Project Background

FGT Professional Development Curriculum

Vignette 1: Exploring Similarity through Paper Folding

Vignette 2: Reasoning about Properties & Area with Tangrams

Project Staff

 

Vignette 1: Exploring Similarity through Paper Folding

Divided into three groups, the twelve teachers explored a multi-step problem that involved paper folding.  One group included the two 9th-grade teachers and the two 8th-grade teachers; a second group included the four 7th-grade teachers; and the third group included the two 5th- and two 6th-grade teachers.  After several scaffolded explorations through paper folding, the problem posed the following:

Doing mathematics and reflecting on it:  Two of the groups explored this part of the problem for a considerable amount of time.  Meanwhile, the 5th and 6th-grade group never found time to explore it, because they were extending a previous part of the problem: "Any rectangular piece of paper can be folded (without using a ruler to measure) so that a square is formed."  These teachers had posed for themselves the question: In how many different ways can we do this?  They had found three ways and were searching for a fourth at the end of the session.

In their work, the 7th-grade teachers compared their observations as they moved point A' along the base of the rectangle:

"We can use the fact that there are parallel lines."

"And we know that vertical angles are equal." 

"I think the angles in all three triangles are the same, because all three have right angles in them, and you've got those corresponding angles.  And doesn't 'angle, angle, angle' mean they’re similar triangles?"

"What are those theorems on similarity?  I haven't thought about them for a while!"

And, in their group, the 8th- and 9th-grade teachers very quickly convinced themselves of similarity, and found themselves wondering if and when two of these triangles would be congruent, not just similar.  They made a few conjectures about this, and spent the remainder of the time exploring them.

Planning for the classroom:  In full group discussion, some of the teachers remarked that it was hard to see "three triangles" in the folded piece of paper.  They were particularly concerned about the impact this would have on their students.  This observation led to the consensus view that they would use transparent paper with their students.  

An animated discussion arose around the nature of the mathematical thinking invited by the problem.   A 6th-grade teacher sparked the discussion by asking: "If the problem is about similarity, don't the students working on it need to have learned about similar triangles?  I'd be reluctant to just drop it on kids who have never studied similar triangles."  The other 6th-grade teacher offered a different point of view: "I don't think it's necessary.  I'll be interested to see how my students think the triangles are related---in what ways they think they are 'the same.'"  As a group, the teachers resolved to try different adaptations of the problem for their respective students.

Analyzing student work.  Next meeting, teachers brought a variety of artifacts.  Some brought collections of individual student papers, one brought a brief transcript of several students' conversation about the problem, and one--an 8th-grade teacher--brought newsprint from student group presentations. 

Displayed around the room, this newsprint work prompted numerous questions and comments from the teachers.  They were particularly curious about how the teacher had set up the problem.  She told them that, believing that the problem could expand the possibilities for geometric reasoning, she rephrased the problem and told her students to go off in groups and to think about what they noticed when they folded the paper.  She encouraged them to generate questions and conjectures and to display them on newsprint.

One group of students experimented with square sheets of paper and made a conjecture about isosceles triangles.  On their newsprint, they displayed 3 folded squares of different sizes to demonstrate their conjecture:

Questions arose in a flurry from the other teachers, such as:

• What led them to investigate isosceles triangles?
• How did they know they had isosceles triangles?
• Why did they choose square sheets?