School and University Partnership for Educational Renewal in An NSF-funded Graduate STEM Fellows in K–12 Education Project University of Hawai‘i Department of Mathematics

Compass and 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 math- ematicians - dates between 600 and 300 BC in ancient Greece. Among the first problems these ancient mathematicians tackled were questions that asked about the relations between different , lines, , 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 that has the same area as a ? 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, , compasses, and without 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 . 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 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]