Lesson video

Hello everyone and welcome to another math lesson with me, Mr. Gratton.

In today's lesson, we will be practising what information is required to describe a reflection as fully as possible.

The keywords are line of reflection and equidistant are super important for today's lesson, but we will address the meaning of these words as and when they become relevant later on in the lesson.

First up is to formalise the language that we use to describe reflections.

The image after a reflection can end up anywhere as you can see here.

Three reflections of the exact same image and they all end up in a different location.

But why do different horizontal reflections of the same object end up in different locations? Well, this dotted line changes where the location of the reflection is.

Can you think of the name of this line? Well, here are some examples of names that you might have heard of before.

Mirror is a very useful, but informal term to describe this line with the phrase line of reflection being the formal language I will try and use consistently throughout the day, and this name does exactly what it says it should.

It is a line that creates reflection.

If the line is in a different place, the reflection ends up in a different place too.

Okay, quick check.

Which of these is not the name for the line that defines the location of reflection, line of reflection, line of equidistance, mirror line, or flipping line? Pause the video to figure out which ones are and are not common names.

So the two that are not common names are line of equidistance and flipping line.

These aren't common names, however, the word flip and equidistance are both very much linked to the process of reflecting and so you can understand why those words might sound familiar in this context.

There are three types of reflection, horizontal, where the image appears either to the left or the right of the object, vertical, where the image appears directly above or directly below the object, and diagonal, which we will put explicit focus to later on in the lesson.

We won't address it much until then.

Three pieces of information are needed to fully define a reflection.

Firstly, you need to state it's a reflection so that it's not mistaken for a different type of transformation such as a rotation or a translation.

Secondly, whether the reflection is horizontal, vertical, or diagonal.

And thirdly, we need some information that describes the location of the line of reflection.

Otherwise, the reflection could end up anywhere.

This information for 0.

3 fixes where the reflection and the image of the reflection end up.

For now, the description of this line of reflection will be two coordinates on the line of reflection as two coordinates is all you need to describe a unique line.

In the future, you may come across different ways to describe the line of reflection, but we will not cover those ways in this lesson.

So as you can see, here's the line of reflection.

Let's put some description to it in practical terms. This is a horizontal reflection.

It is a horizontal reflection as the image is to the right of the object and it's a reflection.

So I have to say the word reflection.

That's already two of the three pieces of information I need to describe this reflection.

The last bit of information is a bit more detailed.

With a line of reflection that passes through the coordinates, 8,0 and 8,3.

However, those two coordinates are not unique.

I could have chosen different coordinates, such as 8,2 or 8,5.

You might be able to spot a pattern.

Each of those examples, four coordinates would've made the description of that line of reflection equally valid.

And a line of reflection can be extended.

We do this in order to help with choosing the coordinates when describing the location of a line of reflection.

For example, this short line is a valid line of reflection, but it's quite difficult to use it to describe the location with two coordinates.

So we can extend the line like this so that the description, a vertical reflection can be then developed in more detail in a line of reflection passing through the coordinates 3,6 and 7,6.

We would've been far more limited with the choice of coordinates had we not extended that line to begin with.

Let's go through a few checks for understanding.

Complete Aisha's description for the reflection.

Remember what the three key features of a reflection are.

Pause the video to fill in that gap.

It is a vertical reflection.

Remember, vertical, horizontal, or diagonal needs to be one of the words used to describe a reflection.

Next up, Lucas.

This time there are two gaps for you to fill.

Pause the video to find both of those gaps to complete its description.

This time it is a horizontal reflection and the line of reflection passes through 9,4.

Last check.

Sam's description has several blanks, some of which you might be able to spot a pattern with.

Pause the video to complete all of their description.

Again, this is a horizontal reflection with one of the coordinates being 6,9, and the other coordinate being well, six anything, such as 6,0 or 6,10, 6,1.

As long as the X part of the coordinate is 6, the Y part of the coordinate can be anything and it will accurately describe the location of that line of reflection.

Okay, onto the independent learning tasks.

On the left are three fully detailed descriptions of reflections.

Match them to the correct diagram that shows the reflection on the right hand side.

Pause the video to match them all.

Question two requires you to complete the descriptions for these two reflections.

Remember the three key features of reflection description, and use that knowledge to fill in all the blanks.

Pause now to fill them all.

Question number three expects you to do exactly the same as question two, but I've given you no guidance on how to describe these reflections.

Pause the video now to construct your full descriptive answers.

And here are the answers.

For question one, description one matches with B.

It is the only horizontal reflection after all.

Two matches with C and three matches with A.

Well done if you've got all of those.

For question two, the first word should describe the direction of the reflection, in this case horizontal as the image is reflected to the right of the object.

For description two, it is a vertical reflection.

Remember to say the word reflection when describing one and the coordinates must have the Y element of 3, but the X element can be any number you want.

For example, 7,3 and 4,3, and here are the answers for question three.

Take some time to check if your response matches the two descriptions shown here.

Pause to make those comparisons now.

So every example and question we have covered so far has given you the line of reflection.

But what if the mirror line wasn't given to you, but the object and reflected image were? This next cycle of the lesson will show you how to find the line of reflection.

To understand the word equidistant, we can break it down into its two parts.

Equi means equal and distant means distance.

So equidistant means of equal distance.

So when used in the context of reflections, equidistant is focused on the corresponding points or corresponding vertices being of equal distance from the line of reflection.

A point on the object is an equal distance away from the corresponding point of the image.

Note that this only works for corresponding points and not just some random pairs of points.

However, it does work for all pairs of corresponding points.

As you can see here, this bottom pair of corresponding points are both four squares away from the line of reflection.

Whilst the top pair of corresponding points are both six squares away from the line of reflection.

This line is a line of reflection as the object and image are both equidistant from that line.

Notice how the closest two vertices are both one square from the line of reflection and the furthest pairs of corresponding points are five squares away from that line of reflection.

This is not an example of a line of reflection as very clearly that line is much closer to the left object than it is to the right image.

And here's a check for you.

Which the following is the best definition for equidistance when talking about reflections? All points on the object are an equal distance from each other.

All points on the image are an equal distance from the line of reflection and corresponding to the axis.

All pairs of corresponding points on the object and image are an equal distance from the line of reflection.

Pause the video to make your decision.

And the answer is C.

It is all about focusing on corresponding points being an equal distance from the line of reflection.

Axes don't come into this at all.

And for A, it mentions all points.

We're only talking about pairs of corresponding points.

The fact of corresponding points is important to consider with every example that we go through.

Answer the next check, which of these are examples of lines of reflection? Pause now to assess the equidistance of each pair of object and image.

So A and D are the lines of reflection as all corresponding points are equidistant from that line of reflection in the middle of both shapes.

So how do we find the lines of reflection? Well, since the line of reflection is equidistant from two corresponding points, we can say that the line of reflection is in the middle of the full distance from one of the two corresponding points to the other.

We can use this idea that the line of reflection is in the middle by step one, picking a pair of corresponding points and joining them with a line segment.

Step two is taking the whole length of its line segment and finding halfway.

In this example, the line segment is four squares, so the midpoint is two squares away from either end of the line.

That is now the midpoint.

Make sure to plot it on that line segment that you draw.

Step three is to repeat that process again for at least one other pair of corresponding points.

Consider the full distance of that line segment and half it then plot that midpoint onto the line segment.

And step four is to take those two points that you've plot, those midpoint and draw a line connecting all the midpoints that you've found.

That line that you've drawn, the last one that you've drawn, that is your line of reflection as it is made from the midpoint or middles of the line segments, connecting at least two pairs of corresponding points between object and image.

We can use this exact same method even if there are no grid lines to help, but it will require a ruler.

Here's an example.

Connect two corresponding points with a line segment.

Use a ruler to measure the length of that line segment.

Find the midpoint by halving the length of that ruler and then repeat that process again.

Draw a line segment, take a ruler, measure the full length, halve that full length and plot that midpoint.

When you're done with that, connect those mid points to create your line of reflection.

Let's check you've got this.

On the left are five words or phrases.

Match three of them with the correct parts labelled on the diagram to the right.

Two of those words or phrases will not match with that diagram.

Pause the video to give yourself some time to do this.

A is a line segment connecting to corresponding points.

That is the first thing that you need to do, drawing that line segment.

B is the midpoint.

Find the midpoint after drawing your line segment and C, well, that's our goal, the line of reflection.

Find this after creating two line segments and two midpoints.

Both of these diagrams show an object and a correctly placed line of reflection, but only in one diagram is the method, the line segment and the midpoint correct.

Pause the video now to consider which diagram shows the correct method and come up with an explanation for why the other one is wrong.

B is correct and A shows random pairs of points being selected, not the corresponding pairs of points that we need to identify like in B, where we have correctly identified corresponding pairs of points to create that line segment between.

Here are four objects and their reflections.

For each object and reflection you will need to draw the line of reflection.

As you complete more questions, you'll have to draw more and more of your method, whether that is a line segment connecting two corresponding points or creating midpoints for a line segment that has already been drawn.

Pause the video to give all four of these a go.

Question number two is exactly the same again, but this time you will need to use your ruler to measure the length of each line segment that is either drawn or that you have to draw yourself.

Pause the video to create or the method to find those lines of reflection.

Here are the answers for question one.

Pay attention to both the lines of reflection themselves, but also the methods that are drawn to get those lines.

And here are some examples of correct answers for question number two.

However, these are only examples as there are more equally correct answers depending on which pairs of corresponding points you choose to draw your line segments between.

However, the locations of the lines of reflection are one unique answer per question.

Those lines of reflection have to be in exactly the same place no matter which pairs of corresponding points you've chosen.

And onto our final cycle where we will finally cover diagonal reflections and look at some examples of what people think are diagonal reflections, but actually they're not.

Let's have a look.

The reason there are so many misconceptions with diagonal reflections is because they are hard to identify, they're hard to analyse, and they're actually very hard to draw.

Which of these do you think is the correct diagonal reflection? Here's important tip number one.

If you have a diagonal reflection on a piece of paper, physically take that piece of paper and rotate it so that the reflection is easier to visualise.

Me personally, I like to take the piece of paper and rotate it so that the reflection is horizontal so that the image and object are to the left and right of each other.

Here's what I mean visually.

Now the reflections are horizontal, thanks to us turning the paper digitally.

We can see that the left hand diagram is the correct reflection.

Whilst the right isn't actually a reflection at all.

We can use our midpoint of a line segment strategy from earlier.

It doesn't matter that the line segment is now diagonal.

For this example, we have grid squares to help us.

This time count the diagonals of the squares connecting the line segment of the corresponding points you've chosen.

Please note, for the purpose of today's lesson, we will only deal with diagonal lines of reflection and diagonal line segments, which are at an angle of 45 degrees from the horizontal.

That is to say each line will pass diagonally from corner to corner of each grid square.

Other lines of reflection do exist, but they are too complicated to cover during this lesson.

The line segment is two diagonals long, so one diagonal from either side is the midpoint, and again, this is four diagonals across.

So two diagonals gives us the midpoint.

Like before, join up your midpoints to create your line of reflection.

There are times however that you need to be careful.

You may need to factor in half diagonals of a square when the line segment connecting corresponding points is an odd number of diagonals across.

Take this long line.

It is 9 diagonals of a square across, and so the midpoint will be 4.

5 diagonals across.

Notice how this midpoint is perfectly in the centre of a square, not on the grid line itself.

This will be the only other type of location for the midpoint you are likely to see.

It is very, very rare that you will see a midpoint in a random location part of the way between a grid line and the centre of a square.

The line of reflection is still drawn in exactly the same way.

For this check, you need to match the diagonal reflection to the same reflection diagram, but now each one has been rotated, so it appears like a horizontal reflection.

Tilt your head to one side if this helps you visualise these reflections.

Pause to match the pairs.

Here are the pairs.

Tilt your head to the right to verify each of these answers.

Next check the answer is a coordinate.

What is the coordinate of the midpoint to the line segment drawn on this diagram? Pause the video to answer that question.

And the midpoint is the coordinate 6,6.

Okay, same question again, but the answer is much trickier to find, what is the coordinate of the midpoint of this line segment? Pause the video and be careful.

Take care of the details to your answer.

Amazing job if you spotted that the midpoint is at the coordinate, 8.

5, 8.

5, which is in the exact centre of one of those grid squares.

Onto the final set of practise questions.

For each of the objects and images below, find which ones are reflections and which ones are not by drawing line segments between corresponding pairs of points, finding the midpoint, and drawing the possible line of reflection and checking if these lines of reflection are in fact equidistant from both the object and the image.

Pause the video and take your time to show your full method for each of the diagrams. The final question similar to the last, but this time I've not given you any grid lines, so you will need to use a ruler and measure accurately to find the midpoint of each line segment connecting corresponding pairs of points.

Pause the video and good luck on identifying which images are reflections of their objects.

So the left and right diagrams are reflections, and these are the lines of reflections to show this is true.

The middle is not a reflection as any method to find the line of reflection would result in a line with no property of equidistance, and the same is true for question two.

Both the left and right are reflections with the line of reflections shown.

The middle one is not a reflection, and so there is no line drawn in that diagram.

Amazing work if you're able to identify these diagonal reflections and draw on the lines of reflection.

You have covered a lot of challenging material in today's lesson, including learning about the use of the line of reflection, it's importance when describing a reflection, and that a line of reflection can be described by any two coordinates on that line.

We've covered in good detail the idea of equidistance and the fact that all lines of reflection must be equidistant from its object and its image.

We've looked at challenging diagonal reflections and diagnosed fake diagonal reflections that so easily trick a lot of people.

I have appreciated all the hard work you've put in today.

Thank you very much for joining me, and I'll see you for another maths lesson.

Have a good day.