Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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Performance objectives

• Construct and interpret histograms.
• Calculate and interpret mean, median, mode, and range.
• Determine experimental probability.
• Use tree diagrams to list possible outcomes.
• Represent and interpret data using pie charts.

Lesson summary

This week, we delve into the fascinating world of Data Handling and Probability. Understanding how to collect, organize, interpret, and present data is crucial in everyday life, from understanding news reports about crime statistics in our communities to making informed decisions about which mobile data package offers the best value. Probability, the chance of something happening, helps us understand risk and make predictions, whether it's predicting the likelihood of rain during the school sports day or understanding the odds in a lottery.

Lesson notes

2.1 Histograms A histogram is a graphical representation of data that is grouped into continuous numerical ranges. Unlike bar graphs, which are used for categorical data, histograms are used for numerical data where the bars touch each other. The area of each bar is proportional to the frequency (or relative frequency) for the class interval.

Class Intervals: Data is grouped into intervals. For example, the heights of learners in a class could be grouped into intervals of 140-149cm, 150-159cm, 160-169cm, etc.

Frequency: The number of data points that fall within each class interval.

X-axis: Represents the class intervals (e.g., heights in cm).

Y-axis: Represents the frequency (number of learners).

Example: A group of Grade 8 learners were weighed.

The data is as follows (in kg): 42, 45, 48, 50, 52, 43, 46, 49, 51, 53, 44, 47, 50, 52, 41, 44, 48, 51, 54, 45 Let's group this data into intervals of 41-44, 45-48, 49-52, and 53-56. | Class Interval (kg) | Frequency | |---|---| | 41-44 | 5 | | 45-48 | 6 | | 49-52 | 6 | | 53-56 | 3 | We can then create a histogram with these class intervals on the x-axis and the corresponding frequencies on the y-axis. Remember to label the axes and provide a title for the histogram. The bars will touch each other as the data is continuous. 2.2 Measures of Central Tendency and Spread Mean: The average of a set of numbers. Sum of all values divided by the number of values.

Formula: Mean = (Sum of all values) / (Number of values)

Median: The middle value in a data set when the data is arranged in ascending order. If there are two middle values (even number of data points), the median is the average of those two values.

Mode: The value that appears most frequently in a data set. A data set can have no mode, one mode, or multiple modes.

Range: The difference between the highest and lowest values in a data set. It represents the spread of the data.

Formula: Range = (Highest Value) - (Lowest Value)

Example: Consider the following data representing the number of hours Grade 8 learners spend on homework per week: 5, 3, 6, 2, 4, 5, 5, 1, 7, 4 Mean: (5+3+6+2+4+5+5+1+7+4) / 10 = 42 / 10 = 4.2 hours Median: First, arrange the data in ascending order: 1, 2, 3, 4, 4, 5, 5, 5, 6,

7. Since there are 10 values (even), the median is the average of the 5th and 6th values: (4+5)/2 = 4.5 hours Mode: The number 5 appears most frequently (3 times), so the mode is 5 hours.

Range: Highest value is 7, lowest value is

1. Range = 7 - 1 = 6 hours 2.3 Experimental Probability Experimental probability is the probability of an event occurring based on the results of an experiment.

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

Example: A coin is tossed 50 times. Heads appears 28 times and Tails appears 22 times. Experimental Probability of Heads = 28/50 = 0.56 or 56% Experimental Probability of Tails = 22/50 = 0.44 or 44% 2.4 Tree Diagrams A tree diagram is a visual tool used to represent all possible outcomes of a series of events. Each branch of the tree represents a possible outcome.

Example: Consider tossing a coin twice.

First Toss: The coin can land on Heads (H) or Tails (T).

Second Toss: After getting Heads on the first toss, the coin can land on Heads (H) or Tails (T). Similarly, after getting Tails on the first toss, the coin can land on Heads (H) or Tails (T).

The tree diagram would look like this: ``` / H (Heads) / H / \ T (Tails) \ / H (Heads) / T / \ T (Tails) ``` The possible outcomes are: HH, HT, TH, TT. If we want to find the probability of getting at least one head, we can see that three out of the four outcomes satisfy this condition (HH, HT, TH). So the probability is 3/4. 2.5 Pie Charts A pie chart (or circle graph) is a circular statistical graphic, which is divided into slices to illustrate numerical proportion. The entire "pie" represents 100%, and each slice represents a percentage of the whole.

To construct a pie chart: Calculate the angle for each category: (Category Value / Total Value) * 360° Draw a circle. Use a protractor to draw the angles for each category. Label each slice with the category and its percentage. Give the pie chart a title.

Example: A survey was conducted to find the favourite sport among Grade 8 learners.

The results are: Football (40 learners), Netball (30 learners), Rugby (20 learners), Other (10 learners). Total number of learners = 40 + 30 + 20 + 10 = 100 Football: (40/100) 360° = 144° Netball: (30/100) 360° = 108° Rugby: (20/100) 360° = 72° Other: (10/100) 360° = 36° You would then draw a circle and use a protractor to draw the slices with these angles, labelling each slice with the sport and its corresponding percentage (e.g., Football: 40%). Title the chart "Favourite Sport Among Grade 8 Learners". Guided Practice (With Solutions)

Question 1: The following data represents the test scores of 20 learners in a Mathematics test: 65, 70, 75, 80, 85, 60, 72, 78, 82, 88, 68, 74, 79, 81, 86, 63, 71, 77, 83, 87.