Lesson video

Well done for choosing to learn using this video.

My name is Ms. Davies, and I'm gonna help you as you work through the lesson.

With that in mind, it might be really useful to be able to pause bits, really think about some of the concepts that we are talking about, and I'm gonna help you by adding in any suggestions, or any hints that might help you with all the different tasks we're gonna be working on.

Let's have a look at this lesson then.

Welcome to our lesson on formalising a sequence.

By the end of the lesson, you'll appreciate that sequence is a succession of terms usually formed according to a rule.

So that's the definition of sequence that we have been working with.

We're also going to have a look at something called a term today.

In a sequence each value or pattern is called a term.

We're gonna start by looking at pictorial sequences.

A sequence can be made from a pattern of objects.

You may have looked at sequences, which are number sequences.

We're gonna look now at sequences which are patterns, or which are objects which form a pattern.

So here's the first pattern in a sequence.

Pause the video, and have a think about what the next pattern could look like.

If you've got square paper, you might want to draw it out.

So here are some of my suggestions.

You might have come up with other suggestions.

Take a look at them.

To be able to draw the next pattern in a sequence, we have to understand what the pattern's going to be, and how that pattern develops.

So there you go.

Here's the first two patterns in a sequence.

Have a think about what you think the next pattern could look like? And if you know that, how could we then describe how the pattern is continued? Pause and have a think.

So I've continued the pattern this way.

So that then is my third pattern in a sequence.

There's lots of ways of describing this pattern.

Three of our students have had a go at doing that.

So Aisha has said that each pattern increases by 1.

Lucas has said each pattern increases by 2.

Sam has said each pattern increases by 3.

Pause the video and have a think about what each pupil might be referring to with their answers.

So they all could be correct depending on what it is they are referring to.

So Aisha said each pattern increases by 1.

What she might have been looking at is the white squares in the middle of the pattern, in which case they are increasing by 1 each time 'cause an extra white square is being added.

She might have also been talking about the width of the pattern.

If you measure the width of the pattern that's going up by 1 each time.

Lucas on the other hand, might have been referring to the number of black squares around the edge of the pattern.

That way we're adding 2 black squares each time, 1 on the top, 1 on the bottom.

He could have also been referring to the perimeter of the pattern, that would increase by 2 each time.

Sam on the other hand, could have been referring to the number of squares overall in each pattern if we combine the black squares, and the white squares, or they could have been referring to the area of the pattern.

Both of those things would increase by 3 each time.

So we can look at individual elements of a pattern to describe how it develops.

It's important that we are precise with which object in the pattern we are referring to.

So for example, in the previous pattern where we're referring to the white squares, the black squares, the overall squares, the area, the perimeter.

Let's have a look at this pattern.

This pattern is made of three solid red lines, and two diagonal dashed blue lines.

They look a little bit like houses.

How would you describe how this pattern is developing? Pause the video, and see if you can put it into words.

We'll have a look at your ideas in a moment.

Let's think about this element.

What do you think is staying the same in this pattern? How could you describe that? You may have said something like each new red section, there are two new blue dashed lines.

You might have said something like, the pattern is staying at the same height, we're just adding on more houses.

Okay, how are the red lines developing from pattern to pattern? You may have said something like two extra red lines are added each time.

Each house shape needs two more red lines, something like that.

That's how the pattern is building.

How are the blue dashed lines developing from pattern to pattern? You may have said something like the two blue dashed lines are added each time, or each house shape needs two more blue lines.

So all of those could have come into your explanation of the description of how that pattern was developing.

So Aisha says, "Each pattern needs two red lines, and two dashed blue lines." Therefore the fourth pattern, the next one, will have eight red lines, and eight blue lines.

What do you think? I wonder if you said something like this.

It's correct that each pattern will have four more lines.

However, the first pattern has an extra red line at the start, therefore she's wrong.

The fourth pattern will not have eight red lines, it'll actually have nine.

Lucas says, "Each how shape needs three red lines, and two blue dashed lines.

For four house shapes then, there will be 20 lines overall." what do you think about his reasoning? I wonder if you have said something similar to this? He's correct that each house needs two blue dashed lines, and I suppose he's looking at that first pattern where he said that, "Each house needs three lines." However, after the first house, the next houses only need two extra red lines 'cause one of them is already there from the previous house.

See if you can apply all that understanding then to this question.

How many red lines will be needed for 10 houses? How many blue dashed lines would be needed for 10 houses? How many lines overall will be needed for 10 houses? What do you think? Before we have a look at your answers to those, have a think about this one.

How many blue lines will there be in the pattern, which has 33 red lines? See if you can use what you've said so far to help you.

Lovely, well done if you said 21 red lines, 2 per house, and then one extra 1 for the first house.

There'll be 20 blue dashed lines, 2 per house.

So overall that's 41 lines.

The number of blue lines is always gonna be one less than the number of red lines.

So 32 blue dashed lines for that last question.

Let's check together then.

So some pupils have drawn the first three patterns of a sequence on squared paper.

You'll see they look a little bit like ladders.

I'd like you to match their statements with the element of the pattern they are describing.

So one person says, "Each pattern increases by 1." Somebody else says, "Each pattern increases by 2," "Each pattern increases by 3," and, "Each pattern increases by 4." Which element of the pattern are they thinking about in each of those cases? All right, well done.

Each pattern increases by one is if you're talking about the number of vertical lines, By two, if you're thinking about the width of the pattern.

By three, if you're thinking about the number of horizontal lines.

And by four, if you're thinking about all the lines overall.

A practise, then.

Andeep is making different patterns that go up in fours.

Alex is making different patterns that go up in threes.

They show one image from each of their patterns.

So A to F is just one image from their pattern.

It could be the first image, or the third image or the fifth image.

It's just one of the images.

Which do you think are Andeep's images? Which do you think are Alex's? It might be possible that you think some could belong to both.

That's fine as well.

Give it a go.

So Sam has drawn a dog by shading in squares.

Their classmate each continues the pattern in different ways.

So you'll see that Sam's dog is in that top right hand corner.

Aisha continues the pattern in the way shown for question two.

Can you describe how the pattern is developing in words, and can you use that to find how many squares would be needed for the 10th dog? Lucas starts with the same dog but continues it in a different way.

See if you can describe how that pattern is developing, and again, use that to see how many squares are needed for the 10th dog.

Try those two and we'll look at the last bit.

Okay, the last way of continuing the dog is Izzy's way.

So Izzy continues it this way.

See if you can describe how her pattern is developing, and then a little bit of a challenge.

Can you work out how many squares needed for the 10th dog this time? Off you go.

Nice.

Well done guys.

So A, I reckon you probably could have put this in Alex's or Andeep's, so I've gone with either.

B, this looks like a pattern that's going up in fours, so that's Andeep's.

C seems to be a pattern increasing by three, so I've gone with Alex.

D, I think was a little bit more ambiguous.

If you are referring to horizontal lines, it's going up in threes, but lines in total is going up in fours.

So I've gone with either.

For E, Andeep seems to be going up in fours.

And with F, I think it's impossible to see whether that's gonna be increasing horizontally or vertically, so I think that could be adding four or adding three.

So either.

Looking at the dogs then.

So Aisha, I've said the body is getting longer by one, and each leg is getting longer by one.

You might have said something similar.

So overall we're adding one to the body, and two, one for each leg to the legs.

How many squares will be needed for the 10th dog? It's gonna be 34.

Well done if you got that right.

Lucas', so I've said the body is getting longer by one.

And then I've called it the neck part.

So I said the neck is getting longer by one.

So one in the body, one in the neck.

Overall that's gonna be 25 for the 10th dog.

This one was a real challenge.

So I've covered my squares a little bit to see if I can show you what I think is happening with this one.

So the legs are both getting longer by one, like the other patterns, so two in total.

The body increases in both directions.

So what you end up with, you end up with a one by one square for the first dog.

But the second dog has a two by two square, so four squares.

The third dog a three by three squares, so nine squares in total, and so on.

I'm gonna use that way of looking at it to work out the 10th dog.

So the 10th dog is gonna have a 10 by 10 square in the middle.

Then it'll have 11 squares on each end for the legs, and then 2 for the head.

So 124 in total.

Well done if you got that final answer.

That was tough.

Okay, so we're now I'm gonna have a look at describing the patterns in pictorial sequences in a bit more detail.

So each value or pattern in a sequence is called a term.

So if we're looking at numerical sequence, each number is called a term.

If we're looking at a pictorial sequence, each pattern is also called a term.

We're gonna start using that language when we're talking about patterns.

So this is our previous pattern, and we've got Term 1, Term 2, Term 3.

We can describe a sequence by looking at how each term relates to the previous one.

How would you describe what is happening to the number of black squares using this language, term? You might have had something like they increase by two from one term to the next, or add two squares to one term to get the next term.

You can also describe a sequence by looking at the relationship between the different objects in the pattern.

How many white squares are there in each term? So how many white squares in the middle? Well one in Term 1, two in Term 2, three in Term 3.

So in this case the white squares is the term number.

So if we wanted term 10, we know that would have 10 white squares in the middle.

That's how we're gonna identify these patterns.

What is the relationship then between the white squares, and the black squares? If I told you the number of white squares, how would you work out the black squares? I'd like you to try this one.

Pause the video.

Okay, I wonder how you went about this.

So I went about it like this, ignoring the three black squares on each end.

Then for each new white square you're gonna get two new black squares.

So double the white squares to get the black squares.

But then we know we're always gonna have three on one end, and three at the other end.

So add on six.

So I've got double the number of white squares, add six.

I wonder if you came up with something similar.

So how many black squares will there be in Term 10? Well the first pattern has eight, so we can add two until we get to Term 10.

Or we can use that second rule.

If we know there's two black squares for every white square, we could do 10 times 2, and then remember to add on 6, the 3 for each side.

It could be helpful to write the number of a certain object in a pattern so that we can see the rule for the sequence.

And rather than just looking at the patterns, actually write down the numbers.

So for this sequence we could count how many lines make up each pattern.

Let's do that then.

So we've got 3 and 9, then 15.

This can now make it easier to work out Term 4 'cause we can see that there is this constant additive pattern of plus 6.

Now we can see Term 4 will be 21.

Instead of looking at the number of lines, let's look at the perimeter.

So counting around the outside.

So we've got 3, 6, 9, and then 12.

And again, we can see this how this sequence is developing maybe a little bit easier from the numbers than we could just from the shapes.

Pause the video, and see if you can put into words what the pattern is in the perimeters.

You might have said something like this, for each term the perimeter increases by three.

Well done if you also notice that the term number multiplied by three gives the perimeter.

Right a check then.

Which picture would come next to make the number of shaded squares have a constant additive pattern? What might help you is to write down the number of shaded squares.

I want it to have a constant additive pattern.

So which one would be the next in the sequence if that was the rule? Lovely, well done if you spotted it was C.

The number of squares so far is 1 and then 5.

So that's adding 4.

So we need the one that's then made up of nine.

Right, let's have a practise.

For each sequence made out of squares, I would like you to fill in the table, and then describe the rule.

You've got two elements, you need to fill in the table for the number of squares, so how many squares in each term, and then write the rule.

And then the perimeter, and then write down the rule.

I filled in some of them to help you.

There's two there.

And then we'll look at the next set.

Well done on those first two.

So the same idea this time you need to count the number of squares, and write down a rule.

And then you need to work out the perimeter on the outside, and write down the rule.

Give it a go.

Okay, a slightly different question this time.

In a school dining hall, six chairs, which we're gonna represent by circles, fit around one rectangular table.

And you can see that over in the top right corner.

Below, A, B, C, and D are four different ways to arrange four tables.

So if I had four tables, those are the different ways I can put the tables.

Notice I haven't put the chairs around the outside.

Here's your question then.

To calculate the number of chairs needed, the caretaker multiplies the number of tables by six.

Which way do you think that means he's planning on arranging the tables? Then for each different arrangement, A, B, C, and D, how many more chairs can be added if an extra table is added? So if I add a fifth table, how many more chairs would that be? Give that a go and we'll look at the next bit.

Final set then.

So we have now decided that the tables are gonna be arranged end to end like the diagrams below.

How many chairs would fit around four tables? You can see I've drawn one, two, and three.

How many would fit around four tables? Can you describe the relationship between the number of chairs, and the number of tables? How many chairs can fit around 10 tables? And then your final challenge.

Jacob has 25 classmates.

How many tables are needed so all 26 of them, Jacob plus 25, can have a chair? Give that a go, and then we'll see whether we agree.

Right, well done.

So first one, your square should be 1, 2, 3, 4, 5, and some rule you could have written it as, increased by one each time, or you might have said that the squares are the same as the term number.

You can describe the rule either way.

The perimeter you've got 4, 6, 8, 10, 12.

You might have said that that is increasing by two each time.

Or if you related it to the term number, you might have said two lines for each square, plus two on each end.

For B, your squares, you've got 1, 3, 5, 7, 9.

So you could have said increased by two each time, or you might have looked at how the sequence is building, and looked at the kind of diagonals, and said it's two squares per new diagonal minus one, because the first diagonal only has the first square.

Perimeter you've got 4, 8, 12, 16, 20.

You could have either written that as increase by four each time, or you might have said it is the term number multiplied by four, and spotted that relationship that's the four times table.

For C, we've got 7, 11, 15, 19, 23.

And our rule for the squares could be increase by four each time, or you might have looked at these eight shapes and said that it's four squares per eight shape, plus the three for the start.

A little bit similar to the house building one we had earlier.

The rule for the perimeter is increase by eight each time.

It starts on 16, you get 16, 24, 32, 40 and 48.

For D, the squares, 1, 3, 6, 10, 15.

This was an interesting sequence.

We increased by two, then three, then four.

You might have also noticed that you just add the term number onto the previous term.

So if I wanted Term 5, I take the previous term, which was 10, and add on 5, 15.

The next one you'd then add on 6, to get 21.

The perimeter you are increasing by four each time.

Or you might notice that it's the term number multiplied by four.

Let's look at question two.

So if the caretaker multiplied the tables by six, he must have been thinking about arrangement D where there's six chairs around each table.

Looking at question B then.

So for A, it's gonna depend where you put the extra table.

So you can either have two extra seats if you put the long sides of the table touching, or four extra seats if you put the short sides of the table touching.

For B, then it's two extra seats.

And for C it's four extra seats.

That final one D, we've already talked about, that's gonna be six extra seats per table.

Last set of questions, how many chairs fit around four tables? That's gonna be 18 chairs.

A relationship between the chairs and the tables.

So each table has four chairs plus one for each end, so plus two for the ends.

How many chairs can fit around 10 tables then? So you've got 40 plus 2 on the end, so 42.

Jacob has 25 classmates.

How many tables are needed so all 26 of them have a chair? Well if you forget about the 2 on the ends, that's 24, so that's 6 tables plus the 2 on the end.

So 6 tables in total.

Well then I hope you really feel like those sequences skills are really beginning to develop.

So we have looked at how a pattern of objects can be made following a rule.

We've talked about describing the rule in words can help us find the next pattern.

And we started using this language of term.

Each value or pattern in sequence is called a term.

And sometimes writing the terms as numerical values rather than pictures can help us spot patterns.

Well done today guys, and I really hope that you choose to join us again.