Lesson 1
Conjecture
Summary
Students will use the
given
GSP drawing to manipulate the vertices of a
triangle and make observations about what type of triangle is formed
when a2 + b2 is equal
to, less
than, or greater than c2 (where c is the
length of the triangle’s longest side). When a2 + b2
= c2,
students can see that they have a right triangle (a2 + b2
= c2
implies ΔABC is right);
conversely,
students can manipulate
the edges of the triangle until one angle is right and see that ΔABC is
right implies a2 + b2 = c2.
Thus, students will make a
conjecture about both the Pythagorean Theorem and its converse.
In addition, students will extend the Pythagorean Theorem to determine
what relationship exists among the squares’ areas when ΔABC is
acute or obtuse, an observation which will be used in Lesson 3.
This activity is adaptable to students of different grade levels and
ability levels. Students who have not yet been introduced to the
Pythagorean Theorem can use the included worksheet to guide them
through the process of making a conjecture about the Pythagorean
Theorem. Students who already know the Pythagorean Theorem, on
the other hand, can use the GSP file not only to gain an appreciation
of the Pythagorean Theorem’s geometric meaning but also to extend it to
see
that similar conclusions can be drawn about acute and obtuse triangles.
GSP is extremely useful in this activity because it gives
students insight into the geometric representation of the Pythagorean
Theorem, and its dynamic nature allows students to easily gather data
when they manipulate the triangle’s vertices. As they move the
vertices to form a triangle that is obtuse, then right, then acute,
students can see a continuous change in the relationship among the
squares’ areas. This makes GSP a more powerful tool for
exploration of this problem than by-hand drawings, which lack this
continuous, dynamic nature.
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