Lesson video

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Hi everyone.

I'm Ms. Jones and today's lesson is all about points of intersection.

Hopefully you have come across that word before, but essentially it means where two lines or several lines cross.

That's all about intersecting.

Before we can begin however, you need to make sure you have a pen and some paper as well as clearing any distractions and making sure if possible you have a nice quiet space to work.

Pause the video here to make sure you have all of that.

And then we will begin.

Okay, let's begin.

So the first thing I would like you to do is draw the graphs of the two equations below on the same axis.

So we've got y equals 2x add one, and we've put x add y equals five.

So draw your set of axis and then plot both of those graphs and those equations onto that.

Pause the video now and let's have a go at that.

So this is what you should have got.

you should have had a wind set of one and a gradient of two for this line here.

And we should have had a gradient of negative one and a wind set of five for this one, or plotting them by finding coordinates on that line, really well done, if you managed to get two lines that look like this.

When straight line graphs meet, they do so at a point of intersection.

So I would like you to use your finger, maybe not your finger, cause it might make your screen greasy, but a pencil or something, point at where you think the point of intersection is.

If you pointed here, well done, this is where it intersects.

And hopefully you've come across that word before.

But I would like you to think about is what will happen to the point of intersection, so this point here, if the gradient of y equals 2X add one increases or decreases.

So we're looking at this line here, what's going to happen if we change the gradient.

So if that gradient were to increase, so gradually getting more and more steep like this, if it were to increase, we can see that that point of intersection with the other line is going to change, it's going to go this way here, which means the x-ordinate is going to get smaller.

And the y-ordinate is going to get larger, and the opposite will happen, if it gets less steep, if the gradient decreases, it's going to go further down that line and the x-ordinate will get larger this time and the y-ordinate smaller.

What will happen to the point of intersection if the y-intercept of y equals 2x add one, increases or decreases? So we talked about the gradient.

What about the wind set? What do you think is going to happen there? So if the y-intercept were to increase, that means this point here is going to go up.

So the line could change to this or this.

And we can see again, that, that point of intersection will be very similar.

The point of intersection is going this way.

So my x-ordinate is getting smaller and my y-ordinate is getting larger.

And again, the opposite will happen.

If my winder set decreases, if it ends up down here we can see that my x-ordinate gets larger and my y-ordinate gets smaller.

So the gradient and the y-intercept is going to affect where the point of intersection is, because it changes the line, doesn't it? So pause the video now to complete your independent task.

The first question is asking you to draw the graph of y equals 3x add four, and y equals negative x add six.

So hopefully you managed to do that either by plotting coordinates, or by using the wind set and gradient.

You are then asked to use the graph that you created to estimate the point of intersection.

The reason it says estimate is because it's not exactly on a whole number where we can see it that's been marked, but hopefully you should have got 0.

5, 5.

5 or something a 10th of that, really well done if you managed to do that.

How could we change the line y equals 3x add four to increase the x-ordinate in the point of intersection.

So remember the x-ordinate are these ones, we want it to go this way, which means I would need to either decrease the gradient, that's becoming less and less steep, like that, or decrease the y-intercept.

So it's going down the y axis like this, just imagine those two lines are parallel.

How could we change the line y equals 3x add four to decrease the y-ordinate in the point of intersection.

So if I want to decrease the y-ordinate, I'm going in this direction.

So that means, again, my line would need to become less steep.

So decrease the gradient, or it need to go lower down.

So decrease at the y-intercept, really well done.

if you managed to get those answers, great job.

I would now like you decided if the following are always sometimes or never true.

I would strongly suggest for this one, trying to use examples.

So maybe drawing out some different graphs to experiment with your answers, because you need to justify your answers.

Two lines intersect at least once.

A set of three lines intersect at two points.

And a set of three lines has one point of intersection.

Are they always, sometimes, or never true? Pause the video to complete that task.

So they were all sometimes true, amazing job if you managed to get those answers, we can see that two lines intersect at least once, is almost always true, except for when we have two parallel lines that we can see here with y equals 2x add five, and y equals 2x subtract five.

They are never going to intersect because they are parallel, and on the same gradient.

For the second one, a set of three lines intersect at two points.

This is sometimes true again, because let's take these three lines here, the purple ones, those three lines only intersect once.

So there's an example where it's not always true, but it is sometimes true.

Sometimes that happens where they intersect at two points.

So for example, if we take two powdered lines and this pink one, that intersects at two points.

That also leads us to the third one, a set of three lines has one point of intersection where we've seen it work with this one, so it is sometimes true at least, but it doesn't always, it's not always the case as we've seen with the two powdered lines and the pink line, those have two points of intersection.

And actually if we took this line, this line, and this line, these actually have three points of intersection.

So it's not always true, it's sometimes true, really, really well done if you manage to get those answers and even more importantly, you managed to justify them with some examples, amazing job today, that was quite a tricky task.

So really, really well done.

Remember to complete the quiz at the end to check your understanding, but great job today.

You must be really proud of yourselves.

Excellent work.

See you next time.