Finding Area on a Geoboard

1. Finding Area on a Geoboard By: Laurie Matecki

2. What is Area? Area is the amount of units an object takes up.

3. What is a Geoboard? This is a type of board with pegs/nails on it, size and materials may vary. Example:

4. Materials Needed for a Geoboard: Well, first of all if you don’t have one, you can make one with a 5 x 5 piece of wood and 25 nails. However, once you have a geoboard, all you need is some rubber bands of various sizes. For more thorough instructions and a pattern for making geoboards click here.

5. How to Find Area on a Geoboard: Therefore, When finding area, consider each little square, one unit. 1 2 3 4 5 6 7 8 is equal to one unit. 9 10 11 12 On this particular board, there are 16 possible units. 13 14 15 16

6. How to Find Area on a Geoboard Con’t: There are three main ways of finding area on a geoboard: • Fill and count • Halving • Surround and uncount

7. Fill and Count In this method, one would fill the area inside the object and count how many squares were included inside the object. • If you had a square, you would fill in the square to see how many smaller squares fit. • As you can see, 4 smaller squares fit inside this big square. 1 2 3 4 4 So what is its area?

8. Halving In this method, diagonals must be present. One counts the number of squares a diagonal is passing through, and then divides that by two, halving it. Or, when a diagonal divides two area units, When you have a triangle whose diagonal goes through one unit, the area of the triangle is equal to area one because 2 / 2 = 1. the area of the triangle is equal to ½ unit.

9. Halving In this method, diagonals must be present. One counts the number of squares a diagonal is passing through, and then divides that by two, halving it. With this triangle, the total area would be 1½ units because 3/2 = 1 ½ . Another example would be when you have a diagonal passing through three area units.

10. Surround and Uncount In this method, one would surround the entire shape and find that whole area. Then one would count the area that is NOT included within the shape, and subtract it from the whole area. However, 4 of the squares are only using half of the unit, so therefore 2 units are outside of the hexagon. If you had a hexagon, you would surround the entire shape, 1 2 3 and then count how many squares are included in that area. 4 5 6 So, 6 – 2 = 4 As you can see, six units are in the whole area. Therefore, the area of the hexagon is 4.

11. Surround and Uncount In this method, one would surround the entire shape and find that whole area. Then one would count the area that is NOT included within the shape, and subtract it from the whole area. surrounding the entire area works better. Another example of this method would be when you have a triangle such as this one. Look at this triangle and tell me what you think the area is. Since you can’t easily fill this triangle, If you said 2 ½ , YOU’RE RIGHT!

12. Surround and Uncount In this method, one would surround the entire shape and find that whole area. Then one would count the area that is NOT included within the shape, and subtract it from the whole area. How did I get 2 ½? Then you uncount the units outside of the triangle using your halving skills. Well, first you count that there are a total of 9 units within this square surrounding the triangle. 1 2 3 3 4 5 6 3 Therefore, 9 – 3 – 3 – ½ = 2 ½ 7 8 9 ½

13. For More Information: Here are some web sites that might be helpful for practice, or for understanding the area of geoboards better. • Area on the Geoboard • A New Algebra: Area on Geoboards • Geoboard • Investigating the Concept of Triangle and the Properties of Polygons: Making Triangles

14. Click Here for an Interactive Geoboard to Test Your New Skills This concludes my instructions on how to find area using a geoboard.