Lesson video

Hi, I'm Mrs. Warehouse and welcome to today's lesson, which is from the unit: Maths in the Workplace.

In this series of lessons we're exploring how maths is used in different careers.

Let's get started.

In today's lesson, you're going to look at some of the ways that maths is used in the media industry.

Now we're gonna be using a particular phrase today, which is forced perspective.

Now, forced perspective is a technique that makes an object appear larger, smaller, farther, or closer than it really is.

Our lesson today's broken into two parts, and we're gonna begin by looking at films. Here's a picture of some people enjoying a park.

What do you notice? Pause the video and record your observations now.

Lucas says, "The person sitting down must be taller than the person walking." Look at it.

We've drawn an arrow to show from the foot to the head of the person sitting down and then say, well, hang on a second.

That's already longer than the person walking over there.

So that person sitting down's got to be taller 'cause when they stand up, they'll be an even greater length.

Now this is not necessarily true.

The person walking is further away from the camera, so they appear smaller.

Let's do a quick check to make sure you've got that.

One of these people is taller than the other.

Is that A, true, because we can see this, B, false, they are the same height, or C, maybe, there's not enough information.

Pause the video and make your choice now.

Welcome back.

You should of course have chosen C.

We can't tell, one of 'em is further away from the other, they're both sitting down.

There's not enough information here at all.

In order to trick a person into seeing an object in a certain way, their perspective can be manipulated.

Now, force perspective is a technique that makes an object appear larger, smaller, farther, or closer than it really is.

Forced perspective is very useful in filmmaking as it's a much cheaper alternative to CGI.

The "Lord of the Rings" films used force perspective to make certain actors seem much taller or shorter than the other actors.

The "Harry Potter" films did the same thing.

The first thing we need to know is how much taller or shorter we want the person to be.

Now Jacob and Sam are the same height.

They want to record a video which makes it seem that Sam is two thirds of Jacob's height.

Jacob stands two metres away from the camera.

In order for Sam to appear to be two thirds of Jacob's height, Sam needs to stand here.

Notice that's three metres away from the camera, and that is what Sam would look like from the camera's perspective.

What do you notice about those distances? Well, if I multiply Jacob's distance from the camera by three over two, I.

e, the reciprocal of two thirds, then I get three which is the distance in metres that Sam is standing from the camera.

There's my two.

There's the two thirds or the reciprocal of two thirds, and there's the three.

So Jacob must stand at a distance from the camera which is two thirds of Sam's distance from the camera.

Jacob and Sam are the same height.

They want to record a video which makes it seem that Jacob is three quarters of Sam's height.

If Jacob stands 12 metres away from the camera, how far away from the camera should Sam be? Pause the video while you work this out now.

Welcome back.

Let's see what you put.

Well, if Jacob stands 12 metres away, then Sam must stand three quarters of the way from the camera and then Jacob will be three quarters of Sam's height.

In other words, Sam must stand nine metres away.

It's now time for you to do an investigation.

You're going to need two rulers of the same size and one tape measure.

You'll need to place your ruler on the shorter edge on a table.

You may need to prop it against some books to get it to stand up and then place the other ruler further away in the same orientation.

Sit so that your eyes are level with the top of the smaller ruler, and use the larger ruler to estimate how tall the smaller ruler is.

In other words, look along that line there.

Now measure the distance between where you sat and each ruler using the tape measure.

In other words, this distance and then this distance.

What is the relationship between the two distances and the perceived height of the smaller ruler? Now you should repeat this investigation multiple times, placing the rulers at different points and then write down what you notice.

Pause the video while you do this investigation now.

Welcome back.

How did you get on? Well, what I'm going to do now is go through just one of the places that I put the rulers, so you can see what kind of values I came up with.

So I placed it so that the larger ruler was 10 centimetres away from where I was and the smaller ruler was 15 centimetres away.

Now I am using quotation marks here around larger and smaller because we don't really have one that's larger than the other.

Remember, they are the same height, they're both measuring 15 centimetres in this case, just one appears larger and the other appears smaller.

Now the larger ruler was two thirds of the distance of the smaller ruler, and the smaller ruler was two thirds of the height of the larger ruler.

Did you notice the same thing? What about when you change positions? You should have noticed that you still had that same relationship.

Well done if you did.

I hope you enjoyed doing that investigation.

Now let's look at how maths is used in games.

Animated films and computer games are created using maths.

For example, describe how the hero can get to point B.

Pause the video and have a chat with the person next to you, or have a think for yourself.

Welcome back.

What sort of things did you say? Well, you might have said, she goes there.

Now that's not a good answer as we have no idea of the path the hero took.

Characters should move smoothly between points, not blink, unless that's their superpower of course.

You could have said she moves to the right and then up.

Goodness me, this is taking a while.

Now, of course that's better.

We have an idea of the path, however with no distances, the hero might end up in the wrong place.

And as you can see, that took a while.

By placing the hero on a coordinate grid, the hero's movement can be described in terms of its start and end position.

In other words, we're going to describe this movement as a translation.

So we've placed our hero on a coordinate grid and we've drawn the X and Y axes.

A corner of the image is marked and I've chosen the bottom right corner.

Now it's possible to describe the movement of the hero to any given point.

So I'd like you to try this please.

Describe the movement of the hero from her starting point to the following points.

So for part A, describe how she gets from the point 4,2 to the point 8,2.

Then for part B, how does she get from the point 4,2 to the point 4,8? In other words, for each part, assume she's starting from her current position, not where she finished up before.

Pause the video and do this now.

Welcome back.

Let's see how you got on.

So to get to the point 8,2, we have a translation of four units right.

There we go.

Part B, she needs to get to the point 4,8 from where she started.

That's a translation of six units up.

Part C, she needs to get to 9,6, which is a translation of five units right, and four units up.

And then to get to the point 10 at negative one, that's a translation of six units right, and three units down.

Now you could of course have described these other way around and said it was a translation, for example of three units down and six units right.

Absolutely fine if you did.

You might have said it's a translation by, and then described your translation using comm vectors.

Absolutely fine if you did that as well.

Now animators use transformations in order to manipulate the characters in animated films and games.

Translation moves a character.

Enlargement changes their size, and you want this when the character is moving close to the screen or farther away.

Rotation allows a character to turn.

Now for our final task, we're going to play a game.

The objective is to reach the goal in three turns.

You are player A.

You need to write down the translation, or translations required to get to the goal.

Now, if you would move to the same space as the computer, you end your movement, unless of course you've reached the goal.

Movement is in one unit increments.

You must make it past the computer, and to the goal in three turns or less.

Now that might seem a little complicated.

So let's have a look at what I mean.

Here's an example.

The goal is indicated by the filled in dot that you can see on the screen.

You are player A and you can see where you start at the top of the grid.

Player B is the computer and you can see them indicated on the right of the grid.

Now their first movement is shown for you, and you need to think about what your first movement could be.

Well, you might say that your first movement is five units down and two units left.

After all that's going to get you to the goal.

But let's see what happens.

Remember, we move in one unit increments.

So A moves one unit down, B moves one unit left.

Then when A moves their second unit down and B moves their second unit left, they collide.

When this happens, they go back to their previous spot and all movement stops.

Player A must now come up with a new translation to reach the goal.

So in this turn, B is going to move four units down and three units left so that they can get to the goal.

So did you spot, B's actually got the wrong translation here to reach the goal.

If they'd said three units down and three units left, they would've made it this turn.

But what are you going to pick as player A, so that you can get there first? Can you capitalise on the computer's mistake? What should you do? Pause the video and have a quick discussion about what you should pick for your translation.

Welcome back.

Did you say four units down and two units left? I mean, you might have said two left and four down.

Doesn't matter because in either case you will win this turn.

Let's see.

So you begin by each player moving their four units down, and now the two and respectively three units left.

and we can see that A has won the game and the computer has lost.

It's now your turn.

So for question one, you can see for the three rounds what the computer player is going to do.

So you need to think about your translations, so that you can get to the goal in those three turns and beat that computer.

Pause the video and work this out now.

Welcome back.

I hope you beat that computer.

Question two or game two.

Off you go again.

And again, you can see the computer player has got their programming for each round put in and you need to decide what your translation's going to be so that you can beat them.

Pause the video and work this out now.

Welcome back.

Hopefully you beat that computer again.

So at question three, you're now going to design your own game.

Decide where A and B will start and where the goal's going to be.

Write your instructions for the computer to follow for those three rounds.

Then swap with a peer, and see if you can beat each other's games.

Good luck with that.

Pause the video and do this now.

Welcome back.

How'd you get on? Well, I'm now gonna go through this.

I'm gonna give an example of some instructions you could have given.

Now, because B, the computer player starts the first round by moving four units left, it's not possible for A to get directly there in round one.

So what we need to do really is get closer, so we're in a good position for round two.

So I suggested for round one, that player A could move one unit left and five units down.

If A does this, they can then move one unit right in round two and beat the computer.

So in other words, you needed at least two rounds to beat the computer.

You might have taken three, but you could do it in two.

For question two or game two, again it's not possible for A to get directly to the dot in round one because that sneaky computer player is going to block them.

That means player A wants to get close enough, so then in round two they can move straight there before B can block them.

In other words, getting to the dot or the goal in two rounds was the most efficient, but you could have taken three.

Well done if you beat that computer.

In question three, I said you needed to design your own game.

Now, I've also pointed out on the screen that you could modify the game by allowing more complicated rules if you wish.

Were you able to beat your friend's game? Were they able to beat yours? Did you design a computer player that was unbeatable? Is that possible with this setup? These are all questions that you could investigate if you're interested.

And it's the sort of questions that people who work in the media, in particular in games are thinking about when they're designing their games.

It's now time to sum up what we've looked at today.

Forced perspective is a technique that makes an object appear larger, smaller, farther, or closer than it really is.

Scaling is used to achieve this.

Transformations are used by animators to describe character movement in films and games, and we saw some examples of that today.

Now, if you're interested and want to learn more about this, you could consider typing into a search engine how animators use transformations.

For example, Pixar used this exact same way of describing movement in order to animate their characters for their animated films, such as "Toy Story." And you can read all about this if that's something that interests you.

Well done today.

I hope you've really enjoyed the lesson and found it interesting to learn some of the ways that maths is used in the media.

I look forward to seeing you again for some more maths.

Goodbye for now.