Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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.

Performance objectives

• Calculate the probability of simple events.
• Represent data using pie charts and histograms.
• Interpret data from pie charts and histograms.
• Use probability to predict the likelihood of events.
• Explain the difference between experimental and theoretical probability.

Lesson summary

Data handling and probability are essential skills in mathematics that help us understand and interpret information around us. In South Africa, these skills are particularly relevant in understanding statistics related to socio-economic issues, planning for future events, and making informed decisions. From understanding crime statistics in our communities to predicting the weather for farming, data handling and probability play a crucial role in our daily lives. This week, we will focus on calculating probabilities of simple events, representing data using different types of graphs, and interpreting the information presented in these graphs.

Lesson notes

2.1 Probability: Probability is the measure of how likely an event is to occur. It's expressed as a number between 0 and 1, where 0 means the event is impossible, and 1 means the event is certain. We can express probability as a fraction, a decimal, or a percentage.

Formula: Probability of an event = (Number of favorable outcomes) / (Total number of possible outcomes)

Example 1: A bag contains 5 red balls and 3 blue balls. What is the probability of randomly picking a red ball? Number of favorable outcomes (red balls) = 5 Total number of possible outcomes (total balls) = 5 + 3 = 8 Probability of picking a red ball = 5/8 = 0.625 = 62.5% Example 2: What is the probability of rolling a 4 on a standard six-sided die? Number of favorable outcomes (rolling a 4) = 1 Total number of possible outcomes (numbers on the die) = 6 Probability of rolling a 4 = 1/6 = 0.1667 (approximately) = 16.67% (approximately)

Example 3: In a class of 40 learners, 25 are girls. If a learner is selected at random, what is the probability that the learner is a boy? Total learners = 40 Number of girls = 25 Number of boys = 40 - 25 = 15 Probability of selecting a boy = 15/40 = 3/8 = 0.375 = 37.5% 2.2 Pie Charts: A pie chart (or circle graph) is a circular chart divided into sectors, illustrating relative magnitudes or frequencies. Each sector represents a category, and the size of the sector is proportional to the percentage of the whole that it represents. The entire pie chart represents 100% of the data.

Key Steps to Construct a Pie Chart: Calculate the angle for each category: (Category Value / Total Value) * 360° Draw a circle. Use a protractor to draw each sector according to the calculated angles. Label each sector clearly with the category name and percentage.

Example 4: A survey of 100 learners asked about their favorite sport. 40 chose soccer, 30 chose rugby, 20 chose netball, and 10 chose cricket. Represent this data in a pie chart.

Soccer angle: (40/100) * 360° = 144° Rugby angle: (30/100) * 360° = 108° Netball angle: (20/100) * 360° = 72° Cricket angle: (10/100) * 360° = 36° You would then draw a circle and use a protractor to create sectors with these angles, labeling each sector with its corresponding sport and percentage. 2.3 Histograms: A histogram is a graphical representation of data that is grouped into numerical ranges. It is similar to a bar graph, but it is used for continuous data, meaning the data can take on any value within a range. The bars touch each other, indicating that the data is continuous. The x-axis represents the numerical ranges (or intervals), and the y-axis represents the frequency (the number of data points that fall within each range). Key Differences between Histograms and Bar Graphs: Histograms represent continuous data (e.g., height, weight, temperature), while bar graphs represent categorical data (e.g., colors, brands, types of fruit). Bars in a histogram touch each other, while bars in a bar graph are separated.

Example 5: The ages of people attending a community event are as follows: 5, 8, 12, 15, 18, 22, 25, 28, 32, 35, 38, 42, 45, 48,

5

2. Group the data into intervals of 10 years (0-9, 10-19, 20-29, 30-39, 40-49, 50-59) and create a histogram. 0-9: Frequency = 2 10-19: Frequency = 3 20-29: Frequency = 3 30-39: Frequency = 3 40-49: Frequency = 3 50-59: Frequency = 1 You would then draw a histogram with these intervals on the x-axis and the corresponding frequencies on the y-axis. The bars would touch each other. 2.4 Experimental vs. Theoretical Probability Theoretical Probability: The probability of an event calculated based on mathematical reasoning and the assumption that all outcomes are equally likely (as seen in sections 2.1). This is what "should" happen.

Experimental Probability: The probability of an event based on actual experiments or observations. This is what "did" happen.

Formula: Experimental Probability = (Number of times the event occurred) / (Total number of trials)

Example 6: You flip a coin 20 times. It lands on heads 12 times.

Theoretical Probability of heads: 1/2 = 50% Experimental Probability of heads: 12/20 = 3/5 = 0.6 = 60% Notice that the experimental probability can differ from the theoretical probability, especially with a small number of trials. As the number of trials increases, the experimental probability usually gets closer to the theoretical probability. Guided Practice (With Solutions)

Question 1: A spinner has 4 equal sections colored red, blue, green, and yellow. What is the probability of landing on blue? Express your answer as a fraction, decimal, and percentage.

Solution: Number of favorable outcomes (blue) = 1 Total number of possible outcomes (sections) = 4 Probability (fraction) = 1/4 Probability (decimal) = 1/4 = 0.25 Probability (percentage) = 0.25 100% = 25%

Commentary: This is a straightforward application of the basic probability formula. It demonstrates how to convert between different representations of probability.