The purpose of this Warm-up is for students to review plotting and determining the coordinates of points in the coordinate plane. Students use repeated reasoning to generalize patterns in the coordinates of points in each quadrant (MP8).
Launch
Arrange students in groups of 4. Assign each person in the a group a different quadrant. Give students 3 minutes of quiet work time to plot and label at least three points, and up to six if they have time, in their assigned quadrant. Follow with a whole-class discussion.
Activity
None
Plot at least 3 points in your assigned quadrant, and label them with their coordinates.
Student Response
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Building on Student Thinking
Activity Synthesis
The focus of this discussion is for students to recognize that the following patterns emerge:
In Quadrants I and IV, the -coordinate of a point (or the first number in an ordered pair) is positive.
In Quadrants II and III, the -coordinate of a point is negative.
In Quadrants I and II, the -coordinate of a point (or the second number in an ordered pair) is positive.
In Quadrants III and IV, the -coordinate of a point is negative.
Invite students to share any patterns they noticed. After each student shares, ask the rest of the class if they noticed the same pattern within their small group. Record and display these patterns for all to see. If possible, plot and label a few example points in each quadrant based on students’ observations.
13.2
Activity
Standards Alignment
Building On
Addressing
6.NS.C.6.b
Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
In this activity, students connect opposite signs in coordinates to reflections across one or both axes. Students investigate relationships between several pairs of points in order to more generally make this connection (MP8).
The use of the word “reflection” is used informally to describe the effect of opposite signs in coordinates. In grade 8, students learn a more precise, technical definition of the word “reflection” as it pertains to rigid transformations of the plane.
Students will learn a more precise, technical definition of the word “reflection” in a future course that may not necessarily be during grade 8, as mentioned in the Activity Narrative.
Launch
Arrange students in groups of 2. Display the first problem from the Task Statement for all to see. Ask students to determine the coordinates for each point, and record their responses for all to see. Then give students 5–6 minutes of quiet work time, and follow with a whole-class discussion.
MLR2 Collect and Display. Circulate to listen for and collect the language that students use as they answer questions about pairs of points. On a visible display, record words and phrases, such as “same distance,” “opposite sides,” “opposite signs,” and “reflection.” During the discussion, invite students to suggest ways to update the display: “What are some other words or phrases we should include?” Invite students to borrow language from the display as needed. Advances: Conversing, Reading
Activity
None
Write the coordinates of each point.
Coordinate plane, origin O, horizontal axis negative 7 through 7 by ones, vertical axis negative 5 through 5 by ones. Point A is 4 units to the right of the origin and 3 units up. Point B is 6 units down from point A. Point C is 1 unit to the left and 2 units down from point B. Point D is 7 units to the left and 2 units up from point C. Point E is 2 units to the left and 6 units up from point D.
Answer these questions for each pair of points.
How are the coordinates the same? How are they different?
How far away are they from the -axis? To the left or to the right of it?
How far away are they from the -axis? Above or below it?
and
and
and
Point has the same coordinates as point , except its -coordinate has the opposite sign.
Plot point in the coordinate plane, and label it with its coordinates.
How far away are and from the -axis?
Point has the same coordinates as point , except its -coordinate has the opposite sign.
Plot point in the coordinate plane, and label it with its coordinates.
How far away are and from the -axis?
Point has the same coordinates as point , except both coordinates have opposite signs. In which quadrant is point ?
Student Response
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Building on Student Thinking
Activity Synthesis
The goal of this discussion is for students to see that coordinates with opposite signs correspond to reflections across the axes. Begin by asking students what patterns they noticed for pairs of points whose -coordinates had opposite signs. Ask students to give specific examples of pairs of points and their coordinates when describing the pattern they saw. Record students’ explanations for all to see. Students may use phrasing like “the point flips across the -axis.” Introduce the word “reflection,” and discuss similarities between reflections across the -axis and reflections in a mirror. Note that rigid transformations, including reflections, will be studied further in a later course, so it is not necessary for students to use this term fluently.
Repeat this discussion for pairs of points where the -coordinates had opposite signs to see that they are reflections across the -axis.
Close by discussing the relationship between points and , where both the - and -coordinates have opposite signs. Ask students how they might describe the relationship between and visually on the coordinate plane. While students may describe the relationship in terms of two reflections (once across the -axis and again across the -axis, or vice versa), it is not expected that students see this relationship in terms of a rotation.
Representation: Internalize Comprehension. Use color coding and annotations to illustrate student thinking. For example, as students describe what they noticed, use color and annotations to scribe their thinking on a display of the coordinate plane that is visible for all students Supports accessibility for: Visual-Spatial Processing
13.3
Activity
Optional
Standards Alignment
Building On
Addressing
7.NS.A.1.c
Understand subtraction of rational numbers as adding the additive inverse, . Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
In this activity, students plot points in a coordinate plane to create a rectangle. They find the lengths of each side by finding the horizontal or vertical distance between points. Students attend to precision in language as they distinguish between distance (which is unsigned) and difference (which is signed) (MP6).
In the digital version of the activity, students use an applet to plot points in a coordinate plane. The digital version may reduce barriers for students who need support with fine-motor skills.
This activity is not optional in this course.
Launch
Arrange students in groups of 2. Give students 5 minutes of quiet work time followed by a partner discussion. Then follow with a whole-class discussion.
Activity
None
Plot and label these points in the coordinate plane: , , , .
Connect the dots in order. What shape is made?
What are the side lengths of figure ?
What is the difference between the -coordinates of and ?
What is the difference between the -coordinates of and ?
How do the differences of the coordinates relate to the distances between the points?
Activity Synthesis
The goal of this discussion is for students to compare the distance between two numbers with the difference between two numbers. Here are some questions for discussion:
“How can you find the difference between two numbers?” (Subtract one number from the other.)
“When finding the difference between two numbers, does the order matter?” (Yes, we need to subtract the final number from the beginning number.)
“How can we find the distance between two numbers?” (We can use a number line and count the distance between the two numbers. We can subtract the two numbers.)
“When finding the distance between two numbers, does the order matter?” (No, we are just looking for the magnitude of the change, not the direction of the change.)
“How can we see this distinction between difference and distance in the points we drew in the coordinate plane?” (When finding the difference between the -coordinates of and , the order we subtracted mattered, but when finding the distance between the two points, the order did not matter.)
If not mentioned in students' explanations, emphasize that differences can be positive or negative (or 0) depending on the order of the numbers subtracted. Distances cannot be negative.
13.4
Activity
Standards Alignment
Building On
6.G.A.3
Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.
Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
In this activity, students practice plotting points in the coordinate plane to make polygons.
In the digital version of the activity, students use an applet to plot points in the coordinate plane. The applet allows students to drag points to their location in the coordinate plane and quickly check their accuracy. The digital version may be helpful for students to quickly plot and adjust points of polygons without needing to erase.
This activity is optional in this course. It provides further practice plotting points on the coordinate plane. Consider using these shapes as an opportunity to reinforce what students have learned about distances on the coordinate plane.
“Which segments in Polygon A are 2 units long?” (segments and , but not segment )
“Which segments in Polygon D are 2 units long?” (none of them)
Students should recognize that they can find horizontal or vertical distances between points by counting grid units, but that a diagonal segment through a grid square is longer than one grid unit.
Activity Synthesis
The purpose of the discussion is to emphasize the connection between numbers, the coordinate plane, and geometry. To highlight these connections, ask:
“How is the coordinate plane related to the number line?” (The coordinate plane has two axes that are both number lines.)
“How are we able to make polygons in the coordinate plane?” (The vertices of a polygon are plotted as points in the coordinate plane.)
Complete the connection by explaining to students that the coordinate plane allows us to describe shapes and geometry in terms of numbers. This is how computers are able to create two- and three-dimensional images even though they can only interpret numbers.
Lesson Synthesis
The purpose of this discussion is to summarize the effect of replacing coordinates with their opposites and how to find horizontal and vertical distances in the coordinate plane. Here are some questions for discussion:
“Without graphing, what can you say about the points and in the coordinate plane?” (The first point is in Quadrant IV, and the second point is in Quadrant I. They are both 3 units from the -axis. They are reflections of each other across the -axis. They are 6 units apart. They are both on the same vertical line.)
“Without graphing, what can you say about and in the coordinate plane?” (The first point is in Quadrant II and the second point is in Quadrant I. They are both 6 units from the -axis. They are reflections across the -axis. They are 12 units apart. They are both on the same horizontal line.)
“Without graphing, what can you say about and in the coordinate plane?” (They are both in Quadrant I. They are both on the same horizontal line because they have the same -value. The second point is 3.5 units to the right of the first point.)
If time allows, challenge students to draw a rectangle with given side lengths and identify its vertices. This will be useful in a future lesson where students explore shapes in the coordinate plane.
Student Lesson Summary
The points , and are shown in the coordinate plane. Notice that they all have almost the same coordinates, except the signs are different. They are all the same distance from each axis but are in different quadrants.
Notice that the vertical distance between points and is 4 units because point is 2 units above the horizontal axis and point is 2 units below the horizontal axis. The horizontal distance between points and is 10 units because point is 5 units to the left of the vertical axis and point is 5 units to the right of the vertical axis.
We can always tell which quadrant a point is located in by the signs of its coordinates.
quadrant
positive
positive
I
negative
positive
II
negative
negative
III
positive
negative
IV
Standards Alignment
Building On
Addressing
6.NS.C.6
Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
Arrange students in groups of 2. Give students 10 minutes of quiet work time, and follow with a whole-class discussion.
Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select 2–3 of the polygons to plot on the coordinate plane. Supports accessibility for: Organization, Social-Emotional Functioning
Activity
None
Here are the coordinates for four polygons. Plot them on the coordinate plane, connect the points in the order that they are listed, and label each polygon with its letter name.
