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Hello, how are you today? My name is Dr.
Shorrock, and I'm very much looking forward to learning with you today.
Today's lesson is from the unit: order, compare and calculate with numbers with up to eight digits.
This lesson is called determine the value of digits in numbers up to 10 million.
As we move through the learning today, we will deepen our understanding of how we read numbers and how we represent them using place value counters and a Gattegno chart.
We will also look at solving problems related to place value.
Now sometimes new learning can be a little bit tricky, but that's okay.
I am here to guide you, and I know if we work really hard together, then we can be successful.
Let's get started then, shall we? How do we determine the value of digits in numbers up to 10 million? These are the key words for our learning today.
We have million and value.
Let's practise saying those aloud together.
My turn, million.
Your turn, fantastic.
My turn, value.
Your turn, brilliant.
So when we say 1 million, well, 1 million is a big number.
It's composed of 1,000 1,000s.
And we write 1 million as a one followed by six zeros.
And when we talk about value, we mean the value of a particular digit is how much that digit is worth within the particular number.
Today's lesson is going to start by looking at the value of digits.
And in the learning today, we will meet Aisha and Lucas.
They are here to help us.
Aisha and Lucas are discussing the value of digits in this numeral, 381,620.
Ah yes, I'm not sure.
What do you think? How do we read this number? That's right.
Thank you, Aisha.
The last three digits are the ones and these are separated from the 1,000s.
There are six hundred and twenty ones, and three hundred and eighty-one thousands.
So we read this number as 381,620.
And we're going to start now by representing this number with place value counters, and we're going to have a go at partitioning it.
So the 300,000 is composed of three 100,000s; 80,000 is composed of eight 10,000s; 1,000 is composed of one 1,000s; 600 are composed of six 100s; the 20 is composed of two 10s.
And then we can determine which digit is in which place.
We've got 3, 8, 1, 6, 2.
But do you notice something? Is there something missing? That's right.
We need to use a placeholder zero because there are no digits in the 1s place.
So we need to have that represented with a zero.
We can now determine the value of the digits.
We can have a think about each digit.
What does the digit 8 represent? That's right.
The digit 8 represents eight 10,000s or 80,000.
What about the value of the digit 3? What do you think the value of that digit 3 is? That's right.
The value of the digit 3 is three 100,000s, or we say 300,000.
What's the value of the digit in the 10s column? Do you know? That's right.
The digit 2 in the 10s column has a value of two 10s or 20.
Let's check your understanding on this so far.
Could you look at this representation of a number then complete the sentences? That digit, mm, is in the 100,000s place.
The 2 represents, mm, or, mm.
The value of the 6 is, mm, or, mm.
Pause the video while you have a go at that.
Maybe compare your thoughts with a friend.
And when you are ready to go through the answers, press play.
How did you get on? Did you say the digit 4 is in the 100,000s place? The 2 represents two 1 millions or 2 million.
And the value of the 6 is six 100s or 600.
How did you get on with those? Well done.
We can also use a Gattegno chart to determine the value of digits.
Let's look at a representation of a number that we've done with place value counters.
And let's represent this on a Gattegno chart.
We can see we've got three 10,000s, so that's 30,000; got five 10s, 50; one 1, and two one-tenths or 0.
2.
And to decide or to help us determine what this number is, we can then recombine those parts.
30,000 add 50, add 1, add 0.
2 is equal to, ooh, I wonder how we read that number.
Let's look at that part of the Gattegno chart and have a go together at recombining the number.
So we know there are three 10,000s, five 10s, one 1, and two 10s in this number.
And we can then read it as 30,000 and 51 and 2 tenths.
Let's check your understanding with that.
Could you look at the Gattegno chart and then complete the equation and sentences? Maybe compare your answers with a friend.
When you are ready to go through the answers, press play.
How did you get on? Did you manage to form an equation? So 600,000, add 4,000, add 60, add 0.
5.
And then did you manage to recombine those parts? And to complete the sentences, there are six 100,000s, four 1,000s, six 10s, five tenths in this number.
And the number is 604,060.
5.
The digit 4 has a value of 4,000s, or we could write that as a 4,000, four with three zeros.
How did you get on with those? Well done.
Let's look at this number then.
Take a moment, have a look.
Could you read this number, do you think? What's the value of the purple digit? Can you see that purple digit 8? What's the value of that 8? Ah, Lucas is wondering, do we have to use the place value counters or the Gattegno chart? It's always a good place to start, but shall we see if we can determine the value of that digit without those resources? What do you think? Well, Aisha is saying, "We know that the purple digit is the lowest value digit in the 1,000s group." And the lowest value digit in the 1,000s group is the 1,000s placed.
So the 8 must represent eight 1,000s.
And the value of the eight then is 8,000, or we could write that as its number 8, 0, 0, 0.
Let's check your understanding with that.
Could you look at this number? What is the value of the purple digit? So could you complete these sentences? The 3 represents, mm.
The value of the 3 is, mm, or, mm.
Pause the video while you do that.
And when you are ready to go through the answers, press play.
How did you get on? Did you say the 3 represents three 10,000s; and the value of the 3 then is 30,000? Or we could write that as a 30 and then three more zeros.
How did you get on with that? Well done.
It's your turn to practise now.
For question one, could you look at these numerals and then complete the given stem sentence for each number? So the value of the 5 is, mm, or mm.
You could use place value counters or a Gattegno chart if you need to.
That is okay.
For question two, could you write a seven-digit number where the digit 5 is worth? So for part A, where the digit five is worth 5 million; for part B, it's worth 5,000; for part C, it's worth 500; for part D, it's worth 50,000; for part E, it's worth 500,000; and for part F, it would be worth five tenths.
Pause the video while you have a go at both of those questions.
And when you are ready to go through the answers, press play.
How did you get on with those questions? You were asked to look at these numerals and then complete the given stem sentence for each number.
Let's have a look at the first number.
520,941, the value of the 5 is 500,000, and we write that as 500 with three more zeros, 905,389, well, the value of the 5 here is 5,000, and we write that as a five and three zeros 651,920, the value of the 5 is 50,000, or we write 50 with three more zeros.
5,876,023, well, the value of the 5 here is 5 million, and we write that as a five and six zeros.
2,120,806.
5, and the value of the 5 here is five tenths, or we write that as 0.
5.
701,003.
15, the value of the 5 here is five hundredths or 0.
05 1,587,900, well, the value of the 5 here is 500,000s, or we write that as 500 with three zeros.
8,002,345, well, the value of the 5 here is five 1s, and we just write that as a 5.
For question two, you were asked to write a seven-digit number whether digit 5 is worth 5 million.
And you might have written a number like this that has a 5 in the millions place.
I wrote 5,670,923.
For part B, you were asked to write a number whether digit 5 is worth 5,000.
You might have written a number like this that has a 5 in the 1,000s place.
I wrote 1,675,923.
Then a number that had a 5 in the 100s place, I wrote 1,670,523.
For part D, you needed to write a number that had a 5 in the 10,000s place.
I wrote 1,650,923.
For part E, you needed to write a number that had a 5 in the 100,000s place.
I wrote 1,510,923.
For part F, you needed to write a number that had a 5 in the tenths place.
I wrote 410,672.
5.
I wonder what numbers you came up with, and I wonder how you got on with those questions.
Well done.
Let's move on to the second part of our learning today.
We're now going to think mathematically about place value.
So we're gonna have a little look at some problems that we are going to work together to solve.
Aisha and Lucas are trying really hard to solve a problem, and that's really important in maths is that we have to try really hard.
If we try really hard, we know we can be successful.
They have been given three counters and a place value chart.
And they need to find two different numbers that add up to, ooh, how do we say that number? That's right, 2,002,002.
Each number should be able to be represented by exactly three counters.
So this is our place value chart.
And Aisha is just gonna think of a number with her three counters.
Ooh, what number has Aisha made? What number it's represented? That's right.
The number that's represented is 1,110,000.
So Aisha has made the number 1,110,000 with her three counters.
So we now need to determine what number we would add to it to get to our target number of 2.
002.
002.
What do you think? And remember, this other number that we find can only be made with three counters.
Do you notice something? Let's represent this in a bar model.
So we know our target number is 2,002,002.
Aisha has made 1,110,000.
What's the missing part going to be? What's that missing addend? Well, we could subtract to find out.
But do you notice something? That's right, we could just estimate.
And if we estimate that difference, we can see that it would have to be between 8 and 900,000.
And so the missing part would have to be represented by more than three counters.
So 1,110,000 won't work as an answer to our problem, okay? Because we would need to use more than three counters.
So let's try working systematically.
Yeah, that's right.
What does it mean to work systematically? It means to work in an orderly way.
And in this way, we make sure we get every answer.
So Lucas and Aisha then use a place value chart again to support 'em to work systematically.
Remember, our target number is 2,002,002.
Oh, good thinking Aisha.
We could start just by halving the target number.
So we would get 1,001,001, and both of those numbers must add to give our target number.
So that's definitely one possible answer, isn't it? And because both of those addends are the same, they're both using three counters only.
So that's one solution.
That are good.
So this is what it means to work systematically.
We're just going to change one digit at a time.
And remembering that that digit sum has to be three for both addends.
This is one of the solutions that we found already.
So we could change one digit each time to get the next possible solutions.
So you can see 1,000,002 added to 1,002,000 would give us our target number.
And then we can change those digits around 1,002,000 and 1,000,002 would give us our target number.
And the same again, we could change a digit working systematically.
We've got 2,000,001 added to 2,001 would give us our target number.
And then we could change those addends around 2,001 added to 2,000,001 would give us our target number.
Ooh, let's check your understanding with this.
There is one pair of possibilities that Aisha and Lucas have not found, which is it? So I've listed there the possibilities that we have found.
Remember, each addend has to have a digit sum of three because we are only allowed to use three counters, and the target has to equal 2,002,002.
So have a look, think about working systematically what might come next and have a go at working out what those pair of possibilities are.
Pause the video while you do that.
You might want to work with a friend for this.
And when you are ready for the answer, press play.
How did you get on? Did you work out that the missing addends must be 2,001,000 added to 1,002.
And then we can swap those addends around.
How did you get on with working systematically? It can be quite tricky.
But if we work in a system, it's usually a little bit more efficient and usually a little bit easier for us.
Well done.
So we did, we worked systematically and found all the possibilities.
It's your turn to practise now.
Question one, these two counters have been placed on the chart so the numbers made have a difference of nine.
Question A, what numbers are represented? And what do you notice about the position of the counters? For the first part of B, could you following the example in part A, use one count to represent each number, find to represent two different numbers that this time have a difference of 90.
So think about the last question, question A, when we had a difference of nine.
But this time, I want you to find two numbers that have a difference of 90.
And at this time, where have you placed your counters, and why? And is there a pattern between where you placed your counters for this and for part A when you placed your counts for nine? For the second part, counters that have a difference of 9,000, or a difference of 90,000, and then a difference of 900,000.
For question two, could you use one counter to represent each number and a place value chart and find at least two different pairs of numbers where: part A, one number is one-thousandth times the size of the other number; and part B, one number is 100 times the size of the other number.
Now pause the video while you have a go at both questions.
And when you are ready to go through the answers, press play.
How did you get on? Let's have a look.
So first, I gave you a place value chart where two counters had been placed on the chart so the numbers had a difference of nine.
The numbers represented a 10 and one, then they have a difference of nine.
And you might have noticed these counters are in consecutive place value columns.
They're next to each other.
For part B, you had to find and represent numbers that had a difference of 90.
And you might have found these two numbers that had a difference of 90.
Remember, each number could only be represented by one place value counter.
So you might have found 100 and 10.
And 100 subtract 10 is 90.
So they have a difference of 90.
And think about where you've placed your counters and was there a pattern between where we place 'em here and where we place them further difference of nine.
And you might have noticed that you placed your counters in consecutive place value places again.
And then you had to do the same thing with a difference of 9,000.
So you might have found these two numbers that had a difference of 9,000.
You got 10,000 subtract 1,000 is equal to 9,000.
You then had to find some numbers that had a difference of 90,000.
You might have identified the numbers 100,000 and 10,000, and the difference between those two is 90,000.
And again, you might have noticed those counters were placed in consecutive place value columns.
Then, you had to think about having a difference of 900,000.
And again, you might have identified two numbers that had a difference of 900,000, which is 1 million and 100,000.
And again, noticing those counters were placed in consecutive place value columns.
So you might have noticed for all of those questions because you were only using powers of 10, the counters had to be in consecutive place value places to produce the given difference.
For question two, you were asked to use one counter to represent each number and use a place value chart to find at least two different pairs of numbers where one number is one-thousandth times the size of the other number.
So you might have found two pairs of numbers like this.
One is one-thousandth times the size of one-thousandth.
You might have found another pair of numbers.
1,000 is one-thousandth times the size of 1 million.
These are all the options that you might have found.
One is one-thousandth times the size of 1,000.
10 is one-thousandth times the size of 10,000.
100 is one-thousandth times the size of 100,000.
1,000 is one-thousandth times the size of 1 million.
10,000 is one-thousandth times the size of 10 million.
Did you work systematically? If you work systematically, you might have found all these possibilities.
Well done.
For part B, you then had to find a number that was 100 times the size of the other number.
You might have found two pairs of numbers like these.
1 million is 100 times the size of 10,000.
You might have found 1,000 is 100 times the size of 10.
And these are all the options that you might have found if you work systematically.
10 million is 100 times the size of 100,000.
1 million is 100 times the size of 10,000.
100,000 is 100 times the size of 1,000.
10,000 is 100 times the size of 100.
1,000 is 100 times the size of 10.
100 is 100 times the size of 1.
Did you work systematically? In which case, you might have got all of those possibilities.
Well done.
Fantastic learning today everyone.
Really impressed with the progress you have made with determining the value of digits in numbers up to 10 million.
We know we can use stem sentences when determining the value of digits in numbers.
We could say that, mm, digit in the, mm, column represents, mm.
We could say the value of the digit in the, mm, column is, mm.
And we know that digits in numbers are grouped in threes to show the ones, thousands, and millions.
And we've also covered the importance of working systematically when problem solving to help us get all of those possibilities.
Really proud of how hard you have tried today.
I have enjoyed learning maths with you, and I look forward to learning maths with you again soon.
