Learn on PengiIllustrative Mathematics, Grade 6Unit 7 Rational Numbers

Lesson 3: The Coordinate Plane

In this Grade 6 lesson from Illustrative Mathematics Unit 7, students extend the coordinate plane to all four quadrants by plotting and identifying ordered pairs with negative x- and y-values. Students practice locating points in Quadrant I, II, III, and IV using coordinates such as (-4, 1) and (-3.5, -3), and explore how the signs of the coordinates determine a point's position relative to the x- and y-axes. The lesson builds fluency with the coordinate plane through activities like graphing on a grid and interpreting coordinates in real-world contexts.

Section 1

Quadrants and Points on the Axes

Property

The axes divide a plane into four regions, called quadrants. The quadrants are identified by Roman numerals, beginning on the upper right and proceeding counterclockwise.

Quadrant I	Quadrant II	Quadrant III	Quadrant IV
$(+, +)$	$(-, +)$	$(-, -)$	$(+, -)$

Points on the Axes
Points with a $y$ -coordinate equal to 0 are on the $x$ -axis, and have coordinates $(a, 0)$ .
Points with an $x$ -coordinate equal to 0 are on the $y$ -axis, and have coordinates $(0, b)$ .

Section 2

Graphing Reflections by Counting

Property

Before memorizing any algebraic formulas, you can always reflect any point or polygon on a grid simply by counting. Find the perpendicular distance from an original vertex to the line of reflection, then count that exact same distance past the line to plot the new vertex. To reflect an entire shape, just repeat this counting process for each corner and connect the new dots.

Examples

Reflecting a Point: Point B is 3 grid squares above the x-axis. To reflect it across the x-axis, count 3 squares down to the axis, then 3 more squares down past the axis. Plot B'.
Reflecting a Polygon: To reflect triangle ABC, do not try to flip the whole triangle in your head!
1. Count distance for A, plot A'.
2. Count distance for B, plot B'.
3. Count distance for C, plot C'.
4. Connect A', B', and C'.

Explanation

Counting is your ultimate backup plan if you forget a formula.But beware of these two micro-traps:

Don't count the dot you start on! Only count the jumps between grid lines.
Count straight across! If your mirror line is vertical, you must count horizontally. If your mirror is horizontal, you must count vertically.

Section 3

Finding Distance Using Absolute Value

Property

To find the distance between two points on the same horizontal or vertical line, use the absolute value of the difference of the coordinates that are not the same.

For two points on a horizontal line, $(x_1, y)$ and $(x_2, y)$ , the distance is $d = |x_2 - x_1|$ .
For two points on a vertical line, $(x, y_1)$ and $(x, y_2)$ , the distance is $d = |y_2 - y_1|$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Quadrants and Points on the Axes

Property

The axes divide a plane into four regions, called quadrants. The quadrants are identified by Roman numerals, beginning on the upper right and proceeding counterclockwise.

Quadrant I	Quadrant II	Quadrant III	Quadrant IV
$(+, +)$	$(-, +)$	$(-, -)$	$(+, -)$

Section 2

Graphing Reflections by Counting

Property

Examples

Reflecting a Point: Point B is 3 grid squares above the x-axis. To reflect it across the x-axis, count 3 squares down to the axis, then 3 more squares down past the axis. Plot B'.
Reflecting a Polygon: To reflect triangle ABC, do not try to flip the whole triangle in your head!
1. Count distance for A, plot A'.
2. Count distance for B, plot B'.
3. Count distance for C, plot C'.
4. Connect A', B', and C'.

Explanation

Counting is your ultimate backup plan if you forget a formula.But beware of these two micro-traps:

Don't count the dot you start on! Only count the jumps between grid lines.
Count straight across! If your mirror line is vertical, you must count horizontally. If your mirror is horizontal, you must count vertically.

Section 3

Finding Distance Using Absolute Value

Property

To find the distance between two points on the same horizontal or vertical line, use the absolute value of the difference of the coordinates that are not the same.

For two points on a horizontal line, $(x_1, y)$ and $(x_2, y)$ , the distance is $d = |x_2 - x_1|$ .
For two points on a vertical line, $(x, y_1)$ and $(x, y_2)$ , the distance is $d = |y_2 - y_1|$ .

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Quadrants and Points on the Axes

Property

The axes divide a plane into four regions, called quadrants. The quadrants are identified by Roman numerals, beginning on the upper right and proceeding counterclockwise.

Quadrant I	Quadrant II	Quadrant III	Quadrant IV
$(+, +)$	$(-, +)$	$(-, -)$	$(+, -)$

Section 2

Graphing Reflections by Counting

Property

Examples

Reflecting a Point: Point B is 3 grid squares above the x-axis. To reflect it across the x-axis, count 3 squares down to the axis, then 3 more squares down past the axis. Plot B'.
Reflecting a Polygon: To reflect triangle ABC, do not try to flip the whole triangle in your head!
1. Count distance for A, plot A'.
2. Count distance for B, plot B'.
3. Count distance for C, plot C'.
4. Connect A', B', and C'.

Explanation

Counting is your ultimate backup plan if you forget a formula.But beware of these two micro-traps:

Don't count the dot you start on! Only count the jumps between grid lines.
Count straight across! If your mirror line is vertical, you must count horizontally. If your mirror is horizontal, you must count vertically.

Section 3

Finding Distance Using Absolute Value

Property

To find the distance between two points on the same horizontal or vertical line, use the absolute value of the difference of the coordinates that are not the same.

For two points on a horizontal line, $(x_1, y)$ and $(x_2, y)$ , the distance is $d = |x_2 - x_1|$ .
For two points on a vertical line, $(x, y_1)$ and $(x, y_2)$ , the distance is $d = |y_2 - y_1|$ .