Reflections and coordinates inquiry
Mathematical inquiry processes: Generate examples; test particular cases; analyse structure. Conceptual field of inquiry: Reflections and other transformations; coordinates.
Michael Joseph, a teacher of mathematics at Haverstock School (Camden, London, UK), devised the prompt after his year 9 class struggled with a 'challenge task' he gave them (below).
Michael explains the origins of the prompt:
When we were using y = x and y = -x as lines of reflection, the students' strategy of 'counting squares across then down on the other side of the mirror line' worked. The challenge task raised a lot of questions. I decided to go back to the beginning and encourage the students to generalise using y = x. Supported by GeoGebra software, the class came up with very interesting observations and questions. (See the picture below from a student’s exercise book.)
Translating the students' statements into mathematical language, this is what they came up with: If coordinate (a,b) is reflected on the line y = x ±k, then the coordinate of the image is (b ± k, a ± k) but this is true only if the gradient is 1.The class then went onto explore what happens if the point is reflected in the line y = mx + c where m ≠ 1.
For me, the prompt, which was inspired by the students' exploration, is a classic example of having less to it and more in it. It can lead to
Exploring the relationship between the coordinates of the object and image under reflection.
Drawing straight line graphs.
Changing the subject of an equation.
Finding the inverse function.
Finding the mid-point between two coordinates and working out the equation of a line that reflects an object onto its image.
Exploring the relationship between the coordinates of the object and image under other transformations, such as rotation.
Exploring the prompt
These are the questions and observations of a year 9 mixed attainment class. The teacher had structured the next part of the inquiry to explore the case of y = x by reflecting quadrilaterals in the line of reflection. Students created their own examples and generalised. The student who asked whether the point (b,a) goes back to (a,b) when reflected in the line y = x ended the first lesson of the inquiry by verifying that that is indeed the case. The second lesson addressed the question, "Could we use different lines of reflection?" Under the direction of the the teacher, the lesson started with an inquiry into the case of y = -x. The students' generalisations are shown in the table below.
The second lesson continued with individual inquiries into different lines with students selecting which lines of reflection they would use.
The teacher collated the results at the end of the lesson and, during a class discussion, the class co-constructed the generalisation (bottom of the right-hand column below).
In the last few seconds of the lesson, a student made this conjecture: If you swap the x and the y in the equation of the line of reflections (x = y + n), then (a,b) becomes (b + n,a - n).
At the start of the third lesson, the class considered the conjecture. The teacher showed that the equation (x = y + n) can be rearranged to give y = x - n. Students chose a value of n and verified that the conjecture is true for each particular case. The teacher moved the inquiry on to look at the change in the coordinates under rotation, as suggested by one student's question at the start of the inquiry. The centre of rotation was fixed at (0,0). The generalisations are in the table below.
The following questions are some that the class did not have time to explore:
What happens if the centre of rotation is not (0,0)?
How do the coordinates change under enlargement?
Could we prove any of the generalisations?