Texas Mathematics Teacher Spring/Summer 2018 - 26

This article describes work done at a summer institute for secondary
mathematics teachers. Participants in the institute were presented with
problems about Pythagorean triples. The problems were sequenced in
a way that allowed for a wide variety of discoveries to develop. The
following observations were made by one of the participants, which the
authors thought would be of interest to secondary mathematics teachers.
Introduction

"

The challenge for
teachers is to
create similar
experiences and
opportunities for their
students that lead
naturally to
important concepts.

"

Pythagoras Unchained

The Rice University School Math Project (RUSMP), established in 1987
through a National Science Foundation grant, has grown into a leading mathematics education center in the
state of Texas, providing support in mathematics education for districts, schools, teachers, and students. In
2016, Rice University (through RUSMP) was awarded a grant through the Robert Noyce Teacher Scholarship
program. The purpose of the grant was to identify and develop 16 Houston Independent School District middle
and high school teachers as Master Teacher Fellows (MTFs). As part of their professional growth, MTFs
participated in the first of two 3-week summer institutes during the summer of 2017. During this summer, the
MTFs investigated the problem sets in the book Applications of Algebra and Geometry to the Work of Teaching
(Kerins, Sinwell, Young, Cuoco, & Stevens, 2015).

Many of the problems in the book centered on an in-depth examination of the properties of Pythagorean triples.
c 2 Participants were introduced to the
Pythagorean triples are whole numbers a, b, and c such that a 2 + b 2 =.
topic by exploring the following introductory problem:
Picture a piece of graph paper. Now picture a dot at each intersection. We'll call this square dot
paper. A 5-by-5 piece of square dot paper would have 5 dots in each direction - also known as
a "geoboard." But the dot paper can be any size, really. We'll say the distance from a dot to its
nearest neighbor is 1. Segments drawn on square dot paper must start and end at dots, but can
be horizontal, vertical, or diagonal at any angle.
Question: On a 6-by-6 piece of square dot paper (see Figure 1), what lengths of segments
are possible?

Figure 1. Dot paper provided to participants for their exploration.

To begin, participants labeled one point as the origin (0,0) and started drawing
segments in the first column, ( 0, 0 ) to (1, 0 ) , ( 0, 0 ) to (1,1) , ( 0, 0 ) to (1, 2 ) ,
( 0, 0 ) to (1,3) , ( 0, 0 ) to (1, 4 ) , and ( 0, 0 ) to (1,5) (see Figure 2).
Figure 2. Drawn segments with an x-coordinate of 1.
Using the Pythagorean theorem, starting at the bottom and working upwards,
participants calculated the lengths of the drawn segments. These lengths were
1, 2, 5, 10, 17, 26 .
26

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http://www.brightcopy.net/allen/txmt/68-01
http://www.brightcopy.net/allen/txmt/67-01
http://www.brightcopy.net/allen/txmt/66-02
http://www.brightcopy.net/allen/txmt/66-01
http://www.brightcopy.net/allen/txmt/65-02
http://www.brightcopy.net/allen/txmt/65-01
http://www.brightcopy.net/allen/txmt/64-02
https://www.nxtbook.com/allen/txmt/64-1
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