Lesson Notes By Weeks and Term v3 - Junior Secondary 3
Whole numbers
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Whole numbers
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Subject: General Mathematics
Class: Junior Secondary 3
Term: 1st Term
Week: 5
Theme: Numbers And Numeration
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Revise basic operations in binary system Convert numbers in binary numbers to other bases and vice versa Solve quantitative aptitude problems on Binary number systems Use computer to do simple mathematical calculations Translate word problems in to numerical expression. Simplify expressions in volving brackets and fractions Solve problems in volving direct and in verse proportions Apply direct and in verse proportions to practical problems; Solve problems on compound in terest; Apply the use of compound in terest in daily life activities.
Teacher Activities: Introduction and Revision (10 mins): Begin by eliciting students' prior knowledge of number bases (specifically base 10), basic arithmetic operations, and simple interest from JSS1/
2. Introduce the concept of a binary system by asking students how computers "count" or "think," linking it to the on/off states of electrical circuits. Briefly outline the importance of the week's topics in daily life (e.g., finance, technology, problem-solving).
Binary System (25 mins): Explain binary system (base 2) and its digits (0, 1). Demonstrate step-by-step basic operations (addition, subtraction, multiplication, division) in binary using simple examples on the whiteboard. Explain and demonstrate binary-to-base 10 conversion and base 10-to-binary conversion methods clearly. Provide simple quantitative aptitude problems involving binary for students to attempt.
Computer/Calculator Use (10 mins): Demonstrate how to use a scientific calculator or a phone's calculator app to perform simple arithmetic calculations. Emphasise order of operations. Allow students who have calculators to practice. Discuss the importance of accuracy.
Word Problems and Expressions (15 mins): Guide students through identifying keywords in word problems that indicate specific mathematical operations. Provide various Nigerian-context word problems and collaboratively translate them into numerical expressions, emphasizing the correct use of brackets.
Simplifying Expressions (15 mins): Review BODMAS/PEMDAS rule thoroughly. Work through examples involving brackets and fractions, explaining each step carefully.
Direct and Inverse Proportion (20 mins): Introduce direct proportion with real-life examples (e.g., cost vs. quantity of goods at a local market). Introduce inverse proportion with examples (e.g., number of workers vs. time to complete a task). Demonstrate solving problems using both unitary method and proportionality constant.
Compound Interest (20 mins): Explain the difference between simple and compound interest using a simple scenario (e.g., money in a cooperative society). Introduce the compound interest formula A = P(1 + R/100)$^n$ and explain each variable. Work through an example calculation on the board. Discuss real-life applications of compound interest (e.g., savings, loans). Guided Practice and Assessment (Remaining time): Lead students through the guided practice questions, providing support and correcting misconceptions. Assign independent practice questions as homework or classwork. Utilise oral questioning throughout the lesson to check for understanding.
Student Activities: Participation: Actively participate in class discussions, answer questions, and contribute ideas during brainstorming sessions.
Note-taking: Copy notes, examples, and formulas from the whiteboard into their exercise books.
Board Work: Volunteer to solve problems on the whiteboard, explaining their steps.
Pair/Group Work: Collaborate with peers to solve practice problems, discuss strategies, and clarify concepts.
Practical Application: Use calculators to verify answers and perform calculations demonstrated by the teacher.
Problem-solving: Attempt guided practice questions individually or in groups and participate in correcting errors.
Independent Practice: Complete assigned independent practice questions outside of class.
Reflection: Reflect on the real-life relevance of each mathematical concept learned. --- The teacher should guide students through these problems, encouraging them to try first before revealing the solution.
Question 1 (Binary Operations): Perform the following binary operation: 11101$_2$ + 1011$_2$.
Solution 1: ``` 11101_2 + 1011_2 101000_2 ``` Step-by-step explanation: Rightmost column (2^0): 1 + 1 = 0, carry
1. Second column (2^1): 0 + 1 + (carry 1) = 0, carry
1. Third column (2^2): 1 + 0 + (carry 1) = 0, carry
1. Fourth column (2^3): 1 + 1 + (carry 1) = 1, carry
1. Fifth column (2^4): 1 + (carry 1) = 0, carry
1. Write down the final carry
1. Commentary: This checks students' understanding of binary addition, especially handling carries. Question 2 (Conversion & Quantitative Aptitude): A local digital clock displays time in binary. If it shows "1011$_2$" hours, convert this time to base 10 hours.
Solution 2: To convert 1011$_2$ to base 10: 1011$_2$ = (1 × 2$^3$) + (0 × 2$^2$) + (1 × 2$^1$) + (1 × 2$^0$) = (1 × 8) + (0 × 4) + (1 × 2) + (1 × 1) = 8 + 0 + 2 + 1 = 11$_{10}$ hours.
Commentary: This assesses binary to decimal conversion in a practical, relatable context (telling time). Question 3 (Word Problem to Expression & Simplification): A farmer in Plateau State harvested 720 bags of Irish potatoes. He sold 2/3 of them. From the remaining, he kept 50 bags for personal use. Write a numerical expression to find the number of bags he has left, and then simplify it.
Solution 3: Expression: Sold: (2/3) 720 Remaining after sale: 720 - (2/3 720)
Kept for personal use: 50 Left over: [720 - (2/3 720)] - 50 Simplification: (2/3 720) = 2 (720/3) = 2 240 = 480 bags (sold) Remaining after sale = 720 - 480 = 240 bags Left over = 240 - 50 = 190 bags.
Commentary: This combines translating a word problem with fractional calculations and subtraction, covering objectives 5 and
6. Question 4 (Direct and Inverse Proportion): a) If a bus travels 180 km in 3 hours, how far can it travel in 5 hours at the same speed? (Direct Proportion) b) If 4 bricklayers can build a wall in 9 days, how many days will it take 6 bricklayers to build the same wall, assuming they work at the same rate? (Inverse Proportion)
Solution 4: a)
Direct Proportion: Distance in 1 hour = 180 km / 3 hours = 60 km/hr Distance in 5 hours = 60 km/hr 5 hours = 300 km b)
Inverse Proportion: Total work units (bricklayer-days) = 4 bricklayers 9 days = 36 bricklayer-days Days for 6 bricklayers = 36 bricklayer-days / 6 bricklayers = 6 days
Commentary: These questions test both types of proportionality with clear, practical scenarios.
Question 5 (Compound Interest): Mr. Okoro deposited ₦200,000 into a fixed deposit account at Access Bank that offers an annual compound interest rate of 5%. Calculate the amount in his account after 2 years.
Solution 5: Principal (P) = ₦200,000 Rate (R) = 5% Number of years (n) = 2 Formula: A = P(1 + R/100)$^n$ A = 200,000 (1 + 5/100)$^2$ A = 200,000 (1 + 0.05)$^2$ A = 200,000 (1.05)$^2$ A = 200,000 1.1025 A = ₦220,500
Commentary: This directly applies the compound interest formula, crucial for financial literacy. --- 0 + 2 + 1 = 27_{10} ``` Base 10 (Decimal) to Binary: Repeatedly divide the base 10 number by 2 and record the remainders. The binary number is formed by reading the remainders from bottom to top.
Example 6: Convert 25$_{10}$ to binary. ``` 25 ÷ 2 = 12 remainder 1 12 ÷ 2 = 6 remainder 0 6 ÷ 2 = 3 remainder 0 3 ÷ 2 = 1 remainder 1 1 ÷ 2 = 0 remainder 1 ``` Reading remainders from bottom to top: 11001$_2$.
Result: 25$_{10}$ = 11001$_2$.
3. Quantitative Aptitude Problems on Binary System: These often involve understanding the value of binary numbers or performing operations to find a result.
Example 7 (Nigeria-focused): A trader records her sales in binary. On Monday, she sold goods worth 1101$_2$ Naira. On Tuesday, she sold goods worth 1010$_2$ Naira. What is her total sales in base 10 Naira for the two days?
Convert Monday's sales: 1101$_2$ = (1x2$^3$) + (1x2$^2$) + (0x2$^1$) + (1x2$^0$) = 8 + 4 + 0 + 1 = 13$_{10}$ Naira.
Convert Tuesday's sales: 1010$_2$ = (1x2$^3$) + (0x2$^2$) + (1x2$^1$) + (0x2$^0$) = 8 + 0 + 2 + 0 = 10$_{10}$ Naira.
Total sales: 13$_{10}$ + 10$_{10}$ = 23$_{10}$ Naira.
Definition: The binary system uses only two digits: 0 and
1. Each position in a binary number represents a power of
2. It is fundamental to digital electronics and computing.
1. Basic Operations in Binary System Addition: 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 0 (carry 1 to the next position) 1 + 1 + 1 (from previous carry) = 1 (carry 1 to the next position)
Example 1: Add 1011$_2$ and 110$_2$. ``` 1011_2 + 110_2 ------- 10001_2 ``` Step-by-step:
1. Rightmost column: 1 + 0 = 1
2. Second column from right: 1 + 1 = 0, carry 1
3. Third column: 0 + 1 + (carry 1) = 0, carry 1
4. Leftmost column: 1 + (carry 1) = 0, carry
1. Write down the carried
1. Result: 10001$_2$.
Subtraction: (Often involves borrowing, similar to base 10) 0 - 0 = 0 1 - 0 = 1 1 - 1 = 0 0 - 1 = 1 (borrow 1 from the next position, making the 0 a 10$_2$ which is 2$_{10}$)
Example 2: Subtract 101$_2$ from 1110$_2$. ``` 1110_2 - 101_2 ------- 1001_2 ``` Step-by-step:
1. Rightmost column: 0 -
1. Borrow 1 from the next position (the 1 becomes 0, the 0 becomes 10$_2$ which is 2$_{10}$). So, 10$_2$ - 1 = 1.
2. Second column from right: Now it's 0 - 0 = 0. (The original 1 became 0 due to borrowing).
3. Third column: 1 - 1 = 0.
4. Leftmost column: 1 - 0 =
1. Result: 1001$_2$.
Multiplication: (Similar to base 10, but sums are binary additions) 0 x 0 = 0 0 x 1 = 0 1 x 0 = 0 1 x 1 = 1 Example 3: Multiply 101$_2$ by 11$_2$. ``` 101_2 x 11_2 ------ 101 (101 x 1) 1010 (101 x 1, shifted left by one place) ------ 1111_2 ``` Step-by-step:
1. Multiply 101$_2$ by the rightmost 1 of 11$_2$: Gives 101.
2. Multiply 101$_2$ by the leftmost 1 of 11$_2$: Gives
1
0
1. Shift this result one position to the left (effectively multiplying by 2). So, 1010.
3. Add the two partial products: 101 + 1010 = 1111$_2$.
Result: 1111$_2$.
Division: (Similar to long division in base 10)
Example 4: Divide 1100$_2$ by 10$_2$. ``` 110_2 _______ 10_|1100_2 -10 --- 10 -10 --- 00 -0 --- 0 ``` Step-by-step:
1. How many times does 10$_2$ go into 11$_2$? 1 time. Write 1 in the quotient. 2. 1 x 10$_2$ = 10$_2$. Subtract 10$_2$ from 11$_2$ to get 1.
3. Bring down the next digit (0) to make 10$_2$.
4. How many times does 10$_2$ go into 10$_2$? 1 time. Write 1 in the quotient. 5. 1 x 10$_2$ = 10$_2$. Subtract 10$_2$ from 10$_2$ to get 0.
6. Bring down the last digit (0) to make 00$_2$.
7. How many times does 10$_2$ go into 00$_2$? 0 times. Write 0 in the quotient.
Result: 110$_2$.
2. Conversion of Numbers Binary to Base 10 (Decimal): Multiply each digit by its corresponding power of 2, starting from 2$^0$ for the rightmost digit, and sum the results.
Example 5: Convert 11011$_2$ to base 10. ``` 11011_2 = (1 × 2^4) + (1 × 2^3) + (0 × 2^2) + (1 × 2^1) + (1 × 2^0) = (1 × 16) + (1 × 8) + (0 × 4) + (1 × 2) + (1 × 1) = 16 + 8 + 0 + 2 + 1 = 27_{10} ``` * Base 10 (Decimal) to Binary: Repeatedly divide the base 10 number by 2 and record the remainders. The binary number is formed by reading the remainders from bottom to top.
Example 6: Convert 25$_{10}$ to binary. ``` 25 ÷ 2 = 12 remainder 1 12 ÷ 2 = 6 remainder 0 6 ÷ 2 = 3 remainder 0 3 ÷ 2 = 1 remainder 1 1 ÷ 2 = 0 remainder 1 ``` Reading remainders from bottom to top: 11001$_2$.
Result: 25$_{10}$ =
Digital Literacy and Technology (Binary System): Application: Understanding binary numbers is foundational to understanding how computers, smartphones, and other digital devices (e.g., smart TVs, payment terminals at supermarkets) process and store information. In Nigeria, with increasing digital transformation, basic knowledge of binary helps students appreciate the technology they use daily, potentially sparking interest in IT careers.
Local Context: Explaining how their mobile phone's memory or a bank's computer system relies on combinations of 0s and 1s makes the concept tangible. Financial Literacy and Economic Planning (Compound Interest & Proportions): Application: Compound interest is vital for personal finance. Students can apply it to understanding how savings grow in bank accounts, how loans accrue interest (e.g., from microfinance banks or cooperative societies common in Nigeria), and the power of long-term investments like pensions or mutual funds. Proportions are used in budgeting (e.g., if fuel price increases, how much less can be spent on other things), market transactions (e.g., scaling quantities of food items, calculating bulk discounts), and small business management (e.g., scaling production, staffing).
Local Context: Discussing saving for JAMB fees, starting a small business (e.g., tailoring, phone accessories), or contributions to "Esusu" (local cooperative savings schemes) where returns might be considered, directly relates these mathematical concepts to students' future or current experiences. Problem-Solving and Critical Thinking (Word Problems, Expressions, Proportions): Application: The ability to translate real-world scenarios into mathematical expressions and solve problems using proportions is a critical life skill. It helps in making informed decisions, planning, and resource management in various fields like agriculture, construction, trade, and even daily household chores.
Local Context: Examples can include calculating the amount of paint needed for a house renovation, estimating the harvest from a farm based on previous yields, determining fair wages for casual labour, or dividing resources among family members during festive seasons. ---