Lesson Notes By Weeks and Term v3 - Primary 6
Percentages
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Percentages
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Subject: General Mathematics
Class: Primary 6
Term: 2nd Term
Week: 3
Theme: Number And Numeration
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This lesson introduces the concept of ratio, specifically focusing on expressing the comparison of two different populations or groups of items as a ratio. While the overarching topic for the week is 'Percentages', understanding ratios is a foundational skill directly linked to grasping percentage concepts, as percentages are a specific type of ratio (a ratio out of 100). This topic is essential for students to develop quantitative reasoning skills applicable in various real-life scenarios within Nigeria, such as comparing community demographics, resource distribution, or even sports statistics.
A. What is a Ratio? A ratio is a way to compare two or more quantities of the same kind. It shows how much of one quantity there is in relation to another quantity. Ratios are used to simplify comparisons and express relationships clearly. For this lesson, the quantities being compared will specifically be populations or groups of items.
B. Ways to Express Ratios: Ratios can be expressed in several ways:
1. Using a colon: `a : b` (read as "a to b") - This is the most common and preferred notation for comparison.
2. As a fraction: `a/b` - While mathematically correct, the colon notation is generally preferred when explicitly referring to ratios.
3. Using the word "to": `a to b`
C. Important Principles of Ratios:
1. Order Matters: The order in which quantities are listed in a ratio is crucial. For example, the ratio of boys to girls (e.g., 2:3) is different from the ratio of girls to boys (e.g., 3:2).
2. Quantities of the Same Kind: Ratios compare quantities that are of the same type. For example, we compare the number of students to the number of students, or the number of trees to the number of trees.
3. No Units: When a ratio is simplified to its lowest terms, it generally does not have units, as the units cancel each other out during the comparison.
4. Simplification: Ratios should always be simplified to their lowest terms, similar to simplifying fractions. This is done by dividing both parts of the ratio by their Highest Common Factor (HCF).
D. Steps to Express Two Populations in a Given Ratio:
1. Identify the Quantities: Clearly identify the two populations or groups to be compared.
2. Write the Initial Ratio: Write the ratio in the order specified in the problem (e.g., first quantity : second quantity).
3. Find the Highest Common Factor (HCF): Determine the largest number that can divide both parts of the ratio without leaving a remainder.
4. Simplify the Ratio: Divide both parts of the ratio by their HCF to express it in its simplest form. Worked Examples (Realistic Nigerian Contexts): Example 1: In a Primary 6 class at Danjuma Primary School, there are 25 boys and 35 girls. Express the ratio of boys to girls in the class.
Solution:
1. Identify the quantities: Number of boys = 25, Number of girls = 35.
2. Write the initial ratio: Boys : Girls = 25 : 35.
3. Find the HCF of 25 and 35: Factors of 25: 1, 5, 25 Factors of 35: 1, 5, 7, 35 The HCF is 5.
4. Simplify the ratio: Divide both parts by 5: (25 ÷ 5) : (35 ÷ 5)
Result: 5 : 7 Therefore, the ratio of boys to girls is 5 :
7. Example 2: A community in Kano State has 120 poultry farmers and 180 rice farmers. What is the ratio of poultry farmers to rice farmers?
Solution:
1. Identify the quantities: Number of poultry farmers = 120, Number of rice farmers = 180.
2. Write the initial ratio: Poultry farmers : Rice farmers = 120 : 180.
3. Find the HCF of 120 and 180: One way to find HCF for larger numbers is to divide by common factors repeatedly: 120 : 180 (Divide by 10) -> 12 : 18 12 : 18 (Divide by 6) -> 2 : 3 The HCF is 10 x 6 = 60.
4. Simplify the ratio: Divide both parts by 60: (120 ÷ 60) : (180 ÷ 60)
Result: 2 : 3 Therefore, the ratio of poultry farmers to rice farmers is 2 :
3. Example 3: In a local market in Oyo, there are 80 traders selling clothes and 60 traders selling food items. Express the ratio of food item traders to clothes traders.
Solution:
1. Identify the quantities: Traders selling clothes = 80, Traders selling food items = 60.
2. Write the initial ratio (careful with order!): Food item traders : Clothes traders = 60 : 80.
3. Find the HCF of 60 and 80: 60 : 80 (Divide : 3 Therefore, the ratio of poultry farmers to rice farmers is 2 :
3. Example 3: In a local market in Oyo, there are 80 traders selling clothes and 60 traders selling food items. Express the ratio of food item traders to clothes traders.
Solution:
1. Identify the quantities: Traders selling clothes = 80, Traders selling food items = 60.
2. Write the initial ratio (careful with order!): Food item traders : Clothes traders = 60 : 80.
3. Find the HCF of 60 and 80: 60 : 80 (Divide by 10) -> 6 : 8 6 : 8 (Divide by 2) -> 3 : 4 The HCF is 10 x 2 = 20.
4. Simplify the ratio: Divide both parts by 20: (60 ÷ 20) : (80 ÷ 20) * Result: 3 : 4 Therefore, the ratio of food item traders to clothes traders is 3 :
4. A.
Introduction (Teacher Activity): Teacher introduces the lesson by asking students to count the number of boys and girls in their own classroom.
Teacher asks: "How can we compare these two groups of students simply?" This prompts students to think about comparison. Teacher explains that "ratio" is a mathematical tool for making such comparisons.
B. Activity 1: Understanding Ratio Notation (Teacher & Student Activities): Teacher: Explains what a ratio is using the classroom example (e.g., "If there are 20 boys and 25 girls, the ratio of boys to girls is 20:25.").
Teacher: Demonstrates on the chalkboard how to write ratios using the colon notation (a:b), fraction form (a/b), and "a to b". Emphasises the colon notation for this topic.
Students: Practice writing simple ratios for pairs of numbers provided by the teacher (e.g., 5 and 10, 12 and 15).
C. Activity 2: Simplifying Ratios (Teacher & Student Activities): Teacher: Reviews the concept of HCF if necessary, reminding students how to find the HCF of two numbers.
Teacher: Explains that ratios must be simplified to their lowest terms using the HCF, similar to simplifying fractions.
Teacher: Demonstrates simplifying the classroom ratio (e.g., 20:25 becomes 4:5 after dividing by HCF of 5).
Students: Work in pairs to simplify ratios like 10:20, 15:25, 12:18, under teacher supervision.
D. Activity 3: Expressing Ratios of Populations (Teacher & Student Activities): Teacher: Presents real-life scenarios involving populations from Nigeria.
For example: "In a village, there are 30 goats and 45 sheep. What is the ratio of goats to sheep?" Teacher: Guides students through the step-by-step process of identifying quantities, writing the initial ratio, finding the HCF, and simplifying.
Students: Solve similar problems presented by the teacher, either individually or in small groups. The teacher circulates, provides support, and checks for understanding.
Teacher: Emphasizes the importance of the order of quantities in the ratio. For instance, comparing the ratio of men to women is different from women to men.
Materials: Whiteboard/Chalkboard Markers/Chalk Class roster (for actual classroom population count) Flashcards with numbers for ratio practice (optional) The teacher should present these questions and guide students through the solutions, ensuring they understand each step.
Question 1: In a primary health care centre in Calabar, 40 children were vaccinated against polio and 60 children were not. What is the ratio of vaccinated children to unvaccinated children?
Solution: Quantities: Vaccinated children = 40, Unvaccinated children =
6
0. Initial Ratio: Vaccinated : Unvaccinated = 40 :
6
0. HCF of 40 and 60: Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40 Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 HCF =
2
0. Simplified Ratio: (40 ÷ 20) : (60 ÷ 20) = 2 :
3. Commentary: Emphasize identifying the correct quantities and maintaining the order as specified in the question.
Question 2: A school has 35 teachers and 700 students. Express the ratio of teachers to students.
Solution: Quantities: Teachers = 35, Students =
7
0
0. Initial Ratio: Teachers : Students = 35 :
7
0
0. HCF of 35 and 700: Recognize that 700 is a multiple of 35 (700 ÷ 35 = 20). HCF =
3
5. Simplified Ratio: (35 ÷ 35) : (700 ÷ 35) = 1 :
2
0. Commentary: This example introduces larger numbers, where recognising multiples can simplify finding the HC
F. The ratio shows that for every 1 teacher, there are 20 students.
Question 3: In a small village in Kaduna, there are 240 men and 360 women. What is the ratio of women to men?
Solution: Quantities: Men = 240, Women =
3
6
0. Initial Ratio (Order is crucial): Women : Men = 360 :
2
4
0. HCF of 360 and 240: 360 : 240 (Divide by 10) -> 36 : 24 36 : 24 (Divide by 12) -> 3 : 2 HCF = 10 x 12 =
1
2
0. Simplified Ratio: (360 ÷ 120) : (240 ÷ 120) = 3 :
2. Commentary: Highlight the importance of reading the question carefully for the order of the ratio (women to men, not men to women).
Community Resource Allocation: Ratios are used by local government agencies to compare populations in different communities (e.g., number of primary school-aged children to the number of available classrooms) to make informed decisions about resource allocation, such as building new schools or health clinics.
Demographic Studies: Understanding the ratio of different age groups (e.g., youths to elderly) or genders (male to female) in a population helps demographers and policymakers understand the structure of a society and plan for social welfare, employment, and healthcare services.
Economic Comparison: Businesses and economists use ratios to compare market segments (e.g., ratio of consumers who prefer local products to imported ones) or the number of people in different occupations (e.g., farmers to artisans) to inform production, marketing, and policy decisions.
Environmental Monitoring: Ratios can be used to compare populations of different species in an ecosystem (e.g., ratio of fish species A to fish species B in a particular river) to assess biodiversity and environmental health.