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Thank you for choosing to learn using this video today.
My name is Ms. Davies, and I'm going to be helping you as you progress through this lesson.
Let's get started.
Welcome to a video on plotting coordinates generated from a rule.
By the end of today's lesson, you'll be able to represent a set of coordinates constructed according to a mathematical rule.
And we're going to look at doing that algebraically, and we're going to look at putting that on a graph.
So we're going to talk loads today about equations.
So an equation is used to show that two expressions are equal.
So, finding a rule from a set of coordinates.
So we can describe coordinates that follow a pattern by looking at the relationship between the x and the y coordinates.
Okay? So, you hopefully know that when you're looking at a coordinate, whether you're plotting it, or when you're reading it, it has an x value and it has a y value.
We're going to be looking at the relationship between those.
So have a look at these four coordinates.
We've got 0, 2 1, 3, 2, 4, and 3,5.
Right.
To start with, what do you notice? So, anything you notice about those coordinates? And then can you take that one step further, and can you describe a relationship between the x and the y? Pause the video and have a think about your answer.
So you might have looked at those four coordinates and thought something like, there's something to do with going up by two, isn't there? You might have noticed that each pair of coordinates have kind of this difference of two, okay? Remember, it's really important that we're talking about the relationship between the values within a single coordinate.
Even better if you added in that mathematical language of x and y.
So you might have said something along the lines of, "The x coordinate plus 2, gives the y coordinate." You could have said that the other way around and said, "The y coordinate minus 2, is the x coordinate." You could have said, "The y coordinate is the x coordinate plus 2." All right, have a look at these three.
So we've got 0, 3, 1, 4, and 2, 5.
Again, pause the video and see if you can come up with a relationship between the x and the y coordinate.
Even better if you can use that language of x and y coordinate.
Off you go.
Right.
Very well done.
If you spotted then that we are now adding three to get from the x coordinate to the y coordinate.
So a nice sentence you might have come up with, "The y coordinate is the x coordinate plus 3." We're now going to have a look at extra coordinate.
So what about the coordinate 6, 3? Does that follow the same rule? Pause the video and think about your answer.
Well, let's try it.
So we're looking for the y coordinate is the x coordinate plus 3.
So if we identify our x coordinate, our x coordinate is 6, is the y coordinate, the x coordinate add 3? Hope you agree with me that, no, this coordinate does not follow that rule.
What about this one? Does the coordinate 3, 6 follow this rule? 3 plus 3, do we get a value of 6? Yes, we do.
So it is really important that our x and our y coordinate are written the correct way around.
It does change the relationship if they are not the correct way around.
Okay, so you've got four ones for you to have a go at yourself.
So have a look at the three coordinates in each set.
Can you describe a relationship? Off you go and then see if you get the same answers as me.
Right.
Very well done with those ones.
So let's start by looking at the first one.
So this first set of coordinates, you probably noticed this relationship of add 5.
Making sure that we're putting that into our words.
The y coordinate is the x coordinate plus 5.
Okay, being careful with this second set, because we've got some negative values, it might help you to draw the arrows onto your coordinate so you can think about what's happening to get from your x to your y.
And well done if you notice that we're adding 1.
So -4 add 1, is -3.
Remember, we're moving up that number line.
So the y coordinate is the x coordinate plus 1.
You may have said something like, "Y coordinate is the x coordinate plus 0.
5." And the very last one.
Good spot if you managed to get this the right way around.
So this time, the y coordinate is the x coordinate -2.
If you wrote that as the x coordinate is the y coordinate add 2, that's absolutely fine as well.
Just make sure that you've got the relationship the right way round for the x and y coordinates.
So now I'm going to start bringing in some of our equation skills.
Hopefully you can see that we've got the y coordinate is the x coordinate -10.
How do you think we could put that into an equation? Okay, I'm just going to add in a little bit more language there.
So the y coordinate is equal to the x coordinate -10.
Our equation would say, y equals, y standing for the y coordinate, so y = x - 10.
We've got another set of coordinates.
See if you can find the rule this time, even better if you can write it as an equation.
Lovely.
So, we might start by looking at the 3 and the 9, we might start by saying, well, we know that we can add 6 to 3 to get 9.
But then you need to check that works through all the coordinates.
So if you add 6 onto 4.
5, do we get 13.
5? I don't think we do.
What about if we add 6 onto 10, do we get 30? So that relationship of add 6 does not work.
I wonder if you found a relationship that did work.
Fantastic.
So it's this idea of multiplying by 3.
So, so far, we've only talked about relationships where we've been plussing our mindset, but here we've got a multiplication linking the x and the y coordinates.
It works in exactly the same way.
So the y coordinate is the x coordinate multiplied by 3.
If we think about our algebra, if we write x times 3, we write that even simpler as y = 3x.
Fantastic if you've got that one straight away.
Don't worry if you're still developing those algebra skills, we're going to pick that up as we move through.
So Jacob and Sophia are going to write an equation for this set of coordinates.
Jacob reckons that the y coordinate is double the x coordinate.
Sophia reckons the y coordinate is half the x coordinate.
Pause the video and have a think about who you think is correct.
I've drawn on the arrows to help me.
So to get from 10 to 5, I'm halving or multiplying by half.
So the y coordinate is half of the x coordinate.
Sophia was right.
Okay, so how could we write an equation for this relationship? So we've looked at how the x coordinate, half, gives us our y coordinate, so we can write that as y equals a half x, or we can write that as y equals x divided by 2.
And I'm going to use that fractional notation to show that division.
So halving a number is the same as multiplying by a half or dividing by 2.
Those two things are the same, which is actually really useful thing to use when you're working with multiplying fractions.
True or false then.
For the coordinates 6, 4, <v ->1, -3,</v> 1, -1, we can write the rule y = x + 2.
So decide whether that's true or false, and you might want to think about why do you think that is true, or why you think that's false? Give that a go and then we'll look at what we think.
Right.
Well done if you spotted that that one is false.
So which of those is the reason then that that is actually false for those coordinates? Okay, so the y coordinate minus 2 is the x coordinate.
Well that's not true either.
The x coordinate minus 2 is the Y coordinate.
Yes, that one is true.
So you could have written it as y = x - 2, or if you wanted to, you could write it as x = y + 2.
Jacob and Sophia are writing a rule for these coordinates, <v ->3, -5,</v> 0, -2, and 4, 2.
The rule Jacob says is y equals a half x.
I'm going to tell you that Jacob is incorrect.
Can you think about how you'd explain that he is incorrect? Sophia says the rule is y = x + 2.
How do you know Sophia is incorrect? Have a go and then we'll come back together.
Right.
Lovely.
So we need to check that this rule works for all the quarters.
So Jacob's rule does work for that last coordinate, 4, 2, y equals a half x.
However, it doesn't work for the other two.
<v ->3 halved is not -5.
</v> Let's check Sophia's as well.
You might have drawn the arrows onto the coordinates as well.
Is the y coordinate the x coordinate plus 2? No, it isn't.
Again, that should be that other way around.
Time for you to have a practise then.
So for each set of coordinates, you've got two bits to fill in.
I would like you to fill in the rest of the relationship, and then see if you can write an equation.
Give that a go and then we'll look at them together.
Right.
Well done on that first set.
So we've got a few more with some different relationships this time.
Off you go and then we'll talk about the answers.
Fantastic.
Well done.
Before that first one, the y coordinate is equal to the x coordinate plus 1.
So B, y = x + 4.
For C, the y coordinate is equal to the x coordinate minus 5.
So we're writing y = x - 5.
And for D, we've got the y coordinate is equal to the x coordinate plus 5.
So we've got y = x + 5.
For that second set, so we've got the y coordinate is double the x coordinate, or you could have written the y coordinate is the x coordinate multiplied by 2.
And that's written as y = 2x.
For F, the y coordinate is 10 times the x coordinate, and that's written as y = 10x.
G, we've got the y coordinate is the x coordinate multiplied by 5, or the y coordinate is 5 lots of the x coordinate, and that's written as y = 5x.
And finding this last one, well done if you have written this correctly using algebraic notation.
So we've got the y coordinate is the x coordinate divided by 10.
So you may have written that as y equals a 10th x, or you can write that as y equals x over 10.
We're using that fractional notation to represent that division by 10.
Right.
Lovely.
Loads of interesting rules there.
So hopefully you'll be able to use those when you're then generating coordinates.
And that's exactly what we're going to have a go at doing now.
So we are going to generate coordinates from a rule.
We can generate coordinates that follow any rule.
Alex wants to write a coordinate where the y value is the x value multiplied by 5.
What coordinates can he write? I'd like you to pause the video and see how many you can think of.
See if you can use some negative numbers or some non-integer values if you want to challenge yourself.
Very well done.
You could have come up with all sorts of different coordinates that follow that rule.
I wonder if you came up with any of the same ones as me.
So I tried some positive integers, so I use 0, and 0 times 5 is 0.
1, 1 times 5 is 5.
But I also tried some negative values.
I wonder if you went for -1 and -5 like me.
And then some non-integer values, so you could have had 0.
5, 2.
5, and ninth, five-ninths.
There's unlimited suggestions that you could have come up with as long as your y value is your x value multiplied by 5.
So this time, Alex wants to write a new coordinate, where the y value is the x value plus 30.
What coordinates could he have this time? Brilliant.
Again, loads and loads of examples.
You might have come up with some of the same ones as me.
Right.
An extra challenge then.
He also wants his coordinate to follow the rule.
The y value is double the x value.
Can you think of a coordinate that will fit both rules? Right.
Very well done if you did spot that one.
So 30, 60 will fit both rules.
So a single coordinate may fit multiple rules, which is why when we're looking at patterns in coordinates, we're looking at two or three, or sometimes four or more coordinates together to see if we can spot that pattern.
Sam wants to write some coordinates, where the y value is the x value minus 7.
They have chosen some x coordinates to use.
How could they work out the y coordinates? Now, this is quite a good way of generating a coordinate, picking an x coordinate and then working your y coordinate from there.
So how would they do that? Well, all they need to do is they need to minus 7 from their x coordinates.
Let's see if we can get what these y coordinate should be then.
So we should have three, 0.
5, <v ->4,</v> and -7.
But what about if we're working the other way around? So now they've chosen some y coordinates to use, how would they work out the x coordinates? So again, we're following the same rule.
We want the x coordinate minus 7 to be the y coordinate.
So we need a number that when we subtract 7, we get 3.
It might be easier to look at the inverse.
So actually, we just need to add 7 to 3 to get our x coordinate.
That give us 10.
And let's just check our rule works, 10 subtract 7, is 3.
For the rest of them then, so we should have 7, 7 subtract 7 is 0, 5, 5 subtract 7 is -2, and 9.
1, 9.
1 subtract 7 is 2.
1.
So we can generate coordinates that satisfy an equation in exactly the same way.
So we've got four different equations that we're going to have a look at.
For each one, I want you to write down some coordinates.
You might want to do pick three or four that satisfy that equation.
We're going to start with y = x + 5.
Off you go.
Right.
Loads of different examples you could come up with.
Here's some of the ones I went with.
0, 5, 1, 6, <v ->5, 0.
</v> Try this one.
Again, loads of different examples you could come up with.
I went with 10, 9, 1, 0, 2.
8, 1.
8.
Well done if you started bringing in those negative values and those non-integer values.
What about y = 7x? Fantastic.
You might have come up with 0, 0.
Remember, I said that's always a nice one to try, seeing what happens when your x value is 0.
It's got 0, 0.
I went with 2, 14, 100, 700, and -5, -35.
Loads of examples you could have come up with as well.
Last one we're going to look at, y equals x over 4.
What can you come up with? So I went with 12, 3, 40, 10, 4, 1, 0, 0, 1, 1/4, and -16, -4.
You might have wanted to make sure that your x value this time was a multiple of 4s to give you some nice answers.
And otherwise, you would've been dealing with decimals and fractions, which is absolutely fine, but as I say, might have made your cases a little bit trickier to do.
Well done if you challenge yourself in that way.
So, which of these coordinates would satisfy the equation, y = x + 2.
1? Pause the video and see which ones you believe that works for.
Okay, so we need to check if we're adding 2.
1, whether we get from the x coordinate to the y coordinate.
So in this case, the x coordinate plus 2.
1 gives the y coordinate for that first one and for that third one.
Right, some negative multiplier now, which is absolutely fine.
We know how to multiply by a negative number.
So which of these coordinates would satisfy the equation, y = -2x? Remember that means -2 times x.
Off you go and see if you get the same as me.
Right, well done if you spotted it's that third one and that fifth one.
The x coordinate multiplied by -2 gives the y coordinate.
Remember, we're multiplying by a negative.
Okay, if we've got positive multiplied by a negative, we're going to get a negative answer.
One lots of -2, is -2.
If we've got a negative number multiplied by a negative number, we'll end up with a positive answer.
So -4 times -2, gives us +8.
Right.
Check then, what is the missing value in each coordinate pair for the equation y = x + 3? Pause the video, write them in, and then see if you've got them right.
Right.
You should have 15, 97.
Well done if you've got this last one.
We've got both a non-integer and we've got a negative, you should have 2.
5.
Perfect for each question then.
You're going to do a very similar thing to what we just done in that check.
I would like you to generate three coordinates that follow each rule.
Some of them I've told you what the x has to be or what the y has to be.
If I haven't, then it's up to you.
You can come up with any values that you like.
Give those four a go and we'll look at the next set.
Okay, this time we're going to need to fill in the equations with three different coordinates for each line, but each coordinate has to satisfy a condition.
I've done the first row for you, so we'll talk through it.
So we've got y = x - 5.
I want you to come up with a coordinate where the y value is positive.
So loads of choices.
I chose 15, 10.
Then I want you to come up with one where the y coordinate is negative.
So I went with -5, -10.
And then this time, the x coordinate has to be a non-integer.
So I've picked any non-integer value, I went 7.
5, take away five, and that gives you 2.
5.
Doesn't matter if it's positive or negative in that last column.
Right, see if you can find a coordinate for all the other missing boxes.
Off you go.
Well done.
Let's look at the answers then.
So for 1a, for the first coordinate, you needed 23.
For your other coordinates, there's loads of examples you can have.
I've put some down there, but check that yours work for y = x + 20.
For b, you should have -9 for the first coordinate, and 18 for the second coordinate.
Your third coordinate, there's plenty of choices.
I've put some examples down.
For c, you should have 0 as your first coordinate.
We've got 0, 0.
And then 3, 300 for the second coordinate.
Plenty of choices for the third coordinate there.
And then for d, you've got 0, 0 again.
And then plenty of choices for your second and third coordinate.
They just need to have the y value as four times the x value.
Well done.
Again, it's going to be very unlikely that you have exactly the same coordinates as me.
So these are just at an example.
So for b, I went with -4, 6, then -20, -10, and 0.
2, 10.
2.
For c, I went with 2, 10, As long as that x coordinate is positive when you multiply it by 5, the y coordinate would be positive.
Then I went with -10, -50.
This time you want your x coordinate to be negative, so when you multiply it by 5, it is still negative.
And then I went with 0.
1, 0.
5.
For d, you've got -3, 9.
Again, you want to choose a negative value for your x coordinate, so when you times it by a negative, it becomes positive.
Then I went with 2, -6.
This time, you want to choose a positive value, so when you times it by a negative, you get a negative.
And then lots of choices for the last one.
I went with 0.
5, -1.
5.
For e, I chose 6 and 5.
98, <v ->5 and -5.
02.
</v> And then I chose 0.
52 because I knew when I subtracted 0.
02, I've got a nice answer of 0.
5.
And finally, so I chose 9 for my x coordinate.
So 9 divided by 3 is 3, so 9, 3.
And I went with -12 divided by 3 is -4.
Again, you're going to want to start with a negative x so that when you divide by 3, your y is negative.
And the last one was really tough.
And I picked 3.
9 because I know that's going to divide by 3 quite nicely.
So 3.
9 divided by 3 is 1.
3.
Right.
Fantastic, guys.
Lots of thinking involved in that activity.
The last section that we're going to have a look at then is plotting coordinates that follow a rule.
You might want to make sure that you've got some square paper in front of you so that you can have a go at drawing these yourself.
So, Laura is plotting the coordinates 7, 1, 7, -2, 7, 0.
2, and 7, -3.
5.
What do you notice about these coordinates? Right, you might have said something along the lines of the x coordinate is 7 in each of these coordinates.
How would you think we could write this as an equation? Well, the y coordinate can be anything.
So we can write this one as x = 7.
'Cause the only stipulation is that the x coordinate has to be 7.
But what do you think this is going to look like if we plot these coordinates? You might want to sketch them out if you've got some square paper, or just have a think about how you would plot these.
Pause the video and see what you come up with.
You may have noticed then that when you plot these coordinates they form a vertical line.
She now wants to plot the coordinates 2, -2, 2, 2, 2, 0, and 2, 3.
1.
What do you notice about these coordinates? Well, straight away, we can say the x coordinate is 2.
If we're writing this as an equation, we can write that as x = 2.
Right.
I'd like you to pause, have a think about this one.
What do you think these coordinates would look like when we plot them? So these coordinates, just like the ones before, also sit on a vertical line.
So coordinates that follow a rule will form a pattern when plotted.
What about these ones, -2, 5, 0, 5, 3, 5, 6, 5, and 7.
002, 5.
What are they going to look like when plotted? So this time, the y coordinate is always 5.
So we could write that as y = 5.
And when we plot them, you'll notice that they form a horizontal line.
Brilliant.
So time for you to have a go at matching some of these rules with some of these lines.
Off you go.
Lovely.
So x = -1 is that one in the bottom, right-hand corner, all those x values are -1.
X = 2.
5 is the one in the top, left-hand corner.
The x value is always 2.
5 Y = -2.
5 is that top, right-hand one.
And the last one, y = -1 is that bottom left.
So Laura has found some coordinates that seem to follow a pattern.
1, 1, 2, 2, 3, 3, and 4, 4.
Jacob says the y value is the x value plus zero, y = + 0.
Jun writes it as y equals x multiplied by 1.
Sophia writes it as y equals x over 1.
Who do you think is correct? Let's try them out then.
So y = x + 0.
Does that work for every coordinate? Yes it does.
So Jacob's rule does work.
Let's look at Jun's.
Y equals x times 1.
Does that work for everyone? Yes it does.
So Jun's rule seems to work as well.
And Sophia's, y was x divided by 1.
1 divided by 1, is 1.
2 divided by 1, is 2.
Yes, her rule seems to work as well.
So let's look at those three rules then.
So it turns out that all three of those are correct.
I wonder if you thought of slightly simpler rule that would also work, because all of these three could be written the same as y = x.
If you look at those coordinates, the y coordinate and the x coordinate are the same, aren't they? 1 and 1, 2 and 2, 3 and 3.
So you can write that as y = x.
When you plot them, it looks like slightly different pattern this time.
We haven't got a horizontal line, we haven't got a vertical line.
We've got a diagonal line.
Laura, then plots 1, -1, 2, -2, 3, -3, and 4, -4.
Will this look the same do you think? Right, now it's not going to look the same because it's a different rule.
I've plotted them for you so you can have a look.
They follow a different rule, and they look like a different pattern.
They do still sit on a straight line though because they follow a rule.
The rule that we would write for this corner is y = -x.
'Cause the y coordinate is the x coordinate times -1.
Laura wants to see the pattern when she plots these coordinates.
So 0, 0, 2, 4, 3, 6, and -1, -2.
Jacob says to her there won't be a pattern, as the x coordinates are all different, and the x and the y coordinates are not the same like the example we looked at before.
So what do you think, will there be a pattern? Because the coordinates do still follow a rule, well done if you spotted that the y coordinates double the x coordinate in all those coordinates, because they follow this rule, the points will still follow a pattern.
And that's the key takeaway from this.
When coordinates follow a rule, when they plotted, they will follow a pattern.
These coordinates will make a straight line when plotted.
True or false? Think about your justification for your answer, and then we'll see what we come up with.
Right.
Well done if you spotted that that was true.
Which one do you think is the correct justification for why that is true? Well, that's that bottom one, "These coordinates all follow a rule," even better if you spotted that rule is y = 3x.
What about these four coordinates? Will these ones make a straight line when plotted? Well done.
If you spotted that this time that is false.
What is the correct justification for your answer? It is true that the x coordinates are all different, but that's not the reason that they won't make a straight line.
The reason they won't make a straight line is there's no rule that works for every coordinate.
Time for you to have a practise then.
So, for each question, I want you to plot all the integer points that satisfy each rule.
See if you can then describe the pattern as well.
Off you go.
Well done, exactly the same for the second set.
I've just written the rule in algebra this time.
Can you plot all the integer points and then describe the pattern? Off you go.
And the last set of questions, I've plotted some coordinates for you on the right-hand side.
You need to complete the table to show which points follow which rules.
I've done the coordinate A for you so you can see the sort of thing you are looking for.
Amazing on that activity.
Well done.
So for the first one, the pattern you should have is a vertical line.
Check that you've got all the integer coordinates.
For B, you're then going to have a diagonal line.
And then for C, you have another diagonal line, slightly less coordinates that fit on the grid this time.
And let's look at those ones with the algebra.
So the first one, you might have noticed that all the coordinates where y = 0, sit on the x-axis.
That's quite a useful thing to know.
The rule for the x-axis is that all the coordinates have a y value of 0.
For the second one, you get another diagonal line.
Check your points the same as mine.
And for F, you get a diagonal line.
You might have said something like it's sloping in the other direction.
Check that you've marked on all the correct points.
Well done for those questions.
Looking at this final set then, I'm going to read the answers out in columns.
So for B, you have yes, yes, no, yes.
For C, no, no, no, yes.
For D, you have no, yes, yes, no.
For E, no, no, yes, yes.
For F, no, no, yes, no.
And finally for G, no, yes, no, no.
Right, as I said before, loads of fantastic skills in that lesson today.
Just to quickly go over then what we've looked at.
So coordinates that follow a pattern can be described with a worded rule or with an equation.
We can then generate coordinates that follow a worded rule or that satisfy an equation.
And finally, this idea that coordinates that follow a rule will form a pattern when plotted.
Well done, guys.
I hope you're feeling very, very proud of yourself and that you will join us for some more lessons in the future.
Thank you very much.
