Transformation properties
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Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about how attributes of shapes change or not under transformation.
Task administration:
This task can be completed with pencil and paper or online (with auto marking displayed to students).
Level:
4
Curriculum info:
Keywords:
Description of task:
This task requires students to indicate the invariant properties of four transformations (translation, reflection, rotation, enlargement).
Curriculum Links:
This resource can be used to help to identify students' understanding of invariant properties under rotation, translation, reflection, and enlargement.
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 (11/2015) | ||
| a) |
All 5 of :
Shape; Direction; Length ; Angle; and Area
|
easy |
| b) |
All 4 of :
Shape; Length; Angle; and Area
|
moderate |
| c) |
Both of:
Direction; Angle;
Partial credit: Includes Shape with Direction; and Angle
|
difficult
difficult
|
| d) |
All 3 of:
Length; Angle; and Area
Partial credit: Includes Shape with Length; Angle ; and Area
|
very difficult
easy
|
Based on a sample of 90 Year 8 students
NOTE: The partial credit scores on Parts c) and d) show how students performed if shape is ignored. The difficulty levels of a) and b) stay the same if the students' responses for Shape are ignored.
The issue of invariance of shape is only an issue for rotation and enlargement.
Teaching and learning:
This resource looks at invariant properties of 2-dimensional objects under four transformations.
The concept of invariance of a 2-dimensional shape can be contentious, and difficult for students. We have defined that the shape of a 2-dimensional object is invariant if it can be moved freely on a 2-dimensional plane so that the object can be exactly super-imposed onto the image. This includes transformations and rotations, but not reflections, as this requires moving the object into the third dimension (i.e., flipping it). This means the shape of the first two objects are invariant, but the shape of the last two objects are not. This would be a fruitful area for a class discussion. The book Flatland: A Romance of Many Dimensions would be a potential resource for this.
Diagnostic and formative information:
| Common error | |
| c) | Students indicate that Angles vary under enlargement |
| c) and d) | Students indicate that Shape as invariant under enlargement or rotation |
Next steps:
Students indicate that Angles vary under enlargement
Get the students to visualize (picture in their heads) if they can move the original shape so matching angles have common vertices. They should the see that the angles are identical. If they have trouble visualising, get them to work with the physical shapes.
For this particular shape (d), the 'angle' is the intersection of a straight line and a curve. This could lead to a mathematical discussion. The concept of 'tangent' may arise from this!
Students indicate that Shape as invariant under enlargement or rotation
Get the students to visualize moving the original shape without lifting it off a flat page (for example Transforming traffic lights). They should find that there is no way to make the original object exactly overlay the reflected image. If the students have trouble visualising, get them to work with two physical shapes.
This could lead to a mathematical discussion, on what objects have their shape invariant under reflection. The discussion could look at how to define that a shape is invariant under transformation (e.g., "The shape can be moved however you want on a flat surface").