aiMOOC

Coordinates translations and reflections - Mathematics Year 5

QR-Code

Introduction

Can you move a triangle without changing its size? Can you describe a mirror image precisely enough for someone else to draw it? Coordinates let you state positions and transformations clearly instead of relying on 'over there' or a picture that merely looks close.

Read and plot ordered pairs, slide every vertex through a common movement, and reflect shapes in stated horizontal or vertical lines. Your evidence includes eight plotted points, complete image-coordinate tables and checks showing what stayed the same.

This editorial Year 5 course follows England's position-and-direction work, using first-quadrant coordinates and boundary axes. Negative-coordinate and diagonal-reflection examples in broader diagrams or films are optional extension context, not required here. Recall Angles and properties of two dimensional and three dimensional shapes - Mathematics Year 5 for shape properties and right angles.

Learning goals

Read x first and y second, checking axis labels and scale.
Plot eight points at intersections, including an axis point with a zero coordinate.
Translate every triangle vertex by the same horizontal and vertical displacement.
Reflect in a stated line parallel to an axis using equal perpendicular distances.
Check coordinates through inverse moves, midpoint distances and preserved shape properties.

All point positions are invented mathematical examples. You may plot on paper, an accessible tactile grid or a suitable geometry tool. The course supplies checkable reasoning, not invented records of a learner's practical plotting performance.

Read and plot ordered pairs

A coordinate grid has two numbered axes. Here the horizontal axis is x and the vertical axis is y. They meet at the origin, written (0, 0). An ordered pair gives x first, then y. The order is part of the position's meaning.

To plot (3, 5), start at the origin, move three units right in the x direction and five units up in the y direction. The point sits on the intersection, not inside a grid cell. Moving up five first and right three reaches the same place, but the written coordinate order remains x then y.

By contrast, (5, 3) lies five right and three up, at a different position. 'Along then up' is a memory aid, not a changeable rule about which written coordinate comes first. Explain the axis labels so you can work without relying on the rhyme.

Read the scale, not only the count of drawn lines

For the original grid below, each horizontal and vertical step is one unit. A different grid might label every interval as two units or half a unit. Start from the numbered origin and follow the stated scale rather than assuming the first visible intersection is one.

y
5 | . . . D . . .
4 | H . F . . . .
3 | . C . . . E .
2 | . . . . . . G
1 | . A . . B . .
0 | O . . . . . .
    0 1 2 3 4 5 6  x

This original letter grid is a symbolic coordinate record, not a calibrated physical measuring instrument. O is the origin. D is (3, 5), while E is (5, 3). H is (0, 4): its x coordinate is zero because it is on the y-axis.

The first quadrant strictly contains points with positive x and positive y. This course also uses the boundary axes, where one coordinate can be zero. Such a point is on an axis rather than strictly inside the quadrant. Negative-coordinate positions in broader diagrams or films are extension material here.

Eight-point plotting task

On a new grid, plot A (1, 1), B (4, 1), C (1, 3), D (3, 5), E (5, 3), F (2, 4), G (6, 2) and H (0, 4). Label x, y, the origin and the step value. Check every point by reading its horizontal and vertical projections back to the axes, then compare with the supplied symbolic grid.

Points with the same y coordinate lie on a horizontal line: A and B both have y equal to one. Points with the same x coordinate lie on a vertical line: A and C both have x equal to one. This provides a check on the sides of triangle ABC before transforming it.

Try it: A point is five units right and three units up. It is E at (5, 3), not D at (3, 5). A point on the horizontal axis at x equal to four is (4, 0), not (0, 4).

Translate every point through one common movement

A translation slides a shape so every point moves by the same amount in the same direction. It does not turn the shape, reflect it or change its size. Describe both the horizontal and vertical displacement, not only where the shape ends up.

The diagram illustrates a slide without giving our numerical coordinates. The copies have different display colours, but colour is not a defining property of a geometric translation. You can label corresponding features instead.

Translate a point

Move point D (3, 5) four units right and two units down. Its x coordinate increases by four, giving seven. Its y coordinate decreases by two, giving three. The image point is (7, 3). Down does not mean subtract from both coordinates.

$3 + 4 = 7$ $5 - 2 = 3$

The reverse translation moves four units left and two up, returning (7, 3) to (3, 5). This checks both components. A description such as 'move it four' is incomplete unless the direction and any vertical movement are clear.

Translate a triangle systematically

Begin again with A (1, 1), B (4, 1) and C (1, 3). This is a separate example, not the four-right-two-down movement above. Translate the whole triangle three units right and two units up. Move each vertex by that same instruction and join corresponding image vertices.

Original vertex	Original coordinates	Translated vertex	Image coordinates
A	(1, 1)	A_t	(4, 3)
B	(4, 1)	B_t	(7, 3)
C	(1, 3)	C_t	(4, 5)

The label suffix t identifies a translated point. It is a label, not a multiplication instruction. Make the new grid large enough to include x equal to seven; the earlier example grid stopping at six does not limit the coordinate system.

Check what stays the same

AB is horizontal and three units long; A_t to B_t is also horizontal and three units long. AC is vertical and two units long; A_t to C_t is also vertical and two units long. The right angle at A is preserved at A_t. All corresponding lengths and angles, hence the size and shape, remain unchanged. The triangle's orientation is preserved too.

Moving only A and then guessing B_t and C_t can distort the triangle. Check every vertex's displacement and the corresponding horizontal and vertical sides. Apply the reverse movement, three left and two down, to recover the original coordinates.

The original Corbettmaths Primary film shows translations through matching movements. Some displayed positions use negative coordinates; the same movement principle applies, but our required tasks use positive coordinates and boundary axes. Focus on corresponding points and the two-part movement rather than copying a different example's coordinates.

Try it: Translate (2, 4) one unit right and three units down. The image is (3, 1). Reverse one left and three up to check. This is a slide; it is not a reflection merely because the point moved downward.

Reflect at equal perpendicular distances

A reflection makes a mirror image in a stated line. Every point and its image lie the same perpendicular distance from that line, on opposite sides unless the point is on the mirror line itself. The joining segment is perpendicular to the mirror and the mirror cuts it halfway.

For a vertical mirror line, compare horizontal distances and keep the y coordinate unchanged. For a horizontal mirror line, compare vertical distances and keep x unchanged. This course uses lines parallel to the axes; diagonal reflection is later extension work.

Reflect a point in a vertical line

The notation x = 5 describes the vertical line through all positions whose x coordinate is five. It is not a single point. Reflect the invented point (2, 3) in this line. Its horizontal distance from the mirror is three units, since five minus two is three. Continue three units on the other side to x equal to eight. The image is (8, 3).

$5 - 2 = 3$ $5 + 3 = 8$

Check that x equal to five is halfway between two and eight and that both y values remain three. Reflecting (8, 3) in the same line returns (2, 3). Merely moving the point onto the mirror at (5, 3) is not its reflection.

Reflect the original triangle, not a previously moved copy

Start again with A (1, 1), B (4, 1), C (1, 3). Reflect this original triangle in x = 5. The suffix r identifies a reflected vertex. Make the grid large enough to include x equal to nine.

Original vertex	Original coordinates	Reflected vertex	Image coordinates
A	(1, 1)	A_r	(9, 1)
B	(4, 1)	B_r	(6, 1)
C	(1, 3)	C_r	(9, 3)

A and C are four horizontal units left of the mirror, so their images are four units right. B is only one unit left, so its image is one unit right. The three points do not move the same horizontal amount: A moves eight units, B two and C eight. This is a reflection, not a common slide.

AB and A_r to B_r are both three units long; AC and A_r to C_r are both two. The right angle and all other corresponding angles are preserved. Size and shape stay the same, while the labelled triangle is mirror-reversed. A shape with mirror symmetry can sometimes coincide with its reflection, so do not insist that every reflected outline must visibly look different.

Horizontal line and fixed points

Reflect (2, 1) in y = 4. The point is three units below the line, so its image is three units above at (2, 7). The x coordinate stays two. A point already on x = 5, such as (5, 2), stays in the same place when reflected in that line because its distance from the mirror is zero.

The original Corbettmaths Primary video explains equal-distance reflection. Focus on its horizontal and vertical mirror-line reasoning. It also includes diagonal lines and negative-coordinate regions beyond this course's required first-quadrant work. The principle of perpendicular distance remains important, but do not copy a diagonal rule into our axis-parallel examples.

Try it: Reflect (3, 2) in y = 4. The image is (3, 6): two units above rather than two below. Reflect it again to recover (3, 2). State the mirror line every time, since 'reflect this point' alone is incomplete.

Check transformations and explain preserved properties

An accurate coordinate answer needs more than a plausible picture. State the original position, the intended operation, the resulting position and a check tied to the operation's defining properties.

A complete translation solution

For triangle ABC at (1, 1), (4, 1), (1, 3), the instruction 'three right and two up' gives images (4, 3), (7, 3), (4, 5). Each x value increases by three and each y value by two. Join the vertices in corresponding order, not in whichever order makes a familiar-looking triangle.

Reverse three left and two down to recover all three originals. Horizontal side AB stays length three and vertical side AC stays length two, preserving the right angle. The same displacement of all vertices explains why the shape is not stretched.

A complete reflection solution

For the same original triangle reflected in x = 5, the images are (9, 1), (6, 1), (9, 3). Each original-image pair has the same y value and an x midpoint of five. Equal distances on opposite sides of the stated vertical mirror provide the check. Reflecting again returns the original triangle.

Do not confuse this table with the translated triangle's table. Labels such as A_t and A_r identify which image is being described. There is no instruction here to reflect the already translated triangle; composing operations would be a different problem with different results.

Compare the preserved properties

Property	Translation	Reflection
Corresponding side lengths	Preserved	Preserved	Corresponding angles	Preserved	Preserved	Size and shape	Preserved	Preserved	Movement of every point	Same displacement	Determined by distance from mirror	Order of labelled vertices around the outline	Preserved	Reversed for a non-degenerate polygon	Point on a mirror line	No mirror is required	Fixed in that reflection

The last row does not say that all translations move every point: the zero translation leaves the whole plane unchanged. Our worked translations are non-zero. Reflection need not visibly alter a symmetric outline, but corresponding labelled points and their mirror distances still follow the rule.

Why one point pair may not identify a transformation

The single point (2, 3) can reach (8, 3) by a translation six units right, or by reflection in x = 5. Seeing only that start and end does not distinguish those operations. An instruction or enough corresponding shape points is needed. For the full triangle, the differing horizontal changes in the reflection rule reject a common-slide explanation.

Repair common errors

Swapping (3, 5) to (5, 3) changes the point. Read the axis names in the required order.
Translating only one vertex and guessing the others can change side lengths. Apply the same two-part movement to every vertex.
Reflecting (2, 3) onto x = 5 does not put it equally far on the other side. The image is (8, 3).
Changing y when reflecting in a vertical line fails the perpendicular-distance rule in these examples.
Counting drawn lines instead of units fails when a grid has a different scale. Read its numbered intervals first.

Final evidence and explanation

Submit the eight-point grid, both three-vertex image tables and labelled copies of the triangles. Write one inverse check and one reflection-distance check. Identify at least two preserved shape properties and explain one misconception. A table and a spoken or tactile explanation can show the same mathematical reasoning when drawing is difficult; do not pretend a practical plotting task was observed if only coordinates were supplied.

Interactive Tasks

Quiz: Test Your Knowledge

What is the coordinate order on the labelled x-y grid? (Horizontal x first, vertical y second) (!Vertical y first, horizontal x second) (!Always the larger number first) (!The order can change without changing the point)

Which coordinates describe three units right and five units up from the origin? (x = 3, y = 5) (!x = 5, y = 3) (!x = 3, y = 2) (!x = 0, y = 5)

Where is a point whose coordinates are x equal to zero and y equal to four? (On the y-axis, four units above the origin) (!On the x-axis, four units right of the origin) (!Strictly inside the first quadrant) (!At the origin)

Move a point at x = 3, y = 5 four right and two down. Where is its image? (x = 7, y = 3) (!x = 7, y = 7) (!x = 1, y = 9) (!x = 3, y = 7)

What must happen to every vertex in one translation? (It receives the same horizontal and vertical displacement) (!It moves only until the outline looks familiar) (!Its distance from a mirror line becomes zero) (!Its coordinates are swapped)

Reflect a point at x = 2, y = 3 in the vertical line x = 5. Where is the image? (x = 8, y = 3) (!x = 5, y = 3) (!x = 2, y = 8) (!x = 3, y = 2)

Reflect a point at x = 2, y = 1 in the horizontal line y = 4. Where is the image? (x = 2, y = 7) (!x = 7, y = 1) (!x = 2, y = 4) (!x = 6, y = 5)

What happens to a point already on the mirror line during that reflection? (It stays fixed) (!It must move one unit away) (!It moves to the origin) (!It doubles both coordinates)

Which properties are preserved by both translation and reflection? (Corresponding side lengths and angles) (!Every point's original coordinates) (!Every point's direction of movement) (!The clockwise order of labelled vertices in every polygon)

One point moves from x = 2, y = 3 to x = 8, y = 3. What identifies the operation? (More information is needed because a slide or a reflection could give that pair) (!It must be a translation) (!It must be a reflection) (!It must change the shape's size)

Memory Game

Origin	The point with both coordinates zero
Ordered pair	Position recorded with x before y
Translation	Common displacement applied to every point
Reflection	Mirror image at equal perpendicular distances
Vertical mirror line	Reflection keeps y unchanged
Horizontal mirror line	Reflection keeps x unchanged

Drag and Drop

D at three right and five up	Coordinates three comma five
E at five right and three up	Coordinates five comma three
A translated three right and two up	Coordinates four comma three
C translated three right and two up	Coordinates four comma five
Point two comma three reflected in x equals five	Coordinates eight comma three

Crossword Puzzle

COORDINATE	A value in an ordered pair specifying a position relative to an axis
ORIGIN	Where the x and y axes meet at zero comma zero
TRANSLATION	A slide with the same displacement for every point
REFLECTION	A transformation defined by a mirror line
VERTEX	A corner point of a polygon that can be transformed
INVARIANT	A property that remains unchanged under a transformation

LearningApps

Optional extra practice. Choose an appropriate activity with your teacher. If the external collection is unavailable, use the native activities and independent tasks in this course.

Cloze Text

Open Tasks

Easy

Read the eight-point record: Read A to H from the symbolic grid. Explain the coordinate order and why H is on an axis rather than strictly inside the quadrant.
Plot eight ordered pairs: Plot all eight supplied points on a fresh labelled unit grid. Check each position by projecting to the two axes and compare with the supplied record.
Translate a single point: Move D four units right and two down. State the image coordinates, then use the reverse movement to check.
Locate a vertical mirror image: Reflect the point two comma three in x equals five. Explain the equal distances and the coordinate that does not change.

Standard

Translate the whole triangle: Move the original ABC three right and two up. List every image coordinate and compare the corresponding horizontal and vertical sides.
Reflect the original triangle: Reflect the original ABC in x equals five. List all image coordinates and check each horizontal midpoint and unchanged y coordinate.
Use a horizontal mirror: Reflect two comma one and three comma two in y equals four. Explain the perpendicular distances and reverse the reflection.
Compare preserved properties: Compare the two transformed triangles with the original. Identify at least two preserved properties and one difference between the transformation rules.

Advanced

Explain ambiguous single-point evidence: Give a translation and a reflection that both move two comma three to eight comma three. Explain what additional information would distinguish the operations.
Repair a distorted triangle: Explain why moving one vertex three right and two up but another four right and two up is not the stated translation. Test corresponding side lengths or displacements.
Investigate fixed points: Explain why points on x equals five stay fixed under reflection in that line. Contrast this with a non-zero translation and the zero translation.
Design and check a new transformation task: Choose three non-collinear first-quadrant points and a stated axis-parallel mirror or common slide. Keep image points within the displayed grid, supply all image coordinates and give an inverse or distance check.

Learning Check

Plot and read eight points: Plot A to H on a labelled unit grid and explain the axis point and the difference between D and E.
Translate all triangle vertices: Apply three right and two up to the original triangle, giving image coordinates and two preserved properties.
Reflect all triangle vertices: Reflect the original triangle in x = 5 and verify every point through mirror distances.
Reflect individual points and fixed points: Reflect (2, 3) in x = 5, (2, 1) in y = 4 and (5, 2) in x = 5, stating the unchanged coordinate.
Check a translation and its inverse: Move (3, 5) four right and two down, then recover it. Explain why subtracting from both coordinates would be wrong.
Explain invariants and insufficient information: State what translation and reflection preserve, and explain why one start-end point pair can be ambiguous.
Correct a mirror-line error: A learner places the reflection of (2, 3) in x = 5 at (5, 3). Identify the first error and correct it.

Worked checks and assessment guidance

The points are A (1, 1), B (4, 1), C (1, 3), D (3, 5), E (5, 3), F (2, 4), G (6, 2), H (0, 4). D and E differ because the coordinate order matters. H is on the y-axis since its x value is zero; it is not strictly inside the positive first quadrant. Verify each position against the stated axis scale rather than counting unspecified lines.
A_t is (4, 3), B_t is (7, 3), and C_t is (4, 5). Each original x increases by three and y by two. The horizontal side remains length three and the vertical side length two; their right angle is preserved. Three left and two down returns every image to its original point.
A_r is (9, 1), B_r is (6, 1), and C_r is (9, 3). The x midpoints are all five and each pair keeps its y value. A and C are four units from the mirror on opposite sides to their images; B is one unit away. The preserved side lengths and angles do not make this a translation, since the horizontal displacements differ.
The images are (8, 3), (2, 7), and (5, 2). Vertical-line reflection keeps y unchanged and places x at equal horizontal distance on the other side. Horizontal-line reflection keeps x unchanged and reverses vertical distance. The last point is already on its mirror line and stays fixed.
The image is (7, 3): x increases by four while y decreases by two. Four left and two up returns (3, 5). Direction instructions apply to different coordinates; moving down does not determine the horizontal change. Both components are necessary for the complete instruction.
Both preserve corresponding lengths, angles, size and shape. A translation uses one common displacement and preserves the order of labelled polygon vertices; a reflection uses a stated mirror and reverses that order for a non-degenerate polygon. The pair (2, 3) to (8, 3) can come from six right or reflection in x = 5, so the operation requires an instruction or more corresponding points.
The proposed image is on the mirror rather than three units to its other side. Since the original is three units left of x = 5, the image is three right at (8, 3). The mirror is halfway between original and image, and their joining segment is horizontal and perpendicular to the vertical mirror. Reflecting again restores the original point.

Learning evidence

Knowledge: Explain axis labels, origin, coordinate order and scale, translation displacement, reflection line and preserved properties.

Skills: Plot and read points, transform all vertices systematically, distinguish mirror distances from a common slide and use inverse checks.

Products: Keep an eight-point grid, translated and reflected triangle coordinate tables, labelled outlines, an inverse translation check and mirror-distance checks.

Assessment: Require every image coordinate and an explanation of the rule, not only a familiar-looking outline. Check corresponding side lengths or angles as well as positions. A table, tactile model or spoken explanation can show the mathematics; any practical plotting observation must be real and separately identified.

Transfer: Identify insufficient information from one point pair, keep operation labels separate and distinguish the positive first quadrant from its boundary axes. All point data are invented; no learner performance records are supplied.

OER and Wikipedia

This original course follows Department for Education: mathematics programmes of study, Year 5 geometry, position and direction. The core uses translations, reflections in lines parallel to the axes and first-quadrant coordinates with boundary-axis examples. The editorial sequence does not require the negative-coordinate or diagonal-line extensions shown in some broader media.

Commons diagrams

2D Cartesian Coordinates.svg, H Padleckas, public domain as stated on the selected file page. Original unchanged. The wider plane includes negative regions; the course identifies its positive first-quadrant focus and provides its own eight-point record.
Translation illustration.svg, Oleg Alexandrov, released to the public domain by the creator as stated on the file page. Original unchanged. The arrow indicates displacement but supplies no numerical scale; our triangle coordinates are separate invented examples.

Original videos

Translations - Primary, original Corbettmaths publication, for common displacement and corresponding points.
Reflections - Primary, original Corbettmaths publication, for mirror-line and equal-distance reasoning. Diagonal mirrors and negative-coordinate regions are extension context here.

Both films are embedded from the original provider without copying video or transcripts. The written worked methods remain available if external media cannot be played. Original coordinate examples, tables and the symbolic letter grid are invented teaching content, not personal learner data. The encyclopedia below is optional reference.