# Lesson video

In progress...

Just a second, finished.

Well, the chapter at least.

I am hooked on this book, and I'm so sorry, I probably shouldn't have had it out already, knowing I was starting a lesson, but I needed to finish that chapter.

Thank you for giving me that time.

The book is aside now, I'm distraction free, but are you? If you are not in a quiet space, if there's noise around you, a television on, you need to make some changes please, because I need your full attention for the next 20 minutes while you focus on your math learning.

Press pause and get yourself sorted in a quiet space, then press play again so that we can start.

In this lesson, we will be using formal written methods for long multiplication.

We're going to start off with an activity where I get you looking for common factors.

Then we will look at some multiplication problems together, drawing out how we can use written multiplication to solve them, and I'll leave you with an independent task to practise those skills.

Things that you're going to need, a pen or pencil, some paper, a pad or book and a ruler.

Press pause, go and collect the items, come back and we will start.

So, starting off with a common factors activity.

Here we have five pairs of numbers, and I would like you to find the common factors across each pair.

So for example, 12 and 30, what are the common factors of those two numbers? Press pause, work through the five pairs, then come back and we'll take a look.

How did you get on? Hold up your paper, let me see your approach.

Let me see how you've been recording, fantastic, good.

And there's different ways that you've highlighted to me there, which are the common factors? Let me show you how I have approached it.

When I focus on a problem like this, I like to be systematic.

Let me show you what I mean.

Now, my screen at the moment, have you seen those extra black dots that have appeared? It looks a little bit like a dirty whiteboard that hasn't scrubbed cleanly and properly.

Bear with me, they'll make sense in a moment, so working systematically.

For the number 12, I think about one and 12 being its smallest and its greatest factor, then I start to work up.

So I think, hey, if one is the smallest factor, could two be a factor? Yes, and the paired factor would be six.

From two, I think would three work? Yeah, three and four.

And now I know there aren't any more factors to find because there aren't any more whole numbers between three and four to try out.

I'm not saying you have to use this approach when you're finding factors of a number, but I'm highlighting to you, one, how you can work systematically, and two, how it can help you to know you have found all of the factors.

So let me show you that then with 30, again, smallest and largest, and then working inwards until I'm at five and six, and I know that there aren't any more whole numbers between those two numbers to try.

So other factors, 16s factors and 34s just compare these to your own lists.

Had you found all of these factors to begin with? If you hadn't then perhaps the common factors you've highlighted might have a few missing.

20 and 50, 42 and 63 and 14 and 56.

The question was though, which of them are the common factors.

I've shown you the list of factors for 10 numbers, but I want us to draw out the ones that are common across those original five pairs.

So now for 12 and 30, notice the pinky reddy colour digits, I've highlighted the common factors, are they the same as the ones you found? Good, and just going through 16 and 34, 20 and 50, 42 and 63, 14 and 56.

So now we've highlighted those common factors across those pairs of numbers.

Really good starts everyone, let's move on.

Here's a problem to help us draw out our focus on long multiplication.

Let's have a read of it together.

Oak Travel wants to start offering customers a car and bike rental service.

They want to offer it in 28 different locations around the world.

Each location will need a minimum of 246 vehicles to meet the demand.

How many vehicles would Oak Travel need to purchase to start their car rental? I'd like you to pause and have a go at answering the three questions about this problem.

What do you know? What do you not know? And what knowledge and skills could you bring to this problem to help you solve it? Press pause, have you got some ideas? Some things you know? Some things you don't know? Often that one, what knowledge and skills do I have? That one's often trickier to answer at the moment from just reading the problem alone.

So let's focus on the pink and green, the reddy colour and green.

Let's think about what we know and what we don't know.

So we know that they want to offer this service in 28 different locations, what else do we know? Each location will need 246 vehicles.

What do we not know? Good, we don't know how many vehicles Oak will need to purchase for all of those locations, all 28 of them.

So thinking about that knowledge and skills that we can bring to this problem, it often helps to use a bar model, or a, some kind of drawn representation of the parts of the problem from the reading and the sentences that are there.

If you want to press pause and have a go at drawing a bar model independently, then play again to watch as I reveal mine, press pause now.

Let's take a look then, let's use what we know.

Let's take what we know from the problem and present it using a bar model.

So we know that each location will need 246 vehicles and how many locations will there be? 28, what do we not know? How many vehicles will be needed in total? So now looking at the bar model, is this helping you to make any connections to knowledge that you've got or skills that you have? Excellent, can you tell me then what you might do next? Good, I'm glad that I didn't have anyone say, you would use any kind of repeated addition, 246 plus 246 plus 246, 28 times, no, more efficient, 246, 28 times, we know that's multiplication and we can use multiplication to solve the problem.

So with this multiplication in mind, how could we solve it? Again, press pause and think about the strategies you could choose from.

What did you come up with? Yeah, say that one again? I was thinking similarly, calculate it mentally, it's an option, but it's not an option I'm going to be picking today.

Written multiplication, I could use an area model or make use of place value grids and counters.

So in this session, we're going to focus in on both the area model and written multiplication, we're not calculating mentally, the numbers are too large.

We will however make an estimate, looking at the numbers as they are, what changes might you make to help you estimate the size of the product? Do that now, what did you get? Yeah, so I had to look, I rounded to the next multiples of 10 in each case and 30 multiplied by 250.

Well, I know three multiplied by 25 and that will help, 750, which will help me with 30 multiplied by 250, 7,500.

So we've got our estimate ready? Let's have a look first of all at the area model.

So I'm showing you here a rectangle with a length of 246 and a width of 28.

And I'm going to multiply those two numbers, which we know is how we find the area of a shape or space by multiplying for a rectangle, length and width.

The only difference here is that I've broken, divided my large rectangle into smaller rectangles.

I breaking down the multiplication.

So I'm going to multiply each part of 246 by 20 and each part of 246 by eight.

Let's start, 200 times 20, two times two might help us, good, 4,000, and 40 times 20, again, four times two might help us, 800.

6 times 20, what will help us with six times 20? 6 times two, 120.

If six, lots of two ones or 12 ones, then six, lots of 20 of two tens, 12 tens.

And then we repeat the same in the bottom row, multiplying each part by eight, 200 multiplied by eight, two eights, 16, 1600.

Four lots of eight will help us with four tens multiplied by eight.

Tell me, good and then six, lots of eight, brilliant, what's next? We've multiplied parts of the numbers from the equation, from the expression 28 multiplied by 246, we want to complete that so it is an equation.

We've multiplied parts, we now need to total 4,800 and 120, totaled, good, and then the bottom row.

If you want to pause while you quickly work out the total at the bottom row you can, do you have a total? What is it? Good, 1,968, then if we combine those two totals, we'll have the overall product at 6,888.

So that is the area model for multiplying, here in this case, three digit number by two digit number, comparing it to our estimate 7,500, not too far off and it gives us an indication that that we're likely to be correct.

How many vehicles then will Oak Travel need to purchase to start their rental? 6,888 vehicles split across those 28 locations.

Let's now think about that same multiplication.

We've solved it with the area model, let's look now at how we can solve it, how we can represent the multiplication that's happening using place value counters.

If you would like to press pause and to have a go, you won't have place value counters at home no doubt, but you could draw them.

If you want to have a go at that independently, then come back and compare, press pause now, otherwise let's work through it.

We're going to multiply 246 by eight and by 20, starting with six multiplied by eight, eight sixes, we can represent the product of that.

48, with four tens and eight ones.

Now 40 multiplied by eight, four multiplied by eight is 32, 32 ones.

So 40 multiplied by eight 32 tens, look 320.

The last part 200 multiplied by eight, two multiply by eight is 16.

So 200 multiplied by eight, 16 hundreds, 1,600, did you see it appear? What it again, 1,600.

So we've multiplied each part of 246 by eight, now we repeat and multiply each part by 20, starting with six multiplied by 20, 6 times two will help us with six times two tens, what is it? 120, 12 tens.

Next part, 40 multiply by 20, what would that be? Four times two would help, good, eight hundreds, 80 tens, four times two is eight ones, 40 times 20 is 80 tens, 800.

And the last part, 200 times 20, what would that be? Good, 40 hundreds, four thousands.

So we can see that the result of multiplying each part of 246 by eight and by 20.

To finish up, similar to in the area model, we need some totals.

We need the totals of the ones and tens, the totals of the hundreds and the thousands.

Notice some regrouping that was happening there.

If I just bring you back to here, to first of all those ones and the tens, we've got some more tens to bring down.

From the hundreds, we've got way more than 10 hundreds there.

So we've regrouped 10 of them as a thousand and the other eight hundreds stay in the hundreds place.

Then we can combine the thousands.

We already had the solution, do you remember what it was? Well, you can see it there now, 6,888.

So we have now used place value counters in a grid.

We've used the area model, same product, of course, because it's the same multiplication.

And this is all to support you with understanding how long multiplication works.

Lots of us are able to do long multiplication without any understanding of why or how it works, but the area model and the place value counters help you to see the why, they're really important, but they are stepping stones.

The end goal is using the long multiplication when appropriate, but with understanding.

If you would like to pause and have a go at completing the long multiplication independently, press pause now, otherwise let's work through it together.

So again, multiplying 246 by eight, and then by 20, each of the parts of 246.

So eight lots of six, six lots of eights, 48.

Eight lots of four, eight lot of four tens, 320, 32 tens, plus the four tens, 36 tens, 360, 8 lots of two, 16, eight lots of two hundreds, 16 hundreds plus the three hundreds, 19 hundreds, 1,900.

So we just repeat that there, ready? So just looking again at where the digits are falling and why.

They're in those places because of the value of the two or the four as hundreds and tens when we're multiplying.

Next row, we're multiplying each part by 20, two multiplied by six, 20 multiply by six, two sixes are 12.

So 20 sixes, 120, see that? 120, watch it appear again, 120.

Two lots of four, 20 times 40, 2 fours are eight.

20 times 40, 800 plus the 100 that we regrouped, 900, then two lots of two, 20 lots of 200, what would that be? 4,000.

And just like we did now in the place value counters on the grid approach or strategy and in the area model, we need to combine the result of multiplying each part by eight, with the result of multiplying each part by 20.

And we end up with, of course that same product, because it's been the same multiplication all the way through.

Now, which of these three do you think, or feel has been most efficient? They all reached the same solution, but which of them would you choose? Where would you say your comfortability? I don't think that's a word.

Where would you say your confidence is at right now with each of those methods? Which of them in the independent task are you likely to be using? Remember, the long multiplication approach or strategy is where you're heading towards with understanding coming from the area model and the place value counters.

Here's another problem, Safari.

The complete Safari trip, which takes one day is 48.

7 kilometres in total.

The Safari tour guides complete 23 safari days every month and has the remaining days as rest days and well-deserved as well.

How far will each guide travel in a month? Press pause and have a go at answering those three questions again, come back when you're ready.

So what do you know? What do you not know? Something we know one day is 48.

7 kilometres in total or not one day, but the safari trip, which takes one day, that trip is 48.

7 kilometres in total.

The guide completes 28 of those trips, 28 days every month, and what we don't know yet is, how far the guide travels in a month.

Now, again, the knowledge and skills, the connections you might be starting to make some, but a drawn representation can really help strengthen and draw out, identify, make really clear what those connections are.

If you would like to pause and draw a bar model to represent what we know and what we don't yet know from this problem, do that now.

Let's take a look at one together.

Using what we know from the problem one day or one trip is 48.

7 kilometres, and there are 23 of those in a month, 23 trips of 48.

7 kilometres.

We don't know the total length, the total distance.

Does this help to reveal some maths, some skills that are going to help us.

Good, tell me the expression that's represented here.

48.

7 multiplied by 23, and we can complete the equation by finding the product of those two factors.

How could you approach solving this equation? Finding the missing, the unnamed product of those two factors.

Press pause and quickly note down strategies that you could use then come back.

So, calculating mentally, did anyone write that down? I would not be choosing calculate mentally to solve this problem, the numbers are too large, they're not friendly enough to work with mentally.

We could to use a written multiplication, a long multiplication because we're multiplying by a two digit number.

We could represent the multiplication using an area model or place value counters on a grid.

We can definitely estimate before we go any further.

What changes are you going to suggest making to those two factors to help you estimate the size of the product? Do that now and come back and I'll share with you the estimate I've chosen, ready to look? So I've rounded to the nearest multiple of 10 in each case, 50 lots of 20, 20 lots of 50, 1,000, 1,000 kilometres every month, 23 trips.

As a tour guide, 1000 kilometres is the estimated distance travelled.

Wow, I'm glad I'm a teacher and not a tour guide right now.

Let's have a look then at the area model for this multiplication.

Why don't you press pause and have a go at filling the area model in, then come back and we can compare the solutions, press pause now.

Should we take a look? So we're multiplying 48.

7 by 23, imagining a rectangle with a length of 48.

7, a width of 23, you're finding the area by multiplying and in this model, we're just breaking that down into smaller parts, multiplying playing each part of 48.

7 by 20, and then each part by three and combining the totals, those smaller products across the rectangle to find the overall product.

So filling in, just hold up for me your area model so I can take a look, looking good.

Your ruler is more important here, you're drawing rectangles and we multiply the area of a rectangle by multiplying length and width, so that accuracy in your shape drawing has been important.

Just to compare then, 40 by 20 and eight by 20, and then 0.

7 by 20, two lots of seven, two lots of seven tenths, 20 lots of seven tenths, 14.

And along the bottom row, you should have these products.

Next we're totaling, totaling 800, 160 and 14, and totalling that bottom row as well, which should give us a combined total.

Well, by combining those two totals, we have the product of 48.

7 multiplied by 23, 1120.

1 kilometres, even more than our estimate, but within the right size, not too far away from our estimate.

So this is looking like it is the correct product.

But wow, that's such a distance for the tour guides to be walking, 1120.

1 kilometres every month.

Next, let's have ago at representing that same multiplication, but using the formal written method.

The area model is there, and I would invite you now to pause and transfer the information from the area model into a long multiplication.

Press pause, and have a go solving the equation, the unknown product using long multiplication, how did you get on? Managed to transfer it across? Can you hold it up so I can take a look? Oooh, you know what I really like about those examples where the digits are aligned in the correct place value columns, that really helps you to keep a track of what you're multiplying and the size of each of those factors or products as you work through, looking good, well done, let's compare then, shall we? So 48.

7 multiplied by 23, multiplying each part 48.

7 by three, three seven is 21.

So three seven tenths, 2.

1, 21 tenths.

Three lots of eight, three lots of eight ones plus the two ones, 24 plus two, 26.

And then three lots of four, 12, three lots of four tens, 120 plus the two tens, 140.

Let me just take that back, watch it again, ready? Looking at where the digits are forming, the size of those digits based on, and the place that they're in is based on what we've multiplied.

It's not three times four, it's three times four tens.

So the solution is 12 tens, but of course we had some tens to add on already.

Second row, we're thinking about each part multiplied by 20, two sevens, fourteen, two seven tenths, 14 tenths, 20 lots of seven tenths, 140 tenths, 14, one 10, four ones.

Let's just go back, watch that again.

So two lots of seven, two lots of seven tenths, 20 lots of seven tenths, 140 tenths, 14, 20 lots of eight, two eights, 16, 20 lots of eight, 16 tens, 16 tens, 160 plus the one 10, 170.

Then, two lots of four, two lots of four tenths, 20 lots of four tens will be eight hundreds, plus the 100, 9 hundreds, product of at this stage 974, but not the final because of course we now need to total those two products.

The product of 48.

7 multiplied by three and the product of 48.

7 multiplied by 20, same overall product as before, 1120.

1 kilometres.

Independent task, I have some questions for you that I would like you to estimate a product of first, then use an efficient strategy to solve the calculation.

When I say efficient strategy, I mean area model, place value grids and counters or long multiplication.

If you are ready for a challenge, then you are going to use more than one strategy for each question, you should have the same product no matter which strategy you use.

So if there's a different product coming out, it tells you that there's an error to check back for.

Press pause, go and complete your activity, then come back and show me how you got on.

So, how did you do? Give me a wave if you were using an area model, place value grid or counters or long multiplication.

So you got a bit of a mixture, hold up your paper so I can take a look, really good.

Whichever method you've used there, I can see you've worked hard to be neat in demonstrating your thinking through your strategy and that's important for helping to check back independently for any mistakes along the way.

Thinking then about the solutions I'm going to show you with long multiplication for each of them, so you can see the estimate that I used rounding to the nearest multiple of 10, and then calculating.

Take a look, compare it to yours.

If you need to press pause to look for a little bit longer than please do, otherwise, I'm moving on to the next question.

Again, I've rounded this time for my estimate to the nearest multiple of 100 and the nearest multiple of 10, and then calculated as well.

So again, press pause if you need to check for any of those regrouped digits, for the size of each of the products compared to yours and the next one.

So this time I have rounded my decimal to the nearest whole and the 506 to the nearest multiple of 10, which is also the nearest multiple of 100, and I then calculated.

So once again, if you want to press pause so you can.

There's a lot of regrouping that's been happening in this calculation so that you can compare against your own, then please do.

A busy session, a focus on long multiplication, but also the area model and place value counters and grids to help us understand how that long multiplication strategy works.

Long multiplication is the outcome.

The goal that you're working towards when appropriate, based on the calculation that you're working with, but we want you to be able to do the maths with understanding.

And the area model and the place value counters can help give you that understanding.