Kig is a free and open source alternative to commercial programs like Geometer's Sketchpad. Originally created by Dominique Devriese, Maurizio Paolini and Franco Pasquarelli (and maintained by Pino Toscano), it continues to evolve into an easy to use and powerful tool intended both for the exploration of geometric figures, and also the construction of geometric diagrams that can be used in other applications.

## Lesson Overview

This lesson uses the freely available Kig software to give students a hands-on introduction to the basics of constructing different geometric forms. Although there is considerable structure in this lesson, students should be encouraged to use the important ideas learned here to make their own constructions. After just a period or two of "the basics", a teacher can differentiate instruction for the remaining lessons by selecting appropriate challenges from the list of challenges as she sees fit for her particular groups of students.

#### GRADE LEVEL

- This activity is appropriate for upper elementary, middle, or even high school students. Although high school students can work through the basic constructions faster, there is still plenty of challenge at the upper level of the challenge list. At the same time, an exceptionally bright elementary student may be able to successfully tackle some of the upper level challenges.

#### DURATION

- The activity could take several weeks to complete. Once students learn the basics of Kig, it will be relatively easy to have this lesson "run itself", as students work their way up the list of challenges.

#### MATERIALS

- Computer lab with the Kig software installed
- Any reference materials you feel are appropriate (mathematical dictionaries)

#### TIPS AND SUGGESTIONS

- Go through this lesson at least once yourself prior to having students do it. Most of the most common questions students will have are those that can be figured out by simply playing around with the Kig software for a bit. Also encourage students to answer their own questions by experimentation.
- It is recommended that you begin by showing the class some of the basics of Kig--on a large screen, perhaps. Shortly after doing so, however, it is recommended that students work in small groups or pairs, with each group having a computer.
- At various times, have students demonstrate some of the easier tasks in front of the class.

### Goals

- Students will learn the basics of using Kig to construct geometric forms
- Students will explore the properties of geometric forms
- Students will understand how smaller components can combine to create a complex whole

### Objectives

- Students will be able to use Kig independently to construct basic geometric forms
- Students will be able to explain the various properties of each geometric construction
- Students will be able to apply basic construction techniques to a much larger range of general geometric construction challenges

## Procedure

In part one, the teacher shows the students how to make a basic construction in Kig. Students can watch this presentation and then immediately begin experimenting on their own to see if they can duplicate what they saw. Most of the basic skills needed to use Kig have already been covered, now they need to practice.

In parts two and three, students work on a more complicated construction by following the instructions below. Here students will learn a few more techniques that they can later apply to the list of construction challenges.

Part four is where students are let loose on the list of construction challenges. Some students will reach this stage faster than others, but from here on out the lesson almost runs itself, as students "climb the ladder" of increasingly difficult challenges. How part four ends is up to the individual teacher, although presentations from groups on each of the constructions is encouraged.

### Part 1

- Begin by showing students how to start Kig on their computers.
- Go to SETTINGS and show students how to hide the grid and axes.
- Show students the OBJECTS menus and the objects toolbar (left side of screen).
Create these objects: a line, a segment, a ray (called a half line), a triangle (by it's vertices), and a regular hexagon see image.

Now show them how these figures are not static, by moving them around a little see image.

- Clear the screen (EDIT/SELECT ALL then DELETE OBJECTS).
- Now Do the main construction for part 1: an equilateral triangle.....
- Begin by using the CIRCLE BY CENTER AND POINT to draw a circle.
Show how to select the two points, right click, and name them A and B, using the SET NAME command see image.

Now press "z" and show how that repeats the last construction. Use this to construct a circle from B to A see image.

- Now show students how to select objects. First, click on a circle to select it (notice it turns red). Now click on the white space to deselect that circle. Show how you can select more than one object by holding down "Ctrl". Now deselect all objects.
Now we want to show the points where the circles intersect. Go to OBJECTS/POINT/INTERSECT (also show where the icon is on the left of the page) and then select each of the circles in turn see image.

- Label on of the new points of intersection C (right click/SET NAME).
Now you could connect the three points with segments, but lets use OBJECT/POLYGONS/TRIANGLE BY ITS VERTICES to then choose the three points and create a triangle see image.

- To prove that it's equilateral, lets create a segment from A to B, then from B to C and C to A (remember to press "z" to begin again the last construction).
Now, highlight and right click each segment, choosing ADD TEXT LABEL/LENGTH. This prints the length of each side. Show how moving the points still produces an equilateral triangle each time see image.

Now have students work in groups to duplicate what you just did. This part will take some time, but the skills learned here will be used in nearly every construction they make from here on.

### Part 2

Students that have completed part 1 can now go on to making a parallelogram, by following these steps:

- Construct a line across the screen.
Now, construct another line, using one of the points on the first line see image.

Now comes the challenge. Using OBJECTS/LINES/PARALLEL, construct the remaining two lines to make a parallelogram see image.

Use OBJECTS/POINTS/INTERSECT to construct the final vertex on the parallelogram see image.

- Now hide everything except the four points on the vertices. Do this by selecting each line, right clicking, and choosing HIDE.
Create segments to form the sides of the parallelogram. To make this easier, press "z" after making each segment and it will automatically start making the next segment. You should end with a true parallelogram. Practice moving it around to show that it stays in the form of a parallelogram see image.

### Part 3

Students that have completed Part 2 successfully can now move on to making a 30-60-90 triangle, by following these steps:

Begin by constructing an equilateral triangle (see Part 1), but only construct the segments, not the internal triangle itself see image.

- Construct a midpoint on one of the side of the triangle (Hint: OBJECT/POINT....).
Now construct a segment from that new midpoint to the opposite vertex see image.

Select almost everything and hide them (right click/HIDE) see image.

- Construct the small side of the triangle by connecting the two vertices with a segment.
- Looks like a 30-60-90 triangle, but prove it.
- Construct three angles by going to OBJECTS/ANGLE BY THREE POINTS. Hint: If you work around the triangle clockwise it works better than counter-clockwise (can you explain why?).
Select each angle and see if you can label the number of degrees in each. Try moving the triangle around to see if they change. If you always get 30, 60 and 90 degrees, you've done it right! Good job see image.

Just for fun, choose EDIT/UNHIDE ALL to see all the geometry behind the 30-60-90 triangle see image.

### Part 4

Students that reach part 4 are now able to climb the ladder of geometric construction challenges! Here you have several choices. Ideally you would make your own ladder to meet the needs of your students. To give an idea of what these may look like, there are two included here. The first is for teachers who insist their students performs the traditional compass and straightedge constructions. They must, therefor, not use some of the features of Kig that have already been used in earlier steps. They may not, for instance, use the shortcut to bisect an angle, but rather have to bisect the angle using the circle constructor and the line constructor. Here's a possible ladder:

====Compass/Straightedge Challenges====

- Construct the perpendicular bisector of a line segment, or construct the midpoint of a line segment.
- Given a point on a line, construct a perpendicular line through the given point.
- Given a point not on the line, construct a perpendicular line through the given point.
- Construct the bisector of an angle.
- Construct an angle congruent to a given angle.
- Construct a line through a given point, parallel to a given line.
- Construct an equilateral triangle, or construct a 60ยบ angle.
- Divide a line segment into n congruent line segments.
- Construct a line through a given point, tangent to a given circle.
- Construct the center point of a given circle.
- Construct a circle through three given points.
- Circumscribe a circle about a given triangle
- Inscribe a circle in a given triangle.
- Construct an octagon from a square.
- Construct a regular pentagon.
- Trisect an angle.

(Note that the last two are jokes -- with only a compass and straightedge, constructing a regular pentagon is very difficult, and trisecting an angle is impossible. Do not assign these to students unless you want to be beaten up by their parents.)

An alternative to this, though, would be to have a more "no holds barred" approach, in which students are able to access all of Kigs features, including some of the more advanced features like differential geometry and transformations. If students are really motivated, they can even write their own macros. Nearly every topic in a geometry class can be explored using Kig. The more you use Kig, the more aware you will become of all the possibilities.