Hands on Exploring Reflections in the Plane Continued
Lesson 5 of 23
Objective: Students will be able to apply relfections using tracing paper to reflect across diagonal lines and then without tracing paper using coordinates.
Warm-up and Homework Review
Ask students to brainstorm for one minute on their own paper everything they know about reflections. This is brainstorming in silence. Then give the youngest member of the group thirty seconds to share what he/she wrote with his/her partner. The older partner gets the second thirty seconds of the second minute to add any additional ideas not already mentioned by the first partner. As a class, brainstorm on the white board for one minute all the ideas generated during partnership time. Next ask student volunteers to put their homework papers (four practice problems at the bottom of page two) under the document camera so students can check their work.
Be explicit about the goal of the lesson today - clearly tell students the lesson today has two main goals. Goal one is to understand the movement and characteristics of a reflection so well that even reflections across diagonal lines are possible. Goal two - Since standardized tests do not allow students to use tracing paper then lets figure out today another context for reflecting that does not require tracing paper, only a pencil.
Depending on your students' ability to graph lines, you may choose to do an expanded opening or alternative opening that includes a review of graphing lines, such as y = 2, x = -4, and y = x.
Continuing the Activity
Ask students to read the directions for the triangle MOP reflection across a diagonal line at the top of page three, question number five. Remind students of the brainstorming on the board and ask them to perform this reflection using their tracing paper and pencil. Be explicit with students that they are to complete the first reflection and follow-up questions in part a but not the second reflection in question six until everyone has discussed question five as a whole class. Move about the room formatively assessing and answering questions. Make sure you are finding or creating experts who can share their work after everyone has finished the reflection. You should expect students to mess this up and try to make it look like a reflection across a horizontal line. Students will also act as if they have never seen tracing paper before, just be patient and ask them questions about how they used the tracing paper yesterday.
Click below to watch a short tutorial video on how to use the tracing paper to perform the reflection in question six.
After everyone has finished, ask your expert groups to present and discuss the follow-up question a as whole group. Some points to discuss include the difficulty in reflecting visually across a diagonal line - it just looks wrong at first. Discuss trusting the tracing paper because the process is not different from the previous day. Discuss how to check the reflection my measuring corresponding veracities distance from line of reflection and angle create as crossing the line of reflection.
Allow students seven to eight minutes to work in their groups to complete the second diagonal reflection in questions six. Move about the room assessing, assisting, and choosing experts to present their work under the document camera to the group.
To introduce questions seven through nine remind students that on standardized tests such as the ACT and SAT reflections are a part of the tested material however tracing paper is not an allowed resource. Therefore, there must be a means of performing reflections without needing tracing paper. Tell students the goal of the next set of questions is to understand exactly what information and what context is important to perform reflections without using tracing paper.
Ask groups to work in partnerships to complete questions seven through nine. As you move about the room assesses and assisting notice if most groups have trouble graphing the line of reflection. If more than 60% of your class is incorrectly graphing the line of reflection then call the group together and either allow an expert group to demonstrate how to graph the line or spend a few minutes group brainstorming on the whiteboard (take notes for the class by scripting) all the important information for graphing a line - example: generate two to three solutions to the equation that make it true and then graph these. The line is the set of all solutions that make the equation true - linear solutions are collinear and form a line.
Again, graphing the line will probably be your students' most difficult task in this reflection question. Once students complete the reflection, allow an expert group to come to the document camera and present.
Extension Activity: There is a lesson extension at the end of the handout if you choose to use it and copy it separately or as an included section. This lesson extension applies reflections across adjacent lines to help students understand line symmetry. Line symmetry or reflectional symmetry is when a line of reflection passes through the interior of a figure. There are a series of extension questions and reflections to be performed in this activity. For homework, I usually assign the extension handout which asks students to draw all the lines of reflectional symmetry within a series of regular polygons. In order to help students see the lines of symmetry, I print the larger regular polygons document and ask students to cut them out and use these large examples to help them find the lines of reflectional symmetry – hint: they fold the polygons and every fold line is a line of reflectional symmetry. The hands on folding really help them to see how the lines of symmetry work and a good website for printing the regular polygons is http://www.math-salamanders.com/shapes-for-kids.html . If I print the large polygons and allow students to fold them, then I usually make a poster for the number of sides and number of lines of reflectional symmetry. I ask students to tape their polygons to the poster and we discuss the patterns noticed – Patterns include: there are as many lines of reflectional symmetry as there are number of sides on the polygon and lines of symmetry in odd numbered polygons connect a vertex to midpoint of opposite side while lines of symmetry in even numbered polygons connect vertex to opposite vertex and midpoint of side to midpoint of opposite side. Refer to the image of an example poster from one of my classes.
Wrapping up the Lesson
Tell students that we not only want to understand how to perform a reflection, but sometimes we need to locate the line of reflection after a reflection has already been performed. In the final questions, which may be homework, the goal is to find the line of reflection that was used to create the reflected image. Encourage students to use their tracing paper wisely to help them locate the line of reflection.
Click below to watch a short tutorial video on how to use the tracing paper to help locate the line of reflection.
Homework: The additional practice at the bottom of page two if not completed during class – mapping the line of reflection as well as reflections practice page. If you do the extension activity, the regular polygons line symmetry page could be the homework instead.