Year 6: Keeping the shape, and two moves
Part of the Year 6 coordinates, reflection and translation chapter. Chapter overview · whole chapter on one page
Keeping the shape, and two moves
A translation or a reflection does not change the size or the shape of a shape. Every side keeps the same length, so the image has the same perimeter and the same area as the original.
Sometimes a shape is moved more than once. Do the moves one at a time, in the order they are given. Check the point after each move before you do the next one.
Key ideas
- A translation or a reflection keeps the lengths of the sides, so the perimeter stays the same.
- Do two moves in the order they are given. Reflecting and then translating can give a different answer from translating and then reflecting.
- Check each step by looking at the numbers that changed.
Key words. The perimeter is the distance all the way round the outside of a shape. A rectangle has four right angles, and its opposite sides are equal.
Method. Count the squares along the side, using only the numbers that change.
- Find the numbers that change. The two points have the same up or down number, so the side runs across.−2 and 1
- Find the length. The two points are on opposite sides of the y-axis, so add how far each one is from the y-axis.2 + 1 = 3
- Write the answer. A length is never negative.3
Check. Count the squares from (-2, 1) to (1, 1) one at a time: there are 3. ✓
Watch out. When the side crosses an axis, add the two distances from the axis. Do not subtract them.
Key words. The perimeter is the distance all the way round the outside of a shape. A rectangle has four right angles, and its opposite sides are equal.
Method. The image has the same side lengths, so add them up.
- Find the image of the rectangle. The rectangle has been reflected in the y-axis, so each corner moves to a new place.(-2, 2), (-6, 2), (-6, 4), (-2, 4)
- Find the sides of the image. A reflection or a translation keeps the length of every side, so the sides are the same as before.4 and 2
- Add the four sides. The perimeter is the distance all the way round.4 + 2 + 4 + 2 = 12
- Write the answer. The perimeter is the same as the perimeter of the original rectangle.12
Check. The original rectangle also has perimeter 12. ✓
Watch out. A reflection or a translation does not change the lengths. Only the position changes.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Do the moves one at a time, in the order they are given.
- Start with the point. Write its coordinates before any move.(3, 2)
- Move 1: reflect in the x-axis. The mirror line is the x-axis, so only the up or down number changes sign.(3, 2) to (3, -2)
y: 2 becomes −2 - Move 2: reflect in the y-axis. The mirror line is the y-axis, so only the across number changes sign.(3, -2) to (-3, -2)
x: 3 becomes −3 - Write the final point. Write across first, then up or down.(-3, -2)
Check. Each move was checked with the number that changed. The final point is (-3, -2). ✓
Watch out. Do the moves in the order they are given. Reflecting and then translating can give a different answer from translating and then reflecting.
Watch out
Common mistake: thinking that a reflected shape is bigger, or that it has a different perimeter. Only its position changes. Another mistake is to do the two moves in the wrong order.
Exercise 11F Practice: keeping the shape, and moves in order
Non-calculatorKey words. A translation or a reflection keeps the size and the shape of a shape. Only its position changes.
Method. A translation does not change the length of a side. Only its position changes.
- Look at the original side. The side is 5 squares long.5 squares
- Look at the move. Moving the side up 3 squares slides it. It does not stretch or shrink.slide: length unchanged
- Write the answer. The image of the side is still 5 squares long.5
Check. The image is still 5 squares long, so 5 is the answer. ✓
Watch out. A slide keeps every length the same.
Key words. The perimeter is the distance all the way round the outside of a shape. A rectangle has four right angles, and its opposite sides are equal.
Method. Add the lengths of all four sides.
- Write the lengths of the four sides. A rectangle has two long sides and two short sides.4, 2, 4 and 2
- Add the four sides. The perimeter is the distance all the way round.4 + 2 + 4 + 2 = 12
- Write the answer. The perimeter is in squares, one for each unit of length.12
Check. 2 × (4 + 2) = 12. ✓
Watch out. Count all four sides, not just the two you can see first.
Key words. The perimeter is the distance all the way round the outside of a shape. A rectangle has four right angles, and its opposite sides are equal.
Method. The image has the same side lengths, so add them up.
- Find the image of the rectangle. The rectangle has been reflected in the x-axis, so each corner moves to a new place.(1, -1), (5, -1), (5, -3), (1, -3)
- Find the sides of the image. A reflection or a translation keeps the length of every side, so the sides are the same as before.4 and 2
- Add the four sides. The perimeter is the distance all the way round.4 + 2 + 4 + 2 = 12
- Write the answer. The perimeter is the same as the perimeter of the original rectangle.12
Check. The original rectangle also has perimeter 12. ✓
Watch out. A reflection or a translation does not change the lengths. Only the position changes.
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. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(2, 4)
- Look at the mirror line. The mirror line is the x-axis. The point flips over it, so the across number does not change.x stays 2
- Flip the up or down number. The point is 4 squares above the x-axis. Its image is the same distance below it.y: 4 becomes −4
- Write the image. Write the across number first, then the up or down number.(2, -4)
Check. (2, -4) is 4 squares below the x-axis, the same distance as (2, 4) is above it. ✓
Watch out. Reflecting in the x-axis changes the up or down number, not the across number.
Key words. A translation or a reflection keeps the size and the shape of a shape. Only its position changes.
Method. Think about what a reflection does to the size of a shape.
- Think about the shape. A reflection flips the shape over a mirror line.flip
- Think about the size. The shape is flipped, not stretched, so every side keeps its length.size: unchanged
- Decide. Sam is wrong. The reflected shape is the same size, only in a different place.No
Check. A reflection keeps every length the same. ✓
Watch out. A reflection moves a shape to the other side. It does not make it bigger.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Do the moves one at a time, in the order they are given.
- Start with the point. Write its coordinates before any move.(-4, 5)
- Move 1: reflect in the x-axis. The mirror line is the x-axis, so only the up or down number changes sign.(-4, 5) to (-4, -5)
y: 5 becomes −5 - Move 2: reflect in the y-axis. The mirror line is the y-axis, so only the across number changes sign.(-4, -5) to (4, -5)
x: −4 becomes 4 - Write the final point. Write across first, then up or down.(4, -5)
Check. Each move was checked with the number that changed. The final point is (4, -5). ✓
Watch out. Do the moves in the order they are given. Reflecting and then translating can give a different answer from translating and then reflecting.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Do the moves one at a time, in the order they are given.
- Start with the point. Write its coordinates before any move.(3, -1)
- Move 1: reflect in the y-axis. The mirror line is the y-axis, so only the across number changes sign.(3, -1) to (-3, -1)
x: 3 becomes −3 - Move 2: translate 2 squares right. Every corner moves the same number of squares, so both numbers change by the move.(-3, -1) to (-1, -1)
x: −3 + 2 = −1; y: −1 + 0 = −1 - Write the final point. Write across first, then up or down.(-1, -1)
Check. Each move was checked with the number that changed. The final point is (-1, -1). ✓
Watch out. Do the moves in the order they are given. Reflecting and then translating can give a different answer from translating and then reflecting.
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. In this chapter the mirror lines are the axes.
Method. Compare the across numbers and the up or down numbers of the two points.
- Compare the across numbers. The across number is -2 for both points, so it did not change.x = -2 for both
- Compare the up or down numbers. The up or down numbers are 3 and -3, which are opposite.3 and -3 are opposite
- Decide. Only the up or down number changes sign, so the mirror line is the x-axis.the x-axis
Check. (-2, 3) and (-2, -3) are both 2 across from the y-axis, and 3 squares from the x-axis on opposite sides. ✓
Watch out. Look at which number changes sign: the up or down number means the mirror line is the x-axis.
Key words. A translation or a reflection keeps the size and the shape of a shape. Only its position changes.
Method. Think about what a translation and a reflection each do to the lengths.
- Look at the translation. Sliding the rectangle 4 squares right keeps its side lengths.slide: lengths unchanged
- Look at the reflection. Reflecting in the y-axis flips the rectangle, and its side lengths stay the same.flip: lengths unchanged
- Decide. Both moves keep every side the same length, so the perimeter does not change.No
Check. The same four sides are added, so the perimeter is the same. ✓
Watch out. Both moves change only where the shape is.
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. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(1, 2)
- Look at the mirror line. The mirror line is the x-axis. The point flips over it, so the across number does not change.x stays 1
- Flip the up or down number. The point is 2 squares above the x-axis. Its image is the same distance below it.y: 2 becomes −2
- Write the image. Write the across number first, then the up or down number.(1, -2)
Check. (1, -2) is 2 squares below the x-axis, the same distance as (1, 2) is above it. ✓
Watch out. Reflecting in the x-axis changes the up or down number, not the across number.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Do the moves one at a time, in the order they are given.
- Start with the point. Write its coordinates before any move.(-2, 3)
- Move 1: translate 5 squares right. Every corner moves the same number of squares, so both numbers change by the move.(-2, 3) to (3, 3)
x: −2 + 5 = 3; y: 3 + 0 = 3 - Move 2: reflect in the y-axis. The mirror line is the y-axis, so only the across number changes sign.(3, 3) to (-3, 3)
x: 3 becomes −3 - Write the final point. Write across first, then up or down.(-3, 3)
Check. Each move was checked with the number that changed. The final point is (-3, 3). ✓
Watch out. Do the moves in the order they are given. Reflecting and then translating can give a different answer from translating and then reflecting.
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. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(-1, 4)
- Look at the mirror line. The mirror line is the y-axis. The point flips over it, so the up or down number does not change.y stays 4
- Flip the across number. The point is 1 squares left of the y-axis. Its image is the same distance right of it.x: −1 becomes 1
- Write the image. Write the across number first, then the up or down number.(1, 4)
Check. (1, 4) is 1 squares right of the y-axis, the same distance as (-1, 4) is left of it. ✓
Watch out. Reflecting in the y-axis changes the across number, not the up or down number.
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. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(-1, -2)
- Look at the mirror line. The mirror line is the y-axis. The point flips over it, so the up or down number does not change.y stays −2
- Flip the across number. The point is 1 squares left of the y-axis. Its image is the same distance right of it.x: −1 becomes 1
- Write the image. Write the across number first, then the up or down number.(1, -2)
Check. (1, -2) is 1 squares right of the y-axis, the same distance as (-1, -2) is left of it. ✓
Watch out. Reflecting in the y-axis changes the across number, not the up or down number.
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. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(-4, -2)
- Look at the mirror line. The mirror line is the y-axis. The point flips over it, so the up or down number does not change.y stays −2
- Flip the across number. The point is 4 squares left of the y-axis. Its image is the same distance right of it.x: −4 becomes 4
- Write the image. Write the across number first, then the up or down number.(4, -2)
Check. (4, -2) is 4 squares right of the y-axis, the same distance as (-4, -2) is left of it. ✓
Watch out. Reflecting in the y-axis changes the across number, not the up or down number.