Imagine you are designing a new level for a video game. The game world is like a giant grid, and every important place in the game has an address: a pair of numbers, like \((3, 5)\). These addresses tell the computer exactly where to draw walls, platforms, and treasure chests. That grid is just like the coordinate plane in math! 🎮
In this lesson, you learn how to:
The coordinate plane is a flat grid made from two number lines that cross at zero.
Every point on the coordinate plane has an ordered pair written like \((x, y)\).
For example, the point \((4, -2)\) is 4 units to the right of the origin and 2 units down.
As shown in [Figure 1], the coordinate plane includes all four quadrants with points labeled by their ordered pairs.
The axes split the plane into four regions called quadrants:
For many 6th-grade problems, you focus on the first quadrant, where all coordinates are positive. That matches real-world situations like maps or floor plans, where distances are usually positive. 🏠
To plot a point like \((5, 2)\):
Always move horizontally first (left or right), then vertically (up or down).
A polygon is a closed shape made of straight line segments. Examples are triangles, rectangles, squares, pentagons, and so on.
On the coordinate plane, you can define a polygon by listing the coordinates of its vertices (corner points). For example, a rectangle might have vertices at:
\((1, 1), (5, 1), (5, 4), (1, 4)\).
When you plot and connect these points in order, you get the shape. [Figure 2] shows how a set of points becomes a polygon when they are connected.
Suppose you are given these points:
To draw the polygon:
You now have a rectangle ABCD.
When you connect two points on the coordinate plane, the segment can be:
If two points share the same first coordinate (x), the segment between them is vertical. If they share the same second coordinate (y), the segment is horizontal.
For example:
On the coordinate plane, the axes and grid make it easy to find distances when the segment is horizontal or vertical.
For a horizontal segment, the y-coordinates are the same. So, the distance depends only on how far apart the x-coordinates are.
If you have points \((x_1, y)\) and \((x_2, y)\), the length of the segment between them is:
\[\textrm{Length} = |x_2 - x_1|\]
The symbol \(|x_2 - x_1|\) means the absolute value of the difference. In 6th grade, you can think of this as “the positive difference between the two x-values.”
For a vertical segment, the x-coordinates are the same. So, the distance depends only on how far apart the y-coordinates are.
If you have points \((x, y_1)\) and \((x, y_2)\), the length of the segment between them is:
\[\textrm{Length} = |y_2 - y_1|\]
Again, this is the positive difference between the y-values.
The grid is made of square units. When you move horizontally, you are counting how many units left or right you go. When you move vertically, you count how many units up or down. Subtracting the coordinates measures that change.
Problem: Find the length of the segment between points P \((2, 5)\) and Q \((9, 5)\).
Step 1: Check if it is horizontal.
Both points have y-coordinate 5. So the segment is horizontal.
Step 2: Use the x-coordinates.
x-values are 2 and 9.
Step 3: Subtract the x-values.
\(9 - 2 = 7\)
Answer: The length of segment PQ is 7 units.
Problem: Find the length of the segment between points R \((4, 1)\) and S \((4, 8)\).
Step 1: Check if it is vertical.
Both points have x-coordinate 4. So the segment is vertical.
Step 2: Use the y-coordinates.
y-values are 1 and 8.
Step 3: Subtract the y-values.
\(8 - 1 = 7\)
Answer: The length of segment RS is 7 units.
Problem: A rectangle has vertices at A \((1, 2)\), B \((6, 2)\), C \((6, 7)\), and D \((1, 7)\). Find the lengths of all sides.
Step 1: Look at side AB.
A \((1, 2)\), B \((6, 2)\).
Same y-coordinate (2), so AB is horizontal.
Length of AB is \(6 - 1 = 5\) units.
Step 2: Look at side BC.
B \((6, 2)\), C \((6, 7)\).
Same x-coordinate (6), so BC is vertical.
Length of BC is \(7 - 2 = 5\) units.
Step 3: Look at side CD.
C \((6, 7)\), D \((1, 7)\).
Same y-coordinate (7), so CD is horizontal.
Length of CD is \(|1 - 6| = 5\) units.
Step 4: Look at side DA.
D \((1, 7)\), A \((1, 2)\).
Same x-coordinate (1), so DA is vertical.
Length of DA is \(|2 - 7| = 5\) units.
Answer: All sides have length 5 units. The polygon is a square. ⭐
Once you know side lengths from coordinates, you can find the area of polygons, especially rectangles and right triangles. This connects directly to real-world problems about area, surface area, and volume.
For a rectangle with length \(l\) and width \(w\), the area is:
\[A = l \times w\]
If the side lengths come from coordinate differences, you plug them into this formula.
Problem: On a map, a rectangular garden has corners at G \((2, 1)\), H \((10, 1)\), I \((10, 6)\), and J \((2, 6)\). Each unit on the grid represents 1 meter. Find the area of the garden.
Step 1: Find the length GH.
G \((2, 1)\), H \((10, 1)\).
Same y-coordinate, so GH is horizontal.
Length of GH is \(10 - 2 = 8\) meters.
Step 2: Find the width HI.
H \((10, 1)\), I \((10, 6)\).
Same x-coordinate, so HI is vertical.
Width of HI is \(6 - 1 = 5\) meters.
Step 3: Use the area formula.
\(A = l \times w = 8 \times 5 = 40\).
Answer: The area of the garden is 40 square meters.
Using coordinates and polygons is not just a math exercise. It helps solve real problems.
Imagine your town wants to design a new rectangular skate park shown on a grid. Each grid square is 1 meter by 1 meter. The corners might be at coordinates like \((0, 0)\), \((12, 0)\), \((12, 8)\), and \((0, 8)\).
[Figure 3] shows a coordinate grid layout for a park with labeled corners and dimensions.
Think of a floor plan of your bedroom drawn on graph paper. The corners might be at \((0, 0)\), \((5, 0)\), \((5, 4)\), and \((0, 4)\).
This helps you decide if a piece of furniture fits or how much carpet is needed.
A coach might draw a soccer field on a coordinate grid. If one corner is at \((0, 0)\) and the opposite corner is at \((100, 50)\), then:
This can represent square meters in a real field.
Once you can find the area of rectangles from coordinates, you are ready to connect this to surface area and volume of 3D shapes.
For example, if you know the floor of a rectangular prism (a box) has length 8 m and width 5 m from coordinates, then:
Even though volume formulas are 3D, they start with accurate 2D measurements from things like coordinate grids.
When working with polygons on the coordinate plane, it helps to check:
