Enviornment and Math Practice Standards
I had to add one more day to the reflections because my students shared such great information during the discussion time that it just took one more day to really finish and discuss reflections in the coordinate plane. Also, some of my students needed a help with graphing horizontal and vertical lines from equations such as x = a number and y = a number. All of this discussion time added one more day to this lesson.
Two strategies I use with my students regularly include: making students owners of their own learning and activating students as resources for each other. One means of helping students take ownership of learning, is by letting them grapple with the math concepts on their own to develop a method of creating solutions. As the students are grappling they are working in partnerships so they are actively helping each other work through the math solution. Click below to watch a short video on each of these strategies from my teacher strategy folder.
Video from strategy folder: Students take ownership of own learning
Video from strategy folder: Activate students as resources for one another
Video from strategy folder: Grouping students homogeneously for partnerships
Because I encourage students to find their own path to the solution, I usually have multiple approaches that all lead to the same correct solution. I like to pull these correct solutions to the front of the room and allow the students who created them time to own their own learning by presenting their method of thinking to the whole class and in turn become a resource to the rest of the class. This method of allowing students to grapple in pairs to find solutions to math questions and then present their thinking to the class clearly aligns to several math practice standards including the following: MP1 make sesnse of problems and persevere in solving them, MP3 construct viable arguments and critique the reasoning of others, MP5 use appropriate tools strategically. In the following student samples, you will see the multiple approaches my students took to answer question #5 from their classwork/homework the day before.
Samples of Student Work
One student presented that he used the tracing paper but added a few extra lines to the diagram. He used a line segment to connect two of the verticies in reflection 5 to the line of
reflection at a 90 degree angle. These short line segments were his guides for placing the image in the correct location (perpendicular to line of reflection) and equal distant on the opposite side of the line of reflection.
A second student completed the reflection without using tracing paper by first tracing the original pre-image to place pencil lead on the figure and then folded his handout along the diagonal line of reflection and rubbed on the back side of the handout to put the pencil lead onto the opposite side of the line of reflection. Then he traced over the faint
impression with his pencil.
A third student used only a ruler to draw straight lines from the orginal pre-image across the line of refleciton. (we discussed that a protractor really needed to be used to ensure these are perpendicular to the line of reflection). Then the student used the ruler to measure the distance each vertex was from the line of reflection. She measured this same distance on the opposite side of the line of reflection but still on the guide line that should be perpendicular.
Because I knew that graphing the lines of reflection would be the most diffiucult part of the final reflections in the handout and in the homework practice page, we spent time reviewing that linear equations have infinite solutions. In order to graph a line you need to find at least two solutions but three solutions are better to graph. We made tables of values to graph for each y = value and x = value equation and some students began to realize the pattern that each type was either a horizontal or vertical line at the given value. I encouraged them to not depend on memorization however because several were trying to find a short cut immediately and were incorrect. Many were graphing x = value equations as horizontal lines because the x-axis is a horizontal line. Also, many were trying to graph y = value equations as verticle lines because the y-axis is a vertical line. We really had to go back to the equation for solutions. Many students relied upon a table of values to really get their mind thinking in a correct graphing direction. We spent the remainder of the time on graphing the lines of reflection before I gave them the homework paper. Students were given time in class to work on reflections with coordinates in quesitons 7, 8, 9, and 10. From walking around the room formatively assessing their progress it was clear that my students understand the movement of a reflection and could correctly perform reflections. I spend time providing feedback to my students that moved their learning forward and in the process created experts who would present their work and thinking at the end of the class period. Click here to watch a short video from my teacher strategies on how to provide feedback that moves learning forward so you create classroom experts.
Video from strategy folder: Providing Feedback to move learning forward and creating experts
The issue was with realizing were to graph the line of reflection so that the correct process was used with a correct line. We presented solutions at the end of class and practice page was assigned for homework practice. It is clear from the work completed during class with graphing linear equations and performing reflections with hands on materials that this lesson really addresses one main common core math standard: 8.G.A.1 Verify experimentally the properties of rotations, reflections, and translations.
Homework: Reflections in the Coordinate Plane Practice Page