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E1.1 · Identify shape properties and transformations in tessellations

Learn to identify shape properties and transformations in tessellations through clear examples and targeted practice.

Ontario Grade 8 Mathematics

Spatial Sense

Ontario Grade 8 Mathematics — E1.1

Look at a tiled floor or a patterned wall. The pieces fit together to cover the surface. A pattern like this can help us notice both shape properties and transformations. In this lesson, you will describe what the shapes are like and explain how copies move or turn to make a repeating pattern.

What you will learn

  • Describe shape properties that help form a tessellation.
  • Recognize when shapes cover a surface without gaps or overlaps.
  • Identify translations, rotations, and reflections in a tessellation.

1. What makes a tessellation?

Imagine covering a table with identical square paper tiles. If the tiles meet edge to edge, cover the whole surface, and do not pile on top of one another, they make a tessellation. A tessellation is a repeating arrangement of shapes that covers a flat surface with no gaps and no overlaps.
A gap is an uncovered space between shapes. An overlap happens when shapes cover the same part of the surface. Both mean that the pattern is not a tessellation of that surface. The shapes may be all alike, or the pattern may use more than one shape.
Before describing a shape, review its basic properties. A polygon is a closed, flat shape made from straight sides. A vertex is a corner where two sides meet. An angle is the opening between two sides that meet. For example, a square has four equal sides and four right angles. A right angle is a square corner.
When identical copies of a shape appear in a pattern, compare their sides, angles, and size. Copies that have the same shape and size are called congruent. A congruent copy can face a different way and still be congruent.
  • A tessellation covers a flat surface with no gaps or overlaps.
  • Describe shapes using properties such as sides, vertices, and angles.
  • Congruent copies have the same shape and size, even if they face different ways.

2. Notice how shapes meet

To check a pattern, first look across the surface. Is any part left uncovered? Do any shapes lie on top of other shapes? Next, look where several corners meet. A well-fitting pattern has no open space at that meeting point and no pieces covering one another.
You can also trace one tile and compare it with nearby tiles. Count its sides and corners. Check whether matching sides and angles stay the same from tile to tile. Then ask whether each copy has been slid, turned, or flipped.
These checks help you describe a pattern without guessing from its appearance alone. A pattern can look regular but still have a small gap. Follow the edges and corners closely.
  • Check the whole surface for gaps and overlaps.
  • Compare sides, corners, angles, and size in repeated shapes.
  • Inspect the points where several shapes meet.

3. Identify transformations

A transformation is a change in a shape's position or orientation. In a tessellation, copies often repeat through slides, turns, or flips.
A translation slides a shape without turning or flipping it. Each point moves the same distance in the same direction. The shape keeps its size and orientation.
A rotation turns a shape around a fixed point. The fixed point is the centre of the turn. The shape keeps its size, but its orientation changes. For example, a quarter-turn changes which way a square's sides point, even though the square still fits the same outline.
A reflection flips a shape across a line, as if the line were a mirror. The reflected copy keeps the same size and shape, but its orientation is reversed. The mirror line is called the line of reflection.
Use clues in the pattern. If a tile appears unchanged and shifted, identify a translation. If it has turned around a point, identify a rotation. If it appears as a mirror image across a line, identify a reflection. Some repeating patterns use more than one transformation.
  • Translation: slide.
  • Rotation: turn around a point.
  • Reflection: flip across a line.
  • These transformations preserve a shape's size and shape.

4. A careful way to describe a pattern

Use a short, ordered description. Name the shape and its properties first. Then say how copies meet and which transformations you can see. Finally, check that the repeated arrangement has no gaps or overlaps.
For example, you might say: “The pattern uses congruent squares. Each square has four equal sides and four right angles. Copies are translated to the left and right, and their edges meet without gaps or overlaps.” This description identifies the shape, its properties, the movement, and the tessellation check.
A drawing does not always show enough detail to identify one exact transformation. If you are unsure, describe only what you can observe. Look for a clear slide, turn, or mirror image before naming it.
  • Name the shape and give observable properties.
  • Describe how the copies move or change orientation.
  • Verify that the surface has no gaps or overlaps.

Worked example

Square tiles on a floor

A floor pattern is made from identical square tiles arranged in straight rows. Describe the shape properties, explain how the pattern repeats, and decide whether it is a tessellation.
Repeated square positions
Repeated square positionsEach outline marks one congruent square; the copies are shifted into a row.positionpositionABCDB′EFC′

Each outline marks one congruent square; the copies are shifted into a row.

  1. Describe one tile
    A square has four equal sides and four right angles. Each tile is a copy of the same shape and size, so the tiles are congruent.
  2. Identify the movement
    A tile beside another tile has been slid across the floor without turning. This is a translation. The same kind of slide can repeat the tiles in rows.
  3. Check the surface
    The tiles meet edge to edge, and the stated arrangement leaves no gaps or overlaps. The pattern is a tessellation.
Answer: The pattern uses congruent squares. Each square has four equal sides and four right angles. The copies repeat by translation and meet without gaps or overlaps, so they form a tessellation.
Check: The conclusion depends on the stated edge-to-edge arrangement covering the surface. If a gap or overlap appeared, the pattern would not meet the definition.

Worked example

A triangle pattern with turns

A pattern is made from congruent equilateral triangles. At each shared corner, six triangle corners meet. Identify a transformation that can repeat the triangles and explain why the arrangement can tessellate.
  1. Name the shape properties
    An equilateral triangle has three equal sides and three equal angles. The pattern uses congruent copies, so each triangle keeps the same size and shape.
  2. Read the corner arrangement
    Six triangle corners meet at each shared point. Together, they fill the space around that point without leaving a gap or covering one another.
  3. Identify a transformation
    A triangle can be turned around a shared corner to match the next triangle's position. That movement is a rotation. Repeating the triangles in different orientations lets their sides meet.
Answer: The pattern uses congruent equilateral triangles. Each has three equal sides and three equal angles. Rotations help place the triangles around shared corners, where six corners meet with no gaps or overlaps. This arrangement forms a tessellation.
Check: The description names a turn around a fixed point, so rotation is the correct transformation. The no-gap, no-overlap condition is also met at the shared corners and along the sides.

Worked example

Recognize a reflection

Two congruent tile shapes lie on opposite sides of a straight line. One tile looks like the mirror image of the other. Explain the transformation and what stays the same.
  1. Compare the tiles
    The tiles are stated to be congruent. That means they have the same shape and size, even though they face different ways.
  2. Use the mirror clue
    A copy that appears reversed across a straight line is a reflection. The line acts as the mirror line.
  3. Connect the tiles to the pattern
    If the reflected tiles fit with their neighbours and the full arrangement has no gaps or overlaps, they can be part of a tessellation. The transformation changes orientation, not size or shape.
Answer: The transformation is a reflection across the straight mirror line. The tiles remain the same size and shape, but their orientations are reversed. They form part of a tessellation if they fit with the surrounding tiles without gaps or overlaps.
Check: A reflection is the transformation indicated by the mirror-image clue. The final tessellation decision must still be checked against the whole arrangement.

Common mistakes and how to avoid them

Calling any repeating picture a tessellation.
Correction: Check that the pattern covers the surface with no gaps and no overlaps.
Thinking a turned or flipped copy cannot be congruent.
Correction: A turn or flip changes orientation, but the copy can keep the same shape and size.
Calling a reflection a translation.
Correction: A translation slides without turning or flipping. A reflection makes a mirror image across a line.
Naming a transformation from a vague visual clue.
Correction: Look for a clear slide, turn around a point, or mirror image before choosing a name.

Lesson summary

  • A tessellation covers a flat surface without gaps or overlaps.
  • Describe a shape using properties such as sides, vertices, and angles.
  • A translation slides; a rotation turns; a reflection flips across a line.
  • Transformations can change orientation while keeping a shape's size and shape.
  • A complete description identifies the shapes, their transformations, and how they meet.

Check your understanding

Question 1

A copy of a tile moves to the right without turning or flipping. Which transformation is shown?
  1. Translation
  2. Rotation
  3. Reflection
  4. correctIndex
Show answer and explanation
Translation
A translation slides a shape without changing its orientation.

Question 2

A pattern leaves uncovered spaces between its tiles. Does it meet the definition of a tessellation?
  1. Yes, because the tiles repeat.
  2. No, because a tessellation has no gaps or overlaps.
  3. Yes, if all the tiles are congruent.
  4. correctIndex
Show answer and explanation
No, because a tessellation has no gaps or overlaps.
Repeating shapes is not enough. The pattern must cover the surface with no gaps or overlaps.

Question 3

A shape appears as a mirror image across a line. Which transformation best describes the change?
  1. Translation
  2. Reflection
  3. Rotation
  4. correctIndex
Show answer and explanation
Reflection
A reflection flips a shape across a line to make a mirror image.

Key terms

Polygon
A closed, flat shape made from straight sides.
Vertex
A corner where two sides meet.
Congruent
Having the same shape and size.
Tessellation
A repeating arrangement of shapes that covers a flat surface with no gaps or overlaps.
Transformation
A change in a shape's position or orientation.
Translation
A slide that does not turn or flip a shape.
Rotation
A turn around a fixed point.
Reflection
A flip across a line that creates a mirror image.

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