6th to 8th,High School
Kris A. Warloe
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:
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:
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:
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:
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
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).
The octagon can be transformed to a pinwheel by pushing each of the
pentagons toward the center. A rotationally symmetric pinwheel will be
formed when each of the pentagons meet in the center.
Have students determine the side lengths and areas of the polygons
that occur when the octagon is transformed to a pinwheel. Compare the
total area of the polygons to the area of the original square.
By placing the pinwheel on top of the original square, students will
see that they have the same area. By inspection, students should notice
that the four uncovered triangles on the square are congruent to the
four green triangles that extend beyond the sides of the square.
Alternatively, students can decompose the pinwheel into an interior
octagon and eight parallelograms and calculate the area of each; the
sum of those areas will equal the area of the original square.
Questions for Students
1. What is the side length of the exterior octagon in terms of the side length of the original square?
[The side length of the original square is 1 unit. The side length of the exterior octagon is ½ unit.]
2. What is the side length of the interior octagon in terms of the side length of the original square?
[The side length of the interior octagon is
What is the ratio of the side length of the interior octagon to the side length of the exterior octagon?
[The ratio of the side lengths is