Paper Folding Activity

## Teacher Lesson Plan

This activity is aligned to NCTM Standards - Grades 6-8: Geometry, Measurement, Problem Solving, Reasoning and Proof, and Communication and California Mathematics Standards Grade 7: Number Sense #1.2, Measurement and Geometry #3.4 and Mathematical Reasoning #1.1, 2.1, 2.4, 2.5, 2.6.

Students fold a sheet of paper (a manipulative) and also simulate folding using the ESCOT Runner software (technology) to observe geometric relations. Students think about the angle measurements and the side measurements of the triangles in order to deduce the relations of the triangles. Ideas of similarity are investigated using both a manipulative and technology. Finally, students are asked to respond to questions in a paper/pencil activity.

It is important to build a bridge between the technology representing the piece of paper and the actual physical piece of paper. Visiting the problem using both techniques addresses a variety of learning styles, brings the abstract into the concrete, and offers interaction with the computer as students investigate, discover, form hypotheses, draw conclusions, and benefit from the quick feedback and the interest a computer provides. Once students have had these experiences it is important to arrive at a synthesis by spending the time necessary to internalize the concepts. Manipulatives can provide space for group work, computers can afford individual explorations, and a synthesis can take place during a full-class discussion. Students can then demonstrate their individual understandings through the writing process.

 Using Manipulatives

Simulating the activity:

1. Pass out the following materials for each student:
• one sheet of blank paper (8.5 in. X 11 in.)
• protractor
• ruler
2. Instruct the students to fold their paper like this:

3. Questions:
Describe the polygons that were created.
Do you know any of the angle measurements or linear measurements by observation? Explain.

4. Instruct the students to label the points of the triangles:

5. Questions:
Name the polygons using the labeling letters.
Name the angle and/or linear measurements using the labeling letters.

 Using Technology

ESCOT Runner software:

Nathalie Sinclair wrote the program for the Paper Folding Activity

Paper Folding Activity

Click on fold

Have the students respond to these questions:

1. What is/are the relation(s) of the angles of triangle A to the angles of triangle B to the angles of triangle C?

2. What do the triangles, EXM, XFJ, GMI, have in common? Explain how you know this.

3. Is there any configuration for which the three triangles are congruent? Explain how you tested for this relation.

4. Is there any configuration for which two triangles are congruent? Explain how you tested for this relation.

5. What are the dimensions of the paper? How did you calculate them?

6. If you fold the paper to the 1/2 mark of line segment EF, what is the property of point I?

7. If you fold the paper to the 1/3 mark of line segment EF, what is the property of point I?

8. What is the relation between the gray shaded area and the striped shaded area?

9. Is there another single fold of the paper that produces three (or even two) similar shapes? Make or draw a model and explain your thinking.

10. What happens when you use another size of paper? For example, what happens when you start with a square and what happens if the rectangle is 8 in. by 9 in.? Do the relations change?

11. If you start with a square and you want to find the one third point on the side of the square, where should you fold the paper to on the top side?

 Synthesizing the Activity

At this point, students have investigated the problem using a manipulatives (a sheet of paper with measuring tools) and technology (the ESCOT Runner simulation).

Once more have the students use their sheet of paper and the vocabulary they have used to identify the vertices, triangles, angles and lengths, to describe the relations that exist between the triangles formed from the fold.

Ask them to summarize the information from their responses to the questions asked above.