Lesson Notes By Weeks and Term v3 - Junior Secondary 1
Multiplications and Divisions of fractions
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Multiplications and Divisions of fractions
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Subject: General Mathematics
Class: Junior Secondary 1
Term: 3rd Term
Week: 4
Theme: Basic Operations
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Solve problems on multiplication of fractions; Solve problems on division of fractions; Solve word problems in volving multiplication and division of fractions.
`2 whole 1/2 × 1 whole 1/5 = 3` cups of garri. 2.
2. Division of Fractions Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is obtained by flipping the numerator and the denominator. Rule for Dividing Fractions (Keep, Change, Flip - KCF): To divide one fraction by another, keep the first fraction as it is, change the division sign to a multiplication sign, and flip (invert) the second fraction (find its reciprocal). Then proceed with multiplication as described above.
Formula: `(a/b) ÷ (c/d) = (a/b) × (d/c)` Steps:
1. Convert Mixed Numbers: If any fraction is a mixed number, convert it to an improper fraction.
2. Keep, Change, Flip: Keep the first fraction. Change the division sign to multiplication. Flip (invert) the second fraction (find its reciprocal).
3. Multiply: Now, multiply the two fractions using the multiplication rules (cross-cancellation, then multiply numerators and denominators).
4. Simplify: Reduce the resulting fraction to its lowest terms. If it's an improper fraction, convert it back to a mixed number if required. Worked
Examples: Example 4: Dividing Proper Fractions Calculate: `(3/4) ÷ (1/2)` Step 1: No mixed numbers.
Step 2: Keep `3/4`, change `÷` to `×`, flip `1/2` to `2/1`. The problem becomes `(3/4) × (2/1)`.
Step 3 (Multiplication): Cross-cancellation: `2` and `4` have a common factor of `2`. `2 ÷ 2 = 1`, `4 ÷ 2 = 2`. The expression becomes `(3/2) × (1/1)`.
Multiply numerators: `3 × 1 = 3` Multiply denominators: `2 × 1 = 2` Step 4: Result: `3/2`.
Convert to a mixed number: `1 whole 1/2`.
Application (Nigerian Context): How many `1/2` litre bottles can be filled from a `3/4` litre container of palm oil? `(3/4) ÷ (1/2) = 1 whole 1/2` bottles. (This means one full bottle and half of another).
Example 5: Dividing a Whole Number by a Fraction Calculate: `8 ÷ (2/3)` Step 1: Convert the whole number to a fraction: `8 = 8/1`. The problem becomes `(8/1) ÷ (2/3)`.
Step 2: Keep `8/1`, change `÷` to `×`, flip `2/3` to `3/2`. The problem becomes `(8/1) × (3/2)`.
Step 3 (Multiplication): Cross-cancellation: `8` and `2` have a common factor of `2`. `8 ÷ 2 = 4`, `2 ÷ 2 = 1`. The expression becomes `(4/1) × (3/1)`.
Multiply numerators: `4 × 3 = 12` Multiply denominators: `1 × 1 = 1` Step 4: Result: `12/1 = 12`.
Application (Nigerian Context): A community project requires `2/3` of a bag of cement for each small construction. How many such constructions can be completed with 8 full bags of cement? `8 ÷ (2/3) = 12` constructions.
Example 6: Dividing Mixed Numbers Calculate: `3 whole 1/3 ÷ 1 whole 2/3` Step 1: Convert mixed numbers to improper fractions: `3 whole 1/3 = (3 × 3 + 1)/3 = 10/3` `1 whole 2/3 = (1 × 3 + 2)/3 = 5/3` The problem becomes `(10/3) ÷ (5/3)`.
Step 2: Keep `10/3`, change `÷` to `×`, flip `5/3` to `3/5`. The problem becomes `(10/3) × (3/5)`.
Step 3 (Multiplication): Cross-cancellation: `10` and `5` have a common factor of `5`. `10 ÷ 5 = 2`, `5 ÷ 5 = 1`. `3` and `3` have a common factor of `3`. `3 ÷ 3 = 1`, `3 ÷ 3 = 1`. The expression becomes `(2/1) × (1/1)`.
Multiply numerators: `2 × 1 = 2` Multiply denominators: `1 × 1 = 1` Step 4: Result: `2/1 = 2`.
Application (Nigerian Context): A mother has `3 whole 1/3` metres of fabric. If each of her children needs `1 whole 2/3` metres of fabric for school uniforms, how many children can she make uniforms for? `3 whole 1/3 ÷ 1 whole 2/3 = 2` children. 2.
3. Solving Word Problems Involving Multiplication and Division of Fractions General Strategy:
1. Read Carefully: Understand the context and what the problem is asking.
2. Identify Key Information: Extract the numerical values (fractions, whole This section provides a detailed explanation of the rules and procedures for multiplying and dividing fractions, including whole numbers and mixed numbers, with relevant examples. 2.
1. Multiplication of Fractions Multiplying fractions involves finding a fraction of another fraction. Conceptually, for example, "1/2 of 1/3" means taking half of an item that has already been divided into three equal parts.
Rule for Multiplying Fractions: To multiply two or more fractions, multiply their numerators together and multiply their denominators together. The resulting fraction should be simplified to its lowest terms.
Formula: `(a/b) × (c/d) = (a × c) / (b × d)` Steps:
1. Convert Mixed Numbers: If any fraction is a mixed number, convert it to an improper fraction first. (Recall: `Whole Number + Numerator/Denominator = (Whole Number × Denominator + Numerator) / Denominator`).
2. Cross-Cancellation (Optional but Recommended): Before multiplying, simplify the fractions by cancelling out common factors between any numerator and any denominator. This makes the multiplication easier and the final simplification simpler.
3. Multiply Numerators: Multiply all the numerators together.
4. Multiply Denominators: Multiply all the denominators together.
5. Simplify: Reduce the resulting fraction to its lowest terms. If it's an improper fraction, convert it back to a mixed number if required. Worked
Examples: Example 1: Multiplying Proper Fractions Calculate: `(2/3) × (4/5)` Step 1: No mixed numbers.
Step 2: No common factors for cross-cancellation (2 and 5 have no common factor, 4 and 3 have no common factor).
Step 3: Multiply numerators: `2 × 4 = 8` Step 4: Multiply denominators: `3 × 5 = 15` Step 5: Result: `8/15`. This is in its lowest terms.
Application (Nigerian Context): A farmer plants maize on `2/3` of his land. If `4/5` of the maize crop survives, what fraction of his total land is successfully cultivated with maize? `2/3` of `4/5` means `(2/3) × (4/5) = 8/15`. So, `8/15` of his total land is successfully cultivated.
Example 2: Multiplying a Fraction by a Whole Number Calculate: `5 × (3/4)` Step 1: Convert the whole number to a fraction: `5 = 5/1`. So the problem becomes `(5/1) × (3/4)`.
Step 2: No common factors for cross-cancellation.
Step 3: Multiply numerators: `5 × 3 = 15` Step 4: Multiply denominators: `1 × 4 = 4` Step 5: Result: `15/4`.
Convert to a mixed number: `3 whole 3/4`.
Application (Nigerian Context): A tailor uses `3/4` metre of fabric for one small Ankara bag. How much fabric does he need for 5 such bags? `5 × (3/4) = 15/4 = 3 whole 3/4` metres of fabric.
Example 3: Multiplying Mixed Numbers (with cross-cancellation)
Calculate: `2 whole 1/2 × 1 whole 1/5` Step 1: Convert mixed numbers to improper fractions: `2 whole 1/2 = (2 × 2 + 1)/2 = 5/2` `1 whole 1/5 = (1 × 5 + 1)/5 = 6/5` The problem becomes `(5/2) × (6/5)`.
Step 2: Cross-cancellation: The numerator `5` and the denominator `5` cancel out to `1`. The numerator `6` and the denominator `2` have a common factor of `2`. `6 ÷ 2 = 3`, `2 ÷ 2 = 1`. The expression becomes `(1/1) × (3/1)`.
Step 3: Multiply numerators: `1 × 3 = 3` Step 4: Multiply denominators: `1 × 1 = 1` Step 5: Result: `3/1 = 3`.
Application (Nigerian Context): A cook uses `2 whole 1/2` cups of garri for a recipe. If she wants to increase the recipe by `1 whole 1/5` times, how much garri will she need? * `2 whole 1/2 × 1 whole 1/5 = 3` cups of garri. 2.
2. Division of Fractions Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is obtained by flipping the numerator and the denominator. Rule for Dividing Fractions (Keep, Change, Flip - KCF): To divide one fraction by another, keep the first fraction as it is, change the division sign to a multiplication sign, and flip (invert) the second fraction (find its reciprocal). Then proceed with multiplication as described above.
Formula: `(a/b) ÷ × 1 = 1` Step 4: Result: `2/1 = 2`.
Application (Nigerian Context): A mother has `3 whole 1/3` metres of fabric. If each of her children needs `1 whole 2/3` metres of fabric for school uniforms, how many children can she make uniforms for? * `3 whole 1/3 ÷ 1 whole 2/3 = 2` children. 2.
3. Solving Word Problems Involving Multiplication and Division of Fractions General Strategy:
1. Read Carefully: Understand the context and what the problem is asking.
2. Identify Key Information: Extract the numerical values (fractions, whole numbers) and keywords that indicate operations (e.g., "of" usually means multiplication, "shared equally" or "how many groups" usually means division).
3. Choose the Operation: Determine whether multiplication or division (or both) is needed.
4. Formulate the Expression: Write down the mathematical expression.
5. Solve: Perform the calculations using the steps outlined above.
6. State the Answer Clearly: Write the answer in the context of the problem, including units if applicable. --- Phase 1: Introduction (5 minutes)
Teacher Activity: Recaps previous knowledge on fractions: identifying proper, improper, mixed numbers, and converting between them. Poses a real-life scenario to spark interest, e.g., "If Mummy gives you half of her `1/2` loaf of bread, what fraction of the whole loaf did you get?" (introduces multiplication concept intuitively). States the lesson objectives clearly.
Student Activity: Respond to recap questions. Listen attentively to the real-life scenario and lesson objectives.
Phase 2: Development of Concepts (25 minutes)
Activity 1: Multiplication of Fractions Teacher Activity: Explains the concept of multiplication of fractions, starting with simple proper fractions using diagrams (e.g., drawing a rectangle, dividing it into thirds, then shading half of that third to show `1/2 × 1/3 = 1/6`).
Introduces the rule: multiply numerators, multiply denominators. Demonstrates how to handle whole numbers (as fractions over 1) and mixed numbers (convert to improper fractions). Emphasizes and demonstrates cross-cancellation for simplification before multiplying. Works through Example 1, 2, and 3 from Section 2.1 on the board, encouraging student input.
Student Activity: Observe and actively participate in diagrammatic representation. Copy notes and worked examples. Ask clarifying questions. Attempt parts of the examples on their own or in pairs as guided by the teacher.
Activity 2: Division of Fractions Teacher Activity: Introduces the concept of division of fractions using a simple scenario, e.g., "How many `1/4` cup servings are in `1/2` cup of rice?" Explains the "Keep, Change, Flip" (KCF) rule with emphasis on the reciprocal concept. Demonstrates how to handle whole numbers and mixed numbers during division (always convert to improper fractions first). Works through Example 4, 5, and 6 from Section 2.2 on the board, involving students in each step.
Student Activity: Listen and grasp the KCF rule. Copy notes and worked examples. Practice inverting fractions. Engage in solving the examples presented.
Phase 3: Solving Word Problems (10 minutes)
Teacher Activity: Guides students on the systematic approach to solving word problems (read, identify, choose operation, formulate, solve, state answer). Presents a word problem that combines both multiplication and division or involves multi-step fractional operations. Facilitates a class discussion to break down the problem and identify the necessary operations. Demonstrates the solution clearly, linking it back to real-life applications discussed.
Student Activity: Participate in dissecting the word problem. Suggest appropriate operations. Work along with the teacher to solve the problem. Confirm understanding.
Phase 4: Class Practice / Guided Practice (15 minutes)
Teacher Activity: Distributes guided practice questions (from Section 4). Monitors students as they work individually or in pairs. Provides immediate feedback and clarification as needed. Selects students to present their solutions on the board.
Student Activity: Attempt guided practice questions. Seek clarification from the teacher or peers. Present solutions and explain their reasoning.
Phase 5: Conclusion (5 minutes)
Teacher Activity: Summarizes key learning points on multiplication and division of fractions. Reinforces the importance of simplifying fractions and converting mixed numbers. Assigns independent practice questions as homework.
Student Activity: Participate in the summary. Note down homework assignment. ---
Example 1: Multiplying Proper Fractions
Calculate: `(2/3) × (4/5)`
Step 1: No mixed numbers.
Step 2: No common factors for cross-cancellation (2 and 5 have no common factor, 4 and 3 have no common factor).
Step 3: Multiply numerators: `2 × 4 = 8`
Step 4: Multiply denominators: `3 × 5 = 15`
Step 5: Result: `8/15`. This is in its lowest terms.
Application (Nigerian Context): A farmer plants maize on `2/3` of his land. If `4/5` of the maize crop survives, what fraction of his total land is successfully cultivated with maize?
`2/3` of `4/5` means `(2/3) × (4/5) = 8/15`. So, `8/15` of his total land is successfully cultivated.
Example 2: Multiplying a Fraction by a Whole Number
Calculate: `5 × (3/4)`
Step 1: Convert the whole number to a fraction: `5 = 5/1`. So the problem becomes `(5/1) × (3/4)`.
Step 2: No common factors for cross-cancellation.
Step 3: Multiply numerators: `5 × 3 = 15`
Step 4: Multiply denominators: `1 × 4 = 4`
Step 5: Result: `15/4`.
Convert to a mixed number: `3 whole 3/4`.
Application (Nigerian Context): A tailor uses `3/4` metre of fabric for one small Ankara bag. How much fabric does he need for 5 such bags?
`5 × (3/4) = 15/4 = 3 whole 3/4` metres of fabric.
Example 3: Multiplying Mixed Numbers (with cross-cancellation)
Calculate: `2 whole 1/2 × 1 whole 1/5`
Step 1: Convert mixed numbers to improper fractions:
`2 whole 1/2 = (2 × 2 + 1)/2 = 5/2`
`1 whole 1/5 = (1 × 5 + 1)/5 = 6/5`
The problem becomes `(5/2) × (6/5)`.
Step 2: Cross-cancellation:
The numerator `5` and the denominator `5` cancel out to `1`.
The numerator `6` and the denominator `2` have a common factor of `2`. `6 ÷ 2 = 3`, `2 ÷ 2 = 1`.
The expression becomes `(1/1) × (3/1)`.
Step 3: Multiply numerators: `1 × 3 = 3`
Step 4: Multiply denominators: `1 × 1 = 1`
Step 5: Result: `3/1 = 3`.
Application (Nigerian Context): A cook uses `2 whole 1/2` cups of garri for a recipe. If she wants to increase the recipe by `1 whole 1/5` times, how much garri will she need?
`2 whole 1/2 × 1 whole 1/5 = 3` cups of garri.
2. 2. Division of Fractions
Agriculture and Land Management: Farmers in rural Nigeria often deal with dividing land or harvests. For example, if a family owns `15` hectares of land and decides to plant cassava on `2/5` of it, students can calculate the exact number of hectares used for cassava (`2/5 × 15 = 6` hectares). If a `3 whole 1/2` hectare plot needs to be divided into `1/4` hectare plots for different crops, students can calculate how many plots are available (`3 whole 1/2 ÷ 1/4 = 14` plots).
Cooking and Catering: Many Nigerian dishes require precise measurements, and often recipes need to be scaled up or down. For instance, if a recipe for Egusi soup serves 8 people and requires `1 whole 1/4` cups of palm oil, students can calculate how much oil is needed for 4 people (`1/2 × 1 whole 1/4 = 5/8` cup) or for 12 people (`1 whole 1/2 × 1 whole 1/4 = 1 whole 7/8` cups).
Small Business and Profit Sharing: In many informal businesses (e.g., market traders, craftsmen), profits are often shared proportionally. If a business makes `₦120,000` profit and partners agree to share it such that one partner gets `3/8` of the profit and another gets `1/4`, students can calculate each partner's share (`3/8 × 120,000 = ₦45,000`, `1/4 × 120,000 = ₦30,000`). If a shared amount needs to be divided into fractional parts, for example, `3/4` of a bale of wrappers to be shared among 5 people, they can calculate each person's share (`3/4 ÷ 5 = 3/20` of a bale). ---