Lesson Notes By Weeks and Term v5 - Grade 10

Lesson Video

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Performance objectives

• Identify and describe patterns in numerical and visual sequences.
• Represent relationships using tables, graphs, and equations.
• Interpret information from tables and graphs.
• Use patterns to make predictions and solve real-world problems.
• Distinguish between linear and non-linear relationships.

Lesson summary

In Mathematical Literacy, understanding patterns, relationships, and representations is crucial. This week, we will focus on how to identify, analyze, and represent relationships between different quantities. This skill is not just about numbers; it's about making sense of the world around us. From budgeting your monthly expenses to understanding how the price of bread changes over time, these skills are essential for informed decision-making in your daily lives. Understanding patterns allows us to predict future trends, which is vital for financial planning, understanding social issues, and even appreciating cultural traditions.

Lesson notes

2.1 What are Patterns? A pattern is a predictable sequence or regularity. Patterns can be found in numbers (numerical patterns), shapes (geometric patterns), or events that repeat. Identifying patterns allows us to predict what comes next and understand the underlying relationships.

Example 1: Numerical Pattern Consider the following sequence: 2, 4, 6, 8, ... What is the pattern? Each number is increasing by

2. What is the next number in the sequence?

1

0. We can represent this pattern with a simple rule: Add 2 to the previous number.

Example 2: Geometric Pattern Imagine a sequence of squares. The first square has a side length of 1 cm, the second has a side length of 2 cm, the third has a side length of 3 cm, and so on. What is the pattern? The side length increases by 1 cm each time.

We can calculate the area of each square: 1 cm 2 , 4 cm 2 , 9 cm 2 , ... The area follows a different pattern (squares of consecutive whole numbers). 2.2 Relationships Between Variables A variable is a quantity that can change. A relationship exists when a change in one variable affects another. We often see these relationships in everyday life. The price of electricity and your electricity bill, the amount of rainfall and the yield of crops, or the number of taxis on the road and traffic congestion are all examples of relationships between variables.

Types of Relationships: Linear Relationship: A relationship where the change between the variables is constant. When graphed, it forms a straight line. The equation is of the form y = mx + c, where 'm' is the constant rate of change (slope) and 'c' is the y-intercept.

Non-Linear Relationship: A relationship where the change between the variables is not constant. The graph is not a straight line; it can be a curve. Examples include exponential and quadratic relationships.

Example 3: Linear Relationship – Taxi Fare A taxi charges a fixed call-out fee of R20 and R10 per kilometer traveled. Let 'x' be the distance traveled (in kilometers) and 'y' be the total fare (in Rand). The relationship can be expressed as an equation: y = 10x +

2

0. This is a linear relationship because for every additional kilometer traveled, the fare increases by a constant R

1

0. If you travel 5 km, the fare is: y = 10(5) + 20 = R

7

0. Example 4: Non-Linear Relationship – Compound Interest Suppose you invest R1000 in a savings account that earns 5% interest per year, compounded annually. Let 'x' be the number of years and 'y' be the total amount in the account.

The relationship can be expressed as: y = 1000(1 + 0.05) x . This is a non-linear (exponential) relationship. The amount of interest earned each year increases as the balance grows. The graph will be a curve. 2.3 Representations: Tables, Graphs, and Equations We can represent relationships in three main ways: Tables: A table organizes data in rows and columns. It shows the corresponding values of the variables.

Graphs: A graph visually represents the relationship between variables. The most common type is a Cartesian graph (x-y plane).

Equations: An equation mathematically describes the relationship between variables.

Example 5: Representing the Taxi Fare (from Example 3) in different ways Table: | Distance (km) | Fare (R) | | :------------ | :------- | | 0 | 20 | | 1 | 30 | | 2 | 40 | | 3 | 50 | | 4 | 60 | Graph: (Imagine a straight line graph with Distance on the x-axis and Fare on the y-axis. The line starts at (0, 20) and goes up at a constant rate.)

Equation: y = 10x + 20 Understanding how to convert between these representations is crucial for analyzing and interpreting data. Guided Practice (With Solutions)

Question 1: The number of houses built in a new township increases each year according to the pattern: 5, 8, 11, 14, ... a) What is the pattern? b) How many houses will be built in the 6th year? c) Write a general rule (equation) for the number of houses built in the nth year.

Solution: a) The pattern is adding 3 to the previous number. b) Following the pattern, the next two numbers are 17 and

2

0. Therefore, in the 6th year, 20 houses will be built. c) Let 'n' be the year number and 'H' be the number of houses built. The equation is H = 3n + 2. (When n=1, H=5; when n=2, H=8, and so on. We can find the '3' by noticing the constant difference between the terms. The '+2' corrects the initial value).

Question 2: A fruit vendor buys apples for R5 each and sells them for R8 each. a) Create a table showing the relationship between the number of apples sold (x) and the profit made (y) for selling 0, 1, 2, 3, 4, and 5 apples. b) Write an equation representing this relationship. c) What is the profit if the vendor sells 20 apples?

Solution: a)

Table: | Apples Sold (x) | Profit (y) | | :--------------- | :--------- | | 0 | 0 | | 1 | 3 | | 2 | 6 | | 3 | 9 | | 4 | 12 | | 5 | 15 | b) The equation is y = 3x (Profit = (selling price – cost price) number of apples sold = (8-5) x = 3x). c) If the vendor sells 20 apples, the profit is y = 3(20) = R60.