Mathematical inquiry processes: Interpret; generate more examples. Conceptual field of inquiry: Enlargements (positive and negative); other transformations.
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.
Structuring inquiry around students' questions
The illustration shows the questions and observations from a year 9 class at Haverstock School (Camden, London, UK). The students had studied rotations two years previously and had knowledge of congruence and similarity. The teacher planned to inquire into enlargement (both scale factors and centres of enlargement) through the prompt. As students made their observations, the teacher introduced the term 'enlargement' when probing the meaning of students' comments.
These are some of the lines of inquiry that could develop from the questions and observations:
Combining transformations to map the red shape onto the blue one;
Moving the position of the red shape to see how the combined transformations change;
Treating the blue shape as the object of the transformation and the red shape as the image;
Enlarging the red shape by a scale factor greater than two; and
Inquiring into the relationship between the linear and area scale factors.
The teacher structured the next few lessons around these lines of inquiry.
Inquiry lesson 1
After the class had given their initial responses to the prompt, the teacher revised the concept of rotation, requiring students to attempt rotations on coordinate axes. The lesson ended with a student (Mohammed) suggesting the class draw axes on the prompt and rotate the red shape 180o about the point (5,8).
Inquiry lesson 2
The teacher prepared a worksheet for the first part of the lesson so that students could develop greater fluency in rotating shapes. The majority of the lesson involved the teacher in introducing the concept of a centre of enlargement. After completing examples given by the teacher with a centre inside, on the perimeter and outside the shape, students generated their own examples.
Inquiry lesson 3
The lesson started with an assessment task. Could students enlarge a shape using different centres of enlargement? After that, the lesson proceeded in a similar way to lesson 2, but this time focusing on how to find a centre of enlargement given the object and its image under enlargement. Once again, the teacher encouraged students to generate their own examples.
The lesson ended when Mohammed was able to complete the combined transformation to map the red shape onto the blue one (see below).
Inquiry lesson 4
Students selected one of three questions to inquire into: Can you change the position of the red shape? (Catarina's question), Could we enlarge the shape and then rotate? (Nafisa's) and Could the scale factor be three or higher? (Hamza's). After a discussion about what the answers might entail, the teacher structured and guided individuals' inquiries as required.
Inquiry lessons 5 and 6
At the start of this pair of lessons, the students were intrigued by the teacher's suggestion that it was possible to map the red object onto the blue image with one transformation.
Students expressed some scepticism before the teacher explained about fractional and then negative scale factors. After practicing with the teacher's examples, students again generated their own. At the end of the lesson, the teacher generated further discussion by selecting some to put under the visualiser. The sixth lesson ended with students using a copy of the prompt to show the single enlargement that maps the object onto the image - scale factor -2, centre (4,9).
Inquiry lesson 7
This lesson focused on Ayoub's question about treating the blue shape as the object and the red as the image. Students worked in pairs to see if they could combine transformations and find a single transformation to map one onto the other. This led into a discussion about comparing the properties of the 'reverse' transformation to the original one.
Inquiry lesson 8
The final lesson of the inquiry considered the relationship between the linear and area scale factors. Through exploration, students derived the formula: (Length scale factor)2 = Area scale factor.
Grade 5 inquiry discussion
Saleshni Cook, a grade 5 teacher at the Beijing City International school, Beijing (China), 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 (illustrated) 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.
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.
The questions and comments from another year 10 class show students attempting to map one shape in the prompt onto the other by combining transformations. Others have noticed that the perimeter of the shape has been enlarged by a scale factor of two, while the scale factor for the area is four.