Transcript Communities of Exemplary Practice

Communities of Exemplary
Practice
Patterns, Formulas, and
Problem-Solving
Summer 2012 Workshop
Mathematicians and scientists
want to understand how and why.
From the Iowa Core, Standards for
Mathematical Practice:
 Make sense of problems and persevere in
solving them.
Mathematicians look to use
patterns to understand
 Look for and make use of structure.
Students who look for patterns in their
environment expect things to make sense
and develop a habit of finding relationships
and making predictions.
Students should investigate patterns in
number, shape, data, change, and chance.
They should be given opportunities to learn
how to represent those patterns numerically,
geometrically and/or algebraically.
How Many Seats?
We have a long skinny room and triangle
tables that we need to arrange in a row with
their edges touching, as shown. Each side
can hold one “seat,” shown with a circle. Can
patterns help us find an easy to answer the
question: How many seats can fit around a
row of triangle tables?
Student Worksheet
Triangle Rule Machine
Input
Rule
Output
Number of
?
Number of
Tables
Seats
Input: # of tables
Output: # of seats
1
3
2
4
3
4
5
6
What patterns do you see?
What formulas can you create?
Input: # of tables n
Output: # of seats S
1
3
2
4
3
5
4
+2
6
5
7
6
8
𝑆𝑛 = 𝑆𝑛−1 + 1
𝑆 =𝑛+2
+
1
Recursive Representation
Exact, closed-form solution
Which pattern/formula do we desire?
The +2 Pattern appears…where?
 Numerically, as shown in the table, going
across
Input: # of tables n
Output: # of seats S
1
3
2
4
3
4
+2
5
6
5
7
6
8
The +2 Pattern appears…where?
 Geometrically
Connections between Geometry
and Algebra
 Encourage students to relate different
representations of the problem
 Consider the classic pool problem:
Pool 1
Pool 2
Pool 3
The Pool Problem
 Find the number of gray tiles in Pool 5.
 Use a table to represent the number of
gray tiles in Pools 1,2,3, and 5.
 Find a formula for the number of gray tiles
in the nth pool.
 Find the a formula for the number of white
tiles in the nth pool.
Pool 1
Pool 2
Pool 3
Many solutions that are based on
the geometry…
 Determine how to “see” the shapes
geometrically so that the formula would
be
𝑊 = 2 𝑛 + 2 + 2𝑛