**Structuring inquiry around students' questions**

The illustration above 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:

(1) Combining transformations to map the red shape onto the blue one;

(2) Moving the position of the red shape to see how the combined transformations change;

(3) Treating the blue shape as the object of the transformation and the red shape as the image;

(4) Enlarging the red shape by a scale factor greater than two; and

(5) 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 180^{o} about the point (5,8).

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**Inquiry lesson 2** The teacher prepared an extended prompt **sheet**** **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 instructed the students 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.

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**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.