Year 6: Positions on the full grid
Part of the Year 6 coordinates, reflection and translation chapter. Chapter overview · whole chapter on one page
Positions on the full grid
A grid can be drawn with an x-axis that goes across and a y-axis that goes up and down. The two axes cross at the origin, which is the point (0, 0).
Points on the grid have coordinates, written as a pair (x, y). The first number is the across number. It is positive to the right of the origin and negative to the left. The second number is the up or down number. It is positive above the origin and negative below it.
The grid is split into four quadrants. They go round anticlockwise, starting at the top right.
| Quadrant | Across number | Up or down number | Example |
|---|---|---|---|
| first (top right) | positive | positive | (3, 2) |
| second (top left) | negative | positive | (-4, 3) |
| third (bottom left) | negative | negative | (-2, -3) |
| fourth (bottom right) | positive | negative | (4, -1) |
Key ideas
- Always write the across number first: (x, y).
- The origin is (0, 0). Points on the x-axis have y = 0. Points on the y-axis have x = 0.
- The first quadrant is top right. The quadrants go round anticlockwise: second, third, then fourth.
- A point on an axis is not in any quadrant.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Read the across number first, then the up or down number, from the origin for point B.
- Start at the origin. The origin is (0, 0), where the two axes cross.(0, 0)
- Read the across number first. Move across along the x-axis. Move left if the number is negative.x = −4
- Read the up or down number. Move up along the y-axis. Move down if the number is negative.y = 3
- Write the coordinates. Always write the across number first, then the up or down number.(-4, 3)
Check. Check on the grid: the point is 4 squares left of the y-axis, and it is 3 squares up from the x-axis. ✓
Watch out. Across is always the first number. Do not swap the two numbers.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Use the signs of the two numbers to find the quadrant.
- Look at the across number. Positive means right of the y-axis. Negative means left of it.x = −2, so left of the y-axis
- Look at the up or down number. Positive means above the x-axis. Negative means below it.y = −3, so below the x-axis
- Name the quadrant. The quadrants go round anticlockwise, starting at the top right.third quadrant
Check. The first quadrant is top right, the second is top left, the third is bottom left and the fourth is bottom right. ✓
Watch out. Look at the signs, not the size of the numbers. Two negative signs make the third quadrant.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Move the across number first, then the up or down number, one move at a time.
- Start at the point. Write the across number and the up or down number of the origin.start = (0, 0)
- Move across first. Moving 3 squares left changes the across number. Take away 3.x: 0 − 3 = −3
- Move up or down. Moving 4 squares up changes the up or down number. Add 4.y: 0 + 4 = 4
- Write the new point. Write the across number first, then the up or down number.(-3, 4)
Check. Move back 3 squares right and 4 squares down and you return to (0, 0). ✓
Watch out. Moving left or down takes away from the number. Moving right or up adds to it.
Watch out
Common mistake: writing the up or down number first, so (3, 2) becomes (2, 3). Always go across first, then up or down. Another mistake is to forget the minus sign for a point to the left of the origin, or below it.
Exercise 11A Practice: reading and naming positions
Non-calculatorKey words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Read the across number first, then the up or down number, from the origin for point A.
- Start at the origin. The origin is (0, 0), where the two axes cross.(0, 0)
- Read the across number first. Move across along the x-axis. Move left if the number is negative.x = 3
- Read the up or down number. Move up along the y-axis. Move down if the number is negative.y = 2
- Write the coordinates. Always write the across number first, then the up or down number.(3, 2)
Check. Check on the grid: the point is 3 squares right of the y-axis, and it is 2 squares up from the x-axis. ✓
Watch out. Across is always the first number. Do not swap the two numbers.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Read the across number first, then the up or down number, from the origin for point D.
- Start at the origin. The origin is (0, 0), where the two axes cross.(0, 0)
- Read the across number first. Move across along the x-axis. Move left if the number is negative.x = 4
- Read the up or down number. Move up along the y-axis. Move down if the number is negative.y = −1
- Write the coordinates. Always write the across number first, then the up or down number.(4, -1)
Check. Check on the grid: the point is 4 squares right of the y-axis, and it is 1 square down from the x-axis. ✓
Watch out. Across is always the first number. Do not swap the two numbers.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Read the across number first, then the up or down number, from the origin for point E.
- Start at the origin. The origin is (0, 0), where the two axes cross.(0, 0)
- Read the across number first. Move across along the x-axis. Move left if the number is negative.x = 0
- Read the up or down number. Move up along the y-axis. Move down if the number is negative.y = 4
- Write the coordinates. Always write the across number first, then the up or down number.(0, 4)
Check. Check on the grid: the point is on the y-axis, and it is 4 squares up from the x-axis. ✓
Watch out. Across is always the first number. Do not swap the two numbers.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Read the across number first, then the up or down number, from the origin for point F.
- Start at the origin. The origin is (0, 0), where the two axes cross.(0, 0)
- Read the across number first. Move across along the x-axis. Move left if the number is negative.x = −5
- Read the up or down number. Move up along the y-axis. Move down if the number is negative.y = −5
- Write the coordinates. Always write the across number first, then the up or down number.(-5, -5)
Check. Check on the grid: the point is 5 squares left of the y-axis, and it is 5 squares down from the x-axis. ✓
Watch out. Across is always the first number. Do not swap the two numbers.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Use the signs of the two numbers to find the quadrant.
- Look at the across number. Positive means right of the y-axis. Negative means left of it.x = 3, so right of the y-axis
- Look at the up or down number. Positive means above the x-axis. Negative means below it.y = 2, so above the x-axis
- Name the quadrant. The quadrants go round anticlockwise, starting at the top right.first quadrant
Check. The first quadrant is top right, the second is top left, the third is bottom left and the fourth is bottom right. ✓
Watch out. Look at the signs, not the size of the numbers. Two negative signs make the third quadrant.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Use the signs of the two numbers to find the quadrant.
- Look at the across number. Positive means right of the y-axis. Negative means left of it.x = −4, so left of the y-axis
- Look at the up or down number. Positive means above the x-axis. Negative means below it.y = 3, so above the x-axis
- Name the quadrant. The quadrants go round anticlockwise, starting at the top right.second quadrant
Check. The first quadrant is top right, the second is top left, the third is bottom left and the fourth is bottom right. ✓
Watch out. Look at the signs, not the size of the numbers. Two negative signs make the third quadrant.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Use the signs of the two numbers to find the quadrant.
- Look at the across number. Positive means right of the y-axis. Negative means left of it.x = −6, so left of the y-axis
- Look at the up or down number. Positive means above the x-axis. Negative means below it.y = 2, so above the x-axis
- Name the quadrant. The quadrants go round anticlockwise, starting at the top right.second quadrant
Check. The first quadrant is top right, the second is top left, the third is bottom left and the fourth is bottom right. ✓
Watch out. Look at the signs, not the size of the numbers. Two negative signs make the third quadrant.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Move the across number first, then the up or down number, one move at a time.
- Start at the point. Write the across number and the up or down number of the origin.start = (0, 0)
- Move across first. Moving 2 squares right changes the across number. Add 2.x: 0 + 2 = 2
- Move up or down. Moving 5 squares down changes the up or down number. Take away 5.y: 0 − 5 = −5
- Write the new point. Write the across number first, then the up or down number.(2, -5)
Check. Move back 2 squares left and 5 squares up and you return to (0, 0). ✓
Watch out. Moving left or down takes away from the number. Moving right or up adds to it.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Move the across number first, then the up or down number, one move at a time.
- Start at the point. Write the across number and the up or down number of the origin.start = (0, 0)
- Move across first. Moving 4 squares left changes the across number. Take away 4.x: 0 − 4 = −4
- Move up or down. Moving 1 square up changes the up or down number. Add 1.y: 0 + 1 = 1
- Write the new point. Write the across number first, then the up or down number.(-4, 1)
Check. Move back 4 squares right and 1 square down and you return to (0, 0). ✓
Watch out. Moving left or down takes away from the number. Moving right or up adds to it.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Move the across number first, then the up or down number, one move at a time.
- Start at the point. Write the across number and the up or down number of the origin.start = (0, 0)
- Move across first. Moving 6 squares left changes the across number. Take away 6.x: 0 − 6 = −6
- There is no move up or down. So the up or down number stays the same.y stays 0
- Write the new point. Write the across number first, then the up or down number.(-6, 0)
Check. Move back 6 squares right and you return to (0, 0). ✓
Watch out. Moving left or down takes away from the number. Moving right or up adds to it.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Compare the signs of each point with the quadrant you need.
- Check the signs of each point. Look at the across number and the up or down number of every point in turn.(3, -4): + across, - up or down (-3, -4): - across, - up or down (-3, 4): - across, + up or down
- Find the point with the right signs. The point (3, -4) has the signs that put it in the fourth quadrant.(3, -4) is in the fourth quadrant
- Write the point. Write the across number first, then the up or down number.(3, -4)
Check. (3, -4) has a positive across number and a negative up or down number. ✓
Watch out. Look at the signs first. Do not choose a point just because its numbers look bigger.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Use the signs of the two numbers to find the quadrant.
- Look at the across number. Positive means right of the y-axis. Negative means left of it.x = 2, so right of the y-axis
- Look at the up or down number. Positive means above the x-axis. Negative means below it.y = −5, so below the x-axis
- Name the quadrant. The quadrants go round anticlockwise, starting at the top right.fourth quadrant
Check. The first quadrant is top right, the second is top left, the third is bottom left and the fourth is bottom right. ✓
Watch out. Look at the signs, not the size of the numbers. Two negative signs make the third quadrant.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Look at the up or down number first, then the across number.
- Look at the up or down number. It is 0, so the point is not above or below the x-axis.y = 0
- Decide where the point is. A point on an axis is on a line, not inside a quadrant.on the x-axis, so no quadrant
- Write the answer. Sam is not right, because the point is not in the third quadrant.No
Check. The quadrants do not include any point on the axes. ✓
Watch out. A point with 0 in it sits on an axis. It is not in a quadrant.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Third quadrant means both numbers are negative. A point 1 square from both axes has 1 as its size in both numbers.
- Use the quadrant. Both numbers are negative in the third quadrant.x < 0 and y < 0
- Find the across number. The point is 1 square from the y-axis, so its across number is -1.x = -1
- Find the up or down number. The point is 1 square from the x-axis, so its up or down number is -1.y = -1
- Write the answer. Write the across number first.(-1, -1)
Check. (-1, -1) is 1 square left of the y-axis and 1 square below the x-axis, in the third quadrant. ✓
Watch out. Both signs are negative in the third quadrant. Do not make either number positive.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Use the signs of the first point to work out the signs of the second.
- Choose a point in the third quadrant. Both numbers are negative. Take P as (-3, -4) to test the rule.P = (-3, -4)
- Keep the across number. Q has the same across number as P, so it is still -3.x = -3
- Change the up or down number. Q has the opposite up or down number, so -4 becomes 4.y = 4
- Find the quadrant of Q. Q is 3 squares left of the y-axis, and 4 squares above the x-axis.second quadrant
Check. Any point in the third quadrant gives the same answer, because the across number stays negative and the up or down number becomes positive. ✓
Watch out. The across number tells you left or right. Changing the sign of the up or down number moves Q above the x-axis.
Watch out
Common mistake: the sign of each number tells you the direction. A point with a negative across number is to the left of the y-axis, even if the number looks big. Check both signs before you name the quadrant.
Exam tip
In a test, write the coordinates with the across number first, and check each sign with a quick picture of the grid.