SWBAT demonstrate geometric definitions and properties using Origami

Students explore theorems about parallelograms through a hands-on activity. By the end of the activity, students will have made an Origami Crane!

8 minutes

To begin today's lesson I ask students to define each of the following in their own words (based on previous lessons and prior knowledge):

- isosceles triangle
- square
- rectangle
- parallelogram
- rhombus

These definitions will be used later in the lesson as students investigate the shapes they fold. After about 5 minutes, we will go over the definitions as a class, with students sharing what they have written and annotating their own definitions with things that they may have forgotten.

These are the definitions I use:

- An
**isosceles triangle**is a triangle with two congruent sides (exclusive definition). - A
**parallelogram**is a quadrilateral with two pairs of opposite sides parallel. - A
**rectangle**is a parallelogram with four congruent angles (or four right angles). - A
**rhombus**is a parallelogram with four congruent sides. - A
**square**is a parallelogram with four congruent sides and four congruent angles.

6 minutes

After we discuss the definitions, I hand each student a piece of Origami paper. I say to the students, “Take the color I give you. If you want a different color, ask someone to swap with you.” I don’t allow students to choose the color of their papers; it takes too much time. Then I ask the students what they notice about the paper. Common responses include:

- It is square.
- It is thin.
- It is smooth.
- It has a color on one side.

Some students will already be familiar with Origami. I ask those students to let others give their ideas first. If no one has said it, I tell the students it is Origami paper.

I then ask, “What is Origami and where does it come from?” A common response is China or Asia, but I like to make sure that students know that Origami is the Japanese art of paper folding. At this point, I ask which students have done Origami before. Usually at least one or two students have done Origami. These students become my helpers later in the lesson.

**Teacher's Note:** If Origami paper is not available, rectangular copy paper can be used. Instruct students to fold the shorter side to the longer side. This will create a triangle with a rectangle beneath it. Students then fold the rectangle up forwards and then backwards a few times. This makes it easier to tear the rectangle off. The paper will now be a square that can be used for today's activities.

22 minutes

In this part of the lesson, students fold an Origami crane and investigate the shapes created at various points in the process. This activity helps students activate prior knowledge and prepares them to write proofs about the properties of quadrilaterals.

For the activity, I have my students arrange their desks in a circle. Depending on how adept students are at Origami and how many students there are, I either sit with them or circulate as I show them how to fold their cranes. While I explain and demonstrate the steps, I often stop and discuss the shapes students make as they fold. I have included a video on Origami Cranes and Geometric Definitions Movie, which demonstrates how I lead the conversation.

**Teacher's Note**: The video can be played in two different ways. One way is to play it non-stop with the sound on. Students can follow along while the teacher circulates. The other option is to stop after every step and/or replay each step. In this case, I’d recommend keeping the sound off.

As a supplement to the activity, I have the students complete a worksheet with an explanation of some of the definitions and properties they were able to demonstrate through Origami (**MP3, MP4**). I often have students can work on the sheet while folding their cranes. But, it can also wait until after they have finished, and some students prefer this. Here is a sample of Student Work on the worksheet.

**Things to Consider as you demonstrate:**

- Although I often hold up my paper when showing students the folds, I tell them to keep their paper flat on the table.
- Some students pick up Origami really quickly. Their folds are neat and they can anticipate the next steps. Other students have great difficulty and need more guidance. I try to place these students next to those who can help them. It helps lessen their frustration.
- Students have difficulty with steps 3, 6 and 8. They may need extra help.

For an extension, I have the students investigate the diagonals of the parallelogram and the properties of a kite (formed in step 7).

10 minutes

As we come to the end of this lesson, we discuss the definitions and properties demonstrated through the folding activity. These properties include:

- the base angles of an isosceles triangle are congruent
- opposite sides of a parallelogram are congruent
- opposite angles of a parallelogram are congruent
- the diagonals of a parallelogram bisect each other

Then, I ask my students to discuss the answers to the questions on their worksheets in pairs. As a class, we will also discuss some of the other mathematical concepts that can be shown with Origami (symmetry, vocabulary, etc.).

At the end of the lesson, I ask students to hand in their worksheets. Those students who would like can take home their cranes. I collect the cranes from any students who do not want to take theirs home and I create a mobile, which I hang in the classroom.