Lesson Notes By Weeks and Term v3 - Primary 6
Ratio of two populations
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Ratio of two populations
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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 fundamental concept of ratio as a means of comparing two quantities of the same kind, specifically focusing on populations. Understanding ratios is critical for learners as it forms the basis for understanding proportions, rates, and percentages, which are essential in everyday life and future mathematical studies. In the Nigerian context, the ability to express comparisons quantitatively is useful in various scenarios, from comparing the number of different types of livestock on a farm to analyzing demographic data within communities or school enrollments.
A. What is a Ratio? A ratio is a way of comparing two quantities of the same kind. It shows how many times one quantity contains another, or how many times one quantity is contained within another. Ratios are used to express relationships between numbers. For example, if there are 3 boys and 2 girls in a group, the ratio of boys to girls is 3 to
2. B. Expressing Ratios Ratios can be expressed in three main ways: Using the word "to": e.g., 3 to 2 Using a colon symbol ":": e.g., 3:2 As a fraction: e.g., $\frac{3}{2}$ (This form is read as "3 to 2" or "3 over 2", not usually as a division in the context of ratio comparison unless specified). When expressing a ratio, the order matters. The ratio of A to B is different from the ratio of B to A. C. What is a Population in this Context? In this topic, "population" refers to the total number of individuals, objects, or items within a defined group. For instance, the "population of boys" in a class refers to the number of boys, and the "population of girls" refers to the number of girls. Similarly, it could be the population of a type of animal, vehicles, or any countable items. D. Simplifying Ratios Ratios should always be expressed in their simplest form. To simplify a ratio, divide both parts of the ratio by their Highest Common Factor (HCF). This is similar to simplifying fractions. For example, the ratio 10:5 can be simplified by dividing both numbers by their HCF, which is 5. 10 ÷ 5 = 2 5 ÷ 5 = 1 So, the simplified ratio is 2:
1. Worked
Examples: Example 1: Boys and Girls in a Classroom In a Primary 6 classroom, there are 20 boys and 15 girls. Express the ratio of boys to girls in its simplest form.
Solution: Identify the two populations: Population of boys = 20 Population of girls = 15 Write the ratio in the given order (boys to girls): 20:15 Find the HCF of 20 and 15: Factors of 20: 1, 2, 4, 5, 10, 20 Factors of 15: 1, 3, 5, 15 The HCF is
5. Divide both parts of the ratio by the HCF: 20 ÷ 5 = 4 15 ÷ 5 = 3 Write the simplified ratio: The ratio of boys to girls is 4:
3. Example 2: Animals on a Farm A Nigerian farmer has 18 goats and 24 chickens. What is the ratio of chickens to goats?
Solution: Identify the two populations: Population of goats = 18 Population of chickens = 24 Write the ratio in the given order (chickens to goats): 24:18 Find the HCF of 24 and 18: Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 Factors of 18: 1, 2, 3, 6, 9, 18 The HCF is
6. Divide both parts of the ratio by the HCF: 24 ÷ 6 = 4 18 ÷ 6 = 3 Write the simplified ratio: The ratio of chickens to goats is 4:
3. Example 3: Traders in a Market In a section of a market in Lagos, there are 45 yam traders and 30 pepper traders. Find the ratio of yam traders to pepper traders.
Solution: Identify the two populations: Population of yam traders = 45 Population of pepper traders = 30 Write the ratio in the given order (yam traders to pepper traders): 45:30 Find the HCF of 45 and 30: Factors of 45: 1, 3, 5, 9, 15, 45 Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30 The HCF is
1
5. Divide both parts of the ratio by the HCF: 45 ÷ 15 = 3 30 ÷ 15 = 2 Write the simplified ratio: The ratio of yam traders to pepper traders is 3:
2. A. Introduction (10 minutes)
Teacher Activity: Begin by engaging pupils with a simple comparison exercise. Ask pupils to count the number of windows and doors in the classroom. "How many windows do we have in our classroom?" "How many doors do we have?" "If we want to compare the number of windows to the number of doors, how can we say it?" (Guide towards "number of windows to number of doors").
Student Activity: Pupils actively count the items and state their counts. They attempt to compare the numbers using everyday language.
Teacher Activity: Introduce the term "ratio" as a mathematical way of comparing two quantities. Explain that "number of windows to number of doors" can be written as a ratio.
B. Lesson Development (25 minutes)
Teacher Activity (Concept Explanation): Clearly define ratio as a comparison of two quantities of the same kind. Introduce the three ways of expressing ratio (using "to", ":", and fraction form).
Emphasize the colon (:) as the primary notation for this lesson. Explain the importance of order in ratios (A:B is not B:A). Explain what "population" means in this context (number of items/individuals in a group). Demonstrate with simple classroom examples (e.g., number of white boards to number of class charts, number of pupils wearing spectacles to those not wearing). Introduce the concept of simplifying ratios to their simplest form using the HCF, similar to simplifying fractions. Use an example like 6 chairs to 9 tables (6:9 simplifies to 2:3). Student Activity (Active Listening and Participation): Pupils listen attentively, ask clarifying questions, and participate by providing numbers for classroom items. They practice writing simple ratios as instructed by the teacher. Teacher Activity (Guided Practice with Examples): Present the worked examples from Section 2 (Boys/Girls, Farm Animals, Market Traders) on the board, explaining each step clearly and soliciting input from pupils. "Let's look at this example: In Mr. Okoro's class, there are 20 boys and 15 girls. We want to find the ratio of boys to girls. What are the two populations?" Guide pupils through identifying the numbers, writing the initial ratio, finding the HCF, and simplifying.
Student Activity (Guided Problem Solving): Pupils work alongside the teacher, solving the examples in their notebooks. They volunteer answers for identifying populations, HCFs, and simplified ratios.
C. Group Activity (15 minutes)
Teacher Activity: Divide the class into small groups (e.g., 4-5 pupils per group). Provide each group with a task card or oral instructions. "In your groups, you will count two specific items in the classroom and find their ratio." Possible Group Tasks:* Group A: Ratio of pupils wearing sandals to pupils wearing covered shoes.
Group B: Ratio of desks to chairs in your group's area.
Group C: Ratio of red books to blue books on the classroom shelf.
Group D: Ratio of light bulbs to ceiling fans in the classroom.
Student Activity: Groups collaborate to count the specified items. They work together to express the counts as a ratio and simplify it to its lowest term. Each group selects a presenter to share their findings with the class.
D. Conclusion (5 minutes)
Teacher Activity: Ask a few pupils to summarize what they have learned about ratio and how to express it.
Reinforce the key concept: Ratio compares two quantities of the same kind, and it must be simplified. Provide a short take-home assignment.
Student Activity: Pupils provide summaries and copy down the assignment.
Instruction: Express each ratio in its simplest form. In a village community meeting, there were 32 women and 24 men. What is the ratio of women to men?
Solution: Population of women = 32 Population of men = 24 Ratio of women to men = 32:24 HCF of 32 and 24: Factors of 32: 1, 2, 4, 8, 16, 32 Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 HCF = 8 Divide both by 8: 32 ÷ 8 = 4; 24 ÷ 8 = 3 Simplified ratio = 4:3
Commentary: Emphasizes identifying the populations and their order before simplifying. A Nigerian school has 60 primary pupils and 45 secondary pupils. Find the ratio of primary pupils to secondary pupils.
Solution: Population of primary pupils = 60 Population of secondary pupils = 45 Ratio of primary pupils to secondary pupils = 60:45 HCF of 60 and 45: Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 Factors of 45: 1, 3, 5, 9, 15, 45 HCF = 15 Divide both by 15: 60 ÷ 15 = 4; 45 ÷ 15 = 3 Simplified ratio = 4:3
Commentary: This example uses larger numbers, requiring careful calculation of the HCF. Mrs. Adedeji's shop has 72 packets of Garri and 48 packets of Rice. What is the ratio of packets of Rice to packets of Garri?
Solution: Population of Garri packets = 72 Population of Rice packets = 48 Ratio of Rice packets to Garri packets = 48:72 (Note the change in order from the problem statement to the final ratio)
HCF of 48 and 72: Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 HCF = 24 Divide both by 24: 48 ÷ 24 = 2; 72 ÷ 24 = 3 Simplified ratio = 2:3
Commentary: This question tests the pupil's attention to the specified order of the ratio (Rice to Garri, not Garri to Rice).
A. Remediation (for struggling learners): Concrete Manipulatives: Provide struggling learners with physical objects (e.g., bottle tops, stones, counters, beans) to represent the populations. Guide them to physically group and compare the quantities before writing the numbers. For example, for 6:9, they lay out 6 stones and 9 stones, then group them into sets of common factors (3 sets of 2 and 3 sets of 3).
Smaller Numbers: Start with very simple ratios involving small numbers that are easy to simplify (e.g., 4:2, 8:4, 6:3) before moving to slightly larger numbers.
Visual Aids: Use number lines or Venn diagrams to help visualize the concept of common factors for simplification.
Pair Work/Peer Tutoring: Pair struggling learners with more capable peers for one-on-one assistance during practice sessions.
Focus on HCF: Provide extra practice on finding the HCF of two numbers, as this is a common stumbling block in ratio simplification.
B. Extension (for high-achieving learners): Three-Part Ratios: Introduce ratios involving three populations (e.g., ratio of boys:girls:teachers in a school, or maize:beans:yams harvested).
Missing Quantities: Present word problems where the total population and a ratio are given, and pupils need to find the individual populations. For example, "The total number of pupils in two classes is
7
0. If the ratio of pupils in Class A to Class B is 3:4, how many pupils are in each class?" Equivalent Ratios: Explore the concept of equivalent ratios and how they are used (e.g., 2:3 is equivalent to 4:6, 6:9, etc.).
Ratio and Proportion: Briefly introduce how ratios are related to proportions (equality of two ratios), setting a foundation for future topics.
Example 1: Boys and Girls in a Classroom
In a Primary 6 classroom, there are 20 boys and 15 girls. Express the ratio of boys to girls in its simplest form.
Solution:
Identify the two populations:
Population of boys = 20
Population of girls = 15
Write the ratio in the given order (boys to girls):
20:15
Find the HCF of 20 and 15:
Factors of 20: 1, 2, 4, 5, 10, 20
Factors of 15: 1, 3, 5, 15
The HCF is
5. Divide both parts of the ratio by the HCF:
20 ÷ 5 = 4
15 ÷ 5 = 3
Write the simplified ratio:
The ratio of boys to girls is 4:
3. Example 2: Animals on a Farm
A Nigerian farmer has 18 goats and 24 chickens. What is the ratio of chickens to goats?
Solution:
Identify the two populations:
Population of goats = 18
Population of chickens = 24
Write the ratio in the given order (chickens to goats):
24:18
Find the HCF of 24 and 18:
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 18: 1, 2, 3, 6, 9, 18
The HCF is
6. Divide both parts of the ratio by the HCF:
24 ÷ 6 = 4
18 ÷ 6 = 3
Write the simplified ratio:
The ratio of chickens to goats is 4:3.
Community Planning and Resource Allocation: Local government areas or communities often conduct population censuses. Ratios can be used to compare the population of different age groups (e.g., children to adults, youth to elders) to help in planning for schools, health centres, or social welfare programs. For instance, if the ratio of children (0-12 years) to adults (18-60 years) in a village is 3:5, it indicates a significant young population needing more educational resources.
Agriculture and Animal Husbandry: Farmers in Nigeria manage various types of livestock and crops. Ratios help them compare the number of different animals (e.g., goats to sheep, chickens to ducks) or the yield of different crops. This comparison can inform decisions on breeding, feeding, or crop rotation, contributing to efficient farm management and food security.
Sports and Team Composition: In local sports events, ratios can be used to compare the number of players from different teams, or the ratio of male athletes to female athletes in a competition. For example, comparing the number of supporters for two rival football clubs in a community (e.g., Kano Pillars vs. Enyimba FC) can give an idea of their relative popularity.