What you'll learn
A pattern with shapes is a sequence that follows a rule you can state in words, and the whole of this topic is about finding that rule and then using it. There are two kinds. A repeating pattern has a small group of shapes that comes round again and again — circle, square, triangle, circle, square, triangle — and you continue it by finding the length of the repeating group. A growing pattern gets bigger by the same amount each step — 4 sticks, then 7, then 10 — and you continue it by finding what is added each time. This topic also covers tessellation (fitting shapes together with no gaps), and sliding, flipping and turning a shape. Geometry carries 6 of the 40 items on the SEA Mathematics paper, and shape patterns are a favourite because they can be asked without any calculation at all.
Key terms and definitions
Repeating pattern — a pattern made by saying the same small group of shapes over and over.
Unit of repeat — the smallest group that repeats. In circle, square, triangle, circle, square, triangle, the unit of repeat is three shapes long.
Growing pattern — a pattern where each step is bigger than the one before, usually by the same amount.
Term — one step or item in a pattern. The 5th term is the 5th one along.
Rule — the instruction that makes the next term, such as "add 3" or "turn a quarter turn".
Difference — how much is added from one term to the next.
Tessellation — a covering made by fitting copies of a shape together with no gaps and no overlaps, the way floor tiles fit.
Slide, flip and turn — moving a shape along without changing it, turning it over as in a mirror, and rotating it about a point.
Core concepts
Repeating patterns and the unit of repeat
Look at this line of shapes, written out in words:
circle, square, triangle, circle, square, triangle, circle, square, triangle
The unit of repeat is circle, square, triangle, 3 shapes long. Everything else is that unit said again.
Once you know the unit's length, you can find any position without writing out the whole line. Divide the position number by the length of the unit and look at the remainder.
- Remainder 1 means the 1st shape in the unit.
- Remainder 2 means the 2nd shape.
- Remainder 0 means the last shape in the unit, because the unit has just finished exactly.
That last rule is the one pupils forget. Remainder 0 does not mean "the first"; it means you landed exactly on the end of a unit.
Patterns that repeat with a turn
Some patterns keep the same shape but change its position. A flag shape might point up, then right, then down, then left, then up again. The unit of repeat is 4 long, and the rule is "turn a quarter turn clockwise each time".
Describe the rule in words before continuing the pattern. "It turns a quarter turn clockwise" is a complete answer and earns the mark.
Growing patterns made of sticks or tiles
A growing pattern is built by adding the same number of pieces at every step.
Suppose squares are made in a row from matchsticks, each new square sharing a side with the one before.
One square uses 4 sticks, two use 7, three use 10 and four use 13.
The differences are 7 − 4 = 3, 10 − 7 = 3 and 13 − 10 = 3. The rule is add 3 each time, because every new square needs three new sticks; the fourth side is already there.
To reach a term further along, either keep adding 3 — safe but slow — or use the shortcut: the number of sticks is three times the number of squares, plus one for the first upright. For 4 squares: 3 × 4 + 1 = 13, which matches.
Patterns that grow by a changing amount
Not every growing pattern adds the same number. Square arrays of tiles go 1, 4, 9, 16, 25 — the square numbers, because a pattern of side 3 holds 3 × 3 = 9 tiles.
Their differences are 3, 5, 7, 9 — the odd numbers. So the differences themselves make a pattern, and looking at them is what you do whenever the first look shows no rule.
Triangular arrangements of dots go 1, 3, 6, 10, 15, with differences 2, 3, 4, 5, because each new row has one more dot than the row above.
Tessellation
A shape tessellates if copies of it fit together to cover a flat surface with no gaps and no overlaps.
| Shape | Tessellates? |
|---|---|
| Square | yes |
| Rectangle | yes |
| Equilateral triangle | yes |
| Regular hexagon | yes |
| Regular pentagon | no |
| Circle | no |
Regular hexagons tessellate, which is why a honeycomb has no wasted space. Circles leave curved gaps at every join, and regular pentagons leave a small wedge, so neither will do.
Any triangle will tessellate if every second one is turned upside down, because two copies of a triangle make a parallelogram, and parallelograms fit together in rows.
Sliding, flipping and turning
Three moves are used to build shape patterns, and the paper may ask you to name them.
- A slide moves the shape along without changing the way it faces. Wallpaper borders usually slide.
- A flip turns the shape over so it faces the other way, like the shape and its reflection in a mirror.
- A turn rotates the shape about a point by a quarter, half or three-quarter turn.
To decide which move was used, ask whether the shape still faces the same way. If yes, it was a slide. If it is now a mirror image, it was a flip. If it has been rotated but not mirrored, it was a turn.
Worked examples
Example 1: The 20th shape in a repeating pattern
Anisa threads beads in this repeating order: circle, square, triangle, circle, square, triangle, and so on. What shape is the 20th bead? What shape is the 30th?
First find the unit of repeat. It is circle, square, triangle, so the unit is 3 long.
For the 20th bead, divide 20 by 3. Since 3 × 6 = 18, we get 6 remainder 2.
A remainder of 2 means the 2nd shape in the unit, which is a square.
For the 30th bead, divide 30 by 3. Since 3 × 10 = 30 exactly, the remainder is 0.
A remainder of 0 means the last shape in the unit, which is a triangle.
Check that a different way. Every bead in position 3, 6, 9, 12 and so on is a triangle, because those are the multiples of 3. As 30 is a multiple of 3, the 30th is a triangle.
Example 2: Matchstick squares
Kern builds a row of squares from matchsticks. Each new square shares one side with the square before it. One square uses 4 sticks, two squares use 7, and three squares use 10. How many sticks does he need for 10 squares?
Look at the differences: 7 − 4 = 3 and 10 − 7 = 3. So the rule is add 3 for each extra square.
The slow way is to keep adding 3 from the third square onwards: 10, 13, 16, 19, 22, 25, 28, 31. That is seven more squares, taking us from 3 up to 10, and the answer is 31.
The quick way uses a shortcut. Each square needs 3 new sticks, and there is one extra stick at the very start, so sticks = 3 × number of squares + 1.
For 10 squares: 3 × 10 = 30, and 30 + 1 = 31 sticks.
Both methods give 31. Test the shortcut on a row you already know: for 2 squares, 3 × 2 + 1 = 7, which matches the question.
Example 3: Growing square patterns of tiles
Devon lays square patterns of tiles. Pattern 1 is a single tile. Pattern 2 is a 2 by 2 square. Pattern 3 is a 3 by 3 square. How many tiles are in pattern 6, and how many more tiles does pattern 6 need than pattern 5?
Each pattern is a square array, so the number of tiles is the side multiplied by itself.
So the patterns hold 1 × 1 = 1 tile, 2 × 2 = 4, 3 × 3 = 9, 4 × 4 = 16, 5 × 5 = 25 and 6 × 6 = 36.
Pattern 6 has 36 tiles. Pattern 5 has 25, so pattern 6 needs 36 − 25 = 11 more tiles.
Look at the differences: 3, 5, 7, 9, 11. They are the odd numbers in order, which is a neat check — the difference after 9 had to be 11, and it is.
Common mistakes and how to avoid them
Getting the unit of repeat wrong. Reading circle, square, triangle, circle as a unit of 4. Fix: the unit is the smallest group that repeats. Stop as soon as the first shape comes round again.
Treating a remainder of 0 as the first shape. Fix: remainder 0 means you landed exactly on the end of the unit, so it is the last shape.
Continuing a pattern by guessing what looks nice. Fix: state the rule in words first — "add 3" or "quarter turn clockwise" — then apply it.
Adding the wrong difference. Fix: work the difference out twice, first to second and second to third. If they disagree, it is not a simple "add the same number" pattern, so look at the differences of the differences.
Forgetting the extra stick. Answering 30 for ten matchstick squares. Fix: test any shortcut on a small case you already know before trusting it.
Saying that circles tessellate. Fix: circles leave gaps wherever three of them meet. Squares, rectangles, equilateral triangles and regular hexagons tessellate; circles and regular pentagons do not.
Mixing up flip and turn. Fix: a flip produces a mirror image; a turn does not. Ask whether the shape has been mirrored.
How parents can help at home
Patterns are everywhere in a Trinidadian home, and spotting them out loud is most of the practice needed.
Find the repeat. Floor tiles, a shirt print, a wire fence — ask "what is the smallest bit that keeps coming back?" That is the unit of repeat, and naming it is the whole skill.
Make a bottle-cap line. Lay out a repeating line of objects, then ask what the 15th or 20th would be without laying that many out. That needs a division and a remainder, which is exactly the exam item.
Build with toothpicks. Make one square, then two, then three, counting the sticks each time and writing the numbers down. Then ask for ten squares. Building it makes the "add 3" rule obvious.
Talk about the tiles. Ask why floors are tiled with squares and not circles. "The circles would leave gaps" is a complete tessellation answer.
When your child is stuck, the prompt is "say the rule out loud". Almost every mistake here comes from continuing a pattern by eye rather than by rule. For a growing pattern, follow up with "write the numbers in a row and find the difference between each pair."
You need not work anything out yourself. Ask them to check by going one step backwards: if the rule is add 3, taking 3 from their answer should give the term before.
Exam technique for patterns with shapes
Decide first whether the pattern repeats or grows. Repeating patterns need the unit length and a division; growing patterns need the difference.
For a repeating pattern, count the unit of repeat and write that number down before dividing.
Divide, and read the remainder carefully. Remainder 0 means the last shape in the unit.
For a growing pattern, write the terms in a row with the differences underneath. If the differences match, the rule is "add that number". If not, look at how the differences themselves change.
Check any shortcut rule on a term you were already given. A rule that works for term 2 and term 3 is almost certainly right.
State rules in words when you are asked to describe a pattern. "Add 3 each time" and "a quarter turn clockwise" are full answers.
Watch the word more. "How many more tiles" wants a subtraction, not the total.
Quick revision summary
- A repeating pattern is built from a unit of repeat; find the smallest group that comes round again.
- To find the shape at a given position, divide the position by the length of the unit and use the remainder.
- Remainder 1 is the first shape in the unit; remainder 0 is the last.
- A growing pattern adds the same amount each step. Find the difference between consecutive terms.
- Matchstick squares in a row: 3 sticks per square, plus 1, so 10 squares take 3 × 10 + 1 = 31.
- Square arrays give 1, 4, 9, 16, 25, 36; their differences are the odd numbers 3, 5, 7, 9, 11.
- Triangular dot patterns give 1, 3, 6, 10, 15, with differences 2, 3, 4, 5.
- Squares, rectangles, equilateral triangles and regular hexagons tessellate. Circles and regular pentagons do not.
- Slide keeps the shape facing the same way; flip makes a mirror image; turn rotates it.
- Always say the rule in words before you continue the pattern.