Systematizing Rhombus Properties

Systematizing Rhombus Properties - Defining a Rhombus

Systematizing Rhombus Properties - Defining a Rhombus
(Suggested Grades 8 - 10)

"... the Socratic didactician would refuse to introduce the geometrical objects by definitions, but wherever the didactic inversion prevails, deductivity starts with definitions." - Hans Freudenthal (1973:416) in Mathematics as an Educational Task.

"To me it appears a radically vicious method, certainly in geometry, if not in other subjects, to supply a child with ready-made definitions, to be subsequently memorized after being more or less carefully explained. To do this is surely to throw away deliberately one of the most valuable agents of intellectual discipline. The evolving of a workable definition by the child's own activity stimulated by appropriate questions, is both interesting and highly educational." Benchara Blandford (1908) quoted in Griffiths & Howson (1974: 216-217), Mathematics: Society and Curricula. Cambridge University Press.

The geometry learning activity below is from my book Rethinking Proof (free to download).

Worksheet & Teacher Notes
Open (and/or download) a guided worksheet and teacher notes to use together with the dynamic sketch below at: Systematizing Rhombus Properties - Defining a Rhombus Worksheet & Teacher Notes.

Prerequisites
Knowledge of the properties of a rhombus - specifically it is expected that you have already completed the following two rhombus activities:
Visually Introducing & Classifying a Rhombus (Grades 1-7)
Exploring Rhombus Properties (Suggested Grades 4 - 9)

Short Description
How would you describe what a rhombus is, over the telephone, to someone who is not yet acquainted with a rhombus?

1. Which of the following descriptions do you think you would be able to use? Circle these descriptions.
a. A rhombus is any quadrilateral with opposite sides parallel.
b. A rhombus is any quadrilateral with perpendicular diagonals.
c. A rhombus is any quadrilateral with two perpendicular axes of symmetry (each through a pair of opposite angles).
d. A rhombus is any quadrilateral with perpendicular bisecting digonals.
e. A rhombus is any quadrilateral with two pairs of adjacent sides equal.
f. A rhombus is any quadrilateral with all sides equal.
g. A rhombus is any quadrilateral one pair of adjacent sides equal, and opposite sides parallel.

One way of testing a description is to construct a figure complying to the description to see if it really gives the desired figure.

Check by Construction
2. In the dynamic sketch below you are provided with 7 different possible constructions. Press each button step by step on each of the seven pages to construct the figures. When each construction is finished, match each page with a descriptions a - g above or in the table given in the Defining a Rhombus Worksheet.
Drag each figure to see if it always remains a rhombus. (Note: Since a rhombus can be dragged into the shape of a square, we regard a square as a special rhombus.) In the table given in the accompanying Defining a Rhombus Worksheet, cross out the names of any pages that construct quadrilaterals that are not always rhombuses.

Checking Descriptions/Definitions by Construction

3. List the descriptions from a – g that you think correctly describe (define) a rhombus.
4. State the description from a–g that you personally think best describes a rhombus. Try to defend your choice with good reasons.

Challenge
5. Start from a description (definition) of your choice as your given assumption and then prove as theorems that a rhombus has each of the other properties not listed in your description (definition). You can use any new theorems that you prove in the subsequent proofs of the other properties.
6. Work through the guided proofs for a particular choice of description/definition as given in the accompanying Defining a Rhombus Worksheet.
7. Formulate at least two other alternative definitions for a rhombus, and check that you can deductively derive all the other properties of a rhombus not included in your definition. Class Discussion
A definition can be seen as an agreement among interested parties about what a specific object is. Although you have now seen it is possible to define a rhombus in many different ways, it can be very confusing if everyone is using a different definition. It is therefore now necessary to choose a common definition that will be acceptable for the whole class.
8. Have a class discussion to decide which definition of a rhombus is most convenient for you.

Notes
a) The main purpose of this activity is to introduce students to the systematization function of proof: the fact that proof is an indispensable tool in the organization of known results into a deductive system of definitions and theorems.
b) Further objectives are
• Developing students’ understanding of the nature of definitions as unproved assumptions, as well as the existence of alternative definitions.
• Engaging students in the evaluation and selection of different formal, economical definitions rather than just providing them with a single ready-made definition.
• Developing students’ ability to construct formal, economical definitions for geometrical concepts.

References
Bennett, Dan. (2012). Constructing Rhombuses, pp. 161-163. Exploring Geometry with Geometer's Sketchpad (free to download). Key Curriculum Press.
Bennett, Dan. (2012). Defining Special Quadrilaterals, pp. 143-145. Exploring Geometry with Geometer's Sketchpad (free to download). Key Curriculum Press.
De Villiers, M. (1998). To Teach Definitions in Geometry or Teach to Define?. In A. Olivier & K. Newstead (Eds), Proceedings of the Twenty-second International Conference for the Psychology of Mathematics Education: Vol. 2. (pp. 248-255). University of Stellenbosch: Stellenbosch, 12-17 July 1998.
De Villiers, M.; Govender, R. & Patterson, N. (2009). Defining in Geometry. In T. V. Craine & R. Rubenstein (Eds.). (2009). Understanding Geometry for a Changing World: NCTM's 71st Yearbook. Reston, VA: NCTM.
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
Govender, R. & De Villiers, M. (2004). A dynamic approach to quadrilateral definitions. Pythagoras, 58, June, pp. 34-45.

Other Rethinking Proof Activities
Other Rethinking Proof Activities

External Links
Van Hiele model (Wikipedia)
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)

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Michael de Villiers, created with WebSketchpad, 2 Nov 2025.

Systematizing Rhombus Properties - Defining a Rhombus (Suggested Grades 8 - 10)

Systematizing Rhombus Properties - Defining a Rhombus
(Suggested Grades 8 - 10)