EDITED Resources. The resources on this page have been aligned with the revised K-12 mathematics TEKS. Necessary updates to the resources are in progress and will be completed Fall 2006. These revised TEKS were adopted by the Texas State Board of Education in 2005–06, with full implementation scheduled for 2006–07.

Clarifying Lessons

Geometry: Classifying Quadrilaterals

activity under revision

Lesson Overview

Students use construction technology to identify and find relationships between the attributes of various quadrilaterals.

Mathematics Overview

Students make and test conjectures about the distinguishing properties of parallelograms, rectangles, rhombuses, squares, kites, and trapezoids.

Set-up (to set the stage and motivate the students to participate)

  1. Read the following description of a square to the students, and have them draw what you have described.

    "My quadrilateral has opposite sides congruent."

  2. Have students compare their drawings with each other and with your square. Have students discuss what all their drawings have in common (they are all parallelograms) and what additional information is necessary to guarantee that they all would draw a square (e.g., all four sides congruent and one right angle). (G.2.A)
  3. Have students use construction technology to explore the properties of each quadrilateral listed in the chart on the worksheet, in order to determine which properties are defining characteristics for which quadrilaterals. (G.2.A, B)
  4. Have students then use the identified properties to develop statements that they believe are true about each quadrilateral, based on their observations. (G.2.B, G.3.D)

Teacher Notes (to personalize the lesson for your classroom)

Guiding Questions (to engage students in mathematical thinking during the lesson)

  • How can you use the properties shown in your chart to make a statement that you believe is true about all parallelograms? (G.2.B, G.3.D)
  • How can you use the properties shown in your chart to make a statement that you believe is true about all rhombuses (including squares)? (G.2.B, G.3.D)
  • How can you use the properties shown in your chart to make a statement that you believe is true about rhombuses, but not about all parallelograms? (G.2.B, G.3.D)
  • How can you use the properties shown in your chart to make a statement that you believe is true about only rhombuses? (G.2.B, G.3.D)
  • Why would this statement be called a definition for a rhombus? (G.1.A) (because it describes a category to which all rhombuses and only rhombuses (including square rhombuses) belong).
  • Use the properties shown in your chart to complete the following statements with "always," "sometimes," or "never." If you choose to answer "sometimes," identify in which cases the property holds. (G.2.B, G.3.D)
    1. A square is __________ a rhombus. (always)
    2. The diagonals of a parallelogram __________ bisect the angles of the parallelogram. (sometimes, when it is a rhombus)
    3. A quadrilateral with one pair of sides congruent and one pair parallel is __________ a parallelogram. (sometimes, it could also be an isosceles trapezoid)
    4. The diagonals of a rhombus are __________ congruent. (sometimes, when it is a square)
    5. A rectangle __________ has consecutive sides congruent. (sometimes, when it is a square)
    6. A rectangle __________ has perpendicular diagonals. (sometimes, when it is a square)
    7. The diagonals of a rhombus __________ bisect each other. (always)
    8. The diagonals of a parallelogram are __________ perpendicular bisectors of each other. (sometimes, when it is a square)

Teacher Notes (to personalize the lesson for your classroom)

Summary Questions (to direct students' attention to the key mathematics in the lesson)

  • How are the properties of rhombuses like the properties of parallelograms in general?
  • How are they different?
  • How is the definition of a rhombus as "a quadrilateral with four congruent sides" displayed in these properties? (G.1.A) (A rhombus is the first category of quadrilateral that has this property, so it is a property that defines a rhombus. The square also has this property, so it is a type of rhombus.)
  • Comparing the properties of the other quadrilaterals and the rhombus, how else might you define a rhombus? (G.1.A, G.2.B) (e.g., as a parallelogram with adjacent sides congruent, or as a quadrilateral with diagonals that are perpendicular bisectors of each other).
  • Which quadrilaterals have exactly one line of symmetry? Exactly two? Exactly three? Exactly four? How are these lines of symmetry related to their properties? (G.1.A, G.2.B, G.5.A)
  • Imagine placing each quadrilateral on a coordinate grid. What patterns do you think you would find in the coordinates of the vertex points in each quadrilateral? Why? (G.1.A; G.2.B; G.5.A; G.7.A, B)

Teacher Notes (to personalize the lesson for your classroom)

Assessment Task(s) (to identify the mathematics students have learned in the lesson)

Make a "family tree" or identify categories in a Venn Diagram to show the relationships between the quadrilaterals you have been investigating.

On the Venn Diagram or "family tree" created, list the properties specific to each of the special quadrilaterals.

A quadrilateral has the following three vertices: A(0, 0), B(3, 4), and C(8, 4). Find the fourth coordinate that will make the quadrilateral a(n):

rhombus (5, 0)

isosceles trapezoid (11, 0)

kite (many answers, for ex. (4, 2) or (6, -2))

Explain how you used the properties of the given quadrilateral to identify the missing vertex.

Teacher Notes (to personalize the lesson for your classroom)

Extension(s) (to lead students to connect the mathematics learned to other situations, both within and outside the classroom)

Have students select one of their statements they have made from exploring the properties of the quadrilaterals with the construction technology and use definitions and properties of parallelism and congruent triangles to prove it.

Have students take each of the quadrilaterals named below, join, in order, the midpoints of the sides and describe the special kind of quadrilateral they get each time.

  1. rhombus (joining the midpoints of the sides forms a rectangle)
  2. rectangle (joining the midpoints of the sides forms a rhombus)
  3. isosceles trapezoid (joining the midpoints of the sides forms a rhombus)
  4. random trapezoid (joining the midpoints of the sides forms a parallelogram)
  5. quadrilateral with no congruent sides (joining the midpoints of the sides forms a parallelogram)
  6. kite (joining the midpoints of the sides forms a rectangle)

Teacher Notes (to personalize the lesson for your classroom)