Lesson 1: Calculations

Introduction to Concepts

In this lesson, we will explore the priority of operations with positive and negative numbers, how to simplify calculations by cancelling, and the importance of using inverse operations in solving mathematical problems.

Key Concepts and Tips

Priority of Operations with Positive and Negative Numbers

The priority of operations is guided by the BODMAS/BIDMAS rule (Brackets, Orders, Division/Multiplication, Addition/Subtraction). When dealing with positive and negative numbers, it’s crucial to follow this order strictly to avoid errors. For example, in the expression

   

, you first perform the multiplication:

   

, then the subtraction and addition in order:

   

.

Example: “Evaluate

   

: First,

   

, then

   

, followed by

   

.”

Misconception: Some students might incorrectly add or subtract before handling multiplication or division. Remember, multiplication and division come before addition and subtraction.

Tip: “Use the BODMAS/BIDMAS rule as a checklist for every calculation involving multiple operations.”

Simplifying Calculations by Cancelling

Cancelling involves reducing fractions or algebraic expressions by dividing the numerator and the denominator by their common factors before performing further operations. For example, in the expression

   

, you can cancel by dividing 4 by 4 and 20 by 4 to get

   

.

Example: “Simplify

   

: Divide 18 and 30 by their common factor of 6, resulting in

   

.”

Misconception: Students might forget to cancel common factors before multiplying or dividing, which can lead to more complex calculations.

Tip: “Always look for common factors to simplify your calculations before carrying out the operations.”

Using Inverse Operations

Inverse operations are used to solve equations or check the correctness of a solution. Addition is the inverse of subtraction, and multiplication is the inverse of division. For instance, if you subtract 5 from a number and get 10, adding 5 back should give you the original number.

Example: “To solve

   

, use the inverse operation of addition (which is subtraction) to find

   

.”

Misconception: Some students might mistakenly apply the same operation instead of the inverse when trying to solve an equation. Remember, the inverse operation is crucial for finding the correct solution.

Tip: “When solving equations, always consider what operation was applied and use its inverse to reverse the effect.”

Interactive Quiz

Below are 10 questions to test your understanding:

1. What is the result of

      

   ? (Multiple Choice)
   1. A) 14
   2. B) 20
   3. C) 12
   4. D) 24
2. When simplifying an expression, addition should be done before multiplication. (True/False)
   1. A) True
   2. B) False
3. What is

      

   when simplified? Answer: ______
4. Simplify

      

   . Answer: ______
5. What is the inverse operation of addition? (Multiple Choice)
   1. A) Multiplication
   2. B) Subtraction
   3. C) Division
   4. D) Exponentiation
6. Inverse operations are used to solve equations. (True/False)
   1. A) True
   2. B) False
7. Solve

      

   using inverse operations.

      

   ______
8. Solve

      

   using inverse operations.

      

   ______
9. Which of the following expressions correctly simplifies

      

   ? (Multiple Choice)
   1. A) 15 + 2
   2. B) 5 \times 5
   3. C) 5 + 3 \times 2
   4. D) 5 \times 3 + 2
10. Simplify the expression

       

    . Answer: ______

Answers

Here are the answers for the quiz questions:

1. A) 14
2. B) False
3. 5
4. 6
5. B) Subtraction
6. A) True
7. 6
8. 8
9. B) 5 \times 5
10. 19

Worksheet

Below are practice problems:

Basic Practice

1. Evaluate

      

   .
2. Simplify

      

   .
3. Solve

      

   using inverse operations.
4. Simplify

      

   .
5. Evaluate

      

   .

Intermediate Practice

1. Simplify

      

   .
2. Solve

      

   using inverse operations.
3. Evaluate

      

   .
4. Simplify

      

   .
5. Solve

      

   using inverse operations.

Advanced Practice

1. Simplify

      

   .
2. Solve

      

   using inverse operations.
3. Evaluate

      

   .
4. Simplify

      

   .
5. Solve

      

   using inverse operations.

Problem Solving

1. Sarah solved the equation

      

   by subtracting 5 and then dividing by 2. What is the value of

      

   ?
2. John wants to simplify

      

   by cancelling. What is the simplified result?
3. Mary needs to evaluate

      

   . What is the correct answer?
4. A car travels

      

   miles in one day. How many miles did the car travel?
5. Tom solved the equation

      

   using inverse operations. What is the value of

      

   ?
6. Emma simplifies

      

   . What is the result?

Worksheet Answers

Here are the final answers for the worksheet problems:

Basic Practice Answers

1. 16
2. 6
3. 12
4. 6
5. 22

Intermediate Practice Answers

1. 12
2. 7
3. 21
4. 6
5. 6

Advanced Practice Answers

1. 6
2. 5
3. 23
4. 36
5. 4

Problem Solving Answers

Here are the answers for the problem-solving tasks:

1. Solution: Sarah subtracts 5:

      

   . Then she divides by 2:

      

   .
   1. Answer:

         

2. Solution: John cancels the common factor:

      

   .
   1. Answer:

         

3. Solution: Mary evaluates the expression:

      

   .
   1. Answer:

         

4. Solution: The car travels

      

   miles.
   1. Answer:

         

      miles
5. Solution: Tom adds 4:

      

   . Then he divides by 3:

      

   .
   1. Answer:

         

6. Solution: Emma simplifies:

      

   .
   1. Answer:

         

Worked Solutions

Below are the step-by-step solutions for each worksheet question and problem-solving task:

Basic Practice Solutions

1. Evaluate

      

   .
   1. Solution: First, multiply:

         

      , then add:

         

      .
2. Simplify

      

   .
   1. Solution: First, simplify:

         

      .
3. Solve

      

   using inverse operations.
   1. Solution: Add 5 to both sides:

         

      .
4. Simplify

      

   .
   1. Solution: First, simplify:

         

      .
5. Evaluate

      

   .
   1. Solution: First, multiply:

         

      , then add:

         

      .

Intermediate Practice Solutions

1. Simplify

      

   .
   1. Solution: First, simplify:

         

      .
2. Solve

      

   using inverse operations.
   1. Solution: Divide both sides by 3:

         

      .
3. Evaluate

      

   .
   1. Solution: First, multiply and divide:

         

      and

         

      . Then add and subtract:

         

      .
4. Simplify

      

   .
   1. Solution: First, simplify:

         

      .
5. Solve

      

   using inverse operations.
   1. Solution: Subtract 7 from both sides:

         

      .

Advanced Practice Solutions

1. Simplify

      

   .
   1. Solution: First, simplify:

         

      .
2. Solve

      

   using inverse operations.
   1. Solution: Add 5 to both sides:

         

      . Then divide by 4:

         

      .
3. Evaluate

      

   .
   1. Solution: First, evaluate the bracket:

         

      . Then multiply:

         

      . Finally, add:

         

      .
4. Simplify

      

   .
   1. Solution: First, simplify:

         

      .
5. Solve

      

   using inverse operations.
   1. Solution: Subtract 3 from both sides:

         

      . Then divide by 5:

         

      .

Problem Solving Solutions

1. Sarah solved the equation

      

   by subtracting 5 and then dividing by 2. What is the value of

      

   ?
   1. Solution: Sarah subtracts 5:

         

      . Then she divides by 2:

         

      .
2. John wants to simplify

      

   by cancelling. What is the simplified result?
   1. Solution: John cancels the common factor:

         

      .
3. Mary needs to evaluate

      

   . What is the correct answer?
   1. Solution: Mary evaluates the expression:

         

      .
4. A car travels

      

   miles in one day. How many miles did the car travel?
   1. Solution: The car travels

         

      miles.
5. Tom solved the equation

      

   using inverse operations. What is the value of

      

   ?
   1. Solution: Tom adds 4:

         

      . Then he divides by 3:

         

      .
6. Emma simplifies

      

   . What is the result?
   1. Solution: Emma simplifies:

         

      .

Interactive Activities and Games

Below are recommended online activities:

• Game: Order of Operations Game. Practice applying the order of operations in this interactive game.
• GeoGebra: Simplifying Calculations. Use this interactive tool to visualize and practice simplifying calculations.
• Khan Academy: Video on Order of Operations. Watch this video to understand the order of operations.
• YouTube Tutorial: Using Inverse Operations Tutorial. Learn how to solve equations using inverse operations.

Visuals

Below is a summary of the visuals included in this lesson:

• Order of Operations: Examples showing how to apply BODMAS/BIDMAS to expressions with positive and negative numbers.
• Simplifying Calculations: Visuals illustrating how to simplify fractions and expressions by cancelling common factors.
• Using Inverse Operations: Examples of solving equations using inverse operations.
• Explanation for Students Who Can’t View Visuals: The visuals in this lesson demonstrate the application of BODMAS/BIDMAS in calculations, the process of simplifying fractions by cancelling, and the use of inverse operations to solve equations. If you can’t view the visuals, the written explanations provide detailed guidance to understand these concepts.