Investigation: hands-on exploration of mathematics - what can you discover? Where do your discoveries lead? What connections are there?
Interaction: collaboration with colleagues and with students - communicating, questioning, listening, gaining knowledge and new ideas
Illumination: discovering principles, connections and applications across the curriculum and in "real life", communicating what you have learned.
The study of quadrilaterals occurs in most geometry courses somewhere in the first semester, usually after a study of triangles and their properties, and after a study of parallel lines and the parallel line theorems.
At the beginning of the study of quadrilaterals, my students do a project that I call "The Quads Project". We spend 3 days in the computer lab on this project. My students are quite familiar with The Geometer's Sketchpad at this point, and can construct figures accurately. If you do not have Sketchpad, the students can use compass and straightedge constructions. I ask the students to work in groups of 3 or 4, using no more than 2 computers per group, with space between the groups if possible. (I form the groups in various ways throughout the year: sometimes they choose their own, at other times I devise some sort of random grouping using my class list or handing out playing cards at random and grouping "all the 10's" etc.)
They are given a printed worksheet which lists the quadrilaterals: parallelogram, rhombus, rectangle, square, trapezoid, and isosceles trapezoid, and the book's definitions of each figure. (You will be able to get a copy of this worksheet using a link below) They have had no other introduction to these figures, and probably do not know their properties. They are asked to construct each quadrilateral so that it follows the definition. For example, to construct a parallelogram, one would follow these steps: 1) Construct a segment. 2) Construct a second segment, "attached" to the previous segment at one endpoint. 3) Select a point at the end of one segment, hold down the Shift Key, then select the other segment (segment not points) and use the Construct menu to construct a line parallel to the segment. 4) Repeat this to complete the parallelogram. 5) Test by dragging each of the four points: the figure should change in size and shape, but remain a parallelogram.
The students then construct each of the rest of the "quads", each in a separate GSP file, each using the Construct menu and the definition of the quadrilateral. Once the students have constructed the quads, with much discussion and a bit of advice from each other and from the teacher, the exploration begins. For example, what appears to be true about the parallelogram? Are the opposite sides parallel? Of course they are - that was the definition, and that was how we constructed it! Are the opposite sides congruent? They do appear to be, even when we drag any vertex any amount. Are we sure? Well, we can measure to check. (Select a segment - not the points - by clicking on it, then use the Measure menu. This should verify that the opposite sides are, indeed, congruent. Are we sure yet? Well, it is very convincing.
Some of the properties are fairly obvious, particularly after dragging a vertex or two. But some of the other properties are not quite so obvious. Do the diagonals of a parallelogram bisect the angles?
Do the diagonals of a parallelogram bisect the angles for any parallelogram? Or is this true only if the parallelogram is a rhombus (or square)?
But (and this is the heart of the matter) in geometry we do not want to accept what we see without verification. And, in mathematics, verification means analysis and proof. Their assignment in this collaborative project is to discover, test, and prove as many properties of each of the quadrilaterals as possible. They have a checklist of possible properties to keep track in the worksheet that is part of the assignment.
You will find a printable copy of the worksheet at the following link:
The students work in their collaborative groups, on and off the computers, for two or three (or more) class periods. They spend some of this time writing two-column proofs to prove their conjectures. Discussion, questioning and brainstorming with their group is a key part of this time, and a valuable activity. In the end, each group turns in one set of papers, the proofs of all the properties that they have discovered.
In my writing-intensive geometry class, the students turn in portfolios at the end of each quarter. They are asked to choose 4 or 5 examples of the work they have done in the quarter - their best work, or the work from which they learned the most, or the work that they feel represents their greatest achievement. Many of the students choose to include The Quads Project in their portfolios, saying that this is a project in which they feel they learned the most. One student, Sean, said "I included this project in my portfolio because I felt like a real mathematician. I felt like I had actually discovered these things myself, like they were my theorems." (He goes on to say, wryly "Well, I knew that they had already been found, and turned out to be right in our book! But I still felt like I had discovered them myself, and I was really proud of that!"
Other students said "... it really makes it easier to remember the theorems when you explore them and figure them out yourselves." And "When I forget if something is true or not, I just picture dragging a vertex and recall what I saw, and also how we argued about how to prove it."
Compare this experience to reading a list of theorems, with (or sometimes without) proofs or any other verification, in a textbook. The value of hands-on investigation, particularly combined with collaboration, is clear. This project combines investigation (constructing and experimenting with GSP), imagination, (the students asking themselves and each other how to construct, trying different methods, then working together to think up ways to prove what they discover), interaction (the students working with each other, hands-on interaction with the software), and illumination (clarifying and communicating their discoveries and their own proofs).
This is one of a number of investigative projects you will see in I-MATH. In each project, I would like you, as a teacher, to put yourself in the place of a student and imagine what the student's experience would be when he or she is working on the project. Then, perhaps you will want to try some of these projects with your own students, perhaps right away, or maybe in the future when you have had more experience with the process. And hopefully, you will want to create some interactive projects of your own, in geometry or in other mathematics classes. There are wonderful applications of The Geometer's Sketchpad for Algebra 2, particularly in Trigonometry and in the Conics. There are, of course, other software, media and methods for creating investigative, exploratory experiences for students in all areas of mathematics.
This process is very much in line with the National Council of Teachers of Mathematics Standards; in particular the process of investigation and discovery.
The NCTM Standards for Geometry say: "to be learned using concrete models, drawings, and dynamic software. With appropriate activities and tools and with teacher support, students can make and explore conjectures about geometry and reason carefully about geometric ideas."
This project also ties in to The NCTM Reasoning and Proof Standard:
"Systematic reasoning is a defining feature of mathematics. Exploring, justifying, and using mathematical conjectures are common to all content areas and, with different levels of rigor, all grade levels. Through the use of reasoning, students learn that mathematics makes sense. Reasoning and proof must be a consistent part of student's mathematical experiences in pre kindergarten through grade 12."
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