Stretching Your Understanding

An Investigation Into Scaling

Guy

Wall

Guy

Shadow

Wall

Guy

Shadow

How many times taller is the shadow than the guy? Let’s call this number the scaling factor for height.

So it seems that the shadow is 3 times as tall as the guy. That is, the scaling factor for height between the guy and the shadow is 3.

Often, we say that the guy’s height has been scaled up by a factor of 3.

Guy

Shadow

Suppose we measure and determine that the guy is 2 meters tall and shadow is 6 meters tall.

Guy

Shadow

Suppose we measure and determine that the guy is 2 meters tall and shadow is 6 meters tall. 6m

2m

Guy

Shadow

Suppose we measure and determine that the guy is 2 meters tall and shadow is 6 meters tall. 6m

2m

How might we determine the scaling factor mathematically?

So, the scaling factor =

shadow’s height

guy’s height

So, the scaling factor =

shadow’s height

guy’s height=

So, the scaling factor =

shadow’s height

guy’s height= 6m

2m

So, the scaling factor =

shadow’s height

guy’s height= 6m

2m = 3

Guy

Shadow

What about the width scaling factor?

Guy

Shadow

What about the width scaling factor?

Height and width scaled equally

Height scaled more than width

Width scaled more than height

Unscaled

Height and width scaled equally

Height scaled more than width

Width scaled more than height

Unscaled

Similar Figures

Height and width scaled equally

Height scaled more than width

Width scaled more than height

Unscaled

Not Similar Figures

What to remember:

How to calculate the scaling factor:

What similarity means in this context:

All dimensions are scaled by the same factor

Scaling factor =Dimension of Big Object

Dimension of Small Object

What you will do:

Part I: Find the scaling factors of the toy car for each dimension (height, length, and width)

What you will do:

Part I: Find the scaling factors of the toy car for each dimension (height, length, and width)

Part II a: Find the area of the parking

space in which the toy car would

park as well as that of the real car. Find

the scaling factor for area. Using

string and tape, mark out this area on

the floor.

What you will do:

Part II b: Make a box for your toy car by

constructing a net using the graph

paper provided. Find the scaling factor for volume.

Remember Similarity?

Similar or not?

Remember Similarity?

Similar or not?

Remember Similarity?

Similar or not?

Remember Similarity?

Similar or not?

Remember Similarity?

Similar or not?

Remember Similarity?

We call this self-similarity.

Now we have self-similarity on two scales.

Now we have self-similarity on two scales.

Now we have self-similarity on two scales.

Now we have self-similarity on three scales.

Now we have self-similarity on three scales.

Now we have self-similarity on three scales.

Now we have self-similarity on three scales.

Here we go.

Let’s build Sierpinski’s Triangle

Let’s build Sierpinski’s Triangle

Let’s build Sierpinski’s Triangle

Let’s build Sierpinski’s Triangle

Let’s build Sierpinski’s Triangle

Let’s build Sierpinski’s Triangle

Let’s build Sierpinski’s Triangle

This thing is self-similar across an infinite number of scales!!!

This thing is self-similar across an infinite number of scales!!!

Here we go again.

Another simple rule, another fractal. This one is known as the Koch snowflake.

Another simple rule, another fractal. This one is known as the Koch snowflake.

Another simple rule, another fractal. This one is known as the Koch snowflake.

Another simple rule, another fractal. This one is known as the Koch snowflake.

Another simple rule, another fractal. This one is known as the Koch snowflake.

A

B

A

B

http://math.rice.edu/~lanius/frac/

http://www.coolmath.com/fractals/gallery.htm

http://www.enchgallery.com/fractals/fracthumbs.htm

Check it out for yourself