Motivated by the goal of engaging more students in doing mathematics, we have been investigating connections between geometry and art. Geometry textbooks include problems that are situated in various art-based contexts such as architecture, crafts, and drawing. In our conversations with geometry teachers, we have found that they welcome opportunities to teach geometry through art. Nevertheless, they ponder how to find artbased contexts that are relevant for their students.
Origami is one way to incorporate art into mathematics lessons. Origami is not only engaging, but useful and beautiful. Aerospace engineers design artifacts based on origami principles to efficiently package them. Architects have been inspired by origami to design beautiful buildings. Fashion designers have turned to origami to create original designs. By opening connections between the art of paper folding, which is practiced in various cultures, and geometry, we can promote students’ deep engagement with mathematics.
When we worked with focus groups with high school geometry teachers, they said that origami can increase students’ engagement. One teacher said, “I like the origami [crane problem], just because [students] get to manipulate it and actually look at those things.” Another teacher said that students have seen origami before, even if they had not made origami pieces, so they can build on their previous knowledge.
Here, we share two activities for teaching congruence criteria for triangles through origami. We take advantage of folding to promote students’ identification of figures with reflection symmetry. The problems exemplify how to use transformation geometry in relation to the definition of congruence.
Christine’s Origami Heart Activity
In my Geometry classroom, I wanted to bridge the prior unit’s construction work to the current unit’s triangle congruence. During the construction unit, students who were normally discouraged became really engaged with the hands-on aspect of the work. I liked the idea of using origami to build on that engagement. In addition, I wanted to elicit students’ prior knowledge about transformations, which we had learned in our first unit. The students followed my modeling and a video to create an origami heart (Figure 1). I then asked them to unfold their origami heart and answer the following questions:
- What lines of symmetry do you see? Highlight three of them.
- What transformations do you see on your paper?
- Are there congruent figures on your paper?
After giving the students time to consider these questions individually, we had a class discussion and began to tie in the mathematical vocabulary, such as the term rigid, that we had used previously. It led into the idea that if a rigid transformation occurs, then the pre-image and the image will be congruent. That was the objective of our following lesson and the foundation of the congruence unit. While this lesson was meant to be an introductory activity, it could be extended into a longer learning task.
The Origami Crane Problem
Another problem uses an origami crane to study the properties of congruent triangles (Figure 2). After demonstrating how to create the crane, instruct students to open the square origami paper to study the crease patterns (Figure 3). Point out that there are many symmetric patterns that appear interesting, but it is unclear how they were created. Ask them to work in pairs to answer the following questions:
- Use color pencils to identify as many pairs of congruent angles as possible
- Choose a pair of congruent triangles and prove that they are congruent.
- Find two figures that are reflections of one another. Explain how you know that they are a reflection.
- Talk to another pair about number 3. Do you agree with their explanation? Why or why not?
Another related activity involves reproducing the “opened” origami crane using GeoGebra (Figure 4). Students could use dynamic geometry and discuss various ways to construct it, including constructions by using reflections. Students could also create art with the diagram as in Figure 5.
Overall, the teachers we worked with anticipated that origami-based problems could provide opportunities to connect with properties of reflections, and they valued that students could share different things that they noticed about the creases.
Gloriana González is Associate Professor of Mathematics Education at the University of Illinois at Urbana-Champaign. Christine Rinkenberger is an alumna of UIUC and works at the Urbana School District.
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