Ratio and Proportion, Direct, Inverse and Family Size Resource Ratio for Primary 6

Class: Primary 6
Term: 1st Term
Week: 6
Age: 11 years
Duration: 45 minutes
Subject: General Mathematics
Curriculum Theme: Numbers and Numeration
Previous Lesson: Demography, Population Meaning, Census and Comparing Population Data.
Topic: RATIO AND PROPORTION
Subject Matter: Simplifying ratios from fractions, direct proportion problems, inverse proportion problems and examples, ratio of family size and resources, ratio of two populations, quantitative reasoning on ratio and proportion

Specific Objectives

By the end of the lesson, pupils should be able to:

Cognitive Domain:

• Define ratio and proportion.
• Simplify ratios involving fractions.
• Solve problems on direct proportion and interpret results.
• Solve problems on inverse proportion.
• Calculate the ratio of family size to resources.
• Calculate the ratio of two populations.
• Solve quantitative reasoning problems involving ratio and proportion.

Affective Domain:

• Appreciate the importance of ratio and proportion in daily life.
• Participate actively in problem-solving discussions.

Psychomotor Domain:

• Demonstrate steps to solve ratio and proportion problems.
• Accurately present solutions to quantitative reasoning problems.

Social Domain:

• Collaborate with peers to solve group problems on ratio and proportion.

Reference Materials

The following resources were used in planning this lesson:

• 9 Years Basic Education Curriculum (Primary 4-6)
• State Unified Scheme of Work (Primary 6)
• New General Mathematics for Primary Schools, Book 6
• https://www.education.gov.ng
• https://www.nigeriancurriculum.gov.ng

Instructional Materials

The teacher will teach this lesson with the aid of:

• Textbook
• Charts showing examples of ratio and proportion
• Posters with quantitative reasoning problems
• Workbook
• Real-life data (e.g., family sizes, population figures)

Rationale for the Lesson

This lesson helps pupils understand how to compare quantities and solve problems involving fair sharing or related changes. Understanding ratio and proportion enables pupils to make sense of comparisons in everyday situations like cooking, budgeting, or understanding statistics.

Prerequisite/Previous Knowledge

Pupils should have prior knowledge of fractions, equivalent fractions, multiplication, and division.

Lesson Content/Board Summary

RATIO AND PROPORTION

What is Ratio?

Ratio is a comparison of two or more quantities of the same kind. It shows how much of one quantity there is compared to another quantity.

Ratios can be written as a:b or a/b. They should always be simplified to their simplest form.

The following are steps to simplifying ratios from fractions:

• Find the Least Common Multiple (LCM) of the denominators of the fractions.
• Multiply each fraction by the LCM to clear the denominators.
• Simplify the resulting whole number ratio.

Example: Simplify the ratio 1/3 : 1/6.

Solution:

Step 1: Find the LCM of the denominators (3 and 6). The LCM is 6.

Step 2: Multiply each fraction by the LCM.

(1/3) * 6 : (1/6) * 6

2 : 1

Answer: The simplified ratio is 2:1.

What is Proportion?

Proportion is a statement that two ratios are equal. If a:b = c:d, then a, b, c, and d are in proportion. This can also be written as a/b = c/d.

Direct Proportion

Direct proportion describes a relationship where an increase in one quantity causes a proportional increase in another quantity, or a decrease in one quantity causes a proportional decrease in another quantity. If x and y are directly proportional, then x = ky, where k is a constant.

The following are characteristics of direct proportion:

• As one quantity increases, the other quantity increases.
• As one quantity decreases, the other quantity decreases.
• The ratio of the two quantities remains constant.

Example 1: If 3 exercise books cost N150, how much will 7 exercise books cost?

Solution:

Step 1: Set up the proportion. Let ‘x’ be the cost of 7 books.

3 books / N150 = 7 books / x

Step 2: Cross-multiply.

3 * x = 150 * 7

Step 3: Simplify.

3x = 1050

Step 4: Solve for x (divide both sides by 3).

x = 1050 / 3

x = N350

Answer: 7 exercise books will cost N350.

Example 2: A car travels 180 km in 3 hours. How far will it travel in 5 hours at the same speed?

Solution:

Step 1: Set up the proportion. Let ‘d’ be the distance traveled in 5 hours.

180 km / 3 hours = d km / 5 hours

Step 2: Cross-multiply.

180 * 5 = 3 * d

Step 3: Simplify.

900 = 3d

Step 4: Solve for d (divide both sides by 3).

d = 900 / 3

d = 300 km

Answer: The car will travel 300 km in 5 hours.

Inverse Proportion

Inverse proportion describes a relationship where an increase in one quantity causes a proportional decrease in another quantity, or a decrease in one quantity causes a proportional increase in another quantity. If x and y are inversely proportional, then x = k/y, where k is a constant, or xy = k.

The following are characteristics of inverse proportion:

• As one quantity increases, the other quantity decreases.
• As one quantity decreases, the other quantity increases.
• The product of the two quantities remains constant.

Example 1: If 6 workers can complete a job in 10 days, how many days will it take 4 workers to complete the same job?

Solution:

Step 1: Identify as inverse proportion (fewer workers mean more days).

Step 2: Set up the relationship: Workers1 * Days1 = Workers2 * Days2

Step 3: Substitute values.

6 * 10 = 4 * x

Step 4: Simplify.

60 = 4x

Step 5: Solve for x (divide both sides by 4).

x = 60 / 4

x = 15 days

Answer: It will take 4 workers 15 days to complete the job.

Example 2: A cyclist travels at 20 km/h and takes 3 hours to reach a destination. How long will it take if the cyclist travels at 30 km/h?

Solution:

Step 1: Identify as inverse proportion (higher speed means less time).

Step 2: Set up the relationship: Speed1 * Time1 = Speed2 * Time2

Step 3: Substitute values.

20 * 3 = 30 * t

Step 4: Simplify.

60 = 30t

Step 5: Solve for t (divide both sides by 30).

t = 60 / 30

t = 2 hours

Answer: It will take 2 hours if the cyclist travels at 30 km/h.

Ratio of Family Size and Resources

This involves comparing the number of family members to available resources (like income, food, or space). It helps in understanding resource distribution per individual.

Example: A family of 4 has a weekly food budget of N8,000. What is the ratio of family members to the budget?

Solution:

Ratio = Family members : Budget

Ratio = 4 : 8,000

Step 1: Divide both sides by the greatest common factor (4).

Ratio = 1 : 2,000

Answer: The ratio of family members to the budget is 1:2,000, meaning N2,000 is available per person.

Ratio of Two Populations

This compares the number of people in two different groups, areas, or demographics. It helps in understanding population distribution or comparison.

Example: Town A has a population of 30,000 people, and Town B has a population of 45,000 people. What is the ratio of the population of Town A to Town B?

Solution:

Ratio = Population of Town A : Population of Town B

Ratio = 30,000 : 45,000

Step 1: Divide both sides by the greatest common factor (15,000).

Ratio = 2 : 3

Answer: The ratio of the population of Town A to Town B is 2:3.

Quantitative Reasoning on Ratio and Proportion

Quantitative reasoning problems involve using numerical data, relationships, and patterns to solve problems. These often include visual representations or word problems that require applying the principles of ratio and proportion.

Example: If 2 baskets represent 10 apples, how many apples do 5 baskets represent?

Solution:

Step 1: Find out how many apples 1 basket represents.

2 baskets = 10 apples

1 basket = 10 / 2 = 5 apples

Step 2: Calculate the number of apples for 5 baskets.

5 baskets = 5 * 5 apples = 25 apples

Answer: 5 baskets represent 25 apples.

Teaching Methods/Instructional Techniques

Discussion, Lecture, Demonstration, Question and Answer, Visual Aids

Instructional Procedures

Step 1: Introduction

Time: 5 minutes

Teaching Skill: Set Induction

Teacher’s Activity: The teacher greets the pupils and asks them to recall how they compare quantities, for example, comparing the number of boys to girls in the class. The teacher then introduces the topic “Ratio and Proportion” as a way to formally compare quantities.

Pupils’ Activity: Pupils respond to questions about comparing quantities and listen attentively to the introduction of the topic.

Learning Point: Pupils recall prior knowledge and are prepared for the new topic.

Step 2: Explanation of Ratio

Time: 8 minutes

Teaching Skill: Explanation/Demonstration

Teacher’s Activity: The teacher defines ratio, explains how to write ratios (a:b or a/b), and demonstrates how to simplify ratios, including those involving fractions, using examples from the board summary. The teacher emphasizes that ratios must be in their simplest form.

Pupils’ Activity: Pupils listen, take notes, and ask questions for clarification. They practice simplifying ratios from fractions as guided by the teacher.

Learning Point: Pupils understand what a ratio is and can simplify ratios, including those with fractions.

Step 3: Direct Proportion

Time: 8 minutes

Teaching Skill: Explanation/Problem Solving

Teacher’s Activity: The teacher defines direct proportion, explains its characteristics, and works through examples of direct proportion problems from the board summary, showing clear steps for solving for unknown quantities. The teacher asks pupils to identify other real-life situations that show direct proportion.

Pupils’ Activity: Pupils listen, observe the problem-solving steps, copy notes, and identify examples of direct proportion from their daily lives.

Learning Point: Pupils can identify and solve problems involving direct proportion.

Step 4: Inverse Proportion

Time: 8 minutes

Teaching Skill: Explanation/Problem Solving

Teacher’s Activity: The teacher defines inverse proportion, explains its characteristics, and works through examples of inverse proportion problems from the board summary, showing clear steps. The teacher guides pupils to identify activities that are inversely related.

Pupils’ Activity: Pupils listen, observe the problem-solving steps, take notes, and contribute examples of inversely related activities.

Learning Point: Pupils can identify and solve problems involving inverse proportion.

Step 5: Ratio of Family Size and Resources

Time: 5 minutes

Teaching Skill: Application/Discussion

Teacher’s Activity: The teacher explains how ratio applies to family size and resources, providing examples to illustrate how to calculate and interpret such ratios. The teacher encourages pupils to think about resource distribution in their own families.

Pupils’ Activity: Pupils listen, discuss, and apply the concept to real-life scenarios, noting down the examples.

Learning Point: Pupils understand how to apply ratios to real-life situations involving family size and resources.

Step 6: Ratio of Two Populations and Quantitative Reasoning

Time: 6 minutes

Teaching Skill: Application/Guided Practice

Teacher’s Activity: The teacher explains how to find the ratio of two populations using examples. The teacher then introduces quantitative reasoning problems related to ratio and proportion, guiding pupils through an example on the board.

Pupils’ Activity: Pupils listen, solve problems with teacher guidance, and ask questions to clarify their understanding of quantitative reasoning.

Learning Point: Pupils can calculate the ratio of two populations and begin to solve quantitative reasoning problems.

Step 7: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

1. Define ratio and proportion.
2. Simplify the ratio 2/5 : 1/10.
3. If 5 oranges cost N100, how much will 8 oranges cost?
4. If 3 painters can paint a house in 12 days, how long will it take 6 painters?
5. Give an example of how ratio is used in relation to family resources.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 2 minutes

Teaching Skill: Summarization

Teacher’s Activity: The teacher summarizes the key points of the lesson, reiterating the importance of ratio and proportion in everyday calculations and problem-solving. The teacher assigns homework for pupils to practice solving more problems.

Pupils’ Activity: Pupils listen to the summary and copy down the assigned homework.

Learning Point: Pupils consolidate their understanding of the topic and are given tasks to reinforce learning.

Home Task: Solve three new exercises on RATIO AND PROPORTION. Show each step and check each answer.

Lesson Keywords

• Ratio – A comparison of two or more quantities of the same kind.
• Proportion – A statement that two ratios are equal.
• Direct Proportion – A relationship where quantities increase or decrease together at the same rate.
• Inverse Proportion – A relationship where one quantity increases as the other decreases, and vice versa.
• Simplest Form – The reduced form of a ratio where the numbers have no common factors other than 1.
• Quantitative Reasoning – The ability to understand and use numerical data and relationships to solve problems.

Differentiation

For pupils who are struggling, the teacher will provide additional simpler examples and work through problems step-by-step with more guidance. Visual aids such as charts and diagrams will be used to illustrate concepts. For advanced pupils, the teacher will provide more complex word problems and encourage them to create their own real-life ratio and proportion scenarios to solve.

Note for teachers using this lesson plan

Teachers should ensure that pupils have a strong foundation in fractions and basic arithmetic before introducing ratio and proportion. Encourage active participation and group work to foster collaborative learning. Emphasize the real-world applications of ratio and proportion to make the lesson more engaging and relevant to pupils’ daily experiences.