Lesson Notes By Weeks and Term v5 - Grade 10

Lesson Video

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Performance objectives

• Identify patterns in tables of values.
• Interpret information presented in graphs.
• Represent relationships using equations (formulas).
• Translate between tables, graphs, and equations.
• Use these skills to solve real-world problems.

Lesson summary

This week, we delve deeper into Patterns, Relationships, and Representations. This is a crucial skill in Mathematical Literacy because it allows us to understand and make sense of the world around us. Whether it's analyzing cellphone data costs, understanding interest rates on a loan, or interpreting statistics about unemployment in South Africa, recognising and working with patterns and relationships is essential for making informed decisions. We will focus on interpreting different representations of relationships, including tables, graphs, and formulas, and on translating between these representations. These skills are vital for success in everyday life and future careers.

Lesson notes

2.1 Patterns and Relationships A pattern is a predictable sequence of events or quantities. Recognizing patterns allows us to make predictions and solve problems. A relationship describes how two or more things are connected or related to each other. We often express relationships mathematically using formulas or equations.

Types of Patterns: Linear Patterns: These patterns show a constant difference between consecutive terms.

The general form for a linear pattern is: `y = mx + c`, where `m` is the constant difference (slope) and `c` is the starting value (y-intercept).

Non-Linear Patterns: These patterns do not have a constant difference. Examples include quadratic patterns (e.g., `y = x^2`), exponential patterns (e.g., `y = 2^x`), and inverse patterns (e.g., `y = 1/x`). We will focus primarily on recognising these non-linear patterns visually, but may touch on simple examples to illustrate the concept. 2.2 Representations A representation is a way of showing or describing information.

Common representations include: Tables: Organized data in rows and columns.

Graphs: Visual representations of data, such as bar graphs, line graphs, scatter plots, and pie charts.

Equations (Formulas): Mathematical statements that express a relationship between variables.

Written Descriptions: Describing the relationships in words. 2.3 Interpreting Tables Tables display information in rows and columns. To interpret a table, look for patterns within the data and identify the relationship between the columns.

Example 1: The table below shows the cost of airtime bundles from a certain network provider: | Data (MB) | Cost (R) | |---|---| | 50 | 10 | | 100 | 20 | | 150 | 30 | | 200 | 40 | Analysis: Pattern: The cost increases by R10 for every 50 MB increase in data.

Relationship: Cost = (Data in MB / 50) 10 or Cost = Data in MB /

5. Equation: Cost = (1/5) Data 2.4 Interpreting Graphs Different types of graphs are used to represent different types of data.

Bar Graphs: Compare different categories.

Line Graphs: Show trends over time.

Scatter Plots: Show the relationship between two variables (correlation).

Pie Charts: Show proportions of a whole.

Example 2: The bar graph below shows the number of learners in each grade at a particular school. (Imagine a bar graph is displayed with Grade 8=200 learners, Grade 9=220 learners, Grade 10=180 learners, Grade 11=150 learners, Grade 12=130 learners).

Analysis: Grade 9 has the most learners (220). Grade 12 has the fewest learners (130). The number of learners generally decreases as the grade increases. This may be due to learners dropping out or failing and repeating grades.

Example 3: A scatter plot shows the relationship between the number of hours spent studying and the marks obtained on a test. (Imagine a scatter plot showing a positive correlation).

Analysis: Trend: As the number of hours spent studying increases, the marks obtained on the test generally increase. This indicates a positive correlation.

Outliers: There might be a few points that don't fit the general trend. For example, a student who studied a lot but still performed poorly. 2.5 Creating and Interpreting Formulas A formula is a mathematical equation that expresses a relationship between variables.

Example 4: A cellphone provider charges R1.50 per minute for calls. Write a formula to calculate the total cost of a call. Let `C` be the total cost of the call. Let `t` be the duration of the call in minutes.

Formula: `C = 1.50 * t` If a call lasts 10 minutes, the total cost would be `C = 1.50 * 10 = R15`. 2.6 Translating Between Representations It is important to be able to translate information from one representation to another. For example, you might be given a table of values and asked to draw a graph, or you might be given a graph and asked to write an equation.

Example 5: Convert the airtime data table from Example 1 into a graph.

Data (MB): x-axis Cost (R): y-axis Plot the points (50, 10), (100, 20), (150, 30), (200, 40) on the graph. Draw a straight line through the points. This line represents the relationship between data and cost. Guided Practice (With Solutions)

Question 1: The table below shows the number of loaves of bread sold at a bakery each day for a week. | Day | Loaves Sold | |---|---| | Monday | 50 | | Tuesday | 60 | | Wednesday | 70 | | Thursday | 80 | | Friday | 90 | What is the pattern in the number of loaves sold each day?

Solution: The number of loaves sold increases by 10 each day. This is a linear pattern with a constant difference of

1

0. Question 2: A taxi charges a base fare of R15 plus R8 per kilometre travelled. Write an equation to represent the total cost of a taxi ride.

Solution: Let `C` be the total cost. Let `d` be the distance travelled in kilometres.

Equation: `C = 8d + 15` Question 3: The following graph (imagine a line graph is described) shows the water level in a dam over a period of 5 months. Describe the trend in the water level.