Lesson video

Hello, My name is Mrs Buckmire and today I'll be teaching you how to recognise linear and non-linear graphs.

Now, first you need something to write with and something to write on and make sure you pause the video whenever I ask you to, but also whenever you need to.

So if you need to pause because you need more time on something, that's more than fine.

It can also be useful sometimes to rewind the video.

So if I say something, you're like, Ooh, I'm not sure what that means, or what's she saying? Just listening again.

And maybe, you might understand it better the second time around.

Okay, Let's start.

so before try this.

I want you to match up these equations with these speech bubbles.

These speech bubbles kind of represent one of the equations each.

And then I also want you to tell me, what is Y if X equals zero, Or is Y if X equals five.

And then what is X if Y equals zero.

Okay, so there's kind of a few tasks there so do pause the video, spend some time doing this.

You can look at this on the video, or on the worksheet.

Just make sure it was in the video.

Pause in three, two, one.

Okay.

So let's see.

What did you say for Y equals five X? Good, Y is five times as big as X and it might have helped you to maybe like substitute values in.

So maybe if X equals one what is Y? Five.

Okay.

So why is five times as big as X? That makes sense.

Here, Y equals, how do we say this? Good! X squared.

So Y is the square of X.

This one? What does that fraction bar mean? Good! Divide.

So X divided by five.

So Y is one fifth of X.

Here? Good.

It's the sum.

So sum is when we add to the sum of X of Y is five and finally Y is always equal to five.

Okay.

So what is Y if X equals zero? So if X equals zero you can substitute it in.

For the first one, Y is zero.

For the next one, it's the same.

For the next one, it's the same! X plus Y equals five, when X equals zero we get Y equals 5.

And when X equals zero.

The next one we get Y equals 5 too Okay.

What about when X equals five? So what is Y? When X equals 5, and Y is five times as big? Good.

It will be 25.

The next one, the square.

What does square mean again? Yes! Times by itself.

So five times five is 25, one fifth.

Why is one fifth of X? What's fifth of five? Good, one.

And the next one.

Good.

When X equals five, Y equals zero.

And finally, Y is always equal to five.

So when X is five, Y is still five.

Okay, and finally, what is X, if Y equals zero? Now we've actually found quite a lot of these answers, so here when Y equals zero X equals zero.

When Y equals zero X equals zero.

When Y equals zero X equals zero.

When Y equals zero X equals five And for this one, what would it be when Y equals zero? It's not possible.

We can't do it.

Y cannot equal zero! Let's even actually do that in red.

Y cannot equal zero.

Well done if you got those answers! Okay.

So looking at the same things, what's the same and what's different.

Pause the video and have a think.

Okay.

So what did you write down? So Y equals five is Y equals X squared.

When X equals five we've got the same answer.

Also the same answer when X equals zero.

Does that mean they're equivalent? Are they the same? No! Cause if X equals two, what happens? For the first one, Y equals 10, And for the second one Y equals 4 so they're not the same thing but just some answers are the same.

What else is the same? Yeah, that's interesting.

So I can see that with this one, there is one X and for this one, there is one X and one Y even.

One X and one Y, One X and one Y And in this one, there's one X, one Y.

And in this one there is an X and Y but this X has a square in it so that's a bit different.

And this one doesn't have an X at all.

So these two seem quite different in that way.

What else? Yeah, there's fives in a lot of places, true.

So this one, there's no five here.

There's lots and lots of things you could say, okay, I'm going to now explore the graphs and see, actually can we get a better understanding of how some of these are alike, and some of them are quite different.

Okay? So I'm using Desmos.

Now, if you Google Desmos graphic calculator, you will come up with this.

So you can have a little play as well.

Absolutely loving using this because it's a really quick way to plot graphs.

So what I'm going to plot is Y equals five X, I'm going to plot, X plus Y equals five.

And I'm going to plot Y equals to X over five.

Now, what do you notice about these? Even going to put on projector mode so everyone can see it.

Yeah, they're all straight lines! Straight diagonal lines.

How is any of them different? Good.

Yes.

The blue one.

It's kind of got a negative gradient.

If you know about that word, well done! Excellent! So let's plot another one.

For now, try this.

How is that different? Good! Y equals X squared is a curve, so all the others, they are linear graphs, but this one why? Because X squared is non-linear, it is not a straight line.

So is not linear.

Hmm What about Y equals five? How is that different? So, Y equals five is horizontal.

It's not diagonal in any way.

So actually, whereas Y equals five X, X plus Y equals five and Y equals X over five.

There's some kind of relationship with X and Y.

Where Y equals five, Y is always five regardless of what X is.

Okay? So actually the other three are linear, but this one is not considered linear because it's not increasing or decreasing by a nonzero amount each step.

Well done.

If you can see that.

Okay.

And we can actually see that kind of equal increment tool increase, by drawing a table as well.

So just want to show you that if it is linear, so let's just complete the table, even though we already know what the graph looks like.

So the easiest one to complete is zero, and whats negative three times five? Good.

Negative 15.

Negative two times five? Yes.

Negative one times five? Yes.

Now what would the next one be? One times five, two times five, Three times five, Excellent.

So maybe you were thinking through it like that, or maybe you're just like, Ugh, I'll just add five each time, because that is true.

So when it is linear, we actually can spot a pattern.

So as long as these ones are increasing by the same amount each time, so here we're plus one each time, then actually these will increase by an equal amount each time.

So there it's plus five.

And we can keep going.

Okay.

So non-linear does that work? So here for non-linear, it would be just five wouldn't it? So, yeah, it's increasing my zero each time, but not by a nonzero.

And what about Y equals X squared? Oh, we've got to be careful here.

So for Y equals X squared It's X times X.

So what's the negative three times negative three? I'll do up here as you can see it really clearly.

Oh, why did I put brackets in? That's to make sure I don't make mistakes.

So negative three times, negative three, positive nine.

Negative two times negative two, positive four.

Negative one times negative one, positive one.

So already, I can see here is decreased by five and then we've decreased by three, so actually it's not increased or decreased by the my same amount each time.

So that's why these two are non-linear.

I could going, this is zero, one, four, nine.

So even goes from decreasing, to then increasing.

That makes sense.

So That was the curved one, Wasn't it? So it was going down first then it was going up.

Interesting.

Okay.

So let's do a little check.

I want to know, is this linear or non-linear? Look really carefully.

You can have 10 seconds to do it.

Here's the first one.

Good.

What should you be looking out for? X and you're making sure that these ones step by the same amount each time.

So then compare that these step by the same amount.

So here we were increasing plus one each time and here where it looks like we're increasing by two each time.

And does it hold true every time? 26, 28, 30 32, 34, 36, 38, Yes! So it is linear! Okay.

What about this one? Linear or non-linear? Non-linear! Uh-uh! So actually it decreases at the start.

So the X is increasing by the same amount each time it's plus two each time, which is fine.

And then actually here we're decreasing by 12 and then we're decreasing by four, so it's not equal.

And then we're increasing by four and then increasing by 12, increasing by 20! Uh, so that's not linear.

Non-linear.

Right, last one.

What did you put? It is linear! So the X value is increasing by 25 each time, which is fine.

And the Y value is increasing by 50.

So when they're increasing, decreasing by the same amount it means they have some kind of proportional relationship there so that it works out and it is linear.

Excellent.

Okay.

So your task, I want you to cite the following equation would produce a linear or non-linear graph.

If linear, I want you to match the equation with the same straight line graph.

So there's more than one, then they match up in some way.

I think a helpful way is a check by testing coordinates.

So just like doing the try this, maybe try zero, try four, try two and just try different co-ordinates out and see oh, do I get the same value, If I put X in, do I get the same value or Y or not? Okay? So you don't have to draw the graph, but I do want you to think about is this linear or not? Maybe if you need to draw bits of it and do a table, it might help you.

So you can do that if it helps you, but just have a go.

So pause the video and have a look at this task now.

Okay.

So the ones in pink here, the one shaded in are all non-linear.

Okay.

And we can see this because here the power of X is two.

So when I say the power of, it means like what the index is.

So what it's raised to that means X times X.

So that is not a linear graph.

That highest power powerful linear graphs is actually just X.

So here what also doesn't work is X to the power of a half.

And so actually you need the highest power to be one.

So this is non-linear as well.

Here is non-linear because actually it'd be Y, equals root X is how it could be rearranged to.

And this one is also non-linear.

Oh, there was one more.

Y equals root, X equals root Y plus two is also non-linear.

Cause actually, I could rearrange this.

So that is X takeaway two all squared, which equals to Y.

So then actually there's a square involved.

Highest power ends up being X squared.

Okay.

So let's see which ones you matched up.

Check those, pause it if you need to and make sure you've got that those ones were the non-linear ones and let's see which ones matched up to the linear ones.

Okay.

So here you have it.

I'm going to go through it quickly.

So two Y equals X, is equal to Y equals X over two.

So if we multiply both sides here by two, we would get that answer.

Y equals X subtract two is equal to Y plus two equals X.

Okay.

So they both produced the same graph.

So sometimes algebra, especially like later on if in life you're modelling situations, you might have it written in different ways.

So it's best to simplify it.

Often we like to say Y equals, but it won't always be in that case, so sometimes you have to change the subjects of the formulas so Y is the subject.

Y take away X equals the two.

This one matches up to Y equals X plus two, Y equals two minus X.

What did he get for that? What did you do to get there? X and add X to both sides? So we have X plus Y equals the two that's how we get from there to there.

And Y equals two X? Yeah.

This one was the easiest.

If you spotted it, it was equal to two X equals Y.

Remember, as long as their equal the sides just have to balance, It doesn't matter which way round it goes.

Well done if you've got those correct! They were all the linear graphs, produce the same linear graphs as each other.

Okay.

So if we explore, so here, I do want you plotting in the graphs.

I want you to make a table of values for X, between negative four and four.

And what I want you to do is do that for each one and then sketch what the graphs look like.

and then maybe comment on it as well.

So some of these graphs, I feel like maybe you've never plotted before.

Okay.

So just having a little go at plotting them and seeing what you get, there is support on the next slide if you need it.

Okay.

So there's some support.

I just thought maybe it can be useful to kind of go through the motion of writing what each one means.

So here Y equals, what's five X? Good.

It's five lots of X so five times X.

So that's how you're going to find each one.

But this next one, good.

You have to square it.

So it's X squared, X times X.

For this one, this one's harder.

So Y squared equals X.

So I want to make Y the subject.

What's the inverse of squaring? Good! Square root.

So Y equals the square root of X.

So each one you want to find the square root of X.

You might not be able to do that for all the numbers, you might need a calculator.

But you don't need to use a calculator.

So just do the ones that you can do and then match them up best you can.

Okay? So, or just predict about where around you think.

So do the ones you can do then just predict where do you think the other ones will be.

But I will go through that.

Y equals X over five, So that is divide X by five.

So X divided by, Yeah, divide X by five, yeah that's right.

Y equals five over X you do five divided by X.

Good! And Y equals five? Hopefully you know how to do that.

Okay.

So hopefully that's helped you.

What you need to do is for each one do a table of values.

Okay? But some of them there might be values that you cannot actually figure out.

So you don't just copy this table, blindly.

Think about, Oh, does that one work? Maybe just write, NA, not available if there's any that don't work.

Okay? But do at least you know, yeah I think you can aim for getting four solid ones done.

Okay? But really try and get all six.

Pause and just have a go.

Okay.

So I've grouped them together.

So this one, Y equals five X you should have got this table, Y equals X squared and Y squared equals X.

I haven't yet done.

So not available.

You cannot do the square root of negatives.

Well, not yet.

You can't do it to get natural numbers.

You get imaginary numbers.

And you get real numbers sorry.

So you can do it and you can get imaginary numbers.

That's A level stuff, but for now it's not available.

There's no answers that we can plot on our real coordinate grid.

So we're going to leave that out.

So, here we could put this root of two and the root of three but it's fine if we just plotted one and two.

So maybe I'm not really sure what that graph looks like yet, but we'll have a look at that on Desmos, but fair.

So which one are linear? Good! So already from the table, you can see these are increasing by five each time.

So this is linear.

And this one is not and we've actually already seen this graph, so we know that this one is linear and is going to look like that, going through the origin.

That is such a bad graph.

I will show you on Desmos, because my sketching is not good here.

Yeah.

There you go.

That should be straight.

And here we're going to have a curve and touches zero, zero.

This one also goes through zero, zero.

And this one, so this is a bit weird.

So that one is one and that four, it's two.

So, so far it looks a bit like that.

But I don't really know the shape because I've only got those two points.

So let's have a look at Desmos and see what it would look like.

Okay.

So I'm going to use Desmos, So remember Desmos graph and calculator, you can check yours as well.

I'm just going to check up on my answers.

So Y equals five X.

Yep.

That looks as I expected.

What was the next one? Y equals X squared? We've already seen that Haven't we? That's the kind of smiley face curve.

And then we have Y squared equals X.

That's the one I'm not really sure of.

What do we think? What are we expecting? Equals, ah, that's interesting.

So what values here? We got one and four, which would fit on the first.

That makes sense.

There's the four value.

And that was the one.

How come it's underneath here as well? Y squared equals X, Oh yeah! When X equals one, Y equals one.

And what happens when X, when, for example, Y equals negative one? When Y equals negative one X will equal one as well.

So that would be this point here.

Can I get it exact, you know, it's around there isn't it.

There that's the one.

There.

So actually both points work out.

So normally when we're sketching, we would only sketch this top half and we would not sketch this bit below the X axes, but it makes sense why it's there.

It's a very interesting graph, and it's definitely non-linear.

So Y equals five X was linear, but why? Cause X squared and Y squared equals X, are Definitely both non-linear graphs.

Okay.

And so the next tables should look like this and pause.

If you need to time to check it.

Now what you will notice I normally write things as fractions, I just think it's easier.

And then maybe when I was doing my Y axis, I probably like put it into fractions as well, so right, oh, zero, one fifth, two fifths, three fifths, to equal like each increment step.

each step, to make it easier to plot as well.

So Y equals X over five? Is it linear or non-linear? Good! It is linear.

So it's going to be some kind of linear graph like that.

That should be a straight line.

Y equals five over X, how does that look? So it goes through zero, zero, and that one is all the way up at five, but then it gets smaller.

So it should look like this.

Hmm.

Let's see.

Let me try and do it with a pen.

Like that I think.

They're not touching the axes there.

And negative.

So when negative one, it goes to negative five and then it keeps getting at negative two it's negative five able to, so like this.

So like that I will show you on Desmos to be sure.

And I'm in this one.

Y equals five.

So plot in zero five, one five, two five.

So good.

Looks like that.

Well, I've got those.

Let's just check it on Desmos quickly.

Okay.

Back on Desmos.

Let's see.

So we had Y equals to X over five, X over five.

Oh.

Not a bad sketch from me.

I even got the kind of low gradient, nicely.

We have Y equals five over X and this is the one.

Yeah! Ah interesting.

Definitely not linear.

So non-linear graph, that one was, for sure.

And finally Y equals five, Also, is that linear or non-linear? It's not non-linear, so even though it's a straight line it's non-linear because there is no X there.

And so it's not increasing or decreasing by the same amount by the same non zero amount each time.

Well done if you've got those! Really well done today, if you had to go at the try this, the connect, the independent tasks and explore, and you've had a go at completing tables and plotting graphs.

I think you should be really proud of yourself.

Hopefully you can now tell the difference between a linear graph and a non-linear graphs and linear non-linear.

And it be great for you to test your knowledge and have a go at the quiz.

And I'd just like to say, thank you so much for your hard work today and hopefully see you in another lesson.

Bye.