Rotations in the Coordinate Plane

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Transition students from the conceptual understanding of rotations with tracing paper to rotations about the origin using coordinates.

Big Idea

Transition students from using tracing paper to perform rotations into using coordinates and realizing the pattern of rotating in multiples of 90 degrees about the origin.


10 minutes

Begin by asking students to pull out their unit organizer and brainstorm with their partner for two minutes the answers to the following two questions:  “What does it mean to rotate a figure?”  “How do we rotate figures?  Give details about what information you need to know and then how you use the given information.”  Put two minutes on a timer either on your desk or through the internet and projector onto your board.  At the end of two minutes, call on several groups to voice answers to each question as you script their answers on the board as notes to use during class today.  

Next, put the bellringer rotation question on the board for groups to discuss.  Allow one minute of discussion and then ask for a class vote by either standing up or raising a hand to vote for example one or example two.  Allow groups to verbalize their reasoning for why they chose either example.  I allow students to come to the board and script their thinking on the board to persuade others to vote for their example.  I then allow a revote just in case anyone has changed his/her vote.   

Tell students the clear goal for today is to figure out how to perform rotations on the coordinate plane using coordinates and a common pattern.  Let students know that we will begin by using tracing paper with the goal of figuring out the pattern with the coordinates as we work through the first several rotations.  By the end of lesson, students should not need the tracing paper for these types of rotations about the origin. 

Beginning the Activity

35 minutes

Pass out the handout Rotations Vignette.  Discuss with students that the first rotation is just for notes and reminders about how to rotate about a point, it is not a question for the students to answer.  Rotations two and four are rotations for the students to work through using tracing paper.  Remind student that each question is followed by what are the coordinate of the new point because we are looking for a pattern in the coordinates that will help us to stop using the tracing paper by the end of the activity.  Give students about 10 minutes to work through questions two through seven within their partner groups.  Questions two through seven allow students to perform two rotations using tracing paper and coordinates, then questions six and seven begin to discuss the possible pattern that hopefully students are beginning to see.  you really need to be moving about the room assessing and assisting. The most difficult part of these first few questions is being precise with coordinates.  If the coordinates of the rotated image are incorrect then the students will never see the correct pattern.  If several groups are having trouble with rotating exactly perfect then during the whole class mini-wrap up time discuss how accurate or inaccurate the tracing paper can be and ask which do students prefer - tracing paper or coordinates?  

It is very important to stop the entire class at question seven and have a good whole group mini-wrap up so far.  Students need to understand if they are moving in the correct direction for finding the pattern, therefore, the coordinates for the first two rotations must be correct on their papers.  I inserted a table in question six simply to organize their thinking by organizing their information.  During the mini-wrap up allow all pattern ideas to be voiced and scripted on the white board.  I usually have groups that keep the pattern simple - the x and y coordinates switch places.  I usually also have students who try to get fancy with changing the sign of the new x-coordinate to the opposite sign.  I script them all for consideration at this point.  

Next, ask students to continue to use the tracing paper, carefully, to rotate point R in question nine, after making an educated guess in question eight.  Allow students to work in student groups to complete questions eight through thirteen.  As students work, move about the room assessing student  progress and creating experts (or finding experts if everyone is working along well on their own).  Once everyone has finished up through question 13 pull the class together for a second mini-wrap up.  During this wrap up the class needs to make a final decision about the pattern that exists among the coordinates when rotating 90 degrees about the origin.  As a class, create a description for how to rotate using coordinates and a clear description of when to use this shortcut.  Script this shortcut on the whiteboard as the class develops it, then once all the editing is finished, tell students to copy this into question 12 on their handout and into their unit organizer for rotations.  This is possibly the final wrap up for class today and will serve as the ending of the main lesson.  

Assign the follow-up practice problems in questions 14 through 18 as class work and then homework if students do not finish during class.  

Wrapping Up the Lesson + Homework

10 minutes

The last mini-wrap up during the main lesson is also the ending of the lesson.  After the mini-wrap up is finished allow students to work together to answer questions 14 - 18 on the handout.  If students do not complete these during class, they can be homework.  A second option for homework is included Rotations Homework Page which uses coordinates to rotate two-dimensional figures.  If your students have accounts, the answers to questions 14 - 18 could be easily discussed or shared through Edmodo.  I even post the solutions myself sometimes just so students can know if they are working the problems correctly.