Lesson Notes By Weeks and Term v3 - Junior Secondary 1
Approximation
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Approximation
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Subject: General Mathematics
Class: Junior Secondary 1
Term: 2nd Term
Week: 3
Theme: Basic Operations
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Teacher Activities: Introduction (10 minutes): Begin by asking students about situations where they don't need exact numbers (e.g., "About how many people are in the school assembly?", "What is the approximate cost of this item in the market?"). Introduce the term "approximation" and "rounding" as ways to make numbers simpler and easier to work with for quick estimation. Briefly revise place values (ones, tens, hundreds, thousands) as a prerequisite for rounding. Use a place value chart. Explanation of Rounding Rules (20 minutes): Rounding to the Nearest 10: Write numbers on the board (e.g., 23, 27, 45, 134, 256).
hundreds digit is 5 or more, add 1 to the thousands digit and replace the hundreds, tens, and ones digits with
0. If the hundreds digit is less than 5, keep the thousands digit as it is and replace the hundreds, tens, and ones digits with
0. Worked Example 3a: Round ₦12,680 to the nearest
1
0
0
0. Step 1: The thousands digit is
2. Step 2: The digit to its right (the hundreds digit) is
6. Since 6 is 5 or greater, round up.
Step 3: Add 1 to the thousands digit (2 + 1 = 3). Replace the hundreds, tens, and ones digits with
0. The ten thousands digit remains unchanged.
Result: ₦12,680 rounded to the nearest 1000 is ₦13,
0
0
0. Worked Example 3b: Round the population of a town, 87,421 people, to the nearest
1
0
0
0. Step 1: The thousands digit is
7. Step 2: The digit to its right (the hundreds digit) is
4. Since 4 is less than 5, round down.
Step 3: Keep the thousands digit as
7. Replace the hundreds, tens, and ones digits with
0. The ten thousands digit remains unchanged.
Result: 87,421 people rounded to the nearest 1000 is 87,000 people. The core rule for rounding depends on the digit immediately to the right of the place value to which the number is being rounded:
1. Identify the target place value: Determine the digit in the place value to which the number needs to be rounded (e.g., tens, hundreds, thousands).
2. Look at the digit immediately to its right: If this digit is 5 or greater (5, 6, 7, 8, 9), round up: Add 1 to the digit in the target place value, and change all digits to its right to zero. If this digit is less than 5 (0, 1, 2, 3, 4), round down: Keep the digit in the target place value as it is, and change all digits to its right to zero.
Example 1: Rounding to the Nearest 10 To round a number to the nearest 10, identify the tens digit. Then, look at the digit in the ones place.
Rule: If the ones digit is 5 or more, add 1 to the tens digit and replace the ones digit with
0. If the ones digit is less than 5, keep the tens digit as it is and replace the ones digit with
0. Worked Example 1a: Round ₦47 to the nearest
1
0. Step 1: The tens digit is
4. Step 2: The digit to its right (the ones digit) is
7. Since 7 is 5 or greater, round up.
Step 3: Add 1 to the tens digit (4 + 1 = 5). Replace the ones digit with
0. Result: ₦47 rounded to the nearest 10 is ₦
5
0. Worked Example 1b: Round 123 km to the nearest
1
0. Step 1: The tens digit is
2. Step 2: The digit to its right (the ones digit) is
3. Since 3 is less than 5, round down.
Step 3: Keep the tens digit as
2. Replace the ones digit with
0. The hundreds digit remains unchanged.
Result: 123 km rounded to the nearest 10 is 120 km.
Example 2: Rounding to the Nearest 100 To round a number to the nearest 100, identify the hundreds digit. Then, look at the digit in the tens place.
Rule: If the tens digit is 5 or more, add 1 to the hundreds digit and replace the tens and ones digits with
0. If the tens digit is less than 5, keep the hundreds digit as it is and replace the tens and ones digits with
0. Worked Example 2a: Round ₦3,450 to the nearest
1
0
0. Step 1: The hundreds digit is
4. Step 2: The digit to its right (the tens digit) is
5. Since 5 is 5 or greater, round up.
Step 3: Add 1 to the hundreds digit (4 + 1 = 5). Replace the tens and ones digits with
0. The thousands digit remains unchanged.
Result: ₦3,450 rounded to the nearest 100 is ₦3,
5
0
0. Worked Example 2b: Round 1,732 students to the nearest
1
0
0. Step 1: The hundreds digit is
7. Step 2: The digit to its right (the tens digit) is
3. Since 3 is less than 5, round down.
Step 3: Keep the hundreds digit as
7. Replace the tens and ones digits with
0. The thousands digit remains unchanged.
Result: 1,732 students rounded to the nearest 100 is 1,700 students.
Example 3: Rounding to the Nearest 1000 To round a number to the nearest 1000, identify the thousands digit. Then, look at the digit in the hundreds place.
Rule: If the hundreds digit is 5 or more, add 1 to the thousands digit and replace the hundreds, tens, and ones digits with
0. If the hundreds digit is less than 5, keep the thousands digit as it is and replace the hundreds, tens, and ones digits with
0. Worked Example 3a: Round ₦12,680 to the nearest
1
0
0
0. Step 1: The thousands digit is
2. Step 2: The digit to its right (the hundreds digit) is
6. Since 6 is 5 or greater, round up. * Step 3: Add 1 to When approximating answers to calculations, the numbers involved are typically rounded before performing the operation. This simplifies the calculation and provides a quick estimate. The degree of accuracy (e.g., nearest 10, 100, 1000) will usually be specified or determined by context. 2.3.
1. Approximation in Addition and Subtraction Strategy: Round each number to the specified degree of accuracy first, then perform the addition or subtraction.
Worked Example 4 (Addition): Approximate the total cost of two items: ₦785 for yam flour and ₦412 for palm oil, to the nearest
1
0
0. Step 1: Round ₦785 to the nearest
1
0
0. Hundreds digit is
7. Digit to its right is 8 (≥5), so round up. 785 ≈ 800 Step 2: Round ₦412 to the nearest
1
0
0. Hundreds digit is
4. Digit to its right is 1 (= 50,000). So, the final step should be to round the calculated approximation.
Step 4: Round ₦80,000 to the nearest 100,
0
0
0. Ten thousands digit is
8. Since 8 is ≥ 5, round up the 0 in the hundred thousands place to 1. 80,000 ≈ 100,000 Final Result: The approximate total earnings are ₦100,
0
0
0. Worked Example 7 (Division): A bus travelled 395 km in approximately 8 hours. What was its approximate average speed (in km/h) to the nearest 10?
Step 1: Round 395 km to the nearest 10. 395 ≈ 400 Step 2: Round 8 hours to the nearest 10 (or keep it as is if it's already a simple number for division). In this case, 8 is a good divisor for
4
0
0. Let's keep it as
8. If it was 7, we might round to 10 for easier division, but 8 is quite close and often easy to work with. For division, a sensible approximation is key. Let's round 8 to 10 for consistency of "nearest 10" although it might make calculation harder.
Let's adjust: "round to a suitable degree of accuracy". Rounding 8 to 10 makes the division 400/10 =
4
0. Let's consider rounding to one significant figure for division. 395 ≈ 400. 8 ≈ 8 (or 10 if we strictly follow nearest 10). If we round 8 to 10, then 400 / 10 =
4
0. If we keep 8: 400 / 8 =
5
0. The actual answer is 395 / 8 = 49.
3
7
5. Rounding 395 to 400 and keeping 8 gives 50, which is closer.
Therefore, the teacher should guide students to make sensible approximations.
Revised Step 2 (for division): Round 8 hours to the nearest whole number (it already is) or a suitable round number. For this example, 8 is a factor of 400, making calculation easy. 8 ≈ 8 (no rounding needed or round to nearest 10 -> 10) Let's stick with rounding to nearest 10 for both for consistency, and then discuss the impact. 8 hours rounded to the nearest 10 is 10 hours.
Step 3: Divide the rounded numbers. 400 km / 10 hours = 40 km/h Step 4: The question asks for approximate average speed to the nearest
1
0. Our calculated 40 km/h is already a multiple of
1
0. Result: The approximate average speed is 40 km/h. (Actual speed is 49.375 km/h. Approximating 395 to 400 and 8 to 10 gives
4
0. If we had approximated 395 to 400 and kept 8, the answer would be 50, which is closer to 49.
3
7
5. This is a good point to discuss flexibility and 'sensible' approximation in class).
Quantitative Reasoning: Involves solving problems that require numerical calculations and logical thinking using approximate values. This includes multi-step word problems where students need to identify quantities, choose appropriate rounding levels, perform operations, and interpret the approximate result within the problem's context.
Qualitative Reasoning: Involves understanding the implications of approximation. This includes discussing why approximation is useful, when it is appropriate (e.g., estimating groceries vs. precise engineering), the loss of precision that occurs, and how rounding up or down affects the estimate (e.g., an 'overestimate' or 'underestimate'). For JSS1, this might involve justifying their choice of rounding or explaining scenarios where approximation is better than exact calculation.
Approximation is deeply embedded in daily life, and connecting it to Nigerian contexts enhances understanding and relevance for students.
Market Transactions and Budgeting: When a Nigerian shopper is buying multiple items at a local market (e.g., tomatoes, pepper, onions, fish), they often mentally approximate the total cost to ensure they have enough cash or to stick to a budget. A market vendor might quickly approximate the change to give back.
Example:* Estimating a grocery list: ₦1,980 for rice, ₦720 for beans, ₦450 for garri. A quick approximation (₦2,000 + ₦700 + ₦500 = ₦3,200) helps the shopper know roughly how much money they need, without needing exact calculation on the spot.
Population and Census Figures: National and state population figures are almost always reported as approximations (e.g., "Nigeria's population is approximately 200 million"). It's impractical and unnecessary to state the exact number of individuals. Understanding these rounded figures is crucial for policy planning, resource allocation, and general knowledge.
Example:* When discussing the population of Lagos (e.g., "about 15 million"), students use approximation to grasp the magnitude without getting lost in exact, rapidly changing figures.
Travel Time and Distance Estimation: When planning a journey by road in Nigeria, people often approximate distances and travel times. "It's about 5 hours from Enugu to Abuja" or "The journey is roughly 300 kilometres." This helps in planning fuel stops, rest breaks, and arrival times.
Example:* A bus driver might estimate that traveling 285 km at an average speed of 85 km/h will take approximately 3 hours (300 km / 100 km/h = 3 hours), knowing that exact calculation isn't necessary for practical planning.