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Primary Maths · Year 4Chapter 10 · Position and direction
Year 4 · Chapter 10

Year 4 Coordinates and Translation: worked examples and questions with answers

By the end of this chapter you will be able to

  • Read and write the coordinates of a point, with the x-coordinate (along) first and the y-coordinate (up) second
  • Plot points on a grid, join them to make a polygon and name the shape
  • Find a missing corner of a square, rectangle, parallelogram, kite or isosceles triangle, and a missing coordinate on a horizontal or vertical line
  • Describe a translation as a number of units left or right and up or down, and write the new coordinates
  • Translate a shape on a grid and describe the move between two shapes
  • Use grid references and the eight compass points to describe where things are on a map
  • Follow and write routes and robot programs using forward, turn left and turn right, and solve treasure-map problems

You should already know

  • Counting along a number line to 12 (Years 1 to 3)
  • Names of 2-D shapes: squares, rectangles, triangles and kites (Chapter 9)
  • Right angles and quarter turns (Chapter 9)
  • Left, right, up and down; North, East, South and West (Chapter 9)
  • Adding and subtracting small numbers (Chapter 2)

Where this leads

  • Statistics: bar charts and graphs with axes (Chapter 11)
  • Reflection and translation of shapes (Year 5)
  • Coordinates in all four quadrants, with negative numbers (Year 6)
  • Drawing and translating shapes on the full coordinate plane (Year 6)

National Curriculum for England, Key Stage 2 mathematics, Year 4. Content: Describe positions on a 2-D grid as coordinates in the first quadrant, Describe movements between positions as translations of a given unit to the left/right and up/down, Plot specified points and draw sides to complete a given polygon.

Printable worksheets. Every lesson in this chapter has a free worksheet and an answer sheet to download and print.

Exercise P  Do you remember?

Non-calculator
1
FHow many right angles are there in a half turn?
2
FStart at 2 on a number line. Count on 3. Then count back 4. What number do you reach?
3
FYou are facing north. You make a quarter turn clockwise. Which way are you facing now?
4
FWhat is the name of the shape with four equal sides and four right angles?
5
FWhat is the opposite direction to north on a compass?
10.1

Reading coordinates

Year 4
Describe positions on a 2-D grid as coordinates in the first quadrant

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Reading coordinatesComing soon

Imagine you are at the front door of a school. To get to your classroom, you walk along the corridor first. Then you go up the stairs. Coordinates work in the same way. They are two numbers that tell you exactly where a point is on a grid: along first, then up.

012345612345xyA (4, 3)along 4up 3
To find A: start at the origin, go 4 along (orange), then 3 up (blue). A is at (4, 3).

The two number lines are called the axes. The x-axis goes along the bottom. The y-axis goes up the side. They meet at the origin, which is the point (0, 0). Every point is counted from the origin. The grids in this chapter show the top-right part of the full coordinate grid, where all the numbers are 0 or bigger. This part is called the first quadrant.

We write the coordinates in brackets with a comma between them: (4, 3). The first number is the x-coordinate (how far along). The second number is the y-coordinate (how far up). On these grids the numbers are written on the lines, not in the squares. A point is where two lines cross.

0123456123456xy(3, 5)(3, 5)0123456123456xy(5, 3)(5, 3)
The order matters. (3, 5) is 3 along and 5 up. (5, 3) is 5 along and 3 up. They are different points.

Some points are on an axis. A point on the x-axis has 0 for its up number, like (5, 0). A point on the y-axis has 0 for its along number, like (0, 4). The origin has both numbers zero: (0, 0).

Key ideas

  • Coordinates are written (along, up), in brackets, with a comma. The first number is the x-coordinate. The second number is the y-coordinate.
  • The origin is (0, 0). The x-axis goes along. The y-axis goes up.
  • Always go along first, then up: “along the corridor, then up the stairs”.
  • Points on the x-axis have y = 0. Points on the y-axis have x = 0.
Worked example 10.1.1
Write the coordinates of point A.
0123456123456xyABC

Key words. Coordinates are two numbers in brackets, like (3, 5). They tell us exactly where a point is. The x-axis is the line going along the bottom. The y-axis is the line going up the side. The origin is the point (0, 0) where the two axes meet.

Method. Start at the origin. Count how far along (the x-axis) and then how far up (the y-axis) to reach the point. Write the two numbers in brackets, along first.

  1. Start at the origin. The origin is the point where the x-axis and the y-axis meet. We count every position from here.
    (0, 0)
  2. Go along the x-axis. The first number tells us how far to go along (think: along the corridor first). Look straight down from the point to the x-axis.
    A is above the number 4 on the x-axis.
    Along = 4
  3. Go up the y-axis. The second number tells us how far to go up (then up the stairs). Look straight across from the point to the y-axis.
    A is level with the number 3 on the y-axis.
    Up = 3
  4. Write the coordinates. We always write the along number first, then the up number, in brackets with a comma between them.
    A = (4, 3)
    0123456123456xyBCA

Check. Start at (0, 0). Go 4 along and 3 up. You arrive at A. ✓

Watch out. Along first, then up. (3, 5) and (5, 3) are two different points.

Worked example 10.1.2
Write the coordinates of point B.
012345612345xyABCD

Key words. Coordinates are two numbers in brackets, like (3, 5). They tell us exactly where a point is. The x-axis is the line going along the bottom. The y-axis is the line going up the side. The origin is the point (0, 0) where the two axes meet.

Method. Start at the origin. Count how far along (the x-axis) and then how far up (the y-axis) to reach the point. Write the two numbers in brackets, along first.

  1. Start at the origin. The origin is the point where the x-axis and the y-axis meet. We count every position from here.
    (0, 0)
  2. Go along the x-axis. The first number tells us how far to go along (think: along the corridor first). Look straight down from the point to the x-axis.
    B is above the number 5 on the x-axis.
    Along = 5
  3. Go up the y-axis. The second number tells us how far to go up (then up the stairs). Look straight across from the point to the y-axis.
    B is on the x-axis, so we go 0 up.
    Up = 0
  4. Write the coordinates. We always write the along number first, then the up number, in brackets with a comma between them.
    B = (5, 0)
    012345612345xyACDB

Check. Start at (0, 0). Go 5 along and 0 up. You arrive at B. ✓

Watch out. Along first, then up. (3, 5) and (5, 3) are two different points.

Worked example 10.1.3
Which letter is at the point (6, 5)?
01234567812345678xyABCDEF

Key words. Coordinates are two numbers in brackets, like (3, 5). They tell us exactly where a point is. The x-axis is the line going along the bottom. The y-axis is the line going up the side. The origin is the point (0, 0) where the two axes meet.

Method. Start at the origin. Go along first, then up. The letter you land on is the answer.

  1. Start at the origin. Every position is measured from the point (0, 0).
    (0, 0)
  2. Go 6 along the x-axis. The first number is the along number.
    (6, 5): along = 6
  3. Go 5 up. The second number is the up number.
    Up = 5
  4. Read the letter at the point. We stop where the two moves meet.
    The letter is C.
    01234567812345678xyABDEFC

Check. Point C is 6 along and 5 up, which is (6, 5). ✓

Watch out. Do not go up first. The first number is always the along number.

Worked example 10.1.4
Mei made a map of her village. The letters show places. What are the coordinates of the school, S?
0123456789101112123456789101112xySTU

Key words. Coordinates are two numbers in brackets, like (3, 5). They tell us exactly where a point is. The x-axis is the line going along the bottom. The y-axis is the line going up the side. The origin is the point (0, 0) where the two axes meet.

Method. Start at the origin. Count how far along (the x-axis) and then how far up (the y-axis) to reach the point. Write the two numbers in brackets, along first.

  1. Start at the origin. The origin is the point where the x-axis and the y-axis meet. We count every position from here.
    (0, 0)
  2. Go along the x-axis. The first number tells us how far to go along (think: along the corridor first). Look straight down from the point to the x-axis.
    S is above the number 9 on the x-axis.
    Along = 9
  3. Go up the y-axis. The second number tells us how far to go up (then up the stairs). Look straight across from the point to the y-axis.
    S is level with the number 7 on the y-axis.
    Up = 7
  4. Write the coordinates. We always write the along number first, then the up number, in brackets with a comma between them.
    S = (9, 7)
    0123456789101112123456789101112xyTUS

Check. Start at (0, 0). Go 9 along and 7 up. You arrive at S. ✓

Watch out. Along first, then up. (3, 5) and (5, 3) are two different points.

Worked example 10.1.5
To reach the point (6, 2) from the origin, how far do you go along and how far up?

Key words. Coordinates are two numbers in brackets, like (3, 5). They tell us exactly where a point is. The x-axis is the line going along the bottom. The y-axis is the line going up the side. The origin is the point (0, 0) where the two axes meet.

Method. The first number says how far to go along. The second number says how far to go up.

  1. Look at the first number. The first number is the along number (the x-coordinate).
    (6, 2): first number = 6
  2. Look at the second number. The second number is the up number (the y-coordinate).
    (6, 2): second number = 2
  3. Write the moves. Start at the origin, go along first, then up.
    Go 6 along and 2 up.
    01234567812345xy(6, 2)

Check. Start at (0, 0), go 6 along and 2 up: you are at (6, 2). ✓

Watch out. Along first, then up.

Worked example 10.1.6
Zainab looks at point P on the grid. She says, “P is at (5, 3).” Is she right? Explain.
0123456123456xyP

Key words. Coordinates are two numbers in brackets, like (3, 5). They tell us exactly where a point is. The x-axis is the line going along the bottom. The y-axis is the line going up the side. The origin is the point (0, 0) where the two axes meet.

Method. Read the point P carefully, along first and then up. Compare with what Zainab wrote.

  1. Read point P. Look down to the x-axis, then across to the y-axis.
    P is above 3 on the x-axis, so along = 3.
    P is level with 5 on the y-axis, so up = 5.
    0123456123456xyP
  2. Write the coordinates. Along first, then up.
    P = (3, 5)
  3. Compare with Zainab’s answer. Zainab wrote (5, 3). That means 5 along and 3 up. This is a different point.
    The point (5, 3) is shown in blue below.
    0123456123456xy(5, 3)

Check. (3, 5) means 3 along and 5 up. That is where P is. ✓

Watch out. Zainab put the up number first. Always say the along number first.

Worked example 10.1.7
Which two points have the same x-coordinate?
01234567812345678xyABCDE

Key words. Coordinates are two numbers in brackets, like (3, 5). They tell us exactly where a point is. The x-axis is the line going along the bottom. The y-axis is the line going up the side. The origin is the point (0, 0) where the two axes meet.

Method. Read the coordinates of all the points. Then look for two that have the same first number (x).

  1. Write the coordinates of every point. We read each point: along first, then up.
    A = (2, 6)
    B = (5, 3)
    C = (2, 2)
    D = (8, 3)
    E = (7, 7)
  2. Look at the x-coordinate (the first number). Two points that have the same number here are on a straight line.
    The x-coordinate 2 is in A and in C.
  3. Write the answer. Points with the same x-coordinate are in a vertical line. Points with the same y-coordinate are in a horizontal line. The orange line joins them.
    A and C
    01234567812345678xyABCDE

Check. A and C both have x = 2. ✓

Watch out. Compare the correct number: first number for x, second number for y.

Watch out

Along first, then up. (2, 6) and (6, 2) are not the same point. Also, the numbers are on the lines, so count the lines from the origin, not the squares. The origin is (0, 0), not (1, 1).

Exercise 10A  Reading and writing coordinates

Non-calculator
1
FWrite the coordinates of each point.
0123456123456xyABCD
(a)Point A
(b)Point B
(c)Point C
(d)Point D
2
FWhich letter is at each point?
01234567812345678xyABCDEFGH
(a)(3, 2)
(b)(0, 5)
(c)(4, 0)
(d)(7, 3)
3
FWrite how far you go along and how far up to reach each point from the origin.
(a)(5, 2)
(b)(3, 6)
(c)(7, 0)
(d)(4, 4)
4
FLook at the grid. A and B are two lines, and C is a point.
012345612345ABC
(a)Which line is the x-axis? Write the letter.
(b)Which line is the y-axis? Write the letter.
(c)What is the point C called? Write its coordinates.
5
FWrite the coordinates of each point as (along, up).
(a)3 along and 4 up
(b)6 along and 1 up
(c)0 along and 5 up
(d)8 along and 8 up
6
FWrite the coordinates of each point.
(a)The origin
(b)A point on the x-axis, 5 along
(c)A point on the y-axis, 3 up
7
FHere is a map of a town. S is the school, P is the park, L is the library and F is the football pitch. Write the coordinates of each place.
01234567891012345678910xySPLF
(a)S: the school
(b)P: the park
(c)L: the library
(d)F: the football pitch
8
FLook at the points on the grid.
01234567812345678xyABCDE
(a)Which two points have the same x-coordinate?
(b)Which two points have the same y-coordinate?
9
RNoah says the coordinates of point A are (6, 2). What mistake has he made? What are the correct coordinates?
012345671234567xyA
10
RWrite true or false for each statement.
(a)(6, 2) is further right than (2, 6).
(b)(5, 1) is higher up than (1, 5).
(c)(3, 4) is higher up than (3, 1).
(d)(2, 5) is further right than (6, 5).
11
PSZainab delivers parcels to four houses, H, J, K and M, on a map.
0123456789101112123456789101112xyHJKM
(a)Write the coordinates of house J.
(b)Write the coordinates of house M.
(c)Which house has the biggest x-coordinate?
12
RWrite always, sometimes or never for each sentence.
(a)The first number in a pair of coordinates tells you how far along to go.
(b)The two numbers in a pair of coordinates are the same.
(c)A point on the x-axis has 0 as its second number.
(d)The point (2, 7) is the same point as (7, 2).
13
RMei has a grid with the numbers 0 to 4 along both axes. How many points on the grid have whole-number coordinates? Include the points on the axes.
012341234xy
10.2

Plotting points and drawing shapes

Year 4
Plot specified points on a grid, Draw sides to complete a given polygon

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Plotting points and drawing shapesComing soon

To plot a point means to mark it on a grid. Here is how to plot the point (3, 2).

  1. Start at the origin, (0, 0).
  2. Go along the x-axis to 3 (the first number).
  3. Go up to 2 (the second number).
  4. Mark the point with a small cross or dot and write its label.
012345612345xyP (3, 2)3 along2 up
Plotting (3, 2): 3 along, then 2 up.

You can make a shape by plotting points and then joining them in order with a ruler. Each corner of the shape is called a vertex. Join the last point back to the first one to close the shape. Then you can name the shape by looking at its sides and its right angles.

012345612345xyABCD
A, B, C and D are plotted and joined in order. The shape has 4 right angles, so it is a rectangle.

Points can also make a straight line. Look at the points below. Every point has a y-coordinate of 3. They make a horizontal (flat) line.

01234567812345xyA (1, 3)B (3, 3)C (5, 3)D (7, 3)y = 3 everywhere01234567812345678xyE (5, 1)F (5, 3)G (5, 5)H (5, 7)x = 5 everywhere
Points on a horizontal line all have the same y-coordinate. Points on a vertical line all have the same x-coordinate.

The points on the second line all have an x-coordinate of 5. They make a vertical (straight up and down) line. A good question to ask is: “What do all the points on this line have in common?”

Key ideas

  • To plot (x, y): start at the origin, go along x, go up y, then mark the point.
  • Join the points in order and join the last one back to the first to make a polygon. Corners are called vertices.
  • Name a shape using its sides and its right angles. Count how long each side is on the grid.
  • Points with the same y are in a horizontal line. Points with the same x are in a vertical line.
Worked example 10.2.1
Plot the point P (3, 2) on the grid.
012345612345xy

Key words. To plot a point means to mark it on a grid with a small cross or dot. A vertex is a corner of a shape (more than one are called vertices). A polygon is a flat shape with straight sides.

Method. Start at the origin. For each point go along first, then up, and mark it with a cross.

  1. Find the origin. Every point is measured from (0, 0), where the axes meet.
    Put your finger on (0, 0).
  2. Plot P (3, 2). The first number is along, the second number is up. Then mark the point with a small cross.
    Go 3 along the x-axis, then 2 up.
  3. Check the picture. Every point should sit where its along number and up number meet.
    012345612345xyP

Check. Read each point back: look along to its x-coordinate and up to its y-coordinate. They match the numbers in the question. ✓

Watch out. Do not mix up the two numbers: along first, then up.

Worked example 10.2.2
Plot the points A (1, 1), B (4, 1) and C (1, 4). Join them in order and join C back to A. What shape have you drawn?
012345612345xy

Key words. To plot a point means to mark it on a grid with a small cross or dot. A vertex is a corner of a shape (more than one are called vertices). A polygon is a flat shape with straight sides.

Method. Start at the origin. For each point go along first, then up, and mark it with a cross.

  1. Find the origin. Every point is measured from (0, 0), where the axes meet.
    Put your finger on (0, 0).
  2. Plot A (1, 1). The first number is along, the second number is up. Then mark the point with a small cross.
    Go 1 along the x-axis, then 1 up.
  3. Plot B (4, 1). The first number is along, the second number is up. Then mark the point with a small cross.
    Go 4 along the x-axis, then 1 up.
  4. Plot C (1, 4). The first number is along, the second number is up. Then mark the point with a small cross.
    Go 1 along the x-axis, then 4 up.
  5. Join the points in order with a ruler and close the shape. Join the last point back to the first one.
    012345612345xyABC
    The shape is a right-angled isosceles triangle.

Check. Read each point back: look along to its x-coordinate and up to its y-coordinate. They match the numbers in the question. ✓

Watch out. Do not mix up the two numbers: along first, then up.

Worked example 10.2.3
Plot the points A (2, 2), B (6, 2), C (6, 5) and D (2, 5). Join them in order. What shape is it?
01234567123456xy

Key words. To plot a point means to mark it on a grid with a small cross or dot. A vertex is a corner of a shape (more than one are called vertices). A polygon is a flat shape with straight sides.

Method. Start at the origin. For each point go along first, then up, and mark it with a cross.

  1. Find the origin. Every point is measured from (0, 0), where the axes meet.
    Put your finger on (0, 0).
  2. Plot A (2, 2). The first number is along, the second number is up. Then mark the point with a small cross.
    Go 2 along the x-axis, then 2 up.
  3. Plot B (6, 2). The first number is along, the second number is up. Then mark the point with a small cross.
    Go 6 along the x-axis, then 2 up.
  4. Plot C (6, 5). The first number is along, the second number is up. Then mark the point with a small cross.
    Go 6 along the x-axis, then 5 up.
  5. Plot D (2, 5). The first number is along, the second number is up. Then mark the point with a small cross.
    Go 2 along the x-axis, then 5 up.
  6. Join the points in order with a ruler and close the shape. Join the last point back to the first one.
    01234567123456xyABCD
    The shape is a rectangle.

Check. Read each point back: look along to its x-coordinate and up to its y-coordinate. They match the numbers in the question. ✓

Watch out. Do not mix up the two numbers: along first, then up.

Worked example 10.2.4
Write the coordinates of the corners of the square.
012345612345xyABCD

Key words. To plot a point means to mark it on a grid with a small cross or dot. A vertex is a corner of a shape (more than one are called vertices). A polygon is a flat shape with straight sides.

Method. Read each corner (vertex) of the shape on its own. For every corner, go along first and then up.

  1. Look at the first corner. Start at the origin for every corner. Look down to the x-axis, then across to the y-axis.
    A: along 2, up 1 → (2, 1)
  2. Do the same for the other corners. Along first, then up, every time.
    B: along 5, up 1 → (5, 1)
    C: along 5, up 4 → (5, 4)
    D: along 2, up 4 → (2, 4)
  3. Write the answer. List the corners in order.
    A (2, 1), B (5, 1), C (5, 4), D (2, 4)
    012345612345xyABCD

Check. Plot the points again: they land on the corners of the shape. ✓

Watch out. Along first, then up, for every corner.

Worked example 10.2.5
The points A (1, 4), B (3, 4) and C (6, 4) are on a straight line. What do all the points on the line have in common?
01234567123456xyABC

Key words. Coordinates are two numbers in brackets, like (3, 5). They tell us exactly where a point is. The x-axis is the line going along the bottom. The y-axis is the line going up the side. The origin is the point (0, 0) where the two axes meet.

Method. Write down the coordinates of all the points. Look for a number that is the same in every pair.

  1. Write the coordinates of every point. Read each point: along first, then up.
    A = (1, 4)
    B = (3, 4)
    C = (6, 4)
  2. Look at the second numbers. The second number is the y-coordinate. Do they match?
    4, 4, 4
    They are all 4.
  3. Say what they have in common. A line of points that share a y-coordinate is horizontal (flat).
    Every point has y = 4, so the points are on a horizontal line.
    01234567123456xyABC

Check. Every point has y-coordinate 4, so the line is flat. ✓

Watch out. The y-coordinate is the second number. A flat line has the same y-coordinate everywhere.

Worked example 10.2.6
The points P (5, 1), Q (5, 3) and R (5, 6) are on a straight line. What do all the points have in common?
012345671234567xyPQR

Key words. Coordinates are two numbers in brackets, like (3, 5). They tell us exactly where a point is. The x-axis is the line going along the bottom. The y-axis is the line going up the side. The origin is the point (0, 0) where the two axes meet.

Method. Write down the coordinates of all the points. Look for a number that is the same in every pair.

  1. Write the coordinates of every point. Read each point: along first, then up.
    A = (5, 1)
    B = (5, 3)
    C = (5, 6)
  2. Look at the first numbers. The first number is the x-coordinate. Do they match?
    5, 5, 5
    They are all 5.
  3. Say what they have in common. A line of points that share a x-coordinate is vertical (straight up and down).
    Every point has x = 5, so the points are on a vertical line.
    012345671234567xyABC

Check. Every point has x-coordinate 5, so the line is straight up. ✓

Watch out. The x-coordinate is the first number. A line going straight up has the same x-coordinate everywhere.

Worked example 10.2.7
Plot A (1, 1), B (5, 1), C (5, 4), D (3, 6) and E (1, 4). Join them in order and close the shape. How many sides does it have, and what is it called?
01234561234567xy

Key words. To plot a point means to mark it on a grid with a small cross or dot. A vertex is a corner of a shape (more than one are called vertices). A polygon is a flat shape with straight sides.

Method. Start at the origin. For each point go along first, then up, and mark it with a cross.

  1. Find the origin. Every point is measured from (0, 0), where the axes meet.
    Put your finger on (0, 0).
  2. Plot A (1, 1). The first number is along, the second number is up. Then mark the point with a small cross.
    Go 1 along the x-axis, then 1 up.
  3. Plot B (5, 1). The first number is along, the second number is up. Then mark the point with a small cross.
    Go 5 along the x-axis, then 1 up.
  4. Plot C (5, 4). The first number is along, the second number is up. Then mark the point with a small cross.
    Go 5 along the x-axis, then 4 up.
  5. Plot D (3, 6). The first number is along, the second number is up. Then mark the point with a small cross.
    Go 3 along the x-axis, then 6 up.
  6. Plot E (1, 4). The first number is along, the second number is up. Then mark the point with a small cross.
    Go 1 along the x-axis, then 4 up.
  7. Join the points in order with a ruler and close the shape. Join the last point back to the first one.
    01234561234567xyABCDE
    The shape is a pentagon (5 sides).

Check. Read each point back: look along to its x-coordinate and up to its y-coordinate. They match the numbers in the question. ✓

Watch out. Do not mix up the two numbers: along first, then up.

Watch out

Plot along first, then up. To plot (4, 2), go 4 along and 2 up, not 2 along and 4 up. When you join points, join them in the order given. Joining them in a different order can make a different shape.

Exercise 10B  Plotting points and drawing shapes

Non-calculator
1
FPlot each point on the grid. Label it with its letter.
0123456123456xy
(a)A (2, 4)
(b)B (5, 1)
(c)C (0, 3)
(d)D (6, 6)
2
FPlot the points A (1, 1), B (5, 1), C (5, 4) and D (1, 4). Join them in order and join D back to A. What shape have you drawn?
012345612345xy
3
FPlot P (1, 1), Q (5, 1) and R (1, 4). Join them in order and join R back to P. What shape have you drawn?
012345612345xy
4
FPlot A (2, 2), B (5, 2), C (5, 5) and D (2, 5). Join them in order. What is the name of the shape?
0123456123456xy
5
FPlot A (1, 1), B (5, 1), C (7, 4) and D (3, 4). Join them in order. What is the name of the shape? (Hint: look at the two pairs of sides that slant the same way.)
01234567812345xy
6
FLook at the shape on the grid.
012345612345678xyABCD
(a)Write the coordinates of A, B, C and D.
(b)What is the name of the shape?
7
FPlot A (2, 1), B (6, 1) and C (4, 5). Join them in order and close the shape. What is the name of the triangle? (Two sides slant the same amount.)
01234567123456xy
8
PSPlot A (1, 1), B (5, 1), C (5, 2), D (2, 2), E (2, 6) and F (1, 6). Join them in order and close the shape. It looks like a letter. Which letter is it? How many sides has it got?
01234561234567xy
9
RWhat do all the points on each line have in common?
(a)The points (1, 6), (3, 6), (5, 6) and (8, 6) are on a line.
(b)The points (2, 1), (2, 4) and (2, 7) are on a line.
10
RThe points (1, 3) and (2, 3) are on a horizontal line. Write the coordinates of two more points on the same line.
012345678123456xy(1, 3)(2, 3)
11
RRavi has been asked to plot (4, 2). He puts a cross at X. What mistake has he made?
0123456123456xyX
12
RJack plots A (1, 2), B (4, 2), C (4, 6) and D (1, 6) and joins them in order. He says, “I have drawn a square.” Is Jack right? Explain how you know.
01234561234567xy
13
RWrite always, sometimes or never for each sentence.
(a)Points with the same y-coordinate are on a horizontal line.
(b)A shape drawn on a grid with four right angles is a square.
(c)The points (1, 2), (2, 3) and (3, 4) are on a horizontal line.
14
PSPlot A (2, 1), B (6, 1), C (6, 4) and D (2, 4) and join them to make a rectangle. Each side of a grid square is 1 cm long. What is the perimeter of the rectangle?
01234567123456xy
10.3

Missing corners and missing numbers

Year 4
Plot specified points and draw sides to complete a given polygon, Describe positions as coordinates

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Missing corners and missing numbersComing soon

Sometimes you are given three corners of a shape and you must find the fourth. You can use what you know about the shape. Here are four clues.

Rectangle and square. The sides go straight across and straight up. So the missing corner lines up with one corner (same x-coordinate) and with another corner (same y-coordinate).

0123456712345xyABCD
D is straight above A (same x-coordinate, 2) and level with C (same y-coordinate, 4). So D is (2, 4).

Parallelogram. Opposite sides are parallel and the same length. Find the move along one side, then use the same move for the opposite side.

01234567812345xyABCD
B to C is 2 right and 3 up. So A to D is also 2 right and 3 up: D = (1 + 2, 1 + 3) = (3, 4).

Kite and isosceles triangle. These shapes are symmetrical. Find the mirror line (the line of symmetry). The missing corner is the mirror image of its partner: the same distance from the line on the other side, at the same height.

01234567812345678xyACDBx = 4
The mirror line is x = 4. D is 2 to the left of it, so B is 2 to the right, at the same height: B = (6, 3).

Missing numbers on a line. If two points are on a horizontal line, they have the same y-coordinate. If they are on a vertical line, they have the same x-coordinate. So (2, 5) and (7, ?) on a horizontal line gives ? = 5.

Key ideas

  • Rectangle or square: the missing corner has the same x as one corner and the same y as another corner.
  • Parallelogram: opposite sides make the same move. Use the move from one side on the opposite side.
  • Kite or isosceles triangle: find the mirror line. The missing corner is the same distance from it on the other side, at the same height.
  • Horizontal line: the y-coordinate stays the same. Vertical line: the x-coordinate stays the same.
Worked example 10.3.1
Three corners of a rectangle are A (2, 1), B (6, 1) and C (6, 4). Find the coordinates of the fourth corner, D.
0123456712345xyABC

Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.

Method. A rectangle has sides that go straight across and straight up. The missing corner lines up with one corner going up or down, and with another corner going across.

  1. Write the coordinates of the corners you know. We use them to work out the missing corner.
    A = (2, 1)
    B = (6, 1)
    C = (6, 4)
  2. Find the side joining D to A. In a rectangle the sides are straight up and down or flat. AD goes straight up or down, so D has the same x-coordinate as A.
    A = (2, 1), so the x-coordinate of D is 2.
  3. Find the side joining D to C. CD is a flat side, so D has the same y-coordinate as C.
    C = (6, 4), so the y-coordinate of D is 4.
  4. Write the missing corner and draw the sides. Put the x-coordinate first, then the y-coordinate. Join it to the other corners.
    D = (2, 4)
    0123456712345xyABCD

Check. The completed shape is a rectangle. ✓

Watch out. Check each side of your shape. It must look like the shape in the question.

Worked example 10.3.2
A, C and D are three corners of a square. A is (1, 1), C is (4, 4) and D is (1, 4). Find the coordinates of B.
012345612345xyCDA

Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.

Method. A square has sides that go straight across and straight up. The missing corner lines up with one corner going up or down, and with another corner going across.

  1. Write the coordinates of the corners you know. We use them to work out the missing corner.
    A = (1, 1)
    C = (4, 4)
    D = (1, 4)
  2. Find the side joining B to C. In a square the sides are straight up and down or flat. CB goes straight up or down, so B has the same x-coordinate as C.
    C = (4, 4), so the x-coordinate of B is 4.
  3. Find the side joining B to A. AB is a flat side, so B has the same y-coordinate as A.
    A = (1, 1), so the y-coordinate of B is 1.
  4. Write the missing corner and draw the sides. Put the x-coordinate first, then the y-coordinate. Join it to the other corners.
    B = (4, 1)
    012345612345xyACDB
  5. Check the sides. A square has four equal sides.
    The sides are 3, 3, 3, 3 long. They are all equal.

Check. The completed shape is a square. ✓

Watch out. Check each side of your shape. It must look like the shape in the question.

Worked example 10.3.3
Three corners of a parallelogram are A (1, 2), B (5, 2) and C (7, 5). Find the fourth corner, D.
012345678123456xyABC

Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.

Method. In a parallelogram the opposite sides are parallel and the same length. So the missing corner is the same move away from its neighbour as the opposite sides show.

  1. Write the coordinates of the corners you know. We use them to work out the missing corner.
    A = (1, 2)
    B = (5, 2)
    C = (7, 5)
  2. Find the move from B to C. In a parallelogram, opposite sides are parallel and the same length. BC and AD are opposite sides, so they are the same move.
    B (5, 2) to C (7, 5): 2 right, 3 up.
  3. Do the same move from A. D is the same move away from A as C is from B.
    A = (1, 2)
    2 right, 3 up
    D = (1 + 2, 2 + 3) = (3, 5)
  4. Draw the sides and check. Join the new corner to the others.
    012345678123456xyABCD

Check. The completed shape is a parallelogram. ✓

Watch out. Move the same way both times: left or right first, then up or down.

Worked example 10.3.4
Three corners of a kite are A (3, 1), C (3, 7) and D (1, 3). Find the fourth corner, B.
012345612345678xyCDA

Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.

Method. A kite is symmetrical about a line through two of its corners. The missing corner is the mirror image of its partner corner.

  1. Write the coordinates of the corners you know. We use them to work out the missing corner.
    A = (3, 1)
    C = (3, 7)
    D = (1, 3)
  2. Find the line of symmetry. A kite is symmetrical. The corners A and C are in a straight up-and-down line, so that line is the mirror line.
    A = (3, 1) and C = (3, 7)
    The mirror line is x = 3.
  3. Reflect D in the mirror line. D is 2 to the left of the line. B is the same distance on the other side, at the same height.
    D = (1, 3) is 2 left of x = 3.
    B is 2 right: x = 3 + 2 = 5
    Same height: y = 3
  4. Write the missing corner and draw the sides. Join it to the other corners.
    B = (5, 3)
    012345612345678xyACDB

Check. The completed shape is a kite. ✓

Watch out. Check each side of your shape. It must look like the shape in the question.

Worked example 10.3.5
A and B are the base corners of an isosceles triangle. A is (2, 1) and B is (8, 1). The top corner C is 5 up. Find the coordinates of C.
0123456789123456xyAB

Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.

Method. The apex of an isosceles triangle is straight above the middle of the base. Find the middle of the base, then use the height.

  1. Write the coordinates of the corners you know. We use them to work out the missing corner.
    A = (2, 1)
    B = (8, 1)
  2. Find the middle of the base. The apex of an isosceles triangle is straight above (or below) the middle of the base AB.
    The base goes from x = 2 to x = 8.
    The middle is (2 + 8) ÷ 2 = 5
  3. Use the height you are told. The question says C is 5 up.
    C = (5, 5)
  4. Draw the sides and check. Join the new corner to A and B. The two sloping sides match.
    0123456789123456xyABC

Check. The mirror line x = 5 cuts the base in half, so the two sloping sides are the same. ✓

Watch out. The middle of the base is halfway between the two x-coordinates, not one of them.

Worked example 10.3.6
The points P (2, 5) and Q (7, ?) are on a horizontal line. What is the missing number?
0123456781234567xyP

Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.

Method. Points on a horizontal line all have the same y-coordinate. So the missing number is the same as the y-coordinate you already know.

  1. Look at the line. P and Q are on a horizontal (flat) line.
    P = (2, 5)
  2. Use what points on a horizontal line have in common. Every point on this line has the same y-coordinate (the second number).
    The y-coordinate is 5.
  3. Fill in the missing number. The missing number is the second number of Q.
    Q = (7, 5)
    0123456781234567xyPQ

Check. Both points have y = 5. ✓

Watch out. On a horizontal line the y-coordinate stays the same. The other number changes.

Worked example 10.3.7
The points P (4, 6) and Q (?, 2) are on a vertical line. What is the missing number?
01234561234567xyP

Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.

Method. Points on a vertical line all have the same x-coordinate. So the missing number is the same as the x-coordinate you already know.

  1. Look at the line. P and Q are on a vertical (straight up) line.
    P = (4, 6)
  2. Use what points on a vertical line have in common. Every point on this line has the same x-coordinate (the first number).
    The x-coordinate is 4.
  3. Fill in the missing number. The missing number is the first number of Q.
    Q = (4, 2)
    01234561234567xyPQ

Check. Both points have x = 4. ✓

Watch out. On a vertical line the x-coordinate stays the same. The other number changes.

Worked example 10.3.8
ABCD is a rectangle. A is (1, 2), B is (6, 2), D is (1, 5) and C is (6, ?). Find the missing number.
01234567123456xyABDC

Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.

Method. A rectangle has straight sides. Use the corners you know to find the missing number in C.

  1. Write the corners. A rectangle ABCD has corners A, B, C, D in order.
    A (1, 2), B (6, 2), C (6, ?), D (1, 5)
  2. Look at the side DC. D and C are joined by a flat side, so they have the same y-coordinate.
    D is (1, 5), so the y-coordinate of C is 5.
  3. Check the side BC. B and C are joined by an upright side, so they have the same x-coordinate.
    B is (6, 2) and C is (6, ?). The x-coordinates match: 6.

Check. C = (6, 5). AB = 5 and DC = 5. AD = 3 and BC = 3. Opposite sides are equal. ✓

Watch out. Flat sides keep the y-coordinate. Upright sides keep the x-coordinate.

Watch out

Do not guess the corner by eye. Use the numbers. For a rectangle, copy one number from each of two corners. For a parallelogram, the missing corner does not line up with the other corners, so use the move (for example 2 right and 3 up) instead.

Exercise 10C  Finding missing corners and missing numbers

Non-calculator
1
FThree corners of a rectangle are A (1, 2), B (5, 2) and C (5, 5). Find the fourth corner, D.
01234561234567xyABC
2
FB, C and D are three corners of a rectangle. B is (7, 1), C is (7, 4) and D is (2, 4). Find the coordinates of A.
01234567812345xyBCD
3
FThree corners of a square are A (3, 2), B (6, 2) and D (3, 5). Find the coordinates of C.
01234567123456xyDAB
4
FThree corners of a parallelogram are A (2, 1), B (6, 1) and C (8, 4). Find the fourth corner, D.
012345678912345xyABC
5
FA, C and D are three corners of a parallelogram. A is (1, 1), C is (6, 5) and D is (2, 5). Find the coordinates of B.
01234567123456xyCDA
6
FThree corners of a kite are A (4, 1), C (4, 7) and D (2, 3). Find the fourth corner, B.
01234567812345678xyCDA
7
FTwo corners of an isosceles triangle are A (1, 2) and B (7, 2). The third corner C is 6 up from the x-axis. Find the coordinates of C.
0123456781234567xyAB
8
FAn isosceles triangle has its top corner C at (4, 6) and a base corner B at (7, 2). The base is flat. Find the coordinates of the other base corner, A.
0123456781234567xyBC
9
FFind the missing number in each pair of points.
(a)P (3, 4) and Q (8, ?) are on a horizontal line. Write the missing number.
(b)P (6, ?) and Q (1, 7) are on a horizontal line. Write the missing number.
(c)P (5, 2) and Q (?, 8) are on a vertical line. Write the missing number.
(d)P (?, 6) and Q (9, 1) are on a vertical line. Write the missing number.
10
RMei has plotted A (1, 1), B (4, 1) and C (4, 5). She wants to make a rectangle. She says the fourth corner, D, is (1, 4). Is Mei right? Explain.
0123456123456xyABCMei’s D
11
RWrite always, sometimes or never for each sentence.
(a)The missing corner of a rectangle has the same x-coordinate as one known corner and the same y-coordinate as another known corner.
(b)The missing corner of a parallelogram is straight above one of the other corners.
(c)Two points on a vertical line have the same x-coordinate.
12
PSThree corners of a rectangle are A (2, 2), B (8, 2) and C (8, 5). Each grid square is 1 cm. Find the fourth corner D. Then find the perimeter of the rectangle.
0123456789123456xyABC
13
RA (3, 2) and B (3, 6) are two corners of a square. The other two corners are on the right of AB. Find the coordinates of the other two corners.
012345678912345678xyAB
10.4

Translating points

Year 4
Describe movements between positions as translations of a given unit to the left/right and up/down

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Translating pointsComing soon

A translation is a slide. When you translate a point, you slide it to a new place. You say how many squares it moves left or right and how many it moves up or down. For example, “3 right, 2 up” means slide 3 squares to the right and then 2 squares up.

01234567123456xyAA′
Point A (2, 3) is translated 3 right and 2 up. The new point is A′ (5, 5). We write a small dash after the letter for the new point.

The coordinates change in a pattern. A move right or left changes the x-coordinate (the first number). A move up or down changes the y-coordinate (the second number).

MoveWhat happens to the coordinates
rightx gets bigger (add)
leftx gets smaller (take away)
upy gets bigger (add)
downy gets smaller (take away)

To describe a translation from one point to another, count how far the point goes across, and then how far up or down. The second picture shows a journey from B (1, 2) to C (6, 5).

01234567123456xyB (1, 2)C (6, 5)5 right3 up
From B to C: 5 right (6 − 1 = 5), then 3 up (5 − 2 = 3).

Key ideas

  • A translation slides a point left or right and up or down. It does not turn or flip.
  • Say the move like this: “3 right, 2 up”. Left or right first, then up or down.
  • Right: add to x. Left: take away from x. Up: add to y. Down: take away from y.
  • To describe a move: compare the x-coordinates (left or right), then the y-coordinates (up or down).
  • To go back, do the opposite move.
Worked example 10.4.1
Point A is at (2, 3). It is translated 4 right and 1 up. What are the coordinates of the new point, A′?

Key words. A translation is a slide. Every point moves the same distance in the same direction. The shape is not turned and not flipped. We describe it with left or right, and up or down.

Method. Left and right moves change the x-coordinate (the first number). Up and down moves change the y-coordinate (the second number).

  1. Write the starting point. Along first, then up.
    A = (2, 3)
  2. Move 4 squares right. Moving right adds to the x-coordinate. Moving left takes away from the x-coordinate. The y-coordinate does not change here.
    x: 2 + 4 = 6
  3. Move 1 square up. Moving up adds to the y-coordinate. Moving down takes away from the y-coordinate. The x-coordinate does not change here.
    y: 3 + 1 = 4
  4. Write the new point. Put the new x-coordinate first and the new y-coordinate second. We write a small dash after the letter for the new point.
    A′ = (6, 4)
    0123456712345xyAA′

Check. Count on the grid from (2, 3): 4 right, 1 up gets to (6, 4). ✓

Watch out. Left and right change the first number. Up and down change the second number. Do not mix them up.

Worked example 10.4.2
Point P is at (7, 6). It is translated 3 left and 4 down. Write the coordinates of the new point, P′.

Key words. A translation is a slide. Every point moves the same distance in the same direction. The shape is not turned and not flipped. We describe it with left or right, and up or down.

Method. Left and right moves change the x-coordinate (the first number). Up and down moves change the y-coordinate (the second number).

  1. Write the starting point. Along first, then up.
    P = (7, 6)
  2. Move 3 squares left. Moving right adds to the x-coordinate. Moving left takes away from the x-coordinate. The y-coordinate does not change here.
    x: 7 − 3 = 4
  3. Move 4 squares down. Moving up adds to the y-coordinate. Moving down takes away from the y-coordinate. The x-coordinate does not change here.
    y: 6 − 4 = 2
  4. Write the new point. Put the new x-coordinate first and the new y-coordinate second. We write a small dash after the letter for the new point.
    P′ = (4, 2)
    0123456781234567xyPP′

Check. Count on the grid from (7, 6): 3 left, 4 down gets to (4, 2). ✓

Watch out. Left and right change the first number. Up and down change the second number. Do not mix them up.

Worked example 10.4.3
Describe the translation that takes A (1, 2) to B (6, 5).

Key words. A translation is a slide. Every point moves the same distance in the same direction. The shape is not turned and not flipped. We describe it with left or right, and up or down.

Method. Compare the x-coordinates to find the move left or right. Compare the y-coordinates to find the move up or down.

  1. Write the two points. Along first, then up.
    A = (1, 2) and B = (6, 5)
  2. Compare the x-coordinates. The x-coordinate tells us how far along we are. If it gets bigger the move is to the right. If it gets smaller the move is to the left.
    x goes from 1 to 6: 6 − 1 = 5
    5 right
  3. Compare the y-coordinates. The y-coordinate tells us how far up we are. If it gets bigger the move is up. If it gets smaller the move is down.
    y goes from 2 to 5: 5 − 2 = 3
    3 up
  4. Write the translation. We say the left or right move first, then the up or down move.
    From A to B: 5 right, 3 up
    01234567123456xyAB

Check. Start at (1, 2) and move 5 right, 3 up: you get to (6, 5). ✓

Watch out. Left and right come from the x-coordinates (first numbers). Up and down come from the y-coordinates (second numbers).

Worked example 10.4.4
Describe the translation that takes C (8, 7) to D (3, 2).

Key words. A translation is a slide. Every point moves the same distance in the same direction. The shape is not turned and not flipped. We describe it with left or right, and up or down.

Method. Compare the x-coordinates to find the move left or right. Compare the y-coordinates to find the move up or down.

  1. Write the two points. Along first, then up.
    C = (8, 7) and D = (3, 2)
  2. Compare the x-coordinates. The x-coordinate tells us how far along we are. If it gets bigger the move is to the right. If it gets smaller the move is to the left.
    x goes from 8 to 3: 8 − 3 = 5
    5 left
  3. Compare the y-coordinates. The y-coordinate tells us how far up we are. If it gets bigger the move is up. If it gets smaller the move is down.
    y goes from 7 to 2: 7 − 2 = 5
    5 down
  4. Write the translation. We say the left or right move first, then the up or down move.
    From C to D: 5 left, 5 down
    012345678912345678xyCD

Check. Start at (8, 7) and move 5 left, 5 down: you get to (3, 2). ✓

Watch out. Left and right come from the x-coordinates (first numbers). Up and down come from the y-coordinates (second numbers).

Worked example 10.4.5
Describe the translation that takes E (2, 6) to F (7, 6).

Key words. A translation is a slide. Every point moves the same distance in the same direction. The shape is not turned and not flipped. We describe it with left or right, and up or down.

Method. Compare the x-coordinates to find the move left or right. Compare the y-coordinates to find the move up or down.

  1. Write the two points. Along first, then up.
    E = (2, 6) and F = (7, 6)
  2. Compare the x-coordinates. The x-coordinate tells us how far along we are. If it gets bigger the move is to the right. If it gets smaller the move is to the left.
    x goes from 2 to 7: 7 − 2 = 5
    5 right
  3. Compare the y-coordinates. The y-coordinate tells us how far up we are.
    y stays 6, so there is no move up or down.
  4. Write the translation. We say the left or right move first, then the up or down move.
    From E to F: 5 right
    0123456781234567xyEF

Check. Start at (2, 6) and move 5 right: you get to (7, 6). ✓

Watch out. Left and right come from the x-coordinates (first numbers). Up and down come from the y-coordinates (second numbers).

Worked example 10.4.6
Point B is translated 2 right and 3 up. It ends at (6, 5). Where did it start?

Key words. A translation is a slide. Every point moves the same distance in the same direction. The shape is not turned and not flipped. We describe it with left or right, and up or down.

Method. Work backwards. Do the opposite of the move starting from the end point.

  1. Write the move and the end point. We know where the point finished.
    Move: 2 right, 3 up
    End: (6, 5)
  2. Undo the move. To go back to the start, do the opposite move: right becomes left, up becomes down (and the other way round).
    The opposite of 2 right, 3 up is 2 left, 3 down.
  3. Change the x-coordinate. Undo the left or right move.
    x: 6 − 2 = 4
  4. Change the y-coordinate. Undo the up or down move.
    y: 5 − 3 = 2
  5. Write the starting point.
    Start = (4, 2)
    01234567123456xystartend

Check. Start at (4, 2) and move 2 right, 3 up: you reach (6, 5). ✓

Watch out. Do the opposite move. Do not do the same move again.

Worked example 10.4.7
Point A is at (1, 1). It is translated 3 right and 2 up. Then it is translated 1 left and 4 up. Where does it end? What one move does the same job?

Key words. A translation is a slide. Every point moves the same distance in the same direction. The shape is not turned and not flipped. We describe it with left or right, and up or down.

Method. Do the moves one at a time. After each move write the new coordinates. Then compare the start and the end to find the one move that does the same job.

  1. Write the start point. We begin here.
    A = (1, 1)
  2. Do move 1: 3 right, 2 up. Change the x-coordinate for left or right. Change the y-coordinate for up or down.
    x: 1 + 3 = 4
    y: 1 + 2 = 3
    Now at (4, 3)
  3. Do move 2: 1 left, 4 up. Change the x-coordinate for left or right. Change the y-coordinate for up or down.
    x: 4 − 1 = 3
    y: 3 + 4 = 7
    Now at (3, 7)
  4. Find the one move that does the same job. Add up all the right and left moves (left counts as taking away). Do the same for up and down.
    Across: 3 + (−1) = 2
    Up: 2 + 4 = 6
    One move: 2 right, 6 up
    01234512345678xyAA′

Check. Start (1, 1), one move of 2 right, 6 up, end (3, 7). ✓

Watch out. Check left and right moves carefully. Left makes the x-coordinate smaller.

Watch out

Left and right change x, up and down change y. Moving a point up does not change its x-coordinate. Also say the move in the right order: (2, 3) moved 3 right and 2 up is (5, 5), not (4, 6). When going backwards, do the opposite move.

Exercise 10D  Translating points

Non-calculator
1
FTranslate each point. Write the coordinates of the new point.
(a)(2, 1): 3 right, 2 up
(b)(5, 4): 2 left, 1 up
(c)(6, 6): 4 left, 3 down
(d)(1, 2): 5 right, 0 up
2
FDescribe each translation. Write left or right first, then up or down.
0123456781234567xyABCD
(a)From A to B
(b)From B to C
(c)From C to D
(d)From D to A
3
FCopy and complete the table. S is the start point and E is the end point.
(a)S = (3, 2); move 2 right, 4 up; E = ?
(b)S = (5, 5); move 3 left, 2 down; E = ?
(c)S = (4, 1); move = ?; E = (7, 5)
(d)S = ?; move 3 right, 2 up; E = (7, 4)
4
FA frog sits at F (2, 2). It jumps 3 squares to the right. Then it jumps 4 squares up.
(a)Where is the frog after the first jump?
(b)Where is the frog after the second jump?
5
RLeo translates the point (2, 3) by 3 right and 2 up. He says the new point is (4, 6). What mistake has Leo made? What is the right answer?
6
FWhich translation takes P (2, 5) to Q (6, 3)? Write the letter.
A: 4 right, 2 down   B: 2 right, 4 down   C: 4 left, 2 up
012345671234567xyPQ
7
PSPoint A (1, 1) is translated to point B (4, 3). C is the point (2, 5). C is translated by the same move. What are the coordinates of the new point?
0123456712345678xyABC
8
PSP is at (1, 2). It is translated 4 right and 1 up. Then it is translated 2 right and 3 up. Where does it finish? Write the one move that goes from the start to the end.
9
PSA bee flies from the hive H (2, 2) to a flower F (7, 5).
012345678123456xyHF
(a)Describe the move from H to F.
(b)Describe the move from F back to H.
10
RWrite always, sometimes or never for each sentence.
(a)When a point is translated up, its x-coordinate stays the same.
(b)When a point is translated right, its y-coordinate changes.
(c)After a translation, both coordinates of a point are bigger than before.
11
RZ is the point (3, 3) on a grid that starts at 0. Zainab says, “If I translate Z 4 squares left, the point is not on the grid.” Is she right? Explain.
0123456123456xyZ
12
RA point moves 4 right and 3 up. It ends at (9, 7). Then it moves 5 left and 2 down.
(a)Where did the point start?
(b)Where does the point finish?
10.5

Translating shapes

Year 4
Describe movements between positions as translations of a given unit to the left/right and up/down, Plot specified points and draw sides to complete a given polygon

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Translating shapesComing soon

You can translate a whole shape, not just one point. A shape is made of corners (vertices). To translate the shape, translate every corner by the same move. Then join the new corners in the same order.

01234567812345xyABCA′B′C′
Triangle ABC (blue) is translated 4 right and 1 up. Every corner moves 4 right and 1 up to make the new triangle (orange). The new corners are A′, B′ and C′.

A translated shape has not changed. It is the same size and the same shape. It faces the same way: it is not turned and not flipped. Only its position is different. Compare these shapes with the first triangle.

01234123xyfirst shape01234123xyslid: a translation01234123xyflipped: not a translation01234123xyturned: not a translation
Only the second shape is a translation. The third is flipped over and the fourth is turned round.

To describe how a shape has moved, choose one corner on the first shape and its matching corner on the second shape. Find the move between them. Then check that another corner makes the same move.

Key ideas

  • To translate a shape, translate every corner by the same move, then join the new corners in order.
  • A translation does not change the size, the shape or the way it faces. It is not turned and not flipped.
  • To describe a move between two shapes, find the move from one corner to its matching corner. Check with a second corner.
  • Corners of the new shape are named with a small dash: A′, B′, C′.
Worked example 10.5.1
Triangle ABC has corners A (1, 1), B (4, 1) and C (1, 3). Translate it 3 right and 2 up. Write the coordinates of A′, B′ and C′.
012345678123456xyABC

Key words. A translation is a slide. Every point moves the same distance in the same direction. The shape is not turned and not flipped. We describe it with left or right, and up or down.

Method. Translate every corner by the same move. Then join the new corners in the same order.

  1. Write the corners of the shape. We move every corner (vertex) by the same move.
    A = (1, 1)
    B = (4, 1)
    C = (1, 3)
  2. Move every corner 3 right, 2 up. Right adds to x, left takes away from x. Up adds to y, down takes away from y.
    A (1, 1) → (1 + 3, 1 + 2) = A′ (4, 3)
    B (4, 1) → (4 + 3, 1 + 2) = B′ (7, 3)
    C (1, 3) → (1 + 3, 3 + 2) = C′ (4, 5)
  3. Plot the new corners and join them in order. The new shape (orange) is the old shape (blue) after the slide. The arrows show the move.
    012345678123456xyABCA′B′C′
  4. Write the new coordinates. The new shape has the same size and shape as the old one.
    A′ (4, 3), B′ (7, 3), C′ (4, 5)

Check. Each new corner is 3 right, 2 up from the old one, and the new shape looks the same as the old shape. ✓

Watch out. Move every corner by the same amount. If one corner is missed, the shape gets squashed.

Worked example 10.5.2
Rectangle ABCD has corners A (5, 4), B (8, 4), C (8, 6) and D (5, 6). Translate it 3 left and 2 down. Write the new coordinates.
01234567891234567xyABCD

Key words. A translation is a slide. Every point moves the same distance in the same direction. The shape is not turned and not flipped. We describe it with left or right, and up or down.

Method. Translate every corner by the same move. Then join the new corners in the same order.

  1. Write the corners of the shape. We move every corner (vertex) by the same move.
    A = (5, 4)
    B = (8, 4)
    C = (8, 6)
    D = (5, 6)
  2. Move every corner 3 left, 2 down. Right adds to x, left takes away from x. Up adds to y, down takes away from y.
    A (5, 4) → (5 − 3, 4 − 2) = A′ (2, 2)
    B (8, 4) → (8 − 3, 4 − 2) = B′ (5, 2)
    C (8, 6) → (8 − 3, 6 − 2) = C′ (5, 4)
    D (5, 6) → (5 − 3, 6 − 2) = D′ (2, 4)
  3. Plot the new corners and join them in order. The new shape (orange) is the old shape (blue) after the slide. The arrows show the move.
    01234567891234567xyABCDA′B′C′D′
  4. Write the new coordinates. The new shape has the same size and shape as the old one.
    A′ (2, 2), B′ (5, 2), C′ (5, 4), D′ (2, 4)

Check. Each new corner is 3 left, 2 down from the old one, and the new shape looks the same as the old shape. ✓

Watch out. Move every corner by the same amount. If one corner is missed, the shape gets squashed.

Worked example 10.5.3
Shape ABCD (blue) is translated to A′B′C′D′ (orange). Describe the translation.
012345678123456xyABCDA′B′C′D′

Key words. A translation is a slide. Every point moves the same distance in the same direction. The shape is not turned and not flipped. We describe it with left or right, and up or down.

Method. Choose one corner of the first shape and find its partner corner on the second shape. Find the move between them, then check with another corner.

  1. Pick one corner and its partner. The corner A on the first shape matches the corner A′ on the second shape.
    A = (1, 2) and A′ = (5, 3)
  2. Find the move. Compare the x-coordinates for left or right. Compare the y-coordinates for up or down.
    x: 1 to 5 is 4 right.
    y: 2 to 3 is 1 up.
  3. Check with a different corner. Every corner must make the same move. If one does not, the shape has not just been translated.
    D (2, 4) to D′ (6, 5): 4 right, 1 up. The same move. ✓
  4. Write the translation.
    The shape moves 4 right, 1 up.
    012345678123456xyABCDA′B′C′D′

Check. Move the first shape 4 right, 1 up and it lands exactly on the second shape. ✓

Watch out. Use matching corners. Do not compare the first corner of one shape with a different corner of the other.

Worked example 10.5.4
Which shape, B, C or D, is a translation of shape A? Write the letter.
01234567812345xyA01234567812345xyB01234567812345xyC01234567812345xyD

Key words. A translation is a slide. Every point moves the same distance in the same direction. The shape is not turned and not flipped. We describe it with left or right, and up or down.

Method. A translation slides a shape. The shape is not turned and not flipped, so it must face the same way as the first shape.

  1. Look at shape A. Notice which way round it faces. For example, find the corner with the right angle.
    01234567812345xy
  2. Check each other shape. A translated shape is only slid. It is not turned and not flipped. It faces the same way as A.
    B: the shape is flipped over (like a mirror picture). No.
    C: same way round, only moved. Yes.
    D: the shape is turned round. No.
  3. Write the answer. Only one shape has been slid without turning or flipping.
    The translation of A is shape C.

Check. Slide A to shape C without turning it and the two shapes match exactly. ✓

Watch out. A flipped shape looks like a mirror picture. A turned shape has been spun round. Neither is a translation.

Worked example 10.5.5
Triangle ABC has corners A (2, 3), B (4, 3) and C (4, 5). After a translation, A moves to A′ (5, 1). Where do B and C move to?

Key words. A translation is a slide. Every point moves the same distance in the same direction. The shape is not turned and not flipped. We describe it with left or right, and up or down.

Method. Find the move from A to A′. Then move the other corners by the same move.

  1. Find the move. Compare the coordinates of A and A′.
    A (2, 3) to A′ (5, 1): x goes up 3, so 3 right; y goes down 2, so 2 down.
    The move is 3 right, 2 down.
  2. Move B and C by the same move. Add 3 to each x-coordinate and take away 2 from each y-coordinate.
    B (4, 3) → (4 + 3, 3 − 2) = (7, 1)
    C (4, 5) → (4 + 3, 5 − 2) = (7, 3)
  3. Check the shape. The new triangle should look the same as the old one.
    0123456781234567xyABCA′B′C′

Check. A′ = (5, 1), B′ = (7, 1), C′ = (7, 3). Every corner moved 3 right and 2 down. ✓

Watch out. Use the same move for every corner.

Worked example 10.5.6
Rectangle ABCD is on a grid where each square is 1 cm. It is translated 3 right and 2 up. What is the perimeter of the new rectangle?
0123456789123456xyABCD

Key words. A translation is a slide. Every point moves the same distance in the same direction. The shape is not turned and not flipped. We describe it with left or right, and up or down. The perimeter is the distance all the way round.

Method. A translation does not change the size of a shape. So the sides of the new shape are the same length as the sides of the old shape.

  1. Find the sides of ABCD. Count squares along each side.
    AB goes from x = 1 to x = 5: 4 cm.
    BC goes from y = 1 to y = 3: 2 cm.
  2. Find the sides of the new shape. Count squares along each side of A′B′C′D′.
    A′B′ goes from x = 4 to x = 8: 4 cm.
    B′C′ goes from y = 3 to y = 5: 2 cm.
    0123456789123456xyABCDA′B′C′D′
  3. Work out the perimeter of the new shape. Add all four sides.
    4 + 2 + 4 + 2 = 12 cm

Check. The old shape has perimeter 4 + 2 + 4 + 2 = 12 cm too. A translation does not change the perimeter. ✓

Watch out. A translation only moves the shape. It does not make it bigger or smaller.

Worked example 10.5.7
Triangle ABC has corners A (2, 1), B (4, 1) and C (2, 4). Can it be translated 3 squares to the left on a grid that starts at 0? Explain.
01234567123456xyABC

Key words. A translation is a slide. Every point moves the same distance in the same direction. The shape is not turned and not flipped. We describe it with left or right, and up or down.

Method. Move every corner by the move, one at a time. Check that no x-coordinate or y-coordinate falls below 0.

  1. Move each corner 3 left. Left takes away from x.
    A: 2 − 3 = −1 (below 0)
    B: 4 − 3 = 1
    C: 2 − 3 = −1 (below 0)
  2. Look at the grid. The grid starts at 0. A and C would be to the left of the y-axis.
    Two corners would be off the grid.
  3. Decide.
    The triangle cannot move 3 left on this grid.
    01234567123456xyABC

Check. The smallest x-coordinate is 2. The shape can move at most 2 squares to the left. ✓

Watch out. Always check the corner closest to the axis.

Watch out

Move every corner, not just one. If you move only some corners the shape changes. Also remember the shape does not turn or flip: it must face the same way. And name each new corner with the same letter and a small dash, so A moves to A′.

Exercise 10E  Translating shapes

Non-calculator
1
FTriangle ABC has corners A (1, 1), B (3, 1) and C (1, 4). Translate it 4 right and 2 up. Write the coordinates of the new corners.
0123456781234567xyABC
2
FRectangle ABCD has corners A (4, 3), B (7, 3), C (7, 5) and D (4, 5). Translate it 3 left and 2 down. Write the coordinates of the new corners.
0123456781234567xyABCD
3
FSquare PQRS has corners P (2, 2), Q (4, 2), R (4, 4) and S (2, 4). Translate it 5 right. Write the coordinates of the new corners.
0123456789123456xyPQRS
4
FThe L-shape ABCDEF has corners A (1, 1), B (4, 1), C (4, 2), D (2, 2), E (2, 4) and F (1, 4). Translate it 3 right and 1 up. Write the coordinates of the new corners.
0123456789123456xyABCDEF
5
FShape ABCD (blue) is translated to A′B′C′D′ (orange). Describe the translation.
012345678123456xyABCDA′B′C′D′
6
FTriangle ABC (blue) is translated to triangle A′B′C′ (orange). Describe the translation.
01234567891234567xyABCA′B′C′
7
FWhich shape, B, C or D, is a translation of shape A? Write the letter.
0123456712345xyA0123456712345xyB0123456712345xyC0123456712345xyD
8
RTriangle ABC has corners A (1, 1), B (4, 1) and C (1, 3). Ravi translates it 3 right. He writes A′ (4, 1), B′ (7, 1) and C′ (1, 3). What has he done wrong? What should C′ be?
01234567812345xyABC
9
FRectangle ABCD has corners A (1, 1), B (5, 1), C (5, 3) and D (1, 3). It is translated so that A moves to A′ (3, 4).
0123456781234567xyABCD
(a)Where does C move to?
(b)Where does D move to?
10
RWrite always, sometimes or never for each sentence.
(a)A translated shape is bigger than the first shape.
(b)A translated shape has the same side lengths as the first shape.
(c)A translation turns the shape round.
(d)When a shape is translated, all its x-coordinates get bigger.
11
PSRectangle ABCD has corners A (2, 2), B (6, 2), C (6, 5) and D (2, 5). Each grid square is 1 cm by 1 cm. The rectangle is translated 3 right and 1 up. What are the perimeter and the area of the new rectangle?
0123456789101234567xyABCD
12
PSTriangle ABC has corners A (2, 1), B (4, 1) and C (2, 4). The grid starts at 0.
0123456123456xyABC
(a)Translate it 3 left. Can it be done on this grid? If so, write the new coordinates.
(b)Translate it 2 left. Can it be done on this grid? If so, write the new coordinates.
(c)Translate it 2 down. Can it be done on this grid? If so, write the new coordinates.
13
RTriangle ABC has corners A (1, 1), B (3, 1) and C (1, 3). It is translated 3 right and 1 up. Then the new triangle is translated 2 left and 4 up. Write the final coordinates of the three corners. What one translation would take ABC to the same place?
0123456123456789xyABC
10.6

Maps and compass directions

Year 4
Describe positions using grid references, Describe direction using compass points

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Maps and compass directionsComing soon

A map is often drawn on a grid of squares. The columns have letters along the bottom. The rows have numbers up the side. A grid reference names a square. We say the letter first, then the number, like F5. It is just like “along first, then up”, but this time it names a whole square, not a point where lines cross.

ABCDEF12345WoodLakeHouseBridgeCastleBoatCaveFlagN
The castle is in square F5. Go along to F, then up to 5.

To say which way to go, we use the compass points. North (N) is usually at the top of the map. East (E) is on the right. South (S) is at the bottom. West (W) is on the left. In between are north-east (NE), south-east (SE), south-west (SW) and north-west (NW). A tip for the order going clockwise: “Never Eat Shredded Wheat”.

NESESWNWNESWthe eight compass pointsSESWNWNESWNENE is opposite SW
North-east is halfway between north and east. The opposite of north-east is south-west.

To find the direction from one place to another, count the squares across (east or west) and the squares up or down (north or south). If you go across and up by the same number of squares, you are going in between, like north-east. A quarter turn clockwise from north is east. A half turn (two quarter turns) from north is south, so south is the opposite of north.

Key ideas

  • A grid reference is the letter first, then the number: C4. Letters go along the bottom. Numbers go up the side.
  • The compass points are N, NE, E, SE, S, SW, W and NW. North is at the top of the map.
  • To find a direction, count squares across (E or W) and up or down (N or S).
  • A quarter turn moves 2 compass points. A half turn moves 4 points and gives the opposite direction.
Worked example 10.6.1
What is the grid reference of the castle?
ABCDEF12345WoodLakeHouseBridgeCastleBoatCaveFlagN

Key words. A grid reference tells you which square of a map something is in. We say the letter (along the bottom) first, then the number (up the side). The compass points are North (N), East (E), South (S) and West (W), and in between them NE, SE, SW and NW.

Method. Look down to the letter along the bottom. Look across to the number up the side. Write the letter first, then the number.

  1. Find the place on the map. Look for the castle.
    The castle is in the square that is highlighted.
    ABCDEF12345WoodLakeHouseBridgeCastleBoatCaveFlagN
  2. Read the letter. The letters are along the bottom of the map. Look straight down from the square. The letter comes first.
    Letter = F
  3. Read the number. The numbers are up the side of the map. Look straight across from the square.
    Number = 5
  4. Write the grid reference. Letter first, then number, with no space or comma.
    F5

Check. Start at F along the bottom and go up to row 5: you reach the castle. ✓

Watch out. The letter comes first. E3 is not the same as 3E.

Worked example 10.6.2
What is in square C2?
ABCDEF12345WoodLakeHouseBridgeCastleBoatCaveFlagN

Key words. A grid reference tells you which square of a map something is in. We say the letter (along the bottom) first, then the number (up the side). The compass points are North (N), East (E), South (S) and West (W), and in between them NE, SE, SW and NW.

Method. Find the letter along the bottom and go up to the number. Look at what is in that square.

  1. Split the grid reference C2. The letter says which column. The number says which row.
    Letter = C (column 3), number = 2 (row 2)
  2. Find the column. Go to column C along the bottom of the map.
    Column C
  3. Go up to the row. Go up column C until you reach row 2.
    This is the square that is highlighted.
    ABCDEF12345WoodLakeHouseBridgeCastleBoatCaveFlagN
  4. Say what is there.
    In C2 there is a bridge.

Check. The bridge is in C2. ✓

Watch out. Letter first, then number.

Worked example 10.6.3
In which direction is the bridge from the lake?
ABCDEF12345WoodLakeHouseBridgeCastleBoatCaveFlagN

Key words. A grid reference tells you which square of a map something is in. We say the letter (along the bottom) first, then the number (up the side). The compass points are North (N), East (E), South (S) and West (W), and in between them NE, SE, SW and NW.

Method. North is at the top of the map. Count how many squares you go across and how many up or down. Then choose the compass point.

  1. Find the two places. Start at the lake. We want to know which way to go to reach the bridge.
    The lake: E4. The bridge: C2.
  2. Count the squares. Count across (east or west). Count up or down (north or south). North is at the top of the map.
    2 squares west and 2 squares south
    ABCDEF12345WoodLakeHouseBridgeCastleBoatCaveFlagN
  3. Say the direction. It goes across and up or down by the same number of squares, so the direction is in between two main directions: south-west.
    The bridge is south-west (SW) of the lake.

Check. Go the other way from the bridge and you are going north-east, back to the lake. ✓

Watch out. Check which way North is. It is at the top of the map, unless the map says something different.

Worked example 10.6.4
In which direction is the castle from the cave?
ABCDEF12345WoodLakeHouseBridgeCastleBoatCaveFlagN

Key words. A grid reference tells you which square of a map something is in. We say the letter (along the bottom) first, then the number (up the side). The compass points are North (N), East (E), South (S) and West (W), and in between them NE, SE, SW and NW.

Method. North is at the top of the map. Count how many squares you go across and how many up or down. Then choose the compass point.

  1. Find the two places. Start at the cave. We want to know which way to go to reach the castle.
    The cave: F2. The castle: F5.
  2. Count the squares. Count across (east or west). Count up or down (north or south). North is at the top of the map.
    3 squares north
    ABCDEF12345WoodLakeHouseBridgeCastleBoatCaveFlagN
  3. Say the direction. It only goes north or south, so the direction is north.
    The castle is north (N) of the cave.

Check. Go the other way from the castle and you are going south, back to the cave. ✓

Watch out. Check which way North is. It is at the top of the map, unless the map says something different.

Worked example 10.6.5
Noah walks north-east. Then he turns round and walks back the way he came. Which direction does he walk now?

Key words. A grid reference tells you which square of a map something is in. We say the letter (along the bottom) first, then the number (up the side). The compass points are North (N), East (E), South (S) and West (W), and in between them NE, SE, SW and NW.

Method. The opposite direction is a half turn away, which is two quarter turns.

  1. Think about a half turn. The opposite direction is a half turn away. That is two quarter turns.
    Start: north-east (NE)
  2. Turn two quarter turns. Each quarter turn moves two compass points.
    NE → SE → SW
  3. Write the answer.
    The opposite of north-east is south-west (SW).

Check. NE and SW are on opposite sides of the compass. ✓

Watch out. North-east is opposite south-west, not south-east.

Worked example 10.6.6
A robot is facing north-east. It makes a quarter turn clockwise. Which way is it facing now?

Key words. A grid reference tells you which square of a map something is in. We say the letter (along the bottom) first, then the number (up the side). The compass points are North (N), East (E), South (S) and West (W), and in between them NE, SE, SW and NW.

Method. A quarter turn is a right angle. On the compass a quarter turn moves two points (for example North to East). Turn one quarter at a time.

  1. Remember what a turn is. A quarter turn is a right angle. It moves two places along the compass, for example N to E. A half turn moves four places.
    Clockwise goes N, NE, E, SE, S, SW, W, NW.
  2. Turn a quarter turn clockwise. 1 quarter turn clockwise. Each quarter turn moves two compass points.
    Start: north-east (NE)
    NE → SE
  3. Write the answer.
    You are now facing south-east (SE).
    NESWNWNESWSE

Check. Turning back the other way, a quarter turn anticlockwise from SE gives NE. ✓

Watch out. Clockwise is the way the hands of a clock go. Anticlockwise is the other way.

Worked example 10.6.7
Start at the boat. Go 2 squares east, then 1 square north. What do you find there?
ABCDEF12345WoodLakeHouseBridgeCastleBoatCaveFlagN

Key words. A grid reference tells you which square of a map something is in. We say the letter (along the bottom) first, then the number (up the side). The compass points are North (N), East (E), South (S) and West (W), and in between them NE, SE, SW and NW.

Method. Start at the place. Follow each direction in turn, counting squares. Write down the grid reference after each leg.

  1. Find the start. Begin at the boat.
    The boat is in D1.
  2. Go 2 squares east. North is up the map, south is down, east is to the right and west is to the left.
    D1 → F1
  3. Go 1 square north. North is up the map, south is down, east is to the right and west is to the left.
    F1 → F2
  4. Look at the square you are in.
    You are at F2. You have reached the cave.
    ABCDEF12345WoodLakeHouseBridgeCastleBoatCaveFlagN

Check. The route ends in F2, where the cave is. ✓

Watch out. Take one direction at a time. East and west change the letter. North and south change the number.

Watch out

Letter first, then number. C2 means column C, row 2. Do not write 2C. Also remember that north is up the page. If you walk down the map you are going south, not north. And a map square is a whole square, not a point: the castle is in F5, not at (6, 5).

Exercise 10F  Maps, grid references and compass points

Non-calculator
1
FWrite the grid reference of each place on the map of the village.
ABCDEF12345SchoolShopParkFarmLakeHouseWoodCastleBridgeN
(a)The school
(b)The lake
(c)The farm
(d)The castle
2
FWhat is in each square of the village map? Write “nothing” if the square is empty.
ABCDEF12345SchoolShopParkFarmLakeHouseWoodCastleBridgeN
(a)E4
(b)C3
(c)D5
(d)B2
3
FWrite the missing compass points in the boxes with a question mark.
?SE??NESW
4
FIn which direction is the second place from the first place? Use the compass points N, NE, E, SE, S, SW, W or NW.
ABCDEF12345SchoolShopParkFarmLakeHouseWoodCastleBridgeN
(a)From the shop to the castle
(b)From the lake to the house
(c)From the wood to the school
5
FWrite the opposite direction.
(a)South (S)
(b)North-east (NE)
(c)West (W)
(d)South-east (SE)
6
FWhich way are you facing at the end?
(a)Facing north (N), make a quarter turn clockwise.
(b)Facing east (E), make a half turn clockwise.
(c)Facing south-west (SW), make a quarter turn anticlockwise.
(d)Facing west (W), make a three-quarter turn clockwise.
7
FStart at the place and follow the directions. What do you find at the end?
ABCDEF12345SchoolShopParkFarmLakeHouseWoodCastleBridgeN
(a)Start at the farm. Go 1 square north, then 1 square east.
(b)Start at the castle. Go 3 squares west, then 3 squares south.
8
PSWrite directions to go from the school to the house. Say how many squares and which direction. (There is more than one right answer.)
ABCDEF12345SchoolShopParkFarmLakeHouseWoodCastleBridgeN
9
RAisha says the grid reference of the lake is 2F. What has she done wrong? What is the correct grid reference?
ABCDEF12345SchoolShopParkFarmLakeHouseWoodCastleBridgeN
10
RMei says, “The castle is south of the shop.” Is she right? Explain.
ABCDEF12345SchoolShopParkFarmLakeHouseWoodCastleBridgeN
11
RWrite always, sometimes or never for each sentence.
(a)In a grid reference the letter comes first, then the number.
(b)If place A is north of place B, then place B is south of place A.
(c)Two squares in the same column have the same letter in their grid reference.
12
RThe treasure on the island is directly north of the boat and directly west of the lake. What is the grid reference of the treasure?
ABCDEF12345WoodLakeHouseBridgeCastleBoatCaveFlagN
10.7

Routes, robots and treasure maps

Year 4
Describe movements between positions, Link position and direction to turns

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Routes, robots and treasure mapsComing soon

Robots follow programs. A program is a list of instructions. Here are three instructions for a robot on a grid:

A half turn is two quarter turns, so we write R R. Remember, left and right are from the robot’s point of view, not yours.

01234512345xySE
The robot starts at S (1, 1) facing north. F3 takes it to (1, 4). R turns it to face east. F2 takes it to E (3, 4).

On a treasure map we use coordinates and compass points together. Going east makes the x-coordinate bigger. Going west makes it smaller. Going north (up) makes the y-coordinate bigger. Going south (down) makes it smaller. A treasure clue can also use a move: “the treasure is 4 right and 3 up from the old tree”.

01234561234567xystartX
Start (2, 2). 3 east, 4 north, 2 west. The treasure X is at (3, 6).

The shortest route along the grid lines from one point to another goes across and then up or down. Add the number of squares across and the number of squares up or down.

Key ideas

  • F n = forward n squares. R = turn right a quarter turn. L = turn left a quarter turn. A half turn is R R.
  • A quarter turn is a right angle. Left and right are from the robot’s point of view.
  • East: x gets bigger. West: x gets smaller. North: y gets bigger. South: y gets smaller.
  • Shortest route along grid lines = squares across + squares up or down.
Worked example 10.7.1
A robot starts at (1, 1) facing north. It follows this program: F3 R F2. Where does it finish, and which way is it facing?

Key words. A program is a list of instructions. Forward means go straight ahead. A turn left or right is a quarter turn (a right angle) unless it says half turn. A half turn is two quarter turns.

Method. Follow the instructions one at a time. F means forward, R means turn right (a quarter turn) and L means turn left (a quarter turn). Write the position after each instruction.

  1. Write the start point and the way the robot faces. We follow the program from here. The robot’s left and right depend on the way it is facing.
    Start at (1, 1), facing north.
  2. Instruction 1: F3. Forward 3 squares north.
    (1, 1) → (1, 4)
  3. Instruction 2: R. Turn right a quarter turn: now facing east.
    north → east
  4. Instruction 3: F2. Forward 2 squares east.
    (1, 4) → (3, 4)
  5. Write where the robot finishes. Look at the last position and the last way it faces.
    The robot ends at (3, 4), facing east.
    01234512345xySE

Check. Draw the path on the grid from (1, 1): it ends at (3, 4). ✓

Watch out. Turn left or right from the way the robot is facing, not from the way you are looking at the page.

Worked example 10.7.2
A robot starts at (5, 1) facing north. It follows this program: F2 L F3 R F1. Where does it finish, and which way is it facing?

Key words. A program is a list of instructions. Forward means go straight ahead. A turn left or right is a quarter turn (a right angle) unless it says half turn. A half turn is two quarter turns.

Method. Follow the instructions one at a time. F means forward, R means turn right (a quarter turn) and L means turn left (a quarter turn). Write the position after each instruction.

  1. Write the start point and the way the robot faces. We follow the program from here. The robot’s left and right depend on the way it is facing.
    Start at (5, 1), facing north.
  2. Instruction 1: F2. Forward 2 squares north.
    (5, 1) → (5, 3)
  3. Instruction 2: L. Turn left a quarter turn: now facing west.
    north → west
  4. Instruction 3: F3. Forward 3 squares west.
    (5, 3) → (2, 3)
  5. Instruction 4: R. Turn right a quarter turn: now facing north.
    west → north
  6. Instruction 5: F1. Forward 1 square north.
    (2, 3) → (2, 4)
  7. Write where the robot finishes. Look at the last position and the last way it faces.
    The robot ends at (2, 4), facing north.
    012345612345xySE

Check. Draw the path on the grid from (5, 1): it ends at (2, 4). ✓

Watch out. Turn left or right from the way the robot is facing, not from the way you are looking at the page.

Worked example 10.7.3
The robot starts at S (1, 1) facing north. Write a program that takes it along the path to E (4, 2).
0123456123456xySE

Key words. A program is a list of instructions. Forward means go straight ahead. A turn left or right is a quarter turn (a right angle) unless it says half turn. A half turn is two quarter turns.

Method. Look at each straight part of the path. If the robot is not facing the right way, turn first. Then write F and the number of squares.

  1. Look at the start and the way the robot faces. We start with the robot at the first point.
    Start at (1, 1), facing north.
  2. Do the path one straight part at a time. For each part: turn to face the right way if needed, then go forward the number of squares.
    (1, 1) to (1, 4): forward 3.
    (1, 4) to (4, 4): turn right (now facing east), then forward 3.
    (4, 4) to (4, 2): turn right (now facing south), then forward 2.
  3. Write the program. Join all the instructions in order.
    Program: F3 R F3 R F2
    0123456123456xySE

Check. Run the program from (1, 1): F3 R F3 R F2 ends at (4, 2). ✓

Watch out. Check which way the robot is facing before you choose R or L.

Worked example 10.7.4
The robot starts at S (2, 2) facing north. It goes up to (2, 5) and then turns round and comes back down to E (2, 3). Write the program.
01234123456xySE

Key words. A program is a list of instructions. Forward means go straight ahead. A turn left or right is a quarter turn (a right angle) unless it says half turn. A half turn is two quarter turns.

Method. Look at each straight part of the path. If the robot is not facing the right way, turn first. Then write F and the number of squares.

  1. Look at the start and the way the robot faces. We start with the robot at the first point.
    Start at (2, 2), facing north.
  2. Do the path one straight part at a time. For each part: turn to face the right way if needed, then go forward the number of squares.
    (2, 2) to (2, 5): forward 3.
    (2, 5) to (2, 3): turn right twice, a half turn (now facing south), then forward 2.
  3. Write the program. Join all the instructions in order.
    Program: F3 R R F2
    01234123456xySE

Check. Run the program from (2, 2): F3 R R F2 ends at (2, 3). ✓

Watch out. Check which way the robot is facing before you choose R or L.

Worked example 10.7.5
A pirate map says: “Start at (2, 2). Go 3 steps east, then 4 steps north, then 2 steps west.” Where is the treasure?

Key words. A program is a list of instructions. Forward means go straight ahead. A turn left or right is a quarter turn (a right angle) unless it says half turn. A half turn is two quarter turns.

Method. Follow the directions one at a time. East and west change the x-coordinate. North and south change the y-coordinate.

  1. Write the start point. The map starts here.
    Start at (2, 2).
  2. Follow each direction. East makes x bigger. West makes x smaller. North makes y bigger. South makes y smaller.
    3 east: (2, 2) → (5, 2) (x gets bigger)
    4 north: (5, 2) → (5, 6) (y gets bigger)
    2 west: (5, 6) → (3, 6) (x gets smaller)
  3. Write where the treasure is. The last point is the place the map shows.
    The treasure is at (3, 6).
    01234561234567xystartX

Check. Count the squares on the grid from (2, 2): you arrive at (3, 6). ✓

Watch out. North is up the grid, so going north makes the y-coordinate bigger.

Worked example 10.7.6
The old tree is at (3, 2). The treasure is 4 right and 3 up from the old tree. What are the coordinates of the treasure?

Key words. A program is a list of instructions. Forward means go straight ahead. A turn left or right is a quarter turn (a right angle) unless it says half turn. A half turn is two quarter turns.

Method. “4 right and 3 up” is a translation. Do it to the old tree’s coordinates.

  1. Write the old tree’s coordinates. Along first, then up.
    Old tree = (3, 2)
  2. Move 4 right. Right adds to x.
    x: 3 + 4 = 7
  3. Move 3 up. Up adds to y.
    y: 2 + 3 = 5
  4. Write the treasure.
    The treasure is at (7, 5).
    012345678123456xyTT′

Check. From (3, 2) count 4 right and 3 up: (7, 5). ✓

Watch out. Right and left change x. Up and down change y.

Worked example 10.7.7
What is the shortest route along the grid lines from A (1, 2) to B (6, 5)? Write the number of squares.

Key words. A program is a list of instructions. Forward means go straight ahead. A turn left or right is a quarter turn (a right angle) unless it says half turn. A half turn is two quarter turns.

Method. Along the grid lines you must go across and then up or down. Add the two distances.

  1. Find how far across. Compare the x-coordinates.
    x: 1 to 6 is 5 squares.
  2. Find how far up or down. Compare the y-coordinates.
    y: 2 to 5 is 3 squares.
  3. Add them. Moving along the grid lines, you go across and then up or down.
    5 + 3 = 8 squares
    01234567123456xyAB

Check. Another route (up first, then across) also has 3 + 5 = 8 squares. ✓

Watch out. Do not count the lines you start on twice. Count the squares between the points.

Watch out

Left and right belong to the robot. If the robot is facing south, turning left makes it face east, not west. Also, do not count the start square as a step: F3 moves the robot 3 squares from where it is. And remember each turn is only a quarter turn.

Exercise 10G  Robots, routes and treasure maps

Non-calculator
1
FA robot follows each program. Where does it finish, and which way is it facing?
(a)Start at (1, 1), facing north. Program: F3 R F4
(b)Start at (2, 5), facing east. Program: F3 R F4 L F1
(c)Start at (6, 6), facing south. Program: F2 R F3 R F1
2
FA robot faces the way shown. It follows the turns. Which way is it facing at the end?
(a)Facing north: R R L
(b)Facing east: L L L
(c)Facing south: R L R R
3
FThe robot starts at S (1, 1) facing north and goes along the path to E (5, 5).
0123456123456xySE
(a)Write a program for the path.
(b)How many right angles does the robot turn through?
4
FThe robot starts at S (5, 1) facing north. Write a program that takes it along the path to E (2, 2).
01234567123456xySE
5
PSThe robot starts at S (1, 1) facing north. Which program takes it to the star at (4, 3)?
A: F3 R F2   B: F2 R F3   C: R F2 L F3
012345612345xyS★
6
FRead each treasure map. Where is the treasure?
(a)Start at (1, 1). Go 4 east, then 3 north.
(b)Start at (7, 6). Go 5 west, then 4 south, then 2 east.
7
FThe treasure is 3 right and 2 up from a rock at (3, 4). Where is the treasure?
8
PSHow many squares is the shortest route along the grid lines from the first point to the second?
(a)(1, 2) to (5, 4)
(b)(0, 0) to (6, 3)
(c)(7, 5) to (2, 1)
9
RA robot starts at (1, 1) facing north. Its program is F2 R F3. Leo says it finishes at (3, 4). Is he right? Explain.
10
RWrite always, sometimes or never for each sentence.
(a)A robot turns right four times. It faces the way it started.
(b)A robot facing north turns right a quarter turn. It now faces west.
(c)When a robot follows F4, its y-coordinate changes.
11
PSA robot starts at (2, 2) facing north. It follows this program: F3 R F3 R F3 R F3. Where does it finish, and which way is it facing? What shape is the path it makes?
12
RA robot starts at (2, 2) facing north. It must reach (6, 5). Write a program with only 3 instructions that does this.

Chapter summary

SkillMethod
CoordinatesWrite (along, up) in brackets. The first number is x (along the x-axis). The second is y (up the y-axis). The origin is (0, 0). Along the corridor, then up the stairs.
PlottingStart at (0, 0). Go along x, go up y, mark the point. Join points in order to make a polygon. Check the sides and right angles to name it.
Missing cornersRectangle: copy x from one corner and y from another. Parallelogram: use the same move on opposite sides. Kite and isosceles triangle: use the mirror line.
LinesA horizontal line: all the points have the same y. A vertical line: all the points have the same x.
Translating pointsRight: add to x. Left: take away from x. Up: add to y. Down: take away from y. Say it as “3 right, 2 up”.
Describing a moveCompare x-coordinates for left or right. Compare y-coordinates for up or down. Check with a second corner for a shape.
Translating shapesMove every corner by the same move. The shape stays the same size and faces the same way.
MapsGrid reference: letter first, then number (for example C4). North is at the top. The eight points: N, NE, E, SE, S, SW, W, NW.
Robots and routesF = forward, R = quarter turn right, L = quarter turn left. East: x up. West: x down. North: y up. South: y down. Shortest route = across + up or down.

Can you…?

  • I can read and write coordinates, along first and then up.
  • I can plot points and join them to make a polygon, and name the shape.
  • I can find a missing corner of a square, rectangle, parallelogram, kite or isosceles triangle.
  • I can find a missing coordinate on a horizontal or vertical line.
  • I can translate a point or a shape and write the new coordinates.
  • I can describe the translation from one point or shape to another.
  • I can use grid references and the eight compass points on a map.
  • I can follow and write robot programs with forward and quarter turns.
  • I can solve treasure-map problems with coordinates and compass directions.

Key terms

Coordinates
Two numbers in brackets, (along, up), that give the position of a point.
x-axis
The line going along the bottom of a grid.
y-axis
The line going up the side of a grid.
Origin
The point (0, 0) where the axes meet.
Plot
To mark a point on a grid.
Vertex
A corner of a shape (plural: vertices).
Polygon
A flat shape with straight sides.
Horizontal
Flat, going across.
Vertical
Straight up and down.
Translation
A slide. The shape does not turn or flip.
Grid reference
The letter then the number that names a square on a map, such as C4.
Compass points
N, NE, E, SE, S, SW, W and NW.
Program
A list of instructions for a robot.
Quarter turn
A right angle. A turn left or right in a robot program.
Perimeter
The distance all the way round a shape.

Exercise 10Z  Check-up (non-calculator)

Non-calculator
1
Write the coordinates of the point P.
01234567123456xyPQR
2
Plot the point A (4, 3) on the grid.
012345612345xy
3
Which letter is at the point (5, 2)?
012345671234567xyABCD
4
A, B and C are three corners of a rectangle: A (1, 2), B (6, 2) and C (6, 5). Find the coordinates of D.
01234567123456xyABC
5
The point A (3, 4) is translated 2 right and 3 up. What are the coordinates of A′?
6
Describe the translation from A (2, 5) to B (6, 2).
01234567123456xyAB
7
Triangle ABC has corners A (3, 3), B (5, 3) and C (3, 5). It is translated 2 left and 1 down. Write the coordinates of the new corners.
012345671234567xyABC
8
What is the grid reference of the shop?
ABCDEF12345SchoolShopParkFarmLakeHouseWoodCastleBridgeN
9
A robot starts at (2, 1) facing north. It follows this program: F2 R F3. Where does it finish?
10
Ben says the coordinates of P are (5, 3). Is he right? Explain how you know.
0123456123456xyP

Total 19 marks. Allow about 19 minutes.

Exercise 10X  SATs-style questions

Non-calculator
1
FWrite the coordinates of the point A.[1 mark]
012345671234567xyABCD
2
FWhich point has the coordinates (2, 6)? Write the letter.[1 mark]
012345671234567xyWXYZ
3
FOn the grid, plot A (1, 1), B (4, 1) and C (4, 3). Plot D so that ABCD is a rectangle. Join the points. Write the coordinates of D.[2 marks]
012345612345xy
4
FTriangle ABC has corners A (1, 2), B (3, 2) and C (1, 5). It is translated 3 right and 2 up. Write the coordinates of A′, B′ and C′.[2 marks]
01234567812345678xyABC
5
FDescribe the translation that takes A (2, 4) to B (7, 2).[1 mark]
012345678123456xyAB
6
PSLook at the map of the island.[2 marks]
ABCDEF12345WoodLakeHouseBridgeCastleBoatCaveFlagN
(a)What is the grid reference of the lake?
(b)In which direction is the lake from the castle? Use a compass point.
7
PSA robot starts at (3, 2) facing east. It follows this program: F2 L F3 L F4. (a) Where does the robot finish? (b) Which way is it facing?[2 marks]
8
RAmira says, “The point (4, 0) is on the y-axis.” Is she correct? Explain how you know.[2 marks]
9
PSA (1, 1), B (5, 1) and C (6, 4) are three corners of a parallelogram ABCD. (a) Find the coordinates of D. (b) Describe the translation that takes A to B.[3 marks]
012345678123456xyABC
10
RA map says: “Start at (2, 3). Go 4 steps east, then 2 steps north, then 3 steps west.” (a) Where is the treasure? (b) What one translation takes the start to the treasure?[3 marks]
← Chapter 9: Shapes, angles and symmetryChapter 11: Statistics →

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