Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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Performance objectives

• Represent data using pie charts and histograms.
• Calculate the probability of simple events.
• Compare different types of graphs.
• Solve problems with independent events.
• Critically analyse data presented in media.

Lesson summary

Data handling and probability are essential skills for navigating the modern world. From understanding news reports about unemployment rates to making informed decisions about buying lottery tickets, these concepts are relevant to South African learners every day. This week, we will focus on representing data using various graphs and interpreting probability, building on previous knowledge of data collection and organization. We will be exploring different types of charts and graphs used to visually represent information. Understanding data handling will help you become critical thinkers and informed citizens.

Lesson notes

2.1 Pie Charts: A pie chart (or circle graph) is a circular statistical graphic, which is divided into slices to illustrate numerical proportion. In a pie chart, the arc length of each slice (and consequently its central angle and area) is proportional to the quantity it represents. Pie charts are excellent for showing the relative sizes of different categories within a single dataset.

How to Construct a Pie Chart: Calculate the Total: Find the sum of all the data values.

Calculate the Angle for Each Category: Divide each category's value by the total and multiply by 360 degrees (since a circle has 360 degrees).

Draw the Circle: Use a compass to draw a circle.

Draw the Slices: Use a protractor to measure and draw each slice based on the calculated angles.

Label the Slices: Clearly label each slice with the category name and its percentage of the total.

Example: A survey was conducted in a Grade 8 class in Soweto to determine their favourite sport.

The results are as follows: Football: 15 students, Netball: 10 students, Rugby: 5 students, Cricket: 5 students, Other: 5 Students. Let's construct a pie chart to represent this data.

Total: 15 + 10 + 5 + 5 + 5 = 40 students Angles: Football: (15/40) 360 = 135 degrees Netball: (10/40) 360 = 90 degrees Rugby: (5/40) 360 = 45 degrees Cricket: (5/40) 360 = 45 degrees Other: (5/40) 360 = 45 degrees and

4. Draw a circle and divide it into sections according to these angles.

Label: Label each slice clearly with the sport and also ideally the percentage. E.g. Football (37.5%) 2.2 Histograms: A histogram is a graphical representation of the distribution of numerical data. It is an estimate of the probability distribution of a continuous variable (quantitative data). The data is divided into intervals (bins), and a rectangle is drawn over each bin with a height proportional to the number of data points (frequency) in that bin. Unlike bar graphs, histograms have bars that touch each other, indicating continuous data.

How to Construct a Histogram: Determine the Range: Find the difference between the highest and lowest values in the data set.

Determine the Number of Bins: Choose an appropriate number of bins (intervals). A good rule of thumb is to use the square root of the number of data points.

Calculate the Bin Width: Divide the range by the number of bins.

Create the Bins: Define the intervals for each bin.

Count the Frequency: Count how many data points fall into each bin.

Draw the Histogram: Draw the bars with heights corresponding to the frequency of each bin. The bars should touch each other.

Example: The following data represents the ages of people attending a jazz festival in Cape Town: 18, 22, 25, 28, 30, 32, 35, 38, 40, 42, 45, 48, 50, 52, 55, 58, 60, 62, 65, 68 Let's construct a histogram with 5 bins.

Range: 68 - 18 = 50 Number of Bins: 5 (given)

Bin Width: 50 / 5 = 10 Bins: 18-27, 28-37, 38-47, 48-57, 58-67, 68-77 (we have 68, so we need the last bin to include it).

Frequency: 18-27: 3 28-37: 4 38-47: 4 48-57: 5 58-67: 4 68-77: 0 (one person, but age is

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8. If there were more older than 67, we might adjust bin widths slightly to better represent the data visually.) Draw a histogram showing these frequencies. 2.3 Probability: Probability is the measure of the likelihood that an event will occur. It is quantified as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

Theoretical Probability: The theoretical probability of an event is calculated as: P(event) = (Number of favourable outcomes) / (Total number of possible outcomes)

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

Favourable outcomes: 1 (rolling a 4)

Total possible outcomes: 6 (1, 2, 3, 4, 5, 6) P(rolling a 4) = 1/6 2.4 Independent Events: Two events are independent if the outcome of one event does not affect the outcome of the other event. To find the probability of two independent events both occurring, you multiply their individual probabilities. P(A and B) = P(A) * P(B)

Example: What is the probability of flipping a fair coin and getting heads, and then rolling a 3 on a standard six-sided die? P(Heads) = 1/2 P(Rolling a 3) = 1/6 P(Heads and Rolling a 3) = (1/2) * (1/6) = 1/12 Guided Practice (With Solutions)

Question 1: A survey was conducted at a local spaza shop to determine the flavours of soft drinks preferred by customers.

The results are as follows: Coke: 40, Fanta: 30, Sprite: 20, Cream Soda:

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0. Represent this data using a pie chart.

Solution: Total: 40 + 30 + 20 + 10 = 100 Angles: Coke: (40/100) 360 = 144 degrees Fanta: (30/100) 360 = 108 degrees Sprite: (20/100) 360 = 72 degrees Cream Soda: (10/100) 360 = 36 degrees Draw a circle and divide it into sections using these angles. Label each slice clearly. Coke (40%), Fanta (30%), Sprite (20%), Cream Soda (10%).

Commentary: This question reinforces the pie chart construction process, focusing on calculating angles and representing proportions.