Approximation

Grade 11 · Mathematics

Semester 1 | Period 1 | Week 5

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Subject: Mathematics

Semester: 1

Period: 1

Week: 5

WEEK 5

Class: Grade 11
Age: 16 years
Duration: 40 minutes of 5 periods
Subject: Mathematics
Topic: Approximation
Focus:

  • Rounding up and down of numbers to significant figures, decimal places, and nearest whole numbers.
  • Application of approximation to everyday life.
  • Percentage error.

SPECIFIC OBJECTIVES:

By the end of the lesson, students should be able to:

  1. Round numbers to significant figures, decimal places, and nearest whole numbers.
  2. Apply approximation methods to solve real-life problems, including financial reports, health, and social data.
  3. Use logarithmic tables and calculators to approximate values.
  4. Calculate percentage error between results obtained from logarithmic tables and calculators.

INSTRUCTIONAL TECHNIQUES:

  • Demonstration
  • Problem-solving sessions
  • Group work
  • Real-life application exercises
  • Guided discussions

INSTRUCTIONAL MATERIALS:

  • Financial reports and budget figures
  • Population figures
  • Logarithmic tables
  • Calculators
  • Data from school records, health sector, economy, etc.

 

PERIOD 1: Introduction to Approximation

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Introduces approximation and its importance in various fields (e.g., budgeting, health statistics). Demonstrates rounding numbers to the nearest hundred, thousand, million, and trillion.

Students listen attentively, take notes, and ask clarifying questions.

Step 2 - Rounding to Significant Figures and Decimal Places

Explains rounding numbers to significant figures and decimal places using real-life examples (e.g., rounding population numbers, financial reports).

Students practice rounding numbers to significant figures and decimal places using given data.

Step 3 - Practical Application

Demonstrates the application of approximation to solve real-world problems (e.g., approximating population figures, financial budgets).

Students work in pairs to solve approximation problems in various contexts (e.g., population, financial reports).

NOTE ON BOARD:

  • Rounding to Significant Figures: Round numbers to the required number of significant figures.
  • Rounding to Decimal Places: Round numbers to the required number of decimal places.
  • Rounding to Nearest Whole Number: Round numbers to the nearest whole number.

EVALUATION (5 exercises):

  1. Round 234567 to 2 significant figures.
  2. Round 987.654 to 2 decimal places.
  3. Round 1523 to the nearest thousand.
  4. Round 2436 to the nearest hundred.
  5. Round 7809 to 3 significant figures.

CLASSWORK (5 questions):

  1. Round 1234567 to 3 significant figures.
  2. Round 58.1369 to 1 decimal place.
  3. Round 342.745 to 2 decimal places.
  4. Round 9520 to the nearest ten.
  5. Round 87563 to 2 significant figures.

ASSIGNMENT (5 tasks):

  1. Approximate the population of a country to the nearest million.
  2. Round 124.567 to 2 decimal places.
  3. Round 654321 to the nearest thousand.
  4. Use the calculator to approximate the result of 987.6543 to 3 decimal places.
  5. Round a given budget value to the nearest hundred.

 

PERIOD 2: Application of Approximation in Everyday Life

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Real-life Scenarios

Introduces different real-life scenarios where approximation is needed (e.g., financial budgeting, population data). Discusses how approximation simplifies complex numbers for easier understanding and decision-making.

Students participate in discussions and identify scenarios where approximation is useful in daily life.

Step 2 - Financial Reports and Budgeting

Demonstrates how approximation is used in financial reports, such as estimating yearly expenses or profits in financial statements.

Students apply approximation to real-life financial scenarios, such as estimating costs for a project or budget.

Step 3 - Health Sector Data

Discusses how health data is approximated for easier analysis (e.g., approximate number of patients in a hospital, average number of visits per year).

Students use provided health data to approximate numbers for reports or presentations.

NOTE ON BOARD:

  • Financial Reports: Approximate yearly income/expenses.
  • Health Sector: Approximate number of patients in hospitals, hospital visits, etc.
  • Population Data: Approximate population numbers of countries and regions.

EVALUATION (5 exercises):

  1. Approximate the annual income of a company given monthly earnings.
  2. Round the total cost of a hospital's monthly expenses to the nearest hundred.
  3. Estimate the population of a country based on census data.
  4. Round the total number of hospital visits in a year to the nearest thousand.
  5. Estimate the average number of students in a school based on the number of classrooms.

CLASSWORK (5 questions):

  1. Approximate the population of a city to the nearest million.
  2. Round the annual budget of a company to the nearest thousand.
  3. Estimate the total number of students in a district.
  4. Approximate the average number of patients visiting a hospital monthly.
  5. Round the number of households in a state to the nearest thousand.

ASSIGNMENT (5 tasks):

  1. Estimate the population of a city given its population density and area.
  2. Round the yearly expenditure of a school to the nearest thousand.
  3. Approximate the total income from sales of a product.
  4. Estimate the total number of visitors to a health clinic in a year.
  5. Approximate the number of books sold by a bookstore in a month.

 

PERIOD 3: Using Logarithm Tables and Calculators for Approximation

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction to Logarithms

Explains the use of logarithms in approximation, including how logarithmic tables are used to simplify complex calculations. Demonstrates finding logarithms using tables and calculators.

Students observe the method and practice using logarithmic tables to find logarithms of various numbers.

Step 2 - Logarithmic Calculation

Demonstrates the steps for approximating complex calculations using logarithmic tables and calculators (e.g., finding the log of a large number).

Students practice using logarithmic tables and calculators for approximating logarithms of given numbers.

Step 3 - Comparing Results

Compares the results obtained from logarithmic tables and calculators. Demonstrates how small differences can arise and how to calculate percentage error.

Students perform calculations using both methods and compare results.

NOTE ON BOARD:

  • Logarithmic Tables: Use logarithmic tables for approximating large numbers.
  • Calculators: Use scientific calculators for logarithmic approximations.
  • Percentage Error: Percentage error = (|True Value - Approximate Value| / True Value) × 100%

EVALUATION (5 exercises):

  1. Use the logarithmic table to find the log of 1000.
  2. Use a calculator to find the log of 1000. Compare the results and calculate the percentage error.
  3. Use the logarithmic table to find the log of 50.
  4. Approximate the square root of 10 using a calculator.
  5. Compare the logarithmic table and calculator results for log 10.

CLASSWORK (5 questions):

  1. Use the logarithmic table to find the log of 500.
  2. Use a calculator to find the log of 500.
  3. Compare results and calculate percentage error.
  4. Use logarithmic tables to find the log of 100.
  5. Approximate the logarithm of a given large number.

ASSIGNMENT (5 tasks):

  1. Use a logarithmic table to approximate the log of 150.
  2. Use a calculator to find the log of 150 and calculate percentage error.
  3. Calculate the percentage error for the log of 200 using both methods.
  4. Use logarithmic tables to find the log of 25.
  5. Use a calculator to find the log of 25 and compare results.

 

PERIOD 4 & 5: Percentage Error and Application of Approximation

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction to Percentage Error

Introduces the concept of percentage error and demonstrates how to calculate it.

Students listen and take notes on the percentage error formula.

Step 2 - Calculating Percentage Error

Guides students through solving problems involving percentage error for real-life data (e.g., comparing measurements from different instruments).

Students calculate percentage error for given scenarios and check their results.

Step 3 - Real-Life Applications

Discusses the importance of understanding percentage error in various fields like health, finance, and education.

Students work in groups to solve problems involving percentage error from different sectors.

NOTE ON BOARD:

  • Percentage Error: Percentage error = (|True Value - Approximate Value| / True Value) × 100%
  • Real-Life Applications: Health, finance, engineering, and education.

EVALUATION (5 exercises):

  1. Calculate the percentage error between the measured and actual value of a product.
  2. Find the percentage error for an estimated population number.
  3. Calculate the percentage error between an approximated cost and actual cost.
  4. Solve percentage error problems involving financial budgets.
  5. Calculate the percentage error for approximated health data.

CLASSWORK (5 questions):

  1. Calculate the percentage error for the cost of a school project.
  2. Solve for the percentage error for an estimated population.
  3. Find the percentage error for a measured quantity in the health sector.
  4. Calculate the percentage error for a financial report approximation.
  5. Solve percentage error problems for various real-life applications.

ASSIGNMENT (5 tasks):

  1. Calculate the percentage error for a population estimate.
  2. Solve percentage error problems for health data approximations.
  3. Calculate the percentage error between two financial estimates.
  4. Calculate the percentage error for approximated school attendance figures.
  5. Solve a real-world problem involving percentage error in a business context.