Lesson Notes By Weeks and Term v5 - Grade 10

Lesson Video

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Performance objectives

• Identify and describe linear number patterns.
• Determine the general rule (nth term) for a linear pattern.
• Use the general rule to find any specific term.
• Solve problems involving linear patterns in real-world contexts.

Lesson summary

Number patterns are sequences of numbers that follow a specific rule or relationship. Understanding number patterns is crucial for developing strong mathematical reasoning and problem-solving skills. This week, we will delve deeper into linear number patterns and arithmetic sequences, focusing on finding the nth term and using it to solve various problems. This is essential not just for mathematics class but also for making informed decisions in everyday life, such as budgeting, understanding data trends, and making predictions. For example, understanding growth patterns can help one understand interest rates on loans, or predict the yield of crops based on consistent inputs.

Lesson notes

A number pattern is a sequence of numbers that follow a specific rule. A linear number pattern, also known as an arithmetic sequence, is a sequence where the difference between consecutive terms is constant. This constant difference is called the common difference (d).

Terms: The individual numbers in a sequence are called terms. We denote them as T1, T2, T3, ..., Tn, where Tn represents the nth term.

General Rule (nth Term): The general rule, or the nth term (Tn), allows us to find any term in the sequence without having to list all the terms before it. The general rule for a linear number pattern is given by: Tn = a + (n - 1)d Where: Tn is the nth term a is the first term (T1) n is the term number (e.g., 1, 2, 3, ...) d is the common difference Finding the Common Difference (d): The common difference is found by subtracting any term from the term that follows it: d = T2 - T1 d = T3 - T2 d = Tn - Tn-1

Worked example

Example 1: Finding the nth term

Consider the number pattern: 5, 8, 11, 14, ...

Identify the first term (a): a = 5

Calculate the common difference (d): d = 8 - 5 = 3

Substitute the values of 'a' and 'd' into the general rule:

Tn = a + (n - 1)d

Tn = 5 + (n - 1)3

Tn = 5 + 3n - 3

Tn = 3n + 2

Therefore, the general rule for this number pattern is Tn = 3n +

2. Example 2: Finding a specific term