Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software.  Specify a sequence of transformations that will carry a given figure onto another.




This is objective is all about performing the transformation. This is a continuation from G.CO.4 where we defined the motions and now we begin to actually perform the transformations. We also extend single transformations to composite transformations of two or more motions in the plane. We investigate the relationships found by reflecting over two parallel lines and reflecting over two intersecting lines. These are not the only patterns to be found; students should be able to given a pre-image and a final image and determine one possible sequence of transformations that maps one onto the other. This emphasizes the properties and definitions of transformations while preparing students for congruence.







(1) The student will be able to perform a reflection, a rotation, and a translation.


(2) The student will be able to construct a rotation, a reflection and a translation.


(3) The student will be able to perform a sequence of transformations.


(4) The student will be able to determine the sequence of transformations performed between a given pre-image and image.


(5) The student will be able to describe which single transformation is the result of two reflections over parallel lines.


(6) The student will be able to describe which single transformation is the result of two reflections over intersecting lines.


(7) The student will be able to identify a transformation by its coordinate rule and then apply it to transform the shape.


(8) The student will demonstrate how some composite transformations are not commutative.




If we can determine a sequence of isometric transformations that maps the pre-image exactly onto the image then they are congruent. Performing an isometric transformation or a sequence of them prepare us for congruence. Congruence gains a new definition than we have used in the past.


Two figures are CONGRUENT if and only if one can be obtained from the other by one or a sequence of rigid motions.






Reflection seems to come easy to students because of its relationship to symmetry. The coordinate rules make sense and are often easy for students to memorize but it seems that the most difficult area for students with reflection is working with non-vertical or non-horizontal lines of reflection. Reflection over the y = x line causes them great trouble. Many of them reflect the shape as if it was a vertical line. This needs extra time and practice.

Rotations are by far the HARDEST of the isometric

transformations. Rotations take more practice and more time - so slow down!! The key to success here is the use of patty paper. When students pin the patty paper at the center of rotation and then begin to spin the paper - rotations come alive. The rules for rotation are also much more complex - it is hard for many students to understand how a y value can be in the x coordinate: for example a rotation of positive 90 degrees generates a rule of (x,y) ---> (-y,x). This is confusing for many students and also hard to remember.

One more big area of difficulty is the positive and negative directions of rotations. Students want clockwise to be positive but it isn’t. Explain how positive rotations are counter-clockwise due to the coordinate grid.


Finally, double reflections over intersecting lines causes’ confusion because the direction (not the size) of the rotation is sometimes hard to determine.  The truth is that rotations can have multiple answers because you could rotate positively to get there or negatively… also that there are an infinite co-terminal rotational values that could work.  This definitely causes some pause and need for discussion.




Students will bring the definitions from G.CO.4 and there past experiences with flip, turn, and slide to this objective. The work in symmetry will also help them visualize reflections and rotations.



As mentioned numerous times, the sequence of isometric transformations is a way to establish congruence. The idea is that if I can map my figure exactly onto another figure by only doing isometric transformations, then those two figures have to be identical or congruent.




MY REFLECTIONS (over line l)

There is a lot in this objective!! (1) Performing single transformations; (2) Developing the rules for those transformations and then (3) Performing sequences of transformations. This takes a number of days. Of the single transformations rotation was the most difficult and translations were the easiest. I would give more time to teaching rotations and less time for translations. I used a lot of hands on teaching techniques such as patty paper which definitely helped.

Concerning sequences of transformations, I found myself a bit rushed and it showed in their understanding. We analyzed double reflections over parallel lines and over intersecting lines. They got the big concepts but struggled with notation and the order of the transformations. I needed more examples to practice with.  Also when asked what the result of double reflection over parallel lines that were horizontal or vertical they froze…. I need to provide some other examples…

I had also hoped for more time to experiment with problems where the pre-image and image are provided and students need to construct a possible sequence of transformations to map one onto the other.