Lesson video

Hello, my name is Mrs Buckmire and today's lesson is on axioms and negative numbers.

So make sure you have a pen and paper, please do pause the video whenever you need to.

So if you need more time or something, just pause it, and when I ask you to as well.

And remember you can go back and rewind the video, If you don't quite understand something and want to hear it again, that can be really helpful to aid your understanding.

Okay, let's start.

So both of you try this.

This an opportunity for you to just practise your multiplication and division with negative numbers as well.

So calculate each of the following from A to F.

And my question is, can you find any calculations that will have the same value? You might even first want to have a little prediction oh, I think this one and this will be the same and this one and this one and this one and this one before actually trying to calculate them.

Pause the video and have a go.

Okay so you can see the answers then.

90, 90 negative 18, negative five, 90 and 90.

So actually A, B, E and F all have the same values, and C and D did not.

Did you predict that? Did you already guess that without actually working them out? Maybe you notice something between A and B.

Maybe you might have even noticed something between F and B.

What can you see the relationship between them? Interesting, okay, so you saw that nine times negative two If we did these first, let me do actually in green.

It's a maybe you saw that doing nine times negative two first, you get to negative 18.

which makes it the same as A.

With A and B there was a rule we're going to talk about it a moment, when you swap things around in multiplication, the order doesn't matter when it's fine.

And maybe in F did you rewrite it? Did you notice that both of them have a negative five in the multiplication part? So it's going to be negative five.

So it's going to be negative five, times negative 10 plus negative eight.

So that equals negative five times negative 18.

So actually it comes the same as B.

Well, did you spot that? We're going to do a little bit more work on this.

Okay, so what we need to do is match the word with the description and an example.

So these words are to do with the axioms of positive numbers.

And I want to see if you can manage to remember them.

So one word in the first column matches the one word in the centre which matches to a calculation at the end with an example.

Okay, so pause the video and do have a good think about this really, really carefully and decide how it matches up.

Okay, it could be useful if you write this out, if there were any way unsure So distributive law which one was that? So distributive law, this one is when actually we get the same answer.

When we multiply number by a group of numbers added together or do each multiplication separately then add them.

So you can see here, it matches to this one.

So we could do five times four plus five times two, or we could just do five an add four plus two first and get five times six.

So that's the distributive law.

Has been really useful when multiplied by the same number to actually then collect them and do five lots of four plus two.

I think there is a video on this.

It's in unit two so if you're not sure about this, some great diagrams there we can actually visualise as well.

And commutativity, what did that mean again? Okay, so that is all about the order.

So the operation can be applied to numbers in any order.

So it is true for adding.

So we see up here, so what we can see here, we've done three plus four plus five, but it can also do three plus five plus four.

What other operation is commutative? You already seen it today.

Axioms in multiplication so three times four equals to four times three.

So for that operation and the numbers can be applied in whatever order and it is the same.

So finally associativity, that is about how we group numbers.

It doesn't matter how we group the numbers like which we calculate first, we end up with the same.

So for example, three plus four can be done first and then add five, or it can be three plus after having done four plus five first.

So three plus four is seven, then plus five is 12, or four plus five is nine, three plus nine is 12.

Okay, feel free to pause the video and write these down with examples.

Okay, so the axioms for positive numbers also hold true for negative numbers too.

Let's see if we can remember the words.

What was A times B equals the B times A, what one was that? Good, it was commutative.

What about A plus B equals B plus A.

That is also commutative.

What about A times B plus A times C equals A times brackets B plus C? Yes, it's the distributive and the last one, A times B done first then time C, or A times B times C first that one's associative.

Well done, so what I want you to do is I want you to generate, let's go for two examples for each one that holds true or using negative numbers as well.

Okay, so have a little go.

So like you might, for example, the first one, try out with negative two times four, that equals the negative 8 and then check it out.

Is it that four times negative two equals negative eight.

Is that true? Think about it.

Yes, it is it does work.

And then maybe try, you could try the same numbers here for the commutative one, but do pause and have a little play trying to come up with a list of two for each one.

Okay, so I wanted to come up with two examples for each.

So I'm just going to go through some, so let's just do a different one here.

Let's use two negative this time.

So negative three times negative five.

Well, that's going to be positive 15 and negative five times negative three what's that? Yes it's also 15.

So it does hold true.

What about here? Let's go for negative six plus negative three.

So I'm going to subtract the additive inverse the additive inverse numbers, the number that sums to zero.

So it's negative six takeaway three, which equals negative nine.

Okay, let's swap these around, negative three plus negative six.

So that's the same as negative three takeaway six, which always equals negative nine.

Did you get the same example? Ah, genius, but let's see this.

Let's use bigger numbers.

Let's use negative three times 14 plus negative three times six.

So that equals or go check out my timetables negative 28 take away number 14.

Is that 42? I think it's negative 42 plus negative 18 which gets me to, what do you get? Negative 60.

So is that the same as negative three brackets 14 plus six makes it three, lots of 14 letter three, lots of six remember there's a fantastic video on this in unit 2 called axioms so do check that out for this distributive one.

And so negative three times 14 plus six is 20.

Yeah, it equals negative 16.

Well, what about this last one.

Amazing if you didn't just use negatives, but decimals as well.

You're way ahead.

I'm going to use smaller numbers again.

Let's use negative two times negative four times negative three.

So negative two times negative four equals eight so eight times negative three equals negative 24 a little of help here 'cause I haven't left a lot of them there.

And then we have negative two times in brackets is negative four times negative three, which is negative two times positive 12, which equals negative 24 as well.

Yay, all works.

So hopefully you found examples for them and you are convinced that the axioms hold true for negative numbers as well.

But let's start using them.

Okay, so I want you to decide if the following are true or false, so you don't have to work them out.

Think about the axioms for this.

So you got your examples that you can refer to as well and just yeah true or false.

Okay, we've done it.

Make sure you paused it.

Make sure you've decided all them true or false.

Are they correct? Are they equal on each side? Okay, I'm going to go through.

So the first one, negative eight plus 12 equals 12 plus negative 8.

Oh, so they'd been swapped.

Does it work out? Yes, that is true.

Maybe already telling me what rule it is.

What about this next one? Negative 42 times negative nine equals negative nine times negative 42.

Yes it is true.

And that's because addition and multiplication are both commutative and maybe you already told me that well done.

Okay, what about C and D? It might help to work these out if you haven't already, if you weren't sure maybe actually work these out because in the end that will tell you if they equal or not, won't they? So negative 80 divided by 10 or that's negative 8.

10 divided by negative 80, if we had 10 things divided by negative 80.

Okay, let's think of instead is what do we have to times by to get negative 80 by to get to 10.

That's how we should think about it.

So negative 80 times something equals to 10.

So if it was a scale factor, it would be negative an eighth.

Oh, that's not so 10 divided by negative 80 is negative eighth.

So no that's not equal.

That's not correct, that is false.

What about the next one? Negative 83 take away 99.

Okay, equals to 99 subtract negative 83.

Let's look at this right hand side.

So if we're subtracting negative 83, I know I can add the additive inverse.

So it would be equal to 99 plus 83.

But this one on the left hand side becoming more negative is going that way, well, actually the right hand side becoming way more positive.

So they're not correct, they're not equal.

That is false as well.

Hey, so division and subtraction are not commutative.

So you can't just swap the order for division and subtraction it won't work out.

Okay, this last one.

So I have negative three times seven, plus negative three times negative nine.

Oh, I recognise this but is it correct? So negative three, lots of seven plus negative nine.

I think this is true, should I check it? Yes check it negative 21 plus positive 27.

So that equals to six or we have negative three times seven plus negative nine is negative two.

Negative three times negative two is positive six.

Yeah, it holds true.

So this one is showing that the distributive law does work for negative and this distributivity perfectly done there.

Well done if you've got those right with true and false.

Okay, so for your independent task, the first question, which pair of calculations are additive inverses? So you can but sometimes it can be helpful.

But if you can use your mathematical sense and your unstable axioms you could use that as well to work bits of this out.

But it's fine to work to actually calculate the values.

And for two sort each of the following calculations into three groups.

Each group must have calculations that are equal.

Again, you can work them all out, but I think it might be better if you can try and think about how can you form an equivalent calculations and find out which ones are equivalent and which ones are not, okay? Do spend some time doing this task independently and just have a go.

Okay, so you could have worked these out, but I'm instead going to use my mathematical sense.

So A which pair isn't additive inverse? So I've got negative a hundred times five.

A hundred times five will be because a hundred is the additive inverse of negative a hundred.

Okay, so what about we have a hundred plus negative five that's or negative a hundred plus five.

These two are going to be additive inverse.

Well, I can quickly see this is going to be 95 and this one's going to be negative 95, okay? Or maybe you noticed that five, the additive inverse is negative five and negative a hundred the additive inverse say after me, the additive inverse, the additive inverse, good.

Anyone get tongue tied like me.

So is positive a hundred so actually because both in the additive inverse in addition, that means you're actually going to create an additive inverse so C and D are pairs.

So finally I'm expecting B and F but let's check it out.

So negative a hundred divided by negative five is positive 20.

Negative a hundred divided by five is negative 20.

Oh, so what do you notice then between a multiplication and division compared to addition? Yes in multiplication and division, you need one of the numbers to be an additive inverse again, negative 5 is additive inverse to five, but this number was the same as this one.

But in addition we need both of them to be additive inverse.

Let's just quickly try that.

If I had minus 30 plus five, how can I create an additive inverse? Good, what would it be say again? Additive inverse of negative 30 is 30.

Additive inverse of five is negative five.

So this sum would be an additive inverse of this one.

Well done, okay, so for question two, now I love this question, especially if you don't actually work out, but just use reasoning.

So I'm going to do it using reason.

So actually if you were is obviously one way you can work out the calculation, but using reasoning so let's just choose a random starting point let's use this.

So negative is six cubed is negative six times negative six times negative six.

Now from associativity, I could do negative six times negative six first.

So I could do 36 times negative six.

Oh look, it looks like H, so H and D above are the same group.

What other one could be? Now this one looks interesting down here.

So this is the distributive.

So 30 lots of negative six and six lots of negative six.

So all together we have negative six times 30 plus six.

So we have negative six times 36 which by our distributivity we know equals the H.

So H, D and I are one group.

Without me even working it out.

Okay, let's do another group.

So let's just start with A then probably more normal.

So 32 now there's a lot of sixteens I see around.

So what times six equals 32.

Good, two so it's negative five times two times 16.

Let's do this part first.

Negative five times two is negative 10 times 16.

Looks exactly the same as the one below.

So A and E are part of the group.

It's one group let's start setting up our next group, A and E.

Okay, so any others? Let's look at B next.

So we have 16 times 10.

I don't know anything else about that, but I know it's not part of that group so B is in its own separate group then.

Let's put B in its own group.

And what about C? So we can have negative eight times negative 20.

Now what I find quite interesting, is I know that's actually the same as eight times 20.

Do you agree with me? Yeah, it ends up with the same answer.

So actually these are equivalent calculations.

And then I know that I can actually do eight times two times 10, and eight times two gives 16 times 10.

So it ends up the same as B.

So this is C, okay? This I'm definitely seeing distributivity here.

So negative five times 30 plus two.

So negative five times 32.

Ah, that's the same as A.

There you go.

So that's F so finally we think G should be part of the third group as if they're all equal groups.

They might not all be equal groups let's see.

So negative 320 divided by negative two.

So that's negative 32 times 10 divided by negative two.

So I can change the all of this and I can do negative 32 divided by two times 10.

So negative 32 divided by two is negative, go on, 16.

So it's negative 16 times 10.

Wait a second I've made a mistake.

Where's my mistake? Oh, negative 30 divided by negative two.

It wasn't a positive two.

So that should be negative 32 divided by negative two is positive 16.

So it does equal the same.

I thought we've created a fourth group there.

No, it's the same as B.

So B, C and D you should be really careful there.

So what I did negative 32 times 10 equals negative 320 divided by negative two.

So to split up that negative 320, and then we know that we can swap the order around here with how we do it so negative 32 divided by negative two first and that equal to my positive 16 after I changed that.

Positive 16 and then it's still timized by 10.

I think saying softly.

I hope that is clear.

Well done if you've got those groups, H, D, I, A, E, F and B, C, G.

Okay, so I went through that so comprehensively, because it's going to support you for your explore.

So for your explore, I'm not actually expecting you to work these out, but you see how with that one, you can work them out or you can actually try and spot relationships.

I want you to try to use your mathematical sense and that relationships and axioms to figure out which of these pair up, okay? There will be one left over, can you work out? Which one is it that is the odd one out and left over and doesn't pair up to be others, okay? So pause the video and have a go.

Okay, so pairing them up.

Now 810 times negative five.

So let's just write it out for now as 81 times 10 times negative five.

Okay seems like most of these are written in terms of 81.

So now actually that then equals the 81 times negative 50.

Is that match up to any, yeah, there you go.

So these two match up.

Okay, let's do another one.

Negative 81 times negative 50.

So that's already in quite interesting ways.

Let's go down to this one.

So this clearly looks like the distributive.

When you see something times something plus something times something, and you can recognise some of the numbers, then it can help so the ones which they're both times by negative 81.

So I've negative 81 times negative five plus negative 45, negative five plus negative 45 is whoops.

Let me move this so I can see it.

Is negative 81 times negative 50, what a guess.

So that equals to this one.

Okay, so we matched up , oh, we got three left.

Okay, what can we do here? Now with this one I suspect maybe we can change the order.

So if we change this to 81 times negative 50 divided by 10, and then perhaps do this first.

So that equals to 81 times negative 50 divided by 10.

What do you get? Negative five oh, now, I know that actually A times negative B, equals negative A times B and that's actually always true.

So actually that equals to negative 81 times five.

So these two are equal to each other.

Now maybe if you want to pause the video and just have a bit of a go checking that statement out.

Because it is always true, but maybe you haven't seen it before.

So which one was the odd one out? Well, hey, it was this one, the middle one.

Well, 81 times negative 5,000.

That was meant to be highlighted.

There we go woe can highlight it.

So that was the odd one out.

I really want that even better if you did that kind of reason, I love that mathematical reason.

Actually let's not work out the actual value.

I don't even know what 81 times 50 is right now, but we can actually have a little go.

I use the axioms using that to understand it applying it to work out what we want.

and what we want was which ones matched together.

We didn't want the actual value.

Well then if you had to go, there's the answers if you want to see them more clearly.

So they're all there.

You can pause it and check if you got them right.

Really, really well done everyone if you had a go at the try this, you did the independent tasks, the true and false work as well, and had to get the explore, Then I think you should be super, super proud of yourself.

I've really enjoyed today's lesson and I hope you have as well.

I think it would be ideal.

And the best thing for you, if you did the exit quiz.

The exit quiz are just a couple of questions, which means gives you feedback as well as it tells you if you're right or wrong and really helps you to assess your understanding, but even better, it helps you to understand more.

When you test yourself, it helps you to remember it and checks if you understand it as well, okay? So do have a go at that quiz and do read the feedback and I would love it if you want to share your work.

So particularly from the day to do with today's work so different way that you might lay out your work and how you worked it out I'd love to see it.

You can ask your parent or carer to share your work on Instagram, Facebook, or Twitter, tag @OakNational and #LearnwithOak.

Have a super, super, super day, and hopefully I'll see you in another lesson, bye.