Lesson video

Hello, My name is Mrs. Buckmire, and today I'll be teaching you about further multiplication.

Make sure you have a pen and paper.

Please do pause the video when I ask you to.

It's just to give you time to think about the ideas and have a go at a question.

And also pause whenever you like.

So if you need more time pause the video and even rewind it.

If I say something and you're like, "Hmm, what was that again?" Listen to it again.

It definitely helps out.

Okay, you ready? Let's go.

For your Try This I want you to use four of the five number cards to fill in the spaces.

So something times three plus something times three.

What is the greatest value you can calculate? What is the least value can calculate? What other possibilities are there? Now you might be strategic with this and think, "Oh, to get the greatest I need two." Or maybe you're just having a go, putting in some numbers and seeing what's the largest and what's the smallest value you can get.

And then if you want to extend yourself a bit, I'd love all of you guys to think about what other possibilities are there? How many different numbers can you get to? Okay, pause the video and have a go.

Okay.

What is the greatest value? Now in a different way to do this, what I got was nine.

And that's because my biggest number is two, my next biggest is one.

So actually nine is going to be our largest value.

Did any of you guys use the distributive law? Do you remember what that is? There's it written out.

Yeah? So actually because we're times-ing them both by three, another thing we can do is think about it as our number plus another number, all of that times by three.

Here we can think of this as two plus one times three to equal to nine.

And actually, yes, that's why it is the greatest 'cause two and one are the biggest two things where we can add them together and get the biggest value and times it by three.

So what was the least value? What do you think that would be then? Thinking about what I just told you.

Yes, it's going to be -2 and -1.

So -2 plus -1 equals -3, times it by three, so stretching it by a scale factor of three, and you're going to get -9.

Did you get more different answers? How many? Good work there were these different answers, let me show you.

Here we go.

So if you didn't get those, maybe you can have a little challenge and pause it and see, how could I get to those different answers? Okay, I'm using GeoGebra.

I think Mastery made this app.

And what I want you to do is predict what will happen as I decrease a.

So first, if I decrease a to one what do you think is going to happen? Okay, so at the moment a is at three and we can see there's three lots of -2 to get to -6, here.

So what about if a was one? Yeah, this arrow, this long one, will become the same length as the -2.

Let's see.

There we go.

And then now, the same length.

I don't know if you predicted that.

Okay, another challenge.

What do you think will happen when a is less than one but greater than zero? Good, so we get into our decimals here, can you see this? Where a is less than one and greater than zero? And it's between zero and the end of that -2 first arrow.

So I'm about to get to zero.

What will it be when it gets to zero? Good, zero lots of -2 is zero.

So it's just represented like this.

Okay, finally, what do you think will happen when a is less than zero? I'm going to make it even smaller, I'm going to decrease a some more.

Excellent, it's going to go in the opposite direction.

So when a is -1, or -0.

5, we get to one.

When a is -1, we get to +2.

So -1 times -2 equals two.

Okay, it's a stretch in the opposite direction.

And it's a stretch by one so it's just the same size.

So if I decrease it some more, to let's say -2, it's a stretch in the opposite direction, but now it's two times bigger.

So two times greater than the absolute value, which is two, to get to four.

So we get -2 times -2 equals four.

Well done if you had a go at those predictions.

Okay, I want you to become super, super confident with this model of multiplying numbers.

We're going to do a lot of practise of this.

So in this example here, I'll just show it to you.

We can see this is the original -16, and each of the arrows below can be connected to -16 using a scale factor, okay? So my first one, I'm going in the opposite direction so I know it's going to be a negative scale factor, and then it is one lot of -16 and a half.

So that means it's one and a half so my scale factor is 1.

5 times a negative, so it's a -1.

5.

That would be my scale factor.

So I want you to go right into this, -1.

5, 1.

5 times -16 equals 24.

So I want you to describe a similar calculation from A to F.

So think about the direction of the arrows.

Do you ? Is it the opposite direction? Is is the same direction? And then how much you have to multiply it by to get to that arrow length.

Okay, pause the video and have a go.

Okay, so I've written a little note here.

A negative scale factor can be interpreted as a stretch in the opposite direction.

So here, -16 stretch, opposite direction it was -1.

5.

So what did you get for A? Good, it's not a negative because it's the same direction.

Yes, so it's going to be +2.

So 2 times -16 is -32.

What about this next one? Good, the stretch is smaller.

So it's actually going to be 0.

5.

0.

5 times -16 to equal -8.

And what about the next one? Good, it's even smaller still.

So 0.

25 times -16 equals -4.

And this one now, D, going the opposite direction.

So it must be a negative scale factor, and actually it's become smaller so it is four.

E? Should've gotten -1.

And F, should've gotten -2 as a scale factor.

Now, can you see a relationship between any of D, E, and F and A, B, C? Yes, so you can see that the D actually is the additive inverse of C.

So actually if we added those two values together we'd get to zero.

And also can we see you've got F and A, they're also additive inverses.

Well done if you spotted that as well.

Okay, so I want you to have lots of practise on this.

So have a go at doing this one completely by yourself.

Page one is what you've done before, and then page two is a multiplication table.

So if you want to pause and do page one now.

If you'd like to do them both on the worksheet just pause, come out of this, look at the worksheet, and then come back to the video when you're completed.

Page two multiplication if you just did page one.

Make sure you show me what you notice.

Okay, so let's go through these.

I'll go slightly faster because you did do the example ones as well.

So the first one should be a scale factor of four.

It's four times bigger.

The next one is a scale factor of two, and then a scale factor of 0.

5.

And all of these scale factors are positive because the arrow is in the same direction as our negative six.

So for D, now the scale factor, what did you get? Did you get it was negative? Yes! Okay, so it is negative and D is actually shorter so it's actually going to be -0.

5.

So -0.

5 times -6 equals the 3.

And you could use the squares to help you with working out those numbers as well.

For E, you get -3.

And for F, -4.

And maybe you notice some similarities, some relationships even, between A, B, C and D, E, F, a couple of them.

Okay, I definitely want you tell me what you noticed here.

I'm going to show you the top line and the bottom line.

Check yours carefully, did you get the same? Why do you think I put these at the same time? Yes, if you look at this relationship, the fact that 3 times -3 is the same as 3 times -3 down here.

So we have a symmetry, as in this one is the same as this one.

This one's the same, it's because they're really the same multiplication.

Multiplication is commutative.

That means that the order does not matter.

Remember that word.

And so actually, it's kind of just backwards.

And also we can see this one's going up in my.

Oh, not going up.

It's going down in my three times table, and this one is going up in my three times table.

So -3 times 3 is -9, -3 times -3 is +9.

We see similar relationships here, as in the kind of reflective, as in this is -6, -4, -2, and this is -6, -4, -2 in the opposite direction.

So going from left to right rather then right to left.

And then we can see this is going up in twos and this one is going down in twos.

And finally, what you guys probably did first, was this line because it's times by one.

So you maybe thought, "Oh, I love that one, that one's easy." Going up in ones and here going down in ones.

Which makes sense 'cause these are times tables here.

And then what on zero? Excellent, all zeroes.

Now there's other things you could notice.

You could have told me, maybe, about the columns and how actually we can also see this is the -3 times table going down.

So each time it's getting bigger by three.

This is getting bigger by two, getting bigger by one, zero, and now it's going down by one, going down by two, going down by three.

So lots of different relationships there.

Oh, did you spot the square numbers? There's always, someone always spots it.

Here they are, this diagonal here.

And here is the negatives of the square numbers.

So -9 is not a square number, but it's the negative of a square number, and so that's why the whole thing has certain symmetries.

Okay, so for your Explore, this task looks very familiar, doesn't it? Okay, I told you we were going to have good practise on this.

So the example.

The scale factor here is -2.

So because it's.

If we say n to be these two squares, well it's four squares, so it's two times bigger, but then it's in the negative direction, the opposite direction even.

So the opposite direction so our scale factor is going to be negative.

So we can say it's -2 times n.

So I want you to write a similar calculation for A to F.

Now, if you're not sure, for your support, what I would recommend is you can actually let n be a number.

So if you, say, change n to 10, maybe that could help you and then you're doing similar to what we did before.

Or maybe.

Yeah, so just look back at the work.

I feel really confident that you can do this.

I know some people get a bit scared of algebra, but I always say that n can be any number.

So it's any type of stretch that we're holding and we're going to compare, a number that we can go compare, and then we're going to stretch it in different ways.

I feel super confident that you can work out the scale factors of these and write a similar calculation to what I've done.

Do have a go.

Okay then, let's go through it.

This first one, I got a scale factor of four because it's four times bigger in the same direction.

Next, I've got a scale factor of three 'cause it's three times bigger, so I'm going to say three times n.

And then for C, what did you get? It's smaller.

Good, so it's between zero and one 'cause it's also the same direction.

Yes, I got 0.

5 as well, so 0.

5 times n.

What about D? It's in the opposite direction, what does that mean? Yes, it must be a negative.

So it's going to be -1 times n.

For E? Also negative, so -1.

5 'cause it's the one and a half.

And for F? Excellent, -2.

5.

Really, really well done if you got that.

Excellent work today everybody! I hope you enjoyed the lesson.

I hope that you feel even more confident with multiplying.

I think it will be very, very, very valuable to do the exit quiz because then you get feedback on your answer and check that you really do understand the work.

Have a lovely day.

Bye.