Problem:

I set out to examine the properties of many different quadrilaterals and discover the relationships between them.  The purpose of the summaries is to present these relationships in a manner that is clear and easy to understand.  For example, answering the question, "Is a square also a rectangle, or is a rectangle also a square," can be confusing.  I set out to discover such relationships and sort them out for the reader.
 

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Part 1

Discovery 1

Summary 1

Part 2

Sketches

Summary 2

Part 3

Summary 3

Part 4

Summary 4

Implications for instruction

Discussion:

Part 1: Determining Properties of Quadrilaterals
 

First, I used The Geometer's Sketchpad to draw a sketch of each quadrilateral.  I used only the minimum properties to draw each sketch.  For example, I knew that a rectangle could have 4 sides of equal length, but it is not a requirement to be a rectangle so I did not include that property when I made my sketch.  Then I added measurements to identify the properties of each quadrilateral.  The following link shows the sketches I made of each, including the measurements, and gives a brief explanation of the properties that were found.
 

Next,  I created the chart in the following link to sort out these minimum characteristics required for each quadrilateral.  The purpose of this chart is to serve as a guide in the next part when I examine relationships between the quadrilaterals.
 

For this part, I again used The Geometer's Sketchpad and created sketches of each quadrilateral.  Instead of just concentrating on the measurements, I manipulated each to see if they could create other quadrilaterals without losing their minimum properties.  By "manipulating," I mean I moved around points and lines to create different angles, lengths and positions as allowed by the stipulations of each shape.  In this way, I was able to determine what quadrilaterals where able to be produced by each and how often.  By examining the sketches and using the properties outlined in the previous chart (go to chart), I was able to make conclusions about quadrilateral relationships.  The following links show:  1.  The general sketch of the quadrilateral being discussed.  2.  Other sketches I was able to create by manipulating the original sketch and keeping the minimum properties in tact.  3.  Summaries explaining what the manipulations show.

Discovery 2:  Quadrilateral Sketches
Discovery 3:  Parallelogram Sketches
Discovery 4:  Rectangle Sketches
Discovery 5:  Square Sketches
Discovery 6:  Rhombus Sketches
Discovery 7:  Trapezoid Sketches
Discovery 8:  Isosceles Sketches

In order to understand the information gathered from the sketches, I created a chart summarizing the findings.  This chart summarizes the information by clearly showing what and how often different quadrilaterals can be produced from each.
 

Part 3:  Other Analyses
 

When analyzing the information gathered by the sketches, I often considered is a quadrilateral could have dual classifications.  For example, one might wonder, "Is a square also a parallelogram and if so, is it so all of the time?"  Using the information I learned in the sketches,  I then created the following chart showing this relationship.
 
 

Note:  The Sketches summary chart and the chart showing other possible classifications are the exact same with their columns and rows inverted.  In other words, if you were to read one chart "column-to-row," it would read the same entries as the other chart read "row-to-column."  This infers a relationship and can be observed from one chart.  For example, if you look as the sketches chart (go to it), you can draw from that the following:

             "A rectangle can produce all squares."

Now, turning that around by inverting the column and row, you can read it as,

             "A square is always (all) a rectangle."

True?  Of course.  Every square has all of the minimum properties required to be a rectangle and has no contradicting properties.  So, every square is, at the same time, a rectangle.  So dual classification does happen.  Is it possible for a quadrilateral to be classified as more than 2 different quadrilaterals?  Yes, the multiple classifications chart shows this.  The square has the most classifications....... every square is also a rhombus, parallelogram, trapezoid, isosceles trapezoid, and rectangle.  Why is this?  Because a square has every minimum property of all the quadrilaterals (see properties chart).

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Part 1

Discovery 1

Summary 1

Part 2

Sketches

Summary 2

Part 3

Summary 3

Part 4

Summary 4

Implications for instruction

Consider the following statement:  A square is a special case of a rectangle.
 

Since every rectangle is not a square but a every square is a rectangle, it follows that a square is a special case of a rectangle.  Specifically, a square is the special case of a rectangle such that the rectangle has 4 equal length sides in addition to it's minimum properties.  Other such cases are discussed in the quadrilateral sketches (go to it).  This idea of special cases may be understood best through a picture. To summarize all of the information gained throughout this investigation, I have created another flow chart to illustrate the relationships between quadrilaterals.
 

Implications for Instruction:
 

1.  When to use such an activity in the classroom:  In the course of a geometry class, sorting out different quadrilaterals for students may be confusing and difficult for their grasp.  Doing such an activity after introducing the different types of quadrilaterals would be beneficial to sorting out the properties of each and how they are related to eachother.

2.  How to use this activity in the classroom:  Guides, directions, groups, inclass or at home?  This activity would be best used as a follow up to a general informative lesson on quadrilaterals.  All properties do not need to be covered beforehand because this activity allows for investigation and discovery of their own.  It would be best started inside class (computer parts) and the analyzing and arranging information into a clear manner would be better as an at-home assignment.  Because this is a very time consuming project, a teacher might consider splitting up the assignment.  For example, one quadrilateral per group and each group could be responsible for the following:

• What properties does your quadrilateral posses?
• What other quadrilaterals can you create from your given quadrilateral without losing it's original properties?
• Put your information in a chart/ presentation for the class.

3.  What can you hope that students will get from this activity?  Hopefully, this activity will push students to explore for themselves the properties of quadrilaterals and be able to express what they learn in a clear and consise way.  Students should gain a sense of being able to discover things for themself and not relying on teachers to tell them everything.  Also, a strong understanding of quadrilaterals can be achieved to help futher explorations in geometry.
 
 
 

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Part 1

Discovery 1

Summary 1

Part 2

Sketches

Summary 2

Part 3

Summary 3

Part 4

Summary 4

Implications for instruction