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Video: Using Properties of Commutativity and Associativity

Tim Burnham

We recap the laws of commutativity and associativity for addition and multiplication, then talk through a series of examples showing how they can help you with your mental math and correctly evaluate expressions.

07:19

Video Transcript

In this video, we’re going to recap the rules of commutativity and associativity for addition and multiplication, then we’re gonna take a look at how you can use them when doing mental math or evaluating expressions. First associativity, when an operation is applied more than once, the result doesn’t depend on how the pairs of terms are grouped. So for example, if you add one and two and three, it doesn’t matter whether you add the one and the two first and then add the three, or whether you add the two and the three first and do one plus the result of that. Either way still gives you the answer six. And likewise for multiplication, four times five times six doesn’t matter whether you do four times five first and then multiply by six, or whether you do four times the result of five times six. It still gives you the result one hundred and twenty. But subtraction and division are not associative. For example, if I do ten minus five minus two, if I group the first two terms together and the second two terms together, I get different answers. And likewise with division, sixteen divided by four divided by two, if I group the first two terms together, I get one answer. If I group the last two terms together, I get quite a different answer. So subtraction and division as we said are not associative. And commutativity is when you apply an operation to two terms and you get the same result regardless of their order.

So for example with addiction, three plus five is eight and five plus three is eight. It doesn’t matter which order you add them together in. And likewise with multiplication, doesn’t matter what order you multiply four and six by. So would you do four times six or six times four, you still get the answer twenty-four. So addition and multiplication are both commutative. But subtraction and division are not commutative. For example, seven take away four is not the same as four take away seven; three isn’t the same as negative three. And with division, you don’t get the same, if you do ten divided by five as you do if you get five divided by ten. Two isn’t equal to a half. So now we’ve reminded ourselves what associativity and commutativity are. Let’s run through a few questions together.

Fill in the missing number: thirty-nine point six times twelve point eight equals something times thirty-nine point six. Well we’ve got thirty-nine point six times something on both sides of our equation. In the first case, it’s thirty-nine point six times twelve point eight. And in the second case, it’s something times thirty-nine point six. And multiplication is commutative, so 𝑎 times 𝑏 is equal to 𝑏 times 𝑎. And this means that thirty-nine point six times twelve point eight must be the same as twelve point eight times thirty-nine point six. So our answer is twelve point eight.

Fill in the missing number: four and a half plus three and two-ninths is equal to three and two-ninths plus something. Well either side of the equation is an addition, and we’ve got three and two-ninths in both cases. So we’ve got four and a half plus three and two-ninths. And we’ve got three and two-ninths plus something. Now addition is commutative, so 𝑎 plus 𝑏 is equal to 𝑏 plus 𝑎. So that means that four and a half plus three and two-ninths must be the same as three and two-ninths plus four and a half.

Fill in the missing blanks: six times four is equal to something times six and that is equal to something times five in parentheses plus four. Well looking at the first part of the equation first, six times four is equal to something times six. We’ve got six in both of those. And multiplication is commutative, so six times four must be equal to four times six. So that first box has a four in it. Now looking at the first and last expressions, they must be equal to each other. So six times four is equal to all that parentheses plus four. So this expression here must be equivalent to six times four. Well we’ve got one, lots of four here and we’ve got five, lots of something here. If that was a four, we’d have five, lots of four plus one, lots of four, which should be six, lots of four. So to answer that bit of the question, we had to use our ideas of commutativity of multiplication, and also this idea that multiplication is repeated addition. So we had five times four added together and then we added another four to make six times four.

Fill in the missing blank: in parentheses, we’ve got eleven point nine plus two point seven, and to that we’re adding nine point eight. And then on the other side of the equation, we’ve got eleven point nine plus, and then in parentheses, we’ve got two point seven plus something. Now addition is associative, so 𝑎 plus 𝑏 plus 𝑐 is equal to 𝑎 plus 𝑏 plus 𝑐, and that is equal to 𝑎 plus 𝑏 plus 𝑐. Now our expressions follow this pattern here. We’ve got 𝑎 plus 𝑏 in the first case plus 𝑐, and we’ve got 𝑎 plus 𝑏 plus 𝑐 in the second case. So 𝑐 is equal to nine point eight.

In this question, we’re asked to complete the table, and we’ve got a set of properties, commutativity and associativity, and we’ve gotta say whether addition, subtraction, multiplication, and division have those properties. We’re told that addition is commutative, subtraction is not associative, multiplication is associative, and division is not commutative. Well we know that addition is commutative and associative, so we can put a tick in here. Subtraction is neither of those, so that’s a cross multiplication in both, so that’s a tick. And division is neither, so that’s a cross.

Now for this question, we’ve got to find the values of 𝑥 and 𝑦. We’ve got two different expressions. We’ve got 𝑥 in the first one and 𝑦 in the second one. So let’s tackle the first one first. Well we know that multiplication is associative, and we’ve got this pattern here. We’ve got 𝑎 times 𝑏 in parentheses times 𝑐 is equal to 𝑎 times 𝑏 times 𝑐 in parentheses. So putting those letters against that pattern, we can see that 𝑥 is the equivalent of 𝑏 and seven point one is the equivalent of 𝑏, so our answer is 𝑥 is equal to seven point one. And in the second part, we’ve got this similar idea, so — but this time it’s addition which is associative. So with addition being associative, now we’ve got this and this situation, although they’re in fact the other way round in the question. So 𝑎 plus 𝑏 plus 𝑐 is equal to 𝑎 plus 𝑏 plus 𝑐. So in this scenario, we can see that the value of 𝑐 is equal to 𝑦, so 𝑦 is equal to seven over two. And there we have it, 𝑦 equals seven over two. So just to summarise, addition and multiplication are associative, but subtraction and division are not associative. Multiplication and addition are commutative, but subtraction and division aren’t commutative. And you can use all of these properties in doing mental math or evaluating the value of expressions more easily.