Lesson video

Hello, how are you today? I trust you are feeling on top form ready for our maths lesson.

My name is Dr.

Shorrock, and we are going to have a lot of fun as we move through the learning together.

Today's lesson is from our unit order, compare and calculate with numbers with up to eight digits.

This lesson is called using patterns in counting sequences.

As we progress through the learning today, we're gonna think really deeply about the composition of numbers and how we can use this composition to help identify patterns, especially in counting sequences.

Now, sometimes new learning can be a little bit tricky, but that's okay.

I'm 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 can we use patterns in counting sequences? These are the keywords that we will use throughout our learning today.

We have sequence and term.

Now you may have heard those words before, but it's always good to practise saying them aloud.

Are you ready? Let's do this together.

My turn, sequence.

Your turn.

Nice.

My turn, term.

Your turn.

Fantastic.

A sequence is a succession of objects, diagrams, or values usually formed according to a rule.

For example, a sequence of three, five, seven, nine, well, it starts at three and increases by two each time.

It's a sequence of odd numbers, and every number in that sequence is called a term.

So each term has a position or a location in that sequence.

For example, in that sequence, three, five, seven, nine, the number five is the second term.

Let's get started with our learning today by looking at how we can use patterns in counting sequences.

And we have Aisha and Lucas to help us today.

Aisha is practising counting on in 100,000s.

Lucas gives her a number to start from.

"Can you count on in 100,000s from 2,500,000?" he is saying to her.

2,500,000, 2,600,000, 2,700,000, 2,800,000, 2,900,000.

Well, let's look at that sequence of numbers that Aisha just said.

Which number will she say next? And how do you know? Have you spotted something? Have you noticed something that is the same or something that changes, something that is different? Lucas spotted a pattern that might help us.

I wonder if you have.

We can start by identifying the part of the number that changing and use our knowledge of the composition of seven digit numbers.

We can see that 2,500,000 is composed of 25 hundred thousands, and we can unitize the number of 100,000s when we count up.

So instead of saying 2,500,000, 2,600,000, 2,700,000, we can say 25 hundred thousand, 26 hundred thousand, 27 hundred thousand, 28 hundred thousand, 29 hundred thousand.

And this then helps us to determine the next term in the sequence.

It must be 30 hundred thousand, or another way we say that is 3,000,000.

This counting pattern can be represented as addition equations.

We're adding 100,000 each time, but why is that? We're adding 100,000 because that is what we are counting up in steps of.

We started at 2,500,000, and by adding 100,000, we took us to 2,600,000 and so on.

Lucas now has a go practising counting back in 100,000s.

He starts from 2,402,159.

Oh, this sounds like it's going to be quite tricky, doesn't it? 2,402,159.

2,302,159.

2,202,159.

2,102,159.

2,002,159.

Let's look at this sequence of numbers that Lucas said.

What number will he say next, do you think? And how do you know? Have a think about that.

What is changing? What is staying the same each time? Can you spot a pattern? Aisha says, we can start by identifying the part of the number that is changing and use the pattern to help us.

And we can also use our knowledge of the composition of seven digit numbers.

We know 2,402,159 is composed of 24 hundred thousands, and then we can unitize the number of 100,000s as we count back.

So we've got 24 hundred thousand, 23 hundred thousands, 22 hundred thousands, 21 hundred thousands.

Do you notice how the rest of the number is remaining the same? 20 hundred thousands.

So what does that mean about the number that comes next? That's right, it must be composed of 19 hundred thousands, and we know that the rest of the number does not change.

So it must be 19 hundred thousands with then 2,159, and another way of saying that is 1,902,159.

So it's much easier, isn't it, if we spot a pattern and use that pattern to help us when we think about what part of the number is changing and what part of the number is staying the same.

This counting pattern can be represented as a subtraction equation this time because we were subtracting 100,000 each time.

And why are we subtracting 100,000 each time? That's because that's what we were counting back in the steps of.

Lucas was counting back in steps of 100,000, wasn't he? And he's counting back, so it's a subtraction.

Let's check your understanding with this.

Could you look at this sequence of numbers? Which number will be the next term in the sequence? Remember, start by identifying the part of the number that is changing.

So have a look what is the same and what is different? Pause the video while you have a go at trying to work out the next term in the sequence.

When you are ready to go through the answers, press play.

How did you get on? Did you spot that pattern? So we had 378,000s, 379,000s 380,000, 381,000.

The 500 stayed the same each time, didn't it? And we can use that pattern because what comes after 381? That's right, 382,000s.

So the next term in the sequence, 382,500.

How did you get on with that? Well done.

It's your turn to practise now.

For question one, if possible work in a pair.

One of you should choose any seven digit number.

You might be able to use a die to help you make one.

For example, you might choose 3,456,000, and the other needs to choose a power of 10, so for example, 1000.

Then could you take it in turns to count up in this power of 10 from the chosen number until you've counted on seven more times? And then take it in turns to count back in this power of 10 from the chosen number until you've counted back seven more times.

It might be helpful for you to write the numbers down as you say them because I know sometimes I can't remember what I've just said.

And that might support you to spot which part of the number is changing and to use the pattern to identify the next term.

For question two, Aisha counts on in 100,000s from 2,902,803.

What number will she say next? And explain how you know.

For question three, could you complete these sequences? Again, stop, have a think.

What do you notice? Is anything changing in each sequence? Is anything staying the same? And use that pattern to help you.

Pause the video while you have a go at those three questions.

When you are ready to go through the answers, press play.

How did you get on? Let's have a look.

So for question one, you were asked if possible to work in a pair and counting on or back from given powers of 10.

So you might have taken it in turns to count up in 1000s and spotted the pattern there.

We had 56,000s, 57,000s, 58,000s, 59,000.

So you might have used that pattern to count up from 3,456,000.

You might also have taken turns to count back in 1000s from 3,456,000.

Question two, Aisha counts on in 100,000s.

What number would she say next after 2,902,803? And explain how you know.

You might have noticed that 2,902,803 is composed from 29 hundred thousands, and because she's counting up in 100,000s, the next number must be composed of 30 hundred thousands.

The next number she says is 3,002,803.

Then you were asked for question three to complete some sequences, and you might have spotted the part of the number that was changing to help you.

So the missing numbers were 6,421,040, 6,431,040.

And did you spot how that sequence was increasing? That's right, it was increasing in 10,000s each time, wasn't it? For the next sequence, the missing terms were 1,998,567, and 1,997,567.

And you might have identified the part of the number that was staying the same, that 567, and the parts of the number that were changing, and using that to help you.

For the next sequence, you were asked to find some terms in the middle of the sequence.

You might have noticed, well, it's increasingly by 100,000s, so 7,830,004, and then 8,130,004, and then 8,330,004.

For the next sequence, you were asked to identify the first term, the third term, the sixth term, and the seventh term.

And you might have noticed what was staying the same here and what was changing.

What was staying the same was that 301, and also the 9,000,000 happened to stay the same here.

And you might have noticed what was changing.

Well, this time the sequence was decreasing, and it was decreasing in 10,000s.

So you were counting down in 10,000s, 159,000, 149,000, 139,000.

How did you get on with those questions? Well done.

Fantastic learning so far.

I know you're trying really hard and you've really deepened your understanding on how we use patterns in counting sequences.

We're now gonna look at some problem solving with these counting patterns.

Aisha and Lucas are looking at this sequence of numbers.

The sequence increases by the same amount each step.

Hmm, we've got to find those three missing terms in the middle of the sequence.

Well, how can we do that? We don't know what the sequence is increasing or decreasing in, do we? Well, actually we can tell it's increasing, can't we? Because it starts at zero and ends at 8,000,000, so it must be increasing.

But how do we work out those missing terms? Good idea, Lucas.

Let's pop this on a number line and see if that helps.

So we've represented this on a number line.

We're starting at zero and we're finishing at 8,000,000, and we know there were three missing terms. Do you notice something? That's right.

There are three missing terms, but actually there are four equal steps after zero to reach 8,000,000.

How does that help us? That means we need to divide the whole by four, and we can use our known facts to help us.

We know eight divided by four is two.

So 8,000,000 divided by four must be 2,000,000.

So the sequence must be increasing in steps of 2,000,000, 2,000,000, 4,000,000, 6,000,000 and 8,000,000.

Let's check your understanding with that.

In the number sequence: zero, blank, blank, blank, 200,000, do the steps go up in 20,000? Is that true or false? Pause the video while you decide, and when you are ready to hear the answer, press play.

How did you get on? Did you say, well, that must be false? But why is it false? Is it because A, there are five equal parts.

Each part must be worth 200,000 divided by five, which is 40,000, so the steps go up in 40,000.

Or is it B, there are four equal steps, each step must be worth 200,000 divided by four, which is 50,000.

The steps go up in 50,000.

Pause the video, maybe talk to someone about this and when you're ready to go through the answer, press play.

How did you get on? Did you say it must be B? There are not five equal parts are there? There are four equal steps.

We go from zero and then we can jump to the first blank, jump to the second blank, jump to the third blank, and jump to the 200,000.

So that's four equal steps.

So we need to divide by four, and we can do that by halving and halving again, 100,000 and then 50,000.

So the steps must go up in 50,000.

And actually this is really a simple way of checking because you could count up in steps of 40,000 or 50,000.

If we count up in 40,000s, we would say zero, 40,000, 80,000, and then 120,000, and then 40,000 after that wouldn't be 200,000.

We wouldn't make it.

So actually you can sanity check your results by checking, by counting up in the steps.

So the number sequence goes zero, 50,000, 100,000, 150,000, 200,000.

Aisha and Lucas are looking at a different sequence of numbers now, and the sequence increases by the same amount each step.

Ooh, do you notice something this time that's slightly different to our previous example? Yes, that's right.

This time the sequence does not start at zero.

Hmm.

How will that affect what we do, do you think? There are still four steps, aren't there, between the given terms, but if we divide the whole, the last term by four, we get 300,000 and that's smaller than our starting point.

So how can we determine there's missing terms in our sequence then if we can't divide 1,200,000 by four? What do we have to do? That's right, but we have to take that starting point into consideration.

So we need to start by finding the difference between the first and last of our given terms, then we'll know what that whole sequence is.

If we find the difference by subtracting, well, we can use our known facts and unitizing to help.

We know 12 ones subtract 4 ones is equal to 8 ones.

So 12 hundred thousand subtract 4 hundred thousand must be equal to 8 hundred thousand.

So that means the whole of our sequence from 400,000 to 1,200,000 is 800,000.

So the difference between our first and last terms is 800,000.

We can now divide the 800,000 by the number of steps to determine the value of each step.

800,000 divided by four because there are four steps is 200,000.

So the sequence is increasing in equal steps of 200,000.

So this is slightly different to the last one when we started at zero.

This has an extra step, doesn't it? We needed to work out that difference first before we could divide to find out what each step was worth.

And you can see if we count up in 200,000s, 400,000, 600,000, 800,000, 1,000,000, 1,200,000, then that gives us our required terms in the sequence.

Let's check your understanding with that.

True or false in the number sequence, 1,800,000, mm, mm, 2,400,000, do the steps go up in 300,000? Is that true or false? Pause the video while you decide, and when you are ready to hear the answer, press play.

How did you get on? Did you say that's false? But why is it false? Is it there are three equal steps? Each step must be worth 2,400,000 divided by three, which is 800,000, so the steps go up in 800,000? Or is it there are three equal steps but first we find the difference, because the sequence is not starting at zero? And this is between the first and the last terms, so we find the difference by subtracting 2,400,000, subtract 1,800,000 and that equals 600,000.

So then we can work out the value of each step by dividing by three, which is equal to 200,000.

So the steps must go up in 200,000.

Pause the video while you decide what the reason is, and when you're ready to hear the answers, press play.

How did you get on? Did you realise that it must be B? Because our first term is not zero, we need to find that difference first before we can then divide.

So if we find the missing terms of the sequence, we know it must be 200,000.

So our missing terms are 2,000,000 and 2,200,000.

Your turn to practise now.

For question one, could you find the missing terms in the sequence? Have a think.

What do you notice? What is the same and what is different? For question two, find the missing terms in these sequences.

What's different here about these ones? Pause the video while you have a go at both those questions.

When you are ready for the answers, press play.

How did you get on? Let's have a look.

For question one, you had to find the missing terms in the sequence.

We can see there are three steps and the sequence starts at zero, so we can divide our last term by three, which is 200,000.

The missing terms are 200,000 and 400,000.

For part B, again, the sequence started at zero, and this time ended at 1,800,000.

Again, there are three steps.

So we can divide our last term by three, which is equal to 600,000.

So the missing terms must be 600,000 and 1,200,000.

Part C, we had four steps, so we know we need to divide the last term by four, which is 120,000.

So the steps must be increasing by 120,000.

For part D, how many jumps did we have? We had one, two, three, four, five, six jumps, so we needed to divide 5,400,000 divided by six.

Did you notice there you could use your times tables for this one and the previous one? 54 divided by six is nine.

So 54 hundred thousand divided by six must be 900,000.

So each step must be increasing by 900,000.

For question two, you had to find the missing terms in these sequences.

Did you notice something different about these sequences? That's right, they didn't start at zero, did they? So we've got that extra step.

We needed to find the difference first.

900,000 subtract 300,000 is 600,000.

We can then divide by the number of steps, which is three, and that means each term is increasing by 200,000, so 500,000, 700,000.

Then for the second part, part B, again, the sequence did not start at zero, so we needed to start by finding the difference by subtracting.

The difference is 1,200,000.

We could then divide that by the number of steps, which is three, and each step is worth 400,000.

So we're counting on in 400,000s.

1,800,000, 2,200,000.

For part C, again, we're not starting at zero, so we need to start by finding that difference.

The difference is 200,000.

This time we've got four steps and we need to divide by four, and that equals 50,000.

So each step is worth 50,000.

550,000, 600,000, 650,000.

For part D, again, it doesn't start at zero.

We need to find that difference first.

If we subtract the difference is 2,500,000.

We then need to divide by five because there are five steps.

Each part is worth 500,000.

So our missing terms are 2,500,000, 3,000,000, 3,500,000 and 4,000,000.

How did you get on with those questions? Well done.

Fantastic learning today.

You've all worked really, really hard and deepened your understanding on how we can use patterns in counting sequences.

We know counting patterns and known facts support calculations with millions.

We know counting patterns can be represented as addition and subtraction equations.

And we know missing values can be calculated using addition and subtraction.

We also know that it's really important to take note of what is the same, what is different, what's changing and what is staying the same, and that helps us to spot patterns and then use those patterns.

So well done.

You should be really proud of how hard you have worked today.

I know I am proud of you.

I look forward to learning with you again soon.