Lesson Notes By Weeks and Term v3 - Primary 5
Measures of central tendency
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Measures of central tendency
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Subject: General Mathematics
Class: Primary 5
Term: 2nd Term
Week: 4
Theme: Everyday Statistics
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find the mode of given data identify the mode as applicable in daily life activities Calculate the mean of a given data Identify mean of a set of data in daily life activities Solve quantitative aptitude problems on mode and mean of data calculate the mean of given data. appreciate the concept of mean of a set of data in daily activities
390 / 5 = 78 Answer: The mean score is
7
8. Worked Example 4 (Mean - Real-life application): A farmer in Benue State harvested the following number of yam tubers from 4 different plots: 45, 52, 48,
5
5. What is the average number of yam tubers harvested per plot?
Step 1: Sum of all values: 45 + 52 + 48 + 55 = 200 Step 2: Number of values: There are 4 plots.
Step 3: Divide the sum by the number of values: Mean = 200 / 4 = 50 Answer: The average number of yam tubers harvested per plot is
5
0. E. Quantitative Aptitude Problems (Mode and Mean): These problems often combine finding the mode or mean with other simple arithmetic or problem-solving skills, sometimes presented in word problem format.
Worked Example 5 (Combined Mode and Mean): The ages (in years) of children in a family are: 4, 7, 2, 7, 5. a) Find the mode of their ages. b) Calculate the mean of their ages.
Part a)
Finding the Mode: Data: 4, 7, 2, 7, 5 Count frequencies: 2 (1 time), 4 (1 time), 5 (1 time), 7 (2 times) The number 7 appears most often.
Answer a): The mode is 7 years.
Part b)
Calculating the Mean: Sum of ages: 4 + 7 + 2 + 7 + 5 = 25 Number of ages: 5 Mean = 25 / 5 = 5 * Answer b): The mean age is 5 years.
A. Data: Data refers to a collection of facts, such as numbers, words, measurements, observations, or even just descriptions of things. In mathematics, we often work with numerical data.
Examples in a Nigerian context: The number of mangoes sold by a vendor each day, the ages of students in a class, the scores obtained by students in a mathematics test, the sizes of yams harvested from a farm.
B. Measures of Central Tendency: These are single values that attempt to describe a set of data by identifying the central position within that set of data. They are a way to find a "typical" or "middle" value. For Primary 5, the focus will be on the Mode and the Mean.
C. Mode: The mode is the value that appears most frequently in a set of data. A data set can have one mode (unimodal), more than one mode (multimodal), or no mode if all values appear with the same frequency. For Primary 5, the focus is predominantly on data sets with a single, clearly identifiable mode.
How to find the Mode:
1. List the data.
2. Count how many times each value appears.
3. The value that appears most often is the mode.
Worked Example 1 (Mode): A small shop in Kaduna sold the following number of bread loaves daily over a week: 15, 12, 18, 15, 20, 15,
1
0. Find the mode of the number of loaves sold.
Step 1: List the data: 15, 12, 18, 15, 20, 15, 10 Step 2: Count the frequency of each number: 10 appears 1 time 12 appears 1 time 15 appears 3 times 18 appears 1 time 20 appears 1 time Step 3: Identify the most frequent value: The number 15 appears 3 times, which is more than any other number.
Answer: The mode is
1
5. Worked Example 2 (Mode - Real-life application): The shoe sizes of 10 students in Primary 5 are: 30, 32, 31, 30, 33, 30, 32, 34, 30,
3
1. What is the modal shoe size?
Step 1: List the data: 30, 32, 31, 30, 33, 30, 32, 34, 30, 31 Step 2: Count the frequency of each size: 30 appears 4 times 31 appears 2 times 32 appears 2 times 33 appears 1 time 34 appears 1 time Step 3: Identify the most frequent value: The size 30 appears 4 times.
Answer: The modal shoe size is
3
0. This means shoe size 30 is the most common among these students.
D. Mean (Average): The mean, often called the average, is calculated by adding all the values in a data set and then dividing by the total number of values in that set. It provides a single value that represents the typical value of the entire data set.
Formula for Mean: Mean = (Sum of all values) / (Number of values)
How to calculate the Mean:
1. Add up all the numbers in the data set.
2. Count how many numbers there are in the data set.
3. Divide the sum (from step 1) by the count (from step 2).
Worked Example 3 (Mean): Five students scored the following marks in a General Mathematics test: 80, 75, 90, 60,
8
5. Calculate the mean score.
Step 1: Sum of all values: 80 + 75 + 90 + 60 + 85 = 390 Step 2: Number of values: There are 5 scores.
Step 3: Divide the sum by the number of values: Mean = 390 / 5 = 78 Answer: The mean score is
7
8. Worked Example 4 (Mean - Real-life application): A farmer in Benue State harvested the following number of yam tubers from 4 different plots: 45, 52, 48,
5
5. What is the average number of yam tubers harvested per plot?
Step 1: Sum of all values: 45 + 52 + 48 + 55 = 200 Step 2: Number of values: There are 4 plots. * Step 3: Divide the sum by the number of values: Mean = 200 /
A. Introduction (10 minutes)
Teacher Activity: Begins by asking questions that elicit data from students: "How many siblings do you have?" "What is your favourite fruit among mango, orange, apple, banana?" "What was your score in the last Math quiz (e.g., out of 10)?" Writes some of the responses on the board, creating a simple data set (e.g., sibling numbers, favourite fruits). Introduces the concept of finding what is "most common" or "average" in the collected data.
States the topic: Measures of Central Tendency (Mode and Mean).
Student Activity: Respond to questions, providing numerical or categorical data. Observe the data being written on the board. Listen attentively to the introduction of the topic.
B. Development (40 minutes)
Phase 1: Understanding Mode (20 minutes)
Teacher Activity: Defines "Mode" using the collected data from the introduction (e.g., from the sibling numbers, identify the number that appears most often). Explains step-by-step how to find the mode, emphasizing counting frequency. Uses a real-life Nigerian example, e.g., "A market woman recorded the number of bags of rice she sold daily for five days: 10, 8, 12, 8,
1
5. What is the mode of bags sold?" Guides students through the example on the board, encouraging them to identify the mode. Emphasizes the applicability of mode in decision making (e.g., market woman knowing which item sells most).
Student Activity: Define mode in their own words after the teacher's explanation. Work individually or in pairs to find the mode for teacher-provided examples. Participate in counting and identifying the most frequent item. Ask questions for clarification.
Phase 2: Understanding Mean (20 minutes)
Teacher Activity: Defines "Mean" as the average, explaining it's the sum divided by the count. Explains the formula for mean and demonstrates the calculation using a new data set, e.g., "The heights (in cm) of four friends are: 120, 115, 125,
1
2
0. Find their mean height." Writes down the steps clearly: Sum values, count values, divide. Uses another relevant Nigerian example, e.g., "Five students scored 15, 18, 12, 17, 13 in a test out of
2
0. Calculate their mean score." Facilitates student involvement in the calculation process. Highlights the use of mean in understanding average performance or quantities.
Student Activity: Recite the definition of mean. Practice adding and dividing numbers as guided by the teacher. Attempt to calculate the mean for given data sets. Record notes and examples in their notebooks.
C. Quantitative Aptitude Problems (15 minutes)
Teacher Activity: Presents mixed problems that require identifying both mode and mean from a single data set, or slightly more complex word problems.
Example: "The daily attendance of Primary 5 class for 5 days was: 35, 38, 35, 39,
3
6. Find the mode and the mean attendance." Guides students through solving these problems, ensuring they apply both concepts correctly. Encourages group discussions for problem-solving strategies.
Student Activity: Work in small groups (2-3 students) to solve quantitative aptitude problems. Discuss how to approach each problem, identifying whether mode, mean, or both are required. Present their solutions and reasoning to the class.
D. Conclusion (5 minutes)
Teacher Activity: Recaps the definitions of mode and mean. Asks students to provide examples of where mode and mean are used in their daily lives. Assigns homework.
Student Activity: Summarize the key concepts learned. Provide real-life examples of mode and mean. Copy down homework. The teacher should present these questions on the board or as handouts and guide students to solve them collaboratively, providing immediate feedback.
Question 1 (Mode): The number of mangoes sold by a fruit vendor at Dugbe Market, Ibadan, over eight hours were: 25, 30, 20, 25, 35, 25, 20,
4
0. Find the mode of the number of mangoes sold.
Solution: List the data: 25, 30, 20, 25, 35, 25, 20, 40 Count the frequency of each number: 20 appears 2 times 25 appears 3 times 30 appears 1 time 35 appears 1 time 40 appears 1 time Identify the most frequent value: The number 25 appears 3 times, which is more than any other number.
Answer: The mode of mangoes sold is
2
5. Commentary: This helps identify the most popular sale quantity, useful for stock management.
Question 2 (Mean): A student saved the following amounts (in Naira) over five days: ₦100, ₦150, ₦80, ₦120, ₦
1
0
0. Calculate the mean amount saved per day.
Solution: Sum of all values: ₦100 + ₦150 + ₦80 + ₦120 + ₦100 = ₦550 Number of values: There are 5 amounts.
Divide the sum by the number of values: Mean = ₦550 / 5 = ₦110 Answer: The mean amount saved per day is ₦
1
1
0. Commentary: This demonstrates how to find an average value, useful for budgeting. Question 3 (Mode and Mean - Quantitative Aptitude): The scores of 6 students in a short spelling quiz (out of 10) are: 7, 9, 6, 7, 8, 5. a) What is the mode of the scores? b) What is the mean score?
Solution: a)
Finding the Mode: List the data: 7, 9, 6, 7, 8, 5 Count frequencies: 5 (1 time), 6 (1 time), 7 (2 times), 8 (1 time), 9 (1 time)
Identify the most frequent value: The number 7 appears 2 times.
Answer a): The mode of the scores is 7. b)
Calculating the Mean: Sum of all values: 7 + 9 + 6 + 7 + 8 + 5 = 42 Number of values: There are 6 scores.
Divide the sum by the number of values: Mean = 42 / 6 = 7 Answer b): The mean score is
7. Commentary: This problem integrates both concepts in a typical quiz context, enhancing problem-solving skills.
Market Traders and Business Owners: Mode: A hawker selling groundnut or a market vendor selling clothes can use the mode to know which product size, colour, or quantity is most popular (the mode). This helps them decide what to stock more of to meet customer demand and reduce unsold inventory. For example, if shoe size 38 is the mode, they'll stock more of it.
Mean: A small business owner can calculate the mean (average) daily or weekly sales to understand their typical earnings and plan for expenses or growth. This helps in making financial projections.
Farming and Agriculture: Mean: Farmers can calculate the mean yield of crops (e.g., kilograms of maize per plot, number of yam tubers per ridge). This helps them assess the productivity of their land, compare yields from different planting methods, or determine average harvest.
Mode: If a farmer sells multiple varieties of a crop (e.g., different types of cassava), identifying the modal type (most preferred by buyers) can guide future planting decisions. School Performance and Classroom Management: Mean: Teachers frequently use the mean to calculate the average score of students in a test or assignment. This gives an overall picture of the class's performance and can help identify if a topic was generally well understood or not.
Mode: A teacher might find the modal age of students in a class or the modal number of siblings. While less directly related to academic performance, it can provide insights into classroom demographics for social activities or grouping.