Lesson Notes By Weeks and Term v3 - Junior Secondary 1
Addition and subtraction of fractions
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Addition and subtraction of fractions
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Subject: General Mathematics
Class: Junior Secondary 1
Term: 2nd Term
Week: 3
Theme: Basic Operations
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Solve given problems on addition and subtraction of fractions Solve word problems in volving addition and subtraction of fractions.
1: Add the whole number parts.
Step 2: Add the fractional parts (using LCM if denominators are different).
Step 3: If the sum of the fractional parts is an improper fraction, convert it to a mixed number.
Step 4: Add this new whole number part to the sum of the original whole numbers.
Step 5: Simplify the resulting fraction.
Example 3 (Nigerian Context): A market woman sold $1\frac{1}{2}$ baskets of oranges in the morning and $2\frac{3}{4}$ baskets in the afternoon. How many baskets of oranges did she sell in total? $1\frac{1}{2} + 2\frac{3}{4}$ Using Method A (Improper Fractions): Convert $1\frac{1}{2}$ to improper fraction: $\frac{(1 \times 2) + 1}{2} = \frac{3}{2}$. Convert $2\frac{3}{4}$ to improper fraction: $\frac{(2 \times 4) + 3}{4} = \frac{11}{4}$.
Now add: $\frac{3}{2} + \frac{11}{4}$. LCM of $2$ and $4$ is $4$.
Equivalent fractions: $\frac{3}{2} = \frac{6}{4}$ (since $4 \div 2 = 2$, and $3 \times 2 = 6$).
Add: $\frac{6}{4} + \frac{11}{4} = \frac{6+11}{4} = \frac{17}{4}$.
Convert back to mixed number: $17 \div 4 = 4$ remainder $1$. So, $4\frac{1}{4}$.
Result: $4\frac{1}{4}$ baskets.
Using Method B (Separate Addition): Add whole numbers: $1 + 2 = 3$.
Add fractions: $\frac{1}{2} + \frac{3}{4}$. LCM of $2$ and $4$ is $4$. $\frac{1}{2} = \frac{2}{4}$. $\frac{2}{4} + \frac{3}{4} = \frac{5}{4}$. Convert $\frac{5}{4}$ to mixed number: $1\frac{1}{4}$. Add this new whole number to the sum of original whole numbers: $3 + 1\frac{1}{4} = 4\frac{1}{4}$.
Result: $4\frac{1}{4}$ baskets.
Explanation: Both methods yield the same result. Method B can be quicker if the fractional parts are easy to combine, but Method A is generally more robust for all cases. 2.3 Subtraction of Fractions Case 1: Fractions with the Same Denominators (Like Fractions) To subtract fractions with the same denominator, subtract the numerators and keep the denominator the same. Simplify the result if possible.
Step 1: Ensure denominators are identical.
Step 2: Subtract the numerators.
Step 3: Keep the common denominator.
Step 4: Simplify the resulting fraction.
Example 4 (Nigerian Context): A child ate $\frac{3}{8}$ of a cake. If the total cake represents $\frac{8}{8}$, what fraction of the cake is left? $\frac{8}{8} - \frac{3}{8}$ Subtract the numerators: $8 - 3 = 5$ Keep the denominator: $8$ Result: $\frac{5}{8}$ Explanation: Similar to addition, when parts are of the same size (eighths), we simply count how many parts remain.
Case 2: Fractions with Different Denominators (Unlike Fractions) To subtract fractions with different denominators, find the LCM of the denominators. Convert each fraction to an equivalent fraction with the LCM as the new denominator. Then, subtract the equivalent fractions as in Case
1. Step 1: Find the LCM of the denominators.
Step 2: Convert each fraction to an equivalent fraction using the LC
M. Step 3: Subtract the new numerators.
Step 4: Keep the common LCM denominator.
Step 5: Simplify the resulting fraction.
Example 5 (Nigerian Context): From a bag of rice, $\frac{1}{2}$ was used for a party. Later, $\frac{1}{5}$ of the original bag was used for family consumption. What fraction of the bag of rice was used more for the party than for family consumption? $\frac{1}{2} - \frac{1}{5}$ Step 1: Find LCM of $2$ and $5$.
Multiples of 2: 2, 4, 6, 8, 10, 12...
Multiples of 5: 5, 10, 15... LCM = $10$.
Step 2: Convert to equivalent fractions with denominator
1
0. For $\frac{1}{2}$: $10 \div 2 = 5$.
Multiply numerator by 5: $1 \times 5 = 5$. So, $\frac{1}{2} = \frac{5}{10}$. For $\frac{1}{5}$: $10 \div 5 = 2$.
Multiply numerator by 2: $1 \times 2 = 2$. So, $\frac{1}{5} = \frac{2}{10}$.
Step 3: Subtract the new numerators: $\frac{5}{10} - \frac{2}{10} = \frac{5-2}{10} = \frac{3}{10}$.
Result: $\frac{3}{10}$ Explanation: We need to express both quantities of rice in terms of equally sized 'small' parts (tenths) before finding the difference.
Case 3: Subtracting Mixed Numbers Again, two main methods: Method A: Convert to Improper Fractions (Recommended for JSS1)** * 2.1 Review of Basic Fraction Concepts Fraction: Represents a part of a whole. It consists of a numerator (top number) and a denominator (bottom number).
Numerator: Indicates the number of parts being considered.
Denominator: Indicates the total number of equal parts the whole is divided into.
Proper Fraction: Numerator is less than the denominator (e.g., $\frac{1}{2}$, $\frac{3}{4}$).
Improper Fraction: Numerator is greater than or equal to the denominator (e.g., $\frac{5}{3}$, $\frac{7}{7}$).
Mixed Number: Consists of a whole number and a proper fraction (e.g., $1\frac{1}{2}$, $2\frac{3}{4}$). 2.2 Addition of Fractions Case 1: Fractions with the Same Denominators (Like Fractions) To add fractions with the same denominator, add the numerators and keep the denominator the same. Simplify the result if possible.
Step 1: Ensure denominators are identical.
Step 2: Add the numerators.
Step 3: Keep the common denominator.
Step 4: Simplify the resulting fraction (reduce to lowest terms or convert to a mixed number if it's an improper fraction).
Example 1 (Nigerian Context): A farmer used $\frac{2}{5}$ of his plot for maize and $\frac{1}{5}$ for yam. What fraction of the plot did he use for both crops? $\frac{2}{5} + \frac{1}{5}$ Add the numerators: $2 + 1 = 3$ Keep the denominator: $5$ Result: $\frac{3}{5}$ Explanation: When parts are of the same size (fifths), we simply count how many of those parts we have in total.
Case 2: Fractions with Different Denominators (Unlike Fractions) To add fractions with different denominators, first find the Least Common Multiple (LCM) of the denominators. Convert each fraction to an equivalent fraction with the LCM as the new denominator. Then, add the equivalent fractions as in Case
1. Step 1: Find the LCM of the denominators.
Step 2: Convert each fraction to an equivalent fraction using the LCM as the new denominator. To do this, divide the LCM by the original denominator, then multiply the result by the original numerator.
Step 3: Add the new numerators.
Step 4: Keep the common LCM denominator.
Step 5: Simplify the resulting fraction.
Example 2 (Nigerian Context): A tailor used $\frac{1}{3}$ of a roll of fabric for a traditional attire and $\frac{1}{4}$ for another garment. What total fraction of the fabric roll was used? $\frac{1}{3} + \frac{1}{4}$ Step 1: Find LCM of $3$ and $4$.
Multiples of 3: 3, 6, 9, 12, 15...
Multiples of 4: 4, 8, 12, 16... LCM = $12$.
Step 2: Convert to equivalent fractions with denominator
1
2. For $\frac{1}{3}$: $12 \div 3 = 4$.
Multiply numerator by 4: $1 \times 4 = 4$. So, $\frac{1}{3} = \frac{4}{12}$. For $\frac{1}{4}$: $12 \div 4 = 3$.
Multiply numerator by 3: $1 \times 3 = 3$. So, $\frac{1}{4} = \frac{3}{12}$.
Step 3: Add the new numerators: $\frac{4}{12} + \frac{3}{12} = \frac{4+3}{12} = \frac{7}{12}$.
Result: $\frac{7}{12}$ Explanation: We need to express both portions of fabric in terms of equally sized 'small' parts before combining them. The smallest common 'small part' here is 'twelfths'.
Case 3: Adding Mixed Numbers There are two main methods: Method A: Convert to Improper Fractions Step 1: Convert each mixed number to an improper fraction. (Whole number $\times$ Denominator + Numerator) / Denominator.
Step 2: If the denominators are different, find the LCM and convert to equivalent fractions.
Step 3: Add the improper fractions.
Step 4: Convert the resulting improper fraction back to a mixed number if necessary and simplify.
Method B: Add Whole Numbers and Fractions Separately Step 1: Add the whole number parts.
Step 2: Add the fractional parts (using LCM if denominators are different).
Step 3: If the sum of the fractional parts is an improper fraction, convert it to a mixed number.
Step 4: Add this new whole number part to the sum of the original whole numbers.
Step 5: Simplify the resulting fraction. * Example 3 (Nigerian Context): A market woman sold $1\frac{1}{2}$ baskets of oranges in the morning and $2\frac{3}{4}$ baskets in the afternoon. How many baskets of oranges did she Multiply numerator by 5: $1 \times 5 = 5$. So, $\frac{1}{2} = \frac{5}{10}$. For $\frac{1}{5}$: $10 \div 5 = 2$.
Multiply numerator by 2: $1 \times 2 = 2$. So, $\frac{1}{5} = \frac{2}{10}$.
Step 3: Subtract the new numerators: $\frac{5}{10} - \frac{2}{10} = \frac{5-2}{10} = \frac{3}{10}$.
Result: $\frac{3}{10}$ Explanation: We need to express both quantities of rice in terms of equally sized 'small' parts (tenths) before finding the difference.
Case 3: Subtracting Mixed Numbers Again, two main methods: Method A: Convert to Improper Fractions (Recommended for JSS1)
Step 1: Convert each mixed number to an improper fraction.
Step 2: If the denominators are different, find the LCM and convert to equivalent fractions.
Step 3: Subtract the improper fractions.
Step 4: Convert the resulting improper fraction back to a mixed number if necessary and simplify.
Method B: Subtract Whole Numbers and Fractions Separately (Requires "Borrowing")
Step 1: Subtract the fractional parts. If the fraction being subtracted is larger than the fraction it's being subtracted from, "borrow" 1 from the whole number part of the minuend (the first number). Convert the borrowed 1 into an equivalent fraction with the same denominator as the fractional part, and add it.
Step 2: Subtract the whole number parts.
Step 3: Combine the results.
Step 4: Simplify the resulting fraction.
Note for teachers: Method B can be confusing for JSS1 learners due to the "borrowing" concept. Method A is generally more straightforward and less prone to errors at this level.
Example 6 (Nigerian Context): A painter had $5\frac{1}{2}$ litres of paint. He used $2\frac{3}{4}$ litres for a project. How much paint is left? $5\frac{1}{2} - 2\frac{3}{4}$ Using Method A (Improper Fractions): Convert $5\frac{1}{2}$ to improper fraction: $\frac{(5 \times 2) + 1}{2} = \frac{11}{2}$. Convert $2\frac{3}{4}$ to improper fraction: $\frac{(2 \times 4) + 3}{4} = \frac{11}{4}$.
Now subtract: $\frac{11}{2} - \frac{11}{4}$. LCM of $2$ and $4$ is $4$. Equivalent fraction for $\frac{11}{2}$: $\frac{11}{2} = \frac{22}{4}$ (since $4 \div 2 = 2$, and $11 \times 2 = 22$).
Subtract: $\frac{22}{4} - \frac{11}{4} = \frac{22-11}{4} = \frac{11}{4}$.
Convert back to mixed number: $11 \div 4 = 2$ remainder $3$. So, $2\frac{3}{4}$.
Result: $2\frac{3}{4}$ litres of paint.
Explanation: Converting to improper fractions simplifies the process by unifying the terms before subtraction. 2.4 Combined Addition and Subtraction of Fractions When problems involve both addition and subtraction, follow the order of operations from left to right. Apply the same principles of finding LCM for unlike denominators and converting mixed numbers to improper fractions.
Example 7: Calculate $\frac{3}{4} + \frac{1}{2} - \frac{1}{8}$.
Step 1 (Add first): $\frac{3}{4} + \frac{1}{2}$ LCM of $4$ and $2$ is $4$. $\frac{1}{2} = \frac{2}{4}$. $\frac{3}{4} + \frac{2}{4} = \frac{5}{4}$.
Step 2 (Subtract next): $\frac{5}{4} - \frac{1}{8}$ LCM of $4$ and $8$ is $8$. $\frac{5}{4} = \frac{10}{8}$. $\frac{10}{8} - \frac{1}{8} = \frac{9}{8}$.
Step 3 (Simplify): Convert to mixed number: $1\frac{1}{8}$.
Result: $1\frac{1}{8}$. 2.5 Solving Word Problems Word problems require careful reading and understanding to identify the operations needed.
Step 1: Read the problem carefully to understand what is being asked.
Step 2: Identify the key numbers and phrases indicating addition ("total", "sum", "altogether", "combined") or subtraction ("left", "remaining", "difference", "how much more/less").
Step 3: Write down the mathematical expression.
Step 4: Perform the calculations using the appropriate methods for adding/subtracting fractions. * Step 5: State the final answer clearly with the correct units (e.g., kilograms, litres, hours). 3.1 Introduction (10 minutes)
Teacher Activity: Begins by reviewing previous knowledge on types of fractions (proper, improper, mixed numbers) and how to convert between them. Asks questions to recall how to find the Least Common Multiple (LCM) of two or more numbers. Presents a simple real-life scenario, e.g., "If I give you $\frac{1}{4}$ of a mango and your friend gives you another $\frac{1}{4}$, how much mango do you have?" (Visual aid: a real or drawn mango divided into parts). Explains that today's lesson will build on this to add and subtract various types of fractions.
Student Activity: Responds to questions on fraction types and LCM. Participates in the mango sharing scenario, offering initial thoughts on combining fractions. 3.2 Lesson Development (40 minutes)
Phase 1: Adding and Subtracting Like Fractions (10 minutes)
Teacher Activity: Explains and demonstrates addition and subtraction of fractions with the same denominators using Example 1 and Example 4 from Key Concepts. Emphasizes adding/subtracting numerators while keeping the denominator constant. Uses visual aids like fraction strips or drawings to reinforce understanding. Provides a quick practice problem on the board for students to solve.
Student Activity: Observes demonstrations and takes notes. Asks clarifying questions. Solves the practice problem individually or in pairs.
Phase 2: Adding and Subtracting Unlike Fractions (15 minutes)
Teacher Activity: Explains and demonstrates addition and subtraction of fractions with different denominators, focusing on finding the LCM and converting to equivalent fractions, using Example 2 and Example
5. Models the step-by-step process of finding LCM and converting fractions. Guides students through another example.
Student Activity: Pays close attention to the LCM and equivalent fraction conversion steps. Participates in guided practice, contributing to finding LCM and numerators. Works through practice problems.
Phase 3: Adding and Subtracting Mixed Numbers and Combined Operations (15 minutes)
Teacher Activity: Explains and demonstrates addition and subtraction of mixed numbers using Method A (converting to improper fractions) as the primary approach for JSS1, using Example 3 and Example
6. Explains how to handle combined addition and subtraction, emphasizing the left-to-right approach as shown in Example
7. Provides a word problem from a Nigerian context, guiding students through the process of extracting information and solving.
Student Activity: Learns the process of converting mixed numbers to improper fractions for operations. Practices combined operations from left to right. Engages in solving the word problem, identifying the operations and numbers. 3.3 Guided Practice (15 minutes)
Teacher Activity: Distributes guided practice questions (see Section 4). Circulates around the classroom, providing individual support, checking understanding, and offering hints. Facilitates peer-to-peer discussion where students can help each other. Selects students to present their solutions on the board, explaining their steps.
Student Activity: Works on guided practice questions. Seeks help from the teacher or peers when needed. Presents solutions and explains reasoning to the class. 3.4 Conclusion (5 minutes)
Teacher Activity: Recap the key steps for adding and subtracting fractions (same/different denominators, mixed numbers, combined operations). Emphasizes the importance of finding LCM and simplifying answers. Assigns independent practice questions for homework (see Section 5). Encourages students to think about where they might use these skills outside of the classroom.
Student Activity: Participates in the recap, confirming understanding. Notes down homework assignment.
Cooking and Recipe Adjustment (Culture/Economy): Scenario: A Nigerian student wants to make a pot of Jollof rice for a family gathering. The recipe calls for $1\frac{1}{2}$ cups of rice, $\frac{3}{4}$ cup of tomato paste, and $\frac{1}{8}$ cup of oil. If they decide to double the recipe, they need to add these fractional amounts together, then multiply by two. Or, if they only have a certain amount of an ingredient, they might need to subtract to see what's remaining. This connects to household responsibilities and the economy of food preparation.
Application: Calculating total ingredient quantities, scaling recipes up or down for different family sizes or events. Resource Allocation and Sharing (Community/Economy): Scenario: In rural communities, land or inherited property might be divided into fractional portions. If a family has a plot of land and gives $\frac{1}{3}$ to one child and $\frac{1}{4}$ to another, they would need to calculate what fraction of the land is remaining or what fraction was distributed in total. This applies to inheritance, community projects, or sharing resources like water from a communal tank.
Application: Determining shares of resources, managing community budgets (e.g., spending $\frac{1}{5}$ of a budget on health and $\frac{2}{7}$ on education), or tracking the usage of common facilities. Measurements in Trades (Economy/Workforce): Scenario: A carpenter might use $2\frac{1}{2}$ metres of wood for a table and $1\frac{3}{4}$ metres for chairs from a total stock of $5$ metres. They need to add the amounts used and subtract from the total to know how much wood is left. Similarly, a mechanic might mix fractional quantities of different engine oils.
Application: Measuring lengths of fabric, wood, or metal; mixing solutions in specific proportions (e.g., chemicals for agriculture, paint for art); calculating fuel consumption (e.g., "I used $\frac{1}{3}$ of my fuel tank to travel from Ibadan to Lagos, then another $\frac{1}{6}$ to visit a relative. How much fuel is left?").