Lesson video

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Hey, everybody! Great to see you again.

Thank you for coming on to Oak National Academy today to continue your learning with me, Mr. Ward.

We're going to be continuing the unit of multiplication division by looking specifically at doubling and halving strategies today, which we've will aid our mental arithmetic and the efficiency of our calculations.

Now it's nice and quiet where I am.

Sun's coming through the curtains.

It put me in a really nice, peaceful, good mood for my learning.

I hope that you were in a similar environment.

If not, can you actually try and be free of distractions? Have everything you need for the lesson.

Okay, I can't wait to get started and nor can you I'm sure.

So let's make a start.

There's always time for our mathematical joke of the day to get you in the mood for learning and to get me cracking up.

This one, I say this every single lesson, this one is the best yet.

I hope you agree.

I recently hired an odd jobs man, to do eight DIY jobs for me around the house.

He was rubbish.

He only did jobs one, three, five, and seven.

I am genuinely laughing.

I hope you are too.

If you are shaking your head and groaning, feel free to send in your own jokes to improve my collection.

And jokes will be shown at the end of today's video lesson.

I can't believe you're not laughing.

It's an absolute belter.

It's an absolute cracker.

Today's lesson, we're going to be looking at doubling and halving.

That's going to be the new learning that we introduced.

Then we're going to be having a go using different strategies, and you're going to talk about and discuss the ideas that you have.

I say this because if you're on your own, that's okay.

You can still have it go working, thinking through your mathematical reasoning.

Then we're going to develop our learning by deriving facts, taking, bringing in information that we already know.

And in fact, we already know to help without doubling and halving.

And then I'm going to hand over to you and you are going to have a gun independent task, and you have to decide which strategies you use to solve the equations.

And finally, the challenge and the quiz, the challenge optional but quiz is something we ask you here.

I would naturally come to me to do it at the end of every lesson.

As always I asked you to be one of, if you can have something to record your work on, a pencil is ideal and a ruler will be useful for presentation also to help with some mental risk taking counting.

You can use paper, doesn't have to be good paper, although that's ideal, it could be anything that you can jot down.

So even if you haven't got pen, paper or notebook, anything back of cereal packets, something that you can just jot down your ideas on, make some notes as we go through.

So if you haven't got any of this, you need to go and get it, pause the video, please go and get what you need.

Get yourself ready for learning, and then resume the video when you are ready to begin.

Today's focus is on doubling and halving and using mental strategies.

So the first thing I'd like to do to help you warm up is to double the numbers in the first box.

And half of them was in the second box.

While you're doing that, I'd like to consider the strategies that you're using.

Would you be able to explain exactly how you've doubled those numbers and how you've halved those numbers and can just think about the strategies you're using hope for the mental that you're using, because they're going to be important in today's lesson.

Pause the videos, but as long as you need doubling and having and then resume when you're ready to share your answers and check that you are correct.

Right.

Very briefly then.

Thank you.

Welcome back.

18 is double nine.

50 is double 25, 260 double is 520, double 18 is 36 and then halving so 24 divided by two is 12 half of 32 is 16, half of 250 is 125 and half of 4,500 is 2,250.

But just going to take some of those questions and briefly go through the quickest strategies you may have used mentally.

So you may have your ideals strategies.

Anyway, any of these strategies are familiar to you.

Fantastic.

You may use something different and that doesn't mean it's wrong if you got the right answer, but it's about trying to identify efficient methods, but also looking at different strategies so that you've got a collection of strategies that you can go to when you're doing your mathematical learning.

So nine times two, we want to double nine.

Well, we know that one away from nine is 10.

So actually because doubling 10 or multiplying by 10 is often one of the quickest ways we can use a multiplication.

It's usually the first one that people remember.

We can multiply 10 twice, make 20.

And we know that one 10 is one away from nine.

So then we can most by two tens, make 20, and then we can take away to ones from 20, 20, 18, or you can see two nines make 18, and I've represented that using a bar model there.

And I've crossed out the 10th one to represent the one from the two last one.

Okay.

There's the older way of usually doing it with your hand.

So we can split nine into two parts.

So nine we can split into five and four.

And what I would do is I split my nine into five and then I have two fives, which make 10.

And I split my nine to five and then four, I have two fours which make eight and then 10 and eight together makes 18.

And you can see that I've shown it with the fingers, but I've also written it down in a number sentence that you can see the calculator that's taking place in an abstract form, but it makes perfect sense when you do it that way.

So that's two different ways or three different ways we've looked at already.

There was a no number fact.

I know that 25 times two make 50.

I think you know that as well.

So the question is how can I use that fact to solve 260 times two? I could we do into 260.

Using that known number fact.

I would think, what would you do? Here's what I did.

So I know that 26 is 25 plus one.

So if I know that 25 times two is 50 and therefore know that one time two is two cause 26 times two will be 52.

Now in less than five, we looked at how multiplying by 10, 100 or 1000 makes the value of a number 10, a hundred or a thousand times greater.

And this is evident here.

If 26 times 10 is 260, I can, I know I can do all that to make 26 times 20 to make 520.

Now I started off by using my 25 times two, and I know that 26 times two is 52.

And then I multiply that by 10.

From that fact I can derive that 260 times two is 520.

Partitioning something we often use with our cubes or resources in the classroom, but we can do this mentally as well.

It can be happening as we're thinking in our head.

So I'm going to break 24 down.

I'm going to break into individual columns.

I'm going to break into pens, but that's 20.

Cause lots of then I'm going to break it into ones.

Four lots of ones.

So I can divide my 20 by two or split my 20 in half.

So I have a 10th and I'm going to divide my four by two to have two.

So I'm going to have a ten and a two.

So therefore collecting one half, It gives me the answer of 12, just by partitioning.

Now I've used a relatively small number that's 24, but partitioning works for hundreds and thousands and ten thousands.

It's a really good way of breaking it into the individual values of columns.

Again, let's look at how we could partition and I've got a different representation here.

I've used counters to demonstrate.

What is half of 32.

There is two ways I can actually partition here half of 32.

The first way is that I split again my number into the value of the specific columns.

So I'm going to split my 33, lots of 10 to tens.

I'm going to split my ones into the ones column.

So that's two ones and three tens.

So half of 30, I know how three tens is 30 half of 30, 15 half of two I know is one.

And then I add that 15 and one together to make 16.

My second strategy, because sometimes actually people struggle a little bit halving 30 because it ends up on an odd number.

If you're more familiar with even numbers, you might split it this way.

You might decide to split 30 to 20, two tens and the remaining 12 into 10 and two ones.

So then they're both even numbers.

It becomes easy to divide by two.

So 20 divided by two is 10 and 12 divided by two is six, if you know, half of 12.

That's where your number going to help.

So, and then again, I add them together and hopefully the answer is 16.

Now here those strategies ideal, if that number happens to be half of three, the 20 had we, you know, the value in greater by 10, you could still do the same strategy.

You could have broken it into 300 or 200 and 120 either way.

So you could just increase the value of each counter by 10.

If the app question was asking about hundreds or thousands and so on and so forth.

So two strategies there, you could have used either one in an efficient and quick manner.

How we can represent our mental strategies while using some jottings.

And yes, that is part of our mathematical work, but there's nothing wrong with showing our working and giving drawings in drawings and demonstrating our strategies.

On your screen you will see one of the most beautiful things in the whole world.

Now you might also recognise it as being looking like part of a fraction model.

And of course, that's essentially what it does.

It shows parts of holes and it gets adapted for fractions and it gets adapted for bar modelling.

When we do calculations, I say, it's beautiful because I sad as I am, I absolutely adore bar modelling.

I use it a lot in all my lessons.

I like to do my own mathematical learning.

So if you're not sure about modelling, you haven't been showing it before.

Please seek out advice from your teachers and how to use bar models, because I think they are absolutely fantastic.

Yes.

I know it makes me a little bit sad, but I love math and that's why I'm here.

And I know you love math too, so it will make your world, it will rock your world.

Should I say, introduce involved, modelling into it.

Question on the board.

What calculation are we doing when we double and double again? So take a number, double it and double it again.

What is the calculation? How could I write that down? Absolutely correct.

You're multiplying by four, two lots of two makes four, so when you double it you time's about two and then you times it by two again.

And the bar model is going to help me represent this calculation.

Nine lots of four.

So nine lots of four.

You can see the four parts of the front here.

So I can start by saying each part's worth nine ,isn't it? Four lots of nine.

So when I double nine, I get 18 times two.

And each level of care of my bar model is going to be multiply by two.

So nine multiply by two is 18.

And then I double that again and that's going to give me 36.

And the beauty of multiplication is that I can use inverse.

So if I wanted to check, I could divide it because one hole here divided by four parts would give me nine.

So I can go both ways with like a bar model.

How about now? What happens if you take that number and you double it, and then you double and then you double again? What is a calculation that we will be doing? Absolutely.

We'll be multiplying by each.

And here it still works in by multiplying by two.

So I've got nine lots of eight.

So I've got eight boxes here and each part is worth nine.

So nine, lots of eight, If nine by two.

Then double the answer by two and do the product with two.

I'm going to get to 72, nine times two is 18, 18 times two is 36 and 36 times two.

Now of course the short version and efficient version is just to multiply by eight.

But if it's a number that you find tricky, or you're not overly familiar with timetables, or it's a larger, one of the parts is large.

One of the factors is a lot larger, the nine will double a hundred.

Then you may decide to use this strategy and break it into chunks.

And you would do that by times by two, times by two, times by two times by two.

What kind of calculation would be looking at multiplication, but what calculation are we doing when we use division? So when you halve, and halve again, how would you write down that as a division, it will be divided by four of course.

You see that model would look down now, it's not in a sense cause it can go both ways.

I can read it both ways, but it looks upside down.

Cause my whole here is 36 and we divide it by four, which is one of my factors.

So first of all, to halve and halve again, I can hold it by two divided by two to my 18 and I can halve 18 again by two to get to nine.

And you can see got four parts of nine.

So that's where my inverse kicks in cause now I can say, well, I've got four parts.

Each part is worth nine, nine times four gives me the total of 36.

We can go either way on this bar model.

And you got to know what the answer is today.

So now, but just make sure we're following and we're keeping on track.

What happens now if I take a number and I halve and halve again and halve it again.

I'm going to write that as eight divide by eight because two lots of two is four and two is eight, two times two times two is eight.

I'm dividing a total by one of my factors would be eight.

And that will give me my other factor.

So 72 divided by two is 36, 36 divided by two is 18 and 18 divided by two is nine.

Of course, a straight forward wants to go 72 divided by eight.

So again, if you're familiar with timetable or your division tables, you probably could have gotten that straight away mentally.

And that's fine, but this is a way of breaking it into manageable chunks.

And again, I can go inverse.

I can multiply because I've got eight parts here and each part's worth nine.

Nine lots of eight gives me 72.

I can go the other way.

With mine we're going to have a go at ask.

You're going to use doubling and halving strategies to complete following calculations.

Now I've got on your page, the bar model to remind you that that is a good strategy.

You do your jottings in, it might be a good idea to draw some anyway way, get habit.

But even if you think you're strong with your mental strategies, by all means, introduce some other strategies that you might use to get to the answers.

We try and try and try, Oh, sorry.

Let's try and try and try and try to use different strategies because then it expands your kind of understanding and it gives you more tools in which to access the mass that you're learning.

So pause the video, have a go at those calculations for doubling and halving, using different strategies we've talked about.

But as long as you need, we're doing the video when you're ready to continue.

Welcome back, everybody.

We won't go through all of the strategies.

You may have used hopefully you use a bar model or you've used some other form of jotting to demonstrate your learning.

Obviously you're taught to ask is about explaining, explaining our work and our strategies.

As much as just getting to the answer.

You're just having your board there, the answers to check that you got the right one.

We have some of them and we doubled some of them.

And of course, when we talk to about four, we double and double again, when we're dividing by four, we would halve, and halve again.

Now that we've established the use of several strategies to help with our mental division and multiplying.

But now we're going to take that concept by introducing derived facts.

Using derived facts means information that we can take from a given fact and known facts.

So kind of building on knowledge, for instance, two, lots of five makes 10.

So therefore a derived factor, two lots of five makes 10 is that two lots of 50 make a hundred because five and 50, 50 is 10 times greater than five.

So that's a derived fact from the very basic two lots of five makes 10.

Oh, it's another bar model.

Look at me.

You can hear my heart skips a beat.

I'm very excited to show you this.

So we're going to look at how we can double numbers and then divide by 10 as a strategy for mental doubling and halving.

Doubling numbers and dividing by 10.

We take this question of 90 divided by five.

And I think actually dividing and multiply by 10 is quicker and more efficient, it is for me.

It might be different for you, but I find a lot of people find it easier and quicker to divide and multiply by 10.

So that's what I want to do.

So on one side you can see a bar model represents 90, into five parts.

And if I doubled both sides, then I'm going to have 180 divided by 10.

And that makes it so much easier for me.

So essentially I take this bar model here and I double it.

Okay.

But because I've doubled the total and I've doubled one of the factors, the other factor will remain the same.

So I worked out that one.

I did it this way.

I know that 18 and 10 make 180, so and therefore an inverse 180 divided by 10 makes 18.

So each part 18.

So then I can just re split that all again, back into it's with your part.

And I split my total over 180 into 90 again, and each part must be worth 18.

Let's look at it from another example, 180 divided by two.

Now, obviously there's a derived fact there.

If it, you may know that 18 divided by two, if you know, 18 divided by two is going to really help.

Cause then you could have just made the total greater by 10 because one of the parts is greater by 10.

As you can see, I've got a bar model here represents 180 divided by two parts.

Okay, now I can use this information to help this side.

Can't I could do this side.

Now again, you may know anything divided by two, and you may know 180 by two, but you may be more familiar for us for instance.

So I can show my bar model on this side by doubling it this time.

So 360 divided by four, 90.

So if you're not 18 divided by two is nine.

You would know 180 divided by two is 90.

So again, you could have done that without the jottings, but if you've used the bar model again, you could keep doubling or halving or adding to help with greater numbers.

So I've doubled that for both sides.

If you're familiar fours and twos, and you can see it's the same representation, just with double the amounts.

We think here again, 145 divide by five again, to remind you tens often the easiest or the quickest way to divide or multiply.

So I'm going to do that.

What I did the first time, I'm going to take my bar model and I am going to double it.

I'm going to put both parts.

So two into 140 becomes 280 and five becomes 10, and now I'm going to be able to solve that.

So I know that 10 lots of threes, 30 and I know 10 lots of 300, 20 away from that so I've got to take two away.

So I know 28 lots of 10 makes 280.

So therefore both parts or each part 10 lots of 28.

So now I'm going to halve that bar model again and half both sides under 40 and five, but the same part is still 28.

Okay.

So false when it makes it 140.

So I've shown that visually, but I've also used some of my known established derived facts about the information in front of me.

We're now going to use the strategies we've just been discussing and how to double the number, double a whole one, the total and divide by 10 to efficiently calculate these two questions.

And then if the box is a my strategy, what I would do, I just want you to spend a couple of moments completing those calculations and then sharing your strategy against mine to see if it was effective and efficient.

If you're comfortable using it.

Pause it.

You need a little bit of time, zoom when you're ready to share strategies.

So I, as I've explained, I bought both sides of that calculation.

The reason I would do that is because I feel more comfortable and quicker and efficient when I'm multiplying, divided by 10 or multiple of 10.

I'm not quite as quick with 15.

So I doubled it so that the 60 divided by 30 multiple of three, sorry, multiples or 10, but I also know that three lots of 12 makes 36 or 36 divided by three makes 12.

And that really helped with my calculation.

I also then therefore know that 360 divided by three would be under the 20.

So therefore 360 divided by 30 would be 12 and below did the same again, 25, hopefully you've known the 25 is a quarter, one fourth of a hundred.

So I multiply that by four, actually not double it, double it and double it again to make a hundred.

Cause it made it a lot easier.

And having done that are less than, and less than five on, you know, dividing by a hundred.

I know that 1,400 divided by a hundred would be a hundred times smaller.

Therefore the answer is of course going to be 14.

Now looking that factor, 25 times four is 100 and 35 times four is 140 and that helped me with my knowledge.

So I didn't have to necessarily try and work out for lots of things.

Cause I know 35 times four was 140, but then at times it by 10 and that gave me 1,400.

It's going to be over to you now to have a go the independent task.

I would like you to use mental strategies for doubling and halving to complete the calculations in front of you.

Now you may need to consider different strategies to help with your questions.

So think about what you know, think about strategy that you are more comfortable using.

Having looked at all of them.

If you want to use a bar model, for instance, you can use a bar model, do your jottings, do drawings downs and then share your answers afterwards.

Pause the video, spend as long as you need to do this task.

And then we'll feed back in a couple of minutes.

Good luck.

Okay.

Are you going to say the answers on your page in red of the brackets? We won't talk about or so it's hard for us to share the strategies, but just be mindful of whatever strategy you use and you've got to the right answer.

That's probably an effective one, but it's also about time.

if you're spending four or five, six minutes on a question, actually that probably suggests that your strategy that you use wasn't effective or efficient in this sense, because it took too long.

So it's about trying to speed up this, you know, calculation and the efficiency of our time.

So you can see there are answers 29, 87 47, 8, 6, 22.

And we'll just do an example for instance, about when we talk about derive facts for 240 or 120 divided by 15, if I double both sides, I've got 240 divided by 30 while the derive fact there is, I know that 24 divided by three or three lots of eight is 24.

So that's the derived factor.

And then I can just kind of make those two parts 10 times greater.

And that helps another example of the tricky one was 210 divided by 35 with a double, both sides 420 divided by 70.

Now we're onto something there because I know my times table.

I know the fact that six lots of seven makes 42.

I know that from a time table, I also know from a division table, is it 42 divided by seven is six.

So therefore, because both sides are great by 10 divided each the out, never find each other.

It becomes 42 divided by seven.

And therefore it gives me the answer of six.

Not quite ready to finish the lesson.

And you still got a little bit of kind of energy left, here is today's challenge.

Slide, pause the video and have it go the reasoning question.

Again it's got one of the most beautiful things in the world, a bar model.

Oh, I do love bar models.

I think you've realised now, feel free to use them.

And I'm really, I'm going to keep pushing the use of bar models in mathematical lessons.

So if you don't use them regularly, please do.

Pause the video, have it go to the challenge.

So I hope you enjoy it.

Are you going to lead today's lesson? We're almost at the end of now, more confident knowing that you've got a number of different strategies that you could actually use to aid your mental arithmetic, feel free to use the ones that you feel most comfortable and confident with, but don't be afraid to try different strategies to kind of improve your efficiency.

Now over to the quiz, as always at the end of the lesson, we would like you to have a go at the questions read very carefully, best of luck.

I know you won't need it.

And when you finish the quiz and you're happy with what you've done, please come back to finish the lesson off for the key messages at the end.

Finally, just a reminder that you can share your fabulous work with us here on Oak National Academy and your incredible jokes to help me out more than anything.

Unfortunately, that brings us to the end of our lesson.

Now I say, unfortunately, because I absolutely love using bar models.

So having the opportunity to use an adaptive bar model to demonstrate or to bring in halving, was a dream come true for me.

And I'm sure you are equally as excited when it comes to using bar modelling.

You've done a really good job that we did cover a lot of information again, and I hope you find it useful.

I just remember using different strategies, whatever works for you.

It's all about your choice and being able to explain what you demonstrate is that strategy.

Okay.

I look forward to seeing some of you again here on Oak National Academy, We've got more lessons that we've got to cover in the unit of most multiplication and division.

So please feel free to tune in and watch all of those videos.

Have a great rest of the day.

And I look forward to seeing you very soon.

My name is Mr. Ward, it's been great working with you.

See you soon.