Lesson Notes By Weeks and Term v3 - Primary 6
Fractions
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Fractions
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Subject: General Mathematics
Class: Primary 6
Term: 1st Term
Week: 3
Theme: Number And Numeration
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Or der fractions; Solve problem on quantitative reasoning; Express decimals as fractions and vice versa.
This section provides a detailed explanation of the core concepts, supported by step-by-step examples relevant to the Nigerian context. A. Ordering Fractions Ordering fractions involves arranging them from smallest to largest (ascending order) or largest to smallest (descending order). This requires comparing the magnitudes of different fractions.
Methods for Ordering Fractions:
1. Fractions with the Same Denominator: When fractions have the same denominator, the fraction with the larger numerator is the larger fraction.
Example: Compare $\frac{3}{7}$ and $\frac{5}{7}$. Since $5 > 3$, then $\frac{5}{7} > \frac{3}{7}$.
2. Fractions with the Same Numerator: When fractions have the same numerator, the fraction with the smaller denominator is the larger fraction. (Think: dividing a whole into fewer, larger pieces).
Example: Compare $\frac{2}{5}$ and $\frac{2}{3}$. Since $3 \frac{2}{5}$.
3. Fractions with Different Denominators (General Method): This is the most common scenario. To compare or order such fractions, they must first be converted into equivalent fractions with a common denominator. The Least Common Multiple (LCM) of the denominators is the most efficient common denominator.
Steps: a. Find the LCM of all the denominators. b. Convert each fraction into an equivalent fraction with the LCM as its new denominator. c. Compare the new numerators. The fraction with the larger numerator is the larger fraction.
Worked Example 1: Ordering Fractions Problem: Arrange the following fractions in ascending order: $\frac{1}{2}$, $\frac{2}{3}$, $\frac{3}{5}$.
Solution:
1. Find the LCM of the denominators (2, 3, 5): The LCM of 2, 3, and 5 is $2 \times 3 \times 5 = 30$.
2. Convert each fraction to an equivalent fraction with a denominator of 30: For $\frac{1}{2}$: $\frac{1 \times 15}{2 \times 15} = \frac{15}{30}$ For $\frac{2}{3}$: $\frac{2 \times 10}{3 \times 10} = \frac{20}{30}$ For $\frac{3}{5}$: $\frac{3 \times 6}{5 \times 6} = \frac{18}{30}$
3. Compare the numerators (15, 20, 18): In ascending order: $15 < 18 < 20$.
4. Write the original fractions in ascending order: $\frac{15}{30}$ corresponds to $\frac{1}{2}$ $\frac{18}{30}$ corresponds to $\frac{3}{5}$ $\frac{20}{30}$ corresponds to $\frac{2}{3}$ Therefore, the ascending order is $\frac{1}{2}, \frac{3}{5}, \frac{2}{3}$. B. Solving Problems on Quantitative Reasoning Quantitative reasoning problems involving fractions require students to understand the problem context, apply logical thinking, and use their knowledge of fractional operations (addition, subtraction, multiplication, division) to find solutions. These problems often involve multiple steps.
Key Strategies: Read the problem carefully to understand what is given and what needs to be found. Identify the operations required. Break down complex problems into smaller, manageable steps. Draw diagrams or models if helpful. Ensure answers are in the required units or form.
Worked Example 2: Quantitative Reasoning Problem: Mr. Chinedu has a plot of land. He cultivated cassava on $\frac{1}{3}$ of the land, maize on $\frac{1}{4}$ of the land, and yam on the remaining portion. What fraction of the land is used for yam?
Solution:
1. Fraction of land for cassava and maize: Add the fractions for cassava and maize: $\frac{1}{3} + \frac{1}{4}$ Find the LCM of 3 and 4, which is 12. $\frac{1 \times 4}{3 \times 4} + \frac{1 \times 3}{4 \times 3} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}$. So, $\frac{7}{12}$ of the land is used for cassava and maize.
2. Fraction of land remaining for yam: The whole plot of land can be represented as 1 (or $\frac{12}{12}$). Subtract the fraction used for cassava and maize from the whole: $1 - \frac{7}{12} = \frac{12}{12} - \frac{7}{12} = \frac{5}{12}$.
Answer: $\frac{5}{12}$ of the land is used for yam.
C. Expressing Decimals as Fractions and Vice Versa
1. Expressing Decimals as Fractions: A decimal number can be written as a fraction by considering its place value.
Steps: a. Write the decimal number without the decimal point as the numerator. b. The denominator will be a power of 10 (10, 100, 1000, etc.) corresponding to the number of decimal places. One decimal place (e.g., 0.X) means denominator
1
0. Two decimal places (e.g., 0.XY) means denominator 100. * Three decimal places (e.g., 0.XYZ) means denominator 1000, and so yam.
C. Expressing Decimals as Fractions and Vice Versa
1. Expressing Decimals as Fractions: A decimal number can be written as a fraction by considering its place value.
Steps: a. Write the decimal number without the decimal point as the numerator. b. The denominator will be a power of 10 (10, 100, 1000, etc.) corresponding to the number of decimal places. One decimal place (e.g., 0.X) means denominator
1
0. Two decimal places (e.g., 0.XY) means denominator
1
0
0. Three decimal places (e.g., 0.XYZ) means denominator 1000, and so on. c. Simplify the resulting fraction to its lowest terms by dividing the numerator and denominator by their greatest common divisor (GCD).
Worked Example 3: Decimal to Fraction Problem: Express $0.75$ as a fraction in its simplest form.
Solution:
1. The decimal $0.75$ has two decimal places.
2. Write $75$ as the numerator and $100$ as the denominator: $\frac{75}{100}$.
3. Simplify the fraction: Both 75 and 100 are divisible by 25. $\frac{75 \div 25}{100 \div 25} = \frac{3}{4}$.
Answer: $0.75 = \frac{3}{4}$.
Worked Example 4: Decimal to Fraction (with whole number part)
Problem: Express $3.125$ as a mixed number fraction in its simplest form.
Solution:
1. Separate the whole number part (3) from the decimal part (0.125).
2. Convert the decimal part $0.125$ to a fraction: $0.125$ has three decimal places. $\frac{125}{1000}$.
3. Simplify $\frac{125}{1000}$: Both 125 and 1000 are divisible by 125. $\frac{125 \div 125}{1000 \div 125} = \frac{1}{8}$.
4. Combine the whole number part with the simplified fraction: $3\frac{1}{8}$.
Answer: $3.125 = 3\frac{1}{8}$.
2. Expressing Fractions as Decimals: A fraction can be converted to a decimal by dividing the numerator by the denominator.
Steps: a.
Perform long division: Divide the numerator by the denominator. b. Add a decimal point and zeros to the numerator if the division does not result in a whole number. Continue dividing until the remainder is zero or a repeating pattern emerges (for Primary 6, focus on terminating decimals).
Worked Example 5: Fraction to Decimal Problem: Express $\frac{3}{8}$ as a decimal.
Solution:
1. Divide 3 by 8 using long division: ``` 0.375 _______ 8 | 3.000 - 0 --- 3 0 - 2 4 ----- 6 0 - 5 6 ----- 4 0 - 4 0 ----- 0 ``` Answer: $\frac{3}{8} = 0.375$.
Worked Example 6: Fraction to Decimal (Common Fraction)
Problem: Express $\frac{1}{5}$ as a decimal.
Solution:
1. Divide 1 by 5: ``` 0.2 _______ 5 | 1.0 - 0 --- 1 0 - 1 0 ----- 0 ``` * Answer: $\frac{1}{5} = 0.2$. --- This section outlines the pedagogical steps for delivering the lesson, involving both teacher and student engagement.
Teacher Activities: Introduction & Review (10 minutes): Begin by briefly reviewing previous concepts of fractions (what a fraction is, types of fractions, addition/subtraction of fractions).
Pose a simple real-life scenario: "Imagine sharing a loaf of bread among three children. One gets $\frac{1}{2}$, another gets $\frac{1}{4}$, and the third gets $\frac{1}{4}$. Are the shares equal? What if one got $\frac{1}{3}$ and another $\frac{1}{2}$? How do we know who has more?" This leads to the concept of ordering.
Ordering Fractions (15 minutes): Demonstrate methods for ordering fractions on the board: Using same denominators (e.g., $\frac{2}{5}, \frac{4}{5}$). Using same numerators (e.g., $\frac{3}{4}, \frac{3}{8}$).
Focus on the general method: Explain finding the LCM of denominators using prime factorization and converting to equivalent fractions. Work through Worked Example 1 with the class, step-by-step. Use fraction strips or drawings to visually represent and compare fractions, especially for visual learners. Quantitative Reasoning Problems (15 minutes): Introduce quantitative reasoning as applying fraction knowledge to solve practical word problems. Read Worked Example 2 aloud, guiding students to identify key information and the steps required to solve it. Emphasize breaking down the problem into smaller, manageable parts (e.g., first find combined fraction, then find remainder). Decimals and Fractions Conversion (20 minutes): Decimals to Fractions: Explain place values (tenths, hundredths, thousandths) using a place value chart. Demonstrate how to write decimals as fractions with powers of 10 as denominators, then simplify. Work through Worked Example 3 and
4. Fractions to Decimals: Explain that a fraction is a division problem. Demonstrate the long division method for converting fractions to decimals. Work through Worked Example 5 and
6. Highlight common fraction-decimal equivalents like $\frac{1}{2}=0.5$, $\frac{1}{4}=0.25$, $\frac{3}{4}=0.75$. Guided Practice & Consolidation (15 minutes): Lead students through a few practice problems that cover all objectives. Provide hints and support as needed. Encourage peer discussion during this phase.
Wrap-up & Assignment (5 minutes): Summarize the key takeaways for the lesson. Assign independent practice questions for homework.
Student Activities: Actively participate in class discussions and question-and-answer sessions. Work in small groups (if feasible) to compare and order fractions using fraction manipulatives (strips, circles, or drawn diagrams). Solve the quantitative reasoning problems presented by the teacher, discussing strategies with group members. Practice converting decimals to fractions and fractions to decimals on their whiteboards or exercise books. Present their solutions and reasoning to the class when called upon. Take notes on key concepts and worked examples. Complete assigned practice exercises independently. --- This section provides scaffolded practice problems that directly target the performance objectives, with detailed solutions for the teacher.
Question 1 (Ordering Fractions): Three friends, Ama, Biodun, and Chika, shared a bag of oranges. Ama got $\frac{2}{5}$ of the oranges, Biodun got $\frac{1}{3}$, and Chika got $\frac{3}{10}$. Arrange their shares from smallest to largest.
Solution: Identify the fractions: $\frac{2}{5}, \frac{1}{3}, \frac{3}{10}$. Find the LCM of the denominators (5, 3, 10): Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30...
Multiples of 5: 5, 10, 15, 20, 25, 30...
Multiples of 10: 10, 20, 30... LCM =
3
0. Convert to equivalent fractions with denominator 30: Ama: $\frac{2}{5} = \frac{2 \times 6}{5 \times 6} = \frac{12}{30}$ Biodun: $\frac{1}{3} = \frac{1 \times 10}{3 \times 10} = \frac{10}{30}$ Chika: $\frac{3}{10} = \frac{3 \times 3}{10 \times 3} = \frac{9}{30}$ Compare numerators and order: Numerators are 12, 10,
9. Smallest to largest: $9 < 10 < 12$. Write the original fractions in ascending order: $\frac{9}{30}$ (Chika) $\rightarrow \frac{3}{10}$ $\frac{10}{30}$ (Biodun) $\rightarrow \frac{1}{3}$ $\frac{12}{30}$ (Ama) $\rightarrow \frac{2}{5}$ Answer: The shares from smallest to largest are $\frac{3}{10}$ (Chika), $\frac{1}{3}$ (Biodun), $\frac{2}{5}$ (Ama).
Question 2 (Quantitative Reasoning): A trader at Onitsha Main Market sold $\frac{3}{5}$ of her bag of rice on Tuesday and $\frac{1}{2}$ of the remainder on Wednesday. If the bag originally contained 50 kg of rice, how many kilograms of rice were sold on Wednesday?
Solution: Amount sold on Tuesday: Fraction sold = $\frac{3}{5}$. Amount sold = $\frac{3}{5} \times 50 \text{ kg} = \frac{150}{5} \text{ kg} = 30 \text{ kg}$.
Remainder after Tuesday's sales: Original amount - amount sold on Tuesday = $50 \text{ kg} - 30 \text{ kg} = 20 \text{ kg}$.
Amount sold on Wednesday: Fraction sold on Wednesday = $\frac{1}{2}$ of the remainder. Amount sold = $\frac{1}{2} \times 20 \text{ kg} = 10 \text{ kg}$.
Answer: The trader sold 10 kg of rice on Wednesday.
Question 3 (Decimal to Fraction): Convert $0.36$ to a fraction in its simplest form.
Solution: Identify place value: $0.36$ has two decimal places (hundredths).
Write as a fraction: $\frac{36}{100}$.
Simplify: Find the greatest common divisor (GCD) of 36 and
1
0
0. Both are divisible by 4. $\frac{36 \div 4}{100 \div 4} = \frac{9}{25}$.
Answer: $0.36 = \frac{9}{25}$.
Question 4 (Fraction to Decimal): Express $\frac{7}{20}$ as a decimal.
Solution: Divide numerator by denominator: $7 \div 20$. ``` 0.35 _______ 20 | 7.00 0 7 0 6 0 1 00 1 00 0 ``` Answer: $\frac{7}{20} = 0.35$. ---
Connecting the abstract concepts of fractions and decimals to tangible situations helps students appreciate their relevance and utility in the Nigerian context.
Marketplace and Trade: Buying and Selling Goods: When buying items like rice, beans, or garri, quantities are often measured in fractions (e.g., "half a bag," "quarter measure"). Prices might involve decimals (e.g., ₦150.50). Traders need to calculate remaining stock, proportions of sales, or discounts using fractions. Students can relate to scenarios where a market seller needs to determine what fraction of their goods has been sold or how much of a product they have left.
Sharing Produce: A farmer might divide a harvest of tubers (yam, cassava) into shares for family members or for sale, using fractional portions.
Resource Management and Allocation: Land Use: Families or communities in rural Nigeria often divide plots of land for different crops or among family members. Understanding fractions is crucial for ensuring fair and accurate division (e.g., "$\frac{1}{3}$ for corn, $\frac{1}{4}$ for vegetables, and the rest for building").
Budgeting and Savings: A household or an individual can use fractions to allocate their monthly income (e.g., $\frac{1}{2}$ for food, $\frac{1}{4}$ for transport, $\frac{1}{8}$ for savings, and the remainder for other expenses). This teaches financial literacy and responsibility.
Cooking and Recipe Proportions: Nigerian cuisine involves precise measurements, even if traditionally done by "eye." Recipes often call for fractional amounts of ingredients (e.g., "$\frac{1}{2}$ cup of palm oil," "$\frac{1}{4}$ teaspoon of salt," "a few slices of yam"). Understanding how to adjust these fractions for larger or smaller servings is a practical application. ---