Lesson video

Hi there, and welcome to another lesson with me, Dr.

Saada.

In today's lesson, we're going to look at finding midpoints.

All you need for this lesson is a pen and a paper.

So make sure you grab these, and when you're ready, let's begin.

Your first task for today's lesson is to try this.

Find the number halfway between the arrows on each number line.

You have four number lines.

In each number line, you have two grey arrows.

I want you to think about what number is right in the middle between the two grey arrows.

So what is halfway between the two? Pause the video and have a go.

Hi, everyone, Ms. Jones here.

I'm just going to go over the solution to these problems for you before handing you back over to your teacher.

So let's look at the first one.

Now, I want you to think about how is the best way to work these out? And what you could have done to find the halfway point is to count one jump each time from either arrow until you get to that midpoint.

And the first point was 10.

10 is halfway between 2 and 18.

Then looking at the next one, to find that midpoint, again, you could jump each time, okay.

Have a think about if there is a more efficient strategy.

And the midpoint would've been -1.

On the third one, the midpoint would've been 15.

And on this one, the vertical number line, the midpoint is -1.

, okay.

So we could've used a strategy of using intervals on our number line, but for some problems we might not be able to always have that number line handy.

Can you look at this and spot any other patterns? Okay, you might be focusing on perhaps looking at the difference between each number, so the difference between 2 and 18, and then halving the difference.

You might've noticed that actually we can sum 2 and 18 and then halve it to get 10.

And that will give us the halfway point.

We're going to be focusing on some of these strategies later in the lesson.

Okay, I'm going to hand you back over to your teacher now.

Now I would like you to look at the image shown here.

What is the same and what is different? Have a little think.

Okay, so you may have noticed that we have a triangle here, and we have another similar triangle.

So ABC is the smaller triangle, and ADE is the bigger triangle, and they are similar triangles.

You may have noticed that I have two parallel lines here and that I have corresponding angles.

Really good job.

Okay, and hopefully you've noticed that AC is equal to CE.

They both are four units long.

And that AC is half of AE, or CE is half of AE, which tells me that this point here, point C, is halfway or is the midpoint of AE.

Anything else? What about the line AD, the line segment AD? What is halfway through it? If I go from A to B, I need to go four to the right and three up.

To go from B to D, what do I need to do? Let's use those triangles that we have been using.

I need to go again four to the right and three up.

So if I'm going to go the same distance from A to B and B to D, this tells me that AB is equal to BD.

Really good.

And this also tells us that this point here, point B, is halfway between A and D.

Or we can say that it's the midpoint of the line segment AD.

So what is a midpoint? A midpoint divides a line segment into two equal parts.

Which of the points could be described as midpoints? Which line segments are these midpoints of? So we already said that C is the midpoint of line segment AE.

It's halfway in the middle of that line segment.

It cuts it into two equal parts.

So AC is equal to CE.

We also said that B is the midpoint of AD.

Now, have a look at this point.

If I tell you that this is a midpoint, what line segment would this point be a midpoint for? Have a little think.

Really good, it is the midpoint for the line segment AC.

It cuts AC into two equal parts.

What about this point? It's the midpoint for, excellent, DE.

It cuts line segment DE into two equal parts.

If I show you here, you can see that this part is equal to this part.

What about this point? Really good job.

It is the midpoint of line segment BC.

It cuts it to 1 1/2 units here and 1 1/2 units there.

Really good.

Okay, now let's have a look at this.

What about this point? Is the midpoint full? Good job, AB.

And this one, the midpoint for BD, really good.

And I'm going to show you using triangles how this works.

So to get from the origin or from A to the midpoint, I need to go 2 to the right and 1 1/2 up.

To get from that midpoint to B, I go two to the right and 1 1/2 up.

So that midpoint is right in the middle.

It has cut the line into two equal parts.

We have the same thing here.

If you follow the orange arrow here, we have from B to the midpoint there labelled with a red circle, we go 2 to the right and then 1 1/2 up.

From that midpoint, to get to D, we do exactly the same thing, so it tells us that that's the midpoint of it.

Okay, have a look at this for me, please.

I have point E here, and look at the line segment AE.

Now, instead of telling, instead of looking at this as the end of the line segment, as E, okay, I want you to think, if I tell you that E is the midpoint.

So imagine A is one end of the line segment, E is the midpoint of the line segment.

Where would the other end be? What is the coordinate of the other endpoint? Have a little think.

It would be at 16, 0 because what we are doing here, we're moving from the zero to eight along the x-axis and then another eight, and that will take us to the coordinate 16, 0.

And now it is time for you to have a go at the independent task.

Please pause the video and have a go at questions number one and two.

When you're ready, press play again so we can mark and correct the work together.

Let's go through the answers to the two questions together.

Question number one, what are the coordinates of the midpoints of the three lines? And you've been given three lines on the grid.

Let's start with line segment AB.

I want to find the midpoint.

Before I do that, I need to look at the point A and think from A to B, what do I need to do? How do I get to it? I would need to go from point one on the x-axis to seven, so that's six to the right, and then up four.

So if I want to find the midpoint, I need to do half of that.

So instead I'm going to go three across and then two up.

Three to the right and two up.

You can see here by drawing triangles that it works.

I've got midpoint has the coordinate 4, 3.

Let's look at CD.

How do I get from point C to D? It really doesn't matter, or D to C.

It does not matter which end you start with.

But to get from C to D, I need to go across to the right one, two, three, four, five places, and then I need to go down from seven to one by six places.

So if I want to find the midpoint, I'm to do half of that.

So instead, I'm going to go up 3 and across 2 1/2, up 3 and across 2 1/2.

That gives me the coordinate of the midpoint, -3.

5, 4.

Did you get the same answer? Well done.

Now let's look at EF.

If I want to get from E to F, I need to move to the right from -7 to 3.

That is 10 places to the right, so half of that is 5.

And I need to move down the y-axis from -3 to -5.

That's two places, so half of that is one.

So I do that.

Five to the right, one down, five to the right, one down.

And this way I'm proving that, yes, it works.

I'm getting exactly the same two triangles on either side of the midpoint.

And I'm finding the coordinate of the midpoint, it's -2, -4.

Huge well done if you had this correct.

Question two.

B is the midpoint of AH.

What are the coordinates of H? So I have the point B now, and point B is now going to become the midpoint for me.

From A to point B, which is the midpoint, I go seven across, seven to the right and four up.

So to get to the next point, I need to do the same, seven to the right and four upwards.

This moves me from point B or the coordinate point B, which is 7, 5.

Add seven because I'm going to the right.

That gives me 14.

And add another four to the five.

That gives me nine.

So the coordinate of point H is going to be 14, 9.

D is the midpoint of CG.

So I have C, D, and I need another endpoint I need to call it G.

What are the coordinates of that point? So let's do the same thing.

From C to get to D, I need to go down by six places and to the right by five places.

So if this is halfway through, I need to do exactly the same thing to get to the other end of the line segment.

So I'm going to go down six and to the right five.

That gets me to this point here.

I label it G.

And the coordinate at this point is 4, -5.

Well done.

And this brings us to the explore task.

Find the midpoint of AB.

Mark this point M.

You've been given line segment AB.

Find the midpoint AM.

Mark this point X.

How does the length AXE compare to AB? If B is the midpoint of AC, what are the coordinates of C? What if A was the midpoint of BC? What other points and midpoints can you come up with? Pause the video and have a go at this.

And now we're going to go through the solutions to the explore task.

Find the midpoint of AB.

Mark this point M.

So if we look at AB, now we'll define the midpoint.

I need to check.

How do I get there? It's eight across and four up, so if I want half of it on the midpoint, I need to go four across, four to the right, and two up.

And that gets me to this point here.

I label it as the question asked me to, and now I need to place the coordinates.

So the coordinate of point M is 2, -1.

Well done if you got that correct.

Find the midpoint of AM, and mark this point X.

Now I'm looking at the smaller line segment AM.

To get from A to M, I went across to the right four places and up two, so I need to do half of that now, which is two to the right and one up.

That gets me to this point, and I'm going to label it X.

Now I'm going to write a coordinate for this point, and that is 0, -2.

How does the length AXE compare to AB? Now let's look at AXE.

To get to AXE, we go two to the right.

But to get to AB, we go eight to the right.

So that's two out of eight, which is, good, well done, that's 1/4.

We need to check that we've not made a mistake by checking also what's happening with that y-coordinate when we're going up.

So to go up to X, we go from A, we go only one up, whereas from A to B to go up, we go four up.

So one out of four, again, that's 1/4.

So this tells me that it is 1/4 of the length.

So AXE is 1/4 of AB.

If B is the midpoint of AC, what are the coordinates of C? Hopefully you got this answer.

A, -2, -3, and B is 6, 1.

So B is eight along and four up from A.

If B is the midpoint of AC, then we go eight along and four up again to C, which will be at 4, sorry, 14, 5.

What if A was the midpoint of BC? From A, we go eight back and four down to C, which will be at -10, -7.

We've done a really good job answering these questions.

What other points or midpoints can you come up with? Can you add your own points now? You have done so much learning today.

Please take a couple of minutes to think about the three most important things that you've learned from today's lesson.

When you're done, please complete the exit quiz.

Thank you, and see you next lesson.

Bye!.