Parallelograms

Contents || Math Lessons

[Materials] [Length] [Area] [Pick's Theorem] [Dot Paper] [Epilogue]

[Lines] [Squares] [Rectangles] [Parallelograms] [Right Triangles] [Bibliography]

    Find all parallelograms

Objectives

• to continue developing the student's sense of area, especially the relation between the area of a parallelogram and the area of the corresponding rectangle
• to realize that all rectangles are parallelograms
• to review some geometric concepts (e.g., congruent, similar) in the context of parallelograms
• to experiment with geometric pattern making

Materials

• 5 x 5 geoboards (the kind that fit together to make a 10 x 10 geoboard) and rubber bands (various sizes and colors)
• at least five geoboards for the teacher and one for each student
• 5 x 5 geoboard dot paper (two or three sheets for each student)
• 10 x 10 geoboard dot paper (for extended activities)
• overhead projector; transparent geoboard or dot paper (with marking pens)

Warm-up activities

1. Make a rectangle with base 2 units on your geoboard. Without removing the rectangle, make a parallelogram with the same base.
2. What is the area of the rectangle in the previous activity? What is the area of the parallelogram? How do you know?
3. Find another parallelogram with the same area as the parallelogram in the previous activity.
4. On a 5 x 5 geoboard, find all parallelograms with a base of 2 units. How many are there? (There are 12 such parallelograms, including four rectangles.)
5. Make a parallelogram with base 1 unit on your geoboard.
6. Find three more parallelograms with base 1 unit having the same area as the parallelogram in the the previous activity. What is the area of each of these parallelograms?
7. Find four more parallelograms with base 1 unit, all having the same area.
8. How many different parallelograms with base 1 unit are there? (There are 16 such parallelograms, including four rectangles.)
9. Find a parallelogram with smallest area. Can you find other parallelograms with this area?
10. Find a parallelogram with next smallest area. Can you find others?
11. Find the parallelogram with largest area.
12. Find the parallelogram with next largest area.

Main activity

Find all possible parallelograms on a 5 x 5 geoboard. (There are 26 non-rectangular parallelograms. Together with the 16 rectangles of Lesson 4, this gives a total of 42 parallelograms.)

Homework

Order each of the parallelograms found in the main activity by area. Here is a summary of the 26 non-rectangular parallelograms:

Extended activities

1. Make a parallelogram that has the same area as the following rectangle:
   

2. Make a parallelogram that is congruent to the one below:
   

3. Make a parallelogram that is similar to the one below:
   

4. Continue this pattern of parallelograms on a 10 x 10 geoboard and complete the table:
   

   

   Compare your results with extended activity 5 of our lesson on rectangles.

5. Continue this pattern of parallelograms on a 10 x 10 geoboard and complete the table:
   

   

   Compare your results with extended activity 6 of our lesson on rectangles.

6. Continue this pattern of parallelograms on a 10 x 10 geoboard and complete the table:
   

   

   Compare your results with extended activity 7 of our lesson on rectangles.

7. Continue this pattern of rectangles on a 10 x 10 geoboard and complete the table:
   

   

   Compare your results with extended activity 8 of our lesson on rectangles.

[Materials] [Length] [Area] [Pick's Theorem] [Dot Paper] [Epilogue]

[Lines] [Squares] [Rectangles] [Parallelograms] [Right Triangles] [Bibliography]

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