Transcript Patterns - TeacherWeb


A pencil and a Highlighter 

A calculator

Your thinking caps!
 You will all be able to:
 Describe what a pattern is
 Discover and analyze patterns found
within different sets of numbers
 Describe what the “Fibonacci
Sequence” is and where it can be
found outside of the classroom.
 Define what a pattern is
 A pattern warm-up
 Talk about the numbers that make up
Pascal’s Triangle and look for patterns
with those numbers to predict future
rows in the triangle
 Learn about the Fibonacci Sequence
and its applications in the real world.
What
is a pattern ?
♦ Something that repeats
♦ Forming a consistent or characteristic
arrangement
Where do we see patterns ?
♦
♦
♦
♦
♦
In fabrics or clothing
The days of the week Mon-Sun
Architecture
Work schedules or Class schedules follow a set pattern
Tire treads
Math can be thought of as the “science of patterns”.
 There are 2 basic types of patterns used in mathematics:

1)
Logic Patterns – categorizing objects based on
characteristics like shape, color, texture, etc…
2) Number Patterns – relationships among different
numerical values/quantities.
 Discovering patterns can help us
predict what will happen next.
 Let’s give it a go in the warm-up!!!
FINDING PATTERNS
Find the missing terms in the patterns below.
(Be able to explain the “rule” you used to find the next term in each sequence.)
1 , 3 , 5 , 7 , ____ , ____ , ____
2) A1 , B1 , A2 , B2 , A3 , B3 , A4 , _____ , _____ , _____
3) 2 , 4 , 8 , 16 , 32 , _____ , _____ , _____
4)  ,  ,  , ______________ , ______________ ,
_____________________
5) A 1 C , E 2 G , I 3 K , M 4 O , ______ , ______ ,
6) 1, 4, 7, ____ , 13 , _____ , 19 , _____ , _____
7) 160 , 80 , 40 , _____ , 10 , _____
8) 17 , 15 , ____ , ____ , ____ , 7
9) 09 , 18 , 27, 36, 45, 54, ____ , _____ , 81 , 90
10) 9 , 98 , 987 , 9876 , _________ , __________ , _____________
1)
1) 1 , 3 , 5 , 7 , 9 , 11 , 13
2) A1 , B1 , A2 , B2 , A3 , B3 , A4 , B4 , A5 , B5
3) 2 , 4 , 8 , 16 , 32 , 64 , 128 , 256
4)  ,  ,  ,   ,   , 

5) A 1 C , E 2 G , I 3 K , M 4 O , Q 5 S , U 6 W
6) 1, 4, 7, 10 , 13 , 16 , 19 ,
22
, 25
7) 160 , 80 , 40 , 20 , 10 , 5
8) 17 , 15 , 13 , 11 , 9 , 7
9) 09 , 18 , 27, 36, 45, 54, 63 , 72 , 81 , 90
10) 9 , 98 , 987 , 9876 , 98765
, 987654 , 9876543
Pascal's triangle is a triangular array of numbers. It is
named after the French mathematician Blaise Pascal,
but other mathematicians studied it centuries before
him in India, Greece, Iran, China, Germany, and Italy.
 The triangle contains many different hidden number
patterns; many of which we will talk about later on.
 The numbers in each row of the triangle are precisely
the same numbers that are the coefficients of
binomial expansions. [ex: (x + y)³ = 1x3 + 3x2y + 3xy2 +

1y3 ]

Its known applications in mathematics extend to
calculus, trigonometry, plane geometry, and solid
geometry.
 HANDOUT:
Fill out as many boxes as you can (pg2)
* What patterns do you notice within the
triangle?
* What method(s) can you use to find the
numbers in the next row?
* Do you notice any patterns that go across the
triangle or diagonally through the triangle?
* Fill in as much of Pascal’s Triangle as you can!
•The diagonals going along the left and
right edges contain only 1's
•The diagonals next to the edge of the
“1’s” diagonals contain the natural
numbers, or counting numbers.
•The 3rd inner set of diagonals are the
“triangular numbers” ; number
amounts that make equilateral triangles
(all sides are the same length)
Highlight
all of the
hexagons
that
contain
odd
numbers.
Odd
numbers:
1,3,5,7,
9 , 11 , etc.

The pattern
obtained by coloring
only the odd
numbers in Pascal's
triangle closely
resembles the fractal
called the Sierpinski
triangle. This
resemblance
becomes more and
more accurate the
more rows you add
to Pascal’s triangle
and the farther you
zoom out.
0,1, 1, 2 , 3 , 5 , 8, 13, 21, 34 , 55 , 89, 144, 233,
1 2
3
5
8
377, 610 , 987, 1597, 2584 , 4181, 6765, 10946,
17711, 28657, 46368 , 75025, 121393 , 196418…
 The next number in the sequence is
found by adding up the two
numbers before it.
 It’s that simple!

When you make squares with the widths 1, 2 , 3 , 5 , 8
and so on you get a nice spiral:

The squares fit neatly together!!!
For example 1 and 1 make 2, 2 and 3 make 5, etc....
 Start with any two numbers you like
 Then add the two previous numbers to
generate the next term.
My example:
1 , 4 , 5 , 9 , 14 , 23 , 37 , 60 , 97 , 157 , 254 …
Trick : The sum of the first ten numbers in your
sequence will automatically be 11 times the
amount of the 7th term in your sequence.
 Let’s see if it works with my sequence

My example:
1 , 4 , 5 , 9 , 14 , 23 , 37 , 60 , 97 , 157 , 254 …
Sum = 407
11 (37) = 407
 IT WORKS!!! COOL 
 Create your own Fibonacci-like sequence.
 Find a general formula for the “Fibonacci #
Trick”. Prove why the trick works from a
mathematical standpoint. (*Hint: call the
first term of your sequence “a” and the
second term of your sequence “b”)
Now you are all expert
mathematical pattern
investigators!!!
YES!!!!! 
Adding the
diagonal rows
of Pascal’s
triangle create
the Fibonacci
Sequence!!!!
How crazy cool
is that?