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.