Lesson video

Hi guys.

Well done for loading this video and deciding to learn with us today.

My name is Ms. Davis and I'm gonna help you as you work through this lesson.

I'm so glad that you decided to join us.

So let's get started.

So welcome to today's lesson where we can have a look at securing your understanding of place value in integers.

By the end of today's lesson, you'll be able to state the value of any digit in a given integer.

So some key words that you would've come across before.

If you're not 100% confident on these, then just pause the video and read through them 'cause we're gonna use them a lot in today's lesson.

So we're gonna start by recognising place value.

Aisha and Jacob are competing to form the integer with the biggest value.

They each draw three cards from a deck of digit cards.

Aisha gets the five, a six and a three, and Jacob gets a seven, a one, and a two.

So then Jacob says, "I have already won because I have the card with the biggest digit." I would like you to have a think about, is Jacob's statement correct? How much do you agree with what he has said? Pause a video and think about your answer.

So in order to know the value of an integer, we have to know the place value of its digits.

If we arrange Jacob's digits in the order given in this place value chart, the seven represents 700.

See we've got our seven in 100s column.

If we arrange the digits in this order, the seven now represents 70 and actually the one represents 100.

So if Jacob was to put his digits in this order, his number would be smaller than any that Aisha can make.

So without knowing which order Jacob is placing his digits in, we cannot tell if he's made the biggest integer.

So let's think of an example then when Aisha's integer has a higher value than Jacob's.

When both players have three digits, like they do in this case, putting the largest digit in the 100s column is needed to give the largest overall value.

If Jacob uses the seven in his 100s column, then he's always gonna win with this set of numbers.

Any example where he doesn't put his largest digit in the 100s column will actually allow Aisha to win because her other three digits are larger than his other two.

So let's give an example then.

Aisha could have 653 and that's larger than 172.

Have a think about this true or false then.

So as long as the largest digit is in the highest place value, the integer will have the highest value possible.

Pause the video.

Think about whether this is true or false and can you justify your answer.

Well done, some good thinking going on there.

You might as thought then this one is false.

Actually all the digits in an integer contribute to its overall value.

So yes, it's important that our largest digit is in our highest place value, but if we're trying to make the largest integer possible, we also need our second largest digit in the second highest place value and so on.

So now we're gonna have a look at writing larger integers by looking at some more columns in our place value chart.

What do you notice about the boxes? Have a think about why they might have been drawn this way.

So what you might have noticed, we've grouped them together in threes.

So we've got the ones, the 10s and the 100s.

Then we've got the 1,000s, 10,000s, 100,000s, millions, 10 millions, 100 millions, billions, 10 billions, 100 billions.

You might even know what comes next after 100 billion.

And that is because that we're following a pattern.

We're grouping them in ones, 10s and 100s, and then we're doing the same for the 1,000s, the millions and the billions.

This becomes really helpful when we're writing our integers down.

So the integers I've written here 4,000.

Notice that when we write that down, we write it as four with three zeros.

If we compare that to a slightly larger number.

So let's look at 21,000, with integers of five or more digits we use spaces to separate the 1,000s.

So if you see how I've written 21,000, there is a gap to separate the 1,000s from the 100s, 10s, and ones.

Notice that when we wrote 4,000, 'cause it's only got four digits, we didn't leave that gap.

And this pattern continues with even larger integers.

So every three digits we're going to leave a space.

So we're gonna separate out the 1,000s.

We're also gonna separate out the millions and the billions.

So you'll see that we've got 5 billion and we've got space to separate our 320 million and then a space, we've got nothing in our 1,000s, no 100s, 10s, or ones.

Have a go yourself then.

So which of these integers is written with the correct spacing, Well done.

Especially if you notice that it was B.

For integers with five or more digits we need the spaces to separate the 1,000s.

Even better if you spotted that that top one's incorrect 'cause where we have four digit integers, we don't need that space.

Fantastic time for you to have a practise then.

So in this task you've got two separate sets of digit cards, for each set you're gonna need to create certain integers.

What that means is for every question you're gonna have two answers, one for each set.

Give those four questions a go and then we'll look at the next lot.

Okay, so now what we're gonna do is we're gonna add another digit card to each set at random.

We dunno what it is.

It's displayed here with a question mark.

What I want you to think about is there a scenario where the second set of digits could now make a larger integer than the first? Then see if you can justify your answer.

And finally we've got June and Lucas.

They're trying to make the smallest values possible this time and they've got to use all of their digit cards.

Off you go and then we'll go through the answers.

Fantastic.

Lots of thinking required in that practise are well done.

The largest integer we could make with both sets.

So the first set, you should have 987,665.

Well done if you put that space as well between your 1,000s and your 100s.

The second set, you should have 653,210.

Then we're making the smallest integer possible.

So for the first set, 566,789.

Then for the second set, it was a little bit trickier.

Well done if you spotted that there should be a zero at the front, the zero digit will make the smallest integer.

However, when we write it down, we don't write that leading zero.

We'll talk about this more in the next part of the lesson.

So you would actually write 12,356.

For the integer closest to 600, you should have 598 and 601.

Then for the last question, 987 and 1023.

Fantastic if you've got all or most of those ones right.

So this one then when we're adding an extra digit.

Good spot, if you notice that, no, there's not a chance that second set of numbers could make a higher integer than that first set of numbers.

The reason being is if we imagine that second set got a nine, it's the biggest digit it could get.

If that second set received a nine, you could make 9 million.

But then the second largest digit is a six.

So you'd make 9,653,210.

However, we look at that first set of cards, we've already got a nine, so we can already have 9 million and then the second largest digit we have is an eight.

So we can already make 9,800,000, whatever that last number is.

Quite a tricky concept that one.

So well done if you wrapped your head around that.

June and Lucas.

So there's quite a subtle bit here.

So you might have spotted that June should have written a gap, should have left a gap when writing his integer.

So 23 gap for 23,357.

So this one was false, because actually Lucas could put two leading zeros before his number and then he would get the integer 889.

And the last one also false, 'cause if June swaps is two for zero, yes he'll have a zero now, but his next largest digit is a three.

So his integer would start zero three, whereas Lucas still has one zero and June's just given him a two.

So he can make 02889, which is gonna be smaller than June's 033527.

Obviously when we write those down, we don't write those leading zeros in our integers.

So the second set of our lesson, we're gonna look at integers as sums of their digits' place value.

Any integer can be written as a sum of the place value of its digits.

Let's see what this looks like then.

So in 526, this could be written as 500 plus 20 plus six, and this works for integers of any size.

So 23,283, 20,000 plus 3000 plus 200 plus 80 plus three.

Let's put this into a place value chart to see what's happening.

So 23,283 and what we're doing is we're separating that out into each of its digits' place value.

So this is 20,000 plus 3000 plus 200 plus 80 plus three.

So how could we write the number below as a sum? Well it would be 5 million plus 80,000 plus 3000 plus 50.

Notice where we have a digit of zero it's more efficient not to include this in our sum.

We can go even further and we can write this sum considering the value of each digit as a multiple of one or 10 or a 100, 1,000 or so on through our place value columns.

So if we return to this number, if you think about 5 million, it can be thought of as five lots of a million or five multiplied by a million.

So this integer could be written as five multiplied by a million plus eight multiplied by 10,000 plus three multiplied by 1,000 plus five multiplied by 10.

And at the moment that might seem like a very long way of writing it, but that's gonna support some of the other maths that we'll be doing later on.

Which of these then show a correct way of writing 8,485? Good thinking then guys.

You might have noticed that it is C.

What you might have also noticed is that I haven't written those values in the order they appear in the integer.

That's because addition is commutative.

We can add them in any order.

So just keep your eye out for that later on.

Which of these show a correct way of writing 1,702? Fantastic.

If you spotted the it is that bottom one.

If you selected the top one, that is also correct, 'cause it is equivalent to D.

However, remember we said that where there's a zero, it is more efficient not to include that in our sum.

It's not necessary.

Fantastic, you've to have a bit of a practise now then.

So for each question you need to fill in the blanks, for each one there's more than one blank to fill in.

Give that a go and then we'll look at the next set.

Fantastic.

This second set then you're doing exactly the same as the first set, but notice this time I might not have written my add-ons in the same orders as they are in the integer.

So just keep your eye open for which digit it is that you are looking for and get the right place value.

Give it a go and we'll have a look through the answers.

Fantastic.

Let's look at how well we did then.

So for the first question, you are missing a 300.

In the second line you are missing a 1,000 and you are missing the seven for seven times 10 to make that 70.

For question two, you are missing 30,000 in the top line.

In the second line you are missing 10,000.

We need three times 10,000 to make that 30,000 and you are also missing the two, for two times 10 to make the 20.

Question three, you are missing the digit of five in the 10s column in your number.

So 858, you're missing that five, therefore you are missing the 50 in the addition.

And then in the second line there's an eight, eight times a 100 to get your 800 and then also a one at the end.

You need eight times one to make the eight.

And that second set then.

So in the top line you have missing 3000.

In the second line you're missing a nine, nine times one to make the nine and you're missing 1,000, three times 1,000 to make the 3,000.

Question five, you're missing the digit of six in the 100s place.

That means you're also missing 600.

In the bottom line, you're missing the seven for seven times 1,000.

Last question.

You are missing a two in the 100s column in the digit in your integer 300,280.

You're missing an 80 in your sum.

Then the bottom line, you're missing a 10.

We need eight times 10 to make 80.

And a two.

We need two times a 100 to make 200.

Fantastic if you've got all or most of those ones right.

Last section then for today, we are looking at rearranging the digits in integers.

In many cases, lots that we've already looked at, moving the digits of an integer into another place value column alters the value of the integer.

If you look below, we've swapped the three and the one digit and what we've actually done is we've increased the value of the integer by moving the largest digit into the 100s place.

How many different integers can you make from these three digits? Pause the video and see if you can write them all down.

Okay, with three unique digits like we have here, we can make six different integers.

Notice that I've written them strategically.

I've got the two different integers you can make with one in the 100s column, 123 and 132.

Then the two integers you can make with two in the 100s column, and then the two integers you can make with three in the 100s column.

And that's a nice way for me to check I haven't missed any out.

Check them against your list, see how many of those you got.

Can you think of a three digit integer now, which will not change its value when its digits are rearranged? There are loads of examples that you could have got for that one.

When two or more digits are the same, we can swap the digits and keep the value the same.

So my example was 242.

When we swap the 100 digits and the one's digit this time, the value of the integer remains the same because the two digits are the same.

How many unique three digit numbers can we actually make this time? This time we can only make three 'cause 422, 242 and 224 are the only unique integers we can make.

We can place a leading zero at the front of an integer to keep the value of the integer the same.

So if I put a digit card of zero in front of my integer, it hasn't changed the value of my integer.

To improve efficiency we do not include the leading zeros when we write our integers.

So we would write that number as 242.

You need to be careful though, because a zero place within or at the end of an integer does change the value of an integer.

So if I place my zero digit card there, that has now changed the value of my integer, so I would need to write the zero, so 2,240.

When I place a digit within a number, like I have done here, again, it's important that I write that in my integer, so 2,204.

Another true or false one to think about then.

So rearranging the digits in an integer always changes the value of the integer.

Pause the video and have a think about your answer and your justification.

Good spot if you notice that one is false.

When two or more digits are the same, we can swap the digits and the value remains the same.

Alright, fantastic.

Another chance for you to practise then.

So for the first question, you want to write down all the unique three digit integers you can make with the digits two, one, and nine.

See if you can get all of those.

For question two, you need to choose your own digits, which can only make three unique three digit integers.

And the last one, you need to choose your own digits that can make exactly one unique three digit integer.

Give that that one a go, and then we'll have a look at the next set.

Fantastic, well done.

So question four, you need to write all the unique four digit integers, which can be made with those digits.

You've got five, a six, a six, and a six.

For question five, Aisha arranges those digits to make the largest integer possible.

You might want to write that down before you then answer the questions.

She then takes a zero digit card and I want you to think about where she can place that zero card so that the first question, her new integer is the largest five digit integer possible.

The second one, her new integer is as close to 61,000 as possible.

And the third one, her new integer is as close to 66,000 as possible.

Fantastic.

Off you go and give that a go and then we'll have a look at it together.

Well done.

So a chance to see if you managed to get all six of these, 129, 122, 219, 291, 912, 921.

Amazing if you managed to get all six.

Question two, so you could have loads of answers for this question.

I went for 121, a one, a two and a one.

Essentially you needed to pick three digits where two of them are the same and one of them is different.

In this case, you can't have the digit zero, 'cause remember, if we put the zero at the front, it'll no longer be a three digit integer.

Question three, you needed three digits that were all the same.

So I went with two, two and two.

Again, you can't have zero 'cause 000 is not a three digit integer.

Loads of thinking with that one.

Let's look at the next set.

You could have made four different integers this time, 5,666, 6,566, 6,656, and 6,665.

Notice the only unique digit is the five.

So there are four place value columns the five could go in, so four unique integers.

Question five, the biggest number Aisha should could make with her four cards was 6,665.

The answer to A then is 66,650.

She's put that zero card at the end to make the largest five digit integer.

B, you needed to put the zero in the 1,000s column to get 60,665.

And for C, you needed the zero in the 100s column, 66,065.

You might have noticed that if we were able to move the digits around, we could actually get a closer number, which is 66,056.

Remember we said Aisha made the largest integer possible and then was put in the zero in place, so we couldn't actually move that five and that six around.

Fantastic.

Let's look at what we've learned today then.

So knowing the place value of its digits helps us understand the value of an in integer.

We looked at how in integers with five digits or more are written with spaces separating the 1,000s.

Any integer can be written as the sum of its place value of its digits.

And then we looked at rearranging the digits of an in integer and seeing how it can alter its value and when it doesn't alter its value.

So those occasions where some of the digits are the same.

All right, guys.

There was lots of good thinking in that lesson today.

I hope you enjoyed learning with me and I look forward to seeing you again.