Becoming More Familiar with the Rules for Transformations
Lesson 2 of 6
Objective: SWBAT perform eight standard transformations fluently and identify these transformations when examining preimages and images.
Today's Do Now is intended to help reinforce the students' understanding of the basic rules for transformations. By using repeated reasoning, we have moved from looking at what happens to specific points to developing general rules (MP 8). The activity is also designed to activate the students' prior knowledge of the equations of lines. We will use this knowledge when we begin to reflect points and figures across non-standard lines of reflection. Idisplay the Do Now on my SMART board, and have the students write their answers on mini-whiteboards. It can also be run off as paper copies.
I have found that the students tend to look at me blankly, in particular, when I ask them to tell me about the line y=x. They easily memorize the rule for reflecting across y=x, but they tend not to really understand what this means. Over and over again, throughout this unit, I ask the students to tell me what the equation y=x represents and to give me examples of points on this line.
I also require the students to describe lines B and C in detail. So often, when asked to describe a horizontal or a vertical line, my students tell me that it is a "straight" line! The words horizontal and vertical need to become meaningful descriptors of commonly used linear relationships.
Going Over the Homework
Next we go over the homework entitled Worksheet on Transformations. I ask multiple students for their answers to each question, and we take the time to listen to different students' reasoning whenever the students disagree on an answer (MP 3).
I give the students the chance to practice using transformations on the mini-whiteboards. I begin by asking them to perform transformations on single points, and then we progress to transforming geometric figures. As we work through these figures, I ask the students questions intended to reinforce their vocabulary and understanding.
- Is your figure an isometry?
- Are your figures congruent?
- If the figures are not congruent, what is their relationship?
- What quadrant is your preimage in? And what quadrant is your image in?
- Did the orientation of the figure change or remain the same? What is the orientation of each?
- Are the corresponding sides of your preimage and image figures parallel or perpendicular or neither?
I also introduce reflections across lines other than y=x, the x-axis, and the y-axis. As the students perform these reflections, I ask them about the line of reflection.
If we connect a preimage point to its image point, what is the line of reflection going to be with respect to that line segment?
I stress that the line of reflection must be the perpendicular bisector of this line segment, and we take the time to discuss once again what a bisector does, as some students have trouble with this concept.
To help demonstrate the role of the perpendicular bisectors in reflections, I hand out graph paper, compasses, and straightedges to my students. I ask that they plot the points A(-2,5) and B(4,-1), and then construct a perpendicular bisector of line segment AB, which should then be the line of reflection. They are all familiar with constructing a perpendicular bisector at this point, but have never before done a construction on graph paper.
When all have completed the construction, I ask the students to find the equation of this line of reflection. This gives us the opportunity to revisit equations of lines in the form y = mx + b, learned in previous grades.
I also ask my students to find the distance between point A and the point (5,6) (a point on the perpendicular bisector), and between point B and (5,6). This allows students to review finding lengths of line segments from the coordinate geometry chapter, and is also a good opportunity to discuss the fact that any point on the perpendicular bisector of a segment will be equidistant from the endpoints of the segment.
On the handout entitled Name That Transformation, I ask the students to identify the transformation or transformations that map A to A'. In support of this, I also ask them to list two points on A and their image points on A', so that the students can understand the transformations visually and in terms of the changes to the coordinates. Looking at the figures in both ways also helps students with differing spatial abilities - some students can easily see the transformation without looking at the coordinates of the points, while others really need to see the coordinates in order to fully understand.
I assign any remaining problems for homework.
I hand out the Quick Transformations Quiz, taking care to alternate the two versions of the quiz. Having different versions of the quiz eliminates the temptation to steal a peak at someone else's paper. This will be today's ticket out the door.
I give a number of these quizzes throughout the chapter. I think it is important for the students to be able to recognize the notations and to perform these transformations fairly automatically. I spend a lot of time with the students developing the rules and making sure that the students understand why the rules are what they are, but in the end I think the students' experience with this unit is a lot more satisfying when they can move effortlessly through many of the problems.