Logical geometry

Logical geometry

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Overview
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Connecting to the Curriculum
Marking Student Responses
Further Resources
This task is about geometric properties of quadrilaterals.
A quadrilateral is a two-dimensional shape with four straight sides. 
Five quadrilaterals are shown below. 
 
geometric-trapezium.png
Trapezium
(At least 1 pair of parallel sides)
geometric-rhombus.png
Rhombus
(2 pairs of parallel sides)
geometric-parallelogram.png
Parallelogram
(2 pairs of parallel sides,
all sides equal length)
geometric-square.png
Square
(2 pairs of parallel sides,
all sides equal length, all angles 90o)    
geometric-rectangle.png
Rectangle
(2 pairs of parallel sides, 
opposite sides equal length, all angles 90o)
 
Following are statements about quadrilaterals. One statement in each pair is true and the other is false.

Question

a)  Select the true statement below. 

    • All rectangles are trapeziums.

    • All trapeziums are rectangles.

Question

b)  Select the true statement below. 

    • All rhombuses are squares.

    • All squares are rhombuses.

Question

c)  Select the true statement below. 

    • All squares are parallelograms.

    • All parallelograms are squares.

Question

d)  Select the true statement below. 

    • All trapeziums are rhombuses.

    • All rhombuses are trapeziums.

Question

e)  Select the true statement below. 

    • All parallelograms are rectangles.

    • All rectangles are parallelograms.

Task administration: 
This task can be completed with pencil and paper or online (with auto marking displayed to students).
Levels:
4, 5
Curriculum info: 
Maths, Geometry and Measurement, Shape
Key Competencies: 
Thinking
Keywords: 
quadrilaterals, shape properties, classification
Description of task: 
Students compare different types of quadrilateral shapes and answer questions about which of a pair is a special case of the other one.
Learning Progression Frameworks
This resource can provide evidence of learning associated with
Geometric thinking, sets 4-5
Geometric thinking, set 6
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
  Y10 (05/2007)
a) All rectangles are trapeziums.   moderate
b) All squares are rhombuses.   moderate
c) All squares are parallelograms.   moderate
d) All rhombuses are trapeziums.   moderate
e) All rectangles are parallelograms.   moderate
a)-e)

3, 4, or 5 correct1

2 or more correct2

2 marks
or
1 mark

moderate

easy
Based on a representative sample of 131 students.  

NOTE:

  1. There was very little difference in the mean ability of students getting 3, 4, or all 5 questions correct in the sample.
  2. Students getting 2 correct had a higher mean ability than students getting 0 or 1 correct. The latter were probably guessing.
  3. About half of the students got each individual part of the question correct. Although this seems close to the number who would get an individual part correct by guessing, in each case students getting that part correct had higher mean abilities than those who were incorrect.