Lesson Notes By Weeks and Term v5 - Grade 7

Lesson Video

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Performance objectives

• Identify and extend numerical and geometric patterns.
• Determine the rule for a given pattern.
• Represent patterns using flow diagrams and tables.
• Use the rule to find any term in a sequence.
• Describe the relationship between variables in a pattern.

Lesson summary

Patterns, sequences, and relationships are fundamental to mathematics. They allow us to predict, generalize, and understand the world around us. From the arrangement of tiles in a Soweto township house to predicting population growth, understanding patterns is essential. In South Africa, where resource management and planning are crucial, recognizing and extending patterns becomes a vital skill. Consider stokvel contributions which operate on a defined sequence. Understanding sequences is useful for budgeting and financial planning. This week, we will focus on identifying, analyzing, and extending numerical and geometric patterns, as well as exploring relationships within sequences.

Lesson notes

2. 1.

Numerical Patterns: A numerical pattern is a sequence of numbers that follow a specific rule. This rule describes how to get from one number to the next. Numerical patterns can be increasing (adding or multiplying) or decreasing (subtracting or dividing).

Example 1: 2, 4, 6, 8, ...

Rule: Add 2 to the previous term. The next term is 10 (8 + 2).

Example 2: 1, 3, 9, 27, ...

Rule: Multiply the previous term by

3. The next term is 81 (27 x 3).

Example 3: 20, 17, 14, 11, ...

Rule: Subtract 3 from the previous term. The next term is 8 (11 - 3). 2.

2. Geometric Patterns: Geometric patterns are patterns that use shapes or figures to create a sequence. They often involve a change in size, shape, or position.

Example 1: Imagine squares. The first square has side length 1cm. The second has side length 2cm. The third has side length 3cm. What is the pattern in the area of the squares?

Area of first square: 1cm x 1cm = 1cm² Area of second square: 2cm x 2cm = 4cm² Area of third square: 3cm x 3cm = 9cm² Pattern: The area is increasing by the square of consecutive whole numbers (1², 2², 3²...). The area of the fourth square would be 4cm x 4cm = 16cm².

Example 2: A pattern is formed using matches to make triangles. 1 triangle needs 3 matches, 2 triangles sharing a side needs 5 matches, 3 triangles sharing a side needs 7 matches. How many matches are needed for 4 triangles sharing a side? Number of triangles | Number of Matches ----------------------|------------------ 1 | 3 2 | 5 3 | 7 4 | 9 Rule: You need 2 matches more for each consecutive triangle.

The pattern is: add 2 to the previous number of matches. For 4 triangles you need 7 + 2 = 9 matches. 2.

3. Rules and Relationships: The rule of a pattern describes the relationship between the term number and the term value. This rule can be expressed in words or symbols. Understanding relationships between variables is crucial to creating these rules.

Example: Consider the pattern 5, 10, 15, 20, ...

Term number (n): 1, 2, 3, 4, ...

Term value: 5, 10, 15, 20, ...

Rule in words: Multiply the term number by

5. Rule in symbols: Term value = 5 n 2.

4. Flow Diagrams and Tables: Flow diagrams and tables are useful tools for representing patterns and relationships.

Flow Diagram: A flow diagram shows the input (term number), the rule (operation), and the output (term value).

Example: Input (n) -> Rule (x 3) -> Output (Term value) 1 -> x 3 -> 3 2 -> x 3 -> 6 3 -> x 3 -> 9 4 -> x 3 -> 12 Table: A table organizes the term number and term value in columns.

Example: | Term Number (n) | Term Value | | --------------- | ---------- | | 1 | 3 | | 2 | 6 | | 3 | 9 | | 4 | 12 | 2.

5. Finding Any Term in a Sequence: Once we have the rule for a pattern, we can find any term in the sequence.

Example: The rule is: Term value = 4 n -

1. Find the 10th term. Substitute n = 10 into the rule: Term value = 4 10 - 1 Term value = 40 - 1 = 39 Therefore, the 10th term is

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9. Guided Practice (With Solutions)

Question 1: Identify the next two terms in the sequence: 3, 7, 11, 15, ...

Solution: The difference between consecutive terms is 4 (7-3 = 4, 11-7 = 4, 15-11 = 4).

The rule is: Add 4 to the previous term.

The next two terms are: 19 (15 + 4) and 23 (19 + 4).

Question 2: What is the rule for the sequence: 2, 6, 18, 54, ...? Express the rule in words.

Solution: Divide each term by the previous term: 6/2 = 3, 18/6 = 3, 54/18 = 3 The rule is: Multiply the previous term by

3. Question 3: Represent the pattern with the rule "Term value = 2n + 1" using a flow diagram for the first 4 terms.

Solution: Input (n) -> Rule (x 2 + 1) -> Output (Term value) 1 -> x 2 + 1 -> 3 2 -> x 2 + 1 -> 5 3 -> x 2 + 1 -> 7 4 -> x 2 + 1 -> 9 Question 4: A pattern is formed by adding beads to a string. The first string has 2 beads, the second has 5 beads, the third has 8 beads. How many beads will the 5th string have?

Solution: Analyze: 2, 5,

8. The difference between consecutive terms is

3. Rule: Add 3 to the previous term. 4th string: 8 + 3 = 11 beads 5th string: 11 + 3 = 14 beads.

Alternatively create a table and extend: String number | Number of beads --------------|---------------- 1 | 2 2 | 5 3 | 8 4 | 11 5 | 14 Question 5: Find the 20th term in the sequence defined by the rule: Term value = 3n - 2 Solution: Substitute n = 20 into the rule: Term value = 3 20 - 2 Term value = 60 - 2 = 58 Therefore, the 20th term is

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8. Independent Practice (Questions Only)

Find the next three terms in the sequence: 1, 4, 9, 16, ...

Describe the rule for the sequence: 100, 50, 25, 12.5, ... in words. Represent the pattern defined by "Term value = n² + 1" using a table for the first 5 terms. A taxi charges a base fare of R15 and then R8 per kilometer. Write a rule to calculate the total fare for any number of kilometers (k). What would be the fare for 12km? Find the 15th term in the sequence defined by the rule: Term value = 5n + 2 Consider the pattern: 1, 8, 27, 64... What is the general rule for this pattern? What is the 7th term?