## October 13, 2012

### Origami Angles

I have so many things that I want to blog about that I have idea lists going in several different locations!  However, I still just can't get caught up.  I'm sure it has nothing to do with restructuring geometry, figuring out pre-calc as I go or having a million things to do outside of school since I bought a house August 30 (best timing ever!).  So this isn't quite a real post, but it's better than nothing (or me whining about how busy I am, since I know you're all busy too).

Over the summer I took a course at the Education Development Center on the CCSS Standards for Mathematical Practice.  We have a follow up session on Monday and I had to write up a lesson plan and a discussion of what math practices it involves.  As long as I wrote it, I figured I'd share:

Objectives: Students will determine angle measurements by reasoning.  Students will write sentences that justify their conclusions.

Materials:
-Origami paper (one per student, plus a few extras)
- Student handout (one per student)

Outline:
Seat students in pairs or small groups.

Give each student one square of origami paper and a student handout.

Instruct them to carefully fold their paper according to the instructions.  If a student gets stuck, ask them to read the step they are on, stopping them mid-sentence to have them point to the part of the origami paper the instructions are referring to.  (Note: for step 2 I find that it’s helpful if I keep a finger on the upper left vertex - holding the paper taut makes it clear when the crease will intersect that point.)  If a student makes an error give them a new origami paper since extraneous creases will be confusing later.

When everyone has folded their square, bring the class together to clarify instructions.  Make sure the students label the points on their origami paper as shown on the handout.  Emphasize that the goal is to come up with reasons that justify the angle measure, not just make a list of numbers.  Depending on the level of the class either set them to work or read the hints together and brainstorm methods of approach.

Circulate the room, encouraging students to work together rather than rely on the teacher for assistance.  Give strategic hints to groups that need them (start with right angles, show students how to fold one person’s paper and use that to find congruent angles on another paper, ask students what types of angles they have studied and where they see examples of those).

Typically students need more than one class period to finish, plan to assign the remainder for homework or allow them to finish on another day.

Note: Idea for lesson and Origami Instructions are taken from the workbook which accompanies the textbook Geometry published by McDougal Littell.

Make sense of problems and persevere in solving them.

One thing I like about this task is that the instructions are straightforward.  Students have some questions about folding the paper properly, but after that they all know exactly what they need to do.  However, it is not at all clear to them how to achieve the goal.  I continue to be surprised how many students struggle to complete this task independently.  As I am circulating the room I will stop by a pair, point to an angle and ask them if they can figure out the measure of it.  Before walking away I might say, “look for other angles like that one” but too often when I check back in with that same group they haven’t made progress.  I use this lesson in early October, and it reminds me that perseverance is a practice my students need to work on.

There are 31 angles to find in the diagram (excluding 180 degree angles but including sums of smaller angles - such as HAC - which students rarely include).  For Geometry CP they need to find at least 20 angles to receive full credit for the assignment.  For Fundamentals of Geometry they need to find at least 15 angles to receive full credit for the assignment.  (The Fundamentals of Geometry students are just as capable as the other students of finding 20 angles, but it takes them longer since they have learning disabilities or gaps in their math background.)  In each case students are asked to find more than the ‘easy’ angles.  They will need to continue working past the point where angle measures are obvious and this is where some students’ ingenuity shines through.

Construct viable arguments and critique the reasoning of others.

The main goal of this lesson is to introduce proof.  I repeatedly ask, “How do you know?”  Asking students to defend their claim that an angle is 90 degrees really highlights the nature of proof.  The conversation usually goes:

Teacher: How big is this angle? (pointing at an angle in a corner)
Student: 90 degrees.
Teacher: How do you know?
Student: It’s a right angle.
Teacher: How do you know?
Student: (getting frustrated) Because it looks like it!
Teacher: Where is it on the paper?
Student: The corner.
Teacher: That’s how you know it’s a right angle. You’re not guessing or assuming, you’re using what you know about origami paper.

When I am grading this assignment I give no credit for correct angle measurements if they do not have an accompanying reason.  So if a student submits their origami square with all the angle measures labeled (correctly or not), they will get back a note asking for their reasons.  To make this point clear I remind students that they need to hand in 20 (or 15) sentences.  In reality I will happily accept a sentence which says, “Angle AHD and Angle GHD are each 90 degrees because I folded a line (180 degrees) in half” so they won’t necessarily have as many sentences as they do angles, but focusing their attention on sentences rather than numbers has the effect I want.

Student Handout

Origami Angles