Area Relationships

by

Aarnout Brombacher

Note to teachers:

This activity is written in two parts which can be used independently and together, while it is hoped that part 1 will provide an insight into the initial stages of part 2 it is by no means a pre-requisite!

In this series of lessons we will be exploring the area relationships between different triangles, between different parallelograms and between various polygons, parallelograms and triangles.

We will be making extensive use of geoboards and will regard the horizontal and vertical distances between two points (pegs) as 1 unit.

Working in pairs complete each of the following activities and the project that follows.

PART 1

Activity 1

TWO DEFINITIONS

1. Interior point of a polygon - any point on the geoboard that falls inside a polygon.
2. Boundary point of a polygon - any point of the polygon that lies on the boundary of the polygon.

In the figure the points A, a, B, b, C, D, c, E, d, F, G and e are all boundary points of polygon ABCD while the points numbered 1 through 11 are the interior points of the polygon

Go to: GSP geoboard no. 1

1. Change the shapes of the various polygons*, and
2. Record the following data for each new polygon you create: the number of interior points, the number of boundary points and the area of the polygon.

* (You should move each vertex to at least 2 new positions!)

Activity 2

1. Using the data gathered in activity 1 determine a relationship (if it exists) between the area of a polygon and its points (both interior and boundary). We will refer to your relationship as your conjecture.

2. Make sure that the conjecture you make (i.e. the relationship that you predicted) works for all your data.

3. Go to GSP geoboard no. 2, create a number of new polygons and test your conjecture.

4. If necessary you may need to modify your conjecture until you are satisfied that it works in all cases.

5. Once you are satisfied that you have a conjecture that works show it to your teacher who will give you the password you need to access Activity 3.

Activity 3

Before clicking on the activity 3 button ensure that you have the required password Go to activity 3.

PART 2

Activity 1

As you work through this activity you may find the observations made during the previous activities useful.

Go to GSP geoboard no. 3

1. For the triangle given; leave the vertices A and B alone and find as many new triangles as possible - each having the same area as the original triangle - by moving vertex C.

2. RECORD YOUR FINDINGS on the dotty paper provided.

3. Repeat the activity moving A (leaving B and C fixed) and moving B (leaving A and C fixed).

4. Reflect on your observations and make some conjecture about triangles with a shared side and which are equal in area.

5. Using GSP geoboard no. 4 draw your own triangle(s) and test your conjecture.

HINT: If you would like some help in getting started with this activity you may wish to go to triangle-animation.

Activity 2

Using GSP geoboard no 5 repeat activity 4 but use parallelograms instead of triangles.

HINT: If you would like some help in getting started with this activity you may wish to go to parallelogram-animation.

Activity 3

You now need to prove the conjectures that you made in Activities 4 and 5. This should not be too difficult provided that you really spent some time on activities 1 and 2 in part 2.