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Exploring Properties using GSP
Mathematical Goals: Teachers will be able to
· Create definitions from examples and nonexamples of geometric figures.
· Understand the difference between partitional and hierarchical definitions.
Pedagogical Goals: Teachers will be able to
· Create activities that allow students to generate their own definitions.
Technological Goals: Teachers will be able to use a technological tool to
· Use common features of dynamic geometry environments (DGEs) including dragging, measuring, and calculating to explore relationships and look for properties of geometric figures.
· Consider how technology can assist students in communicating with each other about geometric ideas.
Mathematical Practices:
· Make sense of problems and persevere in solving them.
· Construct viable arguments and critique the reasoning of others.
· Use appropriate tools strategically.
· Attend to precision.
· Look for and make use of structure.
Length of session: 90 minutes
Materials needed: Computer with Geometer’s Sketchpad, Exploring Properties Participant Handout, ExploringProperties.gsp file, ExploringPropertiesSolution.gsp file
Overview:
In this session participants will use the Geometer’s Sketchpad (GSP) to explore properties of two unknown figures. Participants will then be asked to develop definitions for the unknown figures, which will lead to a discussion about defining in geometry. Participants will be able to apply what they’ve experienced in exploring unknown figures to discuss how GSP could be used to have students explore figures like quadrilaterals.
Estimated # of Minutes / Activity45 minutes / Exploring Properties of Unknown Figures with GSP
· Participants will use GSP to explore the properties of the figures in the ExploringProperties.gsp file. This file consists of five pentagons with specific properties for each that the participants will need to identify. The participants will need to use the measuring tools and dragging features of GSP to identify the properties of the oddly named figures and complete the chart below. Participants should also attempt to create a definition for each pentagon. Note: Here the chart is filled in with the appropriate properties for each figure. On the participant handout, there will be a blank chart for participants to complete.
Shape / Properties / Definition
Glubber / Two pairs of parallel sides
One pair of consecutive congruent sides
One pair of congruent angles / Answers will vary
Slubber / One pair of parallel sides / Answers will vary
Blubber / Two pairs of parallel sides
One pair of congruent sides
Three consecutive right angles / Answers will vary
Flubber / Two pairs of parallel sides
Three consecutive right angles / Answers will vary
Plubber / Two pairs of parallel sides
One pair of congruent angles / Answers will vary
· While the participants are working, they can consider the following questions. Once they are finished, have participants share what they found and discuss the questions as a group. Essentially the purpose of this activity is to have teachers begin to consider how GSP might be used to allow students to explore properties of figures. By providing figures the participants are unfamiliar with, there is a higher chance they will see the value in utilizing GSP for such an activity.
· Questions to consider:
1. Reflect on your thinking processes when generating your definitions. What properties did you consider important or not important? Did you create definitions that did not work? How did you know if a definition “worked” or not? Answers will vary.
2. Compare your definitions with the definitions created by others. Consider similarities, differences, strengths, and weaknesses in the different definitions. Answers will vary.
3. Describe the benefits and drawbacks of allowing students to interact with a constructed figure in a DGE to generate their own definitions versus a teacher providing a formal definition to students. One benefit of allowing students to generate their own definitions includes the possibility of students making connections and linking this new figure to their prior knowledge. More than likely, different ideas and properties will be brought to light. This can be both a benefit and drawback at times. Care must be taken to ensure that ideas are properly pulled together and that the definition encompasses all necessary properties that will distinguish examples of the figure from the non-examples.
4. When students are generating their own definitions, describe how a teacher can bring these different ideas together so that the class is eventually working from a single definition. Mathematical discourse happens when the teacher facilitates a class discussion in which students come to agreement on a definition, procedure, etc. One way that a teacher might do this is to look across the definitions and identify what is common in all of the definitions. The teacher might also highlight what is different among the definitions. As a class, a new definition can be created from those generated by students that the class will all agree to use.
45 minutes / Partitional vs. Hierarchical definitions
· Share with participants the difference between partitional and hierarchical definitions.
· Partitional definitions: considers concepts disjointly from one another; exclusive definition
· Hierarchical definitions: defined as a subset of a more general concept; inclusive definition
· Questions to consider:
1. Consider the following definitions of a trapezoid:
Definition 1: A trapezoid is a quadrilateral with exactly one pair of parallel sides.
Definition 2: A trapezoid is a quadrilateral with at least one pair of parallel sides.
How would you categorize each definition, partitional or hierarchical? Why? Definition 1 is a partitional definition; this definition does not facilitate understanding between relationships of figures. For example, using definition 1, a parallelogram can never be considered a trapezoid. Definition 2 is a hierarchical definition; this definition allows for connections between figures such as considering a parallelogram a trapezoid.
2. Why might students prefer partitional definitions to hierarchical definitions? Students may prefer partitional definitions over hierarchical definitions because they can more easily define a figure by visual inspection, they do not have to consider connections and relationships to other figures, and there is only one correct answer to each question, all of which makes learning through memorization easier.
3. When students are generating their own definitions, describe how a teacher can bring these different ideas together so the class is eventually working from a single definition. One way that a teacher might do this is to look across the definitions and identify what is common in all of the definitions. The teacher might also highlight what is different among the definitions. As a class, a new definition can be created from those generated by students that the class will all agree to use.
4. Look at your properties for the unknown pentagons. Determine which properties are “inherited” from other properties. Use this information to create a hierarchical classification of these pentagons.
5. Describe how you could apply this same approach you experienced to have students investigate properties of quadrilaterals and generate the accompanying sketch. This same approach could be used to have students explore the properties of quadrilaterals. A teacher could have a preconstructed sketch with various quadrilaterals (parallelogram, rhombus, square, kite, trapezoid, rectangle, cyclic quadrilateral) and have students complete a chart similar to the one used here. Then, the teacher could facilitate discussions regarding the development of students’ definitions, which are likely partitional, and move them towards developing a hierarchical understanding of quadrilaterals by having them develop a hierarchical classification.
Exploring properties of figures using GSP
Participant Handout
Puzzling Pentagons
Using the ExploringProperties.gsp file, complete the chart below to identify the properties of and define the oddly named pentagons.
Shape / Properties / DefinitionGlubber
Slubber
Blubber
Flubber
Plubber
Questions to consider:
1. Reflect on your thinking processes when generating your definitions. What properties did you consider important or not important? Did you create definitions that did not work? How did you know if a definition “worked” or not?
2. Compare your definitions with the definitions created by others. Consider similarities, differences, strengths, and weaknesses in the different definitions.
3. Describe the benefits and drawbacks of allowing students to interact with a constructed figure in a DGE to generate their own definitions versus a teacher providing a formal definition to students.
4. When students are generating their own definitions, describe how a teacher can bring these different ideas together so that the class is eventually working from a single definition.
Partitional vs. Hierarchical Definitions
· Partitional definitions: considers concepts disjointly from one another; exclusive definition
· Hierarchical definitions: defined as a subset of a more general concept; inclusive definition
Questions to consider:
1. Consider the following definitions of a trapezoid:
Definition 1: A trapezoid is a quadrilateral with exactly one pair of parallel sides.
Definition 2: A trapezoid is a quadrilateral with at least one pair of parallel sides.
How would you categorize each definition, partitional or hierarchical? Why?
2. Why might students prefer partitional definitions to hierarchical definitions?
3. When students are generating their own definitions, describe how a teacher can bring these different ideas together so the class is eventually working from a single definition.
4. Look at your properties for the unknown pentagons. Determine which properties are “inherited” from other properties. Use this information to create a hierarchical classification of these pentagons.
5. Describe how you could apply this same approach you experienced to have students investigate properties of quadrilaterals and generate the accompanying sketch.
Adapted from: Hollebrands, K. F., & Lee, H. S. (2012). Exploring quadrilaterals. In Preparing to teach mathematics with technology: An integrated approach to geometry (1-22). Dubuque: Kendall Hunt.