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Combined transformations inquiry

If you reflect a shape twice in two different lines of reflection, then you can map the object onto the image with one transformation.
The statement in the prompt is always true. If, for example, the lines are perpendicular, then the reflections can be represented by a 180o rotation with a centre where the lines intersect. Students' initial questions have included:
    • What do 'map', 'object', and 'image' mean?
    • How many (and what) transformations are there?
    • How would you arrange the lines of symmetry?
    • Can you do more than two reflections?
    • Could we use any shape?
    • You can only get back to the object by doing two 'reverse' reflections.
    • Which way would the line of symmetry be facing if you could do the two reflections with one reflection only?
    • Could the shape be three-dimensional?
    • Does it work for any shape?
In the orientation phase, students, with surprising frequency, present a diagram like the one on the right to illustrate the prompt.
The inquiry has proceeded with a discussion of how the two lines might be arranged (and the templates below have proved useful). After that, students might suggest trying different combinations of transformations, which leads to listing all the pairs involving reflection, rotation and translation.
Questions that have arise
n at this point include:
  • Does the order of the transformations matter?
  • Will you be able to use the same transformation for a 90o or 270rotation combined with a translation or reflection as you do for a rotation of 180o?
  • How is the result different if the shape has one or more lines of reflection? (Shall we exclude shapes with lines of symmetry?)
  • Is it possible to do the combined transformation in more than one way?
This inquiry has worked well with all year groups in secondary school. Extra challenge can be developed by including an enlargement as one of the transformations: 
  • Is it possible to map the object onto the image of an enlargement combined with another transformation using a single enlargement that has a negative scale factor?
In this inquiry, students often decide to work in pairs or groups with each one inquiring into specific combinations. Their results can feed into a document of the class's findings (see the examples below).

These are the outcomes of inquiries into the prompt by year 8 all-attainment classes:
Prompt sheet
Templates Pairs of perpendicular and parallel lines for two reflections.
Examples sheet
Results sheet

Discussing alternative prompts 
Paul Aniceto and Kent Nobes (grade 5 teachers from Ontario, Canada) co-planned an inquiry on translation and reflection and then started a discussion on twitter. They posed this question to the pupils as a stimulus to inquiry: If you translate a symmetric trapezoid in any direction, then can you use a different transformation to get to that spot? An alternative prompt would be to present this as a statement, which might lead to students generating their own questions and conjecturing about the truth of the statement: If you translate a symmetric trapezoid in any direction, then you can use a different transformation to get to that spot.
Daniela Vasile - a Head of Mathematics in Hong Kong - suggested another prompt during the discussion: Any composition of two transformations can be replaced by a single one. This suggestion is very interesting because it proposes a conjecture about the general case. Inquiry Maths prompts tend to isolate a particular case for two reasons. Firstly, students can access the prompt without being overloaded by many different possibilities. Secondly, the inductive movement from examples to a generalisation can be carried out by the students themselves. However, a general statement could be appropriate to challenge older students or experienced inquirers.

Paul Aniceto writes a highly-recommended blog on his inquiry classroom. You can follow him on twitter @paul_aniceto. Daniela Vasile is also on twitter @daniela8128.
Guided inquiry in year 8 
Clare Gribben used the prompt with her year 8 class at Bedford Girls' School (Bedford, UK)It was the students' first inquiry and Clare reports that "they struggled at the start and needed lots of guidance." The pictures above show the initial responses of two students. In the top picture, the student has impressively distinguished between (and colour coded) three types of responses: questions, ideas and 'diagrams of what I think we might be doing'. Clare comments, "It was interesting reading the prompt sheets after the lesson as not everything came up in the discussion."

You can follow Clare, a teacher of mathematics, on twitter @CKmaths. Read about the levels of Inquiry Maths (structured, guided and open) here.
Classroom inquiry
These are the initial observations and questions from two mixed-attainment year 7 classes:
Many students went on to explore other combined transformations. As they inquired, students decided whether they needed instruction and structured practice in order to carry out reflections, translations and rotations confidently. The sheets below show some of the results of the exploration phase.