Video Transcript
In this video, we’re gonna be
learning about the distributive property of multiplication, which is sometimes known
as the distributive law. And we’ll be seeing a few examples
of it in action.
First, let’s think about
multiplication as repeated addition and visualize this as rows of objects. Here are four candles. And that’s one row of four
candles. So that’s one times four. Now if I add another row of four
candles, that’s two rows of four candles. So that’s two times four
candles. And if I add a third row, I’ve now
got three times four candles. Now if I add two candles to each
row, I’ve got three lots of four plus two candles. And I could write this as three
times four plus two or, as we would normally write it, three right up against the
parentheses four plus two. But another way of looking at this
would be to think I’ve got three times four candles and three times two candles. So I’m distributing the three into
the parentheses: three times four and three times two.
So the distributive property means
that when you see expressions like this, you can write them like this: three up
against the parentheses four plus two means three times four plus three times
two. Now obviously, we could interpret
this as four plus two is six. So this is three times six. And we’ve got three times six. We get the same answer at the end
of the day, 18 candles. But that’s not the distributive
property. The distributive property is all
about saying that three on the outside is distributed to each term inside the
parentheses.
So the distributive property of
multiplication over addition is that when I’ve got 𝑎 times 𝑏 plus 𝑐, it means I
do 𝑎 times 𝑏, and then I add 𝑎 times 𝑐. And the distributive property of
multiplication over subtraction says that if I’ve got 𝑎 outside the parentheses and
𝑏 minus 𝑐 inside, that equates to 𝑎 times 𝑏 minus 𝑎 times 𝑐. So the distributive property or the
distributive law of multiplication works over addition and subtraction. Okay, let’s see a few examples with
numbers in them and see how they work.
Rewrite the expression five times
nine plus seven using the distributive property.
Well, five times nine plus seven
means five times nine plus five times seven. Well, in fact, that’s it! Mission accomplished. We have just rewritten it using the
distributive property. The question didn’t ask us to
evaluate it or simplify it. It just said rewrite it using the
distributive property. And an alternative way of
expressing that answer is to leave the nine and seven in their own parentheses like
this.
And here’s another question:
Rewrite the expression three times eight minus two using the distributive
property.
So three times eight minus two
means three times eight minus three times two. So we can write that three times
eight minus three times two, and that would be our answer. Again, we’ve just rewritten the
expression using the distributive property. We haven’t evaluated or simplified
it. And again, we could leave the eight
and the two in their individual parentheses. So we’ve got three times eight
minus three times two written that way.
Now, rewrite the expression 𝑥
times seven minus four using the distributive property.
And although this has got a letter
in it, 𝑥 — so it involves a bit of algebra — the principle is just the same as it
was before. It means 𝑥 times seven minus 𝑥
times four. So that’s 𝑥 times seven minus 𝑥
times four. But with algebra, we tend to write
the number first and the letter second, which would give us seven 𝑥 minus four
𝑥.
Now, rewrite the expression 11
times 𝑥 minus five using the distributive property.
This time, the letter is inside the
parentheses, but the process is just the same: 11 times 𝑥 minus 11 times five. And again just tidying up the
algebraic expression, that leaves us with 11𝑥 minus 11 times five. We could go on to simplify that to
11𝑥 minus 55. But remember the question didn’t
ask us to fully simplify. It just asked us to rewrite the
expression. So that’s enough.
And now, we’ve got a sort of
backwards distributive property question: Rewrite the expression seven times 15 plus
seven times 11 using the distributive property.
So we’ve got a common factor of
seven. It’s seven times 15. And then, we’re adding seven times
11. So we can write that as seven times
15 add seven times 11. It’s as simple as that. That’s our answer.
Now in this question, we’re asked
to evaluate the expression six times nine minus seven using the distributive
property.
So distributing the six across the
parentheses is gonna be six times nine minus six times seven. And six times nine is 54 and six
times seven is 42. So we’ve got 54 minus 42, which is
12. Now an alternative way of
evaluating that would have been to evaluate the contents of the parentheses
first. So nine minus seven is two. So that becomes six times two,
which again is 12. Now that was quicker and it did
give us the right answer. However, the question told us to
use the distributive property. So we had to do that to get full
marks.
So just to summarize what we’ve
learned then, the distributive property or the distributive law of multiplication
works over addition and subtraction. And its general form, it’s 𝑎 times
𝑏 plus 𝑐 means 𝑎 times 𝑏 plus 𝑎 times 𝑐. And 𝑎 times 𝑏 minus 𝑐 means 𝑎
times 𝑏 minus 𝑎 times 𝑐. And these are gonna obviously be
written as 𝑎𝑏 plus 𝑎𝑐 and 𝑎𝑏 minus 𝑎𝑐. Just one quick warning before we
go. Be aware of signs.
If we see something like negative
three times two plus five, it’s negative three times two add negative three times
five. And negative three times two is
negative six and negative three times five is negative 15. So we’re adding negative 15, which
means we’re actually taking away 15. And six times negative seven minus
two, it’s negative seven here. So this is six times negative seven
take away six times two. And six times negative seven is
negative 42. And we’re taking away six times
12. So we’re taking away 12.
And let’s look at this extreme
example: negative five times negative two take away negative four. So that’s negative five times
negative two take away negative five times negative four. In fact, I’d probably put a little
pair of parentheses around the negative five as well just so we know that that
negative sign is associated with the five. And negative five times negative
two, negative times a negative makes positive, so that’s positive 10. And negative five times negative
four, well negative times negative makes positive, and five times four is 20. So that’s positive 20. And we’re taking away that positive
20. So that means 10 take away 20.
