Triangle or not?

Triangle or not?

Pencil and paperOnline interactive
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about identifying whether some shapes are triangles and what makes them triangles.

Question

Three children are talking about whether shapes are triangles or not.
 
tom-boy-face-100.png possible triangle
a)  Tom says this is not a triangle.
     What do you think?

    • It's a triangle.

    • It's not a triangle.

Explain why you think this.

Question

sarai-smaller.png possible triangle
b)  Sarai says this is not a triangle.
     What do you think?

    • It's a triangle.

    • It's not a triangle.

Explain why you think this.

Question

boy-face-flattop-smaller.png  possible triangle
c)  Tyler says this is not a triangle.
     What do you think?

    • It's a triangle.

    • It's not a triangle.

Explain why you think this.

Question

d)  Select all the shapes below that are triangles.

    • possible triangle

    • possible triangle

    • possible triangle

    • possible triangle

    • possible triangle

    • possible triangle

    • possible triangle

    •  
      triangle-8b.png

    • possible triangle

Question 1Change answer

possible triangles

e)  Describe how you know a shape is a triangle.

Task administration: 
This task can be completed with pencil and paper or online (with some auto-marking).
Level:
3
Curriculum info: 
Maths, Geometry and Measurement, Shape
Key Competencies: 
Thinking
Keywords: 
2-dimensional shapes, shape properties, triangles
Description of task: 
Student select shapes that are similar but not triangles and explain what makes them triangles or not.
Curriculum Links: 
This resource can help to identify students' understanding of shape properties with justification.
Learning Progression Frameworks
This resource can provide evidence of learning associated with
Geometric thinking, sets 3-4
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
  Y6 (11/2015)
a)
Not a triangle
Sufficient explanation involves identifying the specific feature of the shape that makes it not a triangle, e.g., notes that the top corner has continuing lines that cross over or identifies that there are not only 3 corners.
Examples of responses
Students correctly noted that the shape was not a triangle and responded with reasons (and identified properties):
  • Because triangles have 3 corners 3 sides and can be tall or short but all sides must join together, plus no sides poking out (3)
  • Because it crosses above/over the top (1)
  • Because it has extra lines/sides (1)
  • Because the lines at the top are going pass each other instead of touching each other (1)
  • It is  not  closed properly (1)
  • It is the shape it just has two lines carrying it on (1).
Some of the students noted the crossed lines at the top, but stated that it was still a triangle because it had a triangle (inside) or still had 3 corners or 3 sides.
 easy
b)
It is a triangle
Sufficient explanation involves identifying the specific features of the shape that makes it a triangle, e.g., it has 3 corners and 3 (straight) sides (and is closed).
Examples of responses
Students correctly noted that the shape was a triangle and responded with reasons (and identified properties):
  • Because, like I said before, a triangle has 3 sides, 3 corners and if you look at it upside down it it just a lopsided triangle (3)
  • It has 3 sides and 3 corners (2)
  • Even though it is long it still has 3 corners and is still in the shape of a triangle (2)
  • Because it has 3 points/corners like a normal triangle (1)
  • Because it is a shape with 3 sides (1)
Students mostly noted that a triangle has 3 sides, and fewer identified the 3 corners/points.
 very easy
c)
Not a triangle
Sufficient explanation involves identifying the specific feature of the shape that makes it not a triangle, e.g., the shape is not closed, or does not have 3 corners.
Examples of responses from question c)
Students correctly noted that the shape was not a triangle and responded with reasons (and identified properties):
  • Because a triangle has to have 3 corners all joined up/joined together (1)
  • Because it doesn't have 3 points/corners/suppose to have 3 corners not 2 (1) 
  • Because it is not connected/got an open space, three sides DON'T connect up (1)
  • It has a gap so it is not a triangle/It is broken (1).
 very easy
d) triangle-not-answers.png All 4 correct - moderate
e)
Explanation involves some combination of :
  • closed (2-D) shape,
  • 3 corners/points/angles, and
  • 3 (straight) sides.
And does not include any incorrect additional rules such as sides must be equal, etc
 
Examples of explanations (with number of identified properties)
  • Because all triangles must have 3 sides 3 corners and all must join together (3)
  • If it has connected lines, 3 lines, no gaps and no lines off the top (3)
  • If it has three corners and is triangular (1)
  • Because a triangle has three sides all connected to gether (1) 
NOTE: At this level it might be expected that students provide 1-2 points of a definition.
  
NOTE: A shape only needs to lack one property to disprove that it is a triangle.  However, to show that a shape is a triangle it should have all properties accounted for (e.g., 3 (straight) sides, 3 corners, closed shape).
Teaching and learning: 
This assessment resource encourages students to share their understanding about the properties of shapes. In particular, the shape properties that make up a triangle: a closed plane figure with 3 straight sides and three corners/angles. Over-using one visual representation of a triangle can encourage students to develop a more limited sense of what is (and is not) a triangle. These shapes have been deliberately selected to break open the visual protoype that students may have formed and to explore the actual properties of triangles.
 
 
Diagnostic and formative information: 
Correct answer and insufficient reasons
Some of the responses below are interesting because of the additional ideas they reveal about students' understandings about triangles and shapes in general.
Sometimes students may correctly identify whether a shape is or is not a triangle, but indicate some developing concepts, such as "triangles have to have 3 sides the same", "open shapes aren't shapes they are a just a set of lines", and "triangles have a proper way to look and tend to not be on angles". 
 
Question a)
It is not a triangle ... (correct answer, but insufficient reason given)
  • Because it is uneven/one side is longer than the other side
  • I don't think it is, because all the sides are not the same
  • Because the lines are not even and if you look at the top there's a little cross when there's not meant to be
Question b)
Is a triangle ... (correct answer, but insufficient reason given)
  • Because it has 3 eqal size
  • Because the edges are the same length 
  • Because it is even
  • If it was straight up it would look like a triangle 
  • It has three corners like a normal triangle
  • Just because it is on an angle does not mean it isn't a triangle*
*this response suggests the student is aware of what constitutes a triangle.
 
Question c)
It is not a triangle ... (correct answer, but insufficient reason given)
  • It's not even a shape as the sides aren't join
  • This is not a triangle because, there is lines not connecting and its the wrong way.
  • This isn't a triangle because it's missing on corner making it just a set of lines.
 
Incorrect answer and reasons
Some students made an incorrect answer with reasons:
 
Question a) Is a triangle ...
  • Because it has 3 points and triangles have 3 points 
  • Because it has a triangle in it, even though it has two extra lines it still makes a triangle.
  • Because it is in the shape of a triangle
  • Because it still has 3 sides and 3 corners and triangles can be short and squashed or tall and skinny.
  • Because the main shape is a triangle and only two short lines aren't
  • It has 3 edges and corners
  • It has three corners like a triangle
  • It is the shape of triangle even though it has parts stick out.
Question b)  It is not a triangle ...
  • Because it is not in the proper shape of one
  • Because it's got one side longer than the other side
  • Because two lines are straight and one line is short and the other two lines is longer.
  • Its at an angle. And pointing down
  • This is not a triangle because, the lines are going down, left. And also because triangle has one line across on the bottom and two lines connecting going up.
Next steps: 
Not correctly identifying a triangle
Students who identified that question a) or question c) were triangles may need to explore the definition of a triangle. Students could be asked to work in groups to come up with a definition of  a triangle. The example "triangles" and "almost triangles" from this assessment resource could be used to test their conjectures. The students' responses also included in the Marking and Diagnostic information could be shared and discussed. Discussion will likely flow to open shapes, and on to what defines a corner and a side. These basic ideas might need further clarification. Students could be asked to define a corner (and/or side) and develop examples of what a corner (and/or side) is and is not (and why not). Once they have more awareness about sides and corners they will be more able to determine the important properties for a definition. Ultimately, it is important to get students noticing and considering the properties of shapes, and using this to argue whether a shape is or is not: a triangle, or whatever shape.   
 
Incomplete definition
Students who gave brief and incomplete reasoning for question e) could explore the full definition of a triangle. Using counter examples for any property that is not accounted for in their definition, e.g., if they don't include corners in the definition show a shape with 3 sides, but 2 corners (the letter zed) or an open almost triangle and ask them what else might need to be added to the definition.
 
Further teacher, peer modelling, or sharing of sufficient definitions may also help supplement students' understanding of what is needed in a definition. A definition for triangle could be
  • A plane figure with three straight sides and three angles.
  • A triangle is a closed figure with three straight sides.
  • The plane figure formed by connecting three points in a straight line by straight line segments
  • A three-sided polygon.
For further information about shape properties see the Geometric Thinking Concept Map.