GCSE Maths: Patterns and diagrams (Foundation)
Part of the GCSE Maths sequences and nth term chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Patterns and diagrams
Many sequences come from patterns of shapes, made with matchsticks, tiles or dots. Each pattern in the list uses a few more than the one before. To turn a pattern into a sequence of numbers there are four steps:
- Count how many matchsticks (or tiles, or dots) are in each pattern.
- Write the counts as a list. That list is a sequence.
- Find the rule that takes you from one count to the next.
- Use the rule to carry on and predict later patterns.
Here are the first four patterns made from matchsticks. Count the sticks: 4, 7, 10, 13. The rule is “add 3”.
It is very important to see why the rule works by looking at the picture. Each new square is stuck on the end of the row. It shares one side with the square before it, so it needs only 3 new matchsticks, not 4. That is why the rule is “add 3”. If you can explain the rule from the picture, you can trust it.
Sometimes the number you add changes. Look at these patterns of dots. The first has 1, 3, 6, 10 dots; the differences are 2, 3, 4, so each pattern adds one more dot than the last pattern added. These are called the triangular numbers. The second pattern of dots has 1, 4, 9, 16 dots. The differences are 3, 5, 7. These are the square numbers, because the dots make a square.
You can carry on a rule for a pattern 5th, 6th, 10th and so on. When you want a very large pattern, such as pattern 100, continuing one step at a time takes far too long. In sections 11.4 and 11.5 you will learn to write a formula (the nth term) that jumps straight to any pattern.
Key ideas
- To turn a pattern into a sequence: count, write the list, find the rule, carry on.
- Explain the rule with the picture: new shapes often share a side, so they add fewer than a whole shape.
- If the numbers added go up by the same amount each time, the sequence is not “add the same number”: look at the differences between differences.
- Triangular numbers: 1, 3, 6, 10, …. Square numbers: 1, 4, 9, 16, …
How many matchsticks are needed for pattern 6?
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Count the items in each pattern to make a sequence, find the term-to-term rule, then continue the sequence.
- Count the matchsticks in each pattern shown. Count carefully, one pattern at a time. This turns the pictures into a sequence of numbers.
Pattern 1 2 3 Number of matchsticks 4 7 10 - Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.7 − 4 = 3
10 − 7 = 3 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 3, so the rule is add 3.
- Use the rule to carry on until you reach the pattern you want. Each pattern follows from the one before, so keep applying the rule.Pattern 4: 10 + 3 = 13
Pattern 5: 13 + 3 = 16
Pattern 6: 16 + 3 = 19 - Write the answer. Pattern 6 has 19 matchsticks.
Check. Look at the drawing: The first square needs 4 matchsticks. Each new square is added on the end and shares one side with the square before it, so it needs only 3 new matchsticks. After n squares there are 3 matchsticks for each square, plus 1 extra for the very first side: 3n + 1.
Watch out. Pattern 10 does not have 10 times as many matchsticks as pattern 1. The pattern grows by adding, so check the rule against the counts you already have.
How many matchsticks are needed for pattern 8?
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Count the items in each pattern to make a sequence, find the term-to-term rule, then continue the sequence.
- Count the matchsticks in each pattern shown. Count carefully, one pattern at a time. This turns the pictures into a sequence of numbers.
Pattern 1 2 3 Number of matchsticks 3 5 7 - Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.5 − 3 = 2
7 − 5 = 2 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 2, so the rule is add 2.
- Use the rule to carry on until you reach the pattern you want. Each pattern follows from the one before, so keep applying the rule.Pattern 4: 7 + 2 = 9
Pattern 5: 9 + 2 = 11
Pattern 6: 11 + 2 = 13
Pattern 7: 13 + 2 = 15
Pattern 8: 15 + 2 = 17 - Write the answer. Pattern 8 has 17 matchsticks.
Check. Look at the drawing: The first triangle needs 3 matchsticks. Each new triangle shares a side with the last one, so it needs only 2 new matchsticks. After n triangles there are 2 matchsticks for each triangle, plus 1 extra for the first one: 2n + 1.
Watch out. Pattern 10 does not have 10 times as many matchsticks as pattern 1. The pattern grows by adding, so check the rule against the counts you already have.
How many tiles are in pattern 6?
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Count the items in each pattern to make a sequence, find the term-to-term rule, then continue the sequence.
- Count the tiles in each pattern shown. Count carefully, one pattern at a time. This turns the pictures into a sequence of numbers.
Pattern 1 2 3 Number of tiles 5 9 13 - Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.9 − 5 = 4
13 − 9 = 4 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 4, so the rule is add 4.
- Use the rule to carry on until you reach the pattern you want. Each pattern follows from the one before, so keep applying the rule.Pattern 4: 13 + 4 = 17
Pattern 5: 17 + 4 = 21
Pattern 6: 21 + 4 = 25 - Write the answer. Pattern 6 has 25 tiles.
Check. Look at the drawing: There is always 1 tile in the middle. Then there is an arm going up, down, left and right, and each arm has n tiles. So the number of tiles is 4 arms of n tiles plus the 1 in the middle: 4n + 1.
Watch out. Pattern 10 does not have 10 times as many tiles as pattern 1. The pattern grows by adding, so check the rule against the counts you already have.
How many dots are in pattern 7?
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Count the items in each pattern to make a sequence, find the term-to-term rule, then continue the sequence.
- Count the dots in each pattern shown. Count carefully, one pattern at a time. This turns the pictures into a sequence of numbers.
Pattern 1 2 3 4 Number of dots 1 3 6 10 - Try the differences first. We always look for "add or subtract the same number" before anything else.3 − 1 = 2
6 − 3 = 3
10 − 6 = 4
The differences are not the same. - Look at how the differences change. Sometimes the amount added grows by the same amount each time.2, 3, 4
The differences go up by 1 each time. - Write the rule. The differences increase by 1 each time: add 2, then add 3, then add 4, and so on.
- Use the rule to carry on until you reach the pattern you want. Each pattern follows from the one before, so keep applying the rule.Pattern 5: 10 + 5 = 15
Pattern 6: 15 + 6 = 21
Pattern 7: 21 + 7 = 28 - Write the answer. Pattern 7 has 28 dots.
Check. Look at the drawing: Pattern n has rows of 1, 2, 3, … up to n dots, so the number of dots is 1 + 2 + 3 + … + n. Two copies make a rectangle with n rows and n + 1 columns, so one copy has n(n + 1) ÷ 2 dots.
Watch out. Pattern 10 does not have 10 times as many dots as pattern 1. The pattern grows by adding, so check the rule against the counts you already have.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Make a sequence from the counts, then keep adding the common difference to see if the number is reached.
- Count the matchsticks in each pattern shown. Turn the pictures into a sequence of numbers.
Pattern 1 2 3 Number of matchsticks 4 7 10 - Find the term-to-term rule. Subtract each count from the next.7 − 4 = 3
10 − 7 = 3
Rule: add 3 - Carry on adding until you reach or pass the number. The counts keep going up, so stop as soon as the count is 31 or bigger.P1: 4, P2: 7, P3: 10, P4: 13, P5: 16, P6: 19, P7: 22, P8: 25, P9: 28, P10: 31
- Decide. 31 appears in the list.Yes: pattern 10 has exactly 31 matchsticks.
Check. Pattern 10 follows from 4 by adding 3 a total of 9 times: 4 + 9 × 3 = 31. ✓
Watch out. Stop when you reach or pass the number. If the counts jump over it, there is no such pattern.
Watch out
Do not just multiply. Pattern 3 of the squares uses 10 matchsticks, but pattern 6 does not use 20: it uses 19. The patterns grow by adding the same amount, so doubling the pattern number does not double the count.Exercise 11B Patterns and diagrams
Non-calculatorKey words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Count the items in each pattern to make a sequence, find the term-to-term rule, then continue the sequence.
- Count the matchsticks in each pattern shown. Count carefully, one pattern at a time. This turns the pictures into a sequence of numbers.
Pattern 1 2 3 Number of matchsticks 4 7 10 - Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.7 − 4 = 3
10 − 7 = 3 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 3, so the rule is add 3.
- Use the rule to carry on until you reach the pattern you want. Each pattern follows from the one before, so keep applying the rule.Pattern 4: 10 + 3 = 13
- Write the answer. Pattern 4 has 13 matchsticks.
Check. Look at the drawing: The first square needs 4 matchsticks. Each new square is added on the end and shares one side with the square before it, so it needs only 3 new matchsticks. After n squares there are 3 matchsticks for each square, plus 1 extra for the very first side: 3n + 1.
Watch out. Pattern 10 does not have 10 times as many matchsticks as pattern 1. The pattern grows by adding, so check the rule against the counts you already have.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Count the items in each pattern to make a sequence, find the term-to-term rule, then continue the sequence.
- Count the matchsticks in each pattern shown. Count carefully, one pattern at a time. This turns the pictures into a sequence of numbers.
Pattern 1 2 3 Number of matchsticks 4 7 10 - Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.7 − 4 = 3
10 − 7 = 3 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 3, so the rule is add 3.
- Use the rule to carry on until you reach the pattern you want. Each pattern follows from the one before, so keep applying the rule.Pattern 4: 10 + 3 = 13
Pattern 5: 13 + 3 = 16 - Write the answer. Pattern 5 has 16 matchsticks.
Check. Look at the drawing: The first square needs 4 matchsticks. Each new square is added on the end and shares one side with the square before it, so it needs only 3 new matchsticks. After n squares there are 3 matchsticks for each square, plus 1 extra for the very first side: 3n + 1.
Watch out. Pattern 10 does not have 10 times as many matchsticks as pattern 1. The pattern grows by adding, so check the rule against the counts you already have.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Count the items in each pattern to make a sequence, find the term-to-term rule, then continue the sequence.
- Count the matchsticks in each pattern shown. Count carefully, one pattern at a time. This turns the pictures into a sequence of numbers.
Pattern 1 2 3 Number of matchsticks 4 7 10 - Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.7 − 4 = 3
10 − 7 = 3 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 3, so the rule is add 3.
- Use the rule to carry on until you reach the pattern you want. Each pattern follows from the one before, so keep applying the rule.Pattern 4: 10 + 3 = 13
Pattern 5: 13 + 3 = 16
Pattern 6: 16 + 3 = 19
Pattern 7: 19 + 3 = 22
Pattern 8: 22 + 3 = 25 - Write the answer. Pattern 8 has 25 matchsticks.
Check. Look at the drawing: The first square needs 4 matchsticks. Each new square is added on the end and shares one side with the square before it, so it needs only 3 new matchsticks. After n squares there are 3 matchsticks for each square, plus 1 extra for the very first side: 3n + 1.
Watch out. Pattern 10 does not have 10 times as many matchsticks as pattern 1. The pattern grows by adding, so check the rule against the counts you already have.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Count the items in each pattern to make a sequence, find the term-to-term rule, then continue the sequence.
- Count the matchsticks in each pattern shown. Count carefully, one pattern at a time. This turns the pictures into a sequence of numbers.
Pattern 1 2 3 Number of matchsticks 3 5 7 - Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.5 − 3 = 2
7 − 5 = 2 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 2, so the rule is add 2.
- Use the rule to carry on until you reach the pattern you want. Each pattern follows from the one before, so keep applying the rule.Pattern 4: 7 + 2 = 9
- Write the answer. Pattern 4 has 9 matchsticks.
Check. Look at the drawing: The first triangle needs 3 matchsticks. Each new triangle shares a side with the last one, so it needs only 2 new matchsticks. After n triangles there are 2 matchsticks for each triangle, plus 1 extra for the first one: 2n + 1.
Watch out. Pattern 10 does not have 10 times as many matchsticks as pattern 1. The pattern grows by adding, so check the rule against the counts you already have.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Count the items in each pattern to make a sequence, find the term-to-term rule, then continue the sequence.
- Count the matchsticks in each pattern shown. Count carefully, one pattern at a time. This turns the pictures into a sequence of numbers.
Pattern 1 2 3 Number of matchsticks 3 5 7 - Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.5 − 3 = 2
7 − 5 = 2 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 2, so the rule is add 2.
- Use the rule to carry on until you reach the pattern you want. Each pattern follows from the one before, so keep applying the rule.Pattern 4: 7 + 2 = 9
Pattern 5: 9 + 2 = 11 - Write the answer. Pattern 5 has 11 matchsticks.
Check. Look at the drawing: The first triangle needs 3 matchsticks. Each new triangle shares a side with the last one, so it needs only 2 new matchsticks. After n triangles there are 2 matchsticks for each triangle, plus 1 extra for the first one: 2n + 1.
Watch out. Pattern 10 does not have 10 times as many matchsticks as pattern 1. The pattern grows by adding, so check the rule against the counts you already have.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Count the items in each pattern to make a sequence, find the term-to-term rule, then continue the sequence.
- Count the matchsticks in each pattern shown. Count carefully, one pattern at a time. This turns the pictures into a sequence of numbers.
Pattern 1 2 3 Number of matchsticks 3 5 7 - Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.5 − 3 = 2
7 − 5 = 2 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 2, so the rule is add 2.
- Use the rule to carry on until you reach the pattern you want. Each pattern follows from the one before, so keep applying the rule.Pattern 4: 7 + 2 = 9
Pattern 5: 9 + 2 = 11
Pattern 6: 11 + 2 = 13
Pattern 7: 13 + 2 = 15
Pattern 8: 15 + 2 = 17
Pattern 9: 17 + 2 = 19 - Write the answer. Pattern 9 has 19 matchsticks.
Check. Look at the drawing: The first triangle needs 3 matchsticks. Each new triangle shares a side with the last one, so it needs only 2 new matchsticks. After n triangles there are 2 matchsticks for each triangle, plus 1 extra for the first one: 2n + 1.
Watch out. Pattern 10 does not have 10 times as many matchsticks as pattern 1. The pattern grows by adding, so check the rule against the counts you already have.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Count the items in each pattern to make a sequence, find the term-to-term rule, then continue the sequence.
- Count the tiles in each pattern shown. Count carefully, one pattern at a time. This turns the pictures into a sequence of numbers.
Pattern 1 2 3 Number of tiles 5 9 13 - Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.9 − 5 = 4
13 − 9 = 4 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 4, so the rule is add 4.
- Use the rule to carry on until you reach the pattern you want. Each pattern follows from the one before, so keep applying the rule.Pattern 4: 13 + 4 = 17
- Write the answer. Pattern 4 has 17 tiles.
Check. Look at the drawing: There is always 1 tile in the middle. Then there is an arm going up, down, left and right, and each arm has n tiles. So the number of tiles is 4 arms of n tiles plus the 1 in the middle: 4n + 1.
Watch out. Pattern 10 does not have 10 times as many tiles as pattern 1. The pattern grows by adding, so check the rule against the counts you already have.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Count the items in each pattern to make a sequence, find the term-to-term rule, then continue the sequence.
- Count the tiles in each pattern shown. Count carefully, one pattern at a time. This turns the pictures into a sequence of numbers.
Pattern 1 2 3 Number of tiles 5 9 13 - Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.9 − 5 = 4
13 − 9 = 4 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 4, so the rule is add 4.
- Use the rule to carry on until you reach the pattern you want. Each pattern follows from the one before, so keep applying the rule.Pattern 4: 13 + 4 = 17
Pattern 5: 17 + 4 = 21 - Write the answer. Pattern 5 has 21 tiles.
Check. Look at the drawing: There is always 1 tile in the middle. Then there is an arm going up, down, left and right, and each arm has n tiles. So the number of tiles is 4 arms of n tiles plus the 1 in the middle: 4n + 1.
Watch out. Pattern 10 does not have 10 times as many tiles as pattern 1. The pattern grows by adding, so check the rule against the counts you already have.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Count the items in each pattern to make a sequence, find the term-to-term rule, then continue the sequence.
- Count the tiles in each pattern shown. Count carefully, one pattern at a time. This turns the pictures into a sequence of numbers.
Pattern 1 2 3 Number of tiles 5 9 13 - Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.9 − 5 = 4
13 − 9 = 4 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 4, so the rule is add 4.
- Use the rule to carry on until you reach the pattern you want. Each pattern follows from the one before, so keep applying the rule.Pattern 4: 13 + 4 = 17
Pattern 5: 17 + 4 = 21
Pattern 6: 21 + 4 = 25
Pattern 7: 25 + 4 = 29 - Write the answer. Pattern 7 has 29 tiles.
Check. Look at the drawing: There is always 1 tile in the middle. Then there is an arm going up, down, left and right, and each arm has n tiles. So the number of tiles is 4 arms of n tiles plus the 1 in the middle: 4n + 1.
Watch out. Pattern 10 does not have 10 times as many tiles as pattern 1. The pattern grows by adding, so check the rule against the counts you already have.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Find the common difference using two terms you know, then use it to fill the gaps.
- Find the common difference from two terms you know. The terms in positions 1 and 2 are known. There are 1 step between them, and each step adds the same amount.7 − 4 = 3
Rule: add 3 - Fill in the gaps. Add the common difference to go forwards. Subtract it to go backwards.Term 4 = term 3 + 3 = 10 + 3 = 13
Term 5 = term 4 + 3 = 13 + 3 = 16 - Write the complete sequence. 4, 7, 10, 13, 16
Check. Look at the whole list: the difference between neighbours is 3 every time. ✓
Watch out. If a gap is at the start, work backwards by doing the opposite of the rule.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Find the common difference using two terms you know, then use it to fill the gaps.
- Find the common difference from two terms you know. The terms in positions 1 and 2 are known. There are 1 step between them, and each step adds the same amount.11 − 6 = 5
Rule: add 5 - Fill in the gaps. Add the common difference to go forwards. Subtract it to go backwards.Term 4 = term 3 + 5 = 16 + 5 = 21
Term 5 = term 4 + 5 = 21 + 5 = 26 - Write the complete sequence. 6, 11, 16, 21, 26
Check. Look at the whole list: the difference between neighbours is 5 every time. ✓
Watch out. If a gap is at the start, work backwards by doing the opposite of the rule.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Count the items in each pattern to make a sequence, find the term-to-term rule, then continue the sequence.
- Count the matchsticks in each pattern shown. Count carefully, one pattern at a time. This turns the pictures into a sequence of numbers.
Pattern 1 2 3 Number of matchsticks 6 11 16 - Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.11 − 6 = 5
16 − 11 = 5 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 5, so the rule is add 5.
- Use the rule to carry on until you reach the pattern you want. Each pattern follows from the one before, so keep applying the rule.Pattern 4: 16 + 5 = 21
- Write the answer. Pattern 4 has 21 matchsticks.
Check. Look at the drawing: The first hexagon needs 6 matchsticks. Each new hexagon shares one side with the last, so it needs only 5 new matchsticks. After n hexagons there are 5 for each hexagon, plus 1 extra: 5n + 1.
Watch out. Pattern 10 does not have 10 times as many matchsticks as pattern 1. The pattern grows by adding, so check the rule against the counts you already have.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Count the items in each pattern to make a sequence, find the term-to-term rule, then continue the sequence.
- Count the matchsticks in each pattern shown. Count carefully, one pattern at a time. This turns the pictures into a sequence of numbers.
Pattern 1 2 3 Number of matchsticks 6 11 16 - Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.11 − 6 = 5
16 − 11 = 5 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 5, so the rule is add 5.
- Use the rule to carry on until you reach the pattern you want. Each pattern follows from the one before, so keep applying the rule.Pattern 4: 16 + 5 = 21
Pattern 5: 21 + 5 = 26
Pattern 6: 26 + 5 = 31 - Write the answer. Pattern 6 has 31 matchsticks.
Check. Look at the drawing: The first hexagon needs 6 matchsticks. Each new hexagon shares one side with the last, so it needs only 5 new matchsticks. After n hexagons there are 5 for each hexagon, plus 1 extra: 5n + 1.
Watch out. Pattern 10 does not have 10 times as many matchsticks as pattern 1. The pattern grows by adding, so check the rule against the counts you already have.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Count the items in each pattern to make a sequence, find the term-to-term rule, then continue the sequence.
- Count the matchsticks in each pattern shown. Count carefully, one pattern at a time. This turns the pictures into a sequence of numbers.
Pattern 1 2 3 Number of matchsticks 6 11 16 - Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.11 − 6 = 5
16 − 11 = 5 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 5, so the rule is add 5.
- Use the rule to carry on until you reach the pattern you want. Each pattern follows from the one before, so keep applying the rule.Pattern 4: 16 + 5 = 21
Pattern 5: 21 + 5 = 26
Pattern 6: 26 + 5 = 31
Pattern 7: 31 + 5 = 36
Pattern 8: 36 + 5 = 41 - Write the answer. Pattern 8 has 41 matchsticks.
Check. Look at the drawing: The first hexagon needs 6 matchsticks. Each new hexagon shares one side with the last, so it needs only 5 new matchsticks. After n hexagons there are 5 for each hexagon, plus 1 extra: 5n + 1.
Watch out. Pattern 10 does not have 10 times as many matchsticks as pattern 1. The pattern grows by adding, so check the rule against the counts you already have.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Count the items in each pattern to make a sequence, find the term-to-term rule, then continue the sequence.
- Count the orange tiles in each pattern shown. Count carefully, one pattern at a time. This turns the pictures into a sequence of numbers.
Pattern 1 2 3 Number of orange tiles 8 12 16 - Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.12 − 8 = 4
16 − 12 = 4 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 4, so the rule is add 4.
- Use the rule to carry on until you reach the pattern you want. Each pattern follows from the one before, so keep applying the rule.Pattern 4: 16 + 4 = 20
- Write the answer. Pattern 4 has 20 orange tiles.
Check. Look at the drawing: The blue square in the middle is n tiles by n tiles. The orange border has n tiles along each of the 4 sides, plus 1 tile in each of the 4 corners: 4n + 4.
Watch out. Pattern 10 does not have 10 times as many orange tiles as pattern 1. The pattern grows by adding, so check the rule against the counts you already have.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Count the items in each pattern to make a sequence, find the term-to-term rule, then continue the sequence.
- Count the orange tiles in each pattern shown. Count carefully, one pattern at a time. This turns the pictures into a sequence of numbers.
Pattern 1 2 3 Number of orange tiles 8 12 16 - Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.12 − 8 = 4
16 − 12 = 4 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 4, so the rule is add 4.
- Use the rule to carry on until you reach the pattern you want. Each pattern follows from the one before, so keep applying the rule.Pattern 4: 16 + 4 = 20
Pattern 5: 20 + 4 = 24
Pattern 6: 24 + 4 = 28 - Write the answer. Pattern 6 has 28 orange tiles.
Check. Look at the drawing: The blue square in the middle is n tiles by n tiles. The orange border has n tiles along each of the 4 sides, plus 1 tile in each of the 4 corners: 4n + 4.
Watch out. Pattern 10 does not have 10 times as many orange tiles as pattern 1. The pattern grows by adding, so check the rule against the counts you already have.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Count the items in each pattern to make a sequence, find the term-to-term rule, then continue the sequence.
- Count the orange tiles in each pattern shown. Count carefully, one pattern at a time. This turns the pictures into a sequence of numbers.
Pattern 1 2 3 Number of orange tiles 8 12 16 - Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.12 − 8 = 4
16 − 12 = 4 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 4, so the rule is add 4.
- Use the rule to carry on until you reach the pattern you want. Each pattern follows from the one before, so keep applying the rule.Pattern 4: 16 + 4 = 20
Pattern 5: 20 + 4 = 24
Pattern 6: 24 + 4 = 28
Pattern 7: 28 + 4 = 32
Pattern 8: 32 + 4 = 36
Pattern 9: 36 + 4 = 40
Pattern 10: 40 + 4 = 44 - Write the answer. Pattern 10 has 44 orange tiles.
Check. Look at the drawing: The blue square in the middle is n tiles by n tiles. The orange border has n tiles along each of the 4 sides, plus 1 tile in each of the 4 corners: 4n + 4.
Watch out. Pattern 10 does not have 10 times as many orange tiles as pattern 1. The pattern grows by adding, so check the rule against the counts you already have.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Count the items in each pattern to make a sequence, find the term-to-term rule, then continue the sequence.
- Count the dots in each pattern shown. Count carefully, one pattern at a time. This turns the pictures into a sequence of numbers.
Pattern 1 2 3 4 Number of dots 1 4 9 16 - Try the differences first. We always look for "add or subtract the same number" before anything else.4 − 1 = 3
9 − 4 = 5
16 − 9 = 7
The differences are not the same. - Look at how the differences change. Sometimes the amount added grows by the same amount each time.3, 5, 7
The differences go up by 2 each time. - Write the rule. The differences increase by 2 each time: add 3, then add 5, then add 7, and so on.
- Use the rule to carry on until you reach the pattern you want. Each pattern follows from the one before, so keep applying the rule.Pattern 5: 16 + 9 = 25
- Write the answer. Pattern 5 has 25 dots.
Check. Look at the drawing: Pattern n is a square of dots with n rows and n columns, so it has n × n = n² dots.
Watch out. Pattern 10 does not have 10 times as many dots as pattern 1. The pattern grows by adding, so check the rule against the counts you already have.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Count the items in each pattern to make a sequence, find the term-to-term rule, then continue the sequence.
- Count the dots in each pattern shown. Count carefully, one pattern at a time. This turns the pictures into a sequence of numbers.
Pattern 1 2 3 4 Number of dots 1 4 9 16 - Try the differences first. We always look for "add or subtract the same number" before anything else.4 − 1 = 3
9 − 4 = 5
16 − 9 = 7
The differences are not the same. - Look at how the differences change. Sometimes the amount added grows by the same amount each time.3, 5, 7
The differences go up by 2 each time. - Write the rule. The differences increase by 2 each time: add 3, then add 5, then add 7, and so on.
- Use the rule to carry on until you reach the pattern you want. Each pattern follows from the one before, so keep applying the rule.Pattern 5: 16 + 9 = 25
Pattern 6: 25 + 11 = 36 - Write the answer. Pattern 6 has 36 dots.
Check. Look at the drawing: Pattern n is a square of dots with n rows and n columns, so it has n × n = n² dots.
Watch out. Pattern 10 does not have 10 times as many dots as pattern 1. The pattern grows by adding, so check the rule against the counts you already have.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Count the items in each pattern to make a sequence, find the term-to-term rule, then continue the sequence.
- Count the dots in each pattern shown. Count carefully, one pattern at a time. This turns the pictures into a sequence of numbers.
Pattern 1 2 3 4 Number of dots 1 4 9 16 - Try the differences first. We always look for "add or subtract the same number" before anything else.4 − 1 = 3
9 − 4 = 5
16 − 9 = 7
The differences are not the same. - Look at how the differences change. Sometimes the amount added grows by the same amount each time.3, 5, 7
The differences go up by 2 each time. - Write the rule. The differences increase by 2 each time: add 3, then add 5, then add 7, and so on.
- Use the rule to carry on until you reach the pattern you want. Each pattern follows from the one before, so keep applying the rule.Pattern 5: 16 + 9 = 25
Pattern 6: 25 + 11 = 36
Pattern 7: 36 + 13 = 49
Pattern 8: 49 + 15 = 64 - Write the answer. Pattern 8 has 64 dots.
Check. Look at the drawing: Pattern n is a square of dots with n rows and n columns, so it has n × n = n² dots.
Watch out. Pattern 10 does not have 10 times as many dots as pattern 1. The pattern grows by adding, so check the rule against the counts you already have.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Count the items in each pattern to make a sequence, find the term-to-term rule, then continue the sequence.
- Count the dots in each pattern shown. Count carefully, one pattern at a time. This turns the pictures into a sequence of numbers.
Pattern 1 2 3 4 Number of dots 1 3 6 10 - Try the differences first. We always look for "add or subtract the same number" before anything else.3 − 1 = 2
6 − 3 = 3
10 − 6 = 4
The differences are not the same. - Look at how the differences change. Sometimes the amount added grows by the same amount each time.2, 3, 4
The differences go up by 1 each time. - Write the rule. The differences increase by 1 each time: add 2, then add 3, then add 4, and so on.
- Use the rule to carry on until you reach the pattern you want. Each pattern follows from the one before, so keep applying the rule.Pattern 5: 10 + 5 = 15
- Write the answer. Pattern 5 has 15 dots.
Check. Look at the drawing: Pattern n has rows of 1, 2, 3, … up to n dots, so the number of dots is 1 + 2 + 3 + … + n. Two copies make a rectangle with n rows and n + 1 columns, so one copy has n(n + 1) ÷ 2 dots.
Watch out. Pattern 10 does not have 10 times as many dots as pattern 1. The pattern grows by adding, so check the rule against the counts you already have.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Count the items in each pattern to make a sequence, find the term-to-term rule, then continue the sequence.
- Count the dots in each pattern shown. Count carefully, one pattern at a time. This turns the pictures into a sequence of numbers.
Pattern 1 2 3 4 Number of dots 1 3 6 10 - Try the differences first. We always look for "add or subtract the same number" before anything else.3 − 1 = 2
6 − 3 = 3
10 − 6 = 4
The differences are not the same. - Look at how the differences change. Sometimes the amount added grows by the same amount each time.2, 3, 4
The differences go up by 1 each time. - Write the rule. The differences increase by 1 each time: add 2, then add 3, then add 4, and so on.
- Use the rule to carry on until you reach the pattern you want. Each pattern follows from the one before, so keep applying the rule.Pattern 5: 10 + 5 = 15
Pattern 6: 15 + 6 = 21 - Write the answer. Pattern 6 has 21 dots.
Check. Look at the drawing: Pattern n has rows of 1, 2, 3, … up to n dots, so the number of dots is 1 + 2 + 3 + … + n. Two copies make a rectangle with n rows and n + 1 columns, so one copy has n(n + 1) ÷ 2 dots.
Watch out. Pattern 10 does not have 10 times as many dots as pattern 1. The pattern grows by adding, so check the rule against the counts you already have.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Count the items in each pattern to make a sequence, find the term-to-term rule, then continue the sequence.
- Count the dots in each pattern shown. Count carefully, one pattern at a time. This turns the pictures into a sequence of numbers.
Pattern 1 2 3 4 Number of dots 1 3 6 10 - Try the differences first. We always look for "add or subtract the same number" before anything else.3 − 1 = 2
6 − 3 = 3
10 − 6 = 4
The differences are not the same. - Look at how the differences change. Sometimes the amount added grows by the same amount each time.2, 3, 4
The differences go up by 1 each time. - Write the rule. The differences increase by 1 each time: add 2, then add 3, then add 4, and so on.
- Use the rule to carry on until you reach the pattern you want. Each pattern follows from the one before, so keep applying the rule.Pattern 5: 10 + 5 = 15
Pattern 6: 15 + 6 = 21
Pattern 7: 21 + 7 = 28
Pattern 8: 28 + 8 = 36 - Write the answer. Pattern 8 has 36 dots.
Check. Look at the drawing: Pattern n has rows of 1, 2, 3, … up to n dots, so the number of dots is 1 + 2 + 3 + … + n. Two copies make a rectangle with n rows and n + 1 columns, so one copy has n(n + 1) ÷ 2 dots.
Watch out. Pattern 10 does not have 10 times as many dots as pattern 1. The pattern grows by adding, so check the rule against the counts you already have.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Count the items in each pattern to make a sequence, find the term-to-term rule, then continue the sequence.
- Count the matchsticks in each pattern shown. Count carefully, one pattern at a time. This turns the pictures into a sequence of numbers.
Pattern 1 2 3 Number of matchsticks 7 12 17 - Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.12 − 7 = 5
17 − 12 = 5 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 5, so the rule is add 5.
- Use the rule to carry on until you reach the pattern you want. Each pattern follows from the one before, so keep applying the rule.Pattern 4: 17 + 5 = 22
- Write the answer. Pattern 4 has 22 matchsticks.
Check. Look at the drawing: The first pattern is two squares on top of each other and needs 7 matchsticks. Each new pair of squares adds 2 vertical matchsticks and 3 horizontal ones, which is 5 new matchsticks. After n pairs there are 5n matchsticks plus 2 extra for the first vertical sides: 5n + 2.
Watch out. Pattern 10 does not have 10 times as many matchsticks as pattern 1. The pattern grows by adding, so check the rule against the counts you already have.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Count the items in each pattern to make a sequence, find the term-to-term rule, then continue the sequence.
- Count the matchsticks in each pattern shown. Count carefully, one pattern at a time. This turns the pictures into a sequence of numbers.
Pattern 1 2 3 Number of matchsticks 7 12 17 - Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.12 − 7 = 5
17 − 12 = 5 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 5, so the rule is add 5.
- Use the rule to carry on until you reach the pattern you want. Each pattern follows from the one before, so keep applying the rule.Pattern 4: 17 + 5 = 22
Pattern 5: 22 + 5 = 27 - Write the answer. Pattern 5 has 27 matchsticks.
Check. Look at the drawing: The first pattern is two squares on top of each other and needs 7 matchsticks. Each new pair of squares adds 2 vertical matchsticks and 3 horizontal ones, which is 5 new matchsticks. After n pairs there are 5n matchsticks plus 2 extra for the first vertical sides: 5n + 2.
Watch out. Pattern 10 does not have 10 times as many matchsticks as pattern 1. The pattern grows by adding, so check the rule against the counts you already have.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Count the items in each pattern to make a sequence, find the term-to-term rule, then continue the sequence.
- Count the matchsticks in each pattern shown. Count carefully, one pattern at a time. This turns the pictures into a sequence of numbers.
Pattern 1 2 3 Number of matchsticks 7 12 17 - Find the difference between each pair of neighbouring terms. Subtract the earlier term from the later term. A sequence that goes up has positive differences; one that goes down has negative differences.12 − 7 = 5
17 − 12 = 5 - Look at the differences. They are all the same, so the same number is added each time (a negative difference means we subtract).Every difference is 5, so the rule is add 5.
- Use the rule to carry on until you reach the pattern you want. Each pattern follows from the one before, so keep applying the rule.Pattern 4: 17 + 5 = 22
Pattern 5: 22 + 5 = 27
Pattern 6: 27 + 5 = 32
Pattern 7: 32 + 5 = 37 - Write the answer. Pattern 7 has 37 matchsticks.
Check. Look at the drawing: The first pattern is two squares on top of each other and needs 7 matchsticks. Each new pair of squares adds 2 vertical matchsticks and 3 horizontal ones, which is 5 new matchsticks. After n pairs there are 5n matchsticks plus 2 extra for the first vertical sides: 5n + 2.
Watch out. Pattern 10 does not have 10 times as many matchsticks as pattern 1. The pattern grows by adding, so check the rule against the counts you already have.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Make a sequence from the counts, then keep adding the common difference to see if the number is reached.
- Count the matchsticks in each pattern shown. Turn the pictures into a sequence of numbers.
Pattern 1 2 3 Number of matchsticks 4 7 10 - Find the term-to-term rule. Subtract each count from the next.7 − 4 = 3
10 − 7 = 3
Rule: add 3 - Carry on adding until you reach or pass the number. The counts keep going up, so stop as soon as the count is 31 or bigger.P1: 4, P2: 7, P3: 10, P4: 13, P5: 16, P6: 19, P7: 22, P8: 25, P9: 28, P10: 31
- Decide. 31 appears in the list.Yes: pattern 10 has exactly 31 matchsticks.
Check. Pattern 10 follows from 4 by adding 3 a total of 9 times: 4 + 9 × 3 = 31. ✓
Watch out. Stop when you reach or pass the number. If the counts jump over it, there is no such pattern.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Make a sequence from the counts, then keep adding the common difference to see if the number is reached.
- Count the matchsticks in each pattern shown. Turn the pictures into a sequence of numbers.
Pattern 1 2 3 Number of matchsticks 4 7 10 - Find the term-to-term rule. Subtract each count from the next.7 − 4 = 3
10 − 7 = 3
Rule: add 3 - Carry on adding until you reach or pass the number. The counts keep going up, so stop as soon as the count is 40 or bigger.P1: 4, P2: 7, P3: 10, P4: 13, P5: 16, P6: 19, P7: 22, P8: 25, P9: 28, P10: 31, P11: 34, P12: 37, P13: 40
- Decide. 40 appears in the list.Yes: pattern 13 has exactly 40 matchsticks.
Check. Pattern 13 follows from 4 by adding 3 a total of 12 times: 4 + 12 × 3 = 40. ✓
Watch out. Stop when you reach or pass the number. If the counts jump over it, there is no such pattern.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Make a sequence from the counts, then keep adding the common difference to see if the number is reached.
- Count the matchsticks in each pattern shown. Turn the pictures into a sequence of numbers.
Pattern 1 2 3 Number of matchsticks 4 7 10 - Find the term-to-term rule. Subtract each count from the next.7 − 4 = 3
10 − 7 = 3
Rule: add 3 - Carry on adding until you reach or pass the number. The counts keep going up, so stop as soon as the count is 50 or bigger.P1: 4, P2: 7, P3: 10, P4: 13, P5: 16, P6: 19, P7: 22, P8: 25, P9: 28, P10: 31, P11: 34, P12: 37, P13: 40, P14: 43, P15: 46, P16: 49, P17: 52
- Decide. The counts jump from 49 to 52 and miss 50.No: there is no pattern with exactly 50 matchsticks.
Check. Every count leaves remainder 1 when divided by 3, but 50 leaves remainder 2. ✓
Watch out. Stop when you reach or pass the number. If the counts jump over it, there is no such pattern.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Make a sequence from the counts, then keep adding the common difference to see if the number is reached.
- Count the matchsticks in each pattern shown. Turn the pictures into a sequence of numbers.
Pattern 1 2 3 Number of matchsticks 3 5 7 - Find the term-to-term rule. Subtract each count from the next.5 − 3 = 2
7 − 5 = 2
Rule: add 2 - Carry on adding until you reach or pass the number. The counts keep going up, so stop as soon as the count is 21 or bigger.P1: 3, P2: 5, P3: 7, P4: 9, P5: 11, P6: 13, P7: 15, P8: 17, P9: 19, P10: 21
- Decide. 21 appears in the list.Yes: pattern 10 has exactly 21 matchsticks.
Check. Pattern 10 follows from 3 by adding 2 a total of 9 times: 3 + 9 × 2 = 21. ✓
Watch out. Stop when you reach or pass the number. If the counts jump over it, there is no such pattern.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. Make a sequence from the counts, then keep adding the common difference to see if the number is reached.
- Count the matchsticks in each pattern shown. Turn the pictures into a sequence of numbers.
Pattern 1 2 3 Number of matchsticks 3 5 7 - Find the term-to-term rule. Subtract each count from the next.5 − 3 = 2
7 − 5 = 2
Rule: add 2 - Carry on adding until you reach or pass the number. The counts keep going up, so stop as soon as the count is 30 or bigger.P1: 3, P2: 5, P3: 7, P4: 9, P5: 11, P6: 13, P7: 15, P8: 17, P9: 19, P10: 21, P11: 23, P12: 25, P13: 27, P14: 29, P15: 31
- Decide. The counts jump from 29 to 31 and miss 30.No: there is no pattern with exactly 30 matchsticks.
Check. Every count leaves remainder 1 when divided by 2, but 30 leaves remainder 0. ✓
Watch out. Stop when you reach or pass the number. If the counts jump over it, there is no such pattern.
Key words. A sequence is a list of numbers that follow a rule. Each number is a term. The term-to-term rule tells you how to get from one term to the next.
Method. A term-to-term rule tells you what to do to one term to get the next. Start with the first term and apply the rule again and again.
- Write down the first term. The first term is given, so it is the start of the list.Term 1 = 6
- Apply the rule "add 4" to find each new term. Each term comes from the term before it. Use the answer you have just found as the next starting point.Term 2: 6 + 4 = 10
Term 3: 10 + 4 = 14
Term 4: 14 + 4 = 18
Term 5: 18 + 4 = 22
Term 6: 22 + 4 = 26
Term 7: 26 + 4 = 30
Term 8: 30 + 4 = 34 - Pick out term number 8. The 8th term is 34.
Check. Jump straight to the last term: 6 + 7 × 4 = 34. ✓
Watch out. Do the operations in the order the rule gives them. "Multiply by 2 then add 1" is not the same as "add 1 then multiply by 2".
Method. Find the real number of matchsticks in pattern 10 and compare it with 40.
- Write the counts for the patterns we can see. Count the sticks in patterns 1, 2 and 3.4, 7, 10
- Find the rule. The differences are 3 each time, because each new square shares a side.7 − 4 = 3, 10 − 7 = 3: add 3
- Continue to pattern 10. Add 3 for each new pattern.4, 7, 10, 13, 16, 19, 22, 25, 28, 31 (the 10th number)
- Compare. Rob multiplied, but the pattern is made by adding.Pattern 10 uses 31 matchsticks, not 40. Rob is not right.
Check. Pattern 1 plus nine lots of 3: 4 + 9 × 3 = 31. ✓
Watch out. Multiplying the pattern number only works if the count is a multiple of the pattern number. Here there is a +1 for the first side.
Method. Find the rule, then look at the last digits of the numbers in the sequence.
- Find the rule. Subtract each number from the next.9 − 4 = 5, 14 − 9 = 5, 19 − 14 = 5: add 5
- Write more terms. Add 5 each time.4, 9, 14, 19, 24, 29, 34, 39, 44, …
- Look at the last digits. Adding 5 changes the last digit from 4 to 9 and then from 9 back to 4.The last digit is always 4 or 9, and never 0.
- Answer. 100 ends in 0.There is no pattern with exactly 100 matchsticks: Mia is not right.
Check. The numbers just either side of 100 would be 99 and 104. 99 = 4 + 19 × 5 is a term and 104 is a term; 100 is skipped. ✓
Watch out. Looking at the last digits is a quick way to rule out a number.
Method. Use the way the differences grow.
- Write the first counts and differences. Subtract each count from the next.1, 3, 6, 10, 15, 21
Differences: 2, 3, 4, 5, 6 - Spot the pattern. The difference grows by 1 each time. The difference from pattern 1 to 2 is 2, from 2 to 3 is 3, so from pattern 6 to 7 it is 7.Pattern 7 needs 7 more dots than pattern 6.
- Check with the picture. Pattern 7 has a new row of 7 dots on the end.21 + 7 = 28 dots in pattern 7.
Check. 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28 and 28 − 21 = 7. ✓
Watch out. The new row always has as many dots as the pattern number.
Method. Find the number of matchsticks for the 9th pattern of each sequence, then subtract.
- Sam: write the sequence. Squares use 4, 7, 10, … matchsticks: add 3 each time.4, 7, 10, 13, 16, 19, 22, 25, 28 (the 9th number)
- Tara: write the sequence. Triangles use 3, 5, 7, … matchsticks: add 2 each time.3, 5, 7, 9, 11, 13, 15, 17, 19 (the 9th number)
- Compare. Subtract the smaller from the larger.28 − 19 = 9
Check. Sam has 28 and Tara has 19: 19 + 9 = 28. ✓
Watch out. Make sure you take the 9th term of each list.