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Hi everybody and welcome to our next lesson in the topic of sequences.

Today's lesson is about generalising arithmetic sequences through tracking calculations.

It's a bit of a tongue twister.

Wonder if you could say that five times fast.

Generalising arithmetic sequences through tracking calculations.

Generalising arithmetic sequences through track- okay maybe you can't do it, but either way it sounds harder than it is.

And I'm really looking forward to starting, but before we do can you make sure that you have a pen and some paper.

As per usual, make sure that you turn off any distractions.

And have a nice clear space in front of you.

So pause the video to make sure you've got all of that sorted before we can begin today's lesson.

All right now we are ready to go.

So first of all, what I would like you to do, is imagine that this number grid here that we can see on the right, continues for many rows.

I would like you to write down a number to match each description.

First of all, a number greater than 20 in column A, so this is column A.

And imagine that those rows extending forever and we're trying to see what number there is above 20 in that column A.

A number greater than 20 in column C.

And a number between 100 and 110 in column B.

Once you've done that, I would like you to think of how many ways you can complete the sentence here.

Numbers in column A or B or C or D will always or sometimes or never what? So try and think of an example.

Maybe two or three sentences that you can create that will be true always, sometimes, or never for certain numbers in columns A, B, C, and D.

So remember we need to be thinking, when we've got this grid here, we need to be thinking about multiples.

So that's my big hint to you.

So pause the video to have a go at that.

So for the first one, you could have got any number essentially, any number in this sequence greater than 22.

So 22, 26, 30 and again, we're going up in fours hopefully you noticed that.

So any number going up in fours from 22.

The second one was the one I actually started when I was doing this one myself.

Because C, hopefully we'll notice, that we've actually got the multiples of four here.

So I found that quite a nice starting point for me.

And I knew that I was looking for multiples of four that are above 20.

Now if you wrote 20, you need to just think about that for a second.

It didn't say greater than or equal to 20.

It just said greater than 20.

So we couldn't include 20 in that list.

A number between 100 and 110 in column B.

So this one, you couldn't just have extended this for that long really.

It would have taken you a really long time.

What you could have done, actually it might have been quicker to spot the patterns.

We were looking for multiples of four, but back one.

So you could have been thinking about multiples of four that were above 100, well in between 100 and 110.

Then going back one.

For this part here, there are lots of sentences that you could have gotten.

Not going to go through each and every single one possibility.

But this is an example.

You could have said that numbers in column D will always be one more than multiples of four.

So we've got column D and we got always.

So we needed to pick one of those.

You could have written something about column C being always, being multiples of four or going up in steps of four.

Or we could have said lots and lots of different things.

So if you're unsure about whether your sentence is correct maybe send it to your class teacher or tell somebody at home what you got.

For the connect task.

An arithmetic sequence, so that's a key word there, so you might want to write that down.

An arithmetic sequence is any sequence where the difference between the terms is constant.

So for example, all the columns going down our number grids are examples of arithmetic sequences.

And we'll look at a few more examples in a minute.

We can use tracking calculations to help us find the term in any row.

I would like you to complete the statements and the calculations below.

How would the statements change for the other rows in the grid? So we look at this one first.

I would do this as number one.

Nine, 15, 21, 27, 33.

I know this is an arithmetic sequence because what? So we're looking at this definition at the top.

Looking for the difference between the terms. And we're thinking, what dose it mean by a constant? This is an arithmetic sequence, it said that in the statement, but why is it an arithmetic sequence? I would then move to this one next and fill in the gaps in these rows.

So we're looking at multiples of six here.

And how have we got to this column? And this column is this sequence? So we've got nine, 15, 21, 27, 33.

How have we got to that from multiples of six? And then finally I would go to this one to find numbers in this column.

So we're talking about this column.

In any row, I multiply the row number by what? And add what? So pause the video to complete those sentences and grids? Well hopefully you got the answers here and really well done again if you did.

I know this is an arithmetic sequence because the difference between the terms is constant.

Every time the difference is plus six or add six.

That is the difference.

And it means it's constant because it's the same every time.

With the grid, we were looking at multiples of six here and then we were adding three 'cause we were shifting those multiples of six up three each time.

Here we've got row n.

And this is what we're going to be starting to look at today and this is the generalising bit.

This means that generally we can find some kind of expression for any row.

So it's doesn't have to be a specific row that we've picked this will work for any row.

For row n, which is some row.

We've got six lots of n add three.

To find numbers in this column in any row, I multiply the row number by six and I add three.

So we can see here that we've multiplied this row n by six.

And we've added three.

Pause the video now to complete your independent task.

So for your independent task, it looks something like this.

And the answers are here.

You had to look at this number grid on the right and write down for the first question write down the column and the row that match each tracking calculation.

So the tracking calculation is five lots of five subtract three.

So I would be thinking first of all, if I'm working with multiples of five so I know that it was five multiplied by something.

And that something is going to give me the row.

And I know that my multiples of five are here.

So if I've gone back three I'm going to be in column A.

And the same applies for the other two.

For the second question, I'm going to look at the same number grid and I'm going to write down the tracking calculation that matches each description.

So this time, we're going the other way around.

So if I've been told that I've got column D and I'm looking at row 15.

I know column D is my multiples of five.

So if it's row 15, I've just got five lots of 50.

And similar for these ones.

For question three, I'm going to again look at the same number grid.

And I'm thinking which columns or rows would these numbers fall into? So for this, I actually first considered a tracking calculation.

So I know that this one is five lots of something.

And with multiples of five it's relatively straight forward because we know for a multiple of five, we're either going to end in a five or a zero.

So if I have gotten, got 123, I know that I've gone from a multiple of five and I've subtracted two.

So straight away that's going to tell me my column is B because I've gone back two from my multiples of five.

I then worked out what that was going to be that number in the multiples of five would've been 125 and that would have been five lots of 25.

So that's how I worked that out myself.

And the same applies to the other three.

So really amazing job if you got those correct.

Really, really good work.


Now for the explore task.

Zaki is describing a number grid.

The first row contains the number negative three.

Every column contains odd and even numbers.

So every single one of these columns has to have both odd and even numbers in.

The 10th row contains the number 64.

What could his grid look like? Draw the first two rows.

Once you've managed to get the one, think of how many different grids are possible.

So this again is a little bit of trial and error, a little bit of logical deduction.

So pause the video to have go at this.

So this is an example of a grid that would work.

What needs to happen, is we need to have seven columns.

That's really important.

All of the different grids actually have seven columns 'cause that's the only way we're going to fit negative three on the first row and 64 on the 10th row.

We also needed an odd number of columns to make sure that we've got odd and even in each column.

The other combinations that we could've had would've been where negative three would've shifted one place here.

And if negative three shifts one place to the right, 64 also has to as well.

And all those numbers would've shifted one place to the right.

And we could have also fitted negative three in here.

And 64 in here.

So there were actually three combinations that were possible.

Well done if you managed to get any of those.

Especially if you used trial and error, that was quite a tricky one.

And yeah fantastic job.

Really really well done in today's lesson.

And it brings us to the end.

So again share your work with you class teachers.

I'm sure they'd love to see it.

And also share you work with Oak National.

If you'd like to, please ask your parent or carer to share your work on Instagram, Facebook, or Twitter tagging @OakNational and #LearnWithOak.

Fantastic lesson again.

Well done to all of you.

And hope you have a good rest of your day.