Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Unit 1, order of operations. What is 4 + 3 × 5?
Answer
19. Multiply first: 3 × 5 = 15, then 4 + 15 = 19. - From Unit 1, powers. What is 2³?
Answer
8. Three factors of 2: 2 × 2 × 2. - From Unit 1, factors and multiples. List the first four multiples of 6.
Answer
6, 12, 18, 24. Each is 6 more than the one before.
Words used in this lesson
- pattern
- numbers or shapes that follow a rule. For example: 4, 7, 10, 13.
- term
- one number in a pattern. For example: 7 is the 2nd term of 4, 7, 10.
- sequence
- a list of terms in order. For example: 5, 9, 13, 17.
- core
- the shortest part of a repeating pattern. For example: dot, dot, loop.
- linear
- the same amount is added or subtracted each time. For example: 75, 68, 61.
- gap
- the difference between neighbouring terms. For example: the gaps in 2, 5, 8 are 3 and 3.
Four ways to say one pattern
Bamboo frames for seedling trays are built in a row, and every frame after the first shares a side with its neighbour. One frame needs 4 canes. Two frames need 7, not 8. That small saving is the pattern. A pattern can be shown as a diagram, as a table or sequence of numbers, in words, and later as a rule with a letter. Each way shows the same thing.
Predict first: one frame needs 4 canes and two need 7. Without drawing, how many canes would five frames need? Then slide the number of squares and check.
3 squares in a row need 10 canes: 4 for the first and 3 more for each of the other 2.
Show me the steps
- 1 square needs 4 canes, all four sides.
- 2 squares need 7 canes: the second square shares a side, so it needs 3 new canes, 4 + 3 = 7.
- 3 squares need 10 canes, 7 + 3 = 10. The table reads 4, 7, 10: every step is 3.
- 5 squares need 16 canes: 4 for the first and 3 more for each of the other 4, 4 + 12 = 16.
- 8 squares need 25 canes: 4 + 7 × 3 = 25.
A pattern is a list of numbers or shapes made by a rule. Each number is a term. Ask two questions of any pattern: what stays the same, and what grows? In the frames, the 3 new canes stay the same and the total grows.
A pattern is linear when the same amount is added or subtracted every time, such as 75, 68, 61, … going down by 7. A repeating pattern has a core, the shortest part that repeats. In dot, dot, loop, dot, dot, loop, … the core is dot, dot, loop, so every third symbol is a loop.
Is the growth the same every time?
Numbers that keep growing do not all grow the same way. Look at the gaps between neighbours: if the gap is the same every time, the pattern is linear. Choose a pattern and compare.
5, 9, 13, 17, 21, 25: the gap is +4 every time, so the pattern is linear.
Show me the steps
- Pattern 1, start 5 and add 4: 5, 9, 13, 17, 21, 25. Every gap is +4, so it is linear.
- Pattern 2, start 30 and subtract 3: 30, 27, 24, 21, 18, 15. Every gap is −3. It goes down by the same amount, so it is linear.
- Pattern 3, start 2 and double: 2, 4, 8, 16, 32, 64. The gaps are +2, +4, +8, +16, +32, which are not the same, so it is not linear.
- Pattern 4, the squares: 1, 4, 9, 16, 25, 36. The gaps are +3, +5, +7, +9, +11. It grows, but it is not linear.
- Pattern 5: 1, 2, 4, 7, 11, 16. The gaps are +1, +2, +3, +4, +5. The gaps grow by 1 each time, and the pattern is not linear.
- Pattern 6, start 4 and add 6: 4, 10, 16, 22, 28, 34. Every gap is +6, so it is linear.
A common mistake: “It keeps getting bigger, so it is linear.” The doubling pattern 2, 4, 8, 16 keeps getting bigger, but the gaps are 2, 4, 8, which are not the same. Linear means the same gap every time, whether it goes up or down.
Worked example: the 20th term
How many canes make a row of 20 frames? Drawing 20 frames would take a long time. Use the pattern.
- The first frame needs 4 canes.
- Each of the other 19 frames adds 3 canes, so they add 19 × 3 = 57.
- Total: 4 + 57 = 61 canes. Check the rule on a small case: 3 frames is 4 + 2 × 3 = 10, which matches the table.
Your go. The first step is given. A tiled border starts with 5 tiles and every row after it has 4 more. How many tiles are in the 12th row?
- The first row has 5 tiles, and the 12th row is 11 rows after it.
- Now work out how many tiles those 11 rows add, then the total, yourself.
Check the answer
11 × 4 = 44 tiles added, and 5 + 44 = 49 tiles in the 12th row. The step is added 11 times, not 12, because the first row already has its tiles.
Check before you go on
1. Which description matches the number pattern 7, 11, 15, 19, …?
Your turn
Start at Level 1 and go as far as you can. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. A strip of tiles is pinned to a wall. The first tile needs 5 pins, and each extra tile in the row needs 4 more. How many pins make 7 tiles?
3. A border repeats the core loop, dot, dot (3 symbols). How many dots are in the first 30 symbols?
Level 2 · Building
4. The pattern 40, 34, 28, 22, … goes down by the same amount each time. What is the 7th term?
5. The pattern 2, 5, 8, 11, … goes up by 3 each time. What is the first term greater than 40?
6. Which of these number patterns is NOT linear?
Level 3 · Stretch
7. Matchsticks make a row of triangles: 1 triangle uses 3 sticks, 2 triangles use 5 and 3 triangles use 7. How many sticks make 10 triangles?
8. The pattern 6, 11, 16, 21, … goes up by 5 each time. Which term number is 101?
Something to think about
What if the frames were hexagons instead of squares? One hexagon uses 6 canes, and each new hexagon shares one side with the one before it. How many canes would 3 hexagons need? Can you say how many n hexagons need?
One way to think about it
One hexagon: 6. Each new one adds 5 canes (6 sides, one shared). So 6, 11, 16 for 1, 2, 3 hexagons: 16 canes for 3. For n hexagons: 6 + 5 × (n − 1), which is 5n + 1. The same thinking as the squares, with a step of 5 instead of 3.
Exit ticket
9. Which is the best reason that 2, 4, 8, 16 is not a linear pattern?