Lesson video

It's really nice to see you in this lesson today.

My name is Ms. Davies, and I'm gonna be helping you as you progress through the lesson.

With that in mind, please feel free to pause things so you can really think about the concepts we are discussing.

Let's get started.

Welcome to today's lesson on checking your understanding of sequences.

By the end of today's lesson, you'll appreciate the structure of different patterns and sequences, and we are looking at specific types of sequences today.

So we need to be really clear what we're talking about when we're talking about sequence.

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

Sequences don't have to be formed according to a rule, but the ones that we are gonna be looking at, and we're gonna be looking at the structure of them, are gonna be sequences that have a mathematical pattern.

So we're gonna look at two different types of sequences.

We're gonna look at sequences with a constant additive pattern, and then we're gonna look at sequences with a constant multiplicative pattern.

So lots of opportunities today to work with our numeracy skills.

So we're gonna start with this idea of a sequence that we've already defined, and the first sequence we're gonna look at is this one.

So we've got three, then six, then nine, then 12.

And notice we've got a dot, dot, dot because this sequence is going to continue in the same pattern.

Pause a video and have a think about what you notice about this sequence.

Brilliant, so you may have said something along the lines of if you add three to the previous number, you get the next number.

You might have said that the numbers are going up in threes.

You might have also noticed, this is actually the beginning of the three times table.

One times three is three, two times three is six, three times three is nine.

Let's look at that second idea of times tables for the moment.

So here's the section of the two times table.

I've put it on a number line.

Now when you were learning your two times table, you possibly only went up to 12 times two.

So this might be beyond the multiplication tables that you learnt.

However, we are just carrying on that pattern of the two times table.

What I'd like you to think about is how we would calculate the missing value after 40.

You might be able to tell me straight away what it is, but what I want you to really think about is what process you are doing.

How are we calculating that value? Pause the video and see if you can put it into words.

Brilliant, so what we're seeing is we're seeing that each value is going up by two.

There's this rule of add two between each of our numbers in order to get the next number that we're gonna need to carry on that rule and add two again to get 42.

Might, as I say, seem really simple at the moment, but it's looking at these structures and these processes that's gonna help us when we're looking at more complicated sequences.

So what process would we do to calculate the missing value before 34? Again, pause the video and see if you can put it into words.

Okay well if our rule moving along our sequence is add two, then we need to do the inverse if we're moving back.

So the inverse of adding two is subtracting two.

So to get the next value or the previous value before 34, we do 34 subtract two, and that gives us our 32.

So here's a section of the three times table.

So to calculate the missing value after 129, we can see that we're adding three to get our next value, so we need to add three onto 129, and that gives us 132.

Again if we're adding three, as we're moving to the right, we need to think about our inverse in the other direction.

So our inverse of adding three is minus three, so we need to do 126 minus three to get 123.

So Jacob is writing his four times table.

He's gone through a lot of values at his four times table and he's made it to 224, 228, 230 and 234.

Right, how do you know straight away he has made a mistake? What do you think? Right, we don't particularly want to check that each of those values are in the four times table, but we can see clearly that there's a mistake because numbers in the four times table should increase by four each time, whereas the difference between 228 and 230 is only two.

So multiplication tables are an example of sequences with a constant additive relationship.

We are adding the same value each time.

They're not the only example, but they are an example.

It might seem trivial, but it's actually a really useful thing to spot that successive numbers, so numbers next to each other in the two times table, have a difference of two.

Successive numbers in the three times table have a difference of three, and successive numbers in the four times table have a difference of four.

Right, we're gonna use that then to have a bit of a check for understanding.

So each sequence is a section of the multiplication table.

You don't need to check that.

I'm telling you now that those four sequences are part of a multiplication table.

I would like you to match the sequence with the multiplication table it is part of.

Off you go, and we'll look at our answers.

Fantastic so that top one, hopefully you spotted that we've got a constant additive pattern of add seven each time, so that must be the seven times table if I'm telling you that it's definitely part of a times table.

B, we've got a constant additive pattern of add five each time, so that must be part of a five times table.

And again, we know that because we know values in the five times table end in zero or five.

C are going up in 11 so if they're a part of any times table, it'll be the 11 times table and D, they're going up in nines so if they're part of any times table, it's the nine times table.

Brilliant, so below is a 100 square.

Lucas has shaded all the numbers in the 10 times table that are in 100 square.

If he writes these in ascending order, they will form a sequence with a difference of 10 between each value.

Let's have a look.

Yeah so you'll get 10, then 20, then 30, then 40, and that will be our 10 times table.

This sequence, like the ones we've looked at already, follows a constant additive rule.

You get the next number, you always add 10 to the previous number.

So Jun has shaded a different set of numbers and he says, I have shaded the nine times table.

Just pause the video and think about whether Jun is correct.

Well then if you spotted that, no, he's not correct.

The first number is nine, but the second number is 19, the third number is 29, and they're not in the nine times table.

In order to shade the nine times table, he'd actually end up with a diagonal line across his hundred square.

However, do the numbers Jun has shaded form a sequence with a constant additive rule? It's not the nine times table.

What do you reckon? Yeah, he's shaded nine, 19, 29, 39 and 49.

So although it's not the nine times table, it is still a sequence with a constant additive rule.

We're adding 10 each time, right? Some sequences then are formed by a constant additive rule.

We've seen quite a few already.

Finding the difference between successive numbers can help us spot patterns in a number sequence.

So we have a look at this sequence, we dunno what part of the sequence we're looking at but we know we've got 14 and then the next number is 19.

To find the difference, so to find what our sequence is increasing by, we can do the calculation 19 subtract 14, and that gives us an answer of five.

So we know that moving right in our sequence, we're adding five each time.

It's quite useful when you are looking at structures of sequences to write them out with spaces on your page and then you can use arrows to show the operations to get from one to the next.

We know then that if we want to carry on this pattern, we need to add five to 19 and that'll get us the next value.

Just like we talked about with our multiplication tables, in order to find the previous value, we would need to do the inverse.

So the inverse is subtract five, so 14, subtract five would give us nine.

So let's try it with these different sequences.

So how could we calculate the difference between successive values in these sequences? I'm telling you now that they're all formed by a constant additive rule.

See what you think, we'll look at them together.

Perfect so the top one, we could do 31, subtract 29 to see that our sequence is adding two each time.

Equally, you could have done 33, subtract 31.

If we know it has a constant additive rule, then we can pick any two numbers next to each other to find the difference.

So this time we're gonna do 584, subtract 258.

Equally, we could have done 910, subtract 584.

We just need to subtract successive numbers.

Now that's a little bit trickier to do so I'm gonna use a column subtraction method.

It is always okay to use written methods to help you with trickier calculations.

That's why when we start our mathematics journey, we learn these skills so that we can use them later on.

So please make sure that you write those down as part of your working if you are using those methods.

They're always a good tool for you to use.

So I'm gonna do four, subtract eight.

That gives me a negative value.

So looking at my 10s column, I can make that four into a 14 and then do 14, subtract eight.

That gives me six.

And then I've got seven now in my 10s column, so I've got seven, subtract five is two, and then five subtract two is three, so our difference is 326.

You might wanna check by doing 910, subtract 584 and seeing if you get the same value.

Okay with our decimal calculations, so again we're doing 9.

2, subtract 7.

8.

What I would probably do with this one is start on 7.

8 and count up 'til I get to 9.

2 and see what the difference is.

So this time we're adding 1.

4.

Right, so forming a sequence by adding a negative value will create, a decreasing sequence.

So we look at 23 to 19, we have added negative four, and that has created a sequence that's gonna decrease each time.

We also know that adding negative four is the same as subtracting four, so you might prefer to write that as subtract four.

Either way round is absolutely fine.

To get the next value, we're gonna add negative four again or we're gonna subtract four from 19 and that gets us the value of 15.

Now if we're working in the other direction, we're going to need to subtract negative four, or as I've written on the screen, that's the same as adding four.

We're doing the inverse of whatever we are doing, moving right.

If we add four, get 27.

Right, let's check then.

So the values 38, 34, 30 form a sequence that follows a constant additive rule.

What do you think? Well, I dunno if you spotted that that one is true.

Have a look at these justifications.

Which one do you think supports your argument? Lovely, so 30 subtract 34 is negative four, so the next number can be found by adding negative four.

You could have also said that you are subtracting four each time.

It's the same thing.

Let's have a look at this one, 27, 41, 57 form a sequence that follows a constant additive rule.

What do you reckon? That one is false this time.

Have a look at the justifications.

Why is that false? Well, and if you spotted that the same value hasn't been added each time, the difference between 27 and 41 is 14, but 41 and 57 is 16.

They're not in the same times table but that doesn't mean that they can't form a constant additive rule.

Multiplications are just one type of constant additive sequence, but they're not the only type.

These sequences again continue in equal steps.

Can you work out the previous number in each sequence? Off you go.

All right, we're gonna do a couple of steps to our working.

So we're gonna find that difference so we're adding three each time for the top one, adding 0.

6 each time for the second one and we're adding negative 1.

9 or subtracting 1.

9 for the bottom one.

That means our previous values are 26, 12.

1 and then the bottom one, we're gonna do 4.

8, subtract negative 1.

9 which is the same as doing 4.

8, add 1.

9 and that gets you 6.

7.

Really good chance to practise your negative number skills, today, making sure you're happy with adding and subtracting negative numbers as you move along a numbered line.

Time for practise then.

So these pupils are making number patterns from a rule.

So it starts with Aisha and then it works down the line to Sophia at the end.

Aisha is saying a number, Andeep is adding a number to it.

Each pupil then adds the same number that Andeep has added.

I want you to think about these questions and I want you to justify your answer.

Explain why you know you are correct.

So A, if Aisha says six and Andeep says 10, what then will Izzy say? If Aisha says 6 and Andeep says 10, working down the line, what will Sofia say? Have a go for all those questions, A to E, making sure you think about your justifications.

Well done.

Let's have a look at our answers then.

So if Aisha says 6 and Andeep says 10, Izzy's gonna say 14 as Andeep has added 4 to Aisha's number, so adding another 4 get us to 14.

Well done if your justification was nice and clear.

If Aisha says six and Andeep says 10, what will Sofia say? So Sofia will say 42 as four will have been added another eight times after Andeep says 10.

Okay you could have done that in one step or you could have added four each time until you got to Sophia.

That's absolutely fine at this stage.

So after a different starting number, Sam says 17 and Alex says 19.

They're next to each other in this pattern.

So that means each person is adding on two.

So what we can do is we can subtract two, seven times to get back to Aisha's answer of three, so Aisha must have started on three.

You can always add two down the line to check that you are correct, and Sam does end up saying 17.

If Aisha says four to start and Jun says 40, what will Lucas say? Okay this might take a little bit more thinking.

So Lucas is saying 49, but I dunno if you've got that one correct.

What we've got is we've got one, two, three, four additions between Aisha and Jun.

So that's got to be 36 in total.

36 divided by four is nine, so we know that each pupil adds nine.

You could have also done a bit of trial and error until you got to the right value for Jun.

Lucas then adds nine again so Lucas is 49.

And the last one, well done if you managed to get something that worked for this last one.

So Aisha starts with 10.

Only Lucas's number is another integer.

What could Andeep have said? Well, you could have had any non-integer value which is a multiple of 0.

2.

You could have had 10 plus 0.

4 is 10.

4, or you could have had 10 plus 5.

6 which is 15.

6, or 10 plus 3.

2 which is 13.

2.

But essentially your value needs to have two, four, six or eight in the tenths column, so after your decimal point.

It can't be an integer, otherwise Andy, Izzy and Jacob would also be saying integer values.

So any value 0.

2, 0.

4, 0.

6 or 0.

8 would work for this question.

Well done.

So now we're gonna have a look at sequence with a constant multiplicative pattern.

Jacob has written the values two, four, eight.

Jun says these could be the first three numbers in a sequence.

Luca says, if Jacob was writing a sequence, he has missed out the number six.

Do you agree with either of these statements? Okay, so Lucas is probably thinking of the sequence two, four, six, eight, which is formed by using a constant additive rule, like the ones we've been talking about so far.

However, Jun might be thinking of a different rule that could make this sequence.

Remember, sequences don't necessarily have to follow any mathematical rule, but there are mathematical rules other than adding the same value each time, which we can find and we can use.

So we're gonna look at the multiplicative rule.

So we could continue Jacob's sequence by following a constant multiplicative pattern.

So you might have spotted that we are multiplying by two, multiplying by two and multiplying by two.

So this has a constant multiplicative rule.

We can calculate the next value in the sequence by first calculating the multiplier.

So what we need to know is, what do you multiply two by to get 10, and that is our multiplier.

The way to do this is to calculate 10 divided by two or equally, we could do 50 divided by 10.

If it's got a constant multiplicative pattern, those values should be the same.

So doing that tells us that the multiplier is five.

In order to find the next value in our sequence, then we need to multiply by five again.

Now it's up to you how you want to do 50 multiplied by five.

If I'm doing that in my head, I would do five multiplied by five and then multiplied by 10.

That's 25 times 10, which is 250.

We're gonna look at a different sequence now.

So if we know this sequence has a constant, multiplicative pattern, we can calculate the multiplier.

So I'm gonna tell you that this sequence does have a multiplicative pattern.

To get our multiplier, we need to do 96 divided by 24.

Now that might not be as easy to see straight away, what the multiplier is.

So think about what method you would like to use to do 96 divided by 24.

Personally I would write 24 as two times 12.

I've broken it up into smaller factors, and then I can do 96 divided by two, which I can do nice and easily, that's 48, and then 48 divided by 12, which gives me four.

Whatever your preferred method for dividing larger numbers is absolutely fine, but that's a method that I find quite helpful.

You can also use your calculator or you may wish to check your answers using a calculator if that is available to you.

So in order to get the next value in the sequence, we need to carry on our multiplicative rule so we're gonna do 96 multiply by four.

And again, I would do that by multiplying by two and then multiplying by two again.

Let's get a value of 384.

Feel free to check using your own methods or using your calculator, check that I'm right.

In order to work back to our previous answer, then, we need to do the inverse and multiplying by four.

So if we're multiplying by four to move along our sequence to get a previous value, we're gonna divide by four.

The 24 divided by four gives us a value of six.

All right, let's check.

We're happy with that then.

So this sequence has a constant multiplicative pattern.

Got 0.

3, three, 30 and that carries on.

What is the multiplier? Pause and think about your answer.

Well done if you've got an answer of 10.

We are multiplying by 10 each time.

Okay, what about the multiplier for the sequence that has successive terms 8 and then 104? Pause the video and have a go.

Well done if you spotted this time, it was 13.

You want to do 104 divided by eight.

My personal method doing that, because eight is two times two times two, to do 104 divided by eight, I can do 104 divided by two divided by two divided by two.

That gives you a final answer of 13.

So we have seen how the sequence two, four, eight, 16 has a constant multiplicative pattern.

We're multiplying by two each time.

How could we describe this sequence, do you think? Let's look at a diagram to help us.

We've got a half to start with so if we take this box to be our whole, then that is half, then a quarter, then an eighth, then a 16th and you can see how our sequence would carry on.

What you might have said is something like, the value is halving each time.

To write that as a rule, we can write that as divide by two or multiplied by a half.

I prefer to write my multipliers as multiplication 'cause it makes it easier to do the inverses, so I've written that as multiplied by a half.

So we can have sequences with decimal or fractional multipliers, so let's look at the sequence with success of terms eight and then 12.

Not as easy, straight away to see what our multiplier is, so let's use a method to calculate it.

So we're gonna do 12 divided by eight.

I'm gonna write that division as a fraction, so 12 divided by eight is 12 over eight, and then I'm gonna use my simplifying skills and simplify that down to three over two.

Three over the two is the same as one and a half or 1.

5.

Personally I prefer to work with fractions 'cause I find 'em easier to multiply and divide, so I'm gonna leave my multiplier as a multiplier by three over two.

To get my next value, then I need to do 12 multiplied by three over two.

And again, multiplying by fractions is a really useful skill so I'm gonna do 12 multiplied by three and then divide by two.

So that gives me 36 divided by two, which I can write as a whole, number 18 in this case.

To get the previous answer, I need to do eight divided by three over two.

And again, a really useful check that you're happy with your fraction skills.

So eight divided by three over two is the same as eight multiplied by two thirds.

So that gives me ties by two divided by three, so it gives me 16 divided by three.

I can't write that as a whole number, so I can leave it as 16 over three.

You can choose to write as a mixed number, five and a third if you prefer.

So this sequence has a constant, multiplicative pattern.

Check you are happy with calculating multipliers and then we'll look at your answer.

Right, well if you spotted that the multiplier is times a half, what is the multiplier then to get from five to three? Think about what process you do to work it out.

Off you go.

Perfect.

Work out what you multiply five by to get to three.

You can do three divided by five, so our multiplier is three fifths.

So Jun has written this sequence: one, negative one, one, negative one and he says this is a sequence with a constant additive pattern, as one and negative have a difference of two.

What do you think about his statement? Right, it's a bit of a tricky one.

Well I don't know if you spotted that it's not got a constant additive pattern because it's adding negative two, then two, then negative two, then two.

So it has got some kind of additive pattern, but it's not a constant additive pattern.

Can you see another relationship between successive numbers in this pattern? What do you think? Well done if you spotted this relationship.

We can multiply by negative one.

You can also divide by negative one.

It has the same result.

What you might notice is this sequence ends up alternating between positive and negative values.

We know that when we time the positive by a negative, it's negative, then a negative by a negative becomes positive and a positive by a negative becomes negative again and we have this rule where it ends up alternating between positive and negative values.

Let's look at other sequences with negative multipliers, then.

So let's say this has a constant multiplicative pattern.

If we do negative 30 divided by six, we've got negative five so multiplying by negative five.

If we do negative 30 multiplied by negative five, we get positive 150.

And then if we do six divided by negative five, we'll get our previous answer.

Again I'm gonna use my fraction skills to help me, so I'm gonna write that as six over negative five.

Because I know dividing by 10 is quite simple, I'm going to change that to 12 over negative 10 and then I could write that as negative 1.

2 if I wish to write it as a decimal.

You could have equally left that as a fraction, okay, as negative six over five.

That's absolutely fine as well.

True or false then, a sequence with a constant multiplicative pattern can decrease.

What do you think? Well done if you decided that was true.

Think about which of these justifications fit your answer.

This one's a little bit trickier and quite an important point to raise.

Well done if you notice that it's multiplying by a value between zero and one.

That gives you a number less than the previous one.

It's not any decimal.

Multiplying by any decimal doesn't make your value smaller.

It's multiplying by a decimal between zero and one.

A sequence with a constant, multiplicative pattern can contain both positive and negative numbers.

What do you think? Yeah, we've seen some of those already.

That is true.

Which of these is the correct justification? Perfect, yes this idea that multiplying a positive number by a negative multiplier will then give you a negative number.

Then if you multiply by the negative multiplier, again, it'll give you a positive number and you'll say that alternates between positive and negative.

Time to have a practise.

I'd like you to sort these sequences into the correct column.

So do they have a constant additive relationship or a constant multiplicative relationship? Give it a go and we'll see whether we agree.

Well done on that first set.

This time, I'd like you to find the missing numbers in these sequences, and I'm telling you that they have constant multiplicative patterns.

Sometimes like in A, you are finding the next number in the sequence.

Sometimes like in B, you are finding a number in the middle of a sequence and sometimes like an E, you are finding a previous number in the sequence, remembering they all have constant multiplicative patterns.

Feel free to draw on your values, draw those arrows to show us what the multiplier is to help you.

Well done.

Let's have a look at what we've got in our constant additive relationships.

We've got five, 10, 15, 20; 5.

7, 8.

1, 10.

5, 12.

9; 1.

5, 0.

5 negative 0.

5, negative 1.

5.

We're adding negative one each time, and six, 48, 90 and 132.

That leads in our multiplicative relationships, one, two, four, eight; three, 30, 300, 3,000; 48, 24, 12, six; three, negative six, 12, negative 24; and finally 0.

5, 1.

5, 4.

5, 13.

5.

We're multiplying by three in that last one.

Fantastic if you've got all or some of those correct.

Looking at our missing values then, so A, we are multiplying by two each time, you should have 48.

B, we're multiplying by six each time, you should have 180.

C, we are multiplying by half each time, you should get 20.

D, we're multiplying by negative 10 each time, you should get negative 300.

E, our multiplier is multiplied by three so dividing 21 by three gives you seven.

F, we're multiplying by 0.

2 so the next value in our sequence is 0.

008.

G, we are multiplying by 20 each time so the number after 160 is 3,200.

The number before eight, we're doing eight divided by 20.

That gives you 0.

4.

You could have written that as four tenths or two fifths if you left it as a fraction.

H, our multiplier is eight divided by six or four over three, so eight times four over three is 32 over three.

That gives us a recurring decimal answer, so definitely nicer to leave that as a fraction.

And then we're going to do six divided by four over three and again, you've got choices.

You could leave that as 4.

5 as a decimal, or you could leave that as nine over two if you prefer it as a fraction.

Well done.

Lots of fractions, decimals, and negative number skills there.

So well done if you got most of those correct.

Let's look at what we've learned today then.

So we have looked at sequences with a constant additive pattern.

We've looked at how we can find, what we're adding by each time.

We've looked at how we can continue a sequence and find previous values a sequence.

We've also looked at multiplicative relationships.

We've looked at how we'd find that multiplier by dividing successive terms, and we've looked at finding the next term and the previous term.

Lots of skills in that lesson today.

I hope that you enjoyed some of that.

I hope that you feel like you've developed some of your numeracy as well as your understanding of sequences.

I look forward to working with you again.