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