# Unit

## Constructions with Compass, Straightedge, and GSP

Kate Berryman
cavaleri@uga.edu

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The goal of this unit is for students to learn about constructions using compass and straightedge. Students will then use Geometer's Sketchpad to construct using technology.

Day 1 Day 6
Day 2 Day 7
Day 3 Day 8
Day 4 Day 9
Day 5 Day 10

## Day 1

Key Words:

Construction
Compass
Straightedge
Circle
Perpendicular

Goals:

For students to understand what a construction is and why it is important to mathematics.
To be able to begin making basic constructions with ruler and straightedge.
To begin getting acclimated with and learning how to make constructions using GSP.

Activity 1:

The teacher can begin with giving a brief history of what a construction is and why it is important in mathematics. Introducing Euclid's methods to help students understand why it is not sufficient just to use a ruler and protractor to calculate lengths and angles. Also, introducing the key words and definitions.

Activity 2:

Student will need a straightedge and compass along with plain white paper. Students will be asked to construct a circle given a radius, perpendicular midpoint, perpendicular from a point on a line, perpendicular at the endpoint of a ray, and
divide a segment into n parts. They must think about how to make this constructions with compass and straightedge, without using ruler measure.

Activity 3:

Students will begin to explore GSP.

Day 2

Key Words:

Angle
Bisection

Goals:

For students to understand what an angle is and its importance in mathematics.
To be able to construct given angles with compass and straightedge.
To be able to make angle constructions in GSP.

Activity 1:

The teacher should go over key words and definitions. The students should take a few minutes, going over what they learned on Day 1, reviewing what constructions are and why they are important to mathematics.

Activity 2:

Students should begin making angle constructions with compass and straightedge. These angles should include 30, 45, 60, and 90. Then students could be challenged to construct other angles based on 30, 45, 60, and 90 angles. For example, they can construct 75, 105, 120, 135, and 150 since these are just sums of angles they know how to construct. They can also subtract to get angles also. How many different angles can they construct?

Activity 3:

Students will begin constructions in GSP.

Day 3

Key Words:

Triangle
Equilateral Triangle
Isosceles Triangle
Right Triangle
Obtuse Triangle

Goals:

For students to understand a mathematical definition of a triangle.
To be able to construct different kinds of triangles using compass and straightedge.
To be able to use GSP to construct different triangles.

Activity 1:

The teacher should go over the key words and definitions, placing emphasis on the measure of the interior angles of a triangle sum to 180. Also, it is important to review the previous day's constructions, making sure that students remember how to construct angles (especially 30, 45, 60, and 90).

Activity 2:

The students should begin to construct different triangles. Based on their constructions from the previous day, students should be challenged to see how many different triangles they can construct. They should be able to construct a 30, 60, 90 triangle, equilateral triangle, and a variety of right triangles, just to name a few.

Activity 3:

Students will begin constructions GSP.

Day 4

Key Words:

Centroid
Circumcenter
Incenter
Orthocenter
Circumcircle
Incircle

Goals:

For students to learn about triangle centers.
To be able to construct triangle centers with with ruler and straightedge.
To be able to construct triangle centers in GSP.

Activity 1:

The teacher can begin with going over the key words and definitions. There could be a classroom discussion reviewing how to construct a triangle, perpendicular lines, midpoints, and angle bisectors.

Activity 2:

Students will construct four arbitrary triangles. They are to construct the centroid, circumcenter, incenter, and orthocenter. From here students can also construct the circumcircle and incircle.

Activity 3:

Students will make these constructions in GSP.

Day 5

Key Words:

Euler's line
Pedal Triangle

Goals:

For students to know who Euler is and what Euler's line is.
To understand a proof of Euler's line and to be able to construct Euler's line with compass and straightedge.
To understand how a Pedal triangle is constructed.
To be able to construct Euler's line and a Pedal triangle in GSP.

Activity 1:

The teacher should go over key words and definitions. Here is an time for students to learn about another important mathematician, Euler. Also, reviewing triangle centers from the previous day.

Activity 2:

Students must use compass and straightedge to construct Euler's line and a Pedal Triangle. Students should be able to apply what they know about triangle centers and constructing perpendicular lines.

Activity 3:

Students will begin constructions in GSP.

Day 6

Key Words:

9 Point Circle

Goals:

To understand what the 9 Point Circle is and its proof.
To be able to construct a 9 Point Circle with compass and straightedge.
To be able to use GSP to construct the 9 point circle.

Activity 1:

The teacher can begin by reviewing the previous days activities. Also, it might be useful to review how to construct the midpoint of a line. Next, the teacher could begin to go over what a 9 point circle is and its proof.

Activity 2:

Students must construct a 9 Point Circle using compass and straightedge. Students must be precise and neat in there work.

Activity 3:

Students will begin constructions in GSP.

Day 7

Key Words:

Polygon
Regular Polygon
Square
Pentagon
Hexagon
Octagon

Goals:

For students to understand the mathematical definition of a polygon.
For students to be able to construct regular polygons with compass and straightedge.
For students to make GSP constructions of regular polygon.

Activity 1:

The teacher should begin by going over key words and definitions. Students should understand what a polygon is and what makes a polygon a regular polygon.

Activity 2:

Students can begin to make constructions of a square, pentagon, hexagon, and octagon using compass and straightedge.

Activity 3:

Students will begin constructions in GSP.

Day 8

Key Words:

Rhombus
Trisection
Golden Ratio

Goals:

For students to explore a few more interesting constructions outside of triangles, some involving the golden ratio.
For students to be able to make these constructions with compass and straightedge.
For students to use GSP to make these constructions.

Activity 1:

The teacher can begin by going over key words and definitions. It is important that the students understand what the golden ratio is. Taking time to discuss its history and ways we see the golden ratio in the world around us could be a useful way to have a real life application.

Activity 2:

Students should construct a rhombus, a pentagon using the golden ratio, trisect a line segment, a circle through 3 points, and find the center of a circle.

Activity 3:

Students will begin constructions in GSP.

Day 9

Review Day

Goals:

For students to review the constructions they have done throughout this unit.

Activity 1:

The teacher should take time to review key words and definitions from this unit.

Activity 2:

The class should take time to review constructions they made with compass and straightedge.

Activity 3:

Students should review constructions they made in GSP and make sure they are comfortable with using GSP.

Day 10

Test Day

Goals:

To test the students on there knowledge of the constructions they have learned in this unit.