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Enlargement inquiry

This prompt has generated questions and discussions about how one of the shapes has been transformed into the other. Initially students will often try to describe a combined transformation to map the smaller shape onto the larger. How many combinations are there? Can it be mapped in two transformations? How many ways?
     
In this episode of the inquiry, students might revise the concepts of rotation, reflection and translation and how to describe them fully - or may require, by selecting the appropriate regulatory card, the teacher to instruct them. (It is advisable to have support materials available for this eventuality.) The class will also come across the need for an enlargement. Depending on the first transformation, the centre of enlargement will be different, although the scale factor is the same.
     
Ultimately, students identify (possibly under the teacher's guidance) the one enlargement that maps the object (smaller shape) onto the image (larger shape). If the smaller shape is considered as the image, then the scale factor is different. At this point, a new 
inquiry pathway can open as students seek a link, and attempt to generalise that link, between the scale factors of a negative enlargement and its inverse. 

Notes
Enlargement inquiry discussion This document contains a transcript of a conversation between Caitriona Martin and Andrew Blair, two teachers of inquiry maths, about the enlargement prompt. It raises some important issues about inquiry maths in relation to the curriculum.
Resources
Prompt sheet
Promethean flipchart    download
Smartboard notebook   download
Report sheet

Question and comment
Danny Brown joined a conversation on twitter to pose a question and make a comment that occurred to him on seeing the prompt:
(1) Under what conditions would the large shape contain four copies of the smaller shape?
(2) It makes me think of lominoes.

Danny Brown is Head of Mathematics at Greenwich Free School, London (UK). He is creating a new secondary school curriculum for maths, which can be seen at gfsmaths.com. You can follow Danny on twitter @dannytybrown.


Grade 5 inquiry discussion
Saleshni Cook used the prompt to start a discussion with her grade 5 class. The pupils posted their questions and comments on a wall in Padlet. The rich discussion encompasses a variety of topics, including perimeter, area, scale factors, ratio, and enlargement. Indeed, one pupil is moving towards combined transformations when stating that, "I think it's been doubled and turned around." Saleshni reports that the prompt "sparked an awesome discussion" with "a great vibe in the room," concluding that she was impressed by the pupils’ thinking.
   
The posts (examples on the right) show the pupils attempting to apply their current knowledge to the prompt. They also show how the teacher can intervene during an inquiry to guide the pupils down a particular pathway. With the request for an explanation about what would happen to the next case, Saleshni focusses the pupils' thinking on the relationship between the scale factors for length and area as the shape is enlarged.

Saleshni Cook
teaches grade 5 pupils at the Beijing City International school, Beijing, China. You can follow her on twitter @CookSaleshni.


   
Initial questions and observations  
These are the questions and observations of a year 10 higher ability class. Evidently, the students have knowledge of transformations, including rotations and enlargements. In the suggestions of positive scale factors and the question about the centre of enlargement lie the potential for the teacher to develop an understanding of negative scale factors. After a phase of instruction, requested by students using the regulatory cards, the students moved on to creating their own diagrams illustrating negative scale factors.
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The questions and comments from a different year 10 class show students attempting to map one shape in the prompt onto the other by combining transformationsOthers have noticed that that the perimeter of the shape has been enlarged by a scale factor of two, while the scale factor for the area is four.
  
Reporting on inquiry
At the end of an inquiry, students reported on the combined and single transformations they found to describe the mapping of one shape onto the other.