Visualisation in 2 and 3 Dimensions: 3.5
Supporting materials
Indicator of Progress
Success depends on students being able to rotate mental images of objects.
This is an important part of spatial visualisation ability. It can be thought of in two ways:
- being able to imagine the object rotating, and
- being able to imagine the view if you look at the object from the other side.
The ability to mentally rotate objects develops throughout the primary and junior secondary years.
It is generally more difficult than mentally imaging the effect of a slide or a flip.
Spatial visualisation is an important everyday and workplace skill, and it contributes to general mathematical ability. Read more here about spatial visualisation.
Illustration 1: Levels of difficulty of mental transformations
Look at this picture. It is easy to imagine the Lego tower shifted to the right. This is a mental translation.
It is not too hard to imagine what the Lego tower would look like in a mirror. The horizontal block would be on the right, the yellow tower would be on the left. With good visual imagery you can see the entire picture turned around in a holistic way. Of course, students need plenty of opportunities to develop this skill.
It is generally a little harder to imagine what the Lego tower would be like if you moved around to one side a little way, and even harder to imagine if you moved around 90°.
The hardest mental rotations, beyond this level, combine rotations about different axes, such as looking from near the back and underneath.
Illustration 2
Examples of the types of tasks that would be illustrative of visualisation in 2D and 3D skills, aligned from the Mathematics Online Interview:
- Question 55 (a – c) - Identifying triangles and explain features
- Question 57 (a – c) - Visualise the hidden shape (more than one shape)
- Question 58 - Triads – use ‘prototype’ or ‘properties’ strategy with cards 1, 2 and 3
Teaching Strategies
There are many wonderful learning and teaching activities for improving spatial visualisation. Experience with real objects is essential, but it is not enough. To learn from experience, students also need structured learning activities that will cause them to reflect on what they do, and to learn (usually from each other) some general strategies for dealing with the problem. Reflection is encouraged by writing about solution methods and discussing them. Teachers also need to ensure that students see a variety of successful strategies used by others.
In summary, we need three ingredients: experience - reflection - strategies.
All the activities below address mental rotation around one axis. They all have the same format. Students have to put into sequence a series of pictures of one object taken from different angles. This involves mental rotation about only one axis (vertical axis through the object). The difficulty of all activities can be controlled by showing all or only some of the photos. The photos are provided as PowerPoint presentations.
Activity 1 Lego tower: basic version is appropriate for students working towards the Level 4 Standard, and is also a good introduction to the other activities.
Activity 2 Lego tower (seven photos) uses more photos of the same tower. This makes it more challenging.
Activity 3 Lego vehicle, Activity 4 Stone statue, Activity 5 Triangular box and Activity 6 Billiard Ball are more difficult. The most difficult is the billiard ball, where there are no clues from the shape of the object.
Teachers will also note that students will tend to have less difficulty identifying photos when they are rotated through only a small angle from a known photograph.
Activity 1: Lego Tower - Basic version
This activity can be offered to students with many different degrees of difficulty, by taking photos of objects with more or fewer distinguishing features, or by showing fewer photographs (e.g. only photos P, Q, R and S in the first example).
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Lego tower: basic version
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Some photographs were taken of this Lego tower from different angles. Here are the photographs. The light was shining in the direction of the arrow. Indicate the position on the circle from which each photograph was taken. The positions need not be very accurate, but you need to get the order correct around the circle. |
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Q
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R
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S
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Show this diagram to the students.
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Here is the answer.
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There are many clues in this photo that help locate the positions from which the photos were taken. Photo Q has the light shining on it most directly. Then the position of the yellow tower is helpful. Students will offer many suggestions in a discussion. Click here to get a PowerPoint presentation (PPT - 179Kb) of this activity including answers. |
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Activity 2: Lego tower (seven photos)
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Seven photographs were taken of this Lego tower from different angles. Here is one photograph. The task is to find the angle from which each photograph was taken. All the photographs are here in a PowerPoint presentation (PPT - 277Kb) ready for use, along with the answer. There are many possible points of discussion and observations that students will make to help solve the problem. For example: look at the position of the yellow tower - is it on the right or left of the photograph; is a lot of the shadow being photographed, is the blue base towards or away from the camera? It is important that students realise that these are properties that change with orientation, so they are clues for solving this problem. |
 PPhoto C |
Activity 3: Lego vehicle
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Five photographs were taken of this Lego vehicle with a man on top, from different angles. Here is one photograph. The task is to find the angle from which each photograph was taken. All the photographs are here in a PowerPoint presentation (PPT - 200Kb) ready for use, along with the answer. There are many possible points of discussion and observations that students will make to help solve the problem. For example: look at the position of the man, where is his shadow and the shadow of his arm? It is important that students realise that these are properties that change with orientation, so they are clues for solving this problem. |
 PPhoto D |
Activity 4: Stone statue
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Seven photographs were taken of this stone statue from New Caledonia, from different angles. Here is one photograph. The task is to find the angle from which each photograph was taken. All the photographs are here in a PowerPoint presentation (PPT - 245Kb) ready for use, along with the answer. There are many possible points of discussion and observations which students will make to help solve the problem. For example: is the camera nearest the statue's face or back of head; is the face looking left or right? |
 Photo E |
Activity 5: Triangular box
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Six photographs were taken of this triangular box from different angles. Here is one photograph. The task is to find the angle from which each photograph was taken. All the photographs are here in a PowerPoint presentation (PPT - 235Kb) ready for use, along with the answer. There are many possible points of discussion and observations that students will make to help solve the problem. For example: is the camera nearest the shadow or away from it? Is there only one shadow or more than one? If you looked at the box from the top, what shape is the shadow? |
 PPhoto B |
Activity 6: Billiard Ball
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Seven photographs were taken of a billiard ball near a strong light, from different angles. Here is one photograph. The task is to find the angle from which each photograph was taken. All the photographs are here in a PowerPoint presentation (PPT - 236Kb) ready for use, along with the answer. This is a challenging problem, because there are so few clues. There are a few observations that can help solve the problem. Where is the shadow and how big is it? Can you use the two points of reflection on the billiard ball to assist? |
 PPhoto A |
References
Clements, D.H. & Battista, M.T. (1992) Geometry and Spatial Reasoning. In D. Grouws (ed.) Handbook of Research on Mathematics Teaching and Learning. (pp. 420 - 464) New York: Macmillan.
Donaldson, M. (1978) Children’s Minds. Glasgow : Collins.
Gardner, H. (1983) Frames of Mind: The Theory of Multiple Intelligences. Fontana.
Further Resources
The following resource contains sections that may be useful when designing learning experiences:
Digilearn object *
Building site – students look down on some building towers (a plan view). They build a street-level view of the buildings from a given perspective: front, side or back. They move on to view buildings from a corner angle or build a side view.
(https://www.eduweb.vic.gov.au/dlr/_layouts/dlr/Details.aspx?ID=3273)
Photo hunt [Flash Player version] – students explore visual perspectives of solids such as cylinders, cones and cuboids. They match a 2D photo of a group of 3D objects taken from a different viewpoint. They identify the relative positions of the solids by comparing 2D outlines and colours. Students rotate a base grid until the view matches the original photo.
(https://www.eduweb.vic.gov.au/dlr/_layouts/dlr/Details.aspx?ID=3306)
* Note that Digilearn is a secure site; DEECD login required.