Carla Seeger AF Amistad Academy Middle School, NEW HAVEN, CT
6th Grade Math : Unit #4 - Coordinate Plane : Lesson #3

Reflections on the Coordinate Grid

Objective: SWBAT recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes
Standards: 6.NS.C.6 6.NS.C.8 MP2 MP3 MP4 MP6 MP8
Subject(s): Math
60 minutes
1 Think About It - 7 minutes

Students are given 4 minutes to work independently on the Think About It problems.  

I then cold call on students for each question on this page.  Students identify that the points are located at (4, 5) and (4, -5).  They notice that the points have the same x-coordinate, and that the y-coordinates are opposite of each other. 

I want students to use the language of absolute value throughout this lesson, so I ask them to share out more that the pairs have in common.  If 2-3 more students share out similarities they have noticed, but have not yet talked about distance from the y-axis, I will guide them to get there.  I'll say, "I'm noticing something about distance.  What do both points have in common, when we think about distance?"

Once students recognize that the two points have the same distance from the y-axis, so that the y-coordinates have the same absolute value, I will move into the new material for the lesson.  

More than Just Changing One Sign
Developing a Conceptual Understanding

I spend time in this lesson developing a conceptual understanding of reflections, which is why there are many opportunities to talk about absolute value and distance. 

In 6th grade, students will only be asked to reflect around one of the axes.  It'd be easy enough for me to teach them to simply change the sign of one of the coordinates to represent a reflection.  This wouldn't give my students a deep understanding of the content, though, and they would have difficulties in later years.  Reflections can happen over any central line, and in later grades students will reflect over central lines other than an axis.  When students reflect over x=3, for example, they won't be able to simply change the sign of the x-coordinate to get the answer.  They'll have to understand that the original point and the reflected point must both be the same distance away from x=3.

2 Intro to New Material - 15 minutes

To begin this section of the lesson, I have students help me label the parts of the graph.  I have a copy of the INM materials on the document camera, and I ask for students to give me the names of the important pieces.  We label the origin, the x- and y-axis, and all four quadrants.

Together, we plot the first point, (4, 5) and label it as Point A.  Students annotate the problem by boxing the words 'y-axis.'   My expectation is that they always box the axis over which we're rotating, as a small way to help them not reflect over the incorrect axis.  

For this first point, (4, 5), I ask students to count how many units it takes to get from the y-axis to the point.  I want to spend time on the conceptual idea of reflecting a point, so we don't jump right into simply using the opposite value for the x-coordinate.  Once students name that the point is 4 units away from the axis, I tell them that the reflection is going to be the same distance from the y axis but in the opposite direction.  We know that the y-axis represents 0, so therefore both points have the same absolute value, or distance from 0.  Then we are going to count that same number of units going away from the y-axis and plot and label my point with the coordinate pair (-4, 5).

Once, we've gone through the same process for the other two ordered pairs in this section, I have students turn and talk with their partner about a generalization we can make about reflections. I ask them:

When we reflect a point over the y-axis, what do you notice about the relationship between the x- and y-values of the coordinates of each point?

Students share out, and then I ask for the group to make a conjecture about the relationship of the coordinates when we reflect over the x-axis.

 

3 Partner Practice and Discussion - 15 minutes

Students work on the Partner Practice problem set for 10 minutes.  As they are working, I circulate around the room.  

I am looking for: 

  • Are scholars correctly labeling all of the necessary components of the coordinate graph?
  • Are scholars correctly plotting the coordinate pair on the grid?
  • Are scholars correctly identifying the coordinate pair’s reflection with and without a grid?
  • Are scholars accurately describing what happens when you switch the sign of the values in a coordinate pair?
  • Are scholars correctly describing the similarities and differences between two points that are reflections of one-another?

 

I am asking pairs:

  • How did you know to draw the coordinate pair in that particular place?
  • How did you know that was the correct name of the coordinate pair?
  • What does it mean to reflect a point over an axis?
  • What would happen if you change the signs of the ordered pair?

 

After the 10 minutes of partner work time, students complete the final check for understanding independently.  As a class, we discuss question 14, where students are asked to generalize about any point, (x, y).  

4 Independent Practice - 15 minutes

After our conversation about the final CFU problem, students work on the Independent Practice problems for 15 minutes.  As I am circulating, I am looking for and asking students the same questions that I use for my circulation during partner practice.  

 

5 Closing and Exit Ticket - 10 minutes

At the end of Independent Work time, we discuss Problem_25.  First, I have students show me on their fingers which quadrant Nate's house is in.  This gives me a quick visual check of how students have done with this problem.  The students have to complete a number of reflections to arrive at the location of Nate's house.  It is likely that there will be a variety of answers.  I call on a student with the correct quadrant shown, and ask that student to convince us that her/his answer is correct.  I then give other students the opportunity to add to the response.  I want us to build an exemplar oral response together.  

Students end class by working on the Exit Ticket independently.