Lesson video

Well done and thank you for loading the video for this lesson.

My name is Ms. Davies and I'm going to help you as you work through this topic.

Feel free to pause things, rewind things, so that you are really comfortable with the ideas that we're exploring and I really hope there's bits and pieces that you find really enjoyable and really interesting as we work through.

Well, let's get started then.

Alright, welcome to our lesson on plotting a relationship.

By the end of the lesson, you'll be able to use graphical representation to show all of the points within a range that satisfy a relationship.

We're going to be drawing axes today, so you may want to make sure you've got some square paper or some graph paper that you can draw on at the same time as me.

We're going to look at this idea of substituting today.

So to substitute means to put in place of another.

In algebra, substitution can be used to represent variables with values, and that's what we're going to use it for today.

An integer, you hopefully have come across this idea before, so we're going to plot some values that are integers and some values that are non-integers.

So make sure you are comfortable with that definition of integer.

So there's two parts of what we're going to look at.

We're going to look at plotting lines from coordinates, and then we're going to look at identifying whether points are on a line.

So Aisha draws the axes below.

She then plots these points.

I've done it for you on my axes.

And Aisha then says, "I've plotted all the coordinates where the y coordinate is two." Do you agree with Aisha's statement? Pause the video and come up with your answer.

Well done.

So you might have said something along the lines of Aisha has plotted all the integer coordinates on her axis where y = 2, but let's think about this coordinate -(1.

5,2).

Does that coordinate also fit the rule y = 2? Yeah, it does, doesn't it? It still has a y value of two, so it still fits that rule.

You might be able to think of some others that also fit that rule.

So thinking about where that will go on my graph, how could Aisha show all the points that fit that rule, not just the integer points? What do you think? So this might be a new concept for you, that's absolutely fine, but what she can do is she can draw a line through the points and what that means is she's represented all the non-integer coordinates as well as those set integer coordinates.

When we're talking about this line, we call it y = 2, that's because all the coordinates on that line follow this rule where the y coordinate is two.

With that in mind, what do you think we call this line that's been drawn? Yeah, well done, that should be y = 3, shouldn't it? 'Cause all the y values have a coordinate of three.

So the idea of drawing a line means we can represent all the coordinates that follow a rule.

And the idea with lines is that they go on in either direction past our axes as well.

So when we draw a line, we assume that that line's going to carry on.

What we've done is we've represented the infinite number of points that follow a rule within the set axes that we've drawn.

If you were to plot this using graphing software or technology, the graphing software, the line will actually go on forever.

And you'll be able to scroll and scroll and scroll because that graphing software has the ability to represent the infinite number of points without restricting it to a set range.

So we can draw lines that represent a rule and the best way of doing it is to just choose some coordinates that fit that rule.

So Aisha wants to draw the line that represents all the coordinates that fit the rule x = 4.

How many coordinates do you think she needs to plot? If you haven't come up with your answer yet, this might get you thinking.

How many different ways could you join these two coordinates with a ruler? There's only one way of doing it.

If you're using a ruler, there's only one way of joining those two coordinates.

What that means is the minimum number of coordinates you need to plot is two.

If you plot two coordinates, you should be able to get exactly the correct line.

So Aisha wants to draw the line that represents x = 4.

Her classmates suggest some points.

So Andeep says (3,4).

Laura says (4,10).

and Sophia has (4,2.

1) Have a think about their suggestions.

Can you spot any areas or any difficulties with their suggestions? Off you go.

I don't know if you spotted that Andy has got his coordinates the wrong way around, it's the x coordinate that needs to be four.

He's written a coordinate with a y value of four.

This was a little bit more subtle, so Laura's coordinate does fit the rule, the x value is four, but remember the axes Aisha had only went up to five.

So that coordinate, even though it follows the rule and will be on the line, she's not going to be able to plot it on her graph.

Sophia's is probably the best coordinate.

However, because it's 2.

1, it might be a little tricky to plot it completely accurately.

In this case, an integer coordinate may be better.

It all depends on your axes and what values you've got written in on your axes.

So Aisha goes with (4,2).

I would like you to have a go at suggesting another coordinate to plot for x = 4.

Which other one would you plot? Off you go.

Plenty of suggestions, I wonder if you went with any of these.

So you could have (4,-5), (4,-4), (4,-3), you could have had (4,4) or (4,5).

You'll notice that I haven't said (4,1) or (4,3).

Now there's nothing wrong with those coordinates, it would be absolutely fine for you to have chosen those two, but they're actually visually quite close to (4,2).

What that means is it can make it trickier to line them up with a ruler.

In this case, I don't think it'll be a problem 'cause we've got a grid line to guide us, but if you are looking at trickier equations, you might want to make sure your coordinates are quite far away from each other so you can get the best understanding of where the line will go.

So we did say that two coordinates is the minimum.

However, I do suggest that you go with three or maybe four to get an idea of a pattern.

What it also does is it helps you spot if you maybe plotted one wrong.

So Aisha's going to go for (4,5) and (4,-2) as well.

So she has three coordinates in total.

To show all the points, she joins the coordinates with a ruler and have a look that's been done for you on the graph.

Has she done it then? Has she shown all the coordinates that fit this rule? What do you think? If you haven't come to a decision yet, try this idea.

Does the coordinate (4,-3.

5) fit the rule x = 4? Yeah, I think you'll agree it does, it has an x coordinate of four.

If you look at the graph, does the coordinate (4,-3.

5) sit on Aisha's line? No, it doesn't, does it at the moment? But it should do.

So what Aisha needs to make sure she does is draw the line all the way across the coordinate grid so that all the values that follow that rule are included, not just the ones that she chose as her coordinates.

So I'm going to have a go at showing you how to plot a line and then I'd like you to have a go yourself.

So you're going to need to draw a set of axes that goes from -5 to positive five in the x direction and in the y direction.

So watch what I do first and then draw those axes before you have a go at yours.

So we're going to plot y = x + 3.

The first thing I want to do is pick two or three coordinates that follow the rule.

I do encourage you to pick three so that if you get one wrong you'll be able to spot that.

So I've gone with (0,3), (2,5), (-5-,-2).

They all follow this rule that the y coordinate is the x coordinate plus three.

The next thing you're going to do is plot those coordinates and then you're going to check that they follow a rule.

So in this case we're going to check that they lie in a straight line and then we can draw a straight line with a ruler, making sure it goes all the way across our axes and doesn't just stop at the coordinates I plotted.

Right, there we go, we've drawn the line.

y = x + 3.

Make sure you've got a set of axes ready and then plot a line for the rule y = x - 1.

Off you go.

Well done, you may very well have just drawn your first graph.

So you should have picked two or three coordinates that follow the rule.

They won't necessarily have been the ones I went for, but I went for (0,-1), (5,4) and (-2,-3).

Then you need to plot 'em.

And again, if you chose different coordinates, it won't look exactly the same as mine but it'll sit on the same line as mine.

And then you should have drawn a straight line.

So you need to make sure you are using your ruler and you need to make sure it goes all the way across your page.

Fantastic.

Make any adjustments that you need to make and then we'll move on to the next bit.

Great, so Andeep has tried to plot the line y = 2x.

What I want you to do is see if you can identify any mistakes.

So what he said is he said, I've chosen the coordinates (-2.

5,-5) and (2,4).

Then I've plotted them and I've drawn the line for y = 2x.

Look at what he said again and see if you can identify any mistakes or suggest any corrections for him.

Off you go.

Okay, so the first mistake you might have spotted is he's actually plotted the wrong coordinate.

He's got his negative, non-integer values in a bit of a muddle.

So her has plotted -1.

5 not -2.

5 in the x direction.

To help you, you can use your finger and trace along the axes to make sure you get it in the right place.

So (-2.

5,-5) is one of the issues he has got.

He's found the wrong coordinate.

The other issue is that his line should go all the way across the axes, remember? To represent all the points.

You might have suggested to Andeep that he tries plotting a third point.

If he'd putted a third point, probably one with integer values, he may not have made the same mistake and then he'll have noticed that his point was wrong, 'cause it is not in a line with the others.

So there's his correct line for y = 2x.

Perfect, time for you to give this a go yourself then.

So for each rule you need to fill in the coordinates.

I suggested some values for you, where there's a missing one, you can come up with your own x and y coordinate.

So fill in the coordinates and then I want you to use them to draw the line.

Please make sure that you are drawing a straight line using a ruler.

Well done, you've got three more to look at this time.

Again, if I've put in one of the coordinates, you need to find the other.

If I haven't, then you can come up with any coordinates you like, off you go.

And then the last set.

So the same idea as before.

You must be getting really, really confident with these now.

So I've tried some slightly different relationships for you to look at, off you go.

Congratulations, you've just drawn nine different graphs, that's really well done.

So for A, you are missing coordinates were four and four, 'cause y = 4.

And then once you've drawn your straight line, it should look like a horizontal line.

For the second one, you need -1 as your y value, which actually means you could have put anything in as your x value for the second coordinate.

And then anything as your third coordinate, as long as the y value was -1.

And when you draw that again you should get a horizontal line, it's actually further down the page than your previous one.

So C, you should have -2 and -2 as your x coordinates and then -2 and then any y coordinate you like for the last one.

And this time, you get a vertical line, make sure you've drawn it all the way across your axes.

Second set, y = x.

So you need your x and y values to be the same.

So you need -2 in the first coordinate, five in the second coordinate, And then any coordinate you like where the x and y are the same.

And this time we get a diagonal line, it goes through the origin, notice as well.

Second one, you needed -1 and you needed zero in your first and second coordinates.

And then any coordinate where the y value is the x + 2 and you get another diagonal line.

It doesn't go through the origin this time, though.

And the last one, you've got loads of options for coordinates you could have picked, I've put some up on the screen and then they should all form a nice diagonal line again.

The final set, so y = -x.

So you should have -2 in your first coordinate, zero in your second coordinate.

And then any coordinate where the x and y have that multiplicative relationship of -1 so x times -1 is y.

When you plot them you get a diagonal line.

It's actually sloping in the opposite way to our previous ones.

For H, you need zero.

And then there's all sorts of coordinates you could have come up with.

In order for them to be on the grid, you probably went for (-1,-3) and (1,3).

And then the last one y = 1/2x, you need zero in your first coordinate, <v ->2 in your second coordinate.

</v> And then again any coordinate where y = 1/2x.

Our final graph is another diagonal line, slightly less steep than the previous one.

Brilliant, you have now potted loads and loads of lines and started to sort of investigate what happens when we plot coordinates and turn them into lines.

What we're going to do now then is identify where the points are on a line.

So we're now going to look at this idea of substitution so we can substitute any x value into our rule to work out the corresponding y value.

So far, we've stuck with quite simple algebraic relationships, we're now going to look at some slightly more complex ones.

How do you think we could work out the missing value for the rule y = 4x - 5? If I want the x value to be three, what we can do is we can substitute three for x in our equation.

So the y value has to be four lots of three, subtract five.

And when I've written that, I've put the three in brackets 'cause it's the number with substituting in.

There's a couple of reasons for that, one is it helps me when I've got negative values, not to get in a muddle.

It also means if I'm using a calculator, I can use my arrow keys to just go back and change that number if I want to change my x coordinate.

So using our priority of operations, that gives us 12 - five or seven.

I'd like you to try this one.

So pause and give it a go and then check your answers.

Well done.

So we've got four times five, which is 20.

Then we need to subtract five, which is 15.

So (5,15) is going to be on our line.

What about a negative value now, give this one a go.

So four times -4 gives you -16.

Then we're subtracting five, which gives you -21.

See how I've laid out my work? It makes it really easy for me to follow and it stops me getting in a muddle with my positives and my negative values.

So any coordinate on a line will satisfy the rule for that line.

Let's have a look at what I mean.

So here's the line, y = 2x + 1.

Any coordinate, the y value will be equal to two lots of the x value plus one.

I'm going to try some, and I'm going to pick nice integer coordinates, but it will work for any coordinate.

So the y value three is equal to two lots of one plus one.

Three is two lots of one plus one.

Let's try this one, so the y value is one and that is equal to two lots of zero plus one, two lots of zero are zero plus one is one.

And that works for (-1,-1) and (-2,-3).

So any coordinate fits that rule.

If a coordinate's not on the line, it won't satisfy the rule.

So I've picked a coordinate (2,3), two lots of two plus one does not equal our y value of three.

What that means is we can check if a coordinate is or is not on the line by seeing if the x and y values satisfy the rule.

If we substitute in the x coordinate, if it gives you the y coordinate, then it follows the rule.

So will the coordinate (6,13) be on the line? Well, let's try it.

Two times six is 12, add one is 13.

So that is on the line, y = 2x + 1.

What about the coordinate, (-4,-9), will that be on the line? Let's try it again.

So two times -4 is -8, add one is -7, not -9.

So no, that coordinate will not be on the line.

Which of these coordinates is on the line, y = 5x + 10? So try it for each coordinate and tell me if it is or is not on the line.

Off you go.

So let's try that first one, five times two plus 10 is 20, so it is on the line.

The second one, five times -10 is -50, plus 10 is -40, so it is on the line.

And five times seven is 35, plus 10 is 45, so it is on the line.

And that last one, five times nine is 45, plus 10 is 55, not 65.

So that one wasn't.

Well done if you've got some or all of those correct.

So for each question I'd like you to tell me whether the coordinate lies on the line, if you can explain why that's even better, give those ones a go.

Fantastic.

Alex has started to plot some lines.

He has plotted one point from each line, but then can't remember which point goes with which line.

This is a really silly mistake of Alex's.

If you are going to plot a line, you're going to want to draw in your line before you move on to a different question.

Otherwise you get in a muddle.

So you need to help him out.

Can you match the points, A, B, C, D, and E with the lines that he's tried to plot? Off you go.

Looking at our answers then.

So no, the first one's not on the line.

Four times five is not 18.

Second one, yes, six times -2 is -12.

For C, yes, zero times 3.

5 is still zero.

For D, no, 25 subtract 5.

5 is not 22.

5.

And then some trickier ones at the end, well done if you got this far.

So 11 times three is 33, plus four is 37, so yes, that's on the line.

And the last one, 10 times five is 50, subtract eight is 42, not 50, so that was not on the line.

Hopefully, you were able to help Alex out here.

So y = 20 was the coordinate for E.

B, y = x was the coordinate for A.

C, y = 2x was the coordinate for D.

D, y = x - 10 was the coordinate for C.

And y = x + 10 was the coordinate for B.

That was a real challenge today, so well done for pushing yourself to give that one a go.

Let's look at what we've learned today then.

So we've learned that a line can be drawn using coordinates found following a rule, and that's how you've drawn your first graphs today is by picking coordinates and then drawing a line.

We've seen that we can identify whether points are on a line by substituting the x and y coordinates into the equation for the line.

We've seen that if a point follows a rule, it'll lie on the line.

If a point doesn't follow the rule, it won't lie on the line.

Well done for joining me on that journey today, I hope you learned lots of new bits and pieces and enjoyed finding out how you can actually draw these equations of lines.

Please join us again, goodbye.