## 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.

For some hints on proving your conjectures **click
here**.

**PROJECT **
Each student should complete a minimum of one of
the projects on the **project page**.

**Return to Aarnout's EMT699 page**