How complex

~ Developed by Andrew Derer ~

MathScience Innovation Center, Richmond, VA

Using Student Responders

To respond to a question:

• Wait for polling to be open.

• Select your response while aiming at the receiver.

Aim this…

…at this

What units do we use to measure distance?

56%

0%

0%

0%

11%

28%

0%

6%1. Feet

2. Centimeters

3. Miles

4. Yards

5. Kilometers

6. Nanometers

7. Hands

8. All of the Above

What units do we use to measure area?

11%5%

5%

5%

5%

37%

5%

26%

Square Feet Square Meters

Square Millimeters Square Miles

Acres All of the Above

None of the Above 1, 2, & 4 Only

1. Square Feet

2. Square Meters

3. Square Millimeters

4. Square Miles

5. Acres

6. All of the Above

7. None of the Above

8. 1, 2, & 4 Only

What units do we use to measure mass?

11%

6%

6%

0%

0%

11%

11%

56%

Pounds Kilograms Centimeters

Grams Stone All of the Above

1, 2, & 5 only 1, 2, 4, & 5 only

1. Pounds

2. Kilograms

3. Centimeters

4. Grams

5. Stone

6. All of the Above

7. 1 & 5 only

8. 2 & 4 only

What units do we use to measure complexity?

5%10%

5%

10%

0%

24%14%

33%

Drachm Firkin

Suffolk Whey Fractal Dimension

Meters All of the Above

1, 3, & 4 2 & 5

1. Drachm

2. Firkin

3. Suffolk Whey

4. Fractal Dimension

5. Meters

6. All of the Above

7. 1, 3, & 4

8. 2 & 5

By the end of my visit…

• You have a better understanding of some things that make a shape complex.

• Know some cool things happening in the field of mathematics.

• Use some skills you currently have to solve problems.

• Call math class your favorite subject.

Comparing Polygons• Here are 3 polygons.

• Place them in order from least complex to most complex.

• When you are finished, be prepared to enter your answer and discuss the reason you answered the way you did.

• Let’s take 1 minute to complete this activity.

Which order did you choose?

A)

B)

C)

D)

E)

F)

G) n

one of t

he ab

ove.

17%

28%

11% 11%

22%

6%6%

1. A)

2. B)

3. C)

4. D)

5. E)

6. F)

7. G) none of the above.

Comparing Shapes…Again

• 3 shapes.

• Place them in order from least complex to most complex.

• When you are finished, be prepared to enter your answer and discuss the reason you answered the way you did.

Order from simplest to most complex.

A)

B)

C)

D)

E) n

one

of the

above

.

5%

29% 29%

19%19%

A)

B)

C)

D)

E) none of the above.

40%

30%

10%

5%

15%

A)

B)

C)

D)

E) None of the Above

Can you select the order now?

What can we find out about polygons that can help us with its complexity?

25%

5%

5%

20%

40%

5%0%

Perimeter Area Volume All of the...

1 & 2 2 & 3 1 & 3

1. Perimeter

2. Area

3. Volume

4. All of the above

5. 1 & 2

6. 2 & 3

7. 1 & 3

Ship Shape

• Each of you will have a shape.

• All the shapes are similar.

Similar shapes…

are

alw

ays

the

sam

e ...

are

the

sam

e sh

ape

...

are

the

sam

e si

ze b

u...

are

non

e of t

he ab

ove.

22%

0%

11%

67%1. are always the same

size and shape.

2. are the same shape but may be different sizes.

3. are the same size but may be different shapes.

4. are none of the above.

Let’s explore the perimeter/area relationship.

• Find the perimeter and area of your figure.

• See if you can answer the following question:

What is the ratio of perimeter to area?

Perimeter ÷ Area

What was your ratio?

17%

39%

17%

11%

17% 1. About 0.6

2. About 0.7

3. About 0.8

4. About 1.1

5. None of the above

What is the relationship of perimeter to area?

100%

0%

No matter what si... Once you change t...

1. No matter what size shape the ratio does not change.

2. Once you change the size of the figure, the ratio changes too.

Let’s do some complex exploration

• Let’s start with an equilateral triangle.

What is an equilateral triangle?

0%

88%

12%

1. A triangle with similar sides.

2. A triangle with equal sides.

3. A triangle with no equal sides.

Let’s do some complex exploration

• Let’s start with an equilateral triangle.

• We are going to remove the middle from each of the 3 line segments.

13

How do we find the middle ?13

14%

7%

43%

36%

1. Estimate.

2. Find the length and multiply by 3.

3. Cut each side in half.

4. Find the length and divide by 3.

Let’s do some complex exploration

• Let’s start with an equilateral triangle.

• We are going to remove the middle from each of the 3 line segments.

• Then we will replace it with 2 segments the same length as the original piece removed.

13

Repeating the Pattern

• Now we have an object that looks like a star.

Repeating the Pattern

• Now we have an object that looks like a star.

• Can we remove the middle from each segment again?

• Try it using another piece of triangular graph paper.

• This is called an iteration.

• Try making the 3rd iteration from the 2nd.

13

So an iterative process is a process…

1. that makes triangles into regular polygons.

2. which repeats the same pattern over and over.

3. that you do when your have an itch.

Would you like to see what it would look like if

we kept going?

Koch Snowflake

Complexity

• The figure we made is a fractal called Koch’s snowflake.

• Fractals are one of the newest most exciting fields mathematics.

• Fractals can be used to measure…

Yes, you guessed it –

roughness/complexity!

So could we measure the complexity of this?

Or this?

Or this?

Fractal Dimension

• Finding the fractal dimension (complexity) is relatively easy.

• All we have to do is count boxes.• The math needed to compute it is not

as easy. (You’ll learn it in High School )

• We’ll let the calculator do all the work.

• It’s a program called BOXCNT.

TI-83 Calculators

• Press PRGM Button

• Select BOXCNT and press ENTER .

• Press ENTER again.

• Follow the instructions in the program.

One final question…

Fractals are…

1. One of the newest fields in mathematics.

2. Used to measure complexity (roughness).

3. Formed by iterations (repeating steps).

4. Have dimensions which vary.

5. All of the above.

Thank you!Good-Bye

How many line segments does the figure have now?

3 4 6 8 10 12 14 16

0% 0% 0% 0%0%0%0%0%

1. 3

2. 4

3. 6

4. 8

5. 10

6. 12

7. 14

8. 16

VOTEAnswer Now

Complexity

• We want to look at one type of complexity.

• If we are looking at two-dimensional figures what would give a good measure of its roughness or complexity?

For a 2-D figure, which is best for measuring roughness or complexity?

0% 0% 0%0%0%

1. Perimeter

2. Density

3. Smell

4. Color

5. Sound

:10

The Question is…

Can we measure complexity?

0%0%

1. Yes

2. No

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