Download Saccheri Rectangle and its relationship to Non

Saccheri Rectangle and its relationship to Non-Euclidean Geometries
Authors: Eric Rodriguez
Length of Lesson: 45 - 50 minutes
Grade Level/ Subject: 9-10th grade/ Geometry
TEKS Addressed:
● §111.34. Geometry
○ (b) Knowledge and skills
■ (1) Geometric structure. The student understands the structure of, and
relationships within, an axiomatic system. The student is expected to:
○ develop an awareness of the structure of a mathematical system,
connecting definitions, postulates, logical reasoning, and theorems;
○ recognize the historical development of geometric systems and
know mathematics is developed for a variety of purposes; and
○ compare and contrast the structures and implications of Euclidean
and non-Euclidean geometries.
Concept(s):
● Give an overview of the scientific and/or mathematical concept or concepts that you will
teach through this lesson.
○ The lesson will discuss the importance of the work by Italian mathematician,
Girolamo Saccheri. It will include his work on proving the parallel postulate by
Euclid. Saccheri’s work on proving Euclid’s parallel postulate will be discussed in
full detail. The lesson will explain how Saccheri’s work led to new work in
mathematics that we use to this day.
Objective(s):
● The students will able to:
○ define the parallel postulate from Euclid’s The Elements.
○ identify the problems that mathematicians faced when trying to prove the parallel
postulate.
○ understand the reasoning behind Saccheri’s attempts at proving the parallel
postulate.
○ observe the impact of Saccheri’s work in the mathematics we see today.
Materials:
● Chalkboard
● Projector
● Document Camera
● Handouts
Safety Considerations:
● None
Engagement
Estimated Time: 5-10 minutes
● Before the lesson begins, the objectives of the lesson will be shown, which include:
● define the parallel postulate from Euclid’s The Elements.
● identify the problems that mathematicians faced when trying to prove the parallel
postulate.
● understand the reasoning behind Saccheri’s proof of the parallel postulate.
● observe the impact of Saccheri’s work in the mathematics we see today.
● How will the teacher capture students’ interest and stimulate thinking?
○ I will ask the students to draw two lines that are parallel and explain why they are
parallel. We will give them about 3 minutes.
■ A few students can present their drawing on the document camera.
○ As a class, we will go over details of the explanations such as acute angles, right
angles, intersections, etc.
○ From there I will ask, “From all the explanations we’ve heard, can we write down
an official definition of parallel lines?
■ As a class, we will develop our personal definition of parallel lines.
○ I will then explain to the class that mathematicians have spent nearly their entire
lives working on developing definitions, theorems, and proofs in math.
■ Introduce Euclid and provide a brief history with a picture of him.
○ Considered the “Father of Geometry”
○ His Elements is considered one of the most important works in
geometry and mathematics in general.
Exploration
Estimated Time: 15
• What tasks and questions will help students puzzle through the concept?
● I will present the class the five postulates that Euclid used to derive most of planar
geometry.
● 1) A straight line may be drawn between any two points.
● 2) A piece of straight line may be extended indefinitely.
● 3) A circle may be drawn with any given radius and an arbitrary center.
● 4) All right angles are equal.
● 5) If a straight line crossing two straight lines makes the interior angles on the
same side less than two right angles, the two straight lines, if extended indefinitely, meet
on that side on which are the angles less than the two right angles.
● At this time, I will direct the student’s attention to the 5th postulate.
○ I will now ask the students to provide a drawing (with labels) of how they believe
the postulate is being described.
■ I will go around and see the student’s drawings.
■ I will provide an accurate drawing of the 5th postulate so that students
could compare their drawing to it.
● I will go into detail that Euclid’s 5th Postulate received a lot of criticism from
other geometers at the time because of its complex structure.
○ I will explain that there were several attempts to prove Euclid’s 5th
Postulate.
○ I will ask the class to see if anyone would like to take a guess on how to
prove the 5th Postulate.
■ If they couldn’t prove it, I will let them know that many other
geometers couldn’t prove it either. I will explain that many failed to prove
it because the assumptions being used were equivalent to the 5th Postulate.
(one of the equivalent statements was Playfair’s Axiom)
● Introduce Girolamo Saccheri (provided with a picture)
○ Italian mathematician in the 18th century.
○ Attempted to prove Euclid’s 5th Postulate without using equivalent statements
from Euclid’s work.
• What “big idea” conceptual questions the teacher will use to encourage and/or focus
students’ exploration?
● I will tell the students that Saccheri began his process to prove Euclid’s 5th Postulate by
using a quadrilateral.
○ I will provide a drawing of the quadrilateral (with labels) that Saccheri used.
○ I will now ask the class to take some time alone to see if they can figure out what
Saccheri’s next step in the proof was.
○ After awhile, I will allow the students to discuss within their
groups to see how far along they are in the proof.
○ I will allow a few students volunteer to present their progress in the
proof. I would then transition to explaining the steps that Saccheri took in attempt
of proving Euclid’s 5th Postulate.
Explanation
Estimated Time: 10 minutes
• How can the teacher help students make sense of what they have observed or the
material they've encountered?
● I will present the approaches that Saccheri took while attempting to prove Euclid’s 5th
Postulate.
○ I will explain the 3 approaches from Saccheri.
■ 1) Using a quadrilateral (rectangle) with sides that are equal in
length and perpendicular to the base.
■ 2) Using contradiction, by stating the angles opposite of the base
were larger than right angles.
■ 3) Using contradiction, by stating the angles opposite of the base
were smaller than right angles.
Elaboration
Estimated Time: 10 minutes
• How will students develop a more sophisticated understanding of the concept?
● The students will see that Saccheri’s three attempts at proving Euclid’s 5th Postulate were
not successful but opened doors for mathematicians to produce new work.
○ In Saccheri’s second attempt, he was able to find fundamental statements that hold
true in Non-Euclidean geometry, such as elliptic geometry.
○ In Saccheri’s third attempt, he found seven properties that hold true and would
later become building blocks in hyperbolic geometry.
Evaluation
Estimated Time: 5 minutes
• How will students demonstrate that they have achieved the lesson objective?
○ The students would have developed their definition and drawings of parallel lines
and compared it to their drawing of Euclid’s 5th Postulate.
○ Now knowing Euclid’s 5th Postulate, the students would have seen that there are
many equivalent statements to it, which led to many mathematicians’ mistakes.
○ The students would point out the mistakes that Saccheri made in his three
attempts of solving Euclid’s 5th Postulate.
○ The students saw that Saccheri’s mistakes in his attempts helped other
mathematicians produce new work in Non-Euclidean geometry, such as elliptic
and hyperbolic.
• How can the teacher help students self-evaluate and reflect on what they have learned?
● I can ask the students questions that require thinking and reflecting on what they learned
in the lesson.
○ Questions:
■ Have there been moments in your previous math classes where Euclid’s
5th Postulate was used?
● Have you seen any of his 5 postulates used?