Exploring Angle-Sum Relationships in Triangles and Quadrilaterals Using
The Geometer's
Sketchpad
by Karen S. Ray
Seventh grade Math Teacher
Edison Middle School
Champaign, Illinois
Objective
: To "discover" the angle-sum relationships in triangles and quadrilaterals and
demonstrate through repeated examples that it appears to always be true.
Summary:
Students will use the Geometer's Sketchpad software to create several triangles
and
quadrilaterals, then measure and sum their angles to discover that the angle-sum
for any triangle is 180 degrees and the angle-sum for any quadrilateral is 360
degrees.
Justification for using technology:
Many of you, myself included, have done this activity by drawing triangles and
measuring
their angles with a protractor or cutting pieces of paper and folding them to form
a straight angle. While this activity reaches most of the students, there are still a
good many who get lost in the activity, focusing so much on following the folding
directions or struggling with the protractor, that they do not make the connection
between the activity and the idea
that any triangle will have an angle-sum of 180 degrees. Using the Geometer's
Sketchpad
to construct and then measure the angles of the triangles allows each student to
quickly
create several perfect examples
(how often have you said to a student that his/her model did not work exactly due
to a measuring error) in a class period and still leave enough time for an all
important
class discussion about the results!
Links to NCTM Standards
: This lesson fits into the 5-8 Standards as follows:
Standard 12 Geometry
"In grades 5-8, the mathematics curriculum
should include the study of the geometry of one, two and three
dimensions in a variety of situations so that students can
- 1 identify, describe, compare, and classify geometric figures"
- 5 understand and apply geometric properties and relationships"
Standard 1 Mathematics as Problem Solving
"In grades 5-8, the
mathematics curriculum should include numerous and varied experiences with
problem
solving as a method of inquiry and
application so that students can
- 1 use problem-solving approaches to investigate and understand
mathematical content"
- 3 develop and apply a variety of strategies to solve problems,
with emphasis on multistep and nonroutine problems"
Standard 2 Mathematics as Communication
"In grades 5-8, the
study of mathematics should include opportunities to communicate
so that students can--
- 1 model situations using oral, written, concrete, pictorial,
graphical and algebraic methods"
- 3 develop common understandings of mathematical ideas,
including the role of definitions"
- 5 discuss mathematical ideas and make conjectures and convincing
arguments"
Links to Illinois Learning Standards for Mathematics:
(from June 18, 1997 Draft) This lesson fits into the middle/junior high school
Learning Standards as follows:
State Goal 7
: Estimate, make and use measurements of objects,
quantities and relationships and determine acceptable levels of
accuracy.
"As a result of their schooling, students will be able to:
- A.3a
Measure length, capacity, weight/mass and angles
using sophisticated instruments"
- B.3
Select and apply instruments, including rulers and pro-
tractors, and units of measure to the degree of accuracy
required."
- C.3b
Use concrete and graphic models and appropriate
formulas to find perimeters, areas, surface areas and volumes
of two- and three-dimensional regions."
State Goal 9
: Use geometric methods to analyze, categorize, and
draw conclusions about points, lines, planes and space.
"As a result of their schooling, students will be able to
- A.3a
Draw or construct two- and three-dimensional
geometric figures, including prisms, pyramids, cylinders and
cones."
- B.3
Identify, describe, classify and compare two- and three-
dimensional geometric figures and models according to their
properties."
- C.3a
Construct, develop and communicate logical arguments
(informal proofs)."
Additionally, this lesson provides students an opportunity to work
on skills included in the Applications of Learning section, such as
solving problems, communicating, using technology and working
on teams.
Level
: Grades 6 and above
THE ANGLE-SUM LESSON PLAN
Materials
: Each student or pair of students needs a computer with The Geometer's
Sketchpad
software (or similar software) and a printer.
Procedure:
(N.B. It is assumed that students have a basic, working knowledge of The
Geometer's
Sketchpad software.)
Part I
The teacher can start the activity by challenging the students to discover or find
out through experimentation what the sum of all angles in a triangle will always
equal. If someone in the class calls out "180 degrees", the teacher can respond
by saying "Show me!" Using the software, the students or pairs of students will be
challenged
to construct as many different sized triangles as they can in 15 minutes. The
teacher
should emphasize the need and value of constructing as many "nonroutine"
triangles
as possible, including obtuse and scalene. A quick review of the different types of
triangles may be necessary before proceeding. After time is called, the students
will use the measure function to measure each angle in the triangle and the
calculate
function to sum the measures. The teacher will ask each group to report their
findings
and if they had any exceptions. At this point, the class can come to a consensus
that the sum of all angles in a triangle must equal 180 degrees. The teacher can
confirm this to be true and let students know when in the future they will be able
to verify
this with a proof.
Part II
Now that students have discovered that the sum of all angles in a triangle equal
180
degrees, challenge them to construct quadrilaterals of all sizes and shapes. A brief
discussion about what it means to be a quadrilateral would be helpful--although it
would work to have all different sized squares and rectangles, the activity would
be
more meaningful if other quadrilaterals were also included. (Depending on the
level
of the class, it might be appropriate to compare sums for convex and concave
quadrilaterals.) After students have constructed for 15 minutes, have them
measure and sum the
angles. Poll each student or pair of students and reach a class consensus. Again,
point out that this will always be true and when they will encounter the proof.
Ask the class to look at their quadrilaterals. Is it possible to draw one line segment
in each of the quadrilaterals to get back to an earlier shape that had an agreed
upon sum? Hopefully, someone will think to divide each quadrilateral into two
triangles.
Students can then do that to a few of the quadrilaterals to confirm using the
measure
and calculate functions.
Extras
: These activities could be assigned as enrichment or extensions to the lesson. They
provide connections to other subject areas, also.
Students could be asked to choose their most unusual triangle,
print it out and then contribute to either a scrapbook or bulletin
board. (Art)
The teacher could have a couple of volunteers do this activity using
pencil, paper and compass to compare results. This would
emphasize some good reasons for using the technology and could
generate a discussion on appropriate uses of technology in daily
life. (Graphic drawing and social studies)
Students could write a letter explaining their experiment and
results to a younger student or a grandparent. (Language Arts)
Students could make an entry in their math journal describing the
activity and noting one surprising aspect of the lesson. (Language
Arts)
Students could write a newspaper article describing the lesson to
others. (Language Arts)
Further applications
: A similar lesson format could be used to introduce the relationships among the
angles formed by parallel lines cut by a transversal or perpendicular lines.
I welcome your comments or suggestions regarding this lesson. Please send mail
to Karen S. Ray