Lesson video

Hi, everybody.

It's me, Mr. C.

How are you all? I hope you're well.

Hope you're ready to learn, because I'm certainly ready to get on with a bit of maths and I've been a problem solving today.

We're continuing with our reasoning with patterns and sequences, and we're going to be continuing to develop strategies to plan and solve problems. So let's, without further ado, move straight ahead to see what's coming up.

Well, so here's one for you.

I wonder if you can work out the number I'm thinking of.

I'm going to give you some clues, and hopefully, you'll be able to work the number out from the clues that I give you.

Ready? Okay, here we go.

So, this number has four digits.

It's smaller than 7,000, but bigger than 6,000.

The hundreds digit is smaller than eight, but bigger than six.

The tens digit, well, that's an even number, and it's smaller than 10, but bigger than six.

Now, the ones digit, well that's in the four times table, and it's bigger than five.

And finally, the last two digits are the same.

Wonder if you can work out my number.

Let me read those clues to you one more time.

The number has four digits.

It's smaller than 7,000, but bigger than 6,000.

The hundreds digit is smaller than eight, but it's bigger than six.

The tens digit, well, that's an even number smaller than 10, but bigger than six.

And the ones digit is in the four times table, and it's bigger than five.

The last two digits are the same.

Five seconds, see if you can work that out.

Four, three, two, and one.

Your time is up.

And I just realised, I'm asking you to work with numbers while counting out loud to you.

That's not confusing and off putting at all, is it? Yeah.

Sorry about that.

Maybe that was a distraction technique.

Who knows? Well, let's see what the mysterious number was.

It was 6,788.

So, it's smaller than 7,000, yeah, but it's bigger than 6,000, yes.

The hundreds digit is smaller than eight, but bigger than six.

Well, the only thing it could have been was seven.

The tens digit is smaller than eight, but bigger than six, and it's even, so that must mean that it was eight, yeah.

Digits is four in the four times table, in the ones column.

Yeah, eight is in the four times table, and it's bigger than five, and the last two digits are the same.

6,788.

Brilliant.

All right, let's move ahead.

For the lesson today, you're going to need your pencil, your ruler, printouts from this, if you're using printouts.

Paper, otherwise, is absolutely fine.

Or even a book that you've been working in, and somewhere quiet with no distractions at all.

All right.

Well, our agenda.

We're going to move on to our key vocab in a moment with our number trees warm up, and then a problem solving recap.

We're going to move on problem solving step one, and then we're going to be looking at our main activity, which is the investigation, and the final knowledge quiz to see just how well you've taken it all in.

So, key learning today.

To develop strategies, to plan and solve a problem.

Key vocab's not changed from the last session that we looked at about problem solving.

So, the key words are sequence, patterns, similarities, differences, increasing, decreasing, term, and rule.

Remember increasing going up, decreasing, going down, getting smaller.

Alright, I think I've talked enough there, so let's move on to getting our brains even more warmed up.

Here we are with our old friend, the number tree.

Remember when you add two numbers together like 21 and 13, that will give you the number that comes above.

So, 30 add 17 gave us the 47 here, and so on.

See if you can work out all those missing numbers, and come back to me when you are ready.

See you soon.

Done it? Got those numbers in the right place? Shall we check and see that you've got something that looks like this? Let me zoom in on that for you so you can see those answers a lot more clearly.

All right.

Fantastic work, guys.

If you've got all those in place in the right place, then that's amazing.

You are mathematicians.

All right, moving on then.

So, we'll recognise this layer.

Every time we're looking at the problem solving to do with cube sequences, I'm going to show you it in a very similar setup, okay? So, here at the beginning, this is an up down step.

If I'm walking, I can start here, or let me draw it.

Here we go, I'm going to draw it for you.

This is going to be amazing.

You wait 'til you see this artwork.

I can start here.

I can step up to here.

And then I can step down again, so it's a step up, step down, okay? Similarly, here I can go start one.

Mm, mm, mm, mm, mm.

Mm, mm, mm, mm, okay? Step up, step down.

Make sense? Brilliant.

So, this is term number one in our sequence.

This is term number two.

I wonder what term number three would look like.

Remember, we need to think about what's the same, what's different, what's changed each time, and what's the general statement we can make? So, let's think about the number of blocks.

Here we've got one block.

Here we've got one, two, three, four blocks.

So, do you think the next step we'll have more or less blocks than these two? It's going to be more again, isn't it? So, we can say that it's an increasing pattern.

There is a general statement.

Okay.

What else could I say, I wonder? Have a think, have a look.

What might you be able to say about that sequence? And what would we need to know to be able to work out what comes next? Okay.

Well, here's our next one.

Taking a look.

So, here was our original.

Here's number one, and then we added these blocks.

This one, this one, and this one, these are our new blocks.

They've gone around the outside of the one we started with.

So, if I now go like this, that was part two.

Imagine I've picked up part two and put it here.

Oh, look.

There it is in the middle.

There is shape two.

And can you see now how the new ones have gone around the outside? Yeah? So, this set of blocks, they're all new.

So, here I had one block.

Here, I had one, two, three, four blocks.

And here I've got one, two, three, four, five, six, seven, eight, nine blocks.

One, four, nine.

Okay.

So, we can definitely say that it's an increasing sequence, but how do I get from one to the next? Well, what do I add to one to get to four? Well, here I've added three.

What do I add to four to get to nine? Oh, five.

Hah.

So, this hasn't stayed the same between these two and these two, so something is changing.

Well, worry not.

We're going to explore that together, so don't panic.

We will be exploring together.

We're going to present it again into a table just like we've done before when we've worked with trying to spot sequences.

Tabulating makes them easy to read.

They'll pop out.

So, step one, we had one block.

Step two, we had four blocks.

Step three, what would happen next? Can you complete the sequence? Remember we're not adding the same number each time.

Maybe drawing them is going to help, so see if you can fill this in, and then explain what's happened.

Best of luck.

Off you go.

So, let's see how we did with that, shall we? Let's move straight on.

Hmm, well, here's what I came up with.

We went from one to four.

I'm going to record here that, that's adding three.

Four to nine, that's adding.

Yeah, nine to 16, that's adding.

And I did that by drawing.

I added the extra ones in each time.

I just did it with dots to help me.

Oh, look.

Three, five, seven, nine, 11.

So, the total number of block goes up in steps of odd numbers.

Three, five, seven, nine, 11.

The total number of blocks is, ah, well, I don't want to tell you how to work out the total number of blocks just yet, but we can say that it's going up in steps of odd numbers each time.

Three, five, seven, nine, 11.

So, what would we add next? Yeah, 13, and next? Yeah, 15.

But if we look at these numbers, these are very special numbers.

It will make sense as we move along.

So, just remembering then, what changes each time? Well, we have our original part here, and the new bits arrive around the outside on each of the flat sides there.

So, our next set would go here, here, here, here, and here, and so on.

Okay? So, we're just adding around the outside, so we could add this information to our table, just like this.

So, for step one, we had one new block.

There it is.

And the total number of blocks was one.

For step two, we had one, two, three new blocks, so the total number of blocks was one, two, three, and the original four.

For step three, we've got one, two, three, four, five new blocks.

And our total number of blocks is one, two, three, four, five, six, seven, eight, nine, and so on.

Can you now complete the rest of this table telling me how many new blocks I'm adding each time, and then the total number of blocks? Off you go.

I'm sure you can do this.

Go for it.

You got it? Should we take a look? Excellent.

So, if we look now, our new blocks column is going one three, five, seven, nine, 11, 13, 15.

Okay, and our total blocks is increasing each time, as well.

So, we can say that this is an increasing sequence.

Again, these numbers, the total are very important, very special numbers.

I'm going to circle this and this, just to remind you.

This and this.

Think times tables.

Hmm.

Okay.

Well, that's a little extra clue that I've given you there.

Think times tables.

What do I know about the total blocks in relation to the step number? Hmm, hmm, hmm.

I think I've said too much.

Okay.

Well.

Squared numbers.

That's what you need to know.

Squared numbers.

That's the important bit.

And let me show you what I mean.

You've heard squared numbers before, right? Let's take a look right here again.

Squared numbers.

I wonder where we can see squared numbers on here.

Hmm, look at the ones I've circled.

Two times something is four.

Three times something is nine.

Four times something is 16.

Five times something is 25.

What is that telling us? That's the biggest thing.

Should we just double check? Yeah.

Well, a squared number is when you multiply a number by itself, and that's what's happening here.

The step number, the term number, multiplied by itself gives you the total number of squares needed.

The total number of cubes.

So, three times three, we can call that three squared.

So, three times three can also be written like this.

Three squared.

And that just means three by three.

Laying out those blocks, that would give us nine squares in total.

Five times five, or five squared.

Five times five is 25, yeah? Nine times nine, or nine squared is.

What is it? You all know that.

I'm not going to tell you because this is going to come up in our next bit of the task.

But knowing squared numbers, knowing that a number multiplied by itself is a squared number is what you really need to know for this.

So, let's take a look at your challenge task, shall we? Because that's how we're going to really carry this ahead now.

Our challenge task then is saying, using what we know, how can we work out the number of blocks for any step in a sequence? So, I've given you an example here.

Step one in a sequence, because it's step number one, is one times one, or one squared.

One times one is one.

Step two is two times two, or two squared, so that's four cubes in total.

Step three would be hmm times hmm, or hmm squared, and the answer, and so on.

Can you now work out what those missing values are? Show me the working out in two ways.

So you can either, for example, at number one is one times one, or one squared.

Number two is two times two, or two squared.

Number three will be hmm times hmm, or hmm squared.

Can you fill in that missing info? Good luck.

It's not as tricky as it seems, and when it gets to the 10, the 20, and the 50, start with one, two, and five, and then make your answer 10 times bigger.

That will help.

So, ladies and gentlemen, holding your breath for the big reveal, for I shall now show you the answers to that.

Here they are.

So, for step three, we wrote down three times three, or three squared, which is nine.

For step nine, it would have been nine times nine, or nine squared, which is 81.

Step 10, you could have done one times one, and then made it bigger, or 10 times 10, 10 squared.

That's 100.

20 times 20, or 20 squared.

That's 400.

50 times 50, or 50 squared, that's 2,500.

We can work it out for any number by just timesing that number by itself.

I have to say, I do really enjoy looking at pattern things like this.

I love it when you can start seeing things come together, and it just makes sense.

I like things that are ordered, and then follow a pattern.

It's just how my brain works, and that's how some brains do work.

So, this is the next thing you need to make sure you do.

We zoom off and do our final knowledge quiz, and then when you're done come back, and see us when you're ready.

Well, that is a massive well done from me for all the hard work from today's session.

Remember the squared number thing.

It's a really useful skill to remember.

And what's an interesting thing to do is to look at a multiplication grid and just see where those squared numbers lie.

You'll find that they probably appear in a lovely little diagonal line going down that square for you, and it's just a really quick visual to help you remember them.

But brilliant work today.

So, until next time, from me, Mr. C, I'm saying goodbye.

See you soon.