Lesson video

Hello, I am Mr. Gratton, and thank you so much for joining me in this lesson on similarity and enlargement.

Today we will use our understanding of similarity and scale factors to solve problems in context.

Pause here to have a quick look at some important keywords.

First up, let's have a look at some maps and plans and how they link to similarity.

Similarity exists everywhere.

For example, a map is similar to the real-life location that it shows.

All distances on a map share the same multiplicative relationship to its real location.

And to show these multiplicative relationships, a scale similar to a scale factor is usually given.

This is called a map scale and converts a distance on the map to a distance in real life.

We can use these map scales on a ratio table to show multiplicative relationships and calculate real-life distances.

The ratio table has one column that shows the distance on the map and another that shows the distance in real life.

The first row of the ratio table should always be for the map scale that is given on the map.

For example, the map scale on this map is one unit or one square on the map, and that distance is equivalent to 200 metres in real life.

So the values of one and 200 are placed inside this ratio table.

On the map, the distance between house and bus stop is five units.

We can then use the multiplicative relationship to calculate the real distance between house and bus stop.

5 times 200 equals 1000.

The real distance between house and bus stop is 1000 metres, which can be converted into kilometres at one kilometre.

Here's a different map that shows the distance between a hut and a signpost.

Pause here to fill in this ratio table to find the real distance from hut to signpost.

a and b are the values on the map scale.

One and 30 or one unit and 30 metres.

I count seven squares between the hut and the signpost, so c equals seven.

And using either of these two multiplicative relationships, we can see that the real distance is 210 metres.

Okay, this map is a little bit more tricky.

The map scale does not say one unit to something, rather two units to something.

In this case, two units to 85 metres.

Pause here to complete this ratio table.

The distance on the map is nine units.

a and b remain as the map scale two and 85, and we can use either one of these two multiplicative relationships to get a real distance of 382.

5 metres.

On top of maps, we also have mechanical sketches and designs for buildings.

These sketches and designs are mathematically similar to their real world object or are hopefully similar to the object that they intend to construct or build.

Sometimes two separate bits of information are given on these designs.

The first being a length of a unit on the drawing, and the second, a ratio from the drawing to the real world object.

The ratio of 1 to 42 does not mention any units at all, therefore, the same unit must be used throughout when using this ratio to calculate lengths, for example, whilst using a ratio table.

We can however convert a unit of length after we have used a ratio table and do not intend to use or modify values on the table any further.

So for this ratio table that aims to find the length of the real car, the top row is for the ratio given, 1 to 42, and the bottom row is for the length of the car, both written on the drawing and on the real car.

Here's the length of the drawing at 11 centimetres.

11 squares across on the drawing at one centimetre per square.

It is essential that I acknowledge the unit of measure either in the cell where I write 11 or on the heading at the top.

A multiplicative relationship can then be used to find that the real car is 462 centimetres long.

Importantly, the real car is 462 centimetres, the same units, centimetres, as the lengths on the drawing.

I can then convert this to 4.

62 metres now that the ratio table has been completed.

Okay, here's a scale drawing of a van.

By identifying the correct information on the drawing, pause here to find the values of a and b on this ratio table.

a and b belong to the ratio from the drawing to real life 1 to 51.

And now pause here to find the value of c.

c is the height of the van on the drawing.

The height of the drawing not including the wheels is five centimetres tall.

Now we have enough information to complete the ratio table in order to find the height of the real van.

Pause here to find the height of the real van and give your answer in metres.

We must first keep the height in centimetres as it is consistent with the units on the drawing, also in centimetres.

After finding 255 centimetres, then we can convert it to 2.

55 metres.

Here's a more challenging scale drawing of a living area.

Pause here to find the length of the living area on this scale drawing.

The scale drawing is 12 squares across and each square is three centimetres long, so the total length of the drawing is 12 times 3 or 36 centimetres.

Last check using this drawing.

Pause here to complete the ratio table in order to find the real length of the living area in metres.

36 centimetres goes here as the length of the room on the drawing, and the real living area is 11.

7 metres after converting it from 1,170 centimetres.

Very good.

Onto some independent practise questions.

Pause here to complete this ratio table for question one, and then find the total distance travelled by the golf ball from the teeing area, through the sandpit, and then to hole 2.

And for question two, complete this ratio table to find the length of the plane, and then by constructing your own ratio table, calculate the distance that the plane travels to airport y from airport x.

Pause now for question two.

For question three, complete this ratio table and calculate the distances between different sets of train stations.

Pause now to do this.

And finally, question four.

Pause here to show that the bird flew over 200 metres along the route L to M, M to N, and N back to L.

Great work, everyone, in applying similarity in all of these real world contexts.

For question 1a, hole 1 is 450 metres from the teeing area.

And for question 1b, hole 2 is 1,425 metres from the teeing area after landing first in the sand pit.

For question 2a, the real plane is 4,800 centimetres long, which is 48 metres.

And for question 2b, the plane travelled 666666.

67 metres, which is 667 kilometres after rounding.

For question 3a, stations A and C are 135 kilometres apart.

For question 3b, the difference in the distances between A to H and A to G is 22.

5 kilometres.

Pause here to compare your calculations to the ones on screen.

And for 3c, the two furthest stations from each other are E and G.

On the map they are 21 units apart, meaning that in real life they are 472,500 metres apart.

And finally, question four.

The bird flew 208.

24 metres.

Pause here to compare your calculations to the ones on screen.

Let's now apply some of our understanding of congruence and similarity with different transformations.

If you translate, rotate, or reflect an object, you can check that you've done so correctly by tracing the object using tracing paper, and then placing the tracing paper over the transformation that you just drew.

For example, I translate this object to get this image.

Using tracing paper, I trace over the object and move it over to the image.

The image and the sketch on the tracing paper overlap completely so the shape of my image is congruent to the object.

However, you still need to check that the location of the image is correct.

You cannot check that with tracing paper.

Similarly, let's reflect this object to make this image, and use tracing paper to trace over the object.

However, all reflections change the sense of an object.

This is the same as flipping the tracing paper so the other side of the tracing paper is facing upwards.

I have tried to overlap the tracing paper onto the image perfectly, but I simply cannot do that because the image is not congruent to its object.

The reflection was not drawn correctly.

Rotations work in a similar way to reflections and translations.

It just requires you to rotate the tracing paper after sketching the object onto it.

A better question comes from Jacob who asks whether we can use tracing paper to check if an enlargement has been done correctly.

Whilst it won't tell you for certain if it's been done correctly, it will help you to identify if any angles are incorrect on an enlarged image.

If even one pair of corresponding angles do not match, then the enlargement has been done incorrectly.

For example, this object with this centre of enlargement and these rays from the centre result in this image.

Sam believes that the image is instantly correct because all vertices on the image lie on different rays, so the image is correctly enlarged and similar to the object.

However, I'm not so certain that Sam's claim is correct.

Here's the tracing paper with a sketch of the object.

We can check a pair of corresponding angles, like this pair.

The angle of the enlargement is the exact same size as the object, and Sam thinks that checking one angle is enough, but I disagree.

Let's be rigorous and check all of them, uh, like this angle.

This pair of corresponding angles are clearly not the same size, and Jacob is correct.

If one angle isn't the correct size, usually the other ones are also not correct, like this one.

Therefore, the enlargement has been drawn incorrectly.

Laura thinks that because this pair of corresponding angles are equal, that the object and image are similar.

Pause here to choose the correct statement about what Laura has said.

Laura needs to check all of the angles.

One angle is simply not enough to prove similarity, only disprove it if one pair of angles are different.

Furthermore, checking all of the angles is still not enough to check for similarity as all of the side lengths on the object need to be multiplied by the same scale factor for the enlarged image to be similar.

Okay, Laura has now checked a second pair of angles.

Pause here to identify the correct conclusion.

If even one pair of corresponding angles are not equal, then the image is definitely not similar to the object.

Laura has drawn an enlargement for a different object.

She has also checked all angles with tracing paper like so.

Pause here to identify the correct conclusion.

All corresponding pairs of angles are correct, but you still need to check the lengths of all of its sides as well.

Great stuff.

Onto this practise task.

For question one, grab some tracing paper and identify which of these seven images are either congruent, possibly similar, or definitely not similar to that object.

Pause now to do this.

For question two, enlarge this shape by a scale factor of two from the marked centre.

Pause here to do this, and then use tracing paper to verify that all angles are invariant on the object and your enlarged image.

For question three, do the same as question two, but also measure corresponding side lengths and check that the side lengths on your image are three times the length of the corresponding sides on the object.

Pause now for question three.

And finally question four.

Jun has attempted to enlarge this object.

The image is drawn next to it.

Pause here to identify the correct statements out of these five possible statements.

Lovely work, everyone, on all of your transformations.

For question one, pause here to check that your answers match these on screen.

And for question two, pause here to check that your enlargement looks like this one.

Hopefully the angles that you checked using tracing paper were invariant between object and image.

And for question three, pause here to check that your enlargement looks like this one.

And hopefully all of the angles that you checked using tracing paper were again invariant between object and image.

I have not written any side lengths on screen as the lengths of each side on the object and the image will vary on how big the object was printed off or drawn.

Regardless of the size of the object, the side lengths on the image should have been three times as long.

And finally, question four.

Only b and e are correct statements.

Jun's image is not a correctly drawn enlargement, and therefore the object and image are not similar.

This can be checked by comparing perimeters or by checking that the image is not a square whilst the object is.

The image is six squares wide and five squares high.

Amazing work and problem solving skills, everyone, in a lesson where we have looked at similarity in a range of different real-world contexts, where ratio tables can take information from those contexts, and use multiplicative reasoning to convert lengths on maps or drawings to find real-world measurements.

We have also used tracing paper to check for congruence after a translation, rotation, or reflection, and also used it to help with identifying similarity after an attempted enlargement.

Once again, great work on this problem solving lesson.

I have been Mr. Gratton, and you have been an outstanding student.

Until next time, everyone, take care.

Have an amazing rest of your day, and goodbye.