By constructing a model of an octagon that transforms into a pinwheel, many interesting geometric shapes occur that can be explored. Throughout this lesson, stress mathematical vocabulary to describe the folds and the shapes that are formed. Start each student with two 5½" × 5½" square pieces of paper.
Before attempting this lesson, you absolutely must create an origami pinwheel on your own. You will not be able to offer assistance to students if you are not completely familiar with the folding and combining process for this shape.
In origami, when a fold is made and then unfolded, it leaves a crease. Such a crease is known as a valley fold. Valley folds are used as guidelines for folds that will be made later. During the first part of the process, many valley folds will be made; lead students through these valley folds as follows:
- Begin with a square piece of paper.
- Fold the square along its vertical line of symmetry; that is, from top to bottom, down the middle. Open the fold back to a square.
- Fold the square along each of its diagonal lines of symmetry; that is, from corner to corner. Open the square after each fold.
- Fold the upper right and left corners to the center of the square. Open the folds back to a square.
When all the folds are completed, the square will look like this:
The valley folds allow for a brief mathematical discussion. Ask the students to identify the angle measures formed by the folds without using a protractor. The following prompts can be used to help the students identify the angle measures:
- What is the measure of an interior angle of a square?
- What do the diagonal folds of the square do to the interior angles of the square?
- What does the fold on the vertical line of symmetry do to the square?
When all the angles are correctly identified, the square will look like this:
For the remainder of the lesson, tell the students, "Assume that the side length of the square is 1 unit." With this convention, discussion for the remainder of the lesson will proceed more smoothly.
Ask the students to identify the lengths of the segments made by the folds and on the perimeter of the square, again without using a ruler. Record the segment lengths next to the segments on the square. Stress exact answers—if necessary, record the lengths in simplest radical form. The following prompts can be used to help the students identify the segment lengths:
- Where do the diagonals of a square intersect?
- Use the Pythagorean Theorem to help you calculate the lengths of the diagonals.
- What does the fold along the vertical line of symmetry do to the side of a square?
All segment lengths are identified on the square below:
Ask students to name the shapes formed by the valley folds. There are eight 45‑45‑90 triangles (although two are twice as large as the other six) and two trapezoids. Ask the students to determine the area of each shape. (The original square has an area of 1 × 1 = 1 unit2.) Record the areas on each shape within the square.
It may be necessary for students to consider the area of some shapes in relation to the entire square. For instance, how many red triangles in the diagram below can be placed on the square so that there are no overlaps or gaps? Once a student has determined that it is possible to place 16 red triangles on the square, then the area of the red triangle is 1/16 unit2. The other small triangles are also 1/16 unit2. Further, the larger 45‑45‑90 triangles are equal in size to two red triangles, and the trapezoids are equal to three red triangles.
The area of each piece is identified on the square below:
To make the octagon shape, students will fold the square along the valley folds to create parallelograms. As students are making these folds, they should record the perimeter and area of each shape formed on the Parallelogram activity sheet, which also gives directions for creating the parallelograms.
Have all students fold a total of eight parallelograms, so that every student has four of each color. The remaining parallelograms can be folded as a homework assignment so that every student has eight parallelograms at the beginning of the next class. The eight parallelograms will be used to construct an octagon, as follows:
- Orient the folded edge of one parallelogram as shown below.
- Take a second parallelogram of the other color and rotate it 45° counterclockwise from the first one.
- Place the second parallelogram between the two isosceles triangles on the lower part of the first parallelogram.
Continue to construct the octagon by inserting the rest of the parallelograms, alternating the colors.
Once the octagon is created, the investigation can now focus on determining the segment lengths and the areas of the shapes. In one hand, hold up a 5½ × 5½ square, and in the other hand, hold up a completed octagon. Say to students, "The area of one of the original squares was 1 square unit. What is the area of the octagon that you just made?" Students should attempt to find the area of the octagon, possibly by finding the area of each of the eight congruent pentagons.
(Note that your students may discover that pushing the edges of the octagon towards the center transforms the octagon into a pinwheel, as shown below. This shape will be investigated later, as an extension. For the time being, focus only on the octagon.)
To determine the area of the shapes, it may be necessary to determine the lengths of several segments, as indicated by a‑f below:
The lengths of some of these segments may be difficult for students to determine. For instance, students may incorrectly think that e = 1/8. However, by placing two parallelograms over the original square, as shown below, students can determine the segment lengths by comparison.
To determine the area of the entire octagon, students can use decomposition to determine the area of each of the eight pentagons that occurs within the figure, as shown here:
Determine the area of the red pentagon in terms of the area of the original square. Inspection of the pieces reveals that:
The areas of the small green and red triangles (the tabs) negate each other, so the area of the red pentagon is ¼ − 1/8 = 1/8 square unit.
Since there are eight congruent pentagons, the area of the octagon is equal to 8 × 1/8 = 1 square unit. Said another way, the area of the octagon is equal to the area of the original square. Students will find this result quite surprising (as you may have, too).
