Lesson video

Hello, my name is Mrs. Buckmire Today our lesson title is Interesting Quadratic Patterns.

So we're going to be exploring different patterns.

And make sure you have a pen and paper, you need your brain switched on for this lesson.

Also remember pause whenever you need to and also when I asked you to try my answer, to think about something or do something but also, if you need to pause, if you need more time to process, just pause the video, go at your own pace.

And it can also be really useful to rewind the video.

So if I say something you're not quite sure about it, or you don't quite understand it, just rewind and listen again.

Sometimes hearing it a second, third time can be really, really helpful.

Okay, let's start, so four, we'll try this.

Pick four consecutive numbers, multiply the largest and the smallest and subtract from the product of the remaining two numbers.

Okay, so consecutive, what does that mean? Yeah, they're one off after each other.

Okay, have a look at bins one here.

My set is five, six, seven, eight show T does five times eight.

So smallest times biggest subtracted from the middle two numbers.

Okay, so you try some different sets and just see what do you notice? So do you spot anything? It's 20 patterns.

Okay.

Pause and have a go.

Okay, how did you do? but they're all lots and lots of different answers, but what you should get for all of them is, they all equal two.

I wonder why? Right, I'm going to go through.

I'm going to use five, six, seven, eight as my example.

So five, six, seven, eight.

So, I'm going to let n be my smallest number.

So this is n.

This is n what is six in terms of n? Good as a thrashing can be written as n plus one.

What could seven be? Excellent, and plus two eight, you got this and plus three.

Fantastic, so now it says more applying a large and small is a plaque from the product.

So you can see that it's being done there.

So my five times eight is going to be n times n plus three.

And my six times seven is going to be n plus one times n plus two, and we're subtracting them.

So that is my smallest list times largest subtracted from the product of the remaining two numbers.

Okay, could I simplify this? Yes, by expanding.

Okay, maybe pause the video and have a little go.

But how do we do? So here, I'm going to get n squared plus, when you collect like terms here, we're going to get three n plus two takeaway n squared.

Now it's n times three, it's three n but there's a subtraction here, so take away three n Let's simplify this.

So we have n squared takeaway n squared, well, that makes zero plus three n take away three n, and that's zero right? yes.

There's always going to equal two.

So when we generalise, so by using these ends, we can find it's always two.

So this holds true, for all integers, it will work.

And we picked four consecutive numbers, we're more of licenced, more than platinum from the product of the 2 million numbers.

We will always, always, always get two How amazing is that? Okay, Antoni is exploring the following pattern.

This is your independent task.

These questions are what I want you to do.

I want you to work out each value.

I want you to give three more calculations that follow this pattern.

Tell me what do you notice? Then let the lowest number be n to this pattern and what values in terms of n did the other two numbers take? and then write out the nth term of this pattern.

So that might be, that's really your generalisation.

This is where you're generalising it.

And then I want you to prove that by following this plan, you will always get an answer with a certain property.

So that same property is going to be linked to part C.

Okay, so you might want to pause it and go to work.

So you can make sure you resume back the video, but do pause and have a go at this Okay, so work out each value.

I've shown you the values here, but all scans, three more patterns and calculations.

I followed this pattern Now what I really like to do it just the mathematician in me just likes to do it systematically.

So I had to go, we already had one times four plus five equals nine, and then two times five plus six And then I had to do the next one, three, and then four and then five which given that in six, I missed out seven, so there's no seven there.

And I missed out nine.

I went to eight and then we had 10.

I can see all the answers there.

So do, pause and check.

If yours are not there, then just make sure it's got the same pattern as in what do we notice? Excellent, all of them form square numbers.

So that's three squared, four squared, five squared, six squared, seven squared, eight squared, 10 squared, 12 squared, that's interesting.

So make sure your calculations followed that same rule.

Okay, so now we're trying to let n be for this pattern and the lowest number.

So what bodies do the other two numbers take? So I'm going to let this be our one.

So when you're working out, what you should have done, you're thinking if one is n and in four, So what's four in terms of ones? So I could write one, two, three, four, five.

This is n that's n plus one,n plus two, it's the brackets in so you can see it more clear n plus three, n plus four.

So four is a criminal to n plus three.

Did you get that? Yeah, so n times n plus three plus so five was n plus four.

Okay, so that means I have n brackets n plus three plus n plus four.

So for D it was n plus three and n plus 4 were the answers.

So why the nth terms of this would be the nth term and then show, now I added in, we always get squared numbers.

That was one of the, what do you notice? We always get a square numbers, so let's try it out.

So here we have n times n plus three.

So that's n squared plus three n plus n plus four.

Did you get that? Good, so therefore that equals to n squared plus four n plus four.

So you're saying that's always a square number.

Is that true? Let's see, so we have our square.

I'm going to sure draw one here.

So that's n by n and then this part is four This could be two and this could be two.

So maybe it's n plus two and n plus two.

Let's check that.

n squared plus two n plus two n plus four.

Yes, it works.

That is equal to n plus two squared ,interesting.

So if n is our smallest number, then actually our answer in the end is going to be two more than n squared.

So here we had two, so plus two is four, four squared is 16, it works.

This one, the last number is five plus two is seven squared is 49.

Check it for your example, does it always work? Yeah, and that's why algebra is so awesome.

Maybe you could use this as a bit of trick to someone, like say, give me numbers like this and I can actually tell you what the answer would be in my head really quickly, maybe.

Okay, but you explore, we've got two more patterns here.

And why want you to do is look at the following sets of numbers and pick a set and generates more examples.

Now I think maybe this one is slightly easier? Maybe this one's slightly harder? But it kind of depends on your skills.

So you just choose which one you like.

Okay, can you explain and generalise any patterns you spot? Can you design your own self calculator follow a similar pattern? Now is pretty hard.

I had a go at this to do your quiz and it's pretty tricky, okay.

So that's kind of extension extension, but I think it's quite enjoyable.

So do have a go at that, but definitely the first bit.

So actually finding the values and pick the same examples, just generate some more examples.

Tell me what you noticed, then try and explain and generalise any patterns using what you did and then independent task.

Okay, pause the video and have a go.

Okay, so in this example, what do we see? We always have our lowest number.

So here one times the number two bigger.

So that's, if n is the lowest number, let me have two as n plus one, three is n plus two, plus two times that and bigger number.

So here, we're just picking two numbers.

So we're picking two numbers and we do the lowest times.

And so we know we multiply them together and then we plus two times the highest.

So we do the poor duct of them plus two times the highest.

So it was really important you actually notice what the pattern is.

So it's like here, if you just looked at that, you might think, it's n times two more plus one more times the two more, but that's not why, because actually the two is always here in every case.

Okay, so it's just the one , three is picked here, here two and four are picked here, five and seven are picked here ten and twelve are picked.

So we note it's two between them.

So that's why it's n and n plus two.

Okay, so we have n times and plus two, and then we're going to add two, lots of the big one, which was emphasis too.

There are two ways you can do is you might think, Oh, I spot there's two lots of emphasis twos.

So by the distributive property, it actually equals the n plus two times n plus two, that'd be a very advanced way, that'd be very impressive.

You could have there multiply it out.

As I got n squared plus two n plus two n plus four.

So that ends up equaling to n squared plus four n plus four.

And then you might notice in each case, that actually is the square of the bigger number.

So three, four, seven and twelve So the bigger number here is n plus two.

So that is actually equal to n plus two times n plus two to both methods get the same.

We can obviously write this more efficiently, like all that.

So you should have got, equals the n plus two squared.

So pick two numbers where the difference between them is two And then when you do this pattern, you always get the square of the largest one.

Really, really well done and if you got that.

Okay, so I guess, it was maybe easier than the other one, even if it is, but it definitely is still very tricky.

So while they've got anywhere near that.

Okay.

Okay.

So for this one, we've got eight, 12, 20, 36.

So the part is not necessarily so obvious.

So you probably want to seeing that there's two between them actually like try some more.

So maybe did like 11 square takeaway, nine squared.

You might've done it by square takeaway three squares.

You know how I like to kind of fill in all the gaps.

So I would have done by square takeaway three squared, same square, take away five square.

Just try to explore it some more.

Okay.

So we know that there's two between them.

So if our lowest number is n then our highest number is going to be n plus two.

And what we're doing is we're doing n plus two squared takeaway n squared.

So here we're going to have n squared class for n plus for takeaway and squared.

So we end up with four n plus four, and I can even write that as four, lots of n plus one, or do I do what's the word? Good, I factorised it.

Okay, so what is this saying is saying that if we square, if we pick two numbers where the difference between them is two, and then we do the square, the big one subtract the square, the smaller one.

Then we always get four times the number between them.

Let's try it.

So three square takeaway, one squared.

So the difference, the number between them two, two times four is eight.

It works four and two.

The number between is three times four, it works.

The difference between a middle number is five times four.

It was, so this one was slightly higher.

Maybe the what pattern, like what do you notice? These numbers be a little bit random, but actually they're all multiples of four.

And there are four times the number between them.

Really well done if you spotted that and fantastic if you designed your own set of calculations, I would absolutely love to see you.

And if you would like to share your work with Oak National, and then please ask your parent or care to share your web on Twitter, tag it @OakNational and #learnwithOak.

I would absolutely love to see any of your work that you're proud of from any of the lessons that you've done with me.

And I hope that you bound today find interesting, like I did, please do the exit quizzes, super useful way, my shows in the past, and kind of helped you explore it through that.

So that could be another fun thing to do, but also to help your learning as well and have a wonderful day.

Bye.