Download Compass and Straightedge Constructions - SUPER-M

School and University Partnership for Educational Renewal in Mathematics
An NSF-funded Graduate STEM Fellows in K–12 Education Project
University of Hawai‘i Department of Mathematics
Compass and Straightedge Constructions
1
Introduction
Many aspects of math that we are still learning by high school have been thought about for over 2,000
years! The first recorded pursuit of mathematical knowledge for its own sake - the birth of the first mathematicians - dates between 600 and 300 BC in ancient Greece. Among the first problems these ancient
mathematicians tackled were geometry questions that asked about the relations between different angles,
lines, circles, and polygons.
One class of problems the early Greeks struggled with was what they could create with just a compass
and an unmarked straight edge. For instance, can you cut a line perfectly in half? Can you make a square
that has the same area as a circle? Today we will begin our own investigations into these questions!
2
Grade Levels and Topics
This lesson was designed as a semester-long project for gifted 4th-6th graders to work on once a week, but
has also been used for older grades as well. The first problem is always the most challenging as students will
need to familiarize themselves with the different operations of the straightedge and compass. Constructing
the hexagon will generally take several hours if no hints are given, and bisecting the line about 30 to 45
minutes. This lesson makes connections to critical thinking, reasoning with geometry, problem solving,
and creativity.
3
Objectives
Students will use their creativity and a minimal set of hints to construct basic polygons and other geometry
constructions using just paper, a straightedge, and a compass.
4
Materials
Large sheets of construction paper, pencils, compasses, and straightedges without ruler markings.
5
Exercises
Each exercise centers around the construction of a different geometric idea or object. Working with gifted
4th-6th graders, we completed the first three exercises in about 4 hours spread out over an equal number
of days. Detailed solutions of how to complete each construction are provided on the website Math Open
Reference (http://www.mathopenref.com/tocs/constructionstoc.html)
UHM Department of Mathematics
[email protected]
School and University Partnership for Educational Renewal in Mathematics
An NSF-funded Graduate STEM Fellows in K–12 Education Project
University of Hawai‘i Department of Mathematics
5.1
Hexagon
A hexagon is a regular polygon with six sides. What is a regular polygon you ask? Regular means that all
the interior angles are equal and that all the sides are equal lengths.
5.1.1
Prompt
Can you construct a regular polygon using only compass and straightedge?
5.1.2
Note to Teachers
Most students will try to use only the straightedge to draw a six-sided figure that looks close to a regular
hexagon. You will need some criterion to reject these inexact drawings. One way is to allow the student’s
peers to measure the sides using a ruler and to observe whether each side is in fact equal. Another criterion
would be to require the student to have step-by-step instructions so that anyone with the instructions could
also construct a hexagon.
Other students will try to “cheat” and create an artificial mark on the straightedge so that all sides
of the hexagon can be equal lengths. You will need to emphasize that the straightedge intentionally does
not have any markings.
5.1.3
Hints
• Draw a circle using the compass
• How might we inscribe a hexagon in the circle?
• What do you notice about the lengths of a hexagon’s sides?
5.1.4
Solution
The step-by-step solution can be accessed here: http://www.mathopenref.com/constinhexagon.html.
5.2
Perpendicular Lines
Two lines are considered perpendicular if they form two equal adjacent angles (we would then say that the
two angles are congruent).
5.2.1
Prompt
Can you draw perpendicular lines using only a straightedge and compass?
5.2.2
Hints
• Think of how drawing two circles of the same diameter might help you...
5.2.3
Solution
The step-by-step solution can be accessed here: http://www.mathopenref.com/constbisectline.html.
UHM Department of Mathematics
[email protected]
School and University Partnership for Educational Renewal in Mathematics
An NSF-funded Graduate STEM Fellows in K–12 Education Project
University of Hawai‘i Department of Mathematics
5.3
Bisecting a Line
Bisecting a line means cutting the line into two equal segments.
5.3.1
Prompt
Can you bisect a line using only straightedge and compass?
5.3.2
Hints
• How would you do this if we could use a marked ruler?
• How might knowing 5.2 help us?
5.3.3
Solution
The step-by-step solution can be accessed here: http://www.mathopenref.com/constbisectline.html.
5.4
Center of a Circle
A circle is defined as all the points of equal distance from a given point.
5.4.1
Prompt
Draw a circle. Can you find the center of the circle using only a straightedge and compass?
5.4.2
Hints
• Try drawing a chord across the circle (a chord is a line segment from one point on the circle to
another point on the circle).
• You only need to know 5.2 and 5.3.
UHM Department of Mathematics
[email protected]