View from a different angle II

View from a different angle II

Auto-markingPencil and paperOnline interactive
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about identifying shapes that are identical to a given 3-D shape.

Question

3-d-shape-original.png
Hone had four cubes which were glued together.  The shape is shown above.
Select each picture below that shows Hone's shape from a different angle.
The shape may have been turned sideways or upside down, or rotated.
    • a)
       3-d-shape-A.png

    • b)
      3-d-shape-B.png

    • c)
      3-d-shape-C.png

    • d)
      3-d-shape-D.png

    • e)
      3-d-shape-E.png

    • f)
      3-d-shape-F.png

    • g)
      3-d-shape-G.png

    • h)
      3-d-shape-H.png

    • i)
      3-d-shape-I.png

Task administration: 
This task can be completed with pencil and paper or online (with auto-marking displayed to students).
Level:
4
Description of task: 
Students identify which shapes are identical to a given shape that have been rotated.
Learning Progression Frameworks
This resource can provide evidence of learning associated with within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
  Y8 (10/2009)

a)
b)
c)
d)
e)
f)
g)
h)
i)

True
False
True
False
False
True
True
False
False

9 correct

8 or more correct*

very easy
easy
easy
easy
very easy
very easy
easy
very easy
very easy

difficult
(full credit)
moderate
(partial credit)

Based on a representative sample of 144 Y8 students.  

answers to soma cube question.png

NOTE:
*Students who get 8 correct but get e) incorrect (i.e. they circle it) should not receive partial credit. This is because they do not distinguish that shape c) and shape e) are mirror images so only one of them can be correct.Students who got only 7 shapes correct had a mean ability no higher than students who got 6 or fewer correct, and so were not given partial credit.

Teaching and learning: 
  • This resource is about being able to visualise in three dimensions by recognising which objects are identical with a given one. The objects that are not identical to the given one are mirror images of it.
  • Students find each individual shape easy or very easy, but only a minority can get all nine shapes correct. Only those who got 8 or more correct have higher mean ability than those who got fewer correct.
Diagnostic and formative information: 
  • Shape c) was the hardest to get correct, followed by d), g), b), and e). All of these except  c) are not identical with the original shape.
  • f) was the easiest to get correct followed by a), h), and i). All of these except i) are identical with the original shape and should have been circled.

It was therefore more common for students to see mirror images as identical to the original, but less common to think that an identical shape was different.

Common response Likely errors in visualisation
Circles shapes that are not identical to the original, or Circles all shapes. Sees mirror images as identical to the original shape
Does not circle a), f) or h) Does not recognise identical shapes
This was less common than the above misconception.
Does not circle c) or
Does not circle c) and circles e)
[i.e. transposes responses for c) and e)].
Has issues with one specific shape (c)
Shape c) is the most difficult. This may be that it is hardest to see how to superimpose this particular orientation onto the original shape. A number of students with higher mean ability made these two errors.

The 2-dimensional representation of shapes c) and e) are clearly mirror images in a vertical mirror, so both being circled indicates a lack of higher level geometric visualisation. The same would be the case for the pairs a) & i), and f) & g). Ironically, this mirror image relationship between the original shape and shape b) does not make getting b) correct easier.

Next steps: 
Sees mirror images as identical to the original shape or Does not recognise identical shapes
These students most probably need to build the models with physical blocks to persuade themselves that the shapes are not all identical, but fall into two groups that are mirror images of each other. Soma cubes have shapes like this that can be used.

Soma cubes
These have two shapes like this that can be used. The two mirror-image shapes are shown here. Students will find that they cannot be superimposed on each other by twisting or moving them. One spirals in a clockwise direction, and the other anticlockwise.

"Right screw tetracube".
"Left screw tetracube".

Like the left hand

Like the right hand

They are similar to the diagrams of the two hands, although the terminology is a little inconsistent. Notice how easy it is to see the relationship between the shapes when the mirror line is vertical, and the shapes are side by side. Mirror-images of the other shapes in the Soma cube are not distinct from each other, but can be superimposed on each other by rotation.