Lesson video

All right, 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're 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.

Welcome.

This lesson is on plotting coordinates generated from a rule using technology.

We're going to have a lot of fun today working with technology and graphing software.

By the end of the lesson, you'll be able to use technology to quickly represent a set of coordinates on a graph.

I'm going to use this term "origin" today.

So, if you haven't heard of it before, origin is the point where the x and the y-axis cross, it has the coordinate pair 0,0.

To start with, plotting coordinates in Desmos.

So, graphing software can be used to plot coordinates.

The graphing software that I'm going to use is Desmos, you might want to use the same one as me.

Using graphing software can make it really easy to spot patterns.

Instead of having to draw our own axes and plot our own coordinates accurately, the software will do it for us, which means we can concentrate on looking at patterns and rules, and really exploring what's happening with our coordinates.

It's also really useful to see when a coordinate doesn't fit a pattern, which does allow us to spot our own mistakes, or mistakes in other people's work.

To get started, you're going to want to go to desmos.

com, and then you are looking for the graphing calculators.

There should be a button that says Start Graphing, or it says Graphing Calculator.

When you open it up, your screen should look similar to this one.

So, you have a set of axes in the middle, and then you have a pane on the left-hand side, where you're going to be able to plot coordinates or equations, or lots of other different things as you progress with your algebra skills.

We're going to start by just exploring some of the things we can do with our axes.

So, one of the things we can do is we can change the scale so that they can increase in the required increments.

So you might not want your graph to be going up in ones, you might want it to be going up by a different step.

We're going to make sure, to start with, that we know how to make sure our graph goes up in ones.

So, you're looking at the top right-hand corner of your screen for a button called Graph Settings.

Once you click on that, you end up with lots and lots of options, and you are looking to change the step to 1.

'Cause we want our graph to go up in steps of one.

I'm going to do that on both the x-axis and the y-axis.

If you want to, there's a button at the top, called Minor Gridlines, if you unclick that, it takes out all the minor grid lines.

You'll see what I mean when you unclick it, and you'll see that all those smaller divisions have gone.

So you can put those in or take them out as you want.

So, try having a go at changing the axes so they increase in fives.

Pause the video and make sure you can do that yourself.

Okay, I'll just talk you through the steps.

So you want to make sure you're in that Graph Settings menu, and you are changing the step to 5.

At this point, you may want to put your minor grid lines back in.

You can zoom in and out using the buttons just below the Settings menu.

We might want to see different parts of our graph, so you can drag your screen to see different parts of the axes.

So, try dragging your screen until you can see 200 on the y-axis.

Just check you can do that.

Lovely.

You'll see my example on the screen.

So there's a really useful button there called default viewport.

If you click on that default viewport button, it will centre your screen back around the origin.

So if you've scrolled something, or you've changed the scale of something, and you want to go back to where the origin is, you can press that button.

Right, you can also stretch your axis.

So, if you press Shift, if you're working on a computer, and then move your cursor onto the required axis, you can then stretch it.

So if you try the x-axis, press Shift, move your cursor onto the x-axis, and then drag your mouse left or right, you'll see it changes the scale of that x-axis.

If you do the same with the y-axis, and press Shift, and then move your cursor onto the y-axis, this time drag your mouse up and down, that'll change the scale on your y-axis.

So, try scaling your x-axis to 250 whilst keeping your y-axis the same.

Just like before, if you press that default viewport button, not only does it centre your screen around the origin, it also resets the scale so that the scale's the same on your x and your y-axis.

So, just to check that you're happy with what you can and cannot do on Desmos, the x and y-axis have to have the same step.

Do you think that's true or false? Well done.

That is false.

Which one is the correct justification for your answer? Brilliant, the correct justification is that you can change the step on the axes separately to have different steps.

It's possible to change the step on the y-axis, doesn't have to be the same as the x-axis though.

You can keep the step the same, but change the scale on the x-axis.

Is that one true or false? That one is true.

You can keep the step the same, but actually stretch the x-axis in either direction.

What you might find is that the step has to change, like with our previous example, so that it fits on the screen.

Right, which is the correct justification for why that is true? Perfect.

I hope you've spotted it's that first one.

You can use the cursor and the Shift key to change the scale on either axis.

It's not the case that Desmos will stretch the axes automatically to make the coordinates look good, that's down to you to scale your axes in the way that fits your data or fits your coordinate points.

Right, let's look at some of the other things you can do on Desmos.

So, one of the things you can do is you can plot coordinates.

You can do that by typing those coordinates into left-hand menu.

Remember, coordinates need to have brackets, they also need to have a comma between the x and the y coordinate.

So I've typed in the coordinate 2,4, you may wish to do the same.

You'll notice that the graphing software plots this on your axes.

Yours might not look exactly the same as mine, I'm going to show you how you can change the style of your coordinate.

So, you can change the colour and the type of point by clicking on the edit list button.

So there's a little settings menu there if you click on the edit list button, and then click on the coloured dot next to the coordinate you want to change, and it comes up with all sorts of things you can change.

So you can change the colour, you can also change whether it's plotted as a dot or as a cross.

When we plot coordinates ourselves with a pen and paper, then we're going to use a cross, so you're probably going to want your coordinates to look like a cross as well to replicate what you would be doing on paper.

Okay, I'd like you to have a go at this one now, so you're going to need to pause the video and type these coordinates in before you can answer this question.

So, plot these coordinates.

Which option below best describes the pattern they form? You may have to change your x-axis scale here, give it a go, on which of these do they form? Brilliant.

Hopefully you have something similar to mine.

It might look a little bit different, 'cause you'll see that I've stretched that x-axis so I can see from -200 to 200, but they will form a diagonal line however you have played around with your axes.

Time for you to have a practise, you need to read these instructions carefully and answer the questions.

Well done, I hope you enjoyed playing around with all the things that Desmos can do.

So for that first one, if you change the step on the x-axis to 3 and the y-axis to 4, for most display screens, if you're working on a slightly different device, you might have a slightly different answer, but for most display screens, the x-axis will range from -9 to 9, and the y-axis from -4 to 4.

Then I want you to move your axes until you could see 28 on the y-axis.

So the next value visible below 28 should have been 24.

Then you need to press that default viewport button.

I wonder if your description of what happened is similar to mine? So, I said, "It centres the screen around the origin." That's really important, we've got 0,0 now in the centre of our screen.

Well done, if you also added that it changed the axes so they have the same scale.

Important thing to note, that it still had a step of 3 on the x and a step of 4 on the y-axis.

But you'll see that, actually, the scale is now in proportion.

Okay, so the x and y had the same scale, even though the numbers that are written on the axes are different.

And the last one, what coordinate is between 2,4 and 2,6? You should have the coordinate 2,5.

There's an option to click the label button to actually label your coordinates as well to tell you what you have plotted.

Right, now that you're confident with using Desmos, we're going to start using it to identify coordinates that do not follow a rule.

So, plotting coordinates on Desmos can make it really easy to see whether there is a pattern.

Like we said before, it's going to be a lot easier than drawing them out ourselves on an axis, so where we can use technology, it's a really useful tool.

I'd like you to plot those coordinates, and see what you notice.

Pause the video to give yourself a chance to do that.

Lovely.

Your screen might look similar to mine.

So, looking at mine, I can see a really clear pattern.

So hopefully, if you've plotted the coordinates correctly, you will see a pattern as well.

So, they all lie on a straight line.

Right, a little challenge for you.

Can you find another coordinate that lies on the same line? Pause the video and give that a go.

There are loads of options, perhaps the most obvious is the coordinate 2,2, but if you used non-integer values, you might have been able to find the coordinates that was on that line, or if you went beyond the screen that you can see mine, so if you went further up or down the axes, you may have found other values as well.

So using Desmos can be especially helpful when working with non-integer coordinates.

They're quite tricky to plot if you're doing it by eye.

If you're using a computer, you can plot them more accurately.

Now add those two coordinates to your graph.

I'll give you a second to do that.

Do those coordinates still follow the same rule as before? See if you can think about how you'd explain to someone that you know that they do or do not follow the same rule.

You might have come up with something similar to this.

So, they sit on the same line.

You see how they're all inline with the other points we plotted? That means they follow the same rule as the other points we plotted.

If we actually know the rule linking the coordinates, we can write the equation into Desmos.

I'm going to show you how to do this.

I'm not too worried about you being able to spot the rules at the moment, what I really want you to be able to do is to actually put them into the graphing software.

So, if you click on that left-hand menu where we've plotted the coordinates before, you can type an equation.

So try this equation, try y = 2x-2.

If you haven't got a keyboard, there's a little button at the bottom that you can click on, and it brings up a keyboard to write with.

Just like before, we can change the colour and the style.

So you can either click and hold on that dot, or you can go into that settings menu and click on the dot.

And what that means is you can change the line to a dashed line or a dotted line, or you can change the colour of that line.

So, does the line that I have asked you to draw, y = 2x-2, does that go through all your coordinates? Let's have a look.

If we've typed correctly, yes, indeed, it should go through all our coordinates.

What that means is those coordinates satisfy that rule, y = 2x, subtract 2.

Why do you think Desmos has drawn a straight line instead of just plotting loads and loads of coordinates? See if you can put your answer into words.

Right, the main reason is because there are an infinite number of points that fit this rule.

We have picked a few coordinates that follow this rule, but if we look at decimal values, we could come up with even more.

If we looked at really large integers, you'd find even more, so this rule is not limited to a certain set of coordinates, there are an infinite number of points that fit this rule.

By drawing a line, we can include all the values, including all the little decimal values in between all the integer points.

You might want to spend some time scrolling and moving your screen around, and you'll find that it's not possible to find the end of that line.

What we can do once we've got a rule, like an equation, we can get Desmos to generate some of the coordinates for that rule.

So we've already picked some, but we can get Desmos to show us some more.

Again, if you click on that edit list button, and then we want the create table, and it should bring up this table.

Pause the video and give that a go yourself.

Brilliant.

This table also gives us some examples of coordinates.

We've got the x coordinate in the left-hand column, and the right-hand column is going to be our y coordinate.

You'll notice that it's given us five options there, when x is -2, when x is -1, when x is 0, when x is 1, and when x is 2.

We know that that's just a snapshot of some coordinates that fit that rule.

So have a go now at plotting these coordinates.

You might want to get rid of some of the other ones that you have on your graph so that you can see really clearly where these are.

And again, you may need to move your axes around to find them.

Right, I found 'em on my graph.

So, what do we notice about these points? We could say something like, "They sit on a vertical line." How can we write an equation for this rule? If you look back at your coordinates, what relationship do you notice about all those coordinates? If you then type it into Desmos when you've got an idea, you'll be able to see if it's correct.

Give that a go.

Right, you might have said something along the lines of x = 26.

If you've got that exact equation and typed it in, it should look something like that.

We know it's correct, because our line is going through all those coordinates, so those coordinates must follow the rule, x = 26.

So what we can do is we can check we have the correct rule by typing the rule and the coordinates into Desmos.

So, do these coordinates lie on these lines? Pause the video, have a look at the rules and the coordinates, and see if you agree where they're going to lie on those lines.

You may want to type 'em into Desmos to see if you can prove that's true.

I'm going to look at these coordinates, without Desmos to start with, and then I'll plot them to show you how we know whether they do or don't fit on the line.

So the first one, yes, all these coordinates will sit on the line y = 4, 'cause the y coordinate is 4.

For this second one, all the coordinates will fit on x = -y, 'cause the x-coordinate is the y-coordinate times -1.

And the last one, yes, those coordinates sit on y = 3x, 'cause the y-coordinate is three lots of the x-coordinate.

Let's show that on a graph.

So, there you go, I've typed the coordinates in on the left-hand side, I've then typed in the equation, y = 4, and you'll see that that line goes through my coordinates.

I'll show you the same thing for the next two.

So, there's x = -y, and it goes through all our coordinates.

And finally, y = 3x, and again, it goes through all our coordinates.

Where this becomes really, really useful is that we can check whether a coordinate follows a rule by seeing if it follows that pattern.

So type these five into Desmos, and see if you can find the odd one out, explain to me how you know.

Off you go.

If you've typed 'em into Desmos, it'll look similar to the graph I've put on your screen.

We can identify that 1,3 is the odd one out, hopefully, that's the one you got, because it doesn't sit on a straight line with the other points.

I'm going to show you what happens when we type the equation in, because it gives us an even clearer idea that that one does not sit on the line.

Right, I'll test your skills now.

Do you think you could tell me what the equation is that links those coordinates? Give it a go.

I'm impressed guys, hopefully, you got this y = x+3 Type it in, and then we can see whether it goes through our coordinates.

And you'll notice, it goes through most of the coordinates, but it doesn't go through that 1,3, because 1,3 does not follow that rule of y = x+3.

Now, this is a really cool thing that you can do on Desmos, you can make any point draggable.

So if you click on that edit list button, and then click on the coloured dot next to the coordinate, then you'll see that there's an option to make it draggable.

So if you click on the slider next to drag, you should then be able to click on the coordinate when it's on your axes, and drag it around.

Try doing this for the coordinate 1,3.

Right, now click on the coordinate, and drag it so it's inline with the rest of the coordinates we had.

I also want it to have integer coordinates, you might have to play around until it's inline with those coordinates, and has an integer x and an integer y coordinate.

It should read 1,4.

And that now follows our pattern, doesn't it, of y = x+3? Right, Andeep has plotted these four coordinates, you can see it on my graph below.

He thinks they have the rule, y = x-2.

I've plotted that with a black, dashed line.

Two questions for you.

Do those four coordinates follow a rule? How do you know? And has Andeep written the correct equation linking those four coordinates? How do you know? Give it a go, and we'll see if your explanations are similar to mine.

Well done, if you spotted that, yes, they do follow a rule, because they sit on a straight line, they have that pattern when they are plotted, but the rule is not y = x-2.

We know this because Andeep's line does not go through all his points, so the line he's drawn cannot be the rule for those coordinates.

Right, lovely task for you to have a go at now.

So, using graphing software, I'd like you to plot each set of coordinates.

You want to do all of a first before you move on to b, so that you don't get confused with what coordinates you're looking at.

In each one, I'd like you to identify the coordinate which is the odd one out.

It's really important that you can do this using graphing software.

So, you shouldn't need to try and work out the rules here, okay, do it just from plotting them on your graph.

Off you go.

Lovely, I hope you're enjoying playing around with that graphing software.

So, Izzy has now plotted some coordinates.

Which one do you think she's plotted incorrectly? Again, if you can describe how you know, that's even better.

Once you've done that, I'd like you to try and recreate Izzy's graph.

Then make the incorrect coordinate draggable.

If you've forgotten how to do it, remember, there's instructions on previous slides.

Then I want you to drag the incorrect coordinate so it's inline with the other coordinates, but I want it to have integer values.

When you followed all those instructions, can you jot down what coordinate you have now plotted? Off you go.

All right, let's check our answers then.

So you should have 2,6, 0,7, -4,3.

2, and 1,3/10 as your odd ones out.

For the second set, it's 7,2.

6, which is incorrect, your answer's going to look something like, "It doesn't sit on a straight line with the other four points," or, "It doesn't line up with the other four points." Once you've made your point draggable, if you just drag it vertically, you get the coordinate 7,3.

It's also possible that you could have dragged it all the way over to the coordinate 2,1, that's also inline.

There are many other possibilities not visible on this graph.

Right, final part of the lesson then, we're going to try creating some shapes in Desmos.

So, we can use Desmos to create mathematical shapes.

I would like you to return your axes to the same scale, using that default viewport button, and type in those four coordinates.

I would like you to tell me what shape you have plotted.

I've put some examples of shapes at the bottom to help you if you're not quite sure about your mathematical language for shape.

Right, hopefully, you have plotted a kite.

Try this one.

You've got -2,-3, -6,-4, -7,0, -3,1, the shapes there again to help you.

Well done.

Hopefully, you have plotted a square.

Even though it's tilted, the sides do meet at right angles, they are also the same length, which makes it a square.

Sometimes if you have something like a set square, or the corner of a piece of paper, and you just hold it up to your screen, it's easy to check if two lines meet perpendicular, meet at a right angle.

Right, I'd like to play around now with stretching the scale of your x-axis, so press Shift, and use your mouse.

What happens to your square when you stretch one of the axes? Right, it becomes a parallelogram.

This is going to require a little bit of thinking, so I'd like you to pause the video and think about this answer.

Why do you think that happens? Okay, the reason is, you've changed the distance between the values on one of the axes, but kept the distance between the values the same on the other axis.

That means the coordinates in the x direction and the y direction are no longer in the same proportion.

I'd like you to click on the default viewport button now to bring the axes back to the same scale.

Here are another four coordinates.

What shape do you think they will make? Then I'm going to let you plot them and find out.

You should have plotted a rectangle.

Try changing your scale on the x-axis to make your rectangle look like a square, like I have done.

Off you go.

So, sometimes we do need the scales on the axes to be different, that is okay.

But it's important to be aware that, if you do that, it does change what your points, or shapes, or equations look like.

Let's return to our default viewport button.

I would like you to keep your scales the same for the following activities.

We can use Desmos to help us find the missing vertices of a shape.

So can you plot those three coordinates? And I'd like you to find the fourth coordinate that will make it a square.

Off you go.

Lovely.

You should have got 3,2 for that one.

Well done.

So, what do we think of this then? Changing the scale on one axis will change a shape made out of coordinates.

True or false? True.

Absolutely.

What is the justification for our answer? Right, does changing the scale move all the points to different coordinates? No, it doesn't.

What does happen is it stretches the scale so that they are different on the two axes.

That means the distance between the numbers are different in the x and the y directions.

Right, time for you to have a play around yourself then.

So, for each of these sets of coordinates, type them in, and find the fourth coordinate to make a square.

Off you go.

Right, well done with that one.

This time, I'd like you to plot each set of coordinates, and then determine what type of quadrilateral they make.

I've put some types of quadrilaterals on the screen to help you.

Well done, guys.

So, for that first set you should have 2,8, then -5,4, then 10,4, then -3,1, and then -7,-0.

5.

Lovely, my pictures might help you see the different shapes for this one.

So, I think you should have a rectangle for the first one, a parallelogram for the second one, that was the slightly trickier shape, so, well done, if you got that one.

A kite for the third one.

And then another rectangle, but it's on a tilt this time, it's still a rectangle, for that fourth one.

I really hope you enjoyed playing around with the technology there guys, and that it's something you will be able to return to and use when you're doing other graphing lessons in the future.

So, coordinates can be plotted into graphing software, such as Desmos.

We've also looked at how rules can be entered into Desmos to form an equation, and then what Desmos does is draws a line of infinite coordinates.

We've seen how plotting coordinates using technology makes it really easy to see when coordinates do not follow the rule.

And then lastly, we had some fun playing around with Desmos being used to create mathematical shapes using our coordinates.

Be really lovely to see you again for another video soon.