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Multiplying negative numbers

Multiplying by a negative numbers has applications when students study transformations, use vectors and perform enlargements and stretches.
MICHAEL ANDERSON: So let’s have a look at multiplying with negative numbers.
PAULA KELLY: OK, so we’re going to use this number line to help us. Let’s start off with our first sum. So our first sum is going to be 2 multiplied by 3. So we’ll start with this arrow. So from 0 in the positive direction, between 0 and 2.
MICHAEL ANDERSON: So it’s got a length of 2.
PAULA KELLY: Perfect. If we are multiplying by 3, our arrow is going to become 3 times larger.
MICHAEL ANDERSON: OK, so we’ve got three lots of 2, so that’s going to take us to positive 6.
PAULA KELLY: Yes. What we’ve done here, we’ve stretched through a scale factor of 3 and rotated by zero degrees.
MICHAEL ANDERSON: OK, so the final arrow that we’ve got is 3 times larger than the 2, and we’ve not rotated it at all still facing the direction that we started with. So 3 times 2 gives us 6.
PAULA KELLY: Absolutely, OK. So rather than doing positive 2 times positive 3, this time if we did negative 2 times 3. So this time our arrow will start from 0 but is now pointing the negative direction. So it stretches from 0 to negative 2. We’re multiplying by 3, so our arrow becomes three times longer. So now our arrow start from 0 and it stretches all the way to negative 6.
MICHAEL ANDERSON: Three times larger.
PAULA KELLY: If we’re doing negative 2 times positive 3, we’re stretching by a scale factor of 3 and we’re rotating through 0 degrees.
MICHAEL ANDERSON: OK, so negative 2 multiply by a positive 3 is going to give us negative 6.
PAULA KELLY: Perfect. OK, so let’s move on very slightly. This time, again, we’ll start with positive 2. This time though, we’re going to multiply by negative 3. So we have 2 multiplied by negative 3. So our arrows again will be stretched by a scale factor of 3. Because we’re multiplying by negative 3, our arrows are now stretched in a scale factor of 3, but also rotated through 180 degrees.
MICHAEL ANDERSON: OK, so we end up with an arrow, which has a length of 6 and starts at zero and ends up at negative 6.
PAULA KELLY: That’s perfect, yeah. So if we had 2 multiply by negative 3, our final answer is negative 6.
MICHAEL ANDERSON: So with these types of questions, we almost have to take it in two parts. We’ve got to think about the stretch, and then we’ve got to think about whether we have to rotate it.
PAULA KELLY: That’s a really clear way to think about it, yes, really good. So finally, if we had negative 2 multiplied by negative 3. So thinking about our two steps. We’re multiplying negative 2 by negative 3. So our arrow is stressed by a scale factor of 3.
MICHAEL ANDERSON: OK, so we’re going to go from an arrow that had a length of 2 to an arrow that has a length of 6.
PAULA KELLY: Perfect, think about whether we need to rotate or not. If we’re multiplying by negative, we need to rotate our arrow. So this time we’re going to have our final answer from 0 to 6. So negative 2 multiplied by negative 3 is going to give us a positive 6.
MICHAEL ANDERSON: OK, so we started off with our negative 2 and it was going to the left, which is the negative direction. By multiplying by 3, it’s got three times larger and we’re now facing the opposite direction because of that rotation by 180 degrees, so we have an answer of positive 6.
Multiplying by a negative numbers has applications when students study transformations, use vectors and perform enlargements and stretches.
Students first meet the concept of multiplication as repeated addition. \(3 \times 2\) is talked about as three lots of 2. Students will be aware that \(2 + 2 + 2\) is the same as \(3 \times 2\). Understanding of the mathematical structure may be gained using manipulatives, such as counters, as a concrete form to represent the calculation.
\(3 \times ({-2})\) can be explained using debts. For example, you owe three people two pounds each so in total you owe 6 pounds: \(3 \times ({-2}) = ({-6})\).
Students will, early in their schooling, meet the idea of commutativity, this means \(3 \times 2\) is the same as \(2 \times 3\). The idea of commutativity can be used to tackle \(({-2}) \times 3\) as this is the same as \(3 \times ({-2})\).
Many students will learn the rule: two positives make a positive, a positive and a negative make a negative, two negatives make a positive.
Inquisitive students will question why two negatives make a positive. We all know two wrongs don’t make a right…(!)


Before watching the video consider how you explain why the product of two negative numbers result in a positive number?
In this video Paula and Michael take a different approach to multiplying numbers. They consider multiplication as a combination of a stretch and a rotation on a number line.
This approach develops links to the use in other areas of mathematics and provides a logical explanation as to why the product of two negative numbers results is a positive.

Problem worksheet

Now complete question 3 from this week’s worksheet.
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Maths Subject Knowledge: Understanding Numbers

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