Lesson Notes By Weeks and Term v3 - Junior Secondary 2
Multiplication and division of directed numbers
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Multiplication and division of directed numbers
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Subject: General Mathematics
Class: Junior Secondary 2
Term: 2nd Term
Week: 2
Theme: Basic Operations
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Obtain the squares and square roots of numbers In terpret and use tables, charts, records and schedules; Carryout correct multiplication and division in volving directed number.
Basic Operations ($N^2$)'. To find the square of a number (e.g., 34):
1. Locate the number 34 in the 'Number (N)' column.
2. Read across to the corresponding value in the 'Square ($N^2$)' column.
3. Example: If N=34, $N^2$ will be
1
1
5
6. The table might also be structured with rows for the first digit(s) and columns for the last digit. E.g., to find $3.4^2$, locate 3.4 in the Number column and read off its square. How to read a Square Root Table (e.g., for numbers 1 to 100): The table usually has columns for 'Number (N)' and 'Square Root ($\sqrt{N}$)'. To find the square root of a number (e.g., 64):
1. Locate the number 64 in the 'Number (N)' column.
2. Read across to the corresponding value in the 'Square Root ($\sqrt{N}$)' column.
3. Example: If N=64, $\sqrt{N}$ will be
8. For numbers not perfectly found in the table (e.g., $\sqrt{50}$), students will learn to estimate or use interpolation in higher classes. For JSS 2, focus on direct readings. Sometimes, there are columns for $\sqrt{N}$ and $\sqrt{10N}$ to handle numbers with different magnitudes. Teachers should guide students to use the appropriate column based on the number given (e.g., $\sqrt{2.5}$ vs $\sqrt{25}$). General Interpretation of Tables, Charts, Records, and Schedules: Title: Understand what the table/chart represents.
Headings: Understand the meaning of each row and column.
Units: Pay attention to the units used (e.g., ₦, °C, kg, km).
Data Extraction: Locate the relevant row(s) and column(s) to find specific information.
Pattern Recognition: Identify trends or relationships within the data.
Example (Interpreting a simple record): A family's electricity bill record for 3 months: | Month | Reading (kWh) | Amount (₦) | | :------ | :------------ | :--------- | | January | 150 | ₦4,500 | | February| 180 | ₦5,400 | | March | 160 | ₦4,800 | Question: What was the electricity consumption in February?
Answer: 180 kWh. (By finding 'February' in the 'Month' column and reading across to the 'Reading (kWh)' column).
Question: In which month was the electricity bill highest?
Answer: February (₦5,400). (By comparing values in the 'Amount (₦)' column).
3. Teaching and Learning Activities Teacher Activities:
1. Introduction & Recap (10 minutes): Begin by reviewing directed numbers: definition, representation on a number line, and basic addition/subtraction. Use real-life examples like bank balances (credit/debit), temperature, or altitude (above/below sea level). Explain the relevance of directed numbers in everyday Nigerian life, e.g., economic fluctuations, weather reports.
2. Explanation of Multiplication Rules (15 minutes): Clearly state and explain the rules for multiplying directed numbers (same signs give positive, different signs give negative). Use simple, clear examples for each rule, demonstrating step-by-step solutions on the board. Incorporate practical analogies (e.g., 'enemy of my enemy is my friend' for negative × negative).
3. Explanation of Division Rules (15 minutes): State and explain that division rules are identical to multiplication rules. Provide examples for each rule, solving them step-by-step on the board. Emphasize understanding the sign before performing the numerical operation.
4. Introduction to Squares and Square Roots (10 minutes): Define square of a number as multiplying it by itself, illustrating with both positive and negative directed numbers (e.g., $3^2$ and $(-3)^2$). Define square root, explaining the concept of two roots (positive and negative) and the principal square root symbol.
5. Interpreting and Using Tables (15 minutes): Display a sample mathematical table (e.g., a simple square/square root table) on the board or distribute photocopies if available. Demonstrate how to locate a number and read its square or square root from the table. Discuss how to interpret general tables/charts using simple Nigerian examples (e.g., a bus timetable for a route like Lagos-Ibadan, or a simple farm produce price list). Highlight the importance of understanding headings and units.
6. Guided Practice (15 minutes): Provide 3-5 practice questions covering multiplication, division, and finding squares/square roots from tables. Guide students through the solutions, encouraging participation and asking questions to check understanding.
7. Consolidation and Q&A (5 minutes): understanding weather patterns and issuing advisories for farmers or travelers.
3. Altitude and Sea Level: Directed numbers are used to measure positions relative to sea level. Places above sea level (e.g., Jos Plateau) have positive altitudes, while features below sea level (if any in Nigeria, or in ocean depths) are negative. An aircraft descending at a rate of 100 meters per minute for 5 minutes has its altitude change by $5 \times (-100 \text{m/min}) = -500 \text{m}$.
4. Data Interpretation (Census & Statistics): Understanding population growth rates (positive) or decline (negative) over periods requires knowledge of directed numbers. Demographic tables from the National Population Commission (NPC) or economic indicators from the National Bureau of Statistics (NBS) often use directed numbers to represent changes, and interpreting these tables requires the skills learned in this lesson.
8. Differentiation, Remediation and Extension Differentiation (for Diverse Learners): Visual Learners: Use colour-coded signs (+ in green, - in red) when writing problems on the board. Utilise number lines to visually demonstrate multiplication and division concepts, especially the 'direction' aspect of directed numbers.
Auditory Learners: Encourage verbalizing the rules before solving problems (e.g., "negative times negative equals positive"). Conduct group discussions where students explain concepts to each other. * Kinesthetic Learners: Use physical objects or movement (e.g., walking forward/backward on a number line drawn on the floor) to model operations with directed numbers. Have students create their own real-life scenarios with directed numbers.
Remediation (for Struggling Learners):
1. Revisit Number Line Operations: Go back to addition and subtraction of directed numbers using a number line to solidify the concept of direction and magnitude before moving to multiplication/division.
2. Focus on Sign Rules: Provide extra drills solely on applying the sign rules (e.g., "What is a negative times a positive?"). Use flashcards with operation symbols and signs.
3. Simplified
Examples: Start with very small, single-digit numbers for multiplication and division problems until mastery of sign rules is achieved.
4. Pair Work with Mentorship: Pair struggling learners with stronger peers to work on simplified problems, encouraging peer teaching and immediate feedback.
5. Use of Analogy: Continuously use the 'friend/enemy' or 'debt/credit' analogies to explain the logic behind the sign rules.
Extension (for High-Achieving Learners):
1. Multiple Operations: Introduce problems involving more than two directed numbers or a combination of operations (e.g., $(-2) \times 3 \div (-6)$ or $(-5)^2 + (4 \times -3)$).
2. Order of Operations (BODMAS/PEMDAS): Challenge them to apply BODMAS/PEMDAS rules with directed numbers, including parentheses and exponents.
3. Missing Number Problems: Provide equations with a missing directed number for them to solve (e.g., $(-4) \times \square = -20$).
4. Real-world Problem Creation: Ask them to create complex word problems involving multiplication and division of directed numbers relevant to Nigerian socio-economic contexts (e.g., calculating cumulative profit/loss over several quarters for a small startup, or average rainfall changes over a farming season).
5. Introduction to Integers on Coordinate Plane: Briefly introduce how directed numbers are used in coordinates (e.g., plotting points like (-2, 3)) as a preview for geometry. is
9. Therefore, $\sqrt{81} = 9$.
Commentary: This highlights the practical skill of using mathematical tables as specified in the performance objectives. The teacher would typically provide a visual aid of a table.
5. Independent Practice (Questions Only)
1. Calculate: (-12) × 5
2. Determine the product of -8 and -9.
3. Multiply: 15 × (-4)
4. Find the value of (-7) × 0.
5. Calculate: (-48) ÷ 6
6. Divide: 81 ÷ (-9)
7. Determine the quotient of -100 and -10.
8. What is the square of 13?
9. What is the square of -15?
1
0. Using a mathematical table, find the square of 25.
1
1. Using a mathematical table, find the square root of 144.
6. Evaluation and Assessment Formative Assessment: Observation: The teacher will observe students' participation in discussions, their responses to questions during explanations, and their work during guided practice.
Question and Answer: Pose questions throughout the lesson to check for understanding, focusing on the sign rules for multiplication and division.
Mini Whiteboards/Slates: Students quickly write answers to multiplication/division problems on small whiteboards or slates, displaying them for immediate feedback.
Peer-check: During independent practice, students can exchange notebooks and check each other's work against solutions provided by the teacher (after completion). Summative Assessment (End-of-Lesson Quiz/Homework):
1. Obtain the squares and square roots of given numbers from the table: a. Using a square table, find the square of 37. b. Using a square root table, find the value of $\sqrt{169}$.
Marking Scheme: 1 mark for each correct answer from the table.
Total: 2 marks.
2. Interpret and use given tables, charts, records and schedules: A small provision shop in Aba recorded its profit and loss over a week. | Day | Profit/Loss (₦) | | :-------- | :-------------- | | Monday | +2500 | | Tuesday | -1200 | | Wednesday | +3000 | | Thursday | -800 | | Friday | +4000 | a. On which day did the shop record the highest profit? b. What was the loss recorded on Tuesday? c. What was the net financial position (total profit/loss) for Monday and Tuesday combined?
Marking Scheme: 1 mark for each correct answer by interpreting the table.
Total: 3 marks.
3. Find the product of any two given directed numbers: a.
Calculate: (-18) × 3 b.
Multiply: (-7) × (-11) c. What is the product of 9 and -6?
Marking Scheme: 1 mark for correct sign and 1 mark for correct magnitude.
Total: 6 marks.
4. Determine the quotient of any two given directed numbers: a.
Calculate: (-72) ÷ 9 b.
Divide: 120 ÷ (-10) c. Determine the value of (-96) ÷ (-12)
Marking Scheme: 1 mark for correct sign and 1 mark for correct magnitude.
Total: 6 marks.
Total Marks for Summative Assessment: 17 marks.
7. Real-life Applications / Integration
1. Financial Management (Bank Accounts & Business): Directed numbers are fundamental in understanding bank statements. Deposits are positive, withdrawals are negative. If a customer makes 3 withdrawals of ₦5,000 each, this can be represented as $3 \times (-₦5000) = -₦15,000$, indicating a total reduction in balance. Similarly, profit (positive) and loss (negative) for a local business (e.g., selling yam in the market) can be calculated using multiplication and division of directed numbers to determine average daily earnings or total accumulated wealth/debt.
2. Temperature Readings and Climate: Meteorologists in Nigeria use directed numbers to report temperature changes. For instance, if the temperature at night drops by an average of 2°C per hour for 5 hours, the total change is $5 \times (-2^\circ\text{C}) = -10^\circ\text{C}$. This helps in understanding weather patterns and issuing advisories for farmers or travelers.
3. Altitude and Sea Level: Directed numbers are used to measure positions relative to sea level. Places above sea level (e.g., Jos Plateau) have positive altitudes, while features below sea level (if any in Nigeria, or in ocean depths) are negative. An aircraft descending at a rate of 100 meters per minute for 5 minutes has its altitude change by $5 \times (-100 \text{m/min}) = -500 \text{m}$.
4. Data Interpretation (Census & Statistics): Understanding population growth rates (positive) or decline