Lesson 1
Conjecture
Lesson Plan
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Math
1
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Time: |
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One
50-minute class period, or part of one 90-minute
block
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Goals: |
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Students
will make a conjecture about the Pythagorean
Theorem and its converse. |
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Students
will extend the Pythagorean Theorem to make
a conjecture about the relationship between a2 + b2
and c2 for acute
and obtuse triangles. |
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Students
will gain an appreciation for experimentation and exploration as part
of the mathematical process. |
NCTM Content &
Process Standards
Addressed: |
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Algebra
(use symbolic algebra to represent and explain mathematical
relationships)
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Geometry
(explore relationships among classes of two-dimensional objects, make
and test conjectures about them, solve problems involving them)
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Problem
Solving (build new mathematical knowledge through problem solving)
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Reasoning
and Proof (make and investigate mathematial conjectures)
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Communication
(communicate mathematical thinking coherently to peers, teachers, and
others)
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GPS Content &
Process Standards
Addressed: |
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MM1G2a,b
(make
conjectures; understand the relationship between a statement and its
converse)
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MM1P1a
(build new
mathematical knowledge through problem solving)
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MM1P2b
(make and
investigate mathematical conjectures)
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MM1P4a
(recognize
and use connections among mathematical ideas)
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Supplies and Resources: |
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One
GPS file (see files) |
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Worksheet
(see files) |
Assessment: |
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Worksheet
- due next day
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Classroom
discussion |
Overview: |
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Introduction
(5 minutes)
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What
is a
right triangle? How can you tell if a triangle is right by
looking at angle measures? How can you tell if it is right by
looking at the triangle itself?
Have you heard of the Pythagorean Theorem? What is it?
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Exploration
(40 minutes)
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Students
use the given worksheet (see files)
to guide them toward a conjecture about the Pythagorean Theorem using
the given GSP file (see files).
What sort of if-then
statement can you write if
you're manipulating the triangle until a2 + b2 = c2?
What sort of if-then
statement
can you write if you're manipulating the triangle until it is
right? What is the difference between your "if"s and "then"s in
each of these situations?
Assuming the Pythagorean
Theorem is true (ΔABC is right implies a2 + b2 = c2),
how could you convince yourself that a2 + b2 >
c2 for acute triangles and a2 + b2
< c2 for obtuse
triangles, without using the GSP sketch?
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Wrap-Up (5
minutes)
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Compare
and discuss students' conjectures. (Why is it only a conjecture
so far for us and not yet a theorem?)
Encourage students to begin to think of ways a person might prove this
theorem (not necessarily full proofs, but ideas about how to proceed).
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