Lesson video

Hello everyone, I'm Mr. Grattan and thank you for joining me for another math lesson on physically reflecting objects with a line of reflection using a handful of different techniques.

The main keyword we'll be using today is the phrase line of reflection.

Pause the video here to familiarise self with its meaning.

Our first technique when reflecting objects is using the tracing paper to help us.

As a result, tracing paper is necessary for today's lesson, along with a pencil for drawing the lines of reflection, the object, and the images.

The reason why tracing paper is so useful when reflecting is if I draw an object on one side of the tracing paper, then flip the tracing paper over, I would actually get what the image, the reflection of that object would look like.

When I take an object on some tracing paper and flip it over, the sense of that object has changed.

And when the sense of an object changes, it is identical to that object being reflected in terms of its properties.

The key to successful use of the tracing paper is to do this one step first.

Pick one point on your line of reflection and put a cross through it, this will come in helpful much later when we start using the tracing paper.

Step two, place the tracing paper over your object and make sure the tracing paper covers all of the object, the line of reflection, and that extra point you plot on the line of reflection.

We'll call that point the fixed point for the purpose of simplicity.

Step three, on a tracing paper, draw the object, the line of reflection, and that extra fixed point.

Step four, flip your tracing paper over.

In this example, we will flip it horizontally as we will want to do a horizontal reflection.

Step five, line up the line of reflection on the tracing paper with the line of reflection on the diagram underneath.

Step six, and here's where that extra point on the tracing paper is important, without it, it'll be hard to tell how far up or down you need to align that line of reflection.

With this extra point, you know to overlap the extra point exactly like so.

Step seven, push firmly onto each vertex of your image on the tracing paper.

We do this to leave an imprint of the image on the paper underneath.

That imprint might be the pencil or it might be a little hole by pushing it in quite hard.

Step number eight, when you're happy that all vertices have been imprinted onto the paper underneath, remove the tracing paper, and connect the imprinted dots to draw your full reflection.

First check of this method.

What is the important first step when using the tracing paper method? Pause to think about the advice that was given to you earlier.

The answer, B, always plot an extra fixed point onto the line of reflection itself.

This will make the step of overlaying your tracing paper after reflecting it much, much easier.

Next check, what are the three things you must draw onto your tracing paper? Pause the video until you can name them all.

Well done, the three things are the object or the shape that's already been drawn, the line of reflection, and that extra fixed points.

Okay, final check for this part of the lesson.

In which direction should you flip the tracing paper for this particular example and can you justify why? Pause the video to think of your response.

You'll flip the tracing paper vertically as the object will be reflected vertically downwards.

The brilliance of the tracing paper method is that it's just as simple with a diagonal reflection as it is with the horizontal and vertical ones.

Here's an example, but the method we will use is identical to the one we've used before.

Plot a fixed point onto the line of reflection and it does not matter where.

Place the tracing paper over the top of the object and the line of reflection and draw the object, the line of reflection, and that fixed point.

Since the object is being reflected diagonally, right and downwards, flip the tracing paper by flipping its top left and bottom right corners, the two corners where the line of reflection are not close to, like so.

Line up your tracing paper with a line of reflection and its fixed points.

Push your pencil firmly onto each of the vertices to leave an imprint on the paper underneath.

When you've done that, remove your tracing paper and use its imprints to draw your reflection.

And onto an I do you do, make sure you have a copy of the right hand diagram, do this by either printing off a copy or by sketching this rectangle that has a base length of two squares and height of three squares and drawing a diagonal line of reflection by starting one square down from the bottom right vertex of the object.

Wait for all steps of the method on each slide for the left hand side of the demonstration to finish first before attempting those same steps for your diagram.

Set of steps number one, plot a fixed point onto your line of reflection.

Place tracing paper over the object and trace the object, the line of reflection, and the fixed point onto your tracing paper.

Pause to try on your diagram.

Set of steps number two, flip the tracing paper and line up the line of reflection and the fixed points.

Pause and line everything you can up with your tracing paper after flipping it over.

Set of steps number three, push your pencil firmly onto each vertex to leave an imprint on the paper underneath.

Pause to try this on your diagram.

Push your pencil in firmly without pushing so hard that breaks the pencil or any of the paper.

Final set of steps, remove the tracing paper and draw over the imprints of your vertices to complete your diagram.

Pause to try this on your diagram to draw your final reflected image.

Your reflection should look like this.

It should be a rectangle with a base length of three squares and a height of two squares.

Now onto some independent practise.

For each reflection you must use tracing paper and plot a fixed point onto each line of reflection in order to perform that reflection.

Pause the video to grab some tracing paper and perform those reflections.

Here are those reflections.

Note both the shape of the reflected images as well as their locations.

Onto our second cycle where we will look at counting squares to reflect objects and use the property of equidistance to make it work.

Reflecting using tracing paper is always helpful as reflections through the flipping of the tracing paper works no matter the complexity of the shapes or the locations of the lines of reflections that you are dealing with.

But it can be time consuming and unnecessary if you can visualise where the reflection is and what it looks like before it's even drawn.

Counting squares uses the property of reflection, a point on the object, and its corresponding point on the image are equidistant from the line of reflection, where equidistant means of equal distance away, in this case from the line of reflection.

Take this diagram, I want to reflect the triangle in the line of reflection shown.

How do I do this using the counting squares method? Well choose a vertex of the object, step one, such as the bottom right of this triangle.

With your pencil count the number of squares you travel to get to the line of reflection.

In this case, I counted two squares.

Then you repeat the number of squares again on the opposite side, starting on the line of reflection.

This gives you the corresponding points for the reflected image.

Repeat this process for each vertex of the shape, like so.

I count four squares, then from the line of reflection, I count another four squares.

I will do the same for that last vertex and the corresponding point on my image ends up there.

When all of the vertices have been counted and replicated across the line of reflection, complete the image by joining up all of the vertices, like so.

Okay, a quick few checks.

What is the distance in squares between point A and the line of reflection? Pause to think of your answer.

The answer is four squares across.

Using your answer to this question, what is the location of vertex A after it has been reflected? Pause to consider all of those possible options.

And the answer is B, because the point B is also four squares away from the line of reflection.

Therefore B and the point on the object A are both equidistant from the line of reflection at four squares.

The reason why C isn't correct is because it is too far down.

It does not create one straight, in this case, horizontal line between B, the line of reflection and the point A on the original object.

Whilst this method is potentially already quicker than some of the tracing paper scenarios that you might face, we can make it even quicker still.

If you only need one or two points plot on the page to visualise where the rest of the reflection is, then you can count the squares for one or two of those points and then complete the rest of the reflection yourself using your own visualisation.

Notes, what I'm about to show you is optional as you can always count the squares for as many of the vertices as you need to plot your full reflection.

How do we do this half visualisation method? Step one is to choose one vertex like before.

Count the number of squares to the line of reflection.

In this case, four squares, and then from the line of reflection repeat it again onto the other sides.

With this triangle, I see that the base of the object goes left from my chosen vertex, so in the image the base will go right.

I know this changes because lefts become rights and rights become lefts with a horizontal reflection, so the two squares left becomes two squares right in my reflection.

I have now plot using that pattern the second vertex on the base of the image of my triangle.

From my first chosen vertex, the top of the object will be the same height as the top of the image, this is because height does not change with a horizontal reflection.

Four squares high remains four squares high on my reflection.

This shows that a full reflection can be done with only one vertex used in the counting squares method with the rest of the vertices used by visualising the reflection by self.

There are some challenging checks for understanding.

I have successfully used the counting squares method to reflect A in that line of reflection to get the point A prime.

Which of these two diagrams shows the base of the triangle drawn directly, starting from point A prime.

Pause the video to consider your answer.

The answer is diagram B.

This is because on the object, the base goes left from the point A and so on the image, the base must go right from the point A prime.

Using the correct diagram from the previous question, I have tried to draw the vertical height of the triangle.

In which diagram has this been done successfully? Pause to think about the location of that vertical height.

The answer is diagram A.

This is because that vertical height originates not from the point A, but from the other point on the base of that triangle.

Therefore B is incorrect because that vertical height does originate from point A prime, which is wrong when we look at the object on the left hand side of each diagram.

Here is the completed reflection.

It is clear that diagram A is a correct reflection whilst in diagram B, that is not a reflection at all.

And now a thinking question check.

Where would the reflected image for vertex A be? Pause to review all of these possible responses.

Ah, so there's two answers to this.

Vertex A is a vertex on the line of reflection.

This is the same as saying it is zero squares away from the line of reflection and so its reflected image will also be zero squares away from the line of reflection.

This means it stays on the line of reflection and is zero squares away from vertex A, meaning it overlaps that point.

Time for some independent practise, but this time no tracing paper allowed.

By counting the squares and visually completing the reflections if you feel comfortable, reflect all the objects in the line of reflection closest to it.

Pause the video to practise all six reflections.

And here are the answers.

Again, pay attention to the shape of your reflections as well as the location.

Well done for persevering through all of those horizontal and vertical reflections.

But what about diagonal reflections using the same counting squares technique? This will be what we cover in the next learning cycle.

Diagonal reflections using the counting squares technique is fairly similar to the method using horizontal and vertical reflections.

You just need to count the diagonal distances between the squares rather than the horizontal and vertical distances.

There are two types of options to cover.

Whole number of diagonals.

In this case, three full diagonals.

Or full diagonals with one extra half a diagonal before you reach the line of reflection.

Here we have two diagonals plus an extra half a diagonal as we reach the line of reflection.

Unlike for vertical and horizontal reflection where it was possible to fill in the rest of the reflection by using your own visual understanding of reflection, I completely recommend to count the squares for all the vertices under a diagonal reflection as it is much harder to correctly visualise full diagonal reflections, especially for more complex looking objects.

See here, this vertex is one square, one full diagonal square from the line of reflection.

Let's replicate this one diagonal square on the other side of the line of reflection to get off first reflected vertex.

But this vertex is one diagonal plus an extra half a diagonal extra.

The important advice that you need to focus on is to start by counting the extra half a diagonal first when you go across that line of reflection, as that half diagonal is the first distance that you meet as soon as you cross that line of reflection.

See, we count the extra half our diagonal first before counting that one full extra diagonal.

This is where the second reflected vertex will be.

We'll now do the same for the other vertices of our shape, starting with the far left, two full diagonals across the line of reflection and repeated gets the point up there.

And now for the bottom vertex, it is two and a half full diagonals.

Remember to replicate the half first before replicating the other two full diagonals to get the final reflected vertex.

Join all the reflected vertices to produce your reflected image.

Are all the vertices in the locations that you expected? Remember, you can tilt your head left in this case to help visualise that reflection.

Okay, final three checks for understanding.

By counting the number of diagonal square lengths, describe the distance between vertex A and the line of reflection.

Pause now to count and choose your answer.

The answer is C, two full diagonals and that extra one half as you get to the line of reflection.

Here is another vertex.

By counting the diagonal square lengths to the line of reflection, assess which of these is the correct reflected vertex.

Pause now to consider these options.

And the answer is B because the point B is also two diagonal squares away from the line of reflection and shares the same points on the line of reflection to which it and the vertex A on the object are equidistant.

Whilst C and A are also two diagonals from the line of reflection, they do not share that common point on the line of reflection to which this equidistance occurs.

And one final question of this style using the same diagram.

Which of the following is the reflected vertex of vertex B? Pause the video to assess these possibilities.

And the answer is C.

C is the only sensible answer as it is the only point on the opposite side of the line of reflection that is exactly two and a half diagonals from the line of reflection, the same as point B on the object itself.

Here are your final set of reflection practise questions.

As you can see, they are all diagonal reflections by counting the diagonals and half diagonals if you need to for each vertex of each object, draw your reflections.

Pause now to draw these reflections for each of the five objects and take your time to make sure that you are counting the diagonals and half diagonals correctly.

Here are your answers, the images, check the shape, orientation, and location for each of these reflections.

Amazing work on all of these challenging reflections throughout the lesson.

You have covered so much in this lesson, including the use of tracing paper to reflect objects, the skill of counting squares to and then across the line of reflection to create these reflections vertex by vertex, or by utilising the counting squares method along with your own visual awareness of reflection and that reflecting diagonally needs to account for half diagonals of squares as a frequent hurdle in the process along with the full diagonals of squares.

Thank you so much for all of your attention and effort during this lesson, and I hope to see you very soon for another lesson of maths, have a good day.