The desks in my classroom are arranged in groups of four. The students enter the room and seat themselves in Groups. For each group, there is one piece of paper on the desktop. I ask the groups to brainstorm, thinking all the way back, from fifth grade to their present grade: What are the geometric concepts that you have already learned? I ask that the group list their ideas on the paper.
When the students appear to be running out of ideas, I pull the class together and I go around the room, asking each group to tell me one concept on their list. I record these ideas on a whiteboard and repeat this process until we have exhausted their lists. As I record their ideas, I try to group and consolidate related topics (for example, I write "types of triangles" instead of the specific names of triangles).
I keep this list on the whiteboard throughout the lesson.
Not long ago I was introduced to an activity in which students were asked to investigate how many different shapes can be made by grouping four isosceles triangles. The sides of the triangles can only be matched with sides of equal length, and the shapes must be distinctly different (not reflections or rotations of each other). Examples of this are shown here.
I decided to extend the isosceles right triangle activity to encompass an even wider variety of geometric concepts, and included the concepts of perimeter and area.
I divide my students into groups of four and distribute a large number of isosceles triangles that I had cut out of different colored paper to each group. (I have found that approximately 70 triangles for each group of four is a sufficient number. The use of contrasting colors seems to help some students in making comparisons of the shapes, and also helps to make the task more visually appealing, I think.) I ask the students to work on the task of creating different shapes by taping four triangles together, and explain to them the only stipulations, that adjoining sides must have the same length and that their shapes must be distinctly different. This simple task engages the students, and necessitates that all individuals in a group work and communicate, comparing and differentiating their shapes.
It is possible to create 14 different shapes. When I was first introduced to this activity, I was working with adults and we were clearly focused on finding that number. I have found, however, that my students are not really concerned with the number at all. I am guessing that, as new ninth graders in the opening days of school, they are focused more on working together and communicating. This is one of the benefits of this task – it serves as a their introduction to MP3: Construct viable arguments and critique the reasoning of others.
In this stage of the task, I simply stand back and listen. I have heard students use the terms reflected, flipped, and rotated when they compared their shapes, confirming to me that one of the benefits of this activity is that it helps students recall their prior experiences with transformations. The inclusion of this task in my opening lesson seems highly appropriate to me, as one of the hallmarks of the Common Core Geometry curriculum is that Geometry is taught “from the perspective of geometric transformations.”
Resources: Isosceles Triangles Template, Paper Triangles, tape
When the students appear to have reached the stage where they feel that they have found all the shapes they’re going to find, I ask the groups to categorize their shapes. I deliberately offer neither suggestions nor examples; I simply ask that they divide their shapes into groups and explain that the criteria for grouping are up to them.
Again, I stand back and watch and listen. When it appears that a group has decided on their criteria, I provide that group with a large piece of poster paper and colored markers. (The poster paper that I used was divided into 1 inch squares. This feature becomes important in the next section of the lesson.) I ask them to create a poster by taping their shapes onto the poster paper and explaining the feature or features that they chose to highlight.
When it appears that all the groups have finished with their posters, I ask that the students, still arranged in their groups, take a “gallery walk” and look at each others’ work. We then as a class discuss the all of the different categories chosen. As we discuss the categories, I emphasize vocabulary, asking the students to explain the meanings of the words they’ve chosen to use.
Similar to the first stage of this lesson, this stage provides lots of opportunity for communication, justification, and critique of others’ reasoning. It also facilitates the recall of much prior knowledge – I have seen students divide their shapes into categories by number of sides (using appropriate vocabulary), by convex and concave, by regular and not regular, by “common” and not common geometric shapes, and by number of hypotenuses, just to name a few of the possibilities. Their discussions are rich with good math vocabulary and help to bring to the forefront much of their previous knowledge of Geometry.
In the last few minutes of class, I wrap up our group discussion and set the stage for the next class meeting. I explain that in the next lesson we will be calculating the actual dimensions our shapes, investigating what relationships, if any, might exist between the different polygons. I write a question on the board for them to think about: How are all of your polygons the same, and how are they different?