Coordinates in the First Quadrant

A grid has straight lines that cross to make equal squares. You can see grids on maps, plans, and game boards. The labels around a grid help you find places without searching every square.

Follow a column and a row

A column is a strip of squares going up and down. A row is a strip going across. On many grid maps, columns have letters and rows have numbers. Read the labels on your map to see how it works. Each label sits beside the middle of its strip.

A grid reference names one square. In "B4," the B tells you the column. The 4 tells you the row. Write the letter first, then the number, with no comma.

To find "B4":

1. Put a finger on the label B.
2. Follow column B toward row 4.
3. Follow row 4 across until it meets column B.

The square where they meet is B4. Keep your finger inside that square so you can check its labels.

Give a place its address

On a zoo map, the lion house is in column D, row 2. Its reference is "D2." The café is in "C5." To find it, trace column C and row 5 until they meet.

Now imagine a path and a bench inside D2 too. They share the lion house's reference. A grid reference gives a whole square, so it can name several things inside that square. It does not tell you which tiny spot holds the bench.

If you write "D3" for the lion house, you have picked the next row. Trace the column and row again before writing the reference. A ruler or two fingers can help you stay in the right strips.

A coordinate grid, or coordinate plane, helps you name an exact point. It uses two number lines that meet at a right angle. One number line is an axis. Together, the two lines are the axes.

Meet the axes

The x-axis runs across the bottom. This direction is called horizontal. Its numbers get bigger as you go right.

The y-axis runs up the left side. This direction is called vertical. Its numbers get bigger as you go up.

The axes meet at the origin. Both number lines start at 0 there. We write the origin's position as (0, 0). You will use this starting point when you plot other points.

Numbers belong to lines

On a coordinate grid, each number labels a line. Find 3 on the x-axis and follow its grid line upward. Every point on that line is 3 units to the right of the y-axis. A number written in the middle of a square would leave the exact position unclear.

Keep the steps along each axis equal. The gap from 0 to 1 must match the gap from 1 to 2. Think of the marks on a ruler: equal number changes need equal spacing.

Make your own grid

Use square paper and a ruler.

1. Draw a horizontal line and a vertical line meeting at a corner.
2. Write 0 at their meeting point.
3. Mark equal steps along each line. Number them 1, 2, 3, and onward.
4. Label the horizontal line "x" and the vertical line "y."
5. Draw light grid lines through the number marks.

Put the numbers beside the lines, where you can read them. The first line at the corner is 0; the next line is 1.

Find the first quadrant

The axes divide a whole coordinate plane into four parts, called quadrants. The first quadrant is above the x-axis and right of the y-axis. Inside it, both coordinates are greater than 0.

Your first-quadrant grid also shows the axes along its edges. Points on those edges have a zero coordinate. They lie on an axis at the edge of the quadrant.

Compare the two kinds of address:

Coordinates are two numbers that give a point's exact position. We write them in parentheses with a comma between them, like (5, 3). The first number is the x-coordinate. The second is the y-coordinate. Their order matters, so we call them an ordered pair.

Read across, then up

The x-coordinate tells you how far right the point is from the origin. The y-coordinate tells you how far up it is. To read a plotted point:

1. Look straight down from the point to the x-axis. Read x.
2. Look straight across from the point to the y-axis. Read y.
3. Write both numbers in the order (x, y).

A ruler can help you follow a line without slipping onto its neighbor.

Read point A

Point A lies on the line through 5 on the x-axis. It also lies on the line through 3 on the y-axis.

• Its x-coordinate is 5.
• Its y-coordinate is 3.
• Its coordinates are (5, 3).

You can write the point's name beside the pair: A(5, 3). A useful reminder is "Walk along the hallway, then climb the stairs." Across comes first. The letters help too: x comes before y.

Swapping the numbers changes this point. (3, 5) is closer to the y-axis and higher than (5, 3). When both numbers are equal, swapping them leaves the pair unchanged. For example, (4, 4) stays (4, 4).

Read points on the edges

At (6, 0), the point is 6 units right and has no height above the x-axis. It lies on the x-axis.

At (0, 2), the point has no distance across from the y-axis. It lies on the y-axis, 2 units up.

At the origin, both numbers are 0. Keep both numbers even for an axis point. Writing just "6" loses the information that (6, 0) is on the x-axis.

Use the number on the line that passes through the dot. If you slip to the next line, the answer changes by one grid step. That is 1 unit on an axis counting in 1s, or 5 units on an axis counting in 5s.

To plot a point means to mark its exact place on a grid. The ordered pair gives you two distances from the origin. Begin with the first number, which tells you how far to go across.

Plot B(2, 6)

Use a grid whose axes count in 1s.

1. Start at the origin, (0, 0).
2. Move 2 units right along the x-axis. You reach (2, 0).
3. From there, move 6 units straight up.
4. Mark that spot with a small dot or cross.
5. Write B beside the mark.

B sits where the line through 2 across meets the line through 6 up. Leave the letter beside the dot so it does not hide the exact position.

Count the jumps between grid lines. Starting at 0, the first jump takes you to 1. The second takes you to 2. If you count the starting line as your first jump, you stop too soon.

Check by reading back

Look straight down from B: you should read 2. Look straight across: you should read 6. The pair is (2, 6), which matches the pair you wanted.

If you read (6, 2), you swapped the two distances. Keep the first number with the across move. Across first, then up, helps you keep that order while plotting.

Let zero do its job

To plot C(0, 4), stay at 0 across. Move straight up the y-axis to 4 and mark C there.

To plot D(7, 0), move 7 units right along the x-axis. The second number is 0, so stay on that axis and mark D.

For (0, 0), make the mark at the origin. There is no move in either direction.

Plot more than one point

Start at the origin again for each new ordered pair. Label each point as soon as you mark it. This keeps the names attached to the right dots.

Whole-number coordinates fall on line crossings when the grid counts in 1s. A halfway coordinate can put a dot between lines. Mark the exact place the numbers describe, even when it is inside a square.

A grid step is not always worth 1. The scale tells you how much each equal gap is worth.

Find the value of one gap

An axis labeled 0, 5, 10, 15 at neighboring lines counts in 5s. Each gap is worth 5 units. Three gaps take you from 0 to 15.

Sometimes labels skip a line. Imagine 0, 10, 20, and 30, with one unlabeled line between each pair. From 0 to 10 there are two equal gaps. Each gap is worth 5. The unlabeled line between 20 and 30 is 25.

Read a point on a scaled grid

Both axes count in 2s. Point P lies above the x-axis label 6 and beside the y-axis label 10.

1. Read the across value: 6.
2. Read the up value: 10.
3. Write P(6, 10).

There are three gaps across and five gaps up. Writing (3, 5) would give the gap counts. The coordinates use the labeled values.

Check each axis separately

The x-axis might count in 1s while the y-axis counts in 10s. A point three gaps right and four gaps up is then (3, 40). The same number of paper gaps can have different values in the two directions.

To plot (3, 40) on that grid, go three gaps right. Then go four gaps up, counting 10, 20, 30, 40. Read back from the dot to check both values.

Find a halfway point

A point can sit between grid lines. Halfway between 4 and 6 is 5. On a grid counting in 2s, that is halfway across one gap.

On an axis counting in 1s, halfway between 2 and 3 is 2.5, or two and a half. A point there and 4 units up has coordinates (2.5, 4).

When drawing a grid for values up to 50, steps of 5 or 10 can save space. Keep the same step size along each axis. Label enough lines to make the scale clear.

You can compare points just by reading their coordinates. The first number tells you about left and right. The second tells you about height.

Find the higher point

Compare A(3, 7) and B(8, 2).

• B is farther right because 8 is greater than 3.
• A is higher because 7 is greater than 2.

Neither point has to be both higher and farther right. Look at the coordinate that answers your question. A large first number does not make a point high.

Find points on the same line

The points (4, 1), (4, 3), and (4, 6) share the same x-coordinate. They sit straight above one another on a vertical line. Their heights differ, but their across positions match.

The points (1, 5), (3, 5), and (7, 5) share the same y-coordinate. They sit side by side on a horizontal line. Their across positions differ, but their heights match.

You can use this to picture a point without a drawing. (10, 10) is straight above (10, 0) and level with (0, 10).

Measure between aligned points

For a horizontal distance, subtract the x-coordinates. From (2, 5) to (7, 5), the distance is 7 − 2 = 5 units.

For a vertical distance, subtract the y-coordinates. From (4, 1) to (4, 6), the distance is 6 − 1 = 5 units.

Use the larger value minus the smaller to find the length. The shared coordinate stays the same along that segment.

Look for a repeated move

The pattern (1, 1), (2, 2), (3, 3) goes 1 right and 1 up each time. The next point is (4, 4). These points lie on a straight slanted line.

Another pattern is (1, 2), (2, 4), (3, 6). Here the first number grows by 1, but the second grows by 2. The next point is (4, 8).

Follow both parts of a pattern separately. If you add 1 to both numbers in that second pattern, you get (4, 7). That misses the repeated rise of 2.

A polygon is a flat, closed shape made of straight sides. Its corners are called vertices. One corner is a vertex. Each corner can have its own ordered pair.

Draw a rectangle from four points

Use a ruler and a grid counting in 1s. Plot A(1, 1), B(6, 1), C(6, 4), and D(1, 4).

1. Join A to B. Their second numbers match, so AB runs across.
2. Join B to C. Their first numbers match, so BC runs up.
3. Join C to D, making the top side.
4. Join D back to A to close the shape.

AB and DC are each 5 units long. BC and AD are each 3 units long. The horizontal and vertical sides meet at right angles, making a rectangle.

Keep the joining order. If you join A to C, then B, then D, two sides cross like a bow tie. The correct dots alone are not enough; the joining instructions matter too.

Make a triangle

Plot P(2, 1), Q(8, 1), and R(5, 6). Join P to Q, Q to R, and R back to P.

There are three straight sides, so the shape is a triangle. R is straight above the middle of PQ: 5 is halfway between 2 and 8. The two sides from R to P and Q have the same length. This is an isosceles triangle.

Look at the sides

Two points joined by a straight line make a line segment. It has two ends.

• (1, 3) to (7, 3) gives a horizontal segment.
• (2, 1) to (2, 5) gives a vertical segment.
• (1, 1) to (5, 5) gives a slanted segment.

To name a polygon, count its sides. Then look for right angles, equal sides, and parallel sides. Parallel sides stay the same distance apart and never meet, even if you extend them.

Plot and join (1, 1), (7, 1), (5, 4), and (3, 4), in that order. The bottom side is 6 units long and the top is 2. They are parallel, while the slanted sides are not parallel. This shape is a trapezoid.

If a shape looks wrong, read each corner back from the grid. Swapping one corner's coordinates can turn a straight-across side into a slanted one.

Coordinates give positions. To find the length between two positions, work out their difference. This helps you complete and measure shapes whose sides follow the grid lines.

Measure a side

A horizontal side goes from (2, 3) to (9, 3). Its height stays 3. Its length is 9 − 2 = 7 units.

A vertical side goes from (5, 1) to (5, 8). Its across position stays 5. Its length is 8 − 1 = 7 units.

Count jumps if you check on a grid. On a grid counting in 1s, the lines from 2 through 8 number seven. But there are only six gaps, so the distance is 6 units.

A slanted side needs a different way of measuring. Counting the squares it passes through does not give its length. A diagonal across one square is longer than one side of that square.

Complete a rectangle

Three corners are (2, 1), (8, 1), and (8, 5). The sides run along grid lines.

The missing corner must be straight above (2, 1). That gives it x = 2. It must also be level with (8, 5). That gives it y = 5. Put those clues together: the missing vertex is (2, 5).

Check the size:

• Width: 8 − 2 = 6 units.
• Height: 5 − 1 = 4 units.

Both horizontal sides have length 6. Both vertical sides have length 4.

Complete a square

A square has four equal sides. Its bottom corners are (3, 2) and (7, 2). The bottom side is 7 − 3 = 4 units long.

The top must be 4 units higher than the bottom. Add 4 to the height 2: the top has y = 6. The top corners are (3, 6) and (7, 6).

The middle of the bottom side is (5, 2). Its x-coordinate is halfway between 3 and 7, and its height stays 2.

Find the distance around and the space inside

Return to the rectangle that is 6 units wide and 4 units tall.

Its perimeter is the distance around all four sides: 6 + 4 + 6 + 4 = 20 units.

Its area is the space inside. It has 4 rows of 6 unit squares: 6 × 4 = 24 square units. Use units for length and square units for area.

A translation is a slide. Every point moves the same distance in the same direction. There is no turning or flipping. You can describe the slide with an across part and an up-or-down part.

Compare start and finish

Point A(2, 3) moves to B(7, 5).

First compare the x-coordinates. The number changes from 2 to 7. That is 7 − 2 = 5 units right.

Then compare the y-coordinates. The number changes from 3 to 5. That is 5 − 3 = 2 units up.

The translation is "5 units right and 2 units up." Give the across part first, matching the x-then-y order of coordinates.

Describe left and down

Point C(6, 8) moves to D(1, 4).

The x-coordinate gets smaller, from 6 to 1. That is 6 − 1 = 5 units left. The y-coordinate gets smaller, from 8 to 4. That is 8 − 4 = 4 units down.

Write "5 units left and 4 units down." The subtraction gives the amount. Comparing the start and finish tells you the direction.

Use units carefully

One unit means a change of 1 in a coordinate. On a grid counting in 1s, each gap is one unit. On a grid counting in 2s, each gap is two units.

For example, a horizontal move from x = 2 to x = 8 is 6 units. It spans just three gaps on an axis counting in 2s. Use the coordinate values when you describe a slide.

Notice when one part stays still

From (3, 2) to (3, 9), the x-coordinate stays 3. Only the height changes: 9 − 2 = 7 units. You can say "7 units up."

Check which point is the start. Returning from B to A reverses the earlier slide: it is 5 units left and 2 units down. The amounts stay the same, but both directions reverse.

A description such as "5 squares diagonally" leaves the across and upward moves unclear. Naming each direction and amount tells someone exactly how to make the slide.

You can find a translated point by changing its coordinates. A right or left move changes x. An up or down move changes y.

Add for right and up

Translate P(4, 2) by 3 units right and 5 units up.

1. Add 3 to the x-coordinate: 4 + 3 = 7.
2. Add 5 to the y-coordinate: 2 + 5 = 7.
3. Write the new pair: (7, 7).

On a grid counting in 1s, start at P and count three jumps right. Then count five jumps up. Read the new point back to check your calculation.

Subtract for left and down

Translate Q(9, 6) by 4 units left and 2 units down.

1. Subtract 4 from x: 9 − 4 = 5.
2. Subtract 2 from y: 6 − 2 = 4.
3. The new point is (5, 4).

Keep each move with its own coordinate. Adding an across move to y would change the height, so it would send the point the wrong way.

If you check by counting gaps, look at the scales first. On an axis counting in 2s, a move of 4 units uses two gaps. Use the number values to decide how many gaps to travel.

Stop at an axis

From (2, 5), moving 2 units left lands at (0, 5). The point is now on the y-axis. It has not disappeared: 0 is its new across coordinate.

A move of 3 units left from that starting point would go past the y-axis. It would leave the first-quadrant grid. You can move down only as far as the x-axis while staying on this grid too.

Work backward to the start

A point moved 2 units right and 1 unit up. It landed at (6, 4). To find its start, undo both moves:

• Go 2 units left: 6 − 2 = 4.
• Go 1 unit down: 4 − 1 = 3.

The starting point was (4, 3). Check by going forward again: (4, 3) becomes (6, 4) after 2 right and 1 up.

To translate a shape, slide every vertex the same way. Then join the new vertices in their original order. The new shape is called the image.

Move a triangle

Triangle ABC has A(1, 1), B(4, 1), and C(1, 3). Translate it 5 units right and 2 units up.

• A: add 5 to 1 and 2 to 1. Its image is (6, 3).
• B: add 5 to 4 and 2 to 1. Its image is (9, 3).
• C: add 5 to 1 and 2 to 3. Its image is (6, 5).

Plot those three points and join them in the same order. Compare the bottom sides. AB is 4 − 1 = 3 units long. The new bottom side is 9 − 6 = 3 units long too.

Keep the shape together

Think of sliding a book across a desk. All its corners travel together. The book keeps its size and faces the same way.

A translation keeps every side length and angle unchanged. If one vertex travels a different amount, a side may stretch or tilt. If a shape turns or flips, that movement is not a translation.

Find a slide between two shapes

Match the same corner on each shape. For example, compare the bottom-left corner with the new bottom-left corner.

A square's bottom-left corner moves from (2, 1) to (5, 6). That gives 3 units right and 5 units up. Its top-right corner started at (4, 3). The same slide should put it at (7, 8): add 3 across and 5 up.

Check a second matching pair to make sure both corners give the same slide. Measuring the empty gap between shapes can give a different distance, because the nearest corners may not match.

Combine two slides

You can carry out two translations, one after another. A move of 2 units right followed by 3 units right gives 5 units right altogether. A move of 4 units up followed by 1 unit down gives 3 units up altogether.

So those moves combine into "5 units right and 3 units up." Apply that combined slide to every vertex to get the same final image.

A coordinate map gives places exact positions. A gate might be at (0, 3), on the y-axis. A fountain might be at (5, 4), inside the first quadrant. Use the origin to read each place's coordinates.

Travel between places

On a park map, the playground is at (2, 6). The pond is at (7, 4), and the café is at (7, 1). Paths follow the grid lines.

To go from the playground to the pond, compare the coordinates:

1. Across: 7 − 2 = 5 units right.
2. Down: 6 − 4 = 2 units down.

One route goes right first, then down. From the pond to the café, the across coordinate stays 7. Go 4 − 1 = 3 units down.

Add the parts of the route

The whole route travels 5 units right, 2 down, and another 3 down. Its length is 5 + 2 + 3 = 10 units.

Add every part you walk. A pair such as (7, 1) names the café's position from the origin. It does not give the distance you walked from the playground.

More than one route can reach a place. From (1, 1) to (4, 3), you can go 3 right then 2 up. Or you can go 2 up then 3 right. Both routes reach (4, 3), and both are 5 units long along the grid lines.

Use compass directions

On a map with north at the top:

• Up is north.
• Down is south.
• Right is east.
• Left is west.

The route from the playground to the pond can be called "5 units east and 2 units south." Check the map's north arrow before using those names.

Give an exact target

In a hidden-ships grid game, you can call a target with an ordered pair. Your partner looks for that exact point. Calling (3, 5) when you meant (5, 3) sends your partner somewhere else.

Keep a place and a move separate. "The café is at (7, 1)" tells you where it is. "Go 3 units down" tells you how to reach it from the pond.

Some pairs of numbers belong together. A table can show how many seats go with each number of tables. You can turn each row into an ordered pair and plot it.

Build a growing pattern

One small square table seats 4 people. Push tables together in a row. Each extra table adds 2 seats.

Choose tables for the x-axis and seats for the y-axis. Label them "Number of tables" and "Number of seats." The labels explain what the numbers mean.

Make pairs and plot

Read each row across. Write (tables, seats): (1, 4), (2, 6), (3, 8), and (4, 10).

To plot (3, 8), find 3 across and 8 up. That dot means "3 tables seat 8 people." Swapping it to (8, 3) would mean "8 tables seat 3 people," which changes the story.

Choose scales that fit your largest numbers. If you want to show up to 22 seats, the y-axis could count in 2s from 0 to 24. The x-axis can still count in 1s for tables.

Use the pattern

The plotted points line up. Every extra table takes you 1 unit right and 2 units up. After (4, 10), the next point is (5, 12). It means "5 tables seat 12 people."

The pattern helps you predict a new pair. Explain your answer with both quantities, so someone knows which number counts tables and which counts seats.

Decide whether to join the dots

Whole tables are counted one at a time. In this seating pattern, two and a half tables is not an allowed choice. Leave the points as separate dots. A line between them would include table counts that do not fit the pattern's choices.

In other situations, in-between values can make sense. A graph of hours and miles walked can include part of an hour. Think about what the quantities mean before joining points.

If hours are across and miles are up, a point at (2, 6) means "After 2 hours, 6 miles had been walked." Read the axis labels before telling the point's story.