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The prompt
The prompt
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 of reflection 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 about the prompt include:
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.
Does it work for any shape?
Could the shape be three-dimensional?
In the orientation phase, students often attempt to draw a diagram to illustrate the prompt. Here are two from year 7 students:
After a discussion about how the two lines are arranged, students might explore reflections of their own shapes using the templates in 'Resources'.
In the first phase of the inquiry, the teacher is advised to clarify the use of 'object' and 'image' in the prompt. In fact, the inquiry deals with the second image after the first image has become the object of the second reflection.
New learning
The inquiry is ideal for introducing the concepts of rotation and translation, which arise naturally as a way of mapping the object onto the image with one transformation.
While it is often the case that students are able to to describe rotations and translations informally (as, for example, a 'move' or 'turn') after they have carried out two reflections, the teacher should be prepared to explain the new concepts using formal terms.
Combining different transformations
A second phase of the inquiry starts when students suggest combining different pairs of transformations. Can you map the object onto the image if, for example, you rotate and then translate a shape? Questions that have arisen at this point include:
Does the order of the transformations matter? Will a rotation followed by a reflection give the same image as a reflection followed by a rotation?
Is the result the same for all rotations, regardless of whether the angle is 90o, 180o, or 270o?
Does it matter if the shape has one or more lines of symmetry?
Is it possible to map the object onto the image with different single transformations?
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.
Misconception
A common misconception occurs at this stage of the inquiry when students generalise from one case. They claim it is possible to map an object onto an image with one transformation, even though doing so relies on the shape having a line of symmetry.
In the example below, an isosceles trapezium is reflected (1 to 2) and then rotated by 180o (2 to 3). The object (1) can be mapped onto the image (3) using a single rotation of 180o (1 to 3). However, it is not possible to map the object onto the image when we remove the line of symmetry. To convince a student that the result cannot be generalised to all cases, the teacher might suggest shading in one side of the trapezium.
A line of inquiry
Students have extended the inquiry by looking at three successive transformations (see illustration below).
Structured inquiry
Structured inquiry
The structured inquiry is arranged in four phases. Each phase might last for more than one lesson and, depending on the teacher's direction, students might be working on different phases in the same lesson.
The first three phases contain regulatory cards that give students options for how to proceed through the inquiry. The teacher might ignore the cards and keep the class together.
Phase 1 Students pose questions about the prompt. They notice its properties and wonder about its meaning.
The teacher records the students' comments and refers back to them when appropriate and relevant to do so during the inquiry.
The class then reflects shapes in horizontal, vertical and diagonal lines of reflection. Students' activity ranges from reflecting a given shape once to creating their own examples involving two reflections.
Phase 2 Students learn how to describe translations and rotations using precise mathematical language. They also practise translating and rotating shapes, including by creating their own examples. With this knowledge, they can show that the prompt is always true.
Two additional lines of inquiry deepen students' understanding of the three transformations.
Phase 3 Students describe two transformations and decide if it is possible to map the object onto the image with a single transformation. They make up their own examples, describing them carefully, and recording the results.
Phase 4 Students review their learning by identifying and describing transformations. The teacher leads an evaluation of the inquiry by reviewing students' choices of activities and the habits of mind that arose.
Rahma, a year 7 student, creates examples in phase 1.
Combined transformations - structured inquiry.pptxTesting conjectures
Testing conjectures
Andrea Mendes, a teacher in west London (UK), contacted Inquiry Maths about the inquiry with her year 9 class. After visualising combined transformations, one student, Peter, made a conjecture:
If the lines of reflection go through the shape, then you cannot map the object onto the image in one transformation.
After testing the two cases when the lines of reflection are parallel and when they intersect (see illustration top), he decided that the conjecture is false. In bothe cases the single transformations that map the object onto the image are the same as those when the lines of reflection are outside the shape.
The result is the same for the two cases in the second illustration. The angle between the lines of reflection is 45o. The point of intersection is outside the object in the first case and inside in the second case.
Peter concluded that if the two lines of reflection intersect, then the angle of rotation is twice the angle between the two lines.
He finished his inquiry by returning to the conjecture, speculating that it might be true in three dimensions:
August 2024
Mixed attainment inquiry
Mixed attainment inquiry
The inquiry was carried out by a year 7 mixed attainment class in an inner-city secondary school in the UK.
Below the teacher describes each phase of the inquiry, explaining how inquiry lessons are an excellent way to cohere the class around common aims.
- Question, notice, and wonder
We teach year 7 in mixed attainment classes. The responses to the prompt from the students in my class (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 the inquiry with parallel lines. Another student wonders if the statement applies to all shapes, even irregular ones.
2. First phase of exploration
2. First phase of exploration
Those who felt confident used the templates with perpendicular and parallel lines to reflect their own shapes. A third of the class required more structure. I demonstrated on the visualiser how to use tracing paper to reflect a shape in a line. Students then practised on diagrams I provided. The first lesson ended with a student demonstrating how mapping a shape onto an image with two reflections in perpendicular lines could be carried out with a single 180o rotation. The a centre of rotation would be where the two lines intersect.
3. Extending the line of inquiry
3. Extending the line of inquiry
At the start of the next lesson I used a student’s example to co-construct and broaden knowledge of rotations. The class then had the option to rotate shapes and describe rotations on a structured sheet. More students took up the offer to practise 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.” I 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.
4. Developing knowledge
4. Developing knowledge
I 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 a single translation. I 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.
5. Creating examples systematically
5. Creating examples systematically
In the final two lessons of the inquiry, the whole class created their own examples. As students made claims about combined transformations, I encouraged others to confirm the findings and explain them to each other. When they had convinced each other, they would try to convince me. I would then write their findings on the board. Much excitement was generated as students raced to contribute.
Students’ examples
Students’ examples
Students were keen to present their examples (see below) with as much information as possible and re-submitted them to present a more convincing case.
Promoting student engagement
Promoting student engagement
Raj Vara, a teacher of mathematics in west London (UK), explains how he introduced Inquiry Maths to his department after deciding that his students should be taking the initiative.
I have been teaching for five years. Over the last couple of years, it struck me that I really needed more engagement from my students in class. I felt that they needed to take more ownership of their learning and feel free to explore maths and make mathematical discoveries for themselves. After trialling several different types of activities and resources, Inquiry Maths proved to be the most effective way to address the issue.
As a key stage coordinator in the department, I observed that other teachers were experiencing the same lack of engagement. To tackle the problem, I hosted a training session on the combined transformations inquiry.
The whole department got stuck in and enjoyed the inquiry themselves. We were excited to see the approaches others had taken. After a successful session, all the teachers were on board to implement the prompt with our year 10 classes.
In the classroom, we ran a structured inquiry (see pictures). It was astonishing to see, just within the first 10 minutes, such interesting questions being articulated and then students working collaboratively to answer them. There were lots of 'aha' moments and the whole class was engaged at a level I hadn't seen before.
November 2022
New result
New result
Kenza Hmaimou, one of the teachers in the training session, noticed that the general result for two parallel lines holds if the first shape is drawn in the centre (and not on the left). The translation to the right from shape 1 to shape 3 is twice the distance between the lines of reflection.
Classroom inquiry in year 8
Classroom inquiry in year 8
Question, notice, and wonder
Question, notice, and wonder
In the question, notice, and wonder phase of the inquiry students in year 8 mixed attainment classes responded to the prompt in the ways shown in the pictures below.
Guided inquiry
Guided inquiry
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 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."
March 2017
Inquiry findings
Inquiry findings
Students can use the results sheet to record their findings. These are the collective findings from two year 8 classes (as recorded by their teachers):
Discussion of the prompt
Discussion of the prompt
Don Steward discussed the combined transformation prompt in his presentation at the joint Maths Hubs conference at Villa Park (UK) 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 - from the particular to the general - 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.
Alternative prompts
Alternative prompts
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 more knowledge.
Daniela Vasile (a Head of Mathematics in South Island School, Hong Kong) suggested the prompt. In referring to any case, it contains a general conjecture that might be more appropriate to challenge experienced inquirers.
Paul Aniceto and Kent Nobes (grade 5 teachers from Ontario, Canada) co-planned an inquiry on translation and reflection. As a stimulus to inquiry, they used the prompt as a question to their pupils.
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