Lesson video

Hello everybody, great to see you again.

Thank you for coming onto Oak National Academy, and keeping me company, it's very, very kind of you.

For those that have not had a lesson with me before, my name is Mr. Ward, and I am delivering the unit multiplication and division.

Today we're going to be using derived facts to multiply mentally.

Now the shining outside, but I'd rather be here learning mathematics, 'cause I absolutely love math.

And I hope you do too.

Can I ask that you are free of distraction, you've got everything you need, and you're in a quiet spot, ready to begin your learning.

When all those things have been achieved, and you're ready to start continuing the video.

Let's get started.

For those of you that have watched previous lessons from Mr. Ward on the unit of multiplication, division, you will know that I like to start every lesson, with a mathematical joke to get you smiling.

And today is an absolute belter, this is probably the best one yet.

I know I say it every time, but it is good.

I've been laughing about this all morning, so it must be pretty decent.

Here we go.

Do you know what I find odd? Every other number of course.

And for those of you that went right over your head there, I'm talking about odd and even numbers.

I know, I do apologise if you found it, if you thought that was terrible.

Hopefully you won't find the lesson terrible.

Here's our outlook for today.

We're going to start our new learning by introducing the idea of derived facts, and what we know.

Then when we'll have a go at talk task, when we can discuss and share different multiplication problems, and the strategies we might use to solve them.

Then we're going to develop our learning a bit further, by having the opportunity to look at different strategies, which one would be more suitable, and more effective for each calculation.

Then we're going to hand it over to you, you have a go at the independent task, which we assume is a multiplication calculation, which you have to represent in a series of different ways.

And then it's the end of lesson quiz.

Now hopefully, fingers crossed, you're going to leave the lesson feeling confident that you understand how to use different strategies to calculate mentally.

In order to get the most of out of our lesson, and our learning, we need to make sure we're well equipped.

So here are the following equipment for you to have, pencil, or something to write down on, a ruler, paper, or a notebook that might be provided to you by your school, it doesn't have to be grid paper, it can be line, blank, it could be the back of a cereal box, anything you can jot down ideas on, will be useful.

A rubber is optional in my lesson, because actually I prefer to see work neatly crossed out, with a line, to show that, "Yeah, I got this wrong.

"And I understand why I've got it wrong, "'cause I've understood my misconception." And that shows fantastic mathematical learning in action.

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

If you need the toilet, go now, and then come back when you're ready to begin, and resume the video.

'Cause our lesson is just about to begin.

To get ourselves warmed up for the main learning in the lesson, I have attached one slide called speedy tables.

Feel free to have a go at this challenge, or to skip on if you would like to miss it.

However, if you are watching this, you'd have to pause the video.

The challenge is to complete the speed tables, and see if you can do it within three minutes.

Don't feel like you have to time yourself, you just want to complete the square.

But the extra extension challenge is to time yourself, and see how many you can do within three minutes.

Okay, so pause the video now.

Do as many as you can, and then resume the video when you're ready to share your answers.

Best of luck.

Okay, here are quickly the answers, the complete table.

As always with times tables, I think it's really useful to start on the ones that you find the quickest.

So most people would start on the 10th, for instance, multiples of 10s.

They would probably then do multiples are two.

And because they can do multiples of, sorry and multiples of one, and because they can do multiples of 10, they can probably halve that into multiples of five.

Because they can do multiples of two, they can halve that and do multiples of one, or they can double that and do multiples of four.

And if they can do multiples of four, they might do eight.

So forth, so on, okay? So again, you didn't have to time yourself.

If you completed, that's fantastic.

If there's any errors there, just transport the times table that you need to go away and practise, and become more familiar with, and more efficient with.

Now we're going to move on to our main learning today, which will involve a lot of modelling from me.

So I'm going to demonstrate a lot of different strategies.

We will be using derived facts to multiply and divide.

And that means we're going to be using known facts, simple facts to help us solve more complicated calculations.

What are the different strategies that we could use for this question? There are eight athletes in each race.

There are 10 races, and then another four races, how many athletes competed in total? So we're going to be looking at the different strategies we may use to get the same answer.

First thing we're going to do is, we're going to look at using cuisenaire rods to represent this problem.

Now the problem says there are eight athletes in each race.

And there are 10 races.

So we're going to use these rods to represent, there's eight in total one, two, three, four, five, six, seven, eight.

So one rod represents eight athletes.

And there were 10 races, so I need 10 lots of eight.

Now I haven't got enough of the same colour to show it, so I'm going to use my cuisenaire rods with a bit of artistic licence.

And that's okay, so don't worry about the colour.

What I want you focus on, is the numbers.

So four, five lots of eight.

I've got eight athletes there, and we look really 10, don't we? So six, seven, eight, nine and ten.

Ten lots of eight.

Write 80 underneath, and then move to one side.

And then on the other side, ten lots of eight, these are the first part of the question.

The second question was, there's another four races, and we still got eight athletes.

So again, let's add two more, three more, four more.

So I've got 10, lots of eight here.

And I've got four, lots of eight here.

So I'm going to put that in here.

And I can see that actually, by using cuisenaire rods, I've broken that question into two stages in a sense.

Eight out of 10, and four lots of eight, or rather 10 lots of eight, and four lots of eight to represent them.

Altogether, I can work out the answer of being 112.

That's one way, using cuisenaire rods, I can represent this problem.

The next thing I'm going to do, is do an area model.

So I'm just going to move this out of the way.

We don't need the cuisenaire rods here.

But I am going to show it, just below, okay? And in fact, I'll give you a fresh page, a fresh page, how about that? So I'm going to show an area model.

Okay, these are great when we try to work out area anyway, that's why they called it area models.

But they're just covering the space.

So I thought , I'm just going to jot it down.

I don't need a ruler.

Okay, I'm going to do exactly what I did there my cuisenaire.

I'm going to do it in freehand jotting.

So I know that I've got two boxes, one box is 10 lots of eight, and one box was four lots of eight, okay? In a sense, it's representing, imagine these, it's kind of almost representing what they were, but not to scale, of course.

So my area models is eight lots of 10 in the middle, so I know that's 10 lots of eight, is 8, 10 times greatest, so 80.

And four lots of eight, which I know is 32.

So again, in my area model, I worked out, I can add those two together, put a little cross there to show what I'm going to do, and that gives me the answer of 112.

So that's with two styles, two different representations.

One was the cuisenaire rods to show you visually, and one was a jottings, showing you an area model, which is a really fantastic strategy, I like to use regularly.

So we've looked at two methods now, two ways of representing the problem.

Just going to show you a few more, so that you've got a variety of efficient method that you could decide to use, depending on your choice.

On your screen, you'll see that I'm representing the problem in Dienes, again.

We've used Dienes to show eight lots of 10, 'cause we know that's 80.

And we've got eight rows of four ones.

Each row is 14, 'cause I've got one, 10 block, and four ones, so each row is 14.

So overall, I've got eight lots of 14 there, when I use Dienes.

You may decide to use a number line.

Now ideally, you would like to use an empty number line, 'cause it's actually quick and effective, but you might decide to use a marked number line.

So I've got an example on your board here, with the marks in intervals.

So we know that 10 lots of eight was 80.

So I could do that quick first jump, and then I knew that four lots of eight makes 32, so I can make that next jump.

And I've just connected the 80 and 32 together, to make 112.

So number lines is another way that you can do, and later in the video, I'm going to show you how to quickly do an empty number line, if you're a bit unfamiliar with the concept.

And then we can put that together, all those representations by writing the calculation out, in an abstract number sentence.

Now, this is an example of distributive law.

Distributive law, means we can take a multiplication, and expect we can break it into parts.

So I'm breaking my 14 lots of eight, by breaking it into 10 lots of eight, and four lots of eight.

Now 10 lots of eight, we've proved in our representation makes 80, and four lots of eight makes 32.

When we add them together, it gives us 112, which is the same answer as 14 lots of eight.

So distributive law, when we distribute that sum, across different parts.

Sums it all, add up this works beautifully in multiplication, and I'm going to show you a few more examples as we go through the lesson.

So we just looked at cuisenaire rods, area models, Dienes.

I talked about empty number lines, I showed you marked number lines, and empty number lines, just without the intervals, and the numbers already marked in there, you have to put them in.

And I looked at an abstract number sentence, which showed you distributed law.

I wonder which representation would you prefer to use? Which of these are you familiar with anyway? And which one would you go to, if you have to solve this problem quickly? So, now you're going to have a go.

I want you to choose a representation, read that question.

Choose one of those representations that you would like to have a go at, and you are familiar with, or you are comfortable using, and solve this problem.

And then we'll share all the different strategies we could have used in a couple of moments.

So pause the video, read the question, and have a go doing one of the representations to solve that question.

I'll speak to you in a couple of minutes, good luck.

Right okay, so there was different representation you may have used, I wonder which one you did decide to go for.

I looked at these, and I decided that actually not every strategy is effective each time.

Sometimes it's not the right strategy to use.

So I decided that cuisenaire rods wouldn't be effective, because it will require too many of them.

It would take you a long, long time to get there, because we're starting with 240.

So I would need 24 lots of 10, for instance, and I just haven't got the equipment, and it would take far too long.

The same with Dienes, direct present 10s and ones for 240, that's without multiplying it by eight, would actually be ineffective, it will take too long.

I couldn't represent it.

So I decided those two representations was not appropriate on this occasion, for this question.

But I did decide that the others could be used.

In the athletes village, each apartment has eight athletes living in it, how many athletes are there in 240 apartments.

So there's each partner has eight, eight lots of 240.

You can see what the Dienes, and the rods wouldn't be effective, 'cause the numbers are going to be too high.

Right, so the first strategy we're going to show is the area model, you may have chosen to do this, because I said it's one of my favourites.

We know we've got 240, times by 240, times by eight.

So I'm going to partition it actually, into 240.

So the first model I'm going to do, is this one here, one block here.

I'm going to say 200 at the top, multiply by eight.

And then I'm going to do the second block, which is going to be 40 multiplied by eight.

Now I know two lots of eight 16.

And therefore 20, is 10 times greater, and 100 is 100 times greater.

So I know that 200 times eight is 1600.

And I know that four lots of eight is 32, and 40 is 10 times greater than four, therefore this answer is going to be 10 times greater, So I've got 1600, and I've got 320.

I'm going to add those together.

So my overall answer is going to be 1920.

Okay, I've got that answer by using an area model.

Now I'm going to look at empty number line, I'm just going a little bit further down, how I can show you with the jumps.

Again, my number line is not going to be marked, this time it's empty.

So I don't have to use a ruler, I'm just going to do it for the benefit of looking smart, and printing.

Nice straight number line, and I'm going to start on zero.

First thing I did was I know that eight lots of 200.

Like I said before, I know that eight lots of two is 16, and eight lots of 20, is 160.

So therefore eight lots of 200, is going to get me to 1600.

And then I know that four lots of eight is 32.

So therefore, one of the parts is 10 times greater.

Therefore my answer's is going to be 10 times greater.

So I got that, and then, that's my second jump of 320.

Sorry, I've done that wrong, my second is 320, so I add 320 to that, and that will give me 1920, okay? But you see how easily it is to make a little mistake, but I did correct myself, I realised that actually, I don't add the jump there, the jump is 320 on top of that.

So I jumped from there to there.

And that's where I got.

So that's two ways we've looked at it.

And then we can put this into an abstract number sentence.

And we can do this by doing it in, showing distributive law.

So I might decide to show you that actually, like I said before, 240 times eight, I'm going to break that into 200 times eight, and I'm going to add it into 40 lots of eight, okay? Because again, I know that two lots of eight is 16, but it's 100 times greater, and that's 100 times greater, and then four, eight is 32.

And that's 10 times greater, so 320.

So that's my abstract number line, add them together.

And I would, of course, got to the answer, which we've already established.

So that's three different ways that we've looked at there.

We did the area model, we looked at a blank, empty number line, and we looked at an abstract number sentence.

We're going to move on to a talk task now.

I'm going to ask you to jot down, and represent some of the calculations that you see, using some of the strategies we've looked at.

From the pop myself onto screen, I'm just going to make myself a little bit bigger, there we go.

Okay, so talk to us.

I'm going to model it first, and then you're going to have a go.

If you're working on your own, it's fine.

Just have a go, and think about your mathematics, right? There's some jottings.

If you're lucky enough to be working with a partner, or in a group, fantastic, make sure you're talking about the work that you'll do, as you're demonstrating, and explaining your answers.

So you need to calculate mentally using derived facts.

There are two calculations on the board there.

You need to identify the calculation, and represent with a blank number line, what derived facts that you know, helps solve the problem mentally? Well, first of all, let's think about it, and let's model it.

So I'm going to do 54 times by eight.

Okay, I'm going to break that into 50 and four.

So first of all, I'm just going to quickly do a very sketchy 50 times eight, and four lots of eight.

So collectively, I've got 54 there.

So I know five lots of eight is 40, but it's 10 times greater, 'cause one of the parts is 10 times greater, so therefore it's 400.

And four lots of eight is 32.

So I add those together, I would get myself the answer 432.

How might that look on a number line? Well, it looks a little like this, no ruler this time, that's pretty straight, not too bad actually is it? Start on zero, I took 50 jumps of eight, which was 400.

That's my first jump, which was 50 times eight.

And then I jumped 32.

So I added 32, that's four times eight.

So that's extra 32, I'm just going to put that in brackets, just to make it clear, and I ended up with 432, which was my answer.

So you can see, that's how I've represented that first one.

The second one, I'm just going to do it in a blank page, for this.

Nearly lost my , nearly fell off, I do apologies.

And we're back again.

We've got 124 times by eight.

This time, I'm going to break it into 120.

And I'm going to break it into four, okay? Now you could have broken it actually into 120 and four, which you could do.

I might show you that actually.

So here's my first line, which is 120.

I'm going to times it by eight, and then my second one is going to be four by eight.

Again, I know that 12 times eight is 96.

So therefore, if it's 10 times greater, so is the answer.

And four lots of eight is 32, add those two together, I'm going to have 979, sorry, 992, now what I may have done, I did allude to the fact that actually you may have broken up 124, although it's more parts for you to kind of mentally do.

You can do 100 into 20, into 4, you're going to partition in a sense.

So I could do it this way.

I could have done 100 lots of eight.

I could go 20 lots of eight.

And I could go, as I did, four lots of eight.

So 100 times eight is 80, 20, two is 16, plus 10 times greater, and four is 32.

And then I'll just have to add those together.

You can see why actually, it takes a bit longer, because there's more to do this way, but I could have added it that way.

Okay, so I could see that I did it that way.

That's one way of doing it, and then I take my answer, which is 992, and we just go down my number line again, not quite as straight this time.

Zero, my first jump, I do three jumps to tell you, eight times 100, which is 800.

And then I did 20 lots of eight, which was another jump of 160, so 960.

And then finally, my final jump was four lots of eight, which was 32, so 32 plus that is 992.

So that's how I get to my final answer of 992.

So having modelled that talk task, and I want you to have a go independently.

Calculate mentally using the right facts, I want to represent each problem with an area model, and then record the steps of the strategy on an empty number line.

There are three problems for you to do there, so make sure you're nice and clear in your jottings.

And if you got someone nearby, you can talk and explain your work as you go into.

Pause the video, have a go at the three questions, showing you a representation of the strategies you're using.

And then feedback for some answers in a couple of minutes.

Okay, speak to you very soon.

Right everybody, let's just share our answers.

We obviously can't see what you've done.

But hopefully you've demonstrated with an area model, or an empty number line.

The answers to the first one was 156.

Six lots of 26.

Number two, six lots of 320 was 1920.

And finally, six lots of 834, was 5004, that's now a big question now.

834 multiplied by six.

So we're going to take our learning a bit further now, by looking at how we can be flexible, and we can have a choice in the strategies that we decide to use when we're looking at multiplication.

Look at the two calculations on your screen.

When would you choose a written method of calculation? And when would you choose a mental strategy? I wonder when you would decide to use for short, or long multiplication, when might you choose to use a mental strategy for multiplication? Well first of all, let's take 230 times six.

It's quite a small, it's a small number to start with.

It's also got a multiple of 10 in, so we won't have to regroup at any stage.

So that idea that 230, we can partition that a lot easier, can't we, into 200, and into multiples of 10, which would be 30, so 230.

And that makes it a lot easier for us.

Now I created a word problem with our math stories.

So you can see that those numbers provided me with this question.

In the athletes village, each apartment has six athletes living in it.

How many athletes are there in the 230 apartment.

So I decided, I think you probably would agree, that we should probably do a mental strategy, because it's a multiple of 10, it would be a lot easier for and quicker.

So as discussed, sometimes it's not ideal to use a mental strategy, because there's too much to record, too much information, too much regroup, and whatever.

And that's when written methods of calculation are useful.

Now, there is a lesson nine of the unit, in which we look at short multiplication, but I'm just going to briefly model one here.

So I looked at this, and I thought, actually, I'm going to use written method of calculation, 'cause I think that's the quickest, and most efficient way.

So I can just do it by first, in the ones, four lots of six make 24, obviously, it can only have four ones there.

My two, I'm going to move to the next column.

That becomes 210, so this is 20, so now it's 30 tops to six, 30 lots of six is 180, plus the 20 have moved over, so that becomes 200, okay? But obviously I can't do that, so I put the place hat value, the placeholder, to show that 20, and that becomes, I move that over, went into the hundreds column now.

800 upon six, or six times 8 is 48, but six lots of 800 must be 4800, or 4800, plus the two, gives me 50 hundreds.

So therefore I know my answer is 5004.

So I did that really quickly with a written method of calculation, just to show you, 'cause you might think, "Well, I like these mental strategies." And I'm glad you like the mental strategies, 'cause they are great.

But there's so many moving parts to this.

So I'm going to break that into 800, further into 30, and break that into four.

Now, in order to do my area model.

I'll need to have 800 up here, times by six.

30 times by six, and four times by six.

So already, I've got three different blocks, I've got to work out.

Again, I can I can go with four lots of eight.

Eight lots of six is 48, but then it's 100 times greater.

So it's 4800.

Six lots of three is 18, but it's 10 times greater.

So it's 180.

And four lots of six is 24.

But then I've got to take these three, and I've got to add them together.

So I could do that mentally, but I'm going to just write it here, so I don't make any mistakes.

Four, 10, I can put one at the top or the bottom.

Eight, one, one over, I'll put it at the top to show you, 5004, okay? so yes, I've done it using, and I've got the correct answer with my mental strategy of using an area model.

But actually, it's taken me longer to do the mental strategy, that it was, than it did take me to do the written forms of calculation.

So the point being that sometimes, depending on the number, when you look at it, you might decide that actually, I just need to do it the most efficient and quickest method, which in this case was a written method of calculation, by using short multiplication.

And I did that, because I looked at all three of the columns, and they all add numbers in.

They all had digits in, no placeholder zeros.

So I realised that we're going to have to do quite a lot of regrouping, which we're going to take a little bit of time.

So therefore, that's why I made that decision.

Now that we spent a bit of time looking at different examples of strategies you might use, and modelling those methods, and representing our information in different ways.

I'm going to ask you to have a go at an independent task.

Your job is to complete the calculations, using different representations to show your strategies.

Then, you want to write a word problem, based on a math story for each calculation.

So in effect, I'd like you to do the calculation choosing one or two different representations that we looked at during the course of today's lesson.

And then once you've worked out the answer, you need to think of what the word problem would be, and link it to a math story.

Now, here are the different strategies we've looked at today.

Now, as we've talked about, some are more effective than others.

But of course, in math, you have freedom to choose the method you think is most effective.

So feel free to use any of those strategies, if you feel comfortable doing so, but try and be as efficient as you can in both your time, and what's easy for you to manage.

And then once you've done the calculation, you need to write a word problem.

Pause the video now, take as long as you need on these tasks to complete all of those calculations, and then resume the video when you're ready to check answers.

I'll give you a couple of my examples, to see if you're on the right lines.

Speak to you very soon, good luck with the task.

Welcome back everybody.

Again that's quite an open task.

So I can't give, I can't look at all the answers you may have come up with.

However, here are two examples that I did to show.

So the first one is 430 times eight.

So I actually thought of the work from, I did inverse, basically.

I did the word problem first.

So I was thinking about what sort of numbers involved in the Olympics, would have 430 and eight.

And I came up with this, the athletics lasted for eight days during the Olympics.

Back in the athletes village, all 430 athletes had an eating meal cooked for them after each day of competition.

How many meals were cooked for all the athletes in total.

And to represent that problem, I decided to use an area model, by partitioning my 430, to 400 and 30, times it both by eight, and adding them together.

But it also showed you on a number line, what it would look like on an empty number line.

First doing the 400 times eight, and then doing the 30 times eight, and combining to get an answer of 3440.

Now that is a lot of food.

My second example, would be three times 70.

Now, that's quite a complicated one isn't it, that will obviously be a lot of numbers there.

But I didn't need to use a written method for multiplication, because I know that 63 times seven, seventy is 10 times greater.

So whatever my answer for 63 times seven would be, it would be 10 times greater.

So here's my word problem.

The Olympic stadium had 63 columns of steps, which lead from the bottom to the top of the stadium.

Each column consisted of 70 steps.

How many steps did the Olympic stadium contain in total? But you can say again, and as I talked about in the video, I do really like using area models.

So I did it that way.

Now I did 60 times 70, 'cause I know that 60, I know six times seven is 42.

And I know that 60 times 70 is 420.

So therefore 60 times 70, which is another, no, 10 times greater, would be 4200.

And I also know that seven times three is 21.

Therefore 70, which is 10 times greater, times three would be 210.

And then added them two together to make 4410 steps.

That is a lot of steps, you would definitely be fit if you had to climb them every single day.

But again, I put that information on a blank number line, as an alternative representation.

'Cause again, I really like using both of these models.

I think they're really effective and quick, and you can see the jumps I made.

I could go backwards if I needed to, but actually, I wanted to start at zero, and I wanted to do it this way.

So I wanted to go 60 times 70, and then three times 70.

Alright, okay, that brings us to the end of the work for you to do today, unless of course, you are absolutely going to do the challenge slide, which I know a lot of you are, because you absolutely love your maths like I do.

And that makes me feel great as a maths teacher.

So here is your challenge slide, it's called multiplication madness.

Complete the calculations, and write a word problem for each one using mass vocabulary.

So it's set in formal, but you might decide to do informal mental strategies.

I don't mind which way you do, pause the videos, spend as long as you need on this challenge slide, and I hope you enjoy the task.

We're almost at the end of our lesson, you did a really, really good job again.

We looked at so many different strategies, and over the course of the unit, we've looked at a lot of strategies for doubling, halving, multiplying, dividing mentally, representing it in different ways, using equipment, drawing jottings, all of them.

So you've got lots of options now, when you're calculating mentally.

It's now time to see if you can put that learning into action by completing the end of lesson quiz, which is a custom here at Oak National Academy.

So go find the quiz, read the questions very carefully, see how you got on now.

If you find you need to go back on the video slides, 'cause you missed out something, or didn't quite understand, feel free to do so.

But fingers crossed, after that lesson, and all the hard work that we did together, you're feeling really confident, and you're going to walk away with a really good score in the quiz.

Just trying to remind you once again that we love to receive work here at Oak National Academy.

So if you would like to send in evidence of your work, or share with us some of your mathematical jokes, we would love to hear from you.

Please ask your parent, or carer, to share your work on Twitter, tagging at Oak National, and #LearnwithOak.

Alright everybody that brings us to the end of today's lesson.

You did a very good job again, focusing on all that information, with lots of modelling.

Now I hope you found the modelling of the strategies useful for you to be able to replicate, and do.

I should introduce my best friend in the whole world.

My mini visualizer, this is one of my favourite pieces of kit that I own.

And I hope that by modelling and visualising the strategies, you found it really useful, and you can see specifically what to do.

Now I hope to see many of you again here on Oak National Academy, as we continue our unit on multiplication and division.

But for now, I think deserve a little bit of a break, so go and stretch your legs, and do a little bit of exercise, and get some fresh air.

Thank you once again for your hard work today.

I look forward to seeing you all again soon.

So for me, Mr. Ward, bye for now.