Relate the Pythagorean Theorem to 30-60-90 and 45-45-90 triangles
Day One of this
Web Quest is set up to introduce students to a little bit of Math history and
touches on the treatment of women during the 5th century BCE
and how that contrasts in how women were treated in the Pythagorean cult.
I think this is important to learn and discuss because mathematics has a rich
and interesting history and so often, women are discouraged from the world of
mathematics. It is such a male driven world that needs
diversification. It would be interesting to see if there is time for
discussion at the end of day one or at the end of day two because there is so
much to talk about concerning this activity, both from an historical and
mathematical perspective.
Day Two of the
Web Quest is a very interesting day. As noted above, most students will
need the majority of the time to figure out the geometry and the algebra behind
the proof. It is really important for the teacher to be aware of time and
ensure that students have enough time to solve the theorem. Additionally,
the teacher must be aware of where the students are emotionally because this is
something that can get frustrating after failing to understand what step to
take next (especially after trying several methods). The Web Quest is set
up to guide students gradually through the proof and give them the tools
necessary to solve the proof. However, this does not mean that the proof
will come easy to everyone. I encourage students who succeed to help out
those around them and I too will most certainly help students. My goal is
never to frustrate a student. My goal is to make them think on a deep
level. That involves a delicate dance of giving students just enough
information so that students can get to that “aha moment.”
IL- Illinois
Learning Standards
Learning
Area: Mathematics
Goal: Application Of Learning
Through
Applications of Learning, students demonstrate and deepen their understanding
of basic knowledge and skills. These applied learning skills cross academic
disciplines and reinforce the important learning of the disciplines. The
ability to use these skills will greatly influence students’ success in school,
in the workplace and in the community.
Standard /
Ability: Communicating --
Express and interpret information and ideas.
Everyone must
be able to read and write technical material to be competitive in the modern
workplace. Mathematics provides students with opportunities to grow in the
ability to read, write and talk about situations involving numbers, variables,
equations, figures and graphs. The ability to shift between verbal, graphical,
numerical and symbolic modes of representing a problem helps people formulate,
understand, solve and communicate technical information. Students must have
opportunities in mathematics classes to confront problems requiring them to
translate between representations, both within mathematics and between
mathematics and other areas; to communicate findings both orally and in
writing; and to develop displays illustrating the relationships they have
observed or constructed.
Standard /
Ability: Using Technology
-- Use appropriate instruments, electronic equipment, computers and networks to
access information, process ideas and communicate results.
Technology
provides a means to carry out operations with speed and accuracy; to display,
store and retrieve information and results; and to explore and extend
knowledge. The technology of paper and pencil is appropriate in many
mathematical situations. In many other situations, calculators or computers are
required to find answers or create images. Specialized technology may be
required to make measurements, determine results or create images. Students
must be able to use the technology of calculators and computers including
spreadsheets, dynamical geometry systems, computer algebra systems, and data
analysis and graphing software to represent information, form conjectures,
solve problems and communicate results.
Standard /
Ability: Working on Teams
-- Learn and contribute productively as individuals and as members of groups.
The use of
mathematics outside the classroom requires sharing expertise as well as
applying individual knowledge and skills. Working in teams allows students to
share ideas, to develop and coordinate group approaches to problems, and to
share and learn from each other in communicating findings. Students must have
opportunities to develop the skills and processes provided by team
problem-solving experiences to be prepared to function as members of society
and productive participants in the workforce.
Standard /
Ability: Making
Connections -- Recognize and apply connections of important information and
ideas within and among learning areas.
Mathematics is
used extensively in business; the life, natural and physical sciences; the
social sciences; and in the fine arts. Medicine, architecture, engineering, the
industrial arts and a multitude of occupations are also dependent on
mathematics. Mathematics offers necessary tools and ways of thinking to unite
the concepts, relationships and procedures common to these areas. Mathematics
provides a language for expressing ideas across disciplines, while, at the same
time, providing connections linking number and operation, measurement,
geometry, data and algebra within mathematics itself. Students must have
experiences which require them to make such connections among mathematics and
other disciplines. They will then see the power and utility that mathematics
brings to expressing, understanding and solving problems in diverse settings
beyond the classroom.
Standard /
Ability: Solving Problems
-- Recognize and investigate problems; formulate and propose solutions
supported by reason and evidence.
The solving of
problems is at the heart of "doing mathematics." When people are
called on to apply their knowledge of numbers, symbols, operations,
measurement, algebraic approaches, geometric concepts and relationships, and
data analysis, mathematics’ power emerges. Sometimes problems appear well
structured, almost like textbook exercises, and simply require the application
of an algorithm or the interpretation of a relationship. Other times,
particularly in occupational settings, the problems are non-routine and require
some imagination and careful reasoning to solve. Students must have experience
with a wide variety of problem-solving methods and opportunities for solving a
wide range of problems. The ability to link the problem-solving methods learned
in mathematics with a knowledge of objects and concepts from other academic
areas is a fundamental survival skill for life.
Goal 8: Algebra and Analytical Methods. Use algebraic and analytical methods to
identify and describe patterns and relationships in data, solve problems and
predict results.
Algebra unites
patterns and quantities in patterns with the means of describing change through
the use of variables and functions. Its concepts and analytical methods allow
people to consider general solutions to problems with common characteristics
and develop related formulas. Algebra provides verbal, symbolic and graphical
formats for discussing and representing settings as diverse as the pricing
patterns of merchandise in a store, the behavior of a car as it accelerates or
slows down, the changes in two chemicals as they react with one another, or the
type of variation existing in a comparison of two factors in the economy. All
people must be able to use algebraic methods to construct and examine tables of
values; to interpret the relationships expressed by patterns in these tables;
to relate change and variation in graphs and formulas; to reason about changes
in quantities and the relationships involved in changes; and to find solutions
to everyday problems using algebra’s symbolic manipulation and formulas.
Standard /
Ability B: Interpret and
describe numerical relationships using tables, graphs and symbols.
Grade Level: Early High School
Learning
Benchmark 8.B.4a:
Represent
algebraic concepts with physical materials, words, diagrams, tables, graphs,
equations and inequalities and use appropriate technology.
Goal 9: GeometryUse geometric methods to analyze,
categorize and draw conclusions about points, lines, planes and space.
Geometry
provides important methods for reasoning and solving problems with points,
lines, planes and space. The word "geometry" comes from Greek words
meaning "measurement of the Earth." While we use modern technology
and employ a wider variety of mathematical tools today, we still study geometry
to understand the shapes and dimensions of our world. The applications of
geometry are widespread in construction, engineering, architecture, mapmaking
and art. Historically, geometry is a way to develop skill in forming convincing
arguments and proofs. This goal of developing a means of argument and
validation remains an important part of our reasons for studying geometry
today.
Standard /
Ability C: Construct
convincing arguments and proofs to solve problems.
Grade Level: Early High School
Learning
Benchmark 9.C.4b :
Construct and
communicate convincing arguments for geometric situations.
Learning
Benchmark 9.C.4c :
Develop and
communicate mathematical proofs (e.g., two-column, paragraph, indirect) and
counter examples for geometric statements
