Weaving and repeating patterns
A woven basket or a paper mat is made by going "1 over, 1 under, 1 over, 1 under". The smallest part that repeats is the unit of the pattern.
A pattern is a design that repeats by a rule. The unit is the part that repeats. In weaving each row follows a rule of "over" and "under", and the next row often slides by one box.
Formula: Position in the unit = remainder when the position number is divided by the unit length (if the remainder is 0, it is the last item of the unit).
In the figure: Purple boxes are "over", light blue boxes are "under". Each row is the same rule slid one box to the right.
Rule 1: Find the unit — Read the first row and find the shortest part that repeats. Unit length = over + under.
Weaving and repeating patterns
In the app you can move this figure with your finger.
Open in the app- Easy: "2 over, 1 under" repeats every 2 + 1 = 3 boxes.
- Medium: "1 over, 3 under" repeats every 4 boxes.
- Exam-type: Red, green, blue, red, green, blue … has a unit of 3 colours.
Rule 2: Find any position — Divide the position by the unit length. The remainder tells the place inside the unit.
- Easy: Position 10 of a 3-colour pattern: 10 = 3 × 3 + 1, so it is the 1st colour.
- Medium: Position 12 of a 4-colour pattern: 12 = 3 × 4 + 0, so it is the 4th (last) colour.
- Exam-type: Position 20 of a 3-colour pattern: 20 = 6 × 3 + 2, so it is the 2nd colour.
If the remainder is 0, the position is the LAST item of the unit, not the first.
Counting the unit from 0, or taking position 12 of a 4-unit pattern as the 1st item. 12 ÷ 4 leaves 0, so it is the 4th.
Jump in whole units: from 1 to 10 is 3 full units of 3, so 10 looks like 1. Memory clue: "Divide by the unit, read the remainder."
Spot the difference
| Point | 2 over, 1 under | 1 over, 1 under |
|---|---|---|
| Unit length | 3 | 2 |
| Row 2 slid by 1 → looks like Row 1 after | 3 rows | 2 rows |
Practice questions
- 1Row 1 is "2 over, 1 under …". After how many boxes does it repeat?
- 2Red, green, blue repeat. Which colour is at position 7?
- 3Row 1 is "1 over, 1 under …". Write Row 2 (opposite).
- 4A pattern ■ ● ▲ repeats. What is the 10th shape?
- 5A 4-colour pattern repeats. What is the colour at position 12?
- 6How many "over" boxes in the first 9 boxes of "2 over, 1 under …"?
- 7A weave repeats every 4 boxes and each row slides 1 box. After how many rows does it look like Row 1?
- 8Make your own weave rule with a unit of 5 boxes and tell the number of "over" boxes in 15 boxes.
Show answers (try them yourself first)
- 13
- 2Red (7 = 2 × 3 + 1)
- 31 under, 1 over, 1 under, 1 over …
- 4■ (10 = 3 × 3 + 1)
- 5The 4th colour (12 = 3 × 4)
- 66
- 74 rows
- 8Example: 2 over, 3 under → 3 units × 2 = 6 over
Tiling and tessellation
A bathroom floor is covered with tiles with no gaps and no overlaps. Honeycomb cells fit together in the same way. Covering a surface like this is called tessellation.
A shape tessellates if copies of it fit together around a point with no gap and no overlap. Shapes with all equal sides are regular shapes. A full turn is 360°.
Formula: Around a point: 6 triangles, 4 squares or 3 hexagons make a full turn (360°). Regular pentagons and octagons do not tessellate alone.
In the figure: The shapes meet exactly at the dot. Nothing is left over and nothing overlaps.
Rule 1: Count around the point — Place the shape around a point. If it closes exactly, it tessellates.
Tiling and tessellation
In the app you can move this figure with your finger.
Open in the app- Easy: Equilateral triangle: 6 fit exactly.
- Medium: Square: 4 fit exactly; a fifth would overlap.
- Exam-type: Regular pentagon: 3 fit and leave a gap, so it does not tessellate.
Rule 2: Mix two shapes — Different shapes can tessellate together if their corners add up to 360° at every point.
- Easy: 3 triangles + 2 squares: 3 × 60° + 2 × 90° = 360°.
- Medium: 2 triangles + 2 hexagons: 2 × 60° + 2 × 120° = 360°.
- Exam-type: 2 octagons (2 × 135° = 270°) + 1 square (90°) = 360°.
Corners at a point must add up to exactly one full turn: 360°.
Thinking that any regular shape tessellates. Pentagons (3 × 108° = 324°) and octagons (2 × 135° = 270°) leave gaps that no copy can fill.
Test: 360 ÷ (corner angle) must be a whole number. 360 ÷ 60 = 6, 360 ÷ 90 = 4, 360 ÷ 120 = 3, but 360 ÷ 108 is not whole. Memory clue: "3, 4, 6 fit — 5 and 8 leave gaps."
Spot the difference
| Shape | Corner | Fit around a point |
|---|---|---|
| Triangle | 60° | 6 |
| Square | 90° | 4 |
| Hexagon | 120° | 3 |
| Pentagon | 108° | 3 + gap |
Practice questions
- 1How many equilateral triangles fit around a point?
- 2How many squares fit around a point?
- 3How many regular hexagons fit around a point?
- 4Can 5 squares fit around a point? Why?
- 5Which regular shapes among triangle, square, pentagon, hexagon, octagon do NOT tessellate?
- 63 regular pentagons (108° each) are put around a point. How big is the gap?
- 72 octagons (135° each) leave a 90° gap. Which shape fills it?
- 8Do 3 triangles and 2 squares fit exactly around a point? Check with angles.
Show answers (try them yourself first)
- 16
- 24
- 33
- 4No: 4 squares already fill 360°; a fifth overlaps.
- 5Pentagon and octagon
- 6360° − 324° = 36°
- 7A square (90° corner)
- 8Yes: 3 × 60° + 2 × 90° = 360°
Triangles: equilateral, isosceles, scalene
A roof truss, a samosa and a slice of pizza are all triangles, but they are not all alike. Triangles can be sorted by how many of their sides are equal.
Equilateral: all 3 sides equal. Isosceles: exactly 2 sides equal. Scalene: no sides equal. The angles follow the sides: equal sides have equal angles opposite to them.
Formula: Perimeter of an equilateral triangle = 3 × side. Isosceles: perimeter = 2 × equal side + third side.
In the figure: Small tick marks show equal sides. 3 matching ticks on the first, 2 on the second, none matching on the third.
Rule 1: Sort by sides — Compare the three sides: all equal, two equal or none equal.
- Easy: 5, 5, 5 cm → equilateral.
- Medium: 6, 6, 4 cm → isosceles.
- Exam-type: 3, 4, 5 cm → scalene.
Rule 2: Angles follow sides — Fold a paper triangle to compare angles: isosceles has 2 equal angles, equilateral has 3, scalene has none.
- Easy: Equilateral: all three angles equal.
- Medium: Isosceles: the two angles opposite the equal sides are equal.
- Exam-type: Halving an equilateral triangle gives two scalene triangles.
An equilateral triangle is a special isosceles triangle, but for sorting we call it equilateral.
Calling a triangle isosceles just because two sides look nearly equal. Measure first: 6, 6, 4 is isosceles; 6, 6, 6 is equilateral.
Count equal sides: 3 → equilateral, 2 → isosceles, 0 → scalene. Memory clue: "All equal — equi; two equal — iso; none equal — scalene."
Spot the difference
| Kind | Equal sides | Equal angles |
|---|---|---|
| Equilateral | 3 | 3 |
| Isosceles | 2 | 2 |
| Scalene | 0 | 0 |
Practice questions
- 1Name the triangle with sides 8, 8, 8 cm.
- 2Name the triangle with sides 7, 7, 3 cm.
- 3Name the triangle with sides 4, 7, 9 cm.
- 4How many angles of an isosceles triangle are equal?
- 5Find the perimeter of an equilateral triangle of side 9 cm.
- 6What kind of triangle is each half of an equilateral triangle cut from a corner to the middle of the opposite side?
- 7An isosceles triangle has equal sides of 7 cm and perimeter 20 cm. Find the third side.
- 8Can a triangle have sides 2, 3 and 9 cm? Think about whether the short sides can reach each other.
Show answers (try them yourself first)
- 1Equilateral
- 2Isosceles
- 3Scalene
- 42
- 527 cm
- 6Scalene
- 720 − 14 = 6 cm
- 8No: 2 + 3 = 5 is less than 9, so the two short sides cannot meet.
Quadrilaterals: kite, parallelogram, rectangle, rhombus
A kite in the sky, a window, a door and a tile all have 4 sides, but their sides and corners differ. A 4-sided shape is a quadrilateral.
Kite: 2 pairs of equal adjacent sides. Parallelogram: opposite sides equal (and opposite angles equal). Rectangle: a parallelogram with 4 right angles. Rhombus: all 4 sides equal. Square: equal sides and 4 right angles.
Formula: Perimeter of a rectangle = 2 × (length + width). Rhombus = 4 × side. Kite = 2 × (a + b).
In the figure: Four different 4-sided shapes. Compare the sides and angles of each.
Rule 1: Which shape am I? — Check the sides first (adjacent equal, opposite equal, all equal), then the angles.
- Easy: All sides equal, angles not right → rhombus.
- Medium: All angles right, sides not all equal → rectangle.
- Exam-type: Two pairs of equal adjacent sides → kite.
Rule 2: Special shapes — A rectangle is a special parallelogram, and a square is a special rectangle (and a special rhombus).
- Easy: Every square is a rectangle, but not every rectangle is a square.
- Medium: Perimeter of a 9 cm × 5 cm rectangle = 2 × 14 = 28 cm.
- Exam-type: Perimeter of a rhombus of side 6 cm = 4 × 6 = 24 cm.
A rhombus has equal sides but not necessarily right angles; a rectangle has right angles but not necessarily equal sides.
Mixing up "adjacent" (next to each other) and "opposite" (facing each other). A kite has equal ADJACENT sides; a parallelogram has equal OPPOSITE sides.
Ask 2 questions: "Are all sides equal?" and "Are all angles right?" Yes-Yes: square, Yes-No: rhombus, No-Yes: rectangle. Memory clue: "Kite — neighbours match; parallelogram — opposites match."
Spot the difference
| Shape | Equal sides | Right angles |
|---|---|---|
| Kite | adjacent pairs | no |
| Parallelogram | opposite pairs | no |
| Rectangle | opposite pairs | 4 |
| Rhombus | all 4 | no |
| Square | all 4 | 4 |
Practice questions
- 1How many sides does a quadrilateral have?
- 2Which shape has all sides equal and all angles right?
- 3A rectangle is a special kind of which shape?
- 4Find the perimeter of a rectangle 12 cm long and 7 cm wide.
- 5A kite has adjacent sides 7 cm, 7 cm, 4 cm, 4 cm. Find its perimeter.
- 6A rhombus has a side of 9 cm. What is its perimeter?
- 7Why is every square a rectangle, but not every rectangle a square?
- 8Which shapes among kite, parallelogram, rectangle, square and rhombus always have equal opposite sides?
Show answers (try them yourself first)
- 14
- 2Square
- 3Parallelogram
- 438 cm
- 522 cm
- 636 cm
- 7A square has 4 right angles and equal opposite sides, so it is a rectangle; a rectangle may have unequal sides.
- 8Parallelogram, rectangle, square, rhombus
Circles and tangram
A wheel and a bangle are circles. A tangram is an old puzzle: a square cut into 7 pieces that can be re-arranged into hundreds of pictures.
A diameter passes through the centre and meets the circle at both ends; diameter = 2 × radius. Two diameters joined at their ends make a rectangle (a square if they cross at a right angle). A tangram has 5 triangles, 1 square and 1 parallelogram.
Formula: Diameter = 2 × radius; radius = diameter ÷ 2. Tangram: 2 large + 1 medium + 2 small triangles + square + parallelogram = 7 pieces.
In the figure: Top: the 4 ends of two diameters form a rectangle. Bottom: the 7 tangram pieces — L large triangle, M medium triangle, S small triangle, Sq square, P parallelogram.
Rule 1: Diameter and radius — The centre is the middle of every diameter, so a diameter is two radii end to end.
- Easy: Radius 7 cm → diameter 14 cm.
- Medium: Diameter 18 cm → radius 9 cm.
- Exam-type: Two diameters of a circle of radius 5 cm are 10 cm each, and they bisect each other.
Rule 2: Build with tangram pieces — Pieces of equal size can be swapped or joined: the two large triangles are the same; two small triangles can make a square, a parallelogram or a medium triangle.
- Easy: Total pieces: 5 + 1 + 1 = 7.
- Medium: Two identical isosceles right triangles joined along the longest side make a square.
- Exam-type: Area check: 2 small triangles = 1 square = 1 parallelogram = 1 medium triangle.
Joining the ends of two diameters can never make a kite or an ordinary parallelogram — only a rectangle (or square).
Taking a radius as a diameter. A radius goes from the centre to the circle (half), a diameter goes all the way across.
Doubling and halving: diameter = 2 × radius, radius = half of diameter. Memory clue: "Diameter across, radius half-way."
Spot the difference
| Point | Radius | Diameter |
|---|---|---|
| From … to … | centre → circle | circle → circle through centre |
| Length of a circle of radius 6 cm | 6 cm | 12 cm |
Practice questions
- 1The radius of a circle is 9 cm. Find the diameter.
- 2The diameter of a circle is 26 cm. Find the radius.
- 3How many pieces does a tangram have?
- 4How many triangles are there in a tangram?
- 5Two diameters are drawn in a circle and their ends joined. Which shape is formed?
- 6When do the joined ends of two diameters form a square?
- 7Which two tangram pieces are exactly the same shape and size?
- 8Using 2 small tangram triangles, name 3 shapes you can make.
Show answers (try them yourself first)
- 118 cm
- 213 cm
- 37
- 45
- 5Rectangle
- 6When the diameters cross at a right angle
- 7The two large triangles (also the two small ones)
- 8Square, parallelogram, medium triangle
Cubes and solid shapes
A dice, a sugar cube and a gift box are cubes. Stacking 27 small cubes makes a big cube like a Rubik-style block. Counting faces, edges and corners helps us understand any solid.
A cube has 6 square faces, 12 edges and 8 corners. A 3 × 3 × 3 cube has 27 small cubes. An icosahedron has 20 triangular faces; a dodecahedron has 12 pentagonal faces; both have 30 edges.
Formula: Small cubes in a 3 × 3 × 3 cube = 3 × 3 × 3 = 27. Painted outside: 8 + 12 + 6 + 1 = 27.
In the figure: The 8 corner cubes show 3 painted faces, the 12 edge cubes 2, the 6 face-centre cubes 1, and the hidden middle cube 0.
Rule 1: Painted cube — Sort the small cubes by position: corner, edge, face centre, hidden middle.
- Easy: Corner cubes: 8, each with 3 painted faces.
- Medium: Edge cubes: 12 (one in the middle of each edge), 2 painted faces.
- Exam-type: Face-centre cubes: 6 with 1 painted face; the middle cube: 1 with none.
Rule 2: Count faces, edges, corners — Count systematically: faces by colour, edges by going round each face once and halving (every edge is shared by 2 faces).
- Easy: Cube: 6 faces × 4 edges ÷ 2 = 12 edges.
- Medium: Icosahedron: 20 × 3 ÷ 2 = 30 edges; 5 triangles meet at each corner.
- Exam-type: Dodecahedron: 12 × 5 ÷ 2 = 30 edges; 3 pentagons meet at each corner.
Every edge is shared by exactly 2 faces, so dividing the sum of face sides by 2 gives the edges.
Forgetting the hidden middle cube: 8 + 12 + 6 = 26, but the big cube has 27 small cubes, so 1 cube is unpainted.
For a 3 × 3 × 3 painted cube remember 8 – 12 – 6 – 1. Memory clue: "Corners 8, edges 12, faces 6, heart 1."
Spot the difference
| Solid | Faces | Edges | Corners |
|---|---|---|---|
| Cube | 6 | 12 | 8 |
| Icosahedron | 20 | 30 | 12 |
| Dodecahedron | 12 | 30 | 20 |
Practice questions
- 1How many faces, edges and corners does a cube have?
- 2How many small cubes make a 3 × 3 × 3 big cube?
- 3How many small cubes make a 2 × 2 × 2 cube?
- 4A 3 × 3 × 3 cube is painted outside. How many small cubes have 3 faces painted?
- 5In the same painted cube, how many small cubes have 2 faces painted?
- 6How many small cubes have 1 face painted, and how many none?
- 7How many sticks are needed to make a cube frame?
- 8What shape are the faces of an icosahedron and of a dodecahedron? How many edges do they have?
Show answers (try them yourself first)
- 16, 12 and 8
- 227
- 38
- 48
- 512
- 66 and 1
- 712
- 8Triangles (20 faces) and pentagons (12 faces); both have 30 edges