Download Patterns Part A

Patterns
Part A
“Looking for patterns trains the mind to search out and discover the similarities
that bind seemingly unrelated information together in a whole. A child who
expects things to ‘make sense’ looks for the sense in things and from this sense
develops understanding. A child who does not expect things to make sense and
sees all events as discrete, separate, and unrelated.”
Mary Bratton-Lorton
Math Their Way
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Let’s first look at Natural Numbers and partition them into evens and odds:
A list shows the pattern:
o e o e o e o e…
What are the formulas?
Even
Odd
A sequence is a function whose domain is the natural numbers.
What are the ordered pairs for each of evens and odds?
The graphs?
The domains and ranges?
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Now let’s do this with a range of the Whole Numbers
eoeoeoeo
Formulas?
What changes?
Can we extend this to Integers as a range? Sure!
What are the formulas?
Now let’s make lists of evens and odd and find the difference between them
(Next – Previous)
2
4
6
8
1
3
5
7
Any connection to the “slope” in the formula?
3
Figurate numbers:
What about numbers that just naturally come as “dot patterns”
Square numbers:
Formula
Ordered pairs and graph
Tie in to algebra: parent graph if you switch the domain to the Reals
Square numbers continued
4
Now let’s check out those differences again…subtract twice!
1
4
9
16
25
Why does this work? What’s the connection to the formula?
In Class Exercise with ODDS!
5
Let’s look at another series* now:
*the summation of the terms of a sequence is a series
Let’s look at the sum of the squares of the first “n” odd numbers. Can we discern a
pattern and find the formula?
1, 1
2, 10
3, 35
4, 84
5, 165
Check my numbers and then try differences…
6
In class exercise with a series
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Now let’s go to 3D with the cubic
numbers:
List:
Ordered pairs/parent graph:
Visual:
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Differences again
1
8
27
64
125
216
343
What’s the connection to the formula?
Review where we are on differences
Linear
Quadratic
Cubic
Any idea on why I’m asking about differences?
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Triangular Numbers:
T1
T2
T3
T4
T5
T6
Tie-in to square numbers? Check VISUALLY and compare
10
Triangular numbers and Pascal’s triangle
Formula?
Try the differences
1,1
2,3
3,6
4,10
5,15
6,21
What is the formula to create the triangular numbers?
11
Rectangular Numbers:
List, coordinates, formula
Differences…
1,2
2,6
3,12
4,20
5,30
In Class Exercise with Rectangular Numbers!
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Pentagonal Numbers
1, 5, 12, 22, 35, 51, …
What are the next two after 51?
What does the graph look like?
What’s a good strategy to find the formula?
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Let’s list them and look at the differences and how to find the formula
Which is
n(3n  1)
2
Connection: Every pentagonal number is 1/3 of a triangular number.
This is an if – then statement and doesn’t reverse. How can it be true?
14
Hexagonal Numbers
1, 6, 15, 28, 45, 66, …
What do you think these numbers look like when hexagon shapes are used?
What about a graph?
And a formula?
one idea is to rewrite the sequence multiplicatively
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Differences and formula, visual representation….
In class exercise: Visual Representation!
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Tetrahedral Numbers
– moving into 3 dimensions:
1, 4, 10, 20, 35, 56, …
Try sums to figure out one pattern…then work toward a formula…what’s the
leading power on the formula?
What’s the power on the formula? When will the constant difference show up?
(1/6)(n)(n + 1)(n + 2)
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Exponential Patterns:
Powers of 2 and Pascal’s Triangle
2, 4, 8, 16, 32, 64, …
2n
What is the derivative, second derivative, third derivative for an exponential?
Let’s do differences on these…what happens? Does this explain the weird
derivative?
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Problems:
Group 1
I had a party. The first time the doorbell rang, one guest entered my home. On the
second ring, three guests entered. On the third ring, five guests entered. The
guests continued to arrive following this pattern. During the evening the doorbell
rang 15 times. How many guests arrived on the 15th ring? How many guests came
to my party – the total number not counting me.
Group 2
There are 43 people in the room. Everybody shakes hands with everybody else.
How many unique handshakes are given?
Start with one person and work your way up to a formula…then calculate 43
people.
Group 3
How many unique diagonals are there in an n-sided polygon?
Start with a triangle and work your way up to a formula.
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