Section A — Chapters 1–22
Number Patterns I
Junior Cycle — 1st Year
- ✓By the end of this lesson students will be able to identify and describe patterns in sequences of numbers.
- ✓By the end of this lesson students will be able to distinguish between linear and non-linear patterns.
- ✓By the end of this lesson students will be able to determine the next term in a given number pattern.
- ✓By the end of this lesson students will be able to relate number patterns to real-world situations.
- ✓By the end of this lesson students will be able to formulate a rule for simple linear and non-linear patterns.
Key concepts
A pattern is a predictable arrangement or sequence of numbers, shapes, or objects. It follows a rule that allows us to know what comes next.
A sequence is an ordered list of numbers. Each number in the sequence is called a term.
Each individual number in a sequence is referred to as a term. For example, in the sequence 2, 4, 6, 8, the number 2 is the first term, 4 is the second term, and so on.
A linear pattern (also known as an arithmetic sequence) is a sequence where the difference between any two consecutive terms is always the same. This constant difference is called the common difference. To find the next term, you simply add or subtract the common difference.
A non-linear pattern is a sequence where the difference between consecutive terms is not constant. The pattern might involve operations like multiplication, squaring, or other more complex rules, meaning the terms do not increase or decrease by the same amount each time.
Many situations in the real world can be described using number patterns. Understanding these patterns helps us to make predictions, solve problems, and understand how things change over time. Examples include daily savings, population growth, or the number of seats in an auditorium.
Key facts to remember
- 1A pattern is a predictable arrangement or sequence.
- 2A sequence is an ordered list of numbers, and each number in it is called a term.
- 3In a linear pattern, the difference between consecutive terms is always the same (this is called the common difference).
- 4In a non-linear pattern, the difference between consecutive terms is not constant.
- 5Identifying patterns helps us to predict future terms in a sequence.
- 6Many real-world situations can be modelled and understood using number patterns.
Worked examples
Example 1
Identify the pattern, find the next two terms, and state the rule for the sequence: 3, 7, 11, 15, ...
Answer
The next two terms are 19 and 23. The rule is 'Start with 3 and add 4 to get the next term'.
Always check the difference between all given terms to confirm it's a linear pattern.
Example 2
Identify the pattern, find the next two terms, and describe the rule for the sequence: 1, 4, 9, 16, ...
Answer
The next two terms are 25 and 36. The rule is 'The nth term is n squared' or 'The terms are the square numbers'.
When differences aren't constant, look for multiplication, squaring, or other operations.
Example 3
A plant is 5 cm tall when planted. It grows 2 cm each week. What will its height be after 4 weeks? Write out the sequence of heights for the first 5 weeks (starting from planting).
Answer
The plant will be 13 cm tall after 4 weeks. The sequence of heights for the first 5 weeks is 5, 7, 9, 11, 13.
Be careful to read if the question asks for height after a certain number of weeks, or for the sequence including the starting point.
Common mistakes
- ✗Only checking the difference between the first two terms and assuming the pattern is linear without checking all given terms.
- ✗Making calculation errors when finding the differences between terms or when calculating the next terms.
- ✗Struggling to clearly articulate the rule for a pattern in words.
- ✗Confusing the position of a term (e.g., 3rd term) with the actual value of the term.
Exam tips
- ★Always calculate the differences between *all* given consecutive terms to correctly identify if the pattern is linear or non-linear.
- ★If the first differences are not constant, look for other relationships such as multiplication, squaring, or even the differences of the differences.
- ★Clearly state the rule for the pattern in simple, understandable words.
- ★Show all your working steps, especially when finding the differences and calculating the next terms, as this can earn you partial marks.
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