Student and Teacher Thinking: Vignettes

Vignette 2: Reasoning about Properties & Area with Tangrams

Nine middle-grades teachers worked through a series of transformation and dissection problems that encouraged thinking about properties of shapes. Grouped in pairs and triads according to grade level, teachers built shapes out of tangram pieces and then performed transformations and dissections required to change one shape into another. The problem pushed teachers to think about the properties of the shapes they were dissecting and constructing. After working through several tangram problems, the following dissection problem was given:

Without measuring, find a way to cut this parallelogram (final page of this packet contains a copy for you to cut) into pieces you can rearrange to form a rectangle. You are free to make any number and any type of cuts you would like and do not have to create tangram shaped pieces with your cuts. You should use all the pieces from the original shape and put them together like a puzzle.

a) Describe where you decided to cut and how you rearranged the pieces (use transformation language like “translated (slid), rotated (turned) or reflected (flipped)).

b) Explain how you know you ended up with a rectangle. (Think about the properties of the original parallelogram and the cut(s) you made.)

c) Will your method allow you to transform any parallelogram into a rectangle? Explain.

 

Doing mathematics and reflecting on it : Teachers worked on the math in small groups and then came back together to discuss their approaches to solving various problems. Teachers found several ways to dissect and transform the parallelogram from the dissection problem, but a common response was to cut a right triangle by extending an altitude from either the upper-left or lower-right vertex. The resulting triangle could then be translated, left or right, respectively, to complete the rectangle.

 

Before long, several teachers noticed, “Actually, any altitude will work!”

 

A pair of 6th grade teachers in the group pushed further: “After realizing any altitude would work, we wondered if there were any other ways to dissect the parallelogram to construct a rectangle.” They discovered that they could dissect the parallelogram into two large right triangles, a trapezoid and a small right triangle. Rearranging the shapes led to a rectangle with different dimensions than the one obtained after dropping an altitude. After convincing themselves that this did indeed work, they wondered how to characterize the parallelograms for which this would be possible.

Teachers repeatedly found themselves thinking about the properties of parallelograms and rectangles in the course of this problem. They thought a lot about right angles and how you know you have parallel lines.

“There are different ways to think about determining parallel lines. You can search for equal alternate interior angles. You could focus on the fact that parallel lines are always the same distance apart.”

“Two lines that are each perpendicular to another line are also parallel.”

One teacher wondered, “In order to make these convincing arguments, how essential is the knowledge of properties?”

Planning for the classroom : After discussing how they solved the problems, teachers separated into their grade-level pairs and triads again with the goal of adapting one or more of the problems for their students.

A pair of 6 th grade teachers thought they might present a mini-lesson on transformations before asking students to work on some of the tangram transformation problems. Another teacher mentioned that her students did not have any background in transformations so wanted to stay clear of the terms rotating, translating and reflecting . Her group modified several of the problems such that students were asked to build figures with tangram shapes and describe the resulting figures' properties. Transforming shapes was not a central part of their activity. A group of 7th-grade teachers decided not to focus on properties and instead used transformation and dissections as a context to talk about area and perimeter. This group produced the following modification of the parallelogram dissection problem:

1. Without measuring, find a way to cut this parallelogram (final page of this packet contains copies for you to cut) into pieces you can rearrange to form a rectangle.

Describe where you decided to cut and how you rearranged the pieces (use transformation language like “translated (slid), rotated (turned) or reflected (flipped)).

2. Can you cut parallelogram anywhere else and still make a rectangle?

3. How could you find the area of the rectangles you made?

4. How do these areas compare to the area of the parallelogram?

5. Is there a way to find the area of the parallelogram without cutting it and making a rectangle?

6. How does the perimeter of the parallelogram compare to the perimeter of the rectangle?

Analyzing student work : The next time the teachers met they brought student work from their respective problem adaptations. The adaptation emphasizing area and perimeter produced a particularly interesting example of student work. A representation of that work appears below:

 

6. How does the perimeter of the parallelogram compare to the perimeter of the rectangle?

The perimeter of a parallelogram is a little bit bigger than the perimeter of a rectangle.

   

Above, you can see if we were to extend the parallelogram line out, then it would be a bit taller than the original height.

 

 

After discussing a list of potential geometric habits of mind, or geometric ways of thinking, teachers hypothesized about the possible forms of geometric thinking that may have come into play in this answer. The act of rotating the left side length and creating a 90 degree angle in the parallelogram suggested to many teachers that this student was using visualization skills to aid his/her problem solving. However, this was not the only type of thinking that teachers recognized.

 

“I'm struggling to follow the reasoning but it seems like this student was blending deduction with experimentation,” said one teacher referring to another potential habit of mind.

 

“(He/She) kept the side length the same and moved its position to compare,” commented another teacher, suggesting that this student had the notion of invariant in mind--seeing what changes, what stays the same as the side rotated.

 

Proceeding through other examples of student work, the teachers discussed the types of geometric thinking they recognized in the pieces of student work. On several occasions, teachers saw evidence for a number of habits of mind in a single student response. Besides acknowledging the evidence in this of the complexity that can characterize geometric thinking, the group agreed that the rich pieces of student work they run across are good places to test their opinions of what habits of mind are important to productive geometric thinking.