Lesson video

Hi everyone, it's Mr. Whitehead here.

I'm ready for maths and actually, I'm really ready for maths.

This lesson's focus is something that blew my mind when as a teacher, I learned about it and of course, how to teach it to children.

Still connected to division and remainders, and I hope that for you, perhaps it blows your mind as well.

So really lovely connections in this lesson to other areas of maths.

So please pause and make sure that you are in a place where you're able to focus and absorb and engage and participate.

Once you found yourself in a space for that, come back and join me and we'll kickstart the lesson.

In this lesson, we will be representing remainders in different ways.

We're going to start off with a square number problem.

Before we look at remainders as fractions, remainders as decimals and then I'll leave you to practise those new skills in your independent task.

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

Press pause if you need to go and collect the items, come back when you're ready to start.

So here's our square number problem.

Take a look.

See what's the same, what's different from the top and the bottom examples you're looking for two addends.

In the first three addends, in the second that must be square numbers.

The sums must be square as well.

Press pause.

See how many ways you can satisfy each of those equations.

Come back when you're ready.

How did you get on? How did you approach it? What did you do first? Hold up your paper.

Let me see where all of that thinking led you to, keep it steady, looking good everyone.

So there are a few options.

I asked you what you did first and wonder if anyone was listing square numbers, particularly from the one to 12 times table.

Of course square numbers continue, this is a good place to start.

From there were you able to find 16 and nine is equal to 25? How about 36 and 64 equal to 100? Two square numbers that equal eight squared sum.

In terms of the three addends, we've got the option of 1, 16, 64 totaling 81.

Or two lots of 16 and 4, 36.

How about two lots of 64 and 16 equal to 144.

A little recap as well, please on estimating.

Before we calculate a division or any calculation, estimation is a good place to start.

With these three, I'd like you to have a think about where some rounding can help you with your estimate.

So with this first example, rounding to 8,050 because of the connection between five and 80, between 50 and 800, we're able to mentally work on an estimate.

So carefully selecting what you round the numbers to because of where that's going to lead you with your estimate.

160 because five 10 times greater it's 50.

800, 10 times greater is 8,000.

So the quotient is conserved.

We've got a quotient of 160.

Pause and have a go at the other two.

Come back when you're ready.

Ready to check.

So thinking carefully about your decisions, what are you rounding and why? Can you show me on your paper, your estimates for these two? Oh, some really sensible choices.

Fantastic.

This is what I went for 1,050 of course, relationship between five and 10.

Two fives, 10.

So now if I increase my divisor, sorry, my dividend 10 times keep my divisor the same.

I need to increase my quotient by 10 times.

Whereas this time quotient is conserved because my divisor and dividend have both increased 10 times.

Last one, I chose 1,260 did you? Lovely connection between six and 12 is two.

So six is in 120, 20 and a conserved quotient here.

An estimate of 20, short division.

There will be a remainder.

And I went into focus on how we can represent that remainder in a different way to our four or our 13.

So let's kickstart with an estimate.

You might estimate this by giving two quotients that you think the actual quotient will lie between, or perhaps there's some way of using simulated facts to just to give one.

Go and have a try.

See what you come up with.

Ready to check? So we're looking at the estimate first of all, I've gone for 2000 divided by five, linking nicely to 20 divided by five.

So an estimated equation of 400.

Let's represent the 2,312 using place value counters.

Do we agree that's 2,312? Good so we're dividing by five.

Get to represent it down here as one at the same time, how many groups of five can we make from two thousands? So how about from 23 hundreds? Did you see the exchange? From 23 hundreds instead, what can we get? How many groups of five? Good, four groups of five from 20 of the 23 hundreds.

Three hundreds remain that we exchange for tens.

Noticed them appear, 30 tens.

We've now got 31 tens.

How many groups of five can we make from 31 tens? We can make six.

Good, there we go.

We now have one remaining 10, which we can exchange for 10 ones.

How many fives? How many groups of five, sorry can we make from 12 ones? Two, with two ones remaining.

Now, previously we've recorded that as our two, remainder two.

I'm going to show you two ways to deal with it.

First way is to continue exchanging.

We're in the ones place.

We can move into the next place to the right, which we call that the tenth space.

So I'll show you that again, two ones we can exchange for 20 tenths.

Notice in my recording, how that 20 tenths is represented, how many fives, how many groups of five can we make from 20 tenths? 5, 10, 15, 20.

Four, we can make four groups of five from 20 tenths.

We now haven't got any place value counters left.

So we have divided 2312 by five into five equal groups, into groups of five.

And there are 462.

4 in each group or 462.

4 groups of five, depending on which way we're looking at it.

This is just a division and we would need some context, a problem to help us know whether we've ended up with groups of five or five groups.

But that is the solution either way.

So we've moved from a remainder 0.

4 to 462.

4.

There's another way to deal with that.

So here's that same, so same process, that same division.

So let's just flick through with confident and happy with this part exchanging.

And we're finding how many groups of five we can make from our tens after the hundreds followed by the ones.

And we're at this point with a remainder of two that we've just represented as 0.

4, when we've continued to divide, there's another way to deal with it.

This is where we think about the group size.

What is the group size? Five, we have been creating groups of five.

For two ones we have left, not enough to make a group of five.

Only part of that group.

Only 2/5 of that group size, that remainder of two, we can represent as 2/5, two parts out of the five that the groups were being made of.

And when we think about it 0.

4, 4/10, 2/5 equal to 4/10 equal to 0.

4.

Of course they're related and connected.

Here's one for you to try, press pause.

You can draw the place value counters if you'd like to or just focus on the short division, exchange the remainder for tenths and continue if you need to into the hundredths and keep on dividing.

Come back when you're ready to share.

How did you get on? Hold out your paper, let me see how you've approached it.

Have I got any drawings or place value encounters or just the short division methods? Fantastic, either way.

Let me take you through it.

An estimate, I've gone for lovely 64 and eight.

Nice connection.

Eight in 64, 80 eights in 640.

So in estimate, estimated quotient size of 80.

Looking at the methods.

How many eights are in six? How many eights are in 64? With a one 10 to exchange for 10 ones? How many eights in 14? One with previously a remainder of six.

Let's deal with that six.

Let's continue exchanging six ones, 60 tenths.

How many eights in 60? Times how many groups of eight can we make from 60 tenths? We can make seven with 4/10 remaining that we can exchange for hundredths.

How many groups of eight can we make from 40 hundredths? Five.

And then we haven't got anything remaining.

Now have a go again, but represent the remainder as a fraction.

Press pause, come back when you're ready to check.

Should we have a look? Hold up your paper, let me see how this looks different to the last one.

Good work, good work, well done.

So taking you through those steps, it's the same calculations we're feeling comfortable.

We're good with this first part until we reached this remainder of six.

And instead of exchanging those six ones for tenths, we've done that.

Let's think about it as a fraction, so the group size.

The group size we were trying to create, eight.

A whole group would have eight, but we only have six.

Six of those eight, 6/8.

81 and 6/8.

Or thinking about what we know about equivalent fractions.

6/8 of course is equivalent to 3/4.

So we could have 81 and 6/8 or 81 and 3/4.

Hello everybody, this is Miss Jones.

I hope you don't mind, but I'm going to take you through one more type of problem here before handing you back over to Mr. Whitehead.

Now with him you've already looked at short division and how to express our remainders as both a decimal and a fraction.

So we're going to apply that to another problem now, and this problem is going to have a remainder and I'd like you to express it as a fraction.

But first you're going to need to do some long division in order to calculate this.

I know long division is sometimes a little bit more tricky, but it's really good that we're getting that practise in.

So I want you to have a look at this equation, pause the video now to have a go at solving it.

Okay, hopefully you've had time to have a go at that.

Let's look at it together.

So first of all, I'm going to think about an estimate.

And now I know that if I round each number, I can create an estimate.

So I'm going to just say 6,000 divided by 12 might give me a similar answer to what I'm looking for here.

Now, if I know that 60 divided by 12 is five.

I know that 6,000 divided by 12 is going to be 500.

So I know my answers should be somewhere near 500.

If I get an answer that's really far away, perhaps I've made an error.

Okay, so let's look at our long division here.

So I'm going to start off by looking at my thousands.

Now 13 can't go into six here so I'm going to then look at hundreds.

So when it takes 63 hundreds and I'm going to think about my multiples of 13.

I know that one times 13 is 13, two times 13 will be 26.

I know that 10 times 13 would be 130.

I need to get to 63.

So let's try five times 13, 65.

So it's just slightly too high.

So four times 13 would be 13 less of 52.

So let's put that in and I'm then left with 11 hundreds.

I can put my four in at the top there, as I know four thirteens would get me to that 63 with 11 remaining.

Okay, let's bring the tens and ones down.

So I'm left at 1,159 here.

Okay, so again I think now I'm going to look at the tens.

I've got 115 tens.

Let me think about my 13 times table again.

I know that 10 was 130, so nine would be 117.

Again, just a little bit too high.

So let's think about what eight would be.

Eight would be 104.

Let's put that in.

I'm going to take away that 104 tens from the 115.

And it's going to leave me with 11 tens.

Again, I'm going to bring down my ones now and see what I've got left, 119 ones.

Okay well, I already know that nine lots of 13 was 117.

So do we have anything remaining? We have two.

So we've got the remainder here of two, but of course we can't just put 489.

2 we need to think about that in relation to the number we're dividing by 13.

So I've got 2/13 remaining.

Okay, so my overall answer is 489 and 2/13.

Okay, I'm going to pass you back over to Mr. Whitehead now who's going to go through what you're going to be doing next.

Here is one for you to have a go at, a long division.

I would like you to represent the remainder as a decimal.

Estimate first though.

come back when you're ready to share.

Okay, show me how you got on.

Hold your paper up.

Fantastic, I can see long division estimates.

I can see decimal remainders.

Well done everyone.

Let's compare.

So my estimate, I rounded to 1,500 which I know is 15 lots of 100.

So my estimated quotient size is 100.

In terms of the long division, how many groups of 15 can we make from 1000? How about 14 hundredths? So close.

So what's about 145 tens? Here's our helpful table.

One group of 15, two groups of 15, 30.

10 groups of 15, 150.

Oh, too much.

So I'm thinking it's going to be nine groups of 15, 135.

We can make nine groups of 15 using 135 tens.

There are some tens left.

There are 10 tens left.

There's 100 left.

So we're going to use that now with our eight ones, 108 ones.

We're making groups of 15, how many can we make? Let's go back to our table.

Seven groups of 15, 105.

So we can use 105 ones to make seven groups of 15, with three ones remaining.

Of those three ones, we can make use of those with the tens.

So there around 30 tenths.

Well, there are 30 tenths.

If we're using the three ones.

How many groups of 15 can we make from 30? Two.

So that remainder of three, we have represented as a decimal 0.

2 because we were able to make two more groups of 15 from the 30 tents.

We haven't got anything left.

Our division is complete.

97.

2 compared to our estimate of 100.

You are more than ready now for your independent task, where you have a mixture of long and short division calculations.

For all of them estimate the size of the quotient, use the appropriate short or long division strategy to solve the equation and then represent the remainders as fractions and as decimals.

I've put in brackets to two decimal places at most, just to stop you from spending far too long, continuing to exchange and divide and exchange and divide.

So stop after two decimal places, if necessary.

For some of them, it will naturally stop after one.

Press pause, go and complete the task.

Come back when you're ready to look at some solutions.

Here we go then, let's take a look at the solutions.

Just hold up your paper or papers so I can have a quick look at how you got on.

Some long division, short division estimates for Anne, brilliant I can see decimal remainders, I can see fraction remainders brilliant.

Let me show you then.

So I'm just going to click through the pages.

If you want to pause on any, please pause and you can check carefully against your methods.

You will see the remainder to 42.

06.

Remainder is a decimal and then the fraction to 42 and 1/15 will be on the side.

So just look carefully for both of those.

So there's the first one, second one and question three, of course 8/10 is equal to 4/5.

Question four, well that long wasn't it.

Question five and question six, a busy, busy lesson.

Big sighs of relief I would say now, I haven't completed it because you have worked so incredibly hard using short division, long division with remainders and now representing those remainders as fractions and as decimals.

I'm a really, really proud teacher right now.

And I'm so pleased that there are still lots of smiley faces come the end of the session.

If you want to share any of this learning with Oak National, please ask your parents or carer to share your work on Twitter, tagging @OakNational and #LearnwithOak.

So what did you think? Are you mind blown or did I oversell the lesson a little bit too much? I hope you enjoyed it nonetheless.

And also hope that you're now in a position to be able to continue dividing the remainders by using the decimal places or making connections to fractions.

I found myself particularly when it comes to dividing into the decimal pieces, it all becomes a little bit obsessive and I'm glad that at times there are those rules of right stop after you've gotten to two decimal places.

Because otherwise for some numbers, you will be continuing forever.

Time now for a well deserved break in between this lesson and any other learning that you've got lined up for the day.

I hope you enjoy it.

And I look forward to seeing you again for some more maths.

Bye.