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Four quadrant coordinates and transformations - Mathematics Year 6

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Introduction

How can you move an entire shape without changing its size? Read signed coordinates, translate every vertex by the same movement and reflect a shape in either coordinate axis. Explain the properties that stay unchanged.

Success criteria. Plot eight specified points, complete two full shape transformations, identify axis points, predict a missing vertex and justify an unfamiliar transformation with an independent check.

Use a labelled grid on paper or an accessible coordinate table. The source grid uses one-unit intervals and numbers from minus six to six on each axis. Screen pixels are not physical length units. The original video establishes coordinate order; the chapters extend this to negative coordinates and transformations.

Locate a point using two signed coordinates

Two numbers, two different directions. An ordered pair gives horizontal position first and vertical position second. Start at the origin, where the axes meet. For the point A = (−3, 2), move three units left and then two units up. The negative first number describes left, not below.

The x-axis is horizontal. Positive x lies to the right; negative x lies to the left. The y-axis is vertical. Positive y lies above the origin; negative y lies below. A coordinate describes a position relative to those axes, rather than a sequence of instructions from the previous plotted point.

Four quadrants. Quadrant I is upper right: both coordinates positive. Quadrant II is upper left: x negative, y positive. Quadrant III is lower left: both negative. Quadrant IV is lower right: x positive, y negative. A point on an axis belongs to no quadrant. At (0, 3), x is zero and the point is on the y-axis. At (−4, 0), y is zero and the point is on the x-axis. The origin (0, 0) is on both axes.

Plot these eight points independently. Use equal unit scales, label both axes and give every point its letter. A table is an alternative way to communicate the signed positions if drawing is difficult.

Point	x	y	Position
A	−3	2	Quadrant II
B	1	2	Quadrant I
C	1	5	Quadrant I
D	−3	5	Quadrant II
E	−2	−4	Quadrant III
F	3	−4	Quadrant IV
G	0	3	y-axis
H	−4	0	x-axis

Join A to B to C to D and back to A. The horizontal side AB is four grid units long; BC is three. This is an axis-aligned rectangle. Its area is:

$4 \times 3 = 12$

It covers twelve square grid units. The values 4 and 3 are differences between coordinates, not either vertex's distance from the origin. Its perimeter is fourteen grid units. Connecting the letters in a different order could produce a different drawing; always specify the vertex order.

The original corbettmaths Coordinates - Primary preview places a point at (5, 6): horizontal five, vertical six. Pause and explain that order. The video is a starting model; use this chapter's full grid for the additional negative-coordinate work.

Read a grid scale before counting. Each square in the actual Commons grid represents one coordinate unit. On a different grid a square might represent two units. Counting three squares would then mean six units. Equal printed spacing does not guarantee the same numerical scale on another diagram.

Common error. A learner plots E at (−4, −2) after reading (−2, −4). Both are in Quadrant III, so a quadrant check alone misses the swapped order. Check the horizontal and vertical coordinates separately. A correct method needs more than the right general region.

Translate a whole shape and preserve its measurements

Translation means one fixed movement for every point. Slide a shape without turning, reflecting or resizing it. The source diagram shows a lower-left shape and an upper-right copy with an arrow v. There is no numbered coordinate grid in that image. Do not infer a numerical movement from its pixel distances.

For our separate rectangle ABCD, translate every vertex two units right and three units down. Add two to x and subtract three from y. For A, calculate each coordinate:

$- 3 + 2 = - 1, 2 - 3 = - 1$

The image of A is A′ = (−1, −1). A prime labels a corresponding point after a transformation; it does not mean a new coordinate operation.

Vertex	Original x	Original y	Translated x	Translated y
A	−3	2	−1	−1
B	1	2	3	−1
C	1	5	3	2
D	−3	5	−1	2

Check the entire shape. The translated bottom edge goes from x = −1 to x = 3, so its length is still four units. The vertical edge runs from y = −1 to y = 2, still three. The area remains twelve square grid units, the perimeter fourteen units and the angles right angles. Translation preserves distances, angles, shape, size and orientation. It changes positions.

Use a reverse movement. Move each translated vertex two left and three up. A′ at (−1, −1) returns to (−3, 2). Checking every row prevents a single correct vertex from concealing an incorrect second or third vertex.

Predict before plotting. C has positive x and positive y. After this movement, C′ = (3, 2) remains in Quadrant I. A moves from Quadrant II to Quadrant III. A translation does not require all vertices to change quadrant, or all to remain in the same quadrant.

When is a proposed translation impossible? Suppose one proposal sends A to (−1, −1) but B to (4, −1). A moved two right; B moved three right. These movements disagree and the horizontal side would lengthen. A collection of moved vertices is not a translation merely because the drawing looks shifted.

An unfamiliar inverse problem. E = (−2, −4) must move to J = (1, −2). The horizontal change is plus three and the vertical change plus two. Apply those same changes to F = (3, −4): its image must be (6, −2). Subtracting the changes recovers both originals. The arrow's direction alone would not specify these numbers.

Choose a useful representation. Plotting shows where the shape lies; a coordinate table shows whether every vertex follows the same rule. Use both for a complete explanation. For a large or negative coordinate, a table may be more efficient than drawing an enormous grid, but the signs and axes still have their ordinary meanings.

Reflect in the axes and justify an unfamiliar result

A reflection uses a mirror line. Reflecting in the x-axis keeps horizontal position and reverses vertical position. Points above and below that horizontal line exchange places at equal perpendicular distances. Thus (2, −4) becomes (2, 4), rather than (−2, −4).

Reflect the ORIGINAL rectangle ABCD in the x-axis, independently of the earlier translation. Call these images A″, B″, C″ and D″:

Vertex	Original x	Original y	Reflected x	Reflected y
A	−3	2	−3	−2
B	1	2	1	−2
C	1	5	1	−5
D	−3	5	−3	−5

The first coordinate stays unchanged; only the second changes sign. The midpoint of A and A″ is (−3, 0), on the mirror line. Their vertical distances from it are both two units. Reflecting twice in the same axis returns each original point.

Reflecting in the y-axis is different. Keep y and reverse x. E = (−2, −4) becomes (2, −4); F = (3, −4) becomes (−3, −4). G = (0, 3) lies on this mirror line and stays fixed. A point on the x-axis is fixed by an x-axis reflection. A point on a mirror line does not need to move to qualify as part of a reflected shape.

What stays unchanged? A reflection preserves lengths, angles, perimeter and area. It reverses orientation: for a labelled polygon, clockwise order becomes anticlockwise. The reflected rectangle is congruent to the original. A symmetric unlabelled outline might look unchanged even though labelled vertices have exchanged positions. Distinguish outline appearance from corresponding-point information.

Missing vertex challenge. An axis-aligned rectangle has consecutive vertices R = (−4, −2), S = (2, −2), T = (2, 1) and an unknown U. R and S share the bottom y-value; S and T share the right-hand x-value. U must share R's x and T's y, giving (−4, 1). Its side lengths are six and three units:

$2 - (- 4) = 6, 1 - (- 2) = 3, 6 \times 3 = 18$

Area is eighteen square grid units. Reflecting this rectangle in the y-axis changes each x-sign but preserves those lengths and area. A missing vertex is justified by equal opposite sides and right angles, not by guessing a convenient unused coordinate.

Do operation orders matter? For A = (−3, 2), translate two right and three down, then reflect in the x-axis. The intermediate point is (−1, −1); the result is (−1, 1). Reverse the operations: reflection gives (−3, −2), then translation gives (−1, −5). These different results show that this pair of operations cannot generally be interchanged.

Make a complete spatial argument. State the original points, whether each operation acts on the original or on a previous image, the mirror axis or exact movement, every resulting coordinate and at least two preserved measurements. Check by inverse movement, equal mirror distances or a second calculation. A near-transfer problem can use different values while retaining these principles.

Your assessment route. Submit the eight plotted positions from chapter one; the complete translated rectangle and separately reflected ORIGINAL rectangle; and the missing-vertex or operation-order challenge. Include labelled axes, units, vertex order and an independent check. A coordinate table with a clear spoken or written spatial explanation is an accessible alternative to dragging points.

Interactive Tasks

Quiz: Test Your Knowledge

Where is the point with x equal to minus three and y equal to two? (Three left and two up from the origin) (!Two left and three up from the origin) (!Three right and two down from the origin) (!Two right and three down from the origin)

Which position is on the y-axis? (x zero and y three) (!x three and y zero) (!x minus three and y two) (!x three and y minus four)

Which signs identify Quadrant IV? (Positive x and negative y) (!Negative x and positive y) (!Both coordinates negative) (!Both coordinates zero)

Which statement describes a translation? (Every point moves by the same horizontal and vertical changes) (!Only one vertex must follow the movement) (!The shape must become larger) (!Every coordinate must change sign)

Translate A two right and three down. What are its new coordinates? (x minus one and y minus one) (!x minus five and y five) (!x one and y one) (!x minus three and y minus two)

Reflect the original point with x two and y minus four in the x-axis. What results? (x two and y four) (!x minus two and y minus four) (!x minus two and y four) (!x four and y two)

What happens to a point on the mirror axis? (It stays fixed) (!It must move one grid square) (!It moves to a different quadrant) (!It always becomes the origin)

Which quantity is preserved by both a translation and a reflection? (The length between corresponding vertices) (!The x-coordinate of every vertex) (!The quadrant of every vertex) (!Clockwise labelled orientation in every case)

Why do the proposed movements of A two right and B three right fail a translation check? (The same movement was not applied to both vertices) (!Negative coordinates are forbidden) (!A rectangle cannot cross an axis) (!Only vertical translations are possible)

What does the original Commons arrow v establish without a coordinate scale? (A direction of translation but no specified numerical movement) (!A translation of exactly two right and three down) (!A reflection in the x-axis) (!A physical length measured in centimetres)

Memory Game

Origin	The intersection with both coordinates zero
Ordered pair	Horizontal coordinate followed by vertical coordinate
Quadrant	One of the four regions excluding the axes
Translation	A fixed movement applied to every point
Reflection	A transformation using a mirror line
Invariant	A property preserved by a transformation
Congruent	Having the same shape and size

Drag and Drop

x-axis reflection	Keep x and change the sign of y
y-axis reflection	Keep y and change the sign of x
Two right and three down	Add two to x and subtract three from y
Axis-point check	At least one coordinate is zero
Inverse translation	Undo each horizontal and vertical change

Crossword Puzzle

ORIGIN	Where the two coordinate axes meet
QUADRANT	One of four open regions of the coordinate plane
VERTEX	A corner point of a polygon
REFLECTION	A transformation in a mirror line
TRANSLATION	The same fixed movement of every point
INVARIANT	A property that remains unchanged

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

Eight point plot: Plot A–H from the actual chapter-one table on equal labelled scales. Identify quadrant or axis and check the two coordinates independently.
Rectangle dimensions: Join ABCD in order. Find horizontal and vertical lengths, perimeter and area with suitable grid units.
Single translation: Translate A and B two right and three down. Describe the horizontal and vertical changes in words.
Axis reflections: Reflect E in the y-axis and G in the y-axis. Explain why one moves and one stays fixed.

Standard

Complete translation: Complete all four translation rows. Plot A′B′C′D′ and check that every vertex follows the same movement.
Complete original reflection: Reflect ORIGINAL ABCD in the x-axis. Give every image coordinate and two preserved measurements.
Missing rectangle vertex: Find U from R, S and T. Use shared coordinates and rectangle properties; check lengths and area.
Diagnose the swapped pair: Explain why E plotted at minus four horizontally and minus two vertically passes a quadrant check but is still wrong.

Advanced

Find an unknown movement: Find the translation from E to J given in chapter two. Apply it to F and verify by the inverse.
Compare operation orders: Apply translation then reflection to A, then reflection then translation. Explain the difference with intermediate coordinates.
Preserved and reversed properties: Compare distance, angle, area and labelled orientation under both transformations. Distinguish an unlabelled symmetric outline from point correspondence.
Original transformation dossier: Set a new four-quadrant polygon problem, state original and image coordinates, justify its rule and provide an independent geometric check.

Learning Check

Eight positions: Plot all eight supplied points and label each region or axis.
Two complete transformations: Transform original ABCD by the specified translation and, separately, by reflection in the x-axis.
Inverse and mirror checks: Show one inverse translation and one mirror-distance check.
Unfamiliar rectangle and movement: Check U, rectangle measurements and the E-to-J movement.
Order and misconception: Check both transformation orders and the mistaken E position.
Complete communication: Evaluate a new transformation explanation.

Worked checks and assessment guidance

A and D are in II; B and C in I; E in III; F in IV; G lies on the y-axis; H on the x-axis. Their exact coordinates are in the original task table. A quadrant alone does not establish a correct plotted point.
Translation: A′ = (−1, −1), B′ = (3, −1), C′ = (3, 2), D′ = (−1, 2). Reflection of ORIGINAL ABCD: A″ = (−3, −2), B″ = (1, −2), C″ = (1, −5), D″ = (−3, −5). Both preserve side lengths four and three, perimeter fourteen and area twelve square grid units.
Move A′ two left and three up to recover A = (−3, 2). A and A″ share x = −3, have y-values two and minus two, and lie equally far from the x-axis; their midpoint is on that mirror line.
U = (−4, 1). Rectangle sides six and three give area eighteen square grid units and perimeter eighteen grid units. E to J is three right and two up; F becomes (6, −2). The same differences must hold for all corresponding points.
Translation then x-axis reflection sends A to (−1, 1); reflection then translation sends it to (−1, −5). Intermediate coordinates show why these differ. Swapping E to (−4, −2) changes both component positions even though Quadrant III is unchanged.
Require exact starting points, movement or mirror axis, every image coordinate, labelled scales and vertex correspondence, relevant invariants and a check. Do not assign numerical movement to the source arrow without a scale. Written coordinates and an oral spatial justification may replace a fine-motor drawing task.

Learning evidence

Keep the eight-point plot or accessible table, two complete transformations of original ABCD, an invariant comparison and one unfamiliar checked solution. Preserve mistakes and corrections as learning evidence. This course assesses coordinate reasoning, not observed physical drawing skill or an entire annual curriculum.

OER and Wikipedia

Checked 11 November 2026. Department for Education: mathematics programmes of study, Year 6 position and direction. The full-grid and axis-reflection objectives inform this original lesson.

Original points, tables, inverse checks, misconceptions and assessment feedback are newly authored. Source images are unchanged and do not already contain the course's lettered task points. The arrow v has no numerical scale.

Original embedded Coordinates - Primary, corbettmaths. Its first-quadrant coordinate-order model is extended by the independent teaching here; the video is not falsely described as the complete transformations course or as a Commons-licensed file.

2D Cartesian Coordinates.svg, H Padleckas, Public domain. Unchanged.

Translation illustration.svg, Oleg Alexandrov, Public domain. Unchanged.