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Pinwheel

  • Lesson
6-8,9-12
2
AlgebraGeometry
Kris A. Warloe
Location: unknown

Each student creates parallelograms from square sheets of paper and connects them to form an octagon. During the construction, students consider angle measures, segment lengths, and areas in terms of the original square. At the end of the lesson, the octagon is transformed into a pinwheel, and students discover a surprising result.

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:

1992 pinwheel 1

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:

1992 pinwheel 2

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:

1992 pinwheel 3

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:

1992 pinwheel 4

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.

pdficon  Parallelogram Activity Sheet 

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.
1992 pinwheel 5

Continue to construct the octagon by inserting the rest of the parallelograms, alternating the colors.

1992 side by side

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.)

1992 pinwheel 6

To determine the area of the shapes, it may be necessary to determine the lengths of several segments, as indicated by af below:

1992 pinwheel 7

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.

1992 pinwheel 8

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:

1992 pinwheel 9

Determine the area of the red pentagon in terms of the area of the original square. Inspection of the pieces reveals that:

1992 pinwheel 10

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).

  • 5½" × 5½" Squares of Two Colors (eight squares per student)
  • These squares must be cut accurately to produce a quality project. Astrobright paper can be used; the paper should be cut at a print shop or in the school’s media center. Alternatively, origami squares can be purchased at craft shops. 
  • Parallelogram Activity Sheet 

Assessments 

  1. Use the responses that students give during the class discussion to assess understanding.
  2. Give each student eight square pieces of paper whose side length is twice the length of the original square. Ask students to determine the angle measures, segment measures, and area measures when an octagon is formed with these papers. Students should be able to determine most of the results without actually folding a new shape; for others, they may need to create two or three parallelograms, fit them together, determine the area of one pentagon, and use that to make other calculations.

Extensions 

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.

1992 pinwheel 11

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
1992 root 2 minus 1 over 2 .]

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

1992 ratio root .]

Teacher Reflection 

  • What difficulties did your students encounter when they tried to express the lengths of various segments in terms of the side length of the original square? Hod did you assist them in determining the correct relationships?
  • Were students engaged in the activity? What could be done to engage them more?
  • How did your lesson address auditory, tactile and visual learning styles?
  • Did you find it necessary to make adjustments while teaching the lesson? If so, what adjustments did you make, and were these adjustments effective?
 

Learning Objectives

Students will:

  • Use a unit square of paper to create origami parallelograms.
  • Use the parallelograms to construct an octagon that transforms to a pinwheel.
  • Determine the angle measures, segment lengths, and areas of polygon regions.

Common Core State Standards – Mathematics

Grade 7, Geometry

  • CCSS.Math.Content.7.G.A.2
    Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.