Lesson video

Hello and welcome to today's video.

I'm really glad that you have decided to learn with us today.

My name is Ms. Davies and I'm going to be helping you as you work your way through this lesson.

Let's get started.

This lesson is on features of special number sequences.

So today we're going to look at different types of numbers and investigate some of the sequences that they can form.

By the end of the lesson, you'll appreciate the features of special number sequences such as square, triangular, and cube.

With that in mind then that you need to be happy with your square and your cube numbers.

Pause the video and have a read through the definitions if you are unsure.

Our new keyword today is a triangular number.

We're going to explore this during this lesson.

So to start, we're going to look at types of special number sequences.

A square number is the product of two repeated integers.

Some examples, four is a square number, because it's 2 multiplied by 2.

25 is a square number as it's 5 multiplied by 5.

Just take a second, see if you can work out which of these are square numbers.

Pause the video.

Well done, if you managed to get all five correct and didn't select any of the incorrect numbers.

So 1 is 1 multiplied by 1, 9 is 3 multiplied by 3, 16 is 4 multiplied by 4, 49 is 7 times 7, and 36 is 6 times 6, and that's how we know that they are square numbers.

You could also use dots in an array like on the left-hand side to check.

We can then build some sequences with our square numbers.

Have a look at these three pupils and the sequences they have built.

How could you describe those? What do you think? Right, I wonder whether you said something similar to this.

So for Alex's sequence, we could say that this is the sequence of consecutive square numbers starting at 100 and increasing.

There's different ways of expressing that, but there's a few key points in there that you want to make sure you've got.

Sofia, you might have said this is the sequence of all even square numbers in ascending order, or you might have said it's every other square number.

If you think about the square numbers, they alternate between odd and even.

And Sam's, Sam's was possibly the easiest, actually.

It's the sequence of all square numbers in ascending order.

We're going to have a look at Sam's sequence in more detail.

We're going to use a table to help us.

So you might have seen this before.

We've got our term numbers across the top, so we've got 1, 2, 3, 4, 5.

Then we're going to put in our terms. So the first term was 1, second term was 4, and then 9, 16 and 25.

How could we work out what the 10th term in Sam's sequence is? Pause the video, what would you do? So the 10th term would be the 10th square number, because Sam's sequence is just all the square numbers in ascending order.

So the 10th square number is 10 squared, which is 100.

The rule then for finding any square number, will we square the term number.

So 1 squared is 1, 2 squared is 4, 3 squared is 9, and so on.

So to find any number in this sequence, we just need to square the term number.

So the nth term rule for this sequence would be n squared.

We're going to look now at cube numbers.

So a cube number is the product of three repeated positive integers.

So eight is a cube number, because it's 2 times 2 times 2.

We can see that built out of cubes, if we had eight cubes, we could then build a bigger cube with dimensions 2, 2 and 2.

27 is a cube number, because it's 3 times 3 times 3.

Again, we could build a bigger cube out of 27 cubes, it would have dimensions 3 times 3, that would give you 9, and then times 3, which would give you 27.

Which of these pupils could be writing the sequence of all cube numbers in ascending order? What do you think? Let's have a look.

So Alex's sequence starts with two cube numbers, 8 and 27 that we've just looked at, but 46 is not a cube number.

You might have realised that Alex has actually got an arithmetic sequence which is adding 19 each time.

Sofia has got the first cube number, 1 times 1 times 1 is 1, and the second and the third and the fourth and the fifth.

So that could be the sequence of all cube numbers in ascending order.

Sam's isn't.

Sam has got the multiples of three, not the cube numbers.

So let's have look then at the first five terms at the sequence of all cube numbers.

What do we have to do to work out the 10th term this time? All right, well this time we are cubing our term number to get the term.

So to find the 10th term, we could do 10 cubed, which is 10 times 10 times 10 or 1,000.

That means our nth term rule for this sequence would be n cubed, because to get any term in the sequence, we can cube the term number.

Let's have a check then about your square and cube numbers.

Take your time to put the following numbers in the correct part of the Venn diagram.

Off you go and come back when you're ready for the answers.

Well done.

Check you've got all the values in the correct place.

I'm just going to draw your attention to a few of the interesting ones, 1 and 64 are both square numbers and cube numbers.

They're the only two numbers under 100 that are both square and cube, but there are other numbers that are square and cube larger than that.

Your cube numbers, 8 is 2 cubed, 27 is 3 cubed, and 125 is 5 cubed.

1,000 is 10 cubed.

Check you've got the rest of the numbers correct and then we'll move on.

Right, time for the new bit now.

So we've looked at how dots can be arranged into square arrays to show which numbers are square numbers.

We're going to use that idea to have a look at other types of numbers.

So nine dots, we know nine is a square number.

We can arrange that into a square array.

However, 10 dots cannot be arranged into a square, however you try.

Therefore, 10 is not a square number.

My question to you is what shapes can you arrange 10 dots into? Give yourself some time to play around with this.

You might want to pause the video and see what you come up with.

I'm sure you came up with lots of interesting ways to arrange 10 dots.

Here is one way that's going to lead onto what we're looking at today.

Okay, so 10 dots can be arranged into a triangle.

10 in fact, is an example of a triangular number as 10 dots can be arranged in a triangle and the important thing about these triangles is that the number of dots in each row increases by one each time.

So there's one dot at the top and then two underneath, and then three and then four, and so on, and that's how we're going to build our triangles.

You might also see them built as equilateral triangles with one dot at the top, two underneath, three underneath that with a line of symmetry vertically down the page.

That's absolutely fine as well, it's exactly the same thing.

I think it's easier to spot how this pattern develops with right-angled triangles.

What other numbers do you think are triangular numbers? Pause the video and see what you can come up with.

We'll go through the first few.

So 1 is a triangular number.

Notice it was also one of our square numbers and our cube numbers.

The next triangular number then would be three, 'cause you've got the one dot on the top and then two underneath.

What do you think the next triangular number was? Might be one of the ones you've already written down, it might not be.

Perfect, so we should have six as our next triangular number.

Then we have 10, which is the one we looked at before.

What do we need to do then to calculate the fifth triangular number? What do you think? We need to add another row, and that row is now going to have five dots.

The next triangular number is 15.

So Izzy says, "I think 36 is a triangular number." There's 36 dots, they're currently in a square array, six by six.

I would like you to use the dots to see if Izzy is correct.

Give this one a go.

So we have 1 at the top, then 2, then 3, 4, 5, 6, 7, and 8 fits at the bottom.

So this is the eighth triangular number.

What I want you to think about now is what we could do to then find the seventh triangular number? So we need to work back a step, so we need to take off that row of eight, which gives us 28.

If we wanted to find the ninth triangular number, we'd add a row of nine, so we'd end up with 45 as our ninth triangular number.

So that then is our definition for a triangular number.

I'm hoping you're feeling really confident with this new idea and excited to start playing around with this triangular number sequences.

Alex has spotted something.

Alex says, "That's interesting.

1 plus 3 is 4 and 3 plus 6 is 9.

Both of those sums have square number answers." 4 and 9 are square numbers.

So Izzy says, "Does that mean that adding two triangular numbers equals a square number?" This is what I'd like you to investigate.

So if you can try some other sums with triangular numbers, what do you notice? So I've tried a few sums. 1 plus 3 is 4, 3 plus 6 is 9, 6 plus 10 is 16, 10 plus 15 is 25.

Those all produced square numbers.

I found some that didn't.

1 plus 6 is 7, 1 plus 10 is 11, 3 plus 10 is 13, 6 plus 15 is 21.

So those don't produce square numbers.

I wonder if you came to this conclusion, it's the consecutive triangular numbers that can be added together to give the sequence of square numbers.

We can use our dot patterns to show why.

So if we look at 1 add 3 and we move the 1 over, it's going to equal 4.

3 add 6, we rotate the 3 and slot it in that space, we get 9.

And you'll see that always adding the two consecutive triangles together gives a square.

Let's put that new knowledge to the test then.

So I'd like you to fill in the missing numbers in these triangular number sequences.

Off you go.

Superb.

So the first one, we are missing three, that's our second triangular number.

In B, we're missing 21, and then in C, you are missing 55.

Have a go at this one.

So Sam says, "This dot pattern shows that 16 is a triangular number." Have a look at that dot pattern, why is Sam incorrect? Well done, if you said something like this.

Triangular numbers are built so each new row has one more dot than the row above it.

If you look at Sam's sequence, it adds two dots to every row, so that is not how the triangular numbers are formed.

A chance to have a practise then.

So I'd like you to fill in the missing numbers in those special number sequences.

You are then going to use your triangular square and cube numbers for the following questions.

You want to make sure that you've got those correct.

So look back in your notes or go back in the video if you need to check any of those values.

Give that a go and come back for the next bit.

Superb.

So our next question, Sofia, is having some fun.

So what's Sofia is doing is she's building different sequences using the square, triangular, and cube numbers.

So you want to make sure that you've got those in front of you.

What I would like you to do is suggest what the next number in each sequence could be, but you need to explain your answer, you have to have a reason behind it.

So Sofia's going to help us.

Sofia says some of these were built from more than one type of number.

So what she might have done is she might have added some together, she might have combined the sequences, she might have subtracted them, she might have multiplied them.

She's taken the triangular, the square, and the cube numbers and she's done something with them to make these sequences.

See if you can work as many of these out as you can.

Some of them are a little bit interesting.

Come back when you're ready to check your answers.

Let's have a look at our answers then.

So our triangular numbers 1, 3, 6, 10, 15, 21, 28, and 36.

Check your square and your cube numbers as well.

So the fifth square number is 25, fifth triangular number is 15.

So you get 40 when you add those together.

b, 36 add 64, which gives you 100.

c, you are looking for 1 add 1 which gives you 2.

d, the third square number minus the second cube number.

So you are looking at 9 subtract 8, which is 1.

The 10th cube number minus the 10th square number.

So you've got 1,000 minus 100 which is 900.

And then you've got the 11th triangular number plus the 12th triangular number.

You could use Alex's trick from earlier that would give you the 12th square number, which is 144.

I'm really impressed if you managed to find all of these, 'cause Sofia did some really crazy stuff with some of our sequences.

So a, look to be the square numbers in descending order.

If that was the case, the next one would be 25.

Question b, seemed to be every other triangular number, so the next number would be 45.

This is where it started getting more interesting.

c, look to be the square numbers add the triangular numbers, that gives you 57.

d, the cube numbers minus the square numbers.

That zero was a bit of a clue that maybe something was being subtracted, 'cause all our sequences started on 1.

The next number would then be 180.

This one was strange.

What she was doing here, she was alternating between triangular and square numbers.

So one's the first triangle number and then one's the first square number, three is the second triangular number, four is the second square number, and so on.

And finally, you got the square numbers minus the triangular numbers.

And notice how we end up with the triangular numbers just with zero at the front.

Perfect, let's look at the next bit.

In the second part of the lesson, we're going to look at identifying special number sequences.

So in order to continue any sequence, it's helpful to understand how the sequence grows.

We're going to study the triangular numbers.

Alex says this is not a linear sequence.

We can show that Alex is correct by checking for a common difference.

So our sequence is adding 2, then 3, then 4, then 5 and 6.

So consecutive terms do not have a common difference.

Sofia says this is not a geometric sequence either.

Well let's have a look.

We're looking for a common multiplier.

So 3 divided by 1 is 3, 6 divided by 3 is 2.

So we're timed in by 3, then by 2, then by 5 over 3, so it's definitely not a geometric sequence, we haven't got a common ratio.

So this sequence is not arithmetic or geometric, but we can still describe its term-to-term rule.

How would you describe the term-to-term rule for this sequence? Let's have a look at how it's growing.

So we've got 1, 3, 6, 10, and so on.

By visualising this way, you can see that each term is adding one more than was previously added, adding 2, then 3, then 4, and our next diagonal will have five, and so on.

If we have a look, then at those additions you can see that each time the amount we're adding increases by one.

The differences between consecutive terms form a linear sequence.

So those differences, 2, 3, 4, 5, they're a linear sequence.

We can call that the first difference.

Therefore that's the second difference.

So we can say that has a common second difference of one.

If we want to get the next number, we could add one to the previous difference.

So last time we added five.

So if we add one, we now need to add six.

That would get 21 and that is our next triangular number.

Let's see if you can use this idea.

If the 20th term is 210, what is the 21st term in this sequence of triangular numbers? Right, there's an extra step here that you needed to spot.

So you needed to spot that every time we were adding on the term number.

So to get the 21st term, we're adding on 21 to the 20th term, so that gives us 231.

Well done if you got that.

We're going to explore the square numbers now in a similar way.

Can you describe the term-to-term rule for this one? Think about what we just talked about with triangular numbers.

So we've got 1, 4, 9, 16 and hopefully you can see that this time we're adding the odd numbers, we're adding 3, then 5 and 7, then the next one will be 9.

Let's see if we can use our language of second difference.

This time we have a common second difference of two.

Let's have a look at pentagonal numbers just for a little bit of fun.

So there's some diagrams. Can you use them to write down the first four pentagonal numbers? So we've got 1, 5, 12, 22.

Let's have a look at how this sequence is growing.

Little bit trickier to count this time, but just like the triangular and square numbers, pentagonal numbers have a linear pattern in the differences.

We were adding 4, then 7, then 10.

Each time the difference was increasing by three.

We could say we have a common second difference of three.

Use that fact to work out the next pentagonal number.

So we'd add three to our difference, so this time we need to add on 13, which gives us 35.

So Sam reckons, "The cube numbers do not form an arithmetic or a geometric sequence either." But this time Izzy cannot see a pattern in the differences.

So let's see if we can see a relationship between consecutive terms. So we know how to generate the cube numbers, but can we see a pattern in how the sequence grows? Not obvious at the moment.

Let's see if we can find the differences between those 12, 18, 24, 30.

So they don't have a common second difference.

However, you might see a pattern in those differences.

This time we have a common third difference.

Sam wants to use that fact to work out the seventh term.

So we need to add 6 to get 36.

That means our difference is going to increase by 36.

So our difference this time will be 127.

So the next number is 343.

That's a really long way to go about this and Izzy has spotted that.

So Izzy says, "Because we know the cube numbers can be generated by just cubing the term number, we don't need to use that method." We can actually just cube seven to get the next term in the sequence.

So 7 times 7 times 7 is 343.

Quick check then.

These sequences are made up of either consecutive, triangular, square, pentagonal, or cube numbers.

I would like you to match the sequences to the correct option.

Let's have a look then.

First one has a common second difference of plus two.

They're square numbers, you might have recognised those first three particularly as square numbers.

The second one had a common second difference of one.

These are all triangular numbers.

You might not recognise these, 'cause you might not have come up with triangular numbers quite this high.

c, you might recognise 27 and 64 and 125.

You'll see that they don't have a linear pattern in their differences, but d does.

So d is the pentagonal numbers, 'cause those differences are increasing by three each time.

And c is the cube numbers.

You probably recognised a couple of those.

So if a pattern or scenario produces one of our special number sequences, we can use the key features to make predictions.

So let's have a think about this scenario.

Alex is planning a netball competition for local primary schools.

In the competition, all teams will play each other once.

He does not yet know how many teams are to part.

But what he wants is a quick way to work out how many games there will be once he knows the number of teams. So when we're working out a problem, often it is easiest to pick the simplest case you can come up with.

So what if there was just two teams? So if there were two teams, there would only be one game, A versus B.

Okay, let's think about how many games there'd be if there were three teams. So we'd have A versus B, A versus C, and B versus C.

We wouldn't have B versus A, 'cause that's the same thing as A versus B.

So we're making sure that we don't duplicate any of our answers.

See if you can work out how many there would be if there were four teams. Off you go.

There are 6 games for 4 teams. Notice some of the patterns we are coming across as we are writing out these answers.

That structure we can notice helps us make sure we've not missed any.

So Alex has noticed we've all the games involving A, then all those involving B, and all those involving C, making sure we don't duplicate any of our answers.

So we're going to fill in the table for the number of games for 2, 3, 4, 5, and 6 teams. If you'd like to work this out yourself, pause the video.

So 1, 3, 6, we came up with already.

For five teams, there's going to be 10 games.

For six teams, so A will play all five other teams, B will play all five other teams, but we've already counted the match against A, so that'll be four new games.

Then there's three new games for C, two new games with D, and one new game with E, which would be E versus F.

That gives us 15 games.

You've probably spotted a pattern.

Alex says, "Last year there were eight teams involved." He wants to work out how many games that was.

And you can see he started listing A versus B, A versus C, and then possibly give it up, 'cause that's going to take him ages.

What would be an easier way for Alex to work this out? Have you spotted it? So the easiest way is to notice a pattern.

All our answers so far are triangular numbers.

This pattern is going to continue.

If you think about the way we set out our answers, you can see it's forming triangles, isn't it? With each one adding an extra column with one extra game in it.

So I'd like you to finish off this experiment for Alex.

How many matches will there be for seven teams and eight teams? Alex actually has 10 teams respond, so what's the calculation for that number of matches? And then read Izzy's statement and tell me why she is incorrect.

Well done, we have 21 and 28.

The ninth triangular number, so for 10 teams will be the ninth triangular number, which is 45.

So Izzy says if every team plays nine matches, that should be 10 teams playing 9 matches, that should be 90.

Why was she wrong? Hopefully you spotted it, it's because she's then counted all the games twice, 'cause A versus B is the same as B versus A.

Well done.

We're going to put all those skills together in a bit of an investigation task.

So you need to read the investigation carefully, use any tools that you need to to fill in that table and then reflect on those questions at the end.

This is the second thing I would like you to investigate.

So Alex is playing around with number patterns.

He noticed if he squares the number 3 and subtracts 1 then divides by 8, he gets a square number.

He wants to see if this works for other odd numbers.

So he's squaring his number, subtracting 1, dividing by 8, and he wants to see if he gets a square number each time.

Use the table, tell me what you notice about your answers.

Off you go.

Fantastic, you'll need some paper for this one.

So Sam is investigating the maximum number of intersection points between straight lines drawn in a square.

So they've got a square, so you might want to make sure you've got some squares to hand.

And then they are drawing lines across the square from one side to the other.

They're trying to draw these lines so they intersect the most points possible.

So you only ever want two lines intersecting at once, you don't want all three intersecting at the same point.

What they want to know is how many intersection points there could be for 12 different lines.

But as you can imagine, drawing them is proving difficult.

Sam makes a sensible decision and tries it with three lines first.

So have a look at their drawing with three lines, how many intersection points are there? Then I'd like you to go back and do it with two lines and then do four and five lines as well.

Can you then use that to work out Sam's problem? Final challenge.

This time Sofia has a six by six square grid and she has been challenged to find as many squares in the grid as she can.

Work your way through the questions, see if you can find the final answer.

Come back when you're ready to find the solution to these problems altogether.

I really hope you found some of those interesting, let's have a look now.

So for our cubes, you should add 1, 4, 9, 16, 25.

You should notice that they are the square numbers.

For the 10th pattern then, she would need 100 cubes.

Let's have a look at Alex's problem then.

So squaring 3, subtracting 1 and dividing by 8 gives you 1 as a square number.

Doing the same with 5, gave them 3, then 6, then 10, then 15.

Well then if you spotted that they are the triangular numbers, not the square numbers.

If you apply this calculation to the number 19, 19 is the ninth odd number greater than 1, so it'd be the ninth number in our pattern, and the ninth triangular number is 45.

I wonder if you found this third one, the trickiest.

So for three lines, there were three intersection points.

For two lines, that wasn't too bad, that was just one intersection point.

Four lines was quite tricky to draw.

Have a look at my picture if you're not sure, and there were six intersection points.

For five lines, you should have made 10 intersection points.

Well done if you spotted they were the triangular numbers.

The two lines was the first triangular number, three lines the second triangular number.

So 12 lines should be the 11th triangular number.

That's all the numbers 1 to 11 added together, which is 66.

You can see then why that was quite hard for Sam to draw and to count without using the idea of a sequence.

And finally, so we're going to start by thinking about the different-sized squares.

The biggest-sized square is a six by six.

There's one of those.

The second biggest-sized square is five by five.

There's 1, 2, 3, 4 of those.

There's nine, four by fours, 16, three by threes, 25, two by twos, and 36, one by ones.

They are the square numbers, so adding them all together gives us 91 different squares.

Let's have a look at what we learned today then.

So we've looked at square and cube numbers and how they form sequences.

We have looked at the triangular numbers, which might have been a new type of number to you, even more exciting if it was.

And we've had a look at this idea of a common second difference.

So the square and the triangular numbers have a common second difference.

We can use that to play around with these sequences.

I hope you have fun with some of those parts today.

There's lots of exciting things that produce square and triangular numbers, so keep your eye out for those as you're looking at problems in the future.