### 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 addition,
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.