if we group them this way, then 2 times 2 would be 4 and the 2 times 3 would be 6. And over here, we would have 2 times, 2 times 2 is 4 times 3 is 12. To get the other possibilities we would just change some of the orders around and we would eventually get 3 times 8, 6 times 4, and 12 times 2.
Let’s look at the multiplication table again, but from this perspective. Notice that this diagonal here is your perfect squares; 4 would be 2 times 2, 9 is 3 times 3, and so forth. Now notice that we have this part shaded and this side not, but that’s for a reason. Notice that students should be able to find numbers on one side that match up with numbers on the other. For example, here we have 12 and 12. Now this 12 here comes from multiplying 2 times 6. That’s 2 groups of 6. And this other 12 is actually 6 groups of 2. But they’re both 12. Another combination—this 30 here would be 5 groups of 6 whereas this 30 is 6 groups of 5.
So we can find a lot of these different pairings and notice that they occur along diagonals. In this case we’d have 3 groups of 9 for this 27 and 9 groups of 3 for this other 27. Notice the pattern here, that we have these matching pairs of numbers, the 27s, the 24s, the 21s, and so forth. So, along this column we have matching numbers over here along this row. And notice that really what we have here is the commutative property. All of these pairings—that’s the only difference, that you have your factors in reverse order. So for example, for this 15, this one is 5 groups of 3 as opposed to this other 15 that’s 3 groups of 5.
What about the multiplication patterns dealing with even and odd numbers? Again just using examples, if we take an even times an even, let’s see, that’s 24, 16, 10 times 6 is 60. It looks like an even times an even is going to give us an even number. What about odd times even? Well let’s see, that would be like 5 times 4. That’s 20, 7 times 2 is 14, and so forth. So that pattern seems to be an even result also. Now we need to consider the reversal of this which is just your commutative property, an even times an odd number. Well let’s see, 42, 24 50; looks like the pattern there is even. And then our last combination, our last possibility is odd times odd. Let’s see, 5 times 3 is 15, 7 times 9 is 63; looks like that pattern is coming out to be odd.
If we connect this back to what we were doing with addition, an even times an even number means that our groups are all going to be composed of the same even number ,and we have an even number of them. So when we do our combinations, we’re still going to end up with even numbers and then the same thing here with the an odd number of evens. We still have the same even number repeated over and over, but we have an odd number of them. But that’s okay. No matter how many iterations we were to go down, at the end we’re going to end up with an even plus an even which will still give us an even number for the result.
What about the reverse order, even times odd? This time each of our repeated numbers is an odd number, and we have an even number of those. When we combine two odds, we already know that’s going to give us an even sum. So if we do this, we’ll see the pattern that we’ll still get an even result. And then, last but not least, an odd number times another odd number—so our groups are all going to be the same odd number, but we have an odd number of those. If we keep combining odds with odds, those will give us even sums. But then we’re going to have, at the end, no matter how many iterations, you’re going to end up with an even number plus an odd number which results in another odd.
If we wish to develop little mathematicians, it’s vital that they not only find the patterns, but also figure out why the pattern works and explain and justify their reasoning. Why did it work? So we’ve seen that patterns, both from addition and multiplication, can really be powerful tools because it can do a lot of things. First, it can enhance the pattern recognition skills of the students. We can also use them to practice basic facts without all the drill and kill. In fact, using those patterns in the tables and other places was actually fun.
Students can experience the justification, an explanation of their reasoning, and if they’re doing this in written format with doing journals, that will also help. Working with patterns will also help students learn their fundamental properties of mathematics, especially the commutative and the associative properties. And last but not least, you’ve seen that we can also use these patterns to lay the foundation for more complex topics. In this case, it was for example, the additive inverse and the idea of prime factorization. So done correctly, multiplication patterns and addition patterns can really be useful and very powerful tools.