Lesson video

Hello! Welcome to today's math lesson.

My name is Miss Jones, and I'm going to be taking your lesson today.

I hope you're doing all right today.

I'm really looking forward to doing some pattern spotting today.

Before we get started, let's do a riddle together.

Are you ready? I am the beginning of everything.

I'm the end of everywhere.

I'm the beginning of eternity and the end of time and space.

What am I? I'll say it one more time.

I'm at the beginning of everything and at the end of everywhere.

I'm at the beginning of eternity and the end of time and space.

What am I? Have a little think.

Okay, the answer is the letter E.

All of the words either began or ended with the letter E.

Hope I got you with that one.

Make sure you tell your friends that one.

Okay, I think it's time to get started with today's lesson.

In today's lesson, we're going to be looking at linear number sequences.

We'll explain what we mean by linear later on in the lesson.

Let's have a look at our agenda.

So today we're going to start off by looking at some sequences and thinking how we can best represent them, then we're going to be finding missing terms within that sequence, then you've got a task and a challenge question, and finally a multiple choice quiz.

For this lesson, you'll need a pencil and a piece of paper.

You'll also need today a number line.

Now you don't need to have one of these printed off.

You can draw your own if you want to.

If you haven't got what you need, pause the video now and go and get it.

Okay, let's begin.

I want you to have a look at this sequence and think about how many ways you can describe it.

Take a moment just to think about what the pattern is.

Okay, what I'd like you to do next is think about how you can represent this sequence using a number line.

What might this look like on a number line? If you've got a number line, how would you show this sequence on it? If you haven't got a number line, why don't you try drawing or sketching a number line and showing what the sequence might look like? Once you've done that, think about is there any other or are there any other ways you could represent this sequence? Using objects or perhaps another drawing or sketch.

Pause the video now and draw out or create your representations.

Okay, shall I show you my representation? Here I've got my number line.

Now you can see my number line goes up to 40 here, and because no need for it to go any further at this point, and you can see I've started to represent the sequence.

I've got seven, 12, 17.

And my number line helps me to see the jumps are equal.

Each jump is going up, I can see, in fives.

And it helps me get an idea about where my sequence lies within the number sequence.

So I can see the moment my sequence goes from seven up to 37, and I can start to visualise what it might look like if it continues.

Another representation I used was I made towers out of cubes.

So you can see here the magnitude of each term in the sequence.

You can, again, see we start at seven, so I've got seven cubes here, and then I move on to 12, and you can see quite clearly the difference each time is five cubes.

These representations help us to make sense of the sequence.

They can help us notice patterns.

I want you to compare my two sequence, my two representations here, and perhaps the representations that you created as well if they are different, and think about how they are the same and how they are different.

Take a moment, pause the video, and have a quick think about that.

Okay, how are they the same? Well, we can clearly see in both representations, we can see the jumps of five each time.

But in this representation, it's vertical, and in this representation, it's horizontal, so that's one difference.

Perhaps you found some other differences between the representations.

But both of them help to draw out different, really important bits about the sequence.

Moving on.

Thinking about those representations, I want you to think about which of those representations might support us with understanding some of these things? So each number is two more than a multiple of five.

That's interesting.

So I can see seven is two more than five, 12 is two more than 10.

Now I don't think I'd noticed that using my cubes.

But if I had a number line, which marked the multiples of five, I'd probably be able to see it then.

Ascending in fives but not in the five times table, okay? Again, my number line might help me see that it's going up in fives, in jumps of five, but it's avoiding the number five and the number 10 and all the multiples of five.

The term-to-term rule is add five.

I could really clearly see that using my cubes.

I could see that it was going up five cubes each time.

I could also see that on my number line.

Each number has a two or a seven in the ones place.

Hmm, now I wouldn't be able to notice that using my cubes.

But I think just having my pattern like this, I can notice the ones here each time.

So actually, this representation is probably the best to show that.

The numbers alternate, alternate between odd and even.

Interesting.

Again, I think I'll go to this pattern, this representation of the sequence, in order to see that we've got even, odd, even, odd.

Those are some really interesting observations.

I wonder if you noticed those things too.

Here I've got one more representation I wanted to show you.

Here I've got the same sequence presented on a graph.

You can see this is the first term, and I've plotted seven on the y-axis, the second term, I've plotted 12 on the y-axis, and the third term, 17, as we go up.

And what's really interesting when we plot it on a graph, it creates this line.

If we were to join these out, we'd have a straight line.

How does this graph compare to the sequence and the other representations that we already looked at? What can you say about the points on the graph? Again, they're going up in jumps of five on the y-axis.

Now because it's a straight line, we call this sequence a linear sequence.

Any sequence where there are equal jumps, here we have equal jumps of five, we call a linear sequence.

If we were to plot it on a line graph, we would make a straight line like the one we can see here.

Okay, let's have a look at some sequences together.

I want you to have a think about what you think the term-to-term rule is.

Now when I mean term-to-term, I want you to think about each of these being called a term, so term one, term two.

What's happening as we go from one term to the next and then to the next and to the next? What is the rule of the sequence? Can you do that for each one? If you finish that, have a think about which one of these is the odd one out.

Why? Now there might not just be one right answer.

You might think more than one could be an odd one out.

Have a think about explaining how you know it's the odd one out.

Pause the video now to have a go with that.

Okay, hopefully, you've had a chance to think about the term-to-term rule.

Let's have a look together.

So for this first sequence, I'm starting at 13 and then I go to nine and then to five.

Now I know that 13 and nine have a difference of four.

So what's happening here is that we're subtracting four.

Now we need to check that the same thing's happening each time.

Now if this is a linear sequence, it will increase or decrease by the same amount for each term.

Let's just double-check.

Nine, take away five, that's odd.

Sorry.

Nine and five also have a difference of four.

And then five, take away four is one.

That works.

And now we're going into negative numbers.

One, take away four gets us minus three.

Let's just check that.

One, zero, minus one, minus two, minus three.

Yeah, that works.

And if you want to, I could visualise a number line in my head or draw one to help me check that.

Okay, so we've got here our term-to-term rule, pretty confident, is minus four or subtract four.

Here, we've got some decimal numbers.

I put some commas in here to help us separate each one.

Some of them are a bit close together.

We've got 0.

25, 0.

75, 1.

25.

And if we wanted to again, we could use a number line to help us represent this sequence.

But I know that each time we've got, so I'm looking at my tenths column, we're jumping by five tenths or 1/2.

0.

25 added to five tenths makes 0.

75.

If I add another five tenths, I get 1.

25, then 1.

75.

The rule is add 0.

5 or five tenths.

Here I can see we're increasing.

Now to go from 1/3 to 2 2/3, I need to add 1 1/3.

Let's just check that works for the next term to make sure it's a linear sequence.

So if I add, I'm going to add the third first, which gets me to 2 3/3 or which makes three, and then add the one and I get four.

Then from four, I add one and I get five, and I add the third and I get 5 1/3.

So the answer, the term-to-term rule, is each time we're adding 1 1/3.

Hopefully, you managed to get some of those.

Now the question which one was the odd one out, like I said, there could be more than one answer.

We could say that this sequence is the odd one out because it's the only one that includes negative numbers.

We could say that it's also the odd one out because this one includes only integers.

We don't have any fractions or decimals in this sequence.

We could say that this one is the odd one out because it includes numbers written as a fraction.

Or this one is the odd one out because it includes numbers written as a decimal.

So you could have argued each one, but hopefully you had a go at explaining your reasons why.

Okay, moving on.

Now we've taught about what the term-to-term rule is, and now we know that, we can work out what the next terms in the sequence might be.

So for this one, our term-to-term rule was subtract four, subtract four, subtract four.

The next symbol will be minus seven, subtract four, which I know is minus 11.

If I'm not sure, I could use a number line to help me.

From minus 11, we subtract four again and we get minus 15.

Have a go at deciding what you think the next two terms will be for this sequence and this sequence.

Okay, let's see if you got them.

So we already did this one together, minus 11 and minus 15.

For the next one, let's remember what the term-to-term rule was, we were? That's right, we're adding 0.

5.

So we should get 3.

25 and 3.

75.

And for this final one, we were increasing again by 1 1/3, so we should have got 8 1/3 and 9 2/3.

Okay, now my next question is very similar, but instead of going this way, we're going to be thinking about what the previous terms will be, so what comes before.

Now if, for example, we were taking away or subtracting four, if we're going this way, we're going to need to do the inverse to find out what this term will be.

So I'll have a go with that first one again.

So 13, now I know the terms to rule this way was subtract four, so if I'm going this way, I need to add four.

So this term will be 17, okay? I'll let you figure out what the next one is.

And then once you've done that, have a go at finding the previous terms for these two sequences as well.

Pause the video now to have a quick go.

Okay, let's see if you are right.

So we already did this one, 17, and then if we add another four, we get 21.

Sometimes it's quite nice just to check the sequence by going this way again.

21, 17, 13, nine.

That makes sense to me.

Here, this time, we need to take away or subtract.

5.

Now we were already, we're going to cross zero if we do that.

So we're going to go into negative numbers because there's only two tenths and five hundredths left, so we'd have to go into the negative numbers here, and 0.

75 was the second previous term.

Here we've got zero and then minus 1 1/3.

Hopefully, that gave you a chance to consolidate some of your understanding with negative numbers too doing some of these.

Okay.

For your Let's Explore task, I want you to look at the sequence and explain what the term-to-term rule is and see if you can think about what the next terms or the previous terms are.

Let's just have a look at this one for example.

We've got some explanations to help us have a think about this.

So here she says, "I know that the next term in the sequence will be 14 added to 3.

5." Interesting.

She must think that's the term-to-term rule.

"And that's equal to 17.

5." That seems to make sense.

Here we've got: "I can tell that the term-to-term rule is 'add 3.

5.

'" We can put that in.

"Because the difference between zero and 3.

5 is 3.

5 and I know that 3.

5 add 3.

5 is equal to seven." So she's checked here.

And then to make sure it's a linear sequence, she'd also check the difference here to make sure it's going in equal jumps.

That means it's a linear sequence.

And finally, we've got the same person, just saying, "I know that the previous term in the sequence will be 3.

5 less than zero," which she says is negative 3.

5.

So the previous term, we can write like this, negative or minus 3.

5, okay? I hope that's nice and clear.

Okay, I've got some sequences for you to have a look at.

For each one, tell me what the term-to-term rule is and explain how you know.

Then explain what the next two terms will be and the previous term should be.

Pause the video now to complete your Let's Explore task.

Okay, by now, you should have had a chance to pause the video and complete that task, so we'll go through the answers together.

So in this first sequence, we've got minus seven is the previous term, and we've got 29.

So thinking about the term-to-term rule, I'm going to look here so I can see it nice and clearly, we're adding six.

Of course, I need to check that we're also adding six here.

Okay, and then to get my previous term, I would have had to subtract six.

To get my next term, I would have had to add six.

Let's look at the next one.

Here we go.

That's the term-to-term rule.

We've got 1.

3 and 3.

1, and our term-to-term rule was adding three tenths, 0.

3.

And then our final one, we've got 2 1/4 and negative 3/4 or minus 3/4 as the next term.

Did you get what the term-to-term rule was? Was taking away or subtracting 1/2.

Let's look at that in action.

So we've got here 3/4.

If we take away 1/2, we're going to get 1/4.

If we take away 1/2 again, we're going to negative numbers and we get minus 1/4.

Okay, let's go on to something a little bit different.

I want you to create your own number sequence.

To start off with, pick any number between zero and one.

So for example, 0.

2, 0.

35.

Pick your own number.

Then I want you to create a sequence with five terms using the term-to-term rule plus 0.

2 or two tenths.

I've made my own sequence too, and I'm going to reveal mine on the next slide.

Once I've revealed mine, we're going to compare them.

So what you need to do now is make your own sequence.

Pause the video now to do that.

Okay, hopefully, you've had time to make your own sequence using the term-to-term rule, add two tenths or 0.

2.

Let's have a look at my sequence, and you can see what's the same and what's different with your sequence.

My sequence is, here it comes, I started with five hundredths.

You can see I've got zero, tenths, five hundredths.

I added two tenths, so now I've got 0.

25, 0.

45, 0.

65, you get the pattern, 0.

85.

How does my sequence compare to yours? Question for you to think about.

And you can look at my sequence or your own for this if you want to.

If I continue my sequence, will it contain any integers? Now an integer, remember, is a whole number, such as one, two, three, four.

Will I have any numbers that don't have any tenths or hundredths? Will I have any integers in my sequence? Have a think.

I'm going to do my best to explain why I think in my sequence I don't think I'm going to have any integers.

The reason why is because each time, we're adding two tenths, as we know.

We're not adding any hundredths, which means our hundredths digit is going to stay the same.

I'm always going to have five hundredths.

If I always have five hundredths, I'm never going to have a number with no tenths or no hundredths, such as three for example, because I'm always going to have a five in this column.

Hopefully, I did my best to explain that.

I wonder how you would explain it.

Let's have a look at another.

I'm going to show you another sequence now.

I want you to think about a number that will never appear in this sequence.

Here is my sequence.

Have a think of a number that will never appear in this sequence.

Hopefully, you've noticed that all the numbers in this sequence are ending in seven, two, seven, two.

I wonder why that is.

Let's look at our term-to-term rule.

Each time we're adding five.

Add five, add five.

So each of our numbers is related to a multiple of five.

This one is two more than a multiple of five, two more than five.

This one is two more than 10, two more than 15.

We're never going to have a number that ends in, for example, five because it's always going to be two more.

So my number won't be 15, it would be 17.

This question says will the 25th term also end in a two or seven? I think yes, it will.

All terms in this sequence end in two or seven because they are always two more than a multiple of five.

Lars says, "I think 1,070 will appear in this sequence if I extend it far enough." Do you think Lars is correct? Explain how you know.

I'm going to let you have a think about that.

Think about my explanation before.

Did you manage to explain? Let's have a look together.

We've already answered some of these questions.

We know that the terms and, terms one, three, and five all end in a seven, and terms two, four, and six all end in a two, so all the terms end in two or seven.

Lars was incorrect as the ones column is always two more than a multiple of five.

So it'll always end in two or it will always end in seven.

It's not going to end in zero like Lars said.

For your main task, I want you to look at some linear number sequences.

I want you to think about what the term-to-term rule is, what the next numbers or next two terms in the sequence will be, what the previous two terms in the sequence will be, and in some of them, you've got some missing terms in the middle of the sequence to work out.

Then there's some reasoning-based questions where you can think about the term-to-term rule, have a think about what one of the later terms such as the tenth term might be, and have a think about explaining the following statement.

You've got all of these on your Activity Sheet, so I want you to go off and complete your activity now.

Pause the video and press Play when you're done.

Okay, hopefully, you've had time to complete your independent task.

Let's go over some of the answers together.

So for that first sequence, we can see our previous terms were minus 11 and minus 23 or negative 23, and our next two terms were 37, 49.

So our term-to-term rule was adding 12.

So we're adding 12 here and we're adding 12 here, the same jumps happening each time, meaning it must be a linear number sequence.

Now for our bottom one, we can see we're subtracting as we go down, and the term-to-term rule was minus 1.

8.

So by subtracting 1.

8, 1.

8, this missing term was 10.

9, 1.

8 and you should have got 7.

3 and 5.

5 or five and five tenths.

For this second part of the task, you had to look at this sequence and figure out the term-to-term rule.

A little bit more of a jump this time.

We were subtracting 104.

I'm sure you could have worked that out by finding the difference between two of them and just checking it's a linear sequence by looking at the difference between another two.

What would the tenth term be? It was 2,315.

So you had to do a little bit more investigating to find that.

Hmm, an integer ending in 80 cannot be part of this sequence.

Explain why.

Let's have a look at the pattern.

Now if you extended this to the tenth term, you would have noticed that there is a pattern in our ones, and all of the ones end in one, three, five, seven, nine.

All of these are odd numbers, and there are no even numbers in this sequence.

So you might have included that in your explanation.

I've got a challenge question for you.

You might have seen this on your Activity Sheet, so hopefully you've had a go already.

Okay, this is a linear sequence.

We know it's a linear sequence, so we know the same needs to be happening each time, increasing or decreasing with the same amount.

Can you find out the term-to-term rule and identify the missing numbers? Interesting.

Now I can't see two terms together, so it's a bit more tricky to work out the term-to-term rule.

So what do I know? I know that the first term is 13.

I know that the first, second, third, the fourth term is 28.

So I know that if I do one, two, three jumps, I'm going to go up by the difference between 13 and 28.

Now the difference between 13 and 28 is 15.

I had three jumps, so what's the jump for each one? Well, if 15 is three of the jumps, one jump must be add five.

So 13 add five should get us our second term.

Let's see if it works.

So we've got 13, add five is 18, add five is 23, add five is 28.

Now that works.

Missing numbers were 18 and 23.

Hope you enjoyed your lesson today.

If you want to, you can ask a parent or carer to help you share your work with Oak National.

Now it's time for you to complete your multiple choice quiz.