Read Fractional Coordinates and Reflect Points | Grade 6 Math | LevelKite

A point on the coordinate plane has an address made of two numbers. You already know how to plot a point from its address. Now you will work the other way, and then go further.

Here are the words you need. The coordinate plane is a grid built on two number lines. The x-axis is the horizontal one and the y-axis is the vertical one. They cross at the origin, $(0, 0)$ , and they split the plane into four regions called quadrants. The distance from $0$ to $1$ on either axis is 1 unit.

The address is an ordered pair $(x, y)$ . The first number, $x$ , is the x-coordinate. It tells how far the point is to the right of the y-axis (positive) or to the left of it (negative). The second number, $y$ , is the y-coordinate. It tells how far the point is above the x-axis (positive) or below it (negative).

You will learn to:

read the ordered pair of a point that is already plotted,
plot points whose coordinates are fractions or decimals,
reflect a point across an axis, which means flip it over the axis like a mirror image.

Reading a point is plotting in reverse. When you plot, you are told the moves and you make them. When you read, the point is already there and you count the moves.

To count exactly, you need the grid step. That is the distance between two neighboring grid lines, measured in units. Moving from one grid line to the next is one step. Every figure in this lesson states its grid step, and every point sits exactly where two grid lines cross. So you never have to estimate by eye.

To read a point on a grid like that:

Start at the origin. Move right or left along the x-axis until you reach the vertical grid line through the point. Count the steps. Right makes $x$ positive and left makes $x$ negative.
Move up or down that grid line to the point. Count the steps. Up makes $y$ positive and down makes $y$ negative.
Write $(x, y)$ with $x$ first. If a count is 0 steps, that coordinate is $0$ .

A count is not yet a coordinate. First turn it into a distance in units with the rule below. Then give it the sign for its direction.

\text{distance in units} = \text{number of steps} \times \text{grid step}

The grid step is

1

unit. Point A and Point B each sit where two grid lines cross.

Worked example

Read points on a grid with step 1

In the figure above, the grid step is $1$ unit. Write the coordinates of Point A and Point B.

Find x for Point A. Start at the origin and move left along the x-axis until you reach the vertical grid line through Point A. That takes 4 steps. Each step is 1 unit, so the distance from the y-axis is 4 units. You moved left, so x is negative.
$4 \times 1 = 4, \qquad x = -4$
Find y for Point A. Move up that grid line to the point. That takes 3 steps of 1 unit, so the distance from the x-axis is 3 units. You moved up, so y is positive.
$3 \times 1 = 3, \qquad y = 3$
Write the pair with x first. The order matters: the pair (3, -4) would be a different point, 3 units right and 4 units down. Then check the signs. A negative x and a positive y mean left of the y-axis and above the x-axis, which is where Point A sits.
$(x, y) = (-4, 3)$
Read Point B. It is on the y-axis, so you move 0 steps right or left, and x is 0. Then move 5 steps down the y-axis to reach the point. The distance from the x-axis is 5 units, and you moved down, so y is negative.
$5 \times 1 = 5, \qquad (x, y) = (0, -5)$

Answer

Point A is at $(-4, 3)$ and Point B is at $(0, -5)$ .

A point does not have to sit on whole numbers. A coordinate such as $1.5$ or $-2\tfrac{1}{2}$ names a place between two whole numbers on an axis, just as it does on a number line.

To show such a point exactly, a grid can use a smaller grid step. Suppose the grid step is $0.5$ unit. Then each step is half a unit, and 2 steps make 1 whole unit. Every coordinate that is a multiple of $0.5$ , such as $1.5$ , $-2$ or $2.5$ , lands exactly on a grid line. The rule does not change: 3 steps are $3 \times 0.5 = 1.5$ units.

To plot, you go the other way. You know the distance and you need the number of steps, so you divide by the grid step. Dividing undoes the multiplying: $1.5 \div 0.5 = 3$ because $3 \times 0.5 = 1.5$ .

A fraction and a decimal can name the same number, so they name the same place. For example, $\tfrac{1}{2} = \tfrac{5}{10} = 0.5$ , so $2\tfrac{1}{2} = 2 + 0.5 = 2.5$ . Write a coordinate in the form that matches the grid step.

Worked example

Graph decimal and fraction coordinates

A grid has a grid step of $0.5$ unit. Graph Point C at $(1.5, -2)$ and Point D at $\left(-2\tfrac{1}{2}, \tfrac{1}{2}\right)$ .

Find the horizontal move for Point C. Its x-coordinate is 1.5, so the point is 1.5 units to the right of the y-axis. Divide that distance by the grid step to get the number of steps. Then move that many steps right from the origin.
$1.5 \div 0.5 = 3$
Find the vertical move for Point C. Its y-coordinate is negative 2, so the point is 2 units below the x-axis. Divide by the grid step. Then move that many steps down and mark Point C.
$2 \div 0.5 = 4$
Rewrite the coordinates of Point D as decimals so they match the grid step. The numbers keep their values. Only the way they are written changes.
$-2\tfrac{1}{2} = -2.5, \qquad \tfrac{1}{2} = 0.5$
Find the moves for Point D. Its x-coordinate is negative 2.5, so the point is 2.5 units left of the y-axis. Move 5 steps left from the origin. Its y-coordinate is 0.5, so the point is 0.5 unit above the x-axis. Then move 1 step up and mark Point D.
$2.5 \div 0.5 = 5, \qquad 0.5 \div 0.5 = 1$
Check by reading both points back. Multiply each count by the grid step and give it the sign for its direction. You get the coordinates you started with.
$3 \times 0.5 = 1.5, \quad 4 \times 0.5 = 2, \quad 5 \times 0.5 = 2.5, \quad 1 \times 0.5 = 0.5$

Answer

Point C is 3 steps right and 4 steps down from the origin, at $(1.5, -2)$ . Point D is 5 steps left and 1 step up, at $\left(-2\tfrac{1}{2}, \tfrac{1}{2}\right)$ . The figure below shows both points.

The grid step is

0.5

unit. Point C is at

(1.5, -2)

and Point D is at

(-2.5, 0.5)

. Point E is read in the next example.

Worked example

Read a point on a grid with step 0.5

In the figure above, the grid step is $0.5$ unit. Write the coordinates of Point E.

Find x. From the origin, move left along the x-axis to the vertical grid line through Point E. That takes 3 steps. Multiply by the grid step to get the distance from the y-axis. You moved left, so x is negative.
$3 \times 0.5 = 1.5, \qquad x = -1.5$
Find y. Move down that grid line to the point. That takes 5 steps. Multiply by the grid step to get the distance from the x-axis. You moved down, so y is negative.
$5 \times 0.5 = 2.5, \qquad y = -2.5$
Write the pair with x first. You can also write each coordinate as a mixed number, because one half equals 0.5. Both forms name the same point.
$(-1.5, -2.5) = \left(-1\tfrac{1}{2}, -2\tfrac{1}{2}\right)$

Answer

Point E is at $(-1.5, -2.5)$ , which is the same point as $\left(-1\tfrac{1}{2}, -2\tfrac{1}{2}\right)$ .

Now you can use coordinates to flip a point. A reflection across an axis flips a point over that axis, as if the axis were a mirror. The axis is the mirror line, and the new point is the image.

A mirror keeps distances. The image is straight across the mirror line from the original point, at a right angle to the line. It is exactly as far from the mirror line as the original point, on the other side. A point that is on the mirror line stays where it is.

So a reflection across the x-axis can only move a point straight up or straight down. A reflection across the y-axis can only move it straight left or straight right.

One more word: two numbers are opposites when they are the same distance from $0$ on different sides of it, like $1.5$ and $-1.5$ . The opposite of $0$ is $0$ .

Worked example

Reflect a point across each axis

A grid has a grid step of $0.5$ unit. Point F is at $(2.5, 1.5)$ . Find Point G, the reflection of Point F across the x-axis. Then find Point H, the reflection of Point F across the y-axis.

For Point G the mirror line is the x-axis, so measure Point F from the x-axis. Its y-coordinate is 1.5, so Point F is 1.5 units above the x-axis. On this grid that is 3 steps.
$1.5 \div 0.5 = 3$
Put the image the same distance on the other side: 3 steps below the x-axis. A point 1.5 units below the x-axis has a y-coordinate of negative 1.5, which is the opposite of 1.5.
$3 \times 0.5 = 1.5, \qquad y = -1.5$
Keep the x-coordinate. The point moved straight down, so it is still 2.5 units to the right of the y-axis.
$(x, y) = (2.5, -1.5)$
For Point H the mirror line is the y-axis, so measure Point F from the y-axis. Its x-coordinate is 2.5, so Point F is 2.5 units to the right of the y-axis. That is 5 steps.
$2.5 \div 0.5 = 5$
Put the image 5 steps to the left of the y-axis. A point 2.5 units left of the y-axis has an x-coordinate of negative 2.5, the opposite of 2.5. The point moved straight left, so y stays 1.5.
$(x, y) = (-2.5, 1.5)$

Answer

The reflection of $(2.5, 1.5)$ across the x-axis is Point G at $(2.5, -1.5)$ . The reflection across the y-axis is Point H at $(-2.5, 1.5)$ . The figure below shows all three points.

The grid step is

0.5

unit. Point F is at

(2.5, 1.5)

. Point G at

(2.5, -1.5)

is its image across the x-axis. Point H at

(-2.5, 1.5)

is its image across the y-axis.

Key concept

A coordinate gives a distance and a side

For a point at $(x, y)$ , the number $x$ without its sign is the distance from the y-axis. Its sign gives the side: positive is right, negative is left, and $x = 0$ is on the y-axis. The number $y$ without its sign is the distance from the x-axis: positive is above, negative is below, and $y = 0$ is on the x-axis. This is true for whole numbers, fractions and decimals.

A reflection across the x-axis or the y-axis keeps both distances and moves the point to the other side of the mirror axis only. So one coordinate is replaced by its opposite and the other stays the same. In the formula, $-y$ means the opposite of $y$ and $-x$ means the opposite of $x$ . A point on the mirror axis does not move: the coordinate being replaced is $0$ , and the opposite of $0$ is $0$ .

(x, y) \to (x, -y) \text{ across the x-axis}, \qquad (x, y) \to (-x, y) \text{ across the y-axis}

The reflection rule used on four points

Point	Image across the x-axis	Image across the y-axis
$(2.5, 1.5)$	$(2.5, -1.5)$	$(-2.5, 1.5)$
$\left(-1\tfrac{1}{2}, 2\right)$	$\left(-1\tfrac{1}{2}, -2\right)$	$\left(1\tfrac{1}{2}, 2\right)$
$(-3, -0.5)$	$(-3, 0.5)$	$(3, -0.5)$
$(0, 2.5)$	$(0, -2.5)$	$(0, 2.5)$ , the same point, because it is on the y-axis

Here is the idea to keep. The coordinates of a point tell you its distance from each axis and which side it is on. When the grid step is stated and the point sits where two grid lines cross, each distance is the number of steps times the grid step. Multiply to read a point, and divide to plot one. This works the same way for whole numbers, fractions and decimals. To reflect a point across the x-axis or the y-axis, keep its distance from that axis and put it on the other side: across the x-axis $y$ becomes its opposite, and across the y-axis $x$ becomes its opposite. A point on the mirror axis stays where it is.