Lesson video

Hello, Mr. Robson here.

Superb choice to join me for mathematics today, using dynamic software to explore linear relationships.

This should be wonderful.

Let's get started.

Our learning outcome is that we'll be able to solve a range of problems involving graphical and algebraic aspects of linear relationships using dynamic software.

Key words that I'll be using throughout this lesson, gradient and intercept.

Two parts to today's lesson.

In the first half, we're going to play with changing the gradient.

I'd like you to start by opening up a web browser and going to desmos.

com.

Find and press the graphing calculator button, should be in the centre of your screen and then your screen should look like this.

I'll ask you to pause and just make sure you've caught up to make your screen look like my Desmos screen there.

Once in Desmos, I'd like you to click on the graph settings menu in the top right of your screen.

Upon clicking it, this should come up.

These are your graph settings.

A few amendments I'd like you to make to your graph settings before we start exploring linear relationships.

I'd like you to turn off the minor grid lines.

Can you un-highlight that box? I'd like you to type X in the X axis label box, and Y in the Y axis label box.

This'll label your X and Y axis.

I'd then like you to change the step on both axis to one by typing a one there.

I'll ask you to pause and just make sure your graph settings match mine.

Your screen will look a little like this.

Now I can see the steps of one i.

e, the grid stepping up in ones.

I can see an X axis label.

I can see a Y axis label.

It looks like we're ready to explore linear relationships.

But when we start exploring, we're not going to want to look at this tight little region here.

The values that we plot are going to go way beyond five and negative five in both the X and Y directions, so we need to be able to zoom and drag our screen.

You've got a couple of options for zoom.

You can hit these buttons, the positive and negative in the top right of your screen.

You'll notice that they zoom you in and zoom you out.

Alternatively, if you hold down the shift button on your keyboard and hold down the mouse button, you'll be able to move the mouse back and forth to zoom in and out.

You'll also be able to look at other regions.

We won't always want to see the origin.

We might wanna disappear off way into the first quadrant, way into the third quadrant.

In order to do that, you just click and hold your mouse button without holding shift on the keyboard and you'll be able to drag your axis around the screen.

I'll ask you to just pause this video now and just have a go at zooming in, zooming out and dragging your graph so you can look at different regions.

We're now ready to explore linear relationships.

We're going to start with a very simple one, Y=X, and we're going to look at this region around the origin from values of positive five to negative five on the X and Y axis, so make sure your scale looks a little like mine.

When you type in Y=X there, this will happen.

You'll get a straight line, the line Y=X.

The line Y=X is a gradient of one, intercepting the Y axis at the origin.

Could you change that Y=X to read Y=2X.

Your screen should look like that.

The line Y=2X is a gradient of two intercepting the Y axis at the origin.

You could zoom right the way in, you could zoom miles out and you'll still see a gradient of two everywhere you look on that line.

Let's add a second line.

Just below where you've got Y=2X written, click the area below it and type Y=3X, and your screen should look like that.

You've got 2 lines now.

One is Y=2X, the other is Y=3X.

Y=3X has a gradient of three and also intercepts the Y axis at the origin.

You might want to pause this video and just check that your screen looks just like mine or you might be ready to just continue at this speed.

That's your choice.

Quick check that you can operate Desmos now and make some observations about the lines that it's drawing for you.

You have the lines Y=2X and Y=3X.

I'd like you to add the lines Y=4X and Y=5X.

Once you've done this, you should see four lines on your screen.

I'd like you to then make a comparison of the four lines.

By comparison, I'd like you to observe, what do they have in common and what is different about them.

Pause this video and do that now.

Your screen should have looked like that.

Four lines, Y=2X, 3X, 4X, 5X, and your observations should have been something along the lines of all four lines share the same Y intercept.

They all intersect that Y axis at (0,0).

What's different about them? The gradient, they've got different gradients.

Y=2X, a gradient of 2, Y=3X, a gradient of three, Y=4X, a gradient of four, Y=five X, a gradient of five.

The four lines have the same Y intercept, but different gradients.

Will that be the same for all lines in the form of Y=MX? If I typed in Y=100X, same intercept, goes through the origin, but different gradient.

It feels like that would be right, but we can use Desmos to explore whether there is truth in that statement or not.

I'd like you to delete these four lines and type Y=MX.

Your screen should look like that.

You might wanna pause or you might be okay to keep going at the same speed as this video.

We're going to now click on M and we're going to add a slider just underneath where you've typed in Y=MX, you'll see Desmos offering you the opportunity to add slider.

What do you notice about that M? This is unusual.

Y=2X, Y=3X, Y=4X.

The X was italicised.

The Y was italicised because they were variables, but the two, the three, the four were constant.

The gradient was constantly two or it was constantly three.

Look at that M, it's italicised.

This slider's going to turn M into a variable.

I wonder what's going to happen.

This happened.

You have a slider with values of negative 10 to positive 10 and you've got a play button.

I'd like you to pause this video and then hit that play button on Desmos and enjoy the show.

What happened there was the slider made M, the gradient vary from positive 10 where you saw the line Y=10X down to negative 10 where you saw the line Y=-10X at the other end of the slider.

You saw all the gradients in between.

We saw negative gradients, we saw non-integer gradients.

We saw the gradient change a lot.

When the slider was on M, Y=MX, we saw the gradient change, but the Y intercept did not change.

It didn't move from the origin.

This tells us that M determines our gradient.

It doesn't determine our Y intercept.

M determines the gradient.

You might want to make a note of that.

To study this more closely, we can toggle this slider.

Desmos is a very versatile graphing tool.

What do I mean by toggle the slider? Yours should currently look like this.

I'd like you to click on the negative 10 or the positive 10, the bounds of your slider.

Once you've done that, you can change these bounds.

I'd like you to change the bounds to negative three and positive three.

I'd like to change the steps to one.

Once you've done that, can you pause this video, hit play and try and observe what's different this time.

This time we saw seven different lines because we bound the slider from negative three to positive three inclusive, but made it do steps of one.

We saw the lines Y=3X, 2X, X, 0X, negative X, negative 2X, negative 3X.

The gradient changed in steps of one as the animation rolled.

Could you pause on Y=2X.

Alternatively, just pause the slider and drag the small icon to the two.

You should end up with the line Y=2X.

Your screen should look like mine.

But what does Y=2X mean? So my students having a conversation about this.

Lucas says, Y=2X means a straight line with a gradient of 2.

Laura says the Y coordinate is double the X coordinate.

Sam says, for every change in positive one in X, there is a change of positive 2 in Y.

Who's right.

Pause this video and tell the person next to you who of my three students is correct.

The truth is they're all right.

They're just thinking about the line Y=2X slightly differently.

If you press the edit list icon, we can make some different observations about the line Y=2X.

The edit list icon is that cog there.

When you click it, this should come up.

From there, I'd like you to click on table.

It won't say table, it'll be a small icon of a table.

Click there.

Once you've done that, a table comes up.

Your screen should look less just like mine.

Could you press and hold here to change your coordinates to crosses and resize those crosses so they're nice and visible.

Your screen should now look like mine.

Not only do I have the coordinates for Y=2X, I have those coordinates in a table of values and I have the straight line Y=2X running through all those coordinates.

Think back to what my pupil said about this line.

The Y coordinate is double the X coordinate.

We can see that in the table of values.

Double two to get four, double one to get 2, double negative two to get negative four.

That rule runs right throughout this table of values.

Another pupil said for every change of positive one in X, there's a change of positive 2 in Y.

That's visible in the table also.

Change of positive one and X, change of positive 2 in the Y coordinate that continues on.

Another student said a straight line with a gradient of 2.

Absolutely correct.

We can see that change positive two in both the tabler values and the gradient on our graph.

Quick check you've got that now and you can describe a line in a variety of ways.

I'd like you to move this slider so it no longer reads Y=2X.

I'd like you to move this slider to negative three.

You'll have the line Y=-3X.

Once you have that, I'd like you to make some observations and describe that line.

You might choose to write these descriptions down.

You might choose to tell the person next to you.

Your choice.

Pause now.

Your screen should have looked like that once you move the slider to negative three.

Then you might have said about this line, it has a Y intercept of 0,0.

Don't forget we're changing the gradient.

The Y intercept hasn't changed at all yet.

You might have said a gradient of negative three.

You might have said that's a change of negative three and Y for every positive one change in X.

You can see that on the line.

You can see in the table of values also, for every change of positive one and X, we have a change of negative three in the Y coordinate.

You might also have said the Y coordinate is always negative three times the X coordinate.

Next, I'd like you to change the slider so that we can have a look at some non integer gradients.

Click on either the negative three or the positive three, the bounds of your slider.

Now I'd like you to change the steps to 0.

5.

Once you've done that, pause this video, hit play and enjoy the show.

You would've seen a greater number of lines this time.

Instead of just seeing Y=3X 2X, X, we saw Y=3X, Y=2.

5X, Y=2X, Y=1.

5X.

You saw lots of non integer gradients.

If we pause that slider on negative 0.

5, your screen will look like that.

I'm gonna ask you to undo something I asked you to do at the start of the lesson now because it's going to help us to make closer observations about non-integer gradients.

In the top right, could you click your graph settings again and turn back on the minor grid lines? You might wanna pause and do that now.

Your screen will look like that.

Our grid no longer has a step of one.

We've got our minor grid lines.

This enables us to read all of those coordinates more accurately.

Negative one, 0.

5.

I can see, one, -0.

5 I can see.

All the coordinates in that table of values, I can now read from those minor grid lines.

If you've got this line, Y=-0.

5X on your screen, I'll ask you to pause and have a think about this question.

Which of these is true of the line Y=-0.

5X? Is it a gradient of negative 0.

5? Does it go through (0, -0.

5) as a coordinate? So for Y intercept at the origin.

Does it have a change of -0.

5 in Y for every change of positive one in X? Which of those statements is true for this line? Pause and have a think about that now.

The first one was true, the line does not go through the coordinate (0,-0.

5), but it does go through the origin.

It has a Y interceptor E origin (0,0), and a change of negative 0.

5 in Y for every change of positive one in X is essentially the same as saying it has a gradient of negative 0.

5.

Practise time now, question one.

You are the teacher.

I'd like you to plot the line Y=3X.

Add a table of values once plotted.

Turn the coordinates into crosses and then explain to the Oak pupils the link between the algebra Y=3X, the table of values and the graph.

You are going to use your Dynamic software to complete this task.

You should write a few sentences of explanation to my Oak pupils.

Pause and do that now.

For question two, I'd like to plot the lines Y=-4X and Y equal a quarter X using Dynamic software.

Once you've got those two lines on the same screen, could you write a few sentences to compare them? I'd like a sentence describing something they have in common and I'd like a sentence describing something that is different about those two lines.

For question three, I'd like you to plot the coordinates (2.

43) and (1.

4, 1.

75) And then I'd like you to draw the straight line that goes through both those coordinates and the origin.

The question is what is that line? There's lots of ways you could work it out.

I'll leave that down to you to figure out.

Pause and try those 2 questions now.

For question four, a beautiful task this one.

I'd like you to draw the lines that recreate this pattern.

I will say no more.

I will leave you with a Dynamic software to play around with your screen and see if you can make yours look just like my beautiful pattern here.

Enjoy.

Some feedback now.

You are the teacher.

How exciting.

I asked you to plot the line Y=3X and explain to the Oak pupil's link between the algebra, the table and the graph.

Your screen should have looked like that.

The coordinates for Y=3X visible in that table and on the graph with a straight line Y=3X running through them.

Lots of ways you could have described that.

There's just a couple there.

For question two, two lines to draw and then a couple of sentences to compare them.

If you drew them successfully in Desmos, they should have looked like that.

You might have said in comparison of the two, the thing they have in common is the same Y intercept.

They both have a Y intercept at the origin (0,0).

You might have said in terms of gradients, one has a positive gradient, one has a negative gradient.

You may have been more specific and said they have respective gradients of negative four and one quarter.

Lots of observations you could have made there.

There's a couple.

For question three, plotting those two coordinates and then drawing a straight line that goes through them both and also goes through the origin.

Tricky little problem this one, but if it goes through the origin, it's gonna be in the form Y=MX.

And we know that M is the gradient, so we need to know the gradient of this line.

Did you notice the X coordinates 1.

4 and 2.

4? That's a change of positive one in X.

If we can identify therefore that there's a change from 1.

75 to three in the Y coordinate that will tell you that you see a change of positive one in X, a change of positive 1.

25 in Y.

That tells us a gradient of 1.

25.

Lines in the form of Y=MX go through the origin.

With a gradient of positive 1.

25, this must therefore be the line Y=1.

25X.

You might have typed in Y equals one and a quarter X or Y equals five over 4X.

Both of those would've been perfectly fine.

They would be the same as Y=1.

25X.

For question four, what do we notice about those lines? They all have a wide set at the origins.

They are lines in the form of Y=MX.

We just need to find the M, find the gradients.

You might have noticed that these lines were the same as the ones in question two, or two of these lines were the same as the ones in question two.

They all had a gradient of positive or negative four, positive or negative a quarter.

You should have drawn those six lines.

Second half of the lesson now.

We've changed the gradient.

Now we're going to look at changing the Y intercept.

For every line we've seen so far, we've gone through the origin (0,0) for example, Y=X.

Y intercept at the origin and a gradient of one.

I'd like you to add these lines to this graph and then pause, make a few observations.

Your screen should look like this.

We should have those four lines drawn.

Did you notice there's something similar about them? They all have the same gradient.

That means they're parallel lines.

However, there is something different.

They've got different Y intercepts and different X intercepts, i.

e, where they cut the cut the Y axis, where they cut the X axis has changed.

We're gonna explore this further by adding another slider.

This time, this slide is going to be C.

We're gonna type in Y=X+C and then when we're offered the opportunity to add a slider by Desmos, we're going to take it.

Can you click there to add slider? Your screen should now look like this.

I'd like you to pause this video, hit play on the slider and enjoy the show.

Whilst you were looking at that, two of my Oak pupils did the same thing.

Aisha and Alex hit play on the slider and then made these observations.

Aisha said, I saw the lines moving up and down, whereas Alex said, I saw the lines moving left and right.

Hmm, who do you agree with? Pause this video and tell the person next to you.

Some of you are going to hate me because Aisha is correct.

From X + 1 to X + 2 to X + 3, the lines are translating up and down, they're moving up and down, they're not moving left to right.

Apologies to all of you who said that.

The lines are moving up and down.

In order to convince you that the lines were moving up and down, not left and right, it's going to benefit us to add the coordinate (0, c).

It'll help us to see the truth behind the translation up and down.

I'm gonna ask you to change the bounds on your slider.

Leave the slider in place, but click on the negative 10 or the positive 10 and change those bounds to -3 and positive three.

Also, change the step to one.

Once you've done that, can you type in the coordinate (0, c) and click label? We're going to want to see this coordinate.

If we click label, we'll be able to observe it as it changes.

Once you've done that, hit play and enjoy the show.

You now see something like this or all the variations of this as your slider changed.

I've paused my slider at three.

I've got the line Y=X+3 and I've got the coordinate (0,3).

Can you see as the C changed the coordinate intersecting the Y axis changed? You would've seen all of these things.

Those seven lines would've been drawn by Desmos for you, the line Y=X+3, Y=X+2 Y=X+1, et cetera.

You would've also observed the Y intercept changing like so, and do notice Y=X+3 or Y intercept (0,3).

Just like my screen here.

All the other lines you saw, Y=X+2 intercepted the Y axis at (0,2).

Y=X-2 intersected it at (0,-2).

What was actually happening there when we left C as a slider, Y=X+C had a Y intercept of (0, c).

This enables us to conclude that it is C which determines a Y intercept.

That's really important.

You might wanna repeat that sentence to the screen and write it down.

C determines a Y intercept.

I'd like you to set your Desmos up slightly different now.

Could you set it up to read Y=C-2X? You'll see I've left my slider as it was and I've left the coordinate (0, c) in place.

C should be set up with a slider with steps of one.

Once you've done that, hit play and then tell me what you notice.

Pause this video now and do that.

You would've seen a variety of lines.

You should have observed they had the same gradient, -2.

They were parallel lines with the same gradient.

But what was changing? They had respective Y intercepts of (0, 3), (0, 2), (0 1), (0), (0, 0), (-1), et cetera, and you would've noticed that the (0, c) coordinate was always on the Y axis.

It was always the Y intercept.

I might have mentioned it before, C determines the Y intercept.

Quick check that you've got that now.

Using that same Desmos animation, the line Y=C-2X.

Can you fill in the missing information in this table? Pause, write that down, or tell a person next to you.

For Y intercept of (0,2) we should have had the line Y=2-2X.

For the line Y=1-2X, an intercept of (0,1).

For the line Y=-1-2X, we should have had a Y intercept of (0,-1), and with a Y intercept of (0, -3), we needed the line Y=-3-2X.

Another quick check that you've got all that I've said so far.

Lines in the form Y=MX+C.

It is the M which determines the gradients, whereas the C determines the what? Pause, tell the person next to you.

I do hope you said Y intercept.

For lines in the form Y=MX+C, the M determines the gradient, whereas the C determines the Y intercept.

If the line through the origin is Y=-0.

5X, what is the second line? Three options there.

Pause.

Discuss this with a person next to you.

Have a little think to yourself.

You should have said it was option B, Y=-0.

5X-3.

Practise time now.

We're gonna use Dynamic software to complete these tasks.

Question one, plot these lines and write a sentence or two comparing them.

Pause and do that now.

Question two, plot the lines and complete the table.

Pause and try that now.

Question three, using Desmos, can you draw the four lines that make that beautiful pattern for part A, and can you draw the lines that make that wonderful pattern for part B.

Enjoy.

Feedback time now.

Plot the lines and write a sentence or two comparing them.

If you plotted them, they should have looked just like that.

All three lines and an X coefficient of two, a gradient of 2, they were parallel lines.

What do those lines have in common? The same gradient thus making them parallel.

For Y=2X+3, you should have noticed, it's a translation of the line Y=2X by positive three in the Y direction.

That gave you Y intercept of (0,3).

A translation in the opposite direction for 2X-4 is a translation of the line Y=2X by negative four in the Y direction, giving you a Y intercept of (0, -4).

For question two, the top three lines would look like that when you draw them on Desmos.

And you should have noticed Y=3X+7, a gradient of three.

Y=3X-2, a gradient of three and a Y intercept of (0, -2).

For the line Y=0.

2X+2, a gradients of (0.

2) and a Y intercept of (0,2).

The two lines that we have to discover, gradient of seven, a Y intercept of (0, -3), that's the line Y=7X-3.

A gradient of negative three and a Y intercept of (0,7), that's the line Y=7-3X.

Question three part A, a key observation that would've helped you from the very beginning would've been there's two lines with a Y intercept of (0,4).

There's two lines with a Y intercept of (0, -1).

I just need to notice the gradients now.

The lines that intercept at (0,4), I see a gradient of positive 2 and a gradient of negative 2, so that must mean I need to draw the lines Y=2X+4 and Y=4-2X.

Similar thing for the lines that intercept at (0, -1), I can see a gradient of positive a half and a gradient of negative a half running through that Y intercept, so I would've needed the lines Y equals a half X minus one and Y equals negative one minus a half X.

A similar strategy would've helped you with part B.

If you noticed Y intercepts of (0, 6), (0, 3) (0, -2), when you look at the intercept (0,6) I can see a gradient of positive three and a gradient of negative three on those two lines intercepting there.

That's Y=3X+6 and Y=6-3X.

For the lines which intercept at (0, -2), I can see a gradient of positive one and a gradient of negative one going through there, so I needed lines Y=X-2 and Y=-X-2.

For the line intercepting at (0,3), it has no gradient.

It's just the line Y=3.

If you wrote Y=0X+3, it's fine, but you don't need to write the 0X.

It's just Y=3.

In that lesson, we found that we can use Dynamic software to see algebraic and graphical aspects of linear relationships and problem solve with them.

Dynamics software helped us to see quickly that in Y=MX plus C, it's the M that determines the gradient and the C that determines the Y intercept.

I hope you enjoy this lesson as much as I did, and I hope to see you again soon for more mathematics.

Bye for now.