Lesson video

Hello everyone, my name's Mr. Ward, and thank you for choosing to join me today on Oak National Academy, as we wrap up the unit on multiplication and division.

Today we're going to be reviewing some of the main concepts delivered across the unit of work.

Now wherever you are in the country, I hope that you are well and just as excited as I am about the session that's about to take place.

I ask that you're free of distraction, that you can find a quiet spot to work, and that you're able to focus on the session that's about to take place.

If you are ready to begin, so am I, so let's make a start.

See you in a few moments.

For one final time this unit, I'm going to introduce the mathematical joke of the day to get kickstart the lesson and put a smile on your face.

I hope you enjoy this one, I think this is the best yet.

I've not said that before, I'm sure.

If I had 12 apples in one hand, and 12 oranges in the other hand, what would I have? Enormous hands, of course.

If you think you can do better than my efforts at humour, I will be sharing information at the end of the video on how your parents and carers can share your work or mathematical jokes with us here at Oak National Academy.

Please keep watching.

On your screen, you'll see an overview for today's lesson.

It looks slightly different to the previous lessons you may have watched on Oak National Academy.

Today is a consolidation, or a review of lessons, which means we are returning to some of the concepts we've already taught.

And if any of these are familiar to you, or unfamiliar to you rather, and you not unsure about what they mean, I highly recommend you go back and watch some of the previous lessons in the unit, in which we delivered the learning in more detail to explain what each concept means.

You'll also notice, there's no independent task section.

Instead, I'm going to be introducing a series of smaller problems and questions for you to solve, as we go through each section.

Feel free to pause the video at any stage, to give yourself more time to complete each question or task that you face.

To maximise your learning, you're going to need to have the correct equipment.

Please find yourself a pencil or a pen, or something to record your work with.

A ruler, some paper, or notebook, your books that's been provided to you by your school.

Now the paper doesn't have to be grid paper, that we normally use in math seat.

It can be lined paper, plain paper.

It can be the back of cereal box.

Anything that you need to jot down ideas, in the house.

A rubber, as you see on the screen, is optional.

In fact in Mr. Ward's lessons, I try to encourage people not to rub out, but instead to draw a nice, neat line through the work, to show that they made a mistake, they identified their mistake, and that they have learned from their misconception through the learning delivered.

Without further ado, let's make a start, by returning to the factor and multiples section of our learning in the unit.

Take a moment and look at the two terms multiple and factor.

Can you remember how we defined the terms multiple and factor.

And could you give an example for both? Well, we could say that a multiple is a result of multiplying a number by an integer.

A factor is a whole number, that when multiplied by another factor or factors, makes a given number.

I'm using eight as an example, to identify both.

As a multiple of eight is a multiple of four, two, and one.

Because I can multiply four by two.

I can multiply two by four.

I can multiply one by eight.

Eight can also be a factor of other numbers.

Eight could be a factor of 80, when multiplied by ten.

Eight is a factor of 16, when doubled or multiplied by two.

And eight is a factor of 24, when multiplied by three.

Now, this is just a small selection of possible numbers that eight could be a factor of.

You may think of a lot more examples.

With that in mind then, here's your first mini task of the day.

You'll need to pause the video, after I've finished giving you the instructions.

You'll see 200 squares on your page.

First, I'd like you to circle all the multiples of 13 that you see, in the first 100 square.

And on the second 100 square, I'd like you circle the factors of 60.

All of the factors of 60s that you can identify.

Pause video now, for as long as you need for the task, and then press play when you are ready to check your answers.

All right, briefly If you identified all of the multiples of 13 and the factors of 60 correctly.

Just double check on the answers on the screen.

You'll notice, there are seven multiples of 13.

You'll also notice there are a number of factors of 60 between one and 10, and then they start to become a bit more sparse.

That's because they become factor pairs.

So two lots of 15, for instance makes, two lots of 15 sets, so four lots of 15 makes 60.

Two lots of 30 makes 60.

Three lots of 20 makes 60.

Five lots of 12, six lots of 10.

They act as pairs.

And of course, the primary pair is one and 60.

How did you do? Did you understand the terms multiples and factors? Hopefully, they are familiar with you, and you're quite comfortable with them.

They are actually one of the fundamental concepts we need for multiplication and division.

If we know our multiples and we can identify factors, it makes our mental strategies a lot more efficient.

Look at the pict- your screen.

You'll see a very cute factor bug.

But what is special about this factor bug? Hmm, take a moment to remember, or to think.

What's special about this factor bug? You have noticed the product can be created by a number multiplying by itself.

You have a pair, one, 25, and then one that stood on its own, five.

That means it's a square number.

Because five multiplied by five makes 25.

So when that product can be created by a number multiplied by itself, we call it a square number.

So let me ask you this mini task, this mini question.

Using your knowledge of square numbers, what would the sixth product in this sequence of squared numbers be? What would the sixth product? Pause the video, press play when you've found your answer.

Hopefully, you've identified that the sixth number in the sequence would be seven squared.

Seven, lots of seven.

Which are represented with an array, as well, to show what it looks like.

Seven lots of seven is 49.

You'll notice, when we show a squared number, we use the whole number seven, but show a little two.

And that means seven lots of seven.

Seven times by itself.

It's a squared number, it's a symbol which you'll see regularly in your mathematical learning.

Look at this hyper-factor bug, on your screen.

It's actually called a factor slug.

But what's special about this factor slug? Hopefully, you've identified that it only has one and 13 as factors.

So we've got one pair of factors.

That would be, because this is what we call a prime number.

It shows a whole number, that has exactly two factors only.

One and itself.

When this happens, it's called a prime number.

Here's another mini task for you to pause for a few moments, and complete this task.

Look at the 100 square with four numbers that have been circled.

What number has been incorrectly circled and identified as a prime number? When you're ready to share your answer, resume the video.

Good luck! Now hopefully, you've identified the incorrect circled prime number was 63.

This was because 63 has more than two factors.

One and 63 are factors of 63, however, we can also have three and 21.

We can also have nine and seven.

So there are six factors of 63.

This is why it is not a prime number.

Well, we're going to move on to the next section, which is multiplying, dividing by ten, 100, and 1000.

It is something we've discussed on multiple occasions throughout different lessons across the unit.

Take a minute to read the statement on the screen.

If you're with somebody or working with a pair or a group, then have a discussion about the statements.

And do you agree? What do you think of the statements that you can read? Let's put those statements into action.

When you multiply by ten, you add zero.

When you divide by ten, you take away zero.

Seven lots of ten equals seven.

And 60 divided by ten equals 60.

That's correct, is it not? No, of course it's not.

There's something wrong here.

But I'm doing exactly what it says.

When you multiply by ten, you add zero.

Well I've added zero to seven, and I've got seven.

When you divide by ten, you take away zero.

Well, I've got 60 divided by ten, so when am I, the whole is 60.

I've taken away zero, so of course it's going to remain 60.

That is incorrect.

What we've learned from this, is that we have to be very specific when we're given mathematical instructions.

Zero is nothing of value.

So I can't add zero to a number.

I have to be specific about the math that's taking place.

Let's be more specific here.

Instead, we can write these.

When you multiply by ten, the second part becomes ten times greater.

Now that works.

Seven lots of ten equals 70.

I've multiplied by ten, so therefore my part seven, is ten times greater, becomes 70.

Let's look at division.

The statement now reads, when you divide by ten, each part becomes ten times smaller.

Sixty, ten times smaller than 60 is six.

And ten, ten times smaller than ten is one.

So it would become 60 divided by ten become six, divided by one equals six.

Now that is accurate.

So it's about being very specific, when you're explaining what's happening in the instructions, when you give about your mathematical concepts.

With that information in mind, can you work out which of these following statements are false? Now we've just demonstrated what happens when you divide and multiply by ten.

Ten times greater.

So when you multiply and divide by 100, it becomes 100 times greater, or 100 times smaller.

And when you multiply and divide by 1,000, it becomes 1,000 times greater, or 1,000 smaller.

There are eight statements on your screen.

Pause the video, try to identify which of the statements are incorrect.

Very good, let's just check our answers.

Sixty divided by ten is six.

That is true, that's a true number statement.

Seventy-two times 100 is not 720.

In fact, 72 would become 100 times greater, which would make 7,200.

6,320 divided by 100 does result in 63.

2.

412 times 1,000 does result in 412,000.

Twenty-one times 1,000 is not 2,100, because 21 would need to be 1,000 greater, so that would make 21,000.

And the last false statement is 192 times 100 equals 192,000.

That is incorrect.

Because, you are multiplying by 100.

So therefor 192 needs to become 100 greater.

It would become 19,200.

We're now going to move on to the area of mental strategies.

This is using derived facts that we know, or of already given representations to help with our informal methods of calculations.

When learning some of the lessons, we've talked about partitioning.

This is breaking up sums or amounts, into smaller parts that we can derive facts more quickly.

So for instance, my 48, I can break into 40 and into eight.

And I've represented that using some deans here.

You will see that I've got four rods of ten, which makes 40.

And I've got eight wands, which are broken into four parts of two.

This way, I can divide 40 by four, and becoming ten.

I can divide eight by four to give me two.

I can then add those two parts together, and that will make 12.

Let's try that again.

Sixty-nine, I can break into 60.

I can do that by three lots of 20.

Now I know three times two is six, therefore three times 20 is 60.

I can break three into, I can break nine, sorry, into ones and nine divided by three equals three.

So, in each row is worth 23.

Three lots of 23 equals 69.

Let's look at doubling and halving.

Multiplying by two, or dividing by two, which is usually our most efficient way of quickly multiplying.

You can see that there's a model on your screen, which we can use to help with our mental calculations.

Now, multiplying out by eight is equivalent of doubling and doubling again.

So times by two, times by two, times by two.

The example you can see.

Now it's a bar more that's been adapted to suit this task.

You may also recognise it as looking little like a fraction, or which we use to demonstrate equivalent fractions.

So three, doubled is six, doubled is twelve, doubled is 24.

Three times two, three times, is the same as three lots of eight.

Try to use the model in front of you, to help solve these four calculations.

Pause the video as needed.

Take as long as you need.

If you have strong multiplication, you may ought to be able to multiply by eight and you won't need to double and double and double again.

It's a very good way of doing this mentally and efficiently, without making mistakes.

As multiplying by two is often our quickest way of working out a multiplication.

See you in a few moments.

Very quickly then, the only one exists within the eight times table is seven lots of eight, 56.

The others, you have to use derived facts.

So for instance, 16 lots eight, you need to know ten lots of eight and six lots of eight.

Twenty-two lots of eight, you need to know ten lots of eight, because you're using double that to make 20 lots of eight and two lots of eight and add them together.

Using the model may well help, because as you can see, multiplying by eight is the equivalent of multiplying by two three times, or doubling three times.

We can take that model and we can invert it, and go the other way, because of course, multiplication, division are inverse operations.

So, the ha- the inverse, the opposite of multiplication is dividing.

So I'm going to take that model, flip it around.

I'm now going to divide, or half.

So to half, you divide by two.

And you will see that dividing by eight is the equivalent of dividing by two three times.

Or rather, halving three times.

So take 48, as a whole.

If I divide it by two and halve it, I get 24.

I halve it again, I get 12.

And I halve it again, I get six.

I've halved 48 three times, in succession.

And that is the equivalent of 48 divided by eight.

So once again, pause the video and complete these tasks.

There are four calculations on your screen.

Can you use the model or your knowledge of derived facts, to divide the four numbers by eight? Resume the video when you want to check your answers.

There we are.

Again, using derived facts helps, especially with our mental arithmetic.

So knowing that, for instance, ten lots of eight is 80.

And that 20 lots of eight is going to be 160.

Will help you do 192 for instance.

Because you know that 160 plus 32 makes 192.

So if you know that eight lots of four makes 32, and 20 lots of eight makes 160, that allows to do things mentally.

But again, I've demonstrated this as another example of a strategy you could use to help with long, larger calculations.

There are other models that you may do that are very similar.

For instance, divided by nine, is the same as dividing by three and then divided by three, and then divided by three.

Multiplying by nine is the same as multiplying by three, three times.

We've introduced the category of Distributive Law during the unit.

The Distributive Law is a great tool to be able to break up longer multiplication calculations, to use our derived facts that we know.

And it really aids our mental efficiency.

Take a relatively straight-forward example.

As an example, 14 lots of eight.

Now I know, ten lots of eight, so I can partition 14 into ten and four ones.

I can then multiply ten by eight to create 80.

And I'm going to add that to the multiplication of four lots of eight, which is 32.

Eight plus 32 is 112.

Therefore, the answer to 14 lots of eight is 112.

We can break it into more than two parts.

Take this for example.

I'm going to partition 27 into 20 and seven.

But I'm also going to partition 12 into ten and two, just in case you're not overly familiar with your 12 times tables.

This time, I'm going to multiply 20 by ten, and 20 by two, which gives me 200 and 40.

And I'm going to add that to seven lots of ten and seven lots of two, which is 70 and 14.

If I add those four amounts together, I will get the total of 324, which is the same as 27 times 12, which is as you can see in your abstract calculation.

Distributive Law.

Using that knowledge of Distributive Law, which again, allows you to use derived facts that you know, which of these statements are actually inaccurate? Which of the following number sentences have not used Distributive Law correctly? Pause the video.

This will need to be read very carefully.

So you may need to do some jotting as you go, to check your answers.

When you're happy with what you've done, and you want to check your final answers, resume the video.

Speak to you in a few minutes.

As you will see, that out of the four number sentences, only two of them were correct.

Number one is incorrect.

Thirty lots of ten, should be added to two lots of ten.

Instead, two has been multiplied by 17 incorrectly, and then multiplied by seven.

The number four, we can see, that two again, has been multiplied incorrectly.

Two should be multiplied by seven, and then multiplied by ten.

Or, two should be multiplied by 17 alone.

It's being duplicated and therefore, it's incorrect.

It will give us the incorrect product.

Number two and number three are correct examples of how Distributive Law can be used to break up a multiplication into sections that we are more familiar with, using derived facts.

Our final session, that should conclude today's lesson, is locking and returning to formal written methods.

Across the unit, we looked at formal long and short multiplication and we introduced the algorithm of short division with some remainders.

Whilst demonstrating how to record the formal method of multiplication and division, we've also demonstrated the math taking place, by using pictorial and concrete representations.

An area model is one such example.

You will see on the right hand side, how we would record formally, the algorithm of short multiplication.

The area model demonstrates what's taking place, and allows it to partition.

It shows, when we multiplied six by four ones, it created 24.

Then we multiplied the tens by six to create 60.

We can then add those two parts together to create the product of 84.

You'll see in the written algorithm, that I've shown the 24s being four ones.

The two remaining tens being regrouped into the tens column.

Here, we showed using place value counters.

And we went stage by stage.

This is how it would look in the end of our sum.

You'll see that we multiplied two ones by four.

That created eight ones.

We then multiplied six tens, 60, six tens, by four to create 24 tens.

I wrote four in the tens column and I re-grouped the remaining two tens into the hundreds column.

I then multiplied 400 or four hundreds by four to create 1,600 and then add to re-grouped hundreds, to create 1,800s.

I've write the eight in the hundreds column and I replace the ten tens as 1,000 into the thousand column.

As you can see, the place value counters represent the final product, 1,848.

When multiplying by multiples of ten, we introduced three strategies, with this being the most popular.

We can demonstrate multiplying by multiple of ten by using our knowledge of short multiplication, and our knowledge of multiplying by ten.

So the first thing we do when we multiply 425 by 30, for instance, is we're going to multiply by ten.

You know, when I multiply a number by ten, the answer will be ten times greater than the number being multiplied.

So 425 multiplied by ten creates the product of 4,250.

I then can then take that product 4,250 and multiply by three, using short multiplication.

Three lots of zero units creates zero units.

And I have to write the zero to demonstrate the place value holder.

Three lots of five tens, creates 15 tens, I write five in the tens column, and ten tens remain, gets replaced and regrouped into the hundreds column, as an additional 100.

Three lots of 200 creates 600 plus the re-grouped 100 to create 700.

I write the seven in my algorithm.

Three lots of 4,000 creates 12,000.

I write two in the thousands column and I re-group the 10,000 as a 10,000.

I've demonstrated a private placing with a 10,000 place value counter.

And in my written algorithm, I've written one in the ten-thousands column.

So the final answer, for 425 times 130 is 12,750.

I remind that I wrote, multiplied by ten and then I multiplied that product by three.

And we use those two concepts, long multiplication and short multiplication, to help with our long multiplication.

Because, we can multiply by the unit, and in the ones, and we can also multiply by the tens, and multiples of tens.

We use area models to demonstrate the different stages that take place within the algorithm of long multiplication.

Essentially, we're going to multiply 34 by three, and then we're going to multiply 34 by 20.

So first, three lots of four ones, makes 12 ones.

In my area model, I've wrote the 12 in the section, but in my formal algorithm, I write two and I re-group the ten ones as a ten in the column.

Three lots of three tens is nine tens.

I then add the ten I re-grouped to create ten tens.

I put zero, and I replaced ten tens, as 100 in the hundred column.

I'll write that down.

Then we move onto the second row, which is showing multiplication by 20.

Twenty lots of four ones, create 80.

I can put the zero in the ones and the eight in the tens, to show eight tens.

Twenty lots of three tens, or 20 times 30 is 600.

And you can see I've written the six in the hundreds column, and I've demonstrated it in my area model.

I then have to add the two columns together, product and two columns, to create the overall product the 34 multiplied by 23.

When I add them together, I get 782.

We'll demonstrate it one more time.

We can essentially be multiplying 28 by four and then we're going to multiply 28 by 20.

And we're going to demonstrate that across two columns in our written algorithm.

My area model is here to demonstrate the different parts that are taking place.

And also to help partition if we want to do this informally and quickly, using mentally effective mental efficient strategies.

So four lots of eight ones create 32.

I've write the two in my ones column, and I re-group three tens in the tens column.

I then do four lots of two tens, which creates 80.

I've got eight tens.

By then, I'm going to add the three tens, that I have re-grouped to create 11 tens, which I write down.

Essentially I write one to represent the one ten, and then ten tens are written as one to represent 100.

I'm going to move into the multiplying by 20 now.

Now I can either write zero straight away to demonstrate the fact that I'm multiplying by ten or I can just see it as 20 and multiply it by 20 lots of eight ones, which creates 160 ones.

I write my zero and I write my six in the tens column.

I've got ten tens, which need to be re-grouped as 100.

So, I put that as a little one in my formal, written algorithm.

I then multiply two tens, or 20 by 20, which creates 400.

As you can see, on the area model.

The 400 I add, with the re-grouped 100 to create 500.

And I write down in my formal algorithm.

I then add the total from both rows to create the final product.

The final product is 672, 28 multiplied by 24 equals 672.

The final formal method that we learned about and introduced in the unit of multiply and division, was short division.

And we introduced that the category may be unfamiliar to a lot of people, and that's okay.

But I would like you to become more familiar with you.

So, let me show you again.

The dividend is the amount that is to be divided.

A divisor is a number in which we would divide the dividend.

And the quotient would be the result of the dividend being divided by the divisor.

Along side our formal algorithm of short division, you will see place values counters to demonstrate.

In this one, we shared the dividend of 8,572 into four equal parts, or four equal groups.

There are four parts, all worth 2,143.

How can, can I share 8,000 out amongst four groups? Yes I can, I can share 2,000 to each group.

Can I share 500 out amongst four parts or four groups? I can share 100 in each group, and I had 100 that remained.

And that had to be regrouped into the tens column, to create 17 tens.

Can I share 17 tens amongst four groups.

Yes I can, its group, or each part received four tens.

And I had one ten remaining, which was regrouped into the ones column.

This left 12 ones that had to be shared amongst four groups.

Each group would receive three.

Now as you can see, there are four equal groups, or four equal parts, each worth 2,143.

The short division I've calculated here, shows how we could identify how many groups of six.

So this was division as a form of grouping.

We needed to find out how many groups of six existed within 1,854, the dividend.

The divisor is six.

And the quotient ended up being 309.

Let me take you through how I got to that answer.

How many groups of six thousand exist within- Oh sorry, let me start that again.

How many groups of 600 exist within 1,800? There were three, 300 groups of six that existed within 1,800.

How many groups of 60 existed within 50, five tens? There were zero.

So I'll write my zero as a place holder.

And I re-group the five tens into the ones column, to create 54.

How many groups of six exist within the 54 ones? Nine groups.

There were nine groups of six within 54.

My final answer to my calculation of 1,854 divided by six equals 309.

The quotient was 309.

Now that we've reviewed the three methods of long and short multiplication and short division, I'd like to have a go at the eight calculations you can see on your screen.

Try to use the written formal methods to demonstrate your understanding of those algorithms. Pause the video, spend as long as you need on this, and then once again resume the video to check your answers, when you're happy with the work that you have done.

Best of luck, speak to you in a few moments.

Right everybody! Let's just check our final answers for the final task today.

If you have got the incorrect answers, you may need to check how you made them wrong.

So go back over your algorithms, where you went wrong.

Is there any misconceptions? Perhaps you did not multiply correctly.

Perhaps you didn't add the two totals correctly, if you are doing long multiplication.

Did you not multiply by 20? Or by a multiple of 10? There may be various steps that you made a mistake at.

But hopefully, you got the answers right.

And it doesn't, it's okay to make mistakes, as long as you can identify where you went wrong.

Okay, so, feel free to have a go at these questions again to see if you get the right answer this time, now that you know what you should end up with.

Thank you for completing the task, I hope you enjoyed it and I hope you feel confident with the formal methods that we have taught across this unit.

So we're almost at the end of the unit, and the lesson, but not quite, because as always we're asking you to have a go at the quiz at the end of the note and that sort of lesson.

Read the questions very carefully.

They'll recap on the main concepts that we've reviewed today.

Best of luck and when you finish the quiz, with a smile on face, I hope, please return to the video and finish up with the final few messages of today's session.

A reminder that we would love to receive and see some of the work that's been produced across the country here at Oak National Academy.

If you'd like to share your work or mathematical jokes with us, please ask your parent or carer to share your work on Twitter, tagging @OakNational, and #LearnwithOak.

Right, everybody, that brings us not only to the end of today's lesson, but the end of the unit on multiplication and division.

And what a jam-packed journey that we've taken together.

We've covered everything from area roles to Boolean roles.

Calculating flexibly, estimation, factors, multiples, factor bugs, factor slugs, mental strategies, and formal strategies for recording our multiplication and division.

You name it, we've covered it.

It truly has been a very enjoyable unit to teach, and I thank you for keeping me company and helping on this mathematical journey.

I look forward to seeing you again, in other units of work in Oak National Academy.

But for now, from me, Mr. Ward, thank you for your hard work and I hope to see you again soon.

Have a great rest of the day, so from me, Mr. Ward, bye for now.