PlayingwithPatternsPlayingwithPatterns Andrew Derer MathScience Innovation Center Richmond, Virginia.

Playing with

PatternsAndrew Derer

MathScience Innovation CenterRichmond, Virginia

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

From the list below, which is your favorite class?

Mat

h Cla

ss

Sci

ence

Cla

ss

Soci

al S

tudi

es C

lass

Lan

guage

Arts C

lass

Any

clas

s w

ith m

ath in

it

11% 11%

0%0%

79%1. Math Class

2. Science Class

3. Social Studies Class

4. Language Arts Class

5. Any class with math in it

What is encryption?

Mix

ing u

p num

bers

or...

Mak

ing

a pat

tern

whic

...

A s

pecia

l sys

tem

use

...

A n

ew c

omput

er ru

n...

All

of the

above

.

11%

26%

47%

0%

16%

1. Mixing up numbers or letters in no specific order.

2. Making a pattern which is impossible to decode without the key.

3. A special system used by spies to get into other countries.

4. A new computer running on the Intel Pentium VI chip.

5. All of the above.

What is a cryptex?What is a cryptex?

• A cryptex is a device credited to Leonardo DaVinci which can hold a scroll or secret note.

• It cannot be opened without the correct code.

• The one in the picture below has 5 dials and 26 letters per dial.

• This means there are approximately 11,800,000 combinations.

Encryption is…

• A defined pattern which allows users to make information useless unless decoded properly.

• A method of securing information for transport either physically or electronically.

• A defined topic in mathematics which has helped make internet commerce possible.

Finding the Code

• Your task today is to unlock the code and retrieve the prize.

• You will work in groups, but open the cryptex as a class.

• Each pattern will reveal another piece of the puzzle.

• Good Luck!

• This slide will self destruct in 10 seconds.

What is the next piece in the following sequence?

2, 5, 9, 14, _____

2, 5, 9, 14, __?__

0%

0%

0%

0%

100%

0%

0% 1. 18

2. 19

3. 20

4. 21

5. 22

6. 28

7. 30

What is the next piece in the following sequence?

2, 5, 9, 14, _____

How about this one?

, , , ,

20

Answer?

0%6%

0%

94%

1. 7 sides

2. 8 sides

3. 9 sides

4. 10 sides

con

Can this sequence…

, , , , help us figure out this sequence?

2, 5, 9, 14, _____20

Can you find the nection

What is the link?

The

number

s te

ll ho

...

The

shap

e re

prese

nt...

The

number

is a

mul

ti...

Som

ethin

g els

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6%

29%

65%

0%

1. The numbers tell how much area the shape has.

2. The shape represents how many letters are in the number.

3. The number is a multiple or factor of the number of sides of the shape.

4. Something else.

Hmm?

2, 5, 9, 14, ___

, , , ,

So what do they have in common?

20

Challenge #2: Geometry

• I can make a square with one square.• I can find the perimeter of this square.• I can also find the area.• But what happens to area and

perimeter if I add:– 1 unit to the length and – 2 units to the width?

• What if I did the same thing to this newer, larger rectangle?

• See if you can find a pattern…

Patterns in Architecture

And not just in old buildings…

Challenge #3

• From nature:– Try to draw the next set of branches and

leaves.

The number of branches in the next row is:

0% 10%0%0%0%10%

10%

70%

12 13 14 15 16 17 18 None of the above

1. 12

2. 13

3. 14

4. 15

5. 16

6. 17

7. 18

8. None of the above

The number of leaves in the next row is:

9%

0% 0%

91%

0%0%0%

1. 7

2. 8

3. 9

4. 10

5. 11

6. 12

7. 13

Branches and Leaves

• 21 branches

• 13 leaves

http://www.mcs.surrey.ac.uk/Personal/

R.Knott/Fibonacci/fibnat.html

Fibonacci

• Where else can we find Fibonacci numbers?

• See if you can locate Fibonacci patterns with the objects given…

The 4th is a fun one!

1 1 2 1 3 2 4 1 5 3 6 2 7 4 8 1 9 5 10 3

Can you find the next 3 numbers in the sequence.

Hmm, there’s something unique about this pattern.

Check this out!

1 1 2 1 3 2 4 1 5 3 6 2 7 4 8 1 9 5 10 3

1 1 2 1 3 2 4 1 5 3

Eliminate every number the first time it occurs and the result is…

It’s the same pattern! Whoa!

Fractal Pattern

• A pattern within a pattern within a pattern

• Here are some pictures of patterns within patterns

Sierpinski’s Triangle

No matter how much you zoom, you still get the same

picture.

Fractals are natural…

…and man-made.

Our fifth challenge

• Look at the following sequence.(it uses 1’s and 2’s only).

• Can you find the next 5 numbers?

1 2 2 1 1 2 1 2 2 1 2 …

I’ll let you look at it for a while before I give you a hint

Hint: Check out the length of repetitions of numbers…

Kolakoski Sequence

1, 2, 2, 1, 1, 2, 1, 2, 2, How many numbers should the next group have?

What should the next group of numbers be?

2 questions to answer:

Did we crack the code?

[email protected]

Thank You !Thank You !

Extra Slides

Kolakoski Sequence

• Some questions without answers:– Is there a formula for the nth term?– If a string (e.g. 21221) occurs in the sequence

will it occur again?– Why are hotdogs sold in packs of 10 and

buns in packs of 8?

Kolakoski Sequence

1, 2, 2, 1, 1, 2, 1, 2, 2, How many numbers should the next group have?

What should the next group of numbers be?

2 questions to answer: