Lesson video

Well done for choosing to learn using this video.

My name is Miss.

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 securing understanding of sequences.

Today, we're gonna begin to generalise a sequence.

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

Remember, sequences don't have to have a mathematical rule, but the ones that we're looking at the structures of over this sequence of lessons are ones that are formed using a rule.

We're gonna look at continuing a sequence from a rule, and then we're gonna look at some representations of sequences.

Aisha started the sequence two, four.

She's gonna carry on her sequence, but we don't know what comes next.

Laura says, Aisha's sequence could double each time.

Sam says, Aisha's sequence could add two each time.

Sophia says, Aisha's sequence could add two, then add three, then add four.

Have a think about those three rules.

What will those sequences look like? If we use Laura's rule, we would get the sequence two, four, eight, 16, 32.

Using Sam's rule, we'll get the sequence two, four, six, eight, 10.

And Sophia's rule, we'd have two, we'd add two to get four, then we'd add three to get seven, we'd add four to get 11, then we'd add five to get 16.

So there are many different types of sequences.

In order to continue a sequence, you can use a rule.

So I'm gonna give you the first value as negative eight, and we can continue the sequence so it has the rule plus 10.

That has a constant additive pattern of plus 10 each time.

Let's start with the same value, but continue the sequence so it has the rule multiplied by 10.

And then finally, we could continue the sequence so it has the rule multiplied by negative 0.

5 or negative a half.

So you get positive four, negative two, positive one, negative 0.

5.

Obviously, there's an infinite number of other rules that we could apply to that start number.

Aisha started a new sequence.

She's gone with the values 10, then 30, and then we don't know what comes next.

Laura says the next numbers are 50 and 70.

Sam, the next numbers are 90 and 270.

Sophia, the next numbers are 60 and 100.

And then Alex has gone with 70 and 150.

Little bit of a challenge then, what rules might these pupils be using? Some might be easier to spot than others, see how many you can get.

Laura's rule, you might've said something like, add 20 each time.

Sam's rule, you might've said something like, multiply by three each time.

Sophia's was a little bit harder to explain.

You might've noticed that she added 20, then added 30, then added 40.

So saying something along the lines of, she's adding 10 more than she previously added.

Alex's was really tricky, so well done if you spotted this one.

He's doubling and then adding 10.

So let's look at continuing sequences with certain rules.

So how can you continue the sequence so it has a constant additive rule? We need to start by finding the difference.

So we can do 10 subtract negative two, and that gives us 12.

We know that to get from negative two to 10, we're adding 12.

Then we can carry on adding 12 to carry on our sequence.

That is one way to carry on that sequence.

Let's choose the same start numbers, but this time carry it on so it has a constant multiplicative rule.

So this time we need to know what we multiply negative two by to get 10.

That's 10 divided by negative two, and that gives us negative five.

So the multiplier is negative five.

We can carry that on then by multiplying by negative five again to get negative 50, 250, negative 1,250.

Notice when we have a negative multiplier, our values alternate between positive and negative.

Let's look at some of our fraction skills.

So how can we carry on this sequence so it has a constant additive rule? It might not be easy straight away to see that what we're adding.

One method that can sometimes help is converting to a common denominator.

So I'm gonna convert my 2/5ths into 4/10ths.

I've got the same value, just now it has the same denominator as 1/10th.

Easier for me to see now then that I'm adding 3/10ths each time.

Carrying on, I've got 7/10ths and 10/10ths, which I could write as one.

So now let's carry this on with a constant multiplicative rule.

So we need to find the multiplier by doing 2/5ths divided by 1/10th.

There's several ways of doing this.

For this example, I'm gonna convert to a common denominator again, but you do not have to if you have another method for dividing fractions.

You can see now why I chose to convert to a common denominator.

I get 1/10th and 4/10ths.

That gives me a multiplier of four, times in by four to get the next value.

I can carry on by timesing by four.

So I'm gonna have 8/5ths or 16/10ths, and then 32/5ths or 64/10ths.

Perfect, quick check then.

The pupils below are trying to continue the sequence negative three, then nine.

They've all continued it in their own ways.

Have a read of their next two numbers.

I want to know who has continued it correctly with a constant additive pattern, and who has continued it correctly with a constant multiplicative pattern.

The other people might have other rules, but which one is constant additive, or which one is constant multiplicative? Give it a go.

Well done if you spotted that Sophia has a constant additive rule, the rule would be plus 12 each time.

Sam has continued it with a constant multiplicative pattern.

They have done multiply by negative three each time.

Chance for you to have a practise then.

So for each of these two starting numbers, I'd like you to continue the sequence in two ways.

First with an additive pattern, and I want the next three values.

And then with a constant multiplicative pattern, and I want the next three values.

Once you've done that, I'd like you to describe each of your sequence in words.

Have a go at that, and we'll look at how we've described our sequence in a moment.

Well done.

Let's have a look at the second set.

So exactly the same idea.

We're gonna have a look at maybe some more of our fraction skills and our decimal skills.

Make sure you're leaving your answers in a way that you are happy with, whether that be as a fraction or a decimal, pick which one makes most sense for the context of the question.

Give that a go, and then we'll look at what we've got.

Well done guys.

So the first one, to make it have a constant additive pattern, we'd have five, seven, nine.

Your description might vary from mine, but I've gone something with start on one and then add two each time.

I made it clear what value I was starting on because I know that lots of different sequences could add two each time.

So that's why I've put that in there.

The multiplicative pattern, nine, 27, 81, starting on one but multiplying by three each time.

The good thing about my description here is that anybody would be able to read that description and write the right sequence.

For B, well done if you spotted that you're adding 0.

9.

So you should have 1.

9, 2.

8, 3.

7.

For the multiplicative pattern, we're gonna multiply them by 10.

So you should get 10, 100, 1,000.

Start on 0.

1 and multiply by 10 each time.

For C, this time we're adding 24 if it's an additive pattern.

So we start on negative 16, add 24.

So that gives us eight, then 32, then 56, then 80.

As a multiplicative pattern, we're starting on negative 16 and multiplying by negative a half.

You could say divide by negative two, that's the same thing.

So you should have negative four, two, negative one.

Let's look at that second set.

Let's look at this second set.

I wonder if you found some of these a little bit trickier.

Remember, it's always good to use a written method to help you with any of your calculations.

They should form part of your working out.

Drawing on the arrows like we did in the examples can also help you see what is happening.

So for an additive pattern, we're adding one.

So you should get 2.

5, 3.

5, 4.

5.

For multiplicative pattern, we're multiplying by three.

So you should get 4.

5, 13.

5, 40.

5.

Looking at our fractional sequence for E, we're adding an eighth each time.

It would have been easier to see that if you converted a quarter into 2/8ths.

So you get 3/8ths, 4/8ths, which you might have written as a half, and then 5/8ths.

For our multiplicative pattern, you might have seen that we're doubling each time.

So you'll then get a half, then one, then two.

So starting on an eighth and multiplying by two.

For F, we are subtracting 15, or you might have written that as add negative 15.

So you should get negative five, negative 20, negative 35.

This was quite a tricky one to make into a multiplicative pattern.

Doing 10 divided by 25 will tell us what our multiplier is.

So 10 over 25, which is 2/5ths.

So 10 times 2/5ths is four, times 2/5ths again is 8/5ths, times 2/5ths again is 16/20 fifths.

If you converted those into decimals, they are written underneath.

25, 10, four, 1.

6, 0.

64.

Our rule then was start at 25 and multiply by 2/5ths.

That's the same as 4/10ths, or 0.

4 if you wrote as a decimal.

Well done, that last one especially was really quite tricky, so fantastic working today.

We're going to look now at representations of sequences.

So Izzy is building a sequence with Cuisenaire rods.

If you have any Cuisenaire rods available or Cuisenaire rods on a computer programme, you might want to duplicate her sequence.

Each tower is a value in the sequence.

The first tower is the first value in the sequence.

The second tower is the second value in the sequence.

What could the rule be for her sequence? Pause it and come up with your own answers.

Loads of rules that you can have.

It currently looks like it has a constant additive pattern, the same value being added each time.

You might have given it a value, so you might have said add two each time or add five each time, but we do know that we're adding a single red block each time.

Which sequence then has the same additive pattern as Izzy's sequence? There's your options, pause and have a think.

Lovely, none of the sequences are the same as Izzy's sequence but some have the same additive pattern.

So that bottom left one also adds one red rod each time.

That top one though looks to be doubling each time because first it adds one red rod and then it adds two red rods.

It might then add three red rods or it might double again, we're not sure.

That sequence is adding the larger rod, which is purple, each time.

So that's a different value that's being added each time.

And that bottom one is adding the larger green rod each time.

It's not adding the red rod the same as Izzy's.

Have a look at these four sequences that have been constructed.

Again, each tower is a value in the sequence.

What is the same in these four sequences and what is different in these four sequences? See if you can put it into words.

You might have said something like, they all start on the same value.

They all start with one little white block.

You might have also said that they all have a constant additive pattern.

So far with the first three values, it looks like they're all adding the same value each time.

However, the value that's being added each time within the sequences seem to be different.

The first one's adding a green block, second one a red block, third one another white block and the last one a purple block.

Let's do the same with these four sequences.

What's the same and what is different? Off you go.

You might have said something like, they all have a constant additive pattern.

Each sequence seems to add a green rod each time.

However, the value that the sequence starts on changes.

So they're not the same sequence because they're starting on a different value each time.

So here's Izzy's original sequence again.

Is this the same sequence? What do you reckon? No, well then if you spotted that it has some of the same elements, so it has the same constant additive rule, but it starts on a different value.

What about this one? Is this one the same sequence? What do you think? So no, it has some of the same values in it.

However, it starts on a different value.

So although they're very similar sequences and they will have a lot of the same values, they're not exactly the same sequence because they start on different values.

What do you think about this sequence? Well then if you spotted that yes, this is now the same sequence as Izzy's, both are made of the same values in the same order.

They start on a purple rod and they add a red rod each time.

Have a look at these two sequences as think about what is the same and what is different.

Well done if you said something like they both start on the same value.

The first one is adding one red rod each time and the second one is adding one purple, that longer rod each time.

Okay, quick check then.

These are slightly different types of representations, but can you match the diagram with the rule? So which of those diagrams is showing sequence that adds one each time? Which one adds two each time and so on? Give it a go and we'll look at our answers.

Brilliant, well done.

That first one add one each time matches with the last diagram.

Add two each time matches with the diagram underneath it, the second one.

And four each time matches with the diagram underneath it, the third one.

And multiplied by two each time matches with that left-hand diagram.

Time to put all those skills in for a practise.

So Izzy has built these patterns using blocks.

Which two patterns could she combine? So adding the first two towers together, then the second two towers together, then the third two towers together, so that she has a sequence which increases by five each time.

And then which two could she combine to make a sequence that increases by two each time? Then which two would make a sequence which decreases by one each time? If you have some blocks that you can move around yourself, that might be helpful, okay? Otherwise, you might wanna think about what is happening with each of these sequences and then what is gonna happen when you combine them.

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

Well done.

Second set of questions then.

Laura and Andy are making a repeating pattern from different shaped counters.

You might notice that we get a triangle, three ovals and a square.

That is going to repeat.

Each counter is also numbered.

So we start with one, two, three and four.

And that means we can refer to specific counters as well.

So first question, what shape counter will the number 25 be? If you can explain your answer even better.

B, Laura puts down the counters 144, 145, 146.

What shapes are they? Again, if you can explain how you can tell, then you're developing your mathematical explanation skills.

Andy puts down three counters in a row that are all oval shapes.

They are greater than 50 but less than 60.

What numbers could they be? And finally, they run out of counters when they put down the counter 178.

Can you then use the patterns that you've noticed to work out how many of each shape counter they used in total? Give those a go and we'll look through our answers.

Well done.

That first one then, we want a sequence that increases by five each time.

Now what might have helped if you'd written the pattern for each sequence? So the first one seems to increase by one each time.

So that's A.

And D seems to increase by four each time.

If A increases by one, D increases by four, we combine those patterns, we'll get a sequence that increases by five each time.

So the answer was A and D.

For two, we wanted one that increases by two each time.

This is a little bit trickier.

So B is decreasing by two, and D is increasing by four.

So if we decrease by two but increase by four, we get an overall increase of two each time.

So well done if you spotted that second one was B and D.

And the third one decreases by one each time.

So we're definitely gonna need that decreasing sequence.

And then we need to add A.

So we've got A that's increasing by one, and we've got B that's decreasing by two.

So overall, we get a decrease of one each time.

Looking at Laura and Andeep then.

So shape number 25 is going to be a square.

You might have noticed all multiples of five are squares.

And that is a pattern that's gonna really help us with the rest of our number system.

So Laura puts down the counters 144, 145, and 146.

What shapes are they? Well, we've already suggested that all the multiples of five are going to be squares.

So 145 must be a square.

That means 144 is an oval and 146 is a triangle.

It's really useful in this case to see where the repetitions are.

One, two, three, four, five seems to be one section of the pattern, and then that five repeats each time.

And that gives us the structure where we know that every multiple of five is going to be a square.

We want oval shapes that are greater than 50 but less than 60.

So again, we know the square will be 50, 55, and 60.

So the ovals are gonna be 52, 53, and 54.

Or you could have had 57, 58, 59.

Even better if you managed to work out both of those different answers.

And the last one, again, I would use our multiples of five to help.

So if we get 178, so if we think about 180, 180 would be, so if you do 180 divided by five, that's 36.

So if we'd all got up to 180, we would have 36 squares.

We didn't quite get up to 180, so we'd only have 35 squares.

In terms of our triangles then, notice that we always have a triangle for before we have a square.

So the number of triangles match up with the number of squares.

But remember, we didn't quite reach 180.

We only got to 178.

So there's actually gonna be one more triangle than there will be a square because you'll have a triangle at 176, one more than a multiple of five, but you won't then get this square at 180 because we ran out at 178.

So 35 squares, an extra triangle, 36 triangles.

In terms of the ovals then, you're going to have 107.

Again, for each square, you're gonna have three ovals before it.

So three lots of 35, which gives you 105.

But then 177 and 178 are also ovals.

So we've got another two more to add on.

Well done today.

Let's have a look at what we've learnt.

If we know a rule for a sequence, we know we can continue a sequence from a starting value.

We've also looked at describing sequences in words.

We know that we have to have enough values to do that in order to spot the pattern if there is a pattern.

There can be more than one rule linking to successive numbers in a sequence.

And remember that there are sequences that don't follow a mathematical rule at all.

And then we can demonstrate certain rules with diagrams as well.

Using a diagram can be helpful to see how a pattern is developing.

And if you carry on working with your sequences skills, you may find more and more diagrams become more and more useful the more skills you develop.

Thank you for joining us today.

And I really look forward to working with you again.