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Spatial and non-verbal sequences
Spotting the rule that makes a visual pattern grow.
Some sequences are made of pictures instead of numbers: matchsticks, tiles, dots or shapes. In this lesson you will learn how to:
- turn a picture pattern into a number sequence by counting
- spot the new part that each pattern adds
- see the rule in the picture, so you can find Pattern 20 or Pattern 100 without drawing it
- work with patterns that grow faster and faster
- find a shape far along a repeating pattern
- spot what changes in a sequence of shapes, and what comes next
1. Picture patterns
Example: matchstick squares
| Pattern | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Matchsticks | 4 | 7 | 10 | 13 |
| New each time | +3 | +3 | +3 |
Each new pattern adds one more square. That takes 3 more matchsticks (the red ones), so the rule is add 3. Pattern 4 has 10 + 3 = 13 matchsticks.
Tables and chairs
Each new table brings 2 more chairs: 4, 6, 8. So 4 tables seat 8 + 2 = 10.
Matchstick triangles
Each new triangle needs 2 more matchsticks: 3, 5, 7. Pattern 4 has 9.
Grow the pattern
- Pick a pattern. Look at how it grows, then say how many pieces the next pattern will have.
- If you are right, the pattern grows and the new part turns red. Keep going up to Pattern 6.
2. Seeing the rule in the picture
Example: how many matchsticks in Pattern 20?
- Look at Pattern 4. There is 1 red matchstick on the left, then a group of 3 for each square:1 + 3 + 3 + 3 + 3 = 13
- So the rule is:3 times the pattern number, then add 1
- Pattern 20 has 20 squares:3 × 20 = 60, 60 + 1 = 61
- Check it with a pattern you can count:Pattern 3 is 3 × 3 + 1 = 10
This is the times-table trick from Lesson 5.1. The picture shows you why it works: the 3 times table counts the groups, and the 1 is the extra matchstick.
A garden path
Every grey tile has a white tile above it and below it: 2 for each grey tile. Each end has 3 more, so that is 6 extra. The rule is 2 times the pattern number, then add 6. Pattern 10 has 2 × 10 + 6 = 26 white tiles.
Backwards: which pattern uses 46 matchsticks?
Undo the rule. Take away the extra: 46 − 1 = 45. Then share the rest into groups of 3: 45 ÷ 3 = 15. It is Pattern 15.
Pattern detective
- Step 1: find what each pattern adds. Step 2: pick the rule in words. Step 3: use it.
- Two with no slips moves you up a level. Level 1 finds a pattern up to Pattern 12, level 2 goes as far as Pattern 100, and level 3 works backwards: which pattern uses this many pieces?
3. Patterns that grow faster
Example: squares of tiles
| Pattern | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Tiles | 1 | 4 | 9 | 16 | 25 |
| New each time | +3 | +5 | +7 | +9 |
Each new square adds an L shape of tiles (red). The L shapes have 3, 5, 7 and 9 tiles, going up by 2 each time. Pattern 5 has 16 + 9 = 25 tiles.
There is a quicker way too. Pattern 5 is 5 rows of 5, and 5 × 5 = 25. These are the square numbers from Lesson 1.7. Pattern 10 has 10 × 10 = 100 tiles.
Triangles of dots
Each new row is one dot longer: add 2, then 3, then 4. Pattern 5 adds a row of 5: 10 + 5 = 15. The numbers 1, 3, 6, 10, 15 are called triangle numbers.
Rectangles of tiles
Each pattern is one row more and one column more. Pattern 3 is 3 rows of 4. So Pattern 10 is 10 rows of 11: 10 × 11 = 110 tiles.
Build the next one
- The blue part is the pattern you have. Tap the empty spaces to build the next pattern, then press Check. Tap a red one again to take it away.
- Then count it. Watch how the new part grows each time.
4. Repeating patterns
Example: what is the 20th shape?
- The block is circle, triangle, square:3 shapes long
- Divide the position by 3 (Lesson 2.3):20 ÷ 3 = 6 remainder 2
- 6 whole blocks take you to the 18th shape. Then 2 more:the 2nd shape in the block
Remainder 0
21 ÷ 3 = 7 remainder 0. The 21st shape is the end of the 7th block, so it is the last shape in the block: a square.
How many triangles in the first 20?
Each block has 1 triangle. 6 whole blocks give 6 triangles. The last 2 shapes are a circle and a triangle: 1 more. That makes 7.
Repeat detective
- Step 1: tap the last shape of the first block. Step 2: divide. Step 3: answer the question.
- Two with no slips moves you up a level. Level 1 has short blocks, level 2 has longer blocks that use a shape twice, and level 3 asks how many of one shape there are.
5. Shapes that turn and change
Example: what comes next?
- What changes? The arrow turns each time:a quarter turn clockwise
- After left, a quarter turn clockwise points:up
- Four quarter turns make a whole turn, so it repeats every 4. The 22nd arrow?22 ÷ 4 = 5 remainder 2: it points right
The shaded part moves round
The shaded quarter moves clockwise round the square, one quarter each time. Next it is back at the top left.
One more side each time
3 sides, 4 sides, 5 sides: next is a shape with 6 sides, a hexagon.
Two things change at once
One more side each time, and the shapes take turns: dark, white, dark. So the next shape has 6 sides and is white: a white hexagon. Check both changes before you pick an answer.
What comes next?
- Work out what changes, then tap the picture that fits the gap.
- Two with no slips moves you up a level. Level 1 has one change, level 2 has eighth turns, anticlockwise turns and gaps in the middle, and level 3 has two changes at once or a picture far along.
6. Practice: pick your level
Start with Foundation if you are in Year 4 or just starting. Move up when you are getting most of them right.
The mistakes that cost marks
- Counting a shared side twice. Saying two matchstick squares in a row use 8 matchsticks. the side in the middle is one matchstick for both squares: 7.
- Using the first pattern as the step. Saying the squares go up by 4 because Pattern 1 has 4. the step is what each new pattern adds: 7 − 4 = 3.
- Doubling to get a pattern twice as far along. Saying Pattern 20 has 2 × 31 = 62 matchsticks. use the rule for Pattern 20 itself: 3 × 20 + 1 = 61.
- Forgetting the extra part. Saying Pattern 20 has 3 × 20 = 60 matchsticks. add the extra matchstick on the left: 61.
- Expecting every pattern to add the same amount. Saying the square after 1, 4, 9 has 12 tiles. the new parts grow: 3, 5, then 7. So it is 9 + 7 = 16.
- Forgetting what remainder 0 means. Saying the 21st shape of circle, triangle, square is a circle. 21 ÷ 3 = 7 remainder 0, so it is the last shape of a block: a square.
- Mixing up clockwise and anticlockwise. Turning an arrow from up to left when the sequence turns clockwise. clockwise goes the way the hands of a clock go: up, right, down, left.
- Only checking one change. Picking a hexagon of the wrong colour when the shapes also take turns dark and white. check every thing that changes before you choose.
Quick checklist
Tick each one when you can do it without help.
All ticked. Brilliant work! You have finished Lesson 5.2.
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