Video Transcript
In this video, we’ll learn how to
identify when multiplication is required, how to multiply directed numbers — these
are numbers that have a direction and size — and how to apply this process to
real-life situations.
Now, we already have various
strategies for multiplying integers; those are, of course, whole numbers. We can use times tables, use the
grid method, or the column method. So, for example, five times seven
means what have we got in total if we add five lots of seven. Well, we know from our times tables
that this is 35. We also say though that
multiplication is commutative; it can be done in any order. So, five times seven is the same as
seven times five. This is a really useful fact to
remember if, for example, you know your five times tables better than you do your
sevens.
But what about if I were to
calculate five times negative seven? Well, this time, the question is
saying what have we got in total if we add five lots of negative seven. Well, that’s negative seven add
negative seven add negative seven add negative seven add negative seven. Well, if we add a negative number,
we move further down the number line. So, negative seven add another
negative seven is negative 14. We add another negative seven and
move seven more down the number line. That takes us to negative 21. If we continue moving in multiples
of seven down the number line, we end up at negative 35. Okay, so, five times negative seven
is negative 35.
Similarly, if I were to calculate
negative five times seven, I would say that this is the same as seven lots of
negative five and add negative five and negative five and negative five and so
on. Once again, that gives me negative
35. So, what’s actually happened? Well, if we think about it in terms
of a number line, five times seven looks like this. We move up the line in multiples of
seven five times. Five times negative seven looks
more like this. We move down the number line five
times.
Notice how the introduction of the
negative sign changes the direction in which we move on that number line. For this reason, if we say that
when we multiply a positive number by another positive number, we get a positive
result. We can say that multiplying a
positive number by a negative number or a negative number by a positive number
changes direction and gives us a negative result. But what happens if we multiply a
negative times a negative?
This time, we want to calculate
negative five times negative seven. We know that a negative sign
changes the direction in which we move along the number line. Five times negative seven took us
in the negative direction. So, adding another negative into
our sum changes the direction again and we move in the positive direction. Negative five multiplied by
negative seven is, therefore, 35. And we see that when we multiply
two negative numbers together, we get a positive answer. So, let’s formalize this.
The product of two positive
integers or two negative integers is a what integer.
We begin by recalling what the word
“product” means. If we’re finding the product of two
numbers, we’re timesing them together. So, what is the product of two
positive integers — remember those are whole numbers — or two negative integers? Well, we already know that a
positive number times another positive number is a positive number. Similarly, if we find the product
of two negative numbers, we also get a positive number. And so, we say that the product of
two positive integers or two negative integers is a positive integer.
At this stage, it’s really
important that we’re careful with how we word these rules. We simply must not say, “a negative
and a negative make a positive.” That can cause us issues when
calculating something like negative three plus negative two. We know that’s negative five. But if we read the statement “a
negative and a negative make a positive,” we might think the answer is meant to be
positive five. Instead, we say a negative number
multiplied by a negative number gives a positive number.
The product of a negative integer
and a positive integer is a what integer.
We remember, of course, that the
product of two numbers is the result we get when we multiply those two numbers
together. Here, we’re finding the product of
a negative integer and a positive integer and that the introduction of a negative
symbol in a multiplication problem changes the direction in which we move along the
number line. Since we know that a positive
number multiplied by another positive number gives us a positive result, when we
multiply a negative by a positive number, we change the direction. Instead of moving up the number
line, we move down the number line. And we see that gives us a negative
result.
Of course, we can do this in any
order; multiplication is commutative. A positive times a negative gives
us another negative. And so, the product of a negative
integer and a positive integer is a negative integer.
Let’s now look at calculating some
answers to these kinds of problems.
Negative two times six is equal to
what.
To answer this problem, we begin by
simply calculating two times six. That’s two lots of six or six lots
of two. We know from our times tables that
two times six is equal to 12. We also know that the product of a
negative number and a positive number — that’s a negative times a positive — is a
negative number. That means negative two times six
must be negative. And so, the answer is negative
12.
Negative two times negative six is
equal to what.
To answer this question, we begin
by simply calculating two times six. And of course, two times six is
equal to 12. We also know that the product of
two negative integers — that’s the negative integer times another negative integer —
is a positive integer. That means negative two times
negative six must have a positive result; it has to be 12. Negative two times negative six is
12.
We’ll now look at finding the
product of more than two numbers.
Calculate negative three times
seven times 10.
Now, these pairs of parentheses or
brackets might make this look a little bit strange. But all this means is negative
three times seven times 10. Now, to begin with, we’re just
going to perform the multiplication of three, seven, and 10. Now, we can do this in any
order. It does make sense though to save
the 10 for last. And so, we begin by calculating
three times seven. That’s three lots of seven or seven
lots of three. And we know from our times tables
that three times seven is equal to 21.
We then calculate the product of
three times seven and 10. So, that’s now 21 times 10. And of course, when we multiply by
10, we move the digits to the left one space. So, 21 times 10 is 210. So, we’ve calculated three times
seven times 10 to be equal to 210. But we were actually calculating
negative three times seven times 10. And so, we recall our rules for
working with directed numbers. We know that a negative integer
multiplied by a positive integer gives a negative result. So, since three times seven is 21,
negative three times seven is negative 21. We then multiply negative 21 by
10. So, we’re multiplying a negative by
a positive, and that gives us another negative. We get negative 210. So, negative three times seven
times 10 is negative; it’s negative 210.
In our next example, we’ll look at
how of what we’ve learned can help us calculate factor pairs.
Find three different pairs of
integers, where each pair has a product of negative 24.
The product of two numbers is the
value we get when we multiply them together. In this question, we need integers;
those are whole numbers. So, we’re simply going to begin by
finding the factor pairs of 24. Remember, factors of a number are
numbers that divide in without leaving a remainder. We know that 24 divided by one is
24. So, one and 24 is a factor pair of
24. Similarly, two times 12 is 24. So, two and 12 is a factor
pair. Three and eight is another factor
pair, as is four and six.
But we’re trying to find a pair of
integers whose product is negative 24. And so, we recall that a negative
integer times a positive integer, or, of course, the other way round, gives a
negative result. So, to get negative 24, we could
use negative one and 24 or one and negative 24. We could use negative two and 12 or
two and negative 12. We could use negative three and
eight or three and negative eight, or, finally, negative four and six or four and
negative six. Any of these pairs will work. We have a total of eight different
pairs we could use. We’re going to use three and
negative eight, two and negative 12, and six and negative four.
Let’s consider one further
example.
𝑥 divided by negative 13 is
negative 879. Work out the value of 𝑥.
We’ve been given an equation. What this equation tells us is that
when we divide 𝑥 by negative 13, we’re left with negative 879. To find the value of 𝑥, we’re
going to perform an inverse operation. Remember, that means opposite. We’re going to do the opposite of
dividing by negative 13, which is multiplying by negative 13. Well, 𝑥 divided by negative 13
timesed by negative 13 is 𝑥. That’s why we chose to perform this
step. And we’re left with 𝑥 equals
negative 879 times negative 13.
Well, let’s begin by simply
calculating the value of 879 times 13. We could choose to use the grid
method, for example, or the column method. Let’s look at the column
method. To begin, we multiply each of the
digits of 879 by three. Nine times three is 27. So, we put a seven here and carry
the two. Seven times three is 21, and 21 add
this two is 23. Then, eight add [times] three is 24. When we add this two, we get
26. Next, we multiply each of the
digits 879 by one. However, we’re actually multiplying
by 10. So, we add a zero here. Nine times one is nine, seven times
one is seven, and eight times one is eight.
Our final step is to add these two
four-digit numbers. When we do, we get 11427. So, we’ve calculated 879 times
13. But we wanted to work out negative
879 times negative 13. And so, we recall the rules of
directed numbers. We know that the product of two
negative integers, that’s a negative integer multiplied by another negative integer,
is a positive integer. This means 𝑥, which is the product
of two negative numbers, negative 879 and negative 13, will be a positive
number. So, it’s positive 11427.
In this video, we’ve learned that
the product of two positive integers or the product of two negative integers is a
positive integer. We also learned that the product of
a negative integer and a positive integer or the other way around is a negative
integer. We saw that we can perform these
sorts of calculations by initially calculating the value of the product of two
positives and then considering the sign of our answer.
