Lesson Notes By Weeks and Term v5 - Grade 7
Patterns, sequences and relationships – Week 9 focus
Download the Lessonotes Mobile South Africa app for faster lesson access on Android and iPhone.
Patterns, sequences and relationships – Week 9 focus
Download the Lessonotes Mobile South Africa app for faster lesson access on Android and iPhone.
Subject: Mathematics
Class: Grade 7
Term: 2nd Term
Week: 9
Theme: General lesson support
This page supports the lesson note with a companion video and a short classroom-ready summary.
For class groups and homework, share this lesson page so learners also get the summary, objectives, and full lesson context.
Patterns, sequences, and relationships are fundamental to mathematics and are found everywhere in the world around us. Understanding these concepts allows us to predict future events, solve problems efficiently, and develop critical thinking skills. In South Africa, patterns can be seen in traditional art forms like Ndebele house paintings, in the arrangement of houses in a township, and even in financial planning when saving money for university. This week, we will focus on identifying, describing, and extending number patterns, geometric patterns, and relationships represented in tables and flow diagrams. This builds a foundation for algebra and more advanced mathematics.
2.1 Number Patterns A number pattern (or sequence) is an ordered list of numbers. Each number in the pattern is called a term. Patterns are formed based on a rule that describes how to get from one term to the next. These rules involve addition, subtraction, multiplication, division, or a combination of these operations.
Types of Number Patterns: Arithmetic Sequences: These sequences have a constant difference between consecutive terms (addition or subtraction).
Geometric Sequences: These sequences have a constant ratio between consecutive terms (multiplication or division).
Other Patterns: Some patterns may involve squares, cubes, or other mathematical operations.
Identifying the Rule: Look for the relationship between consecutive terms.
Ask yourself: "What operation (or operations) do I need to perform on one term to get the next term?" Example 1 (Arithmetic Sequence): Consider the sequence: 3, 7, 11, 15, ...
Find the difference: 7 - 3 = 4, 11 - 7 = 4, 15 - 11 =
4. The difference between consecutive terms is
4. State the rule: The rule is "Add 4 to the previous term." Find the next term: To find the next term, add 4 to 15: 15 + 4 =
1
9. So, the next term is
1
9. Example 2 (Geometric Sequence): Consider the sequence: 2, 6, 18, 54, ...
Find the ratio: 6 / 2 = 3, 18 / 6 = 3, 54 / 18 =
3. The ratio between consecutive terms is
3. State the rule: The rule is "Multiply the previous term by 3." Find the next term: To find the next term, multiply 54 by 3: 54 * 3 =
1
6
2. So, the next term is
1
6
2. Example 3 (Pattern Involving Squares): Consider the sequence: 1, 4, 9, 16, ...
Recognize the terms: These are the squares of consecutive whole numbers: 1² = 1, 2² = 4, 3² = 9, 4² =
1
6. State the rule: The rule is "The nth term is n squared" (or n²).
Find the next term: The next term is 5² = 25. 2.2 Relationships in Tables and Flow Diagrams A relationship shows how two variables are connected. A variable is a quantity that can change. We can represent these relationships using tables or flow diagrams. These are also useful in representing functions.
Tables: A table shows corresponding values for two variables. One variable is usually considered the input (or x), and the other is the output (or y).
Flow Diagrams: A flow diagram shows the input, the rule that transforms the input, and the resulting output.
Example 4 (Table): Consider the following table: | Input (x) | Output (y) | | --------- | ---------- | | 1 | 3 | | 2 | 5 | | 3 | 7 | | 4 | 9 | Find the relationship: Notice that each output is two more than twice the input: (2 1) + 1 = 3, (2 2) + 1 = 5, (2 3) + 1 = 7, (2 4) + 1 =
9. State the rule: The rule is "Multiply the input by 2 and add 1" or y = 2x +
1. Example 5 (Flow Diagram): Input --> x --> Multiply by 5 --> Add 2 --> Output y Interpret the flow diagram: This diagram shows that to get the output, you must first multiply the input by 5 and then add
2. Write the rule: The rule can be expressed as y = 5x +
2. Finding Inputs and Outputs: If the input x=3, then y = (5*3) + 2 =
1
7. If the output is y=22, then to find x, we solve: 22 = 5x+2, meaning 20=5x, and x=
4. Example 6 (South African Context - Cellphone Data): Consider a cellphone contract where you pay R50 per month plus R1 per gigabyte (GB) of data used.
Table Representation: | Data Used (GB) (x) | Monthly Cost (R) (y) | | --------------------- | ---------------------- | | 0 | 50 | | 1 | 51 | | 2 | 52 | | 5 | 55 | Flow Diagram Representation: Input (Data Used in GB) --> x --> Multiply by 1 --> Add 50 --> Output (Monthly Cost in R) y Rule: The rule can be expressed as y = x +
5
0. Guided Practice (With Solutions)
Question 1: Find the next two terms in the sequence: 5, 10, 15, 20, ...
Solution: Identify the pattern: The difference between consecutive terms is 5 (10-5=5, 15-10=5, 20-15=5).
State the rule: The rule is "Add 5 to the previous term." Extend the pattern: 20 + 5 = 25, 25 + 5 =
3
0. Answer: The next two terms are 25 and
3
0. Question 2: A table shows the relationship between the number of workers (x) and the number of houses built in a week (y). Determine the rule that relates x and y. | Workers (x) | Houses Built (y) | | ----------- | ---------------- | | 2 | 4 | | 3 | 6 | | 4 | 8 | | 5 | 10 | Solution: Analyze the relationship: Notice that the number of houses built is always twice the number of workers.
State the rule: The rule is "Multiply the number of workers by 2" or y = 2x.
Question 3: Complete the following flow diagram: Input (x) --> x --> Multiply by 3 --> Subtract 1 --> Output (y) If x = 4, what is the value of y?
Solution: Apply the rule: The rule is to multiply the input by 3 and then subtract
1. Calculate the output: If x = 4, then y = (3 * 4) - 1 = 12 - 1 =
1
1. Answer: The value of y is
1
1. Question 4: Determine the rule for the following number pattern: 1, 8, 27, 64...
Solution: Recognize the pattern: These are cubes of consecutive whole numbers: 1³ = 1, 2³ = 8, 3³ = 27, 4³ =
6
4. State the rule: The rule is "The nth term is n cubed" (or n³).