The purpose of this Warm-up is to connect coordinates with transformations.
There are many ways to express a translation because a translation is determined by two points and once we know that is translated to . There are many pairs of points that express the same translation. This is different from reflections which are determined by a unique line and rotations which have a unique center and a specific angle of rotation.
Launch
Ask students how they describe a translation. Is there more than one way to describe the same translation? After they have thought about this for a minute, give them 2 minutes of quiet work time followed by a whole-class discussion.
Activity
None
Select all of the translations that take Triangle T to Triangle U. There may be more than one correct answer.
A. Translate to .
B. Translate to .
C. Translate to .
D. Translate to .
Student Response
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Building on Student Thinking
Students may think that they need more information to determine the translation. Remind them that specifying one point can determine the distance and direction all of the other points move in a translation.
Activity Synthesis
Remind students that once you name a starting point and an ending point, that completely determines a translation because it specifies a distance and direction for all points in the plane. Appealing to their experiences with tracing paper may help. In this case, we might describe that distance and direction by saying “all points go up 2 units and to the right 4 units.” Draw the arrow for the two correct descriptions and a third one not in the list, like this:
Point out that each arrow does, in fact, go up 2 and 4 to the right.
4.2
Activity
Standards Alignment
Building On
8.G.A.1
Verify experimentally the properties of rotations, reflections, and translations:
The goal of this activity is for students to work through multiple examples of specific points reflected over the -axis and then generalize to describe where a reflection over the -axis takes any point. They also consider reflections over the -axis as they leverage the structure of the coordinate grid (MP7).
Monitor for students who use these different strategies:
Count units or measure distances away from the axis of reflection to plot the new point
Reason about the coordinate plane to determine the new coordinate
The key connection here is that when reflecting over the vertical or horizontal axis, one coordinate value must stay the same and one will be the opposite.
The goal of the activity is not to create a rule that students memorize. The goal is for students to notice the pattern of reflecting over an axis changing the sign of the coordinate without having to graph. The coordinate grid can sometimes be a powerful tool for understanding and expressing structure and this is true for reflections over both the -axis and -axis.
This activity uses the Compare and Connect math language routine to advance representing and conversing as students use mathematically precise language in discussion.
Launch
Tell students that they will have 5 minutes of quiet think time to work on the activity, and tell them to pause after the second question.
Select 2–3 students to share their strategies. Start with students who are measuring distances of points from the -axis or counting the number of squares a point is from the -axis and then counting out the same amount to find the reflected point. While these strategies work, they overlook the structure of the coordinate plane. Point out the role of the coordinate plane by selecting a student who noticed the pattern of changing the sign of the -coordinate when reflecting over the -axis.
Use MLR7 Compare and Connect when students present their strategies for reflecting points using the -axis as the line of reflection, such as those described in the Activity Narrative, before continuing on. After all strategies have been presented, lead a discussion comparing, contrasting, and connecting the different approaches and representations. Ask the following:
“What is the same and what is different about the strategies?”
“Did anyone solve the problem the same way, but would explain it differently?”
These exchanges strengthen students’ mathematical language use and reasoning of reflections along the -axis and -axis.
After this initial discussion, give 2–3 minutes of quiet work time for the remaining questions, which ask them to generalize how to reflect a point over the -axis.
Engagement: Provide Access by Recruiting Interest. Use visible timers or audible alerts to help learners anticipate when to pause and prepare to transition to sharing strategies. Supports accessibility for: Organization, Attention
Activity Synthesis
Display a blank coordinate grid.
Questions for discussion:
"What was the same and what was different about reflecting over the -axis instead of the -axis?"
"When you have a point and an axis of reflection, how do you find the reflection of the point?"
"How can you use the coordinates of a point to help find the reflection?"
"Are some points easier to reflect than others? Why?"
"What patterns have you seen in these reflections of points on the coordinate grid?"
4.3
Activity
Standards Alignment
Building On
8.G.A.1
Verify experimentally the properties of rotations, reflections, and translations:
In this activity, students rotate a segment on a coordinate plane.
In general, it is difficult to use coordinates to describe rotations unless the center of the rotation is and the rotation is 90 degrees (clockwise or counterclockwise).
Unlike translations and reflections over the - or -axis, it is more difficult to visualize where a 90-degree rotation takes a point. Tracing paper is a helpful tool, as is an index card.
Launch
Demonstrate how to use tracing paper in order to perform a 90-degree rotation. It is helpful to put a small set of perpendicular axes (a + sign) on the piece of tracing paper and place their intersection point at the center of rotation. One of the small axes can be lined up with the segment being rotated and then the rotation is complete when the other small axis lines up with the segment.
An alternative method to perform rotations would be with the corner of an index card, which is part of the geometry toolkit.
Activity
None
Apply each of the following transformations to segment .
Rotate segment counterclockwise around center . Label the image of as . What are the coordinates of ?
Rotate segment counterclockwise around center . Label the image of as . What are the coordinates of ?
Rotate segment clockwise around . Label the image of as and the image of as . What are the coordinates of and ?
Compare the two counterclockwise rotations of segment . What is the same about the images of these rotations? What is different?
Activity Synthesis
Ask students to describe or demonstrate how they found the rotations of segment . Make sure to highlight these strategies:
Use tracing paper to enact a rotation through a 90 degree angle.
Use an index card by placing the corner of the card at the center of rotation, aligning one side with the point to be rotated, and finding the location of the rotated point along an adjacent side of the card. (Each point's distance from the corner needs to be equal.)
Use the structure of the coordinate grid so that a 90-degree rotation with center at the intersection of two grid lines will take horizontal grid lines to vertical grid lines and vertical grid lines to horizontal grid lines.
The third strategy should only be highlighted if students notice or use this in order to execute the rotation, with or without tracing paper. This last method is the most accurate because it relies on the structure of the coordinate grid.
If some students notice that the three rotations of segment are all parallel, this should also be highlighted.
MLR8 Discussion Supports. Display sentence frames to support students in explaining the similarities and differences of the segment rotations for the last question: “ and are the same/alike because . . .” or “ and are different because . . . .” Advances: Speaking
Engagement: Develop Effort and Persistence. Break the class into small discussion groups and then invite a representative from each group to report back to the whole class. Supports accessibility for: Language; Social-emotional skills; Attention
Lesson Synthesis
By this point, students should start to feel confident applying translations, reflections over either axis, and rotations of 90 degrees clockwise or counterclockwise to a point or shape in the coordinate plane.
To highlight working on the coordinate plane when doing transformations, ask:
"What are some advantages to knowing the coordinates of points when you are doing transformations?"
"What changes did we see when reflecting points over the -axis? -axis?"
"How do you perform a 90-degree clockwise rotation of a point with center ?"
If time allows, ask students to apply a few transformations to a point. For example, "What is the image of when it is . . ."
"Reflected over the -axis?"
"Reflected over the -axis?"
"Rotated 90 degrees clockwise with center ?"
Student Lesson Summary
We can use coordinates to describe points and find patterns in the coordinates of transformed points.
We can describe a translation by expressing it as a sequence of horizontal and vertical translations.
For example, segment is translated right 3 and down 2.
Quadrilateral on a coordinate plane. Horizontal axis scale negative 3 to 5 by 1’s. Vertical axis scale negative 2 to 3 by 1’s. Quadrilateral A B B prime A prime has coordinates A(negative 2 comma 1), B(1 comma 2), B prime(4 comma 0) and A prime (1 comma negative 1).
Reflecting a point across an axis changes the sign of one coordinate.
For example, reflecting the point whose coordinates are across the -axis changes the sign of the -coordinate, making its image the point whose coordinates are . Reflecting the point across the -axis changes the sign of the -coordinate, making the image the point whose coordinates are .
3 points on a coordinate plane. Horizontal axis scale negative 3 to 5 by 1’s. Vertical axis scale negative 2 to 2 by 1’s. The points have these coordinates: A(2 comma negative 1), A prime(2 comma 1) and A double prime (negative 2 comma negative 1).
Reflections across other lines are more complex to describe.
We don’t have the tools yet to describe rotations in terms of coordinates in general. Here is an example of a rotation with center in a counterclockwise direction.
Point has coordinates . Segment is rotated counterclockwise around . Point with coordinates rotates to point whose coordinates are .
Segment A B rotated on a coordinate plane, origin O. Horizontal axis scale negative 4 to 4 by 1’s. Vertical axis scale negative 2 to 4 by 1’s. The segments have these coordinates: A(0 comma 0), B(2 comma 3) and B prime (negative 3 comma 2). Angle B A B prime is a right angle.
Standards Alignment
Building On
8.G.A.1
Verify experimentally the properties of rotations, reflections, and translations:
Plot each point and label each with its coordinates.
Using the -axis as the line of reflection, plot the image of each point.
Label the image of each point with its coordinates.
Include a label using a letter. For example, the image of point should be labeled .
If the point were reflected using the -axis as the line of reflection, what would be the coordinates of the image? What about ? ? Explain how you know.
The point has coordinates .
Without graphing, predict the coordinates of the image of point if point were reflected using the -axis as the line of reflection.
Check your answer by finding the image of on the graph.
Label the image of point as .
What are the coordinates of ?
Suppose you reflect a point using the -axis as the line of reflection. How would you describe its image?
Student Response
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Building on Student Thinking
If any students struggle getting started because they are confused about where to plot the points, refer them back to the Warm-up and practice plotting a few example points with them.
