Reflection and translation
By the end of this chapter you will be able to
- Describe a translation by the squares across (left or right) and then the squares up or down
- Draw a translation by moving every corner the same number of squares
- Explain that a reflection flips a shape, and that each point and its image are the same distance from the mirror line, on opposite sides
- Draw a reflection in a vertical or a horizontal mirror line, and find a mirror line from a point and its image
- Explain that a translation or a reflection keeps the size and shape, so the shape has not changed
You should already know
- Position and direction on a grid, and first-quadrant coordinates (Year 4, Chapter 10)
- Right angles, quarter turns and 2-D shapes (Year 4, Chapter 9 and Year 5, Chapter 9)
- Negative numbers on a number line (Year 5, Chapter 1)
Where this leads
- Coordinates in all four quadrants, and reflecting and translating shapes on the coordinate plane (Year 6, Chapter 11)
- Shapes, angles and circles, with the properties of the shapes that are moved (Year 6, Chapter 10)
National Curriculum for England, Key Stage 2 mathematics, Year 5. Content: Identify and describe the position of a shape following a translation, Identify, describe and represent the position of a shape following a reflection, using the appropriate language, Represent the position of a shape after a translation by drawing it on a grid, Know that the shape has not changed after a reflection or translation.
Exercise P Do you remember?
Non-calculatorMethod. Start at 0. Move 4 steps right, then 6 steps left.
- Start at 0. The number line starts at 0.0
- Move 4 steps right. Moving right adds to the number.0 + 4 = 4
- Move 6 steps left. Moving left takes away. We go past 0, so the answer is negative.4 − 6 = −2
- Write the answer. −2
Check. Start at -2, move 6 steps right, and you reach 4. ✓
Watch out. Going below 0 gives a negative number. The minus sign goes in front of the number.
Method. Add the four sides. A square has four equal sides.
- Write the length of one side. Each side of the square is 6 cm.6 cm
- Add the four sides. The perimeter is all four sides added together.6 + 6 + 6 + 6 = 24
- Write the answer. 24 cm
Check. 4 × 6 = 24. ✓
Watch out. Count all four sides, not just two.
Method. Use the compass and turn clockwise.
- Start facing north. North is at the top of the compass.facing north
- Turn 90 degrees clockwise. A quarter turn clockwise moves you a quarter of the way round the compass.north to east
- Write the answer. east
Check. A half turn from north is south. ✓
Watch out. Clockwise is the way the hands of a clock turn.
Method. Count across first, then up.
- Count across from the left edge. The point is 3 squares across from the edge.3 across
- Count up from the bottom edge. The point is 2 squares up from the bottom.2 up
- Write the pair, across first. We always write across first, then up.(3, 2)
Check. Start at the corner, go 3 across and 2 up, and you are at the point. ✓
Watch out. Across comes first, then up.
Method. Count the squares in each row, then multiply by the number of rows.
- Count the squares in one row. The rectangle is 5 squares long, so each row has 5 squares.5 squares
- Count the rows. It is 2 squares high, so there are 2 rows.2 rows
- Multiply. Rows times the squares in a row gives the total.5 × 2 = 10
- Write the answer. 10 squares
Check. 2 × 5 = 10 too. ✓
Watch out. Count the squares inside the rectangle, not only the edges.
Sliding shapes: translations
A translation slides a shape from one place to another. Every point of the shape moves the same distance, in the same direction. The shape does not turn, flip or change size. The new shape is called the image. We name the corners of the image with a dash, so the image of corner A is A'.
To describe a translation, count the squares across (to the right or to the left), then count the squares up or down. Always say the across move first.
| Word | What it means | Example |
|---|---|---|
| right | across, to the right: the shape moves on the right-hand side | 3 squares right |
| left | across, to the left: the shape moves on the left-hand side | 2 squares left |
| up | the shape moves higher up the grid | 4 squares up |
| down | the shape moves lower down the grid | 1 square down |
Key ideas
- A translation slides a shape. It keeps its size, its shape and the way it faces.
- Every point moves the same distance, in the same direction.
- Describe a move as the squares across (right or left), then the squares up or down.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Count the squares across, then up or down, from one corner. Every corner moves the same way.
- Pick one corner of the shape. Use the same corner (A) on the start shape and on the image.corner A
- Count 3 squares to the right. The matching corner of the image is 3 squares right.3 squares right
- Count 1 square down. The matching corner of the image is 1 squares down.1 squares down
- Check another corner. Every corner moves the same way, so the shape only slides.✓ the same move
- Write the answer. 3 squares to the right and 1 square down
Check. Count from another corner and you get the same move. ✓
Watch out. Count the squares on the grid. Do not count the sides of the shape.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Start at the first number, then add or take away the move.
- Start at the first number. The counter starts 2 squares from the edge.2
- Move 5 squares right. Moving right or up adds. Moving left or down takes away.2 + 5 = 7
- Write the answer. 7
Check. Move 5 squares the other way to get back to 2. ✓
Watch out. Moving left takes away. Count carefully across the squares.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Count the squares across, then up or down, from one corner. Every corner moves the same way.
- Pick one corner of the shape. Use the same corner (A) on the start shape and on the image.corner A
- Count 2 squares to the left. The matching corner of the image is 2 squares left.2 squares left
- Count 2 squares down. The matching corner of the image is 2 squares down.2 squares down
- Check another corner. Every corner moves the same way, so the shape only slides.✓ the same move
- Write the answer. 2 squares to the left and 2 squares down
Check. Count from another corner and you get the same move. ✓
Watch out. Count the squares on the grid. Do not count the sides of the shape.
Watch out
A common mistake is to count the sides of the shape instead of the squares on the grid. Count the squares between the start and the image. Also, a move with no up or down part is just a move across, so do not invent an up or down move.
Exercise 10A Describing translations
Non-calculatorKey words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Count the squares across, then up or down, from one corner. Every corner moves the same way.
- Pick one corner of the shape. Use the same corner (A) on the start shape and on the image.corner A
- Count 3 squares to the right. The matching corner of the image is 3 squares right.3 squares right
- Check up and down. There is no move up or down, so the corner stays at the same height.0 squares up or down
- Check another corner. Every corner moves the same way, so the shape only slides.✓ the same move
- Write the answer. 3 squares to the right
Check. Count from another corner and you get the same move. ✓
Watch out. Count the squares on the grid. Do not count the sides of the shape.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Count the squares across, then up or down, from one corner. Every corner moves the same way.
- Pick one corner of the shape. Use the same corner (A) on the start shape and on the image.corner A
- Check across. There is no move across, so the corner stays in the same column.0 squares across
- Count 2 squares up. The matching corner of the image is 2 squares up.2 squares up
- Check another corner. Every corner moves the same way, so the shape only slides.✓ the same move
- Write the answer. 2 squares up
Check. Count from another corner and you get the same move. ✓
Watch out. Count the squares on the grid. Do not count the sides of the shape.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Count the squares across, then up or down, from one corner. Every corner moves the same way.
- Pick one corner of the shape. Use the same corner (A) on the start shape and on the image.corner A
- Count 4 squares to the left. The matching corner of the image is 4 squares left.4 squares left
- Count 1 square down. The matching corner of the image is 1 squares down.1 squares down
- Check another corner. Every corner moves the same way, so the shape only slides.✓ the same move
- Write the answer. 4 squares to the left and 1 square down
Check. Count from another corner and you get the same move. ✓
Watch out. Count the squares on the grid. Do not count the sides of the shape.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Count the squares across, then up or down, from one corner. Every corner moves the same way.
- Pick one corner of the shape. Use the same corner (A) on the start shape and on the image.corner A
- Check across. There is no move across, so the corner stays in the same column.0 squares across
- Count 3 squares up. The matching corner of the image is 3 squares up.3 squares up
- Check another corner. Every corner moves the same way, so the shape only slides.✓ the same move
- Write the answer. 3 squares up
Check. Count from another corner and you get the same move. ✓
Watch out. Count the squares on the grid. Do not count the sides of the shape.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Start at the first number, then add or take away the move.
- Start at the first number. The counter starts 4 squares from the edge.4
- Move 3 squares right. Moving right or up adds. Moving left or down takes away.4 + 3 = 7
- Write the answer. 7
Check. Move 3 squares the other way to get back to 4. ✓
Watch out. Moving left takes away. Count carefully across the squares.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Start at the first number, then add or take away the move.
- Start at the first number. The counter starts 4 squares from the edge.4
- Move 2 squares left. Moving right or up adds. Moving left or down takes away.4 − 2 = 2
- Write the answer. 2
Check. Move 2 squares the other way to get back to 4. ✓
Watch out. Moving left takes away. Count carefully across the squares.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Start at the first number, then add or take away the move.
- Start at the first number. The counter starts 4 squares from the edge.4
- Move 6 squares right. Moving right or up adds. Moving left or down takes away.4 + 6 = 10
- Write the answer. 10
Check. Move 6 squares the other way to get back to 4. ✓
Watch out. Moving left takes away. Count carefully across the squares.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Find the new distance across and the new distance up, one at a time.
- Find the new distance from the left edge. The square starts 1 square from the left edge and moves 4 squares to the right.1 + 4 = 5
- Find the new distance from the bottom. It starts 1 square from the bottom and moves 3 squares up.1 + 3 = 4
- Write the answer. 5 squares from the left edge and 4 squares from the bottom
Check. Move back 4 to the left and 3 down, and you reach the start again. ✓
Watch out. Moving right or up adds. Do not add the 4 and the 3 together.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Look at the way each shape faces, not only where it is.
- Look at the blue shape. Its corner, where the two arms meet, is at the bottom left.corner at the bottom left
- Look at the orange shape. Its corner is at the bottom right, so the shape has turned.corner at the bottom right
- Check the rule for a translation. A translation only slides a shape. It never turns it.turned, so not a slide
- Decide. Ruby is wrong, because a shape that has been turned is not a translation.No
Check. A translated shape faces the same way as the original. ✓
Watch out. Check the way a shape faces, not only where it is.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Add the two moves: one across to the right and then back to the left.
- Write the first move. The shape moves 3 squares to the right.+3
- Write the second move. Then it moves 3 squares to the left, back the way it came.−3
- Add the moves together. Moves in opposite directions cancel each other out.3 − 3 = 0
- Write the answer. The total move is 0 squares, so the shape ends where it started.back where it started
Check. The total move is 0 squares. ✓
Watch out. Two equal moves in opposite directions cancel out.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Add the across moves and add the up and down moves separately.
- Add the across moves. Two squares to the right and then one square to the right.2 + 1 = 3 squares to the right
- Add the up and down moves. One square up and then three squares down.1 − 3 = −2, so 2 squares down
- Write the single move. One move with 3 across and 2 down does the same job.3 squares to the right and 2 squares down
Check. 3 squares right and 2 squares down is the same as the two moves. ✓
Watch out. Add the across moves, then the up and down moves. Do not mix them up.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. A translation only slides a shape.
- Look at A. Turning changes the way a shape faces, so it is not a translation.turn: not a slide
- Look at B. Sliding the shape 3 squares to the right is a translation.slide: yes
- Look at C. Flipping over a line is a reflection, not a translation.flip: not a slide
- Write the answer. B
Check. Only B slides the shape without turning or flipping it. ✓
Watch out. A translation keeps the shape facing the same way.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Find the new position of the left side and the right side, then measure from the left edge.
- Find the left side. The left side starts on the left edge, at 0 squares. It moves 3 squares to the right.0 + 3 = 3
- Find the right side. The house is 4 squares wide, so its right side starts 4 squares from the edge.4 + 3 = 7
- Write the answer. 7 squares
Check. The house is still 4 squares wide: 7 − 3 = 4. ✓
Watch out. The width of a shape does not change when it is translated.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Find the gap between the two triangles.
- Write the first move. Tom moves the triangle 4 squares down.4 squares down
- Write the second move. Sam moves the same triangle 4 squares up.4 squares up
- Find the gap. The two places are on opposite sides of the start, so the gap is the two moves added together.4 + 4 = 8
- Write the answer. No, they are 8 squares apart
Check. The two triangles are 8 squares apart. ✓
Watch out. Moving up and moving down can be added to find the gap between two places.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Think about what a translation does to every point.
- Check Ali. In a translation every point moves the same distance and in the same direction.the same for every point
- Check Dana. The top corner and the bottom corner move the same amount, so the top corner does not move further.top and bottom move the same
- Decide. Ali is right, and Dana is wrong.Ali
Check. Every corner of a slid shape moves the same way. ✓
Watch out. In a slide, the whole shape moves together.
Drawing a translation
To draw a translation, move every corner the same number of squares. Start with one corner. Count the squares across, then count the squares up or down, and put a dot. Then do the same for every other corner, and join the dots in the same order.
Check your drawing: the new shape must be the same size and shape as the old one.
Key ideas
- To draw a translation, move every corner the same number of squares.
- Count across first, then up or down.
- Check that the new shape is the same size and shape as the old one.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Find the new position of the bottom-left corner, one move at a time.
- Start at the corner. The bottom-left corner is 1 square from the left edge and 0 squares from the bottom.1 across, 0 up
- Move 2 squares to the right. Moving right adds to the across distance.1 + 2 = 3
- Move 3 squares up. Moving up adds to the height.0 + 3 = 3
- Write the answer. 3 squares from the left edge and 3 squares from the bottom
Check. The new corner is 2 squares right and 3 squares up from the old one. ✓
Watch out. Add the moves to the numbers. Do not write the moves on their own.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Find the new position of the bottom-left corner, one move at a time.
- Start at the corner. The bottom-left corner of the house is 4 squares from the left edge and 4 squares from the bottom.4 across, 4 up
- Move 3 squares to the left. Moving left takes away from the across distance.4 − 3 = 1
- Move 2 squares down. Moving down takes away from the height.4 − 2 = 2
- Write the answer. 1 square from the left edge and 2 squares from the bottom
Check. Go back 3 squares to the right and 2 squares up to reach the start. ✓
Watch out. Left and down take away. Right and up add.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Count the squares across, then up or down, from one corner. Every corner moves the same way.
- Pick one corner of the shape. Use the same corner (A) on the start shape and on the image.corner A
- Count 2 squares to the right. The matching corner of the image is 2 squares right.2 squares right
- Count 2 squares down. The matching corner of the image is 2 squares down.2 squares down
- Check another corner. Every corner moves the same way, so the shape only slides.✓ the same move
- Write the answer. 2 squares to the right and 2 squares down
Check. Count from another corner and you get the same move. ✓
Watch out. Count the squares on the grid. Do not count the sides of the shape.
Watch out
A common mistake is to move some corners and not others. Every corner must move the same number of squares, in the same way. When you have finished, check that the new shape is still the same size.
Exercise 10B Drawing a shape after a translation
Non-calculatorKey words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Move each corner 3 squares to the right. Start with the left side.
- Find the left side now. The left side is 1 square from the left edge.1 square
- Move it 3 squares to the right. Moving right adds to the across distance.1 + 3 = 4
- Write the answer. The new left side is 4 squares from the left edge.4 squares
Check. The new house is still 4 squares wide. ✓
Watch out. Count from the left edge of the grid, not from the shape.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Move the top corner up 2 squares, then count from the bottom edge.
- Find the top corner now. The top corner is 3 squares above the bottom edge.3 up
- Move it 2 squares up. Moving up adds to the height.3 + 2 = 5
- Write the answer. 5 squares
Check. The triangle is 3 squares tall, so the new bottom is 2 squares up. ✓
Watch out. Count the squares from the bottom edge of the grid.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Move the bottom-left corner 2 squares right and 1 square down.
- Find the across distance. The corner starts 1 square from the left edge.1 + 2 = 3
- Find the height. The corner starts 3 squares from the bottom.3 − 1 = 2
- Write the answer. The corner is 3 squares from the left edge and 2 squares from the bottom.3 squares from the left edge and 2 squares from the bottom
Check. Go back 2 squares left and 1 square up to reach the start. ✓
Watch out. Down takes away from the height. Right adds to the across distance.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Find where the tip is, then move it 4 squares to the left.
- Find the tip now. The tip of the arrow is 9 squares from the left edge.9 across
- Move it 4 squares to the left. Moving left takes away from the across distance.9 − 4 = 5
- Write the answer. 5 squares
Check. Move 4 squares right from 5 and you are back at the tip. ✓
Watch out. The tip is the point on the right, so start there.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. The bottom side of the house is 1 square from the bottom edge. Only the up move changes its height.
- Find the bottom side now. The bottom side is 1 square from the bottom edge.1 square
- Look at the move up. A move to the right does not change the height. A move up adds to the height.up 0
- Find the new height. Add the up move to the height.1 + 0 = 1
- Write the answer. 1 square
Check. The house is still the same height, so its bottom side only moves up by 0. ✓
Watch out. A move to the right does not change how high a shape is.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. The bottom side of the house is 1 square from the bottom edge. Only the up move changes its height.
- Find the bottom side now. The bottom side is 1 square from the bottom edge.1 square
- Look at the move up. A move to the right does not change the height. A move up adds to the height.up 2
- Find the new height. Add the up move to the height.1 + 2 = 3
- Write the answer. 3 squares
Check. The house is still the same height, so its bottom side only moves up by 2. ✓
Watch out. A move to the right does not change how high a shape is.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. The bottom side of the house is 1 square from the bottom edge. Only the up move changes its height.
- Find the bottom side now. The bottom side is 1 square from the bottom edge.1 square
- Look at the move up. A move to the right does not change the height. A move up adds to the height.up 2
- Find the new height. Add the up move to the height.1 + 2 = 3
- Write the answer. 3 squares
Check. The house is still the same height, so its bottom side only moves up by 2. ✓
Watch out. A move to the right does not change how high a shape is.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Look at the direction of the move.
- Check the move Sam was given. Three squares to the right is a move across only.across 3, up 0
- Check what Sam did. Sam moved the triangle up, so it is in the wrong place.up instead of across
- Write the answer. He moved it up instead of across.
Check. Three squares to the right has no up or down part. ✓
Watch out. Read the direction of the move before you draw.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Find the bottom-right corner now, then move it.
- Find the bottom-right corner now. The rectangle is 4 squares wide, so the bottom-right corner is 4 squares from the left edge.4 across, 0 up
- Move it 3 squares to the right. Moving right adds to the across distance.4 + 3 = 7
- Move it 1 square up. Moving up adds to the height.0 + 1 = 1
- Write the answer. 7 squares from the left edge and 1 square from the bottom
Check. The new rectangle is still 4 squares wide and 2 squares high. ✓
Watch out. Find the corner first, then move it.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Move the corner across, then down.
- Find the across distance. The top-right corner is 5 squares from the left edge. It moves 2 squares to the left.5 − 2 = 3
- Find the height. The corner is 6 squares from the bottom. It moves 4 squares down.6 − 4 = 2
- Write the answer. 3 squares from the left edge and 2 squares from the bottom
Check. Move 2 squares right and 4 squares up from the new corner to reach the old one. ✓
Watch out. Left and down both take away from the numbers.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Try both orders and compare the place each one ends in.
- Try the first order. Move 2 squares right, then 3 squares up.2 right, then 3 up: ends at 2 across and 3 up
- Try the second order. Move 3 squares up, then 2 squares right.3 up, then 2 right: ends at 2 across and 3 up
- Decide. Both orders end at the same place, so the order does not change the answer.Yes
Check. Adding the across moves and the up moves in any order gives the same total. ✓
Watch out. Each move is a shift, so the shift can be done in either order.
Key words. A translation or a reflection keeps the size and shape of a shape. Only its position (and, for a reflection, the way it faces) can change.
Method. Think about what changes and what stays the same in a slide.
- Look at the three choices. The position changes when the shape moves.position: changes
- Look at the size. A slide keeps the same size.size: stays the same
- Look at the direction. A slide keeps the way the shape faces.direction: stays the same
- Write the answer. position
Check. Only the position changes in a translation. ✓
Watch out. A translation changes where the shape is, not what it is like.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Find the left-hand corner of the old triangle, then add the across move.
- Find the old corner. The left-hand corner of the blue triangle is 1 square from the left edge.1 across
- Add the across move. The new triangle is 3 squares to the right.1 + 3 = 4
- Write the answer. 4 squares
Check. The up move of 2 does not change the across distance. ✓
Watch out. Use only the across part of the move for this question.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Add the across and up moves to the numbers from the edges.
- Start with the counter. The counter is 4 squares from the left edge and 5 squares from the bottom.4 across, 5 up
- Move it 3 squares to the right. Moving right adds to the across distance.4 + 3 = 7
- Move it 1 square down. Moving down takes away from the height.5 − 1 = 4
- Write the answer. 7 squares from the left edge and 4 squares from the bottom
Check. Go back 3 squares to the left and 1 square up to reach the start. ✓
Watch out. Left and down take away, right and up add.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Find the move that undoes the first move.
- Write the first move. The shape moved 4 squares to the right.+4
- Write the move that undoes it. To go back, move the same number of squares the other way.4 squares to the left
- Write the answer. 4 squares to the left
Check. 4 squares right then 4 squares left is a total move of 0. ✓
Watch out. Undo a move by moving the same number of squares the other way.
Flipping shapes: reflections
A reflection flips a shape over a mirror line. The mirror line is shown as a dashed red line. Every point of the image is the same distance from the mirror line as the point it came from, but on the other side.
The image is the same size and the same shape as the original. It faces the other way round, like your reflection in a mirror. A mirror line can be vertical (up and down) or horizontal (across).
Key ideas
- A reflection flips a shape over a mirror line.
- The image is the same distance from the mirror line as the original, on the other side.
- The image is the same size and shape, but it faces the other way round.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Count the squares from the point to the mirror line, then count the same number on the other side.
- Find the mirror line. The dashed red line is the mirror line.the dashed red line
- Count the squares from the point to the line. The point is 2 squares left the line.2 squares
- Count the same number on the other side. A reflection puts the image the same distance from the line, on the other side.2 squares right the line
- Write the answer. 2 squares to the right of the mirror line
Check. The point and its image are both 2 squares from the line. ✓
Watch out. The image is on the other side of the line, not on the same side.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Count the squares from the point to the mirror line, then count the same number on the other side.
- Find the mirror line. The dashed red line is the mirror line.the dashed red line
- Count the squares from the point to the line. The point is 5 squares below the line.5 squares
- Count the same number on the other side. A reflection puts the image the same distance from the line, on the other side.5 squares above the line
- Write the answer. 5 squares above the mirror line
Check. The point and its image are both 5 squares from the line. ✓
Watch out. The image is on the other side of the line, not on the same side.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Count the squares from the point to the mirror line, then count the same number on the other side.
- Find the mirror line. The dashed red line is the mirror line.the dashed red line
- Count the squares from the point to the line. The point is 3 squares left the line.3 squares
- Count the same number on the other side. A reflection puts the image the same distance from the line, on the other side.3 squares right the line
- Write the answer. 3 squares to the right of the mirror line
Check. The point and its image are both 3 squares from the line. ✓
Watch out. The image is on the other side of the line, not on the same side.
Watch out
A common mistake is to put the image on the same side of the mirror line as the original. The image must be on the other side. Also, a reflection is not a slide: the image is the same distance from the line on the other side, so count from the line, not from the original shape.
Exercise 10C Mirror lines and distances
Non-calculatorKey words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Count the squares from corner A to the dashed mirror line.
- Find the mirror line. The dashed red line is the mirror line, at 6 squares from the left edge.line at 6
- Find corner A. Corner A is 4 squares from the left edge.A at 4
- Count the squares between them. The distance is the mirror position minus the position of A.6 − 4 = 2
- Write the answer. 2 squares
Check. The image A is on the other side, and it is also 2 squares from the line. ✓
Watch out. Count the squares between the corner and the line, not the squares of the whole house.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Count the squares from the bottom of the triangle to the horizontal mirror line.
- Find the mirror line. The dashed red line is 5 squares above the bottom of the grid.line at 5
- Find the bottom of the triangle. The bottom of the triangle is on the bottom of the grid.bottom at 0
- Count the squares between them. Subtract the bottom from the line.5 − 0 = 5
- Write the answer. 5 squares
Check. The image of the bottom is 5 squares on the other side of the line. ✓
Watch out. The mirror line is horizontal, so count up and down, not across.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Count the squares from the point to the mirror line, then count the same number on the other side.
- Find the mirror line. The dashed red line is the mirror line.the dashed red line
- Count the squares from the point to the line. The point is 2 squares left the line.2 squares
- Count the same number on the other side. A reflection puts the image the same distance from the line, on the other side.2 squares right the line
- Write the answer. 2 squares to the right of the mirror line
Check. The point and its image are both 2 squares from the line. ✓
Watch out. The image is on the other side of the line, not on the same side.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Count the squares from the point to the mirror line, then count the same number on the other side.
- Find the mirror line. The dashed red line is the mirror line.the dashed red line
- Count the squares from the point to the line. The point is 2 squares right the line.2 squares
- Count the same number on the other side. A reflection puts the image the same distance from the line, on the other side.2 squares left the line
- Write the answer. 2 squares to the left of the mirror line
Check. The point and its image are both 2 squares from the line. ✓
Watch out. The image is on the other side of the line, not on the same side.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Count the squares from the point to the mirror line, then count the same number on the other side.
- Find the mirror line. The dashed red line is the mirror line.the dashed red line
- Count the squares from the point to the line. The point is 3 squares above the line.3 squares
- Count the same number on the other side. A reflection puts the image the same distance from the line, on the other side.3 squares below the line
- Write the answer. 3 squares below the mirror line
Check. The point and its image are both 3 squares from the line. ✓
Watch out. The image is on the other side of the line, not on the same side.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. The mirror line is in the middle, so the image is the same distance on the other side.
- Find the mirror line. The mirror line is 4 squares from the left edge.line at 4
- Find the image. The image is 7 squares from the left edge.image at 7
- Count the distance from the line to the image. The point is the same distance on the other side of the line.7 − 4 = 3
- Write the answer. The point is 1 square from the left edge, which is 3 squares from the line.3 squares
Check. The point is at 1, and 4 − 1 = 3. ✓
Watch out. Count the distance from the line, not from the left edge.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Count the squares from B to the mirror line, then the same number on the other side.
- Find B. Corner B is 1 square from the left edge, at the top of the L.B at 1
- Count the squares from B to the line. The mirror line is 5 squares from the left edge.5 − 1 = 4
- Count the same number on the other side. The image of B is 4 squares on the other side of the line.5 + 4 = 9
- Write the answer. 4 squares
Check. B is 4 squares from the line, and so is B'. ✓
Watch out. B and its image are the same distance from the line.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Think about what a reflection does to the distance.
- Check the distance of the point. The point is 3 squares from the line.3 squares
- Check the distance of the image. The image must be the same distance on the other side.3 squares
- Decide. Ali is not right, because the image is also 3 squares from the line.No
Check. A reflection keeps the same distance from the line. ✓
Watch out. Both the point and its image are the same distance from the line.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Count the squares from the point to the line.
- Find the mirror line. The mirror line is 5 squares from the left edge.line at 5
- Find the point. The point is 2 squares from the left edge.point at 2
- Count the squares between them. Subtract the smaller position from the larger one.5 − 2 = 3
- Write the answer. 3 squares
Check. The image is also 3 squares from the line. ✓
Watch out. Count to the line, not all the way to the edge.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Count the distance to the mirror line, then count the same distance on the other side.
- Find the mirror line. The mirror line is 5 squares from the left edge.mirror line at 5
- Count the squares from the point to the line. The point is 3 squares from the line.3 squares
- Count the same number on the other side. The image is the same distance from the line, on the other side.5 + 3 = 8
- Write the answer. 8 squares from the left edge
Check. The point and its image are both 3 squares from the line. ✓
Watch out. Count from the left edge every time, so you do not mix up the two sides.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Count the squares from the point to the mirror line, then count the same number on the other side.
- Find the mirror line. The dashed red line is the mirror line.the dashed red line
- Count the squares from the point to the line. The point is 6 squares above the line.6 squares
- Count the same number on the other side. A reflection puts the image the same distance from the line, on the other side.6 squares below the line
- Write the answer. 6 squares below the mirror line
Check. The point and its image are both 6 squares from the line. ✓
Watch out. The image is on the other side of the line, not on the same side.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Count the distance to the mirror line, then count the same distance on the other side.
- Find the mirror line. The mirror line is 3 squares from the bottom edge.mirror line at 3
- Count the squares from the point to the line. The point is 1 square from the line.1 squares
- Count the same number on the other side. The image is the same distance from the line, on the other side.3 − 1 = 2
- Write the answer. 2 squares from the bottom edge
Check. The point and its image are both 1 squares from the line. ✓
Watch out. Count from the bottom edge every time, so you do not mix up the two sides.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Count the squares from the point to the mirror line, then count the same number on the other side.
- Find the mirror line. The dashed red line is the mirror line.the dashed red line
- Count the squares from the point to the line. The point is 1 square left the line.1 squares
- Count the same number on the other side. A reflection puts the image the same distance from the line, on the other side.1 squares right the line
- Write the answer. 1 square to the right of the mirror line
Check. The point and its image are both 1 squares from the line. ✓
Watch out. The image is on the other side of the line, not on the same side.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Count the squares from the point to the mirror line, then count the same number on the other side.
- Find the mirror line. The dashed red line is the mirror line.the dashed red line
- Count the squares from the point to the line. The point is 3 squares right the line.3 squares
- Count the same number on the other side. A reflection puts the image the same distance from the line, on the other side.3 squares left the line
- Write the answer. 3 squares to the left of the mirror line
Check. The point and its image are both 3 squares from the line. ✓
Watch out. The image is on the other side of the line, not on the same side.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Reflect the shape once, then reflect the image again in the same line.
- First reflection. The shape moves to the other side of the line.the other side
- Second reflection. The image is reflected back across the same line, to the other side again.back to the first side
- Write the answer. Two flips in the same mirror line put the shape back where it started.back where it started
Check. Two reflections in the same line make the original again. ✓
Watch out. Two flips cancel each other out, like two equal moves.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. A point on the mirror line is 0 squares away from the line.
- Count the distance. The point is on the line, so it is 0 squares from the line.0 squares
- Count the same distance on the other side. The image is 0 squares away on the other side, so it is on the same point.0 squares
- Write the answer. On the same point
Check. A point that is on the line does not move. ✓
Watch out. A point on the mirror line does not move at all.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. The mirror line is halfway between a point and its image.
- Add the two distances from the edge. The point is 2 squares from the edge and the image is 8 squares from the edge.2 + 8 = 10
- Halve the total. The mirror line is in the middle, so halve the total.10 ÷ 2 = 5
- Write the answer. 5 squares from the left edge
Check. The point is 3 squares from the line (5 − 2), and its image is 3 squares on the other side. ✓
Watch out. The mirror line is always halfway between a point and its image.
Drawing reflections and finding the mirror line
To draw a reflection, take each corner in turn. Count the squares from the corner to the mirror line. Then count the same number of squares on the other side, and put a dot. Join the dots in the same order as the original shape.
If you know a point and its image, the mirror line is halfway between them.
Key ideas
- Count the squares from each corner to the mirror line, then the same number on the other side.
- The mirror line is halfway between a point and its image.
- The image has the same size and shape, and faces the other way.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Count the squares from the point to the mirror line, then count the same number on the other side.
- Find the mirror line. The dashed red line is the mirror line.the dashed red line
- Count the squares from the point to the line. The point is 5 squares left the line.5 squares
- Count the same number on the other side. A reflection puts the image the same distance from the line, on the other side.5 squares right the line
- Write the answer. 5 squares to the right of the mirror line
Check. The point and its image are both 5 squares from the line. ✓
Watch out. The image is on the other side of the line, not on the same side.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. The mirror line is halfway between the point and its image.
- Add the two distances from the edge. The point is 2 squares from the edge, and its image is 8 squares from the edge.2 + 8 = 10
- Halve the total. The mirror line is in the middle, so halve the total.10 ÷ 2 = 5
- Write the answer. the mirror line is 5 squares from the left edge
Check. From the line, 5 − 2 = 3 and 8 − 5 = 3, the same distance. ✓
Watch out. Check the answer: the distance on both sides must be the same.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Count the distance to the mirror line, then count the same distance on the other side.
- Find the mirror line. The mirror line is 3 squares from the bottom edge.mirror line at 3
- Count the squares from the point to the line. The point is 1 square from the line.1 squares
- Count the same number on the other side. The image is the same distance from the line, on the other side.3 − 1 = 2
- Write the answer. 2 squares from the bottom edge
Check. The point and its image are both 1 squares from the line. ✓
Watch out. Count from the bottom edge every time, so you do not mix up the two sides.
Watch out
A common mistake is to draw the image the wrong distance from the line, for example one square too close. Count the squares from the line for each corner, and then count the same number on the other side.
Exercise 10D Drawing reflections and finding the mirror line
Non-calculatorKey words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Count the squares from A' to the mirror line. The image is the same distance on the other side.
- Find the mirror line. The dashed red line is 6 squares from the left edge.line at 6
- Find the image corner. A' is 11 squares from the left edge.A' at 11
- Count the squares from A' to the line. Subtract the line from the image corner.11 − 6 = 5
- Write the answer. 5 squares
Check. A is 5 squares from the line too: 6 − 1 = 5. ✓
Watch out. Count from the image to the line. Its distance matches the original.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Count the squares from corner B to the mirror line, then find the image corner the same distance above the line.
- Find B. Corner B is at the bottom right of the blue rectangle, 3 squares below the mirror line.B is 3 below
- Find the image corner. The image corner is 3 squares above the line.B is at 5, the line is at 8
- Check the image corner. B' is at the top right of the orange rectangle, and it is 11 squares up from the bottom.B' at 11
- Write the answer. It is 11 minus the line at 8, so it is 3 squares above the line.11 − 8 = 3 B'
Check. B is 3 squares below the line, and B' is 3 squares above it. ✓
Watch out. Check the other side of the line, not the same side.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Count the distance to the mirror line, then count the same distance on the other side.
- Find the mirror line. The mirror line is 5 squares from the left edge.mirror line at 5
- Count the squares from the point to the line. The point is 1 square from the line.1 squares
- Count the same number on the other side. The image is the same distance from the line, on the other side.5 + 1 = 6
- Write the answer. 6 squares from the left edge
Check. The point and its image are both 1 squares from the line. ✓
Watch out. Count from the left edge every time, so you do not mix up the two sides.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Count the distance to the mirror line, then count the same distance on the other side.
- Find the mirror line. The mirror line is 5 squares from the left edge.mirror line at 5
- Count the squares from the point to the line. The point is 3 squares from the line.3 squares
- Count the same number on the other side. The image is the same distance from the line, on the other side.5 + 3 = 8
- Write the answer. 8 squares from the left edge
Check. The point and its image are both 3 squares from the line. ✓
Watch out. Count from the left edge every time, so you do not mix up the two sides.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Count the distance to the mirror line, then count the same distance on the other side.
- Find the mirror line. The mirror line is 5 squares from the left edge.mirror line at 5
- Count the squares from the point to the line. The point is 2 squares from the line.2 squares
- Count the same number on the other side. The image is the same distance from the line, on the other side.5 − 2 = 3
- Write the answer. 3 squares from the left edge
Check. The point and its image are both 2 squares from the line. ✓
Watch out. Count from the left edge every time, so you do not mix up the two sides.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Draw the image on the other side of the line, one corner at a time.
- Find the mirror line. The dashed red line is 5 squares from the left edge.line at 5
- Reflect each corner. Each corner is the same distance from the line on the other side.corners move across the line
- Measure the width of the image. A reflection keeps the same size and shape, so the width does not change.3 squares wide
- Write the answer. 3 squares
Check. The original L is 3 squares wide, and the image is too. ✓
Watch out. A reflection never changes the width of a shape.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Count the distance to the mirror line, then count the same distance on the other side.
- Find the mirror line. The mirror line is 4 squares from the bottom edge.mirror line at 4
- Count the squares from the point to the line. The point is 2 squares from the line.2 squares
- Count the same number on the other side. The image is the same distance from the line, on the other side.4 − 2 = 2
- Write the answer. 2 squares from the bottom edge
Check. The point and its image are both 2 squares from the line. ✓
Watch out. Count from the bottom edge every time, so you do not mix up the two sides.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Find the distance from the edge to the point, then to the line.
- Find the total. The point is 2 squares from the bottom and its image is 8 squares from the bottom.2 + 8 = 10
- Halve it. The mirror line is halfway between them.10 ÷ 2 = 5
- Write the answer. 5 squares
Check. The point is 3 squares below the line (5 − 2), and the image is 3 squares above it. ✓
Watch out. Halve the total distance, not the distance from one edge only.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Think about where the image of a horizontal reflection goes.
- Look at the mirror line. A horizontal mirror line flips a shape up and down.up and down
- Look at the image. The image is above or below the shape, and it is not to the right.above or below
- Decide. So Sam is not right. A horizontal mirror line does not move the image to the right.No
Check. Only a vertical mirror line puts the image to the side. ✓
Watch out. Think about the direction of the mirror line first.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. The point and its image are the same distance from the line on opposite sides.
- Find the distance from the point to the line. The point is 3 squares from the line.3 squares
- Find the distance from the image to the line. The image is also 3 squares from the line, on the other side.3 squares
- Add the two distances. The gap between the point and its image is both distances added together.3 + 3 = 6
- Write the answer. 6 squares apart
Check. The point and its image are 6 squares apart. ✓
Watch out. Add the two distances, one each side of the line.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. The image is the same distance on the other side, so the point is the same distance on the other side too.
- Find the distance of the image. The image is 6 squares to the right of the mirror line.6 squares
- Find the distance of the point. The point is the same distance from the line, on the other side.6 squares
- Write the answer. 6 squares
Check. The point and its image are both 6 squares from the line. ✓
Watch out. The point is on the other side of the line from its image.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Count the distance to the mirror line, then count the same distance on the other side.
- Find the mirror line. The mirror line is 6 squares from the left edge.mirror line at 6
- Count the squares from the point to the line. The point is 4 squares from the line.4 squares
- Count the same number on the other side. The image is the same distance from the line, on the other side.6 + 4 = 10
- Write the answer. 10 squares from the left edge
Check. The point and its image are both 4 squares from the line. ✓
Watch out. Count from the left edge every time, so you do not mix up the two sides.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Count from each corner to the line, and then from the line to each image.
- Count from A to the line. A is at the right-hand corner of the blue triangle, 3 squares from the left edge.6 − 3 = 3
- Count from A' to the line. A' is 9 squares from the left edge, so it is 3 squares to the right of the line.9 − 6 = 3
- Decide. Both are the same distance from the line.Yes
Check. Each is 3 squares from the dashed line. ✓
Watch out. Count from the same line each time.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Count the squares from the point to the mirror line, then count the same number on the other side.
- Find the mirror line. The dashed red line is the mirror line.the dashed red line
- Count the squares from the point to the line. The point is 4 squares above the line.4 squares
- Count the same number on the other side. A reflection puts the image the same distance from the line, on the other side.4 squares below the line
- Write the answer. 4 squares below the mirror line
Check. The point and its image are both 4 squares from the line. ✓
Watch out. The image is on the other side of the line, not on the same side.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Look at the two sides of the mirror line.
- Look at the left side of the line. The rectangle is the same on each side of the line.the same on each side
- Look at the corners. Each corner lands on a matching corner in the same place.matching corners
- Explain. A shape whose two halves match across the line is symmetrical about it, so its image lands on itself.The rectangle is symmetrical about the line
Check. A rectangle with the line through its middle is symmetrical. ✓
Watch out. A shape that lands on itself must match itself across the line.
The shape has not changed
When a shape is translated or reflected, its size and its shape stay the same. We say that the shape has not changed. A translation slides the shape. A reflection flips it over a mirror line. Neither one makes the shape bigger, smaller, longer or shorter.
A shape that is made bigger, or stretched so that one side is longer, has changed. It is not a translation or a reflection of the first shape.
Key ideas
- A translation or a reflection keeps the size and shape of a shape.
- Only the position changes, and for a reflection the way the shape faces also changes.
- A shape that is bigger, or stretched, has changed.
Key words. A translation or a reflection keeps the size and shape of a shape. Only its position (and, for a reflection, the way it faces) can change.
Method. Compare the sizes and the shapes of the blue and the orange house.
- Compare the width. The blue house is 4 squares wide, and the orange house is 4 squares wide.4 = 4
- Compare the height. The blue house is 4 squares high, and the orange house is 4 squares high.4 = 4
- Compare the shape. The two houses have the same corners in the same order, just in different places.same shape
- Decide. The house has only moved, so its size and shape have not changed.No, it has not changed
Check. Only the place of the house has changed. ✓
Watch out. Check the size and the shape, not only where the shape is.
Key words. A translation or a reflection keeps the size and shape of a shape. Only its position (and, for a reflection, the way it faces) can change.
Method. Compare the two triangles side by side.
- Compare the bottom sides. The blue bottom side is 3 squares long, and the orange bottom side is 3 squares long.3 = 3
- Compare the other sides. The other sides match too, because the orange triangle is a mirror image.the same
- Decide. A reflection keeps the size and shape, although it faces the other way round.No, the shape has not changed
Check. The two triangles are exactly the same size and shape. ✓
Watch out. A reflection can change the way a shape faces, but not its size.
Key words. A translation or a reflection keeps the size and shape of a shape. Only its position (and, for a reflection, the way it faces) can change.
Method. Compare the lengths of the sides of the two triangles.
- Measure the blue bottom side. The blue bottom side is 3 squares long.3 squares
- Measure the orange bottom side. The orange bottom side is 6 squares long, which is twice as long.6 squares
- Decide. The sides are not the same length, so the shape has changed.Yes, the shape has changed
Check. 6 is twice 3, so the orange shape is twice as big. ✓
Watch out. Compare the same side of each shape.
Watch out
A common mistake is to think a shape has changed because it is in a different place or faces a different way. For a translation or a reflection, the size and shape stay the same. Only a shape that is made bigger or stretched has changed.
Exercise 10E Has the shape changed?
Non-calculatorKey words. A translation or a reflection keeps the size and shape of a shape. Only its position (and, for a reflection, the way it faces) can change.
Method. A translation only moves a shape, so check the size and the shape.
- Look at the size. The house is the same size after it has moved.same size
- Look at the shape. The corners are in the same order, so the shape is the same.same shape
- Write the answer. It has only moved, so it has not changed.No
Check. The house has only slid to a new place. ✓
Watch out. A new place does not mean a new shape.
Key words. A translation or a reflection keeps the size and shape of a shape. Only its position (and, for a reflection, the way it faces) can change.
Method. Think about what a reflection does to the size and shape.
- Look at the two triangles. The orange triangle is the blue triangle flipped over the mirror line.flipped
- Check the size. Flipping does not stretch or shrink the shape.same size
- Decide. The shape has not changed size or shape. It faces the other way round.No
Check. A flipped shape has the same size and shape. ✓
Watch out. A flipped shape faces the other way, but it is still the same shape.
Key words. A translation or a reflection keeps the size and shape of a shape. Only its position (and, for a reflection, the way it faces) can change.
Method. Compare the lengths of the sides.
- Measure the blue side. The bottom side of the blue triangle is 3 squares long.3 squares
- Measure the orange side. The bottom side of the orange triangle is 6 squares long.6 squares
- Decide. Each side is twice as long, so the orange triangle is a bigger shape.Yes
Check. The orange triangle is twice as big, so it has changed. ✓
Watch out. A shape made bigger has changed, so it is not a reflection or a translation.
Key words. A translation or a reflection keeps the size and shape of a shape. Only its position (and, for a reflection, the way it faces) can change.
Method. Look at each choice and ask if the size and shape are kept.
- Check A. A slide keeps the size and the shape.A: has not changed
- Check B. A stretch makes the shape wider, so its shape has changed.B: has changed
- Check C. Making a shape twice as big changes its size.C: has changed
- Write the answer. A
Check. Only A keeps the same size and shape. ✓
Watch out. Only a slide or a flip keeps the shape the same.
Key words. A translation or a reflection keeps the size and shape of a shape. Only its position (and, for a reflection, the way it faces) can change.
Method. A translation keeps the lengths of the sides the same.
- Look at the blue rectangle. The blue rectangle is 5 squares long and 3 squares high.5 long, 3 high
- Look at the orange rectangle. Sliding it does not change its sides.the same sides
- Write the answer. 5 squares long and 3 squares high
Check. The orange rectangle has the same sides, so 5 and 3 are right. ✓
Watch out. A slide keeps every side the same length.
Key words. A translation or a reflection keeps the size and shape of a shape. Only its position (and, for a reflection, the way it faces) can change.
Method. Think about what a reflection does to the size.
- Look at Sam. Sam says a reflection changes the size of a shape.Sam: size changes
- Look at Dana. Dana says a reflection does not change the size.Dana: size stays
- Decide. A reflection only flips the shape, so its size stays the same. Dana is right.Dana
Check. A reflection keeps the size and shape, so Dana is right. ✓
Watch out. Flipping does not make a shape bigger or smaller.
Key words. A translation or a reflection keeps the size and shape of a shape. Only its position (and, for a reflection, the way it faces) can change.
Method. Add the four sides of the rectangle.
- Write the sides. A rectangle 5 squares by 3 squares has two sides of 5 and two sides of 3.5 + 3 + 5 + 3
- Add them up. The perimeter is all four sides added together.5 + 3 + 5 + 3 = 16
- Write the answer. The translated rectangle has the same sides, so its perimeter is the same.16 squares
Check. 2 × (5 + 3) = 16. ✓
Watch out. A translation keeps the perimeter the same.
Key words. A translation or a reflection keeps the size and shape of a shape. Only its position (and, for a reflection, the way it faces) can change.
Method. A reflection keeps every side the same length.
- Write the perimeter of the shape. The shape has a perimeter of 14 squares.14 squares
- Think about the image. Each side of the image is the same length as the side it came from.same sides
- Write the answer. 14 squares
Check. The image has the same sides, so the perimeter is still 14. ✓
Watch out. The perimeter of a reflection is the same as the original.
Key words. A translation or a reflection keeps the size and shape of a shape. Only its position (and, for a reflection, the way it faces) can change.
Method. Translating and reflecting both keep the size of the shape.
- Write the area of the shape. The shape covers 9 squares of the grid.9 squares
- Move and flip the shape. A translation and a reflection keep the size of the shape.same size
- Write the answer. 9 squares
Check. The image still covers 9 squares. ✓
Watch out. Moving and flipping a shape does not change how many squares it covers.
Key words. A translation or a reflection keeps the size and shape of a shape. Only its position (and, for a reflection, the way it faces) can change.
Method. Think about what it means to be the same shape.
- Check Tom. A reflection makes an image with the same size and shape as the original.same size and shape
- Check the difference. The image only faces the other way round.faces the other way
- Decide. So Tom is right: the two shapes are the same, but one is the mirror image of the other.Yes
Check. A shape and its mirror image are the same size and shape. ✓
Watch out. The same shape can face a different way.
Key words. A translation or a reflection keeps the size and shape of a shape. Only its position (and, for a reflection, the way it faces) can change.
Method. Think about what a translation changes.
- Check the place. The new shape is in a different place on the grid.position changes
- Check the size and shape. The size and shape are the same, because the shape has only slid.size and shape stay
- Decide. Ruby is not right. The shape looks the same, only in a new place.No
Check. Only the position changed. ✓
Watch out. A new place does not change the shape.
Key words. A translation or a reflection keeps the size and shape of a shape. Only its position (and, for a reflection, the way it faces) can change.
Method. Count the corners of the original, then the corners of the image.
- Count the corners of the original. The original shape has 6 corners.6 corners
- Count the corners of the image. Each corner of the original has one image, so the image also has 6 corners.6 corners
- Write the answer. 6 corners
Check. A reflection keeps the same number of corners. ✓
Watch out. Every corner has a matching corner in the image.
Key words. A translation or a reflection keeps the size and shape of a shape. Only its position (and, for a reflection, the way it faces) can change.
Method. Find the shape that is not like the other two.
- Look at the slide. A slide keeps the size and shape.same size
- Look at the flip. A flip keeps the size and shape.same size
- Look at the bigger shape. A shape made twice as big has a different size.different size
- Write the answer. The bigger shape is the odd one out, because the other two keep the same size and shape.the shape made twice as big
Check. The other two keep the same size and shape. ✓
Watch out. Look at what each move does to the size.
Chapter summary
| Skill | Method |
|---|---|
| Translation | A slide. Every point moves the same distance, in the same direction. Describe it as squares across (right or left), then up or down. |
| Reflection | A flip over a mirror line. Each point and its image are the same distance from the line, on opposite sides. |
| Mirror line | The line the shape is flipped over. In these pictures it is dashed and red. |
| Image | The new shape after a move. Its corners are named with a dash, so the image of A is A'. |
| Same size and shape | A translation or a reflection keeps the size and shape. The shape has not changed. |
| Changed | A shape made bigger or stretched has changed, so it is not a translation or a reflection of the first shape. |
| Drawing a translation | Move every corner the same number of squares across and up or down. |
| Drawing a reflection | Count from each corner to the line, then the same number of squares on the other side. |
Can you…?
- I can describe a translation by the squares across and the squares up or down.
- I can draw a translation by moving every corner the same way.
- I know a reflection flips a shape over a mirror line, and the image is the same distance from the line on the other side.
- I can draw a reflection in a vertical or a horizontal mirror line.
- I can find the mirror line when I know a point and its image.
- I know that a translation or a reflection keeps the size and shape, so the shape has not changed.
- I can explain the difference between a translation and a reflection.
Key terms
- Translation
- A move that slides a shape. Every point moves the same distance in the same direction.
- Reflection
- A move that flips a shape over a mirror line.
- Mirror line
- The line that a shape is flipped over in a reflection.
- Image
- The new shape made by a translation or a reflection.
- Across
- Sideways, to the right or to the left.
- Vertical
- Going up and down.
- Horizontal
- Going across, from side to side.
- Corner
- A point where two sides of a shape meet.
Exercise 10Z Check-up (non-calculator)
Non-calculatorKey words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Count the squares across, then up or down, from one corner. Every corner moves the same way.
- Pick one corner of the shape. Use the same corner (A) on the start shape and on the image.corner A
- Count 3 squares to the right. The matching corner of the image is 3 squares right.3 squares right
- Count 2 squares down. The matching corner of the image is 2 squares down.2 squares down
- Check another corner. Every corner moves the same way, so the shape only slides.✓ the same move
- Write the answer. 3 squares to the right and 2 squares down
Check. Count from another corner and you get the same move. ✓
Watch out. Count the squares on the grid. Do not count the sides of the shape.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Count from the mirror line to the image, on the other side.
- Find the mirror line. The dashed red line is 6 squares from the left edge.line at 6
- Find the image corner. A' is 8 squares from the left edge.A' at 8
- Count the squares from the line. Subtract the line from the image corner.8 − 6 = 2
- Write the answer. 2 squares
Check. A is 2 squares from the line too: 6 − 4 = 2. ✓
Watch out. Count from the line, not from the edge of the grid.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Move the bottom side up 2 squares, then count from the bottom edge.
- Find the bottom side now. The bottom side of the house is 1 square from the bottom edge.1 up
- Move it 2 squares up. Moving up adds to the height.1 + 2 = 3
- Write the answer. 3 squares
Check. The house moves up by 2, so its bottom is 3 squares up. ✓
Watch out. Count from the bottom edge of the grid.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Count the squares from the point to the mirror line, then count the same number on the other side.
- Find the mirror line. The dashed red line is the mirror line.the dashed red line
- Count the squares from the point to the line. The point is 4 squares left the line.4 squares
- Count the same number on the other side. A reflection puts the image the same distance from the line, on the other side.4 squares right the line
- Write the answer. 4 squares to the right of the mirror line
Check. The point and its image are both 4 squares from the line. ✓
Watch out. The image is on the other side of the line, not on the same side.
Key words. A translation or a reflection keeps the size and shape of a shape. Only its position (and, for a reflection, the way it faces) can change.
Method. Compare the new shape with the old one.
- Look at the size. A reflection keeps the size of the shape.same size
- Look at the shape. A reflection keeps the shape. It only faces the other way.same shape
- Write the answer. No, it has not changed size or shape
Check. A reflected shape has the same size and shape. ✓
Watch out. A reflected shape faces the other way, but it is not a new shape.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Find the bottom-right corner, then move it.
- Find the bottom-right corner. The rectangle starts 1 square across, and it is 4 squares wide, so its bottom-right corner is at 5 across.1 + 4 = 5 across, 0 up
- Move it 3 squares right. Moving right adds to the across distance.5 + 3 = 8
- Move it 2 squares up. Moving up adds to the height.0 + 2 = 2
- Write the answer. 8 squares across and 2 squares up
Check. The rectangle moves 3 across and 2 up, so its bottom-left corner is now at 4 across and 2 up. ✓
Watch out. Find the corner first. Then move it, one move at a time.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Count the distance to the mirror line, then count the same distance on the other side.
- Find the mirror line. The mirror line is 6 squares from the left edge.mirror line at 6
- Count the squares from the point to the line. The point is 5 squares from the line.5 squares
- Count the same number on the other side. The image is the same distance from the line, on the other side.6 + 5 = 11
- Write the answer. 11 squares from the left edge
Check. The point and its image are both 5 squares from the line. ✓
Watch out. Count from the left edge every time, so you do not mix up the two sides.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Think about what a translation does.
- Look at the first move. Sam moves the shape 3 squares to the right.+3
- Look at the second move. Then he moves it 3 squares to the left, back the other way.−3
- Decide. The two moves cancel, so the shape is back where it started.Yes
Check. 3 − 3 = 0, so the total move is 0. ✓
Watch out. Two equal moves in opposite directions cancel out.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Move the corner across, then up or down.
- Start at the corner. The corner is 2 squares across and 5 squares up.2 across, 5 up
- Move it 4 squares right. Moving right adds to the across distance.2 + 4 = 6
- Move it 3 squares down. Moving down takes away from the height.5 − 3 = 2
- Write the answer. 6 squares across and 2 squares up
Check. Go back 4 to the left and 3 up to reach the start. ✓
Watch out. Down takes away from the height. Right adds to the across distance.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Compare what each move does to the shape.
- Think about a translation. A translation slides the shape. It does not turn or flip it.slides
- Think about a reflection. A reflection flips the shape over a mirror line.flips
- Write the answer. A translation slides a shape, and a reflection flips it over a mirror line.
Check. The two moves are different: one slides, the other flips. ✓
Watch out. Ask whether the shape is slid or flipped.
Total 22 marks. Allow about 22 minutes.
Exercise 10X SATs-style questions
Non-calculatorKey words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. A slide is a translation.
- Look at the words. Turning and flipping are not slides.turn, flip
- Write the answer. slide
Check. A translation slides the shape. ✓
Watch out. A translation slides the shape without turning it.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Count the squares from the left-hand corner.
- Find the left-hand corner. The left-hand corner of the blue triangle is 1 square from the left edge.1 across
- Find the image corner. The image corner is 5 squares from the left edge.5 across
- Write the answer. The shape has moved 4 squares to the right and no squares up or down.4 squares to the right
Check. 5 − 1 = 4. ✓
Watch out. There is no up or down move, so do not write one.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Start at the first number, then add or take away the move.
- Start at the first number. The counter starts 2 squares from the edge.2
- Move 5 squares right. Moving right or up adds. Moving left or down takes away.2 + 5 = 7
- Write the answer. 7
Check. Move 5 squares the other way to get back to 2. ✓
Watch out. Moving left takes away. Count carefully across the squares.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Count the squares from the point to the mirror line, then count the same number on the other side.
- Find the mirror line. The dashed red line is the mirror line.the dashed red line
- Count the squares from the point to the line. The point is 3 squares left the line.3 squares
- Count the same number on the other side. A reflection puts the image the same distance from the line, on the other side.3 squares right the line
- Write the answer. 3 squares to the right of the mirror line
Check. The point and its image are both 3 squares from the line. ✓
Watch out. The image is on the other side of the line, not on the same side.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. The image is the same distance from the line, on the other side.
- Write the distance to the line. The point is 4 squares from the mirror line.4 squares
- Find the image. The image is on the other side, the same distance from the line.4 squares
- Write the answer. 4 squares
Check. The point and its image are both 4 squares from the line. ✓
Watch out. The image is the same distance on the other side.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Reverse each part of the move.
- Write the move. Ruby moved the shape 3 squares to the left and 2 squares up.3 left, 2 up
- Reverse the across move. To go back, move 3 squares to the right.3 right
- Reverse the up move. To go back, move 2 squares down.2 down
- Write the answer. 3 squares to the right and 2 squares down
Check. 3 right and 2 down brings the shape back to the start. ✓
Watch out. Reverse each part of the move: left becomes right, up becomes down.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. Count the distance to the mirror line, then count the same distance on the other side.
- Find the mirror line. The mirror line is 5 squares from the bottom edge.mirror line at 5
- Count the squares from the point to the line. The point is 3 squares from the line.3 squares
- Count the same number on the other side. The image is the same distance from the line, on the other side.5 − 3 = 2
- Write the answer. 2 squares from the bottom edge
Check. The point and its image are both 3 squares from the line. ✓
Watch out. Count from the bottom edge every time, so you do not mix up the two sides.
Key words. A translation slides a shape. Every point moves the same distance, in the same direction. The shape does not turn, flip or change size.
Method. Find the top-right corner first, then move it.
- Find the top-right corner now. The rectangle starts 2 squares from the left edge and is 6 squares wide, so its right side is 8 squares from the edge.2 + 6 = 8
- Find its height. It starts 1 square from the bottom and is 2 squares high, so its top is 3 squares from the bottom.1 + 2 = 3
- Move it 3 squares to the right. Moving right adds to the across distance.8 + 3 = 11
- Move it 4 squares up. Moving up adds to the height.3 + 4 = 7
- Write the answer. 11 squares from the left edge and 7 squares from the bottom
Check. The rectangle is still 6 squares wide: 11 − 3 − 2 = 6. ✓
Watch out. Find the corner before the move, then add the moves to it.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides.
Method. The mirror line is halfway between a point and its image.
- Look at the point and its image. The image is 2 squares to the right of the line.image: 2 right
- Find the point. The point must be the same distance on the other side of the line.point: 2 left
- Decide. So Ali is right: the point is 2 squares to the left of the line.Yes
Check. Each point and its image are the same distance from the line, on opposite sides. ✓
Watch out. Check the distance on both sides, and the side of the line.
Key words. A translation or a reflection keeps the size and shape of a shape. Only its position (and, for a reflection, the way it faces) can change.
Method. Think about the size and shape of each move.
- Look at Ruby. A translation and a reflection both keep the size and shape of the shape.Ruby: size stays
- Look at Dana. A reflection only flips the shape. It cannot make it larger.Dana: wrong
- Give an example. A rectangle 3 squares by 2 squares, reflected in a line, is still 3 squares by 2 squares.3 by 2 stays 3 by 2
- Decide. So Ruby is right, and Dana is wrong.Ruby is right
Check. A flipped rectangle is still 3 squares by 2 squares. ✓
Watch out. A reflection flips a shape. It does not stretch it.