Lesson Notes By Weeks and Term v3 - Junior Secondary 1
Use of symbols
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Use of symbols
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Subject: General Mathematics
Class: Junior Secondary 1
Term: 2nd Term
Week: 3
Theme: Algebra Processes
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solve problems expressed in open sentences identify the relationship between addition and subtraction; multiplication and division; use letters to represent symbols or shapes in open sentences. Solve open sentence problems in volving two arithmetic operations; Solve word problems in volving use of symbols; Solve quantitative aptitude problems on the use of symbols
This section provides a detailed breakdown of the core concepts for the teacher. 2.
1. Symbols in Mathematics In mathematics, a symbol is a character or mark used to represent a mathematical object, operation, or relationship. In the context of "Use of Symbols" for JSS1, symbols primarily refer to placeholders for unknown numerical values.
These placeholders can be: Letters: Commonly used (e.g., `x`, `y`, `a`, `b`, ``m`).
Shapes: Geometric shapes (e.g., `□`, `∆`, `○`, `☆`).
Empty Boxes: Simple visual placeholders (e.g., `[ ]`, `( )`).
Example: `x + 5 = 12` (Here, 'x' is the symbol representing an unknown number) `□ - 3 = 7` (Here, '□' is the symbol) 2.
2. Open Sentences An open sentence is a mathematical statement that contains one or more unknown quantities (represented by symbols) and whose truth value (true or false) cannot be determined until the unknown quantities are replaced by specific values.
Example: `x + 5 = 12` is an open sentence. It becomes true if `x = 7` and false if `x = 10`. `3 × y = 21` is an open sentence. It becomes true if `y = 7`. The goal of solving an open sentence is to find the value(s) of the symbol(s) that make the sentence true. This value is called the solution. 2.
3. Relationship Between Inverse Operations Understanding inverse operations is fundamental to solving open sentences. Inverse operations "undo" each other.
Addition and Subtraction: If a number is added to an unknown, subtraction can be used to find the unknown. If a number is subtracted from an unknown, addition can be used to find the unknown.
Rule: If `A + B = C`, then `A = C - B` and `B = C - A`.
Rule: If `A - B = C`, then `A = C + B`.
Rule: If `B - A = C`, then `A = B - C`.
Example 1 (Addition): `x + 7 = 15` To find `x`, subtract 7 from both sides (or use the rule `x = 15 - 7`). `x = 15 - 7` `x = 8` Example 2 (Subtraction): `□ - 9 = 13` To find `□`, add 9 to both sides (or use the rule `□ = 13 + 9`). `□ = 13 + 9` `□ = 22` Example 3 (Subtraction with unknown minuend): `20 - y = 8` To find `y`, subtract 8 from 20 (or use the rule `y = 20 - 8`). `y = 20 - 8` `y = 12` Multiplication and Division: If an unknown is multiplied by a number, division can be used to find the unknown. If an unknown is divided by a number, multiplication can be used to find the unknown.
Rule: If `A × B = C`, then `A = C ÷ B` and `B = C ÷ A`.
Rule: If `A ÷ B = C`, then `A = C × B`.
Rule: If `B ÷ A = C`, then `A = B ÷ C`.
Example 1 (Multiplication): `4 × p = 28` To find `p`, divide both sides by 4 (or use the rule `p = 28 ÷ 4`). `p = 28 ÷ 4` `p = 7` Example 2 (Division): `∆ ÷ 6 = 5` To find `∆`, multiply both sides by 6 (or use the rule `∆ = 5 × 6`). `∆ = 5 × 6` `∆ = 30` Example 3 (Division with unknown divisor): `36 ÷ m = 9` To find `m`, divide 36 by 9 (or use the rule `m = 36 ÷ 9`). `m = 36 ÷ 9` `m = 4` 2.
4. Solving Open Sentences with Two Arithmetic Operations When an open sentence involves two operations, students must apply the inverse operations in the reverse order of operations (PEMDAS/BODMAS). Usually, addition/subtraction are "undone" first, followed by multiplication/division.
General Strategy:
1. Isolate the term containing the symbol by undoing any addition or subtraction.
2. Then, isolate the symbol itself by undoing any multiplication or division.
Example 1: `2x + 5 = 17` Step 1 (Undo addition): Subtract 5 from both `m = 36 ÷ 9`). `m = 36 ÷ 9` `m = 4` 2.
4. Solving Open Sentences with Two Arithmetic Operations When an open sentence involves two operations, students must apply the inverse operations in the reverse order of operations (PEMDAS/BODMAS). Usually, addition/subtraction are "undone" first, followed by multiplication/division.
General Strategy:
1. Isolate the term containing the symbol by undoing any addition or subtraction.
2. Then, isolate the symbol itself by undoing any multiplication or division.
Example 1: `2x + 5 = 17` Step 1 (Undo addition): Subtract 5 from both sides. `2x + 5 - 5 = 17 - 5` `2x = 12` Step 2 (Undo multiplication): Divide both sides by 2. `2x / 2 = 12 / 2` `x = 6` Example 2: `(y - 4) / 3 = 5` Step 1 (Undo division): Multiply both sides by 3. `(y - 4) / 3 × 3 = 5 × 3` `y - 4 = 15` Step 2 (Undo subtraction): Add 4 to both sides. `y - 4 + 4 = 15 + 4` `y = 19` 2.
5. Solving Word Problems Involving Symbols This involves translating a real-world scenario into an open sentence, then solving it.
Steps:
1. Read the problem carefully to understand the context.
2. Identify the unknown quantity and represent it with a symbol (e.g., `x`, `y`, `□`).
3. Identify the known quantities and operations involved.
4. Formulate an open sentence (equation) that represents the problem.
5. Solve the open sentence for the unknown symbol.
6. State the answer in the context of the word problem.
Example: A market woman bought some oranges. She sold 35 of them and was left with 15 oranges. How many oranges did she buy originally?
Step 1: Understand the problem: We need to find the initial number of oranges.
Step 2: Let `x` be the original number of oranges she bought.
Step 3: Known: sold 35, left with
1
5. Operation: subtraction.
Step 4: Open sentence: `x - 35 = 15` Step 5: Solve: `x - 35 + 35 = 15 + 35` `x = 50` Step 6: Answer: The market woman originally bought 50 oranges. 2.
6. Quantitative Aptitude Problems on Use of Symbols These problems often involve patterns, coded operations, or logical reasoning where symbols are used in non-standard ways. The key is to decipher the rule or relationship defined by the symbols.
Example 1 (Symbolic representation): If `∆` represents the number 4 and `○` represents the number 7, what is the value of `∆ + ○ - 2`?
Solution: Replace symbols with their given values. `4 + 7 - 2` `11 - 2 = 9` The value is
9. Example 2 (Coded operation): If `a b` means `(a + b) / 2`, what is `6 10`?
Solution: Apply the defined operation. `6 * 10 = (6 + 10) / 2` `= 16 / 2` `= 8` The value is 8. --- basic quantitative aptitude question. Circulate among groups, providing support, clarification, and monitoring progress.
Student Activity: Collaborate within their groups to solve the assigned problems. Discuss different approaches and verify solutions. Prepare to present one of their solutions to the class. 3.
5. Conclusion (5 minutes)
Teacher Activity: Ask students to summarize the key concepts learned: what symbols are, what open sentences are, how inverse operations help solve them, and how to approach word problems. Assign homework from the Independent Practice section.
Student Activity: Participate in the summary discussion. * Note down homework assignments. --- 3.
1. Introduction (10 minutes)
Teacher Activity: Begin by writing simple arithmetic problems with a missing number represented by a blank space or a shape on the board (e.g., `5 + ___ = 12`, `10 - [ ] = 4`, `3 × ∆ = 18`). Ask students to quickly identify the missing numbers. Explain that these missing numbers are "unknowns" and in mathematics, we often use symbols (letters, shapes) to represent them. Introduce the term "open sentence" as a statement with an unknown that needs to be solved.
Student Activity: Solve the introductory problems orally or on mini-whiteboards. Engage in a brief discussion about how they found the missing numbers. 3.
2. Presentation and Explanation of Concepts (25 minutes)
Teacher Activity: Topic: Use of Symbols: Define symbols, open sentences, and the purpose of using symbols. Use examples from Section 2.1 and 2.
2. Topic: Inverse Operations (Performance Objective 2): Explain the concept of inverse operations using real-life examples (e.g., adding money vs. spending money).
Demonstrate with number examples: `5 + 3 = 8`, so `8 - 3 = 5` and `8 - 5 = 3`. Similarly, for multiplication and division: `4 × 5 = 20`, so `20 ÷ 5 = 4` and `20 ÷ 4 = 5`. Write down the general rules for addition/subtraction and multiplication/division as shown in Section 2.
3. Topic: Solving Open Sentences with One Operation (Performance Objective 1): Work through examples of open sentences involving one operation (addition, subtraction, multiplication, division), using both shapes/boxes and letters. Emphasize using inverse operations to isolate the symbol. (Refer to Examples in 2.3).
Topic: Using Letters to Represent Symbols (Performance Objective 3): Explain that while shapes are good for starters, letters are more commonly used in algebra. Present open sentences with shapes and demonstrate how to rewrite them using letters (e.g., `□ + 5 = 12` becomes `x + 5 = 12`).
Student Activity: Listen attentively and take notes. Answer questions posed by the teacher. Participate in oral drills on inverse operations. Practice converting shape-based open sentences to letter-based ones. 3.
3. Guided Practice and Application (25 minutes)
Teacher Activity: Topic: Solving Open Sentences with Two Operations (Performance Objective 4): Present examples involving two operations (e.g., `2x + 3 = 11`, `(y - 4) / 2 = 3`). Guide students step-by-step, emphasizing the order of undoing operations (reverse BODMAS). Solve 2-3 examples collaboratively on the board.
Topic: Solving Word Problems (Performance Objective 5): Present a word problem relevant to Nigerian context (e.g., market, farming). Guide students through the 6-step process of translating the word problem into an open sentence and solving it (refer to Section 2.5). Work through one word problem example together.
Topic: Quantitative Aptitude Problems (Performance Objective 6): Introduce simple quantitative aptitude problems involving symbols. Guide students to understand the underlying pattern or definition of the symbolic operation (refer to Section 2.6). Solve one simple example as a class.
Student Activity: Work on problems presented by the teacher, either individually or in pairs. Share their thought processes and solutions with the class. Practice translating word problems into mathematical sentences. 3.
4. Group Activity / Collaborative Learning (15 minutes)
Teacher Activity: Divide the class into small groups (e.g., 4-5 students per group). Provide each group with a set of mixed problems: One-step open sentences (using shapes and letters). Two-step open sentences. A simple word problem. A basic quantitative aptitude question. Circulate among groups, providing support, clarification, and monitoring progress.
Student Activity: Collaborate within their groups to solve the assigned problems. Discuss different approaches and verify solutions. Prepare to present one of their solutions to the class. 3.
5. Conclusion (5 minutes)
Teacher Activity: Ask students to summarize the key concepts learned: what symbols are, what open sentences are, how inverse operations help solve them, and how to approach word problems. Assign homework from the Independent Practice section.
Student Activity: *
Market Transactions and Budgeting (Community/Economy): Application: Students can relate the use of symbols to everyday buying and selling. For example, if a parent gives a child ₦500 to buy items, and the child buys bread for ₦300, how much change (`x`) should the child receive? (`500 - 300 = x`). Or, if a trader sells a bag of garri for ₦8,000, and makes a profit of ₦1,500, what was the cost price (`C`)? (`C + 1500 = 8000`). This directly integrates with financial literacy and understanding local economic activities.
Integration: Encourage students to bring scenarios from their local markets or household budgeting. Role-play scenarios where students are buyers and sellers, having to calculate unknowns. Farming and Resource Management (Environment/Economy): Application: A farmer needs to distribute fertilizer. If he has 120 kg of fertilizer and needs to apply 5 kg per plot, how many plots (`P`) can he cover? (`P × 5 = 120`). Or, if a community well serves `N` households, and each household consumes 50 litres of water daily, how many households can be served by a 1000-litre tank daily? (`N × 50 = 1000`). This teaches resource allocation and planning.
Integration: Discuss local farming practices (e.g., yam, cassava, maize) and how farmers might need to estimate yields or inputs. Students can create simple word problems based on these scenarios. Simple Engineering and Construction (Community/Technology): Application: When building a fence, if a carpenter uses planks of uniform length, and a total length of 20 metres is needed, with each plank being 2.5 metres, how many planks (`N`) are required? (`N × 2.5 = 20`). Or, if a builder buys `X` bags of cement and uses 10 bags for a project, having 5 bags left, how many did he buy? (`X - 10 = 5`). This introduces foundational concepts used in vocational trades.
Integration: The teacher can use realia like measuring tapes, blocks, or drawings of local structures to make these problems tangible. ---