Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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Performance objectives

• Represent data using histograms and frequency polygons.
• Determine the mode and median of ungrouped data.
• Determine the mean of ungrouped data.
• Calculate experimental probability.
• Predict the likelihood of events using experimental probability.

Lesson summary

Data handling and probability are essential mathematical tools that help us understand and interpret the world around us. In South Africa, understanding data is crucial for making informed decisions about issues like resource allocation, healthcare, and education. Being able to calculate probabilities allows us to assess risks and make predictions in everyday situations, from sports to business. For example, understanding crime statistics in different areas can inform personal safety choices, and understanding the probability of rainfall impacts agricultural planning.

Lesson notes

Histograms: A histogram is a graphical representation of data grouped into intervals. It uses bars to show the frequency (or count) of data values within each interval. Unlike bar graphs, the bars in a histogram touch each other, representing the continuous nature of the data. The width of each bar represents the interval, and the height represents the frequency.

Example: Suppose we surveyed 30 Grade 8 learners about the amount of time they spend on social media each day (in minutes).

The data is as follows: 20, 35, 40, 50, 60, 25, 30, 45, 55, 65, 30, 40, 50, 60, 20, 35, 45, 55, 25, 30, 40, 50, 60, 35, 40, 45, 55, 25, 30, 40 To create a histogram, we first group the data into intervals, for example: 20-29, 30-39, 40-49, 50-59, 60-

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9. We then count how many learners fall into each interval: 20-29: 6 30-39: 8 40-49: 7 50-59: 6 60-69: 3 We then draw bars for each interval, with the height corresponding to the frequency. The X-axis represents the time intervals (in minutes), and the Y-axis represents the frequency (number of learners).

Frequency Polygons: A frequency polygon is a line graph that connects the midpoints of the top of each bar in a histogram. It provides a visual representation of the distribution of data. The endpoints of the frequency polygon are usually connected to the x-axis to close the polygon. To create a frequency polygon from the histogram above, we mark the midpoint of each bar's top (e.g., the midpoint of the 20-29 bar is 24.5) and connect these midpoints with straight lines. The polygon is closed by connecting the first and last points to the x-axis, effectively at the midpoints of intervals before the first interval and after the last interval containing data.

Mode: The mode is the value that appears most frequently in a data set. A data set can have one mode (unimodal), two modes (bimodal), or more than two modes (multimodal). If all values appear with the same frequency, there is no mode.

Example: Consider the shoe sizes of 10 learners: 6, 7, 7, 8, 8, 8, 9, 9, 10,

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0. The mode is 8 because it appears three times, which is more than any other shoe size.

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

Example 1 (Odd number of values): Consider the ages of 7 siblings: 5, 7, 9, 11, 13, 15,

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7. The median age is 11 because it is the middle value when the ages are arranged in ascending order.

Example 2 (Even number of values): Consider the test scores of 8 learners: 60, 70, 75, 80, 85, 90, 95,

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0. The median is the average of the two middle values (80 and 85): (80 + 85) / 2 = 82.

5. Mean: The mean (or average) is calculated by adding up all the values in a data set and dividing by the total number of values.

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

Example: Consider the heights (in cm) of 5 learners: 150, 155, 160, 165,

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0. The mean height is: (150 + 155 + 160 + 165 + 170) / 5 = 800 / 5 = 160 cm.

Experimental Probability: Experimental probability is the probability of an event occurring based on observed data from an experiment or real-world observation. It is calculated as the number of times the event occurs divided by the total number of trials.

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

Example: A coin is flipped 100 times, and heads appears 45 times. The experimental probability of getting heads is 45/100 = 0.45 or 45%. Guided Practice (With Solutions)

Question 1: The following data represents the number of customers visiting a local spaza shop each hour for 8 hours: 10, 12, 15, 18, 20, 16, 14,

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2. Determine the mode of this data.

Solution: The mode is the value that appears most frequently. In this data set, the number 12 appears twice, which is more than any other number.

Therefore, the mode is

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2. Question 2: The following data represents the ages of 9 learners in a Grade 8 class: 13, 14, 13, 15, 14, 13, 14, 15,

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3. Determine the median age.

Solution: First, arrange the data in ascending order: 13, 13, 13, 13, 14, 14, 14, 15,

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5. The median is the middle value. Since there are 9 values, the middle value is the 5th value, which is

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4. Therefore, the median age is

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4. Question 3: A die is rolled 20 times, and the number 6 appears 3 times. What is the experimental probability of rolling a 6?

Solution: Experimental Probability = (Number of times the event occurs) / (Total number of trials) = 3 / 20 = 0.15 or 15%.

Question 4: The following data shows the number of hours spent studying by 10 learners during a week: 5, 6, 7, 8, 9, 5, 6, 7, 8,

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0. Calculate the mean number of hours spent studying.

Solution: Mean = (Sum of all values) / (Number of values) = (5 + 6 + 7 + 8 + 9 + 5 + 6 + 7 + 8 + 10) / 10 = 71 / 10 = 7.1 hours.