Lesson Notes By Weeks and Term v5 - Grade 7
Patterns, sequences and relationships – Week 6 focus
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Patterns, sequences and relationships – Week 6 focus
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Subject: Mathematics
Class: Grade 7
Term: 2nd Term
Week: 6
Theme: General lesson support
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Patterns, sequences, and relationships are fundamental building blocks in mathematics. Understanding them allows us to predict, generalise, and solve problems effectively. This week, we'll delve deeper into recognizing, describing, and creating numerical and geometric patterns. This is not just about numbers; it's about understanding how things change and relate to each other in a predictable way. Think about the patterns in the traditional Ndebele art, the repeating structures in a honeycomb, or even the way stokvels (informal savings groups common in South Africa) calculate contributions and payouts – all these involve patterns and relationships!
2.1 Numerical Patterns A numerical pattern is a sequence of numbers that follows a specific rule. There are two main types we will focus on this week: Linear Patterns: These patterns have a constant difference between consecutive terms. This constant difference is crucial.
Non-linear Patterns: The difference between consecutive terms is not constant. 2.1.1 Linear Patterns - The Constant Difference The key to linear patterns is the common difference. To find the common difference, subtract any term from the term that follows it. If the difference is the same between any two consecutive terms, it’s a linear pattern.
Example 1: Identifying a linear Pattern Consider the sequence: 2, 5, 8, 11, 14... 5 - 2 = 3 8 - 5 = 3 11 - 8 = 3 14 - 11 = 3 Since the difference is always 3, this is a linear pattern with a common difference of
3. Example 2: Finding the next terms in a linear Pattern Continuing the sequence from Example 1 (2, 5, 8, 11, 14...), what are the next two terms? Since the common difference is 3, we add 3 to the last term to get the next term. 14 + 3 = 17 17 + 3 = 20 Therefore, the next two terms are 17 and
2
0. Example 3: Finding the Rule (General Term) of a Linear Pattern Let’s find the rule for the pattern: 4, 7, 10, 13, 16… First, find the common difference: 7 - 4 = 3 The general term for a linear pattern is often represented as: `Tn = an + b` (where Tn is the nth term, a is the common difference, n is the term number, and b is a constant). We know `a = 3`. So, `Tn = 3n + b` To find `b`, substitute the values from the first term (n = 1, T1 = 4) into the equation: `4 = 3(1) + b` Solve for `b`: `4 = 3 + b` => `b = 1` Therefore, the rule for the pattern is `Tn = 3n + 1`.
Example 4: Non-Linear Pattern Identification Consider the sequence: 1, 4, 9, 16, 25... 4 - 1 = 3 9 - 4 = 5 16 - 9 = 7 25 - 16 = 9 The differences are not constant (3, 5, 7, 9), so this is a non-linear pattern. (This is the sequence of square numbers: 1², 2², 3², 4², 5²...) We won't focus on explicitly finding rules for non-linear patterns this week, but recognizing them is important. 2.2 Geometric Patterns Geometric patterns involve shapes that are arranged in a specific order or sequence. These patterns can grow or shrink, and we often look for a rule that describes how the pattern changes from one step to the next.
Example 5: Geometric Growing Pattern Imagine a pattern made with matches: Step 1: One square (4 matches)
Step 2: Two squares in a row (7 matches)
Step 3: Three squares in a row (10 matches)
Describe the pattern: Each step adds one more square to the row.
Numerical representation: The number of matches forms the sequence: 4, 7, 10... This is a linear pattern with a common difference of
3. Rule: Similar to Example 3, the rule is `Tn = 3n + 1`. 2.3 Flow Diagrams and Tables Flow diagrams and tables are useful ways to represent the relationship between the term number (input) and the term value (output).
Example 6: Flow Diagram for the pattern Tn = 2n + 3 Input (n): 1 -> 2 -> 3 -> 4 -> ...
Rule: x 2 + 3 Output (Tn): 5 -> 7 -> 9 -> 11 -> ...
Example 7: Table for the pattern Tn = 5n - 1 | Term Number (n) | Term Value (Tn) | |-------------------|-------------------| | 1 | 4 | | 2 | 9 | | 3 | 14 | | 4 | 19 | | 5 | 24 | Guided Practice (With Solutions)
Question 1: Identify the type of pattern (linear or non-linear) and find the next two terms: 6, 11, 16, 21… Solution: Calculate the differences: 11-6 = 5, 16-11 = 5, 21-16 =
5. The difference is constant (5), so it is a linear pattern.
Next two terms: 21 + 5 = 26, 26 + 5 =
3
1. The next two terms are 26 and
3
1. Question 2: Find the rule for the linear pattern: 3, 8, 13, 18… Solution: Find the common difference: 8 - 3 =
5. So, `a = 5`.
General term: `Tn = 5n + b` Substitute the first term (n=1, T1 = 3): `3 = 5(1) + b` Solve for b: `3 = 5 + b` => `b = -2` The rule is: `Tn = 5n - 2`.
Question 3: Represent the pattern `Tn = 4n + 1` using a flow diagram for the first four terms.
Solution: Input (n): 1 -> 2 -> 3 -> 4 Rule: x 4 + 1 Output (Tn): 5 -> 9 -> 13 -> 17 Question 4: A pattern is formed using small stones. The first step has 3 stones, the second step has 5 stones, and the third step has 7 stones. How many stones will be in the 5th step?
Solution: The pattern is 3, 5, 7… Find the common difference: 5 - 3 =
2. The rule is `Tn = 2n + b`.
Substitute the first term: `3 = 2(1) + b` => `b = 1`. The rule is `Tn = 2n + 1`. To find the 5th term, substitute n = 5: `T5 = 2(5) + 1 = 10 + 1 = 11`. There will be 11 stones in the 5th step. Independent Practice (Questions Only) Identify the type of pattern (linear or non-linear): 1, 3, 6, 10, 15… Find the next two terms in the pattern: 9, 13, 17, 21… Determine the rule (general term) for the pattern: 2, 7, 12, 17… Represent the pattern `Tn = 3n - 2` using a table for the first five terms. A gardener plants roses in rows. The first row has 4 roses, the second row has 7 roses, and the third row has 10 roses. How many roses will be in the 8th row?