Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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Performance objectives

• Construct and interpret various graphs (bar, histogram, pie, line).
• Calculate the probability of single events.
• Understand the difference between theoretical and experimental probability.
• Analyze data to draw conclusions.

Lesson summary

Data handling and probability are crucial skills in the 21st century. We are constantly bombarded with information, and understanding how to interpret, analyze, and present data allows us to make informed decisions. From understanding the latest COVID-19 statistics in South Africa to analyzing the chances of winning the Lotto, these skills are highly relevant to everyday life.

Furthermore, these concepts are foundational for many other areas of mathematics and statistics encountered in later grades and tertiary education.

Lesson notes

2.1 Types of Graphs and Their Uses: Bar Graphs: Used to compare different categories of data. The height of each bar represents the frequency or quantity of each category.

Single Bar Graph: Represents one set of data.

Double Bar Graph: Compares two sets of data side-by-side for each category.

Example: A survey was conducted in a Grade 8 class about their favourite sports.

Here are the results: | Sport | Number of Students | |------------|--------------------| | Soccer | 15 | | Netball | 10 | | Rugby | 8 | | Cricket | 7 | This data can be represented by a single bar graph. The x-axis would represent the sports, and the y-axis would represent the number of students. Each bar's height would correspond to the number of students who chose that sport.

Histograms: Similar to bar graphs but used to represent continuous data grouped into intervals (classes). The bars touch each other to show the continuous nature of the data.

Example: The heights of students in a Grade 8 class are measured and grouped into intervals: | Height (cm) | Number of Students | |-------------|--------------------| | 140-149 | 5 | | 150-159 | 12 | | 160-169 | 8 | | 170-179 | 3 | A histogram would represent this data. The x-axis would represent the height intervals, and the y-axis would represent the number of students. Notice that the bars representing each interval touch each other.

Pie Charts: Used to show how a whole is divided into parts. Each "slice" of the pie represents a percentage or proportion of the whole.

Example: A household budget is divided as follows: Rent (40%), Food (30%), Transport (20%), Entertainment (10%). A pie chart would clearly show the proportion of the budget allocated to each category. The "Rent" slice would take up 40% of the pie. To calculate the angle for each slice, multiply the percentage by 3.6 (since a circle has 360 degrees). For example, the angle for "Rent" would be 40 * 3.6 = 144 degrees.

Line Graphs: Used to show trends over time. Points are plotted and connected with lines to show how a variable changes over a period.

Example: The daily temperature in Johannesburg is recorded for a week: | Day | Temperature (°C) | |---------|--------------------| | Monday | 20 | | Tuesday | 22 | | Wednesday| 25 | | Thursday | 23 | | Friday | 21 | | Saturday| 24 | | Sunday | 26 | A line graph would show how the temperature changes each day. The x-axis would represent the days of the week, and the y-axis would represent the temperature. 2.2 Probability: Probability is a measure of how likely an event is to occur. It is expressed as a fraction, decimal, or percentage between 0 and 1 (or 0% and 100%). Probability = (Number of favorable outcomes) / (Total number of possible outcomes)

Example 1: What is the probability of rolling a 4 on a standard six-sided die?

Favorable outcome: Rolling a 4 (1 outcome)

Total possible outcomes: 1, 2, 3, 4, 5, 6 (6 outcomes) Probability = 1/6 Example 2: A bag contains 3 red balls and 5 blue balls. What is the probability of picking a red ball at random?

Favorable outcome: Picking a red ball (3 outcomes)

Total possible outcomes: 3 red + 5 blue = 8 balls (8 outcomes) Probability = 3/8 2.3 Theoretical vs.

Experimental Probability: Theoretical Probability: The probability of an event based on mathematical calculations and assumptions. It's what should happen in an ideal situation.

Experimental Probability: The probability of an event based on actual experiments or observations. It's what actually happens when you repeat an experiment multiple times.

Example: Flipping a coin.

Theoretical Probability of getting heads: 1/2 (50%)

Experimental Probability: You flip a coin 20 times and get heads 12 times.

Experimental Probability of getting heads: 12/20 = 3/5 (60%) The experimental probability may not always match the theoretical probability, especially with a small number of trials.

However, as the number of trials increases, the experimental probability tends to get closer to the theoretical probability (Law of Large Numbers). Guided Practice (With Solutions)

Question 1: A survey was conducted asking Grade 8 learners what their favourite South African music genre is.

The results are: Amapiano (30), Gqom (20), Kwaito (15), Afrikaans Pop (10), Other (5). Draw a pie chart to represent this data.

Solution: Calculate the total number of learners: 30 + 20 + 15 + 10 + 5 = 80 Calculate the angle for each sector: Amapiano: (30/80) 360 = 135 degrees Gqom: (20/80) 360 = 90 degrees Kwaito: (15/80) 360 = 67.5 degrees Afrikaans Pop: (10/80) 360 = 45 degrees Other: (5/80) 360 = 22.5 degrees Draw the pie chart using a compass and protractor, or using software. Label each sector clearly with the genre and the corresponding percentage (e.g., Amapiano: 37.5%).

Question 2: A bag contains 5 green marbles, 7 blue marbles, and 3 yellow marbles. What is the probability of picking a blue marble at random?