Triangles and rectangles

Triangles and rectangles

Pencil and paperOnline interactive
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about the area of two triangles and two rectangles.

Question

rectangles-and-triangles.png
 
The two rectangles above have the same area.
 
a)  Which of these statements is true about  triangle X and triangle Y?

    • The area of triangle X is larger than the area of triangle Y.

    • The area of triangle X is the same as the area of triangle Y.

    • The area of triangle X is smaller than the area of triangle Y.

    • The area of triangle X may be larger than, equal to, or smaller than the area of triangle Y.

Explain why you decided on your answer
Task administration: 
This task can be completed with pen and paper or online (with SOME auto marking).
Level:
5
Curriculum info: 
Maths, Geometry and Measurement, Shape
Key Competencies: 
Using language, symbols, and texts
Keywords: 
triangles, rectangles, area
Description of task: 
Students compare the area of two different triangles.
Curriculum Links: 
Key competencies
This resource involves communicating why two triangles have the same area. This relates to the Key Competency: Using language, symbols and text.
For more information see https://nzcurriculum.tki.org.nz/Key-competencies
 
Learning Progression Frameworks
This resource can provide evidence of learning associated with
Geometric thinking, set 6
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
    Y10 (04/2016)
a)
"The area of triangle X is the same as the area of triangle Y." [Option 2]
Explanation that is based on two key points:

  1. Each triangle is half the area of the rectangle that it is enclosed by,  and
  2. Both rectangles have the same area.

Examples:

  • Because the area of a triangle in a rectangle is half the area of the rectangle so if both rectangles are the same size then both triangles will be too. 
  • Because both triangles are half the area of the rectangles. Formula for area of triangle is half base x height.
  • Since the rectangles have the same area, the triangles would as well because it's just halfing the amount.
  • Because the area of the triangle will always be half of the area of the rectangle in this situation and both of these rectangles have the same area.
moderate
very difficult
Based on an online sample of 137 Y10 students.
Teaching and learning: 
This resource is about following a line of reasoning using properties of shapes rather than using given dimensions of shapes to compare their area .
Diagnostic and formative information: 
Common incomplete or incorrect response Next Steps
Incomplete explanation given
Correctly identifies that both triangles have the same area, but just states that the rectangles have the same area.
Examples:
  • The statement at the top says "The two rectangles have the same area".
  • Because the two rectangles have the same area.
  • Because it said the two rectangles have the same area.
 
Students need to include the information that a triangle is half the area of the enclosing rectangle (or parallelogram) as well as that the rectangles have the same area.
Some may have known this. These students need to learn to give a complete argument.
Thinks that there is insuffiicient information
Selects "It could be smaller, the same, or larger" [Option 4] and sees this as insufficient information.
Examples:
  • Because all it said was that the 2 rectangles had the same area. It never said about the area of the triangles.
  • The only piece of information they give you is the area of the rectangles, not the triangles.
  • Each rectangle does have the same area but they are not drawn the same.
 
Students may need to explore the relationship between the area of a triangle and the enclosing rectangle/parallelogram (½ b.h compared with b.h). They then need to integrate this information across both shapes.
Students want actual dimensions of shapes
Student selects "It could be smaller, the same, or larger" [Option 4] or "They are the same" [Option 2], but thinks that actual dimensions (i.e., base and height) need to be given.
Examples:
  • I can only estimate the width, length and area of each triangle ...
  • We don't know the dimensions of X or Y.
  • Since I haven't been given any measurements I just went on what I saw and assumed they were the same.
 
Students need be challenged to use other information than dimensions to make inferences. They may also need to explore the relationship between the area of a triangle and of the enclosing rectangle/parallelogram (½ b.h compared with b.h). They may then retry the question.
Using visual estimation
Students base their answer on a visual comparison.
Example:
  • The area of both X and Y looks the same.
  • Because its hard to tell on the different areas.
 
Students may need to explore multiple examples of  the area of rectangles and triangles, that lead to the general rules for their areas and then to see the relationship between them.