Combined transformations inquiry
Mathematical inquiry processes: Generate examples; test particular cases; analyse structure. Conceptual field of inquiry: Reflections, rotations and translations; combined transformations.
The statement in the prompt is always true. If the lines are perpendicular, then the two reflections can be represented by a 180o rotation about the point at which the lines intersect. If they are parallel, the two reflections can be represented by a translation. Students' initial questions and observations 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.
Could the shape be three-dimensional?
Does it work for any shape?
Should the shape have a line of symmetry?
In the orientation phase, students often attempt to draw a diagram to illustrate the prompt.
After a discussion about how the two lines are arranged, students might explore reflections of their own shapes using the templates below. Depending on prior learning, the teacher could decide (or respond to students' requests) to explain how the combined reflections can be represented by one transformation. This leads naturally into the concepts of rotation (for reflections in perpendicular lines) and translation (for reflections in parallel lines).
A second stage of the inquiry starts when students suggest combining different pairs of the three transformations. Questions that have arisen at this point include:
Does the order of the transformations matter?
Will you be able to use the same transformation for a 90o or 270o rotation 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?
Groups of students or the whole class might generate a list of the possible combinations and then divide up the list to explore. The findings can feed into a class record of the inquiry.
Students have extended the inquiry by looking at three transformations (see illustration below). Including an enlargement as one of the transformations also provides greater challenge. The class might decide if it is possible to represent an enlargement with a positive scale factor and a rotation by a single enlargement using a negative scale factor.
Mixed attainment inquiry
The inquiry was carried out by a year 7 mixed attainment class in a state secondary school in the UK. Andrew Blair spoke to the teacher who explained how inquiry lessons are an excellent way to cohere the class around common aims.
Questioning, noticing and wondering
Each year 7 class in mathematics contains students from across the range of prior attainment in the cohort. Their responses to the prompt (above) show different levels of mathematical sophistication. The majority are trying to find meaning in the statement, including a link to reflection in physics. A pair of students speculate about the arrangement of the two lines and another suggests a starting point for the inquiry. Another student wonders if the statement applies to all shapes, even irregular ones.
First phase of exploration
Those who felt confident used the templates with perpendicular and parallel lines (see 'Resources' below) to reflect their own shapes. A third of the class required more structure. The teacher demonstrated how to use tracing paper to reflect a shape in a line before students practised on diagrams provided for them. The lesson ended with one student demonstrating how mapping a shape onto an image with two reflections in perpendicular lines could be carried out with a single 180o rotation using a centre of rotation where the two lines intersect.
Extending the line of inquiry
The next lesson started with the teacher using the student’s example to co-construct and broaden knowledge of rotations. Once again, students had the option to rotate shapes and describe rotations on a structured sheet. More students took up the offer this time. However, half the class continued on the main line of inquiry by creating examples with two reflections in parallel lines. They noticed that the object and image looked the same “just in a different place.” The teacher suggested varying the distance between the parallel lines to see how far the shape moved to the right. At the end of the lesson students were able to generalise, claiming that the shape moved twice the distance between the lines. The teacher explained why this is so to conclude the lesson.
Developing knowledge of translations
The teacher started the third lesson by inviting the class to discuss two of the examples with parallel lines. What was the same and what was different about the examples? The task was designed to lead the class into translations, which very few students had met before. This time students could choose between two sheets, one using words to describe the translation and the other using column vectors. Some students began to extend the idea of two transformations by carrying out consecutive translations. They noticed that mapping the object onto the image could be accomplished with one single translation. The teacher suggested they explore their word descriptions or column vectors to see if they could come up with a rule connecting the two translations with the single translation.
Creating examples systematically
The final two lessons of the inquiry saw the whole class in creating their own examples. As students made claims about combined transformations, the teacher encouraged others to confirm the findings and explain them to each other. When they had convinced each other, they would try to convince the teacher who would then write their findings on the board. Much excitement was generated as students raced to contribute.
Classroom inquiry in year 8
Questioning and noticing
These are the initial observations and questions from two mixed-attainment year 8 classes.
Students went on to explore other combinations of 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 show some of the results of the exploration phase.
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 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. Clare comments, "It was interesting reading the prompt sheets after the lesson as not everything came up in the discussion."
Discussion of the prompt
Particular to general
Don Steward discussed the combined transformation prompt in his presentation at the joint Maths Hubs conference at Villa Park in June 2017. In his presentation, Don explained the process he follows for designing mathematical tasks. Starting with an exam question, which focuses on a particular case, he explores how it could be generalised
By following the same process in the classroom, students appreciate the general mathematical structure underlying a particular question. Don showed how the reasoning that was required to solve a question on an Edexcel GCSE paper (paper 2, June 2017) can be developed through the combined transformation inquiry. You can see the section of Don’s presentation related to the inquiry here. It starts with the examination question and then considers initial pathways the inquiry might take.
Three alternative prompts
(1) Transformation 1 followed by transformation 2, gives the same image as transformation 2 followed by transformation 1.
Dan Pearcy (the Head of Mathematics at The International School of Lausanne) used this prompt at end of a unit on transformations. In being more general than the combined transformations prompt, it assumes knowledge of more transformations. The original prompt is accessible to students who have come across only reflections. The teacher can introduce new knowledge about rotations and translations in order to confirm the prompt is true for specific cases of combined reflections.
(2) Any composition of two transformations can be replaced by a single one.
Daniela Vasile - a Head of Mathematics in Hong Kong - suggested another prompt: 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.
(3) If you translate a symmetric trapezoid in any direction, then you can use a different transformation to get to that spot.
Paul Aniceto and Kent Nobes (grade 5 teachers from Ontario, Canada) co-planned an inquiry on translation and reflection. They posed this question to their 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.