Grade 7
  1. Number System  1.1. Integers  1.1.1. Properties of Integers  1.1.2. Operations on Integers  1.1.3. Additive and Multiplicative Inverses  1.1.4. Applications of Integers  1.2. Rational Numbers  1.2.1. Definition of Rational Numbers  1.2.2. Properties of Rational Numbers  1.2.3. Operations on Rational Numbers  1.2.4. Standard Form of Rational Numbers  1.3. Decimals  1.3.1. Operations on Decimals  1.3.2. Converting Between Fractions and Decimals  1.4. Powers and Exponents  1.4.1. Laws of Exponents  1.4.2. Simplifying Expressions with Exponents  1.4.3. Scientific Notation  1.5. Roots  1.5.1. Square Roots  1.5.2. Cube Roots
  2. Algebra  2.1. Expressions  2.1.1. Algebraic Expressions  2.1.2. Simplifying Expressions  2.1.3. Evaluating Expressions  2.2. Linear Equations  2.2.1. Solving Simple Equations  2.2.2. Applications of Linear Equations  2.3. Algebraic Identities  2.3.1. Expanding Using Identities  2.3.2. Simplifying Using Identities
  3. Ratio and Proportion  3.1. Ratios  3.1.1. Understanding Ratios  3.1.2. Simplifying Ratios  3.1.3. Equivalent Ratios  3.2. Proportion  3.2.1. Concept of Proportion  3.2.2. Direct Proportion  3.2.3. Inverse Proportion  3.3. Unitary Method  3.3.1. Solving Problems Using the Unitary Method
  4. Geometry  4.1. Lines and Angles  4.1.1. Types of Angles  4.1.2. Pairs of Angles  4.1.3. Properties of Parallel Lines and a Transversal  4.1.4. Angle Bisectors  4.2. Triangles  4.2.1. Types of Triangles  4.2.2. Angle Sum Property  4.2.3. Exterior Angle Property  4.2.4. Pythagoras Theorem  4.3. Congruence of Triangles  4.3.1. Criteria for Congruence  4.3.2. Applications of Congruence  4.4. Quadrilaterals  4.4.1. Types of Quadrilaterals  4.4.2. Properties of Parallelograms  4.4.3. Special Quadrilaterals  4.4.4. Properties of Rectangles and Rhombuses
  5. Mensuration  5.1. Perimeter and Area  5.1.1. Perimeter of Plane Figures  5.1.2. Area of Plane Figures  5.1.3. Area of a Trapezium  5.1.4. Area of a Parallelogram  5.1.5. Area of a Circle  5.1.6. Area of Composite Figures  5.2. Surface Area and Volume  5.2.1. Cubes and Cuboids  5.2.2. Surface Area of Cylinders  5.2.3. Volume of Prisms and Cylinders  5.2.4. Volume and Surface Area of Cones
  6. Data Handling  6.1. Data Collection  6.1.1. Organizing Data  6.1.2. Frequency Distribution  6.2. Graphical Representation of Data  6.2.1. Bar Graphs  6.2.2. Double Bar Graphs  6.2.3. Pie Charts  6.2.4. Histograms  6.2.5. Line Graphs  6.3. Probability  6.3.1. Simple Events  6.3.2. Experimental Probability  6.3.3. Theoretical Probability
  7. Practical Geometry  7.1. Construction of Triangles  7.1.1. Constructing Triangles Given Side Lengths  7.1.2. Constructing Triangles Given Side and Angle  7.1.3. Constructing Right-Angled Triangles  7.2. Construction of Quadrilaterals  7.2.1. Quadrilaterals Given Side Lengths and Angles  7.2.2. Constructing Parallelograms and Rectangles

Grade 7 → Number System → Integers ↓


Operations on Integers


It is important for students to understand operations on integers as they advance in mathematics. Integers are whole numbers that include positive numbers, negative numbers, and zero. The four basic mathematical operations are addition, subtraction, multiplication, and division. Each of these operations can be performed on integers in different ways, and it is important to understand how these operations affect integers.

Addition of integers

Adding integers involves combining the values. The main thing to remember is how to handle positive and negative numbers:

  • Positive + Positive: Numbers added together become more positive.
  • Negative + Negative: Numbers added together become more negative.
  • Positive + negative (or negative + positive): The numbers essentially cancel each other out, and you take the difference. The sign of the result will be the sign of the larger absolute value.

Examples of totals

Example 1: 5 + 3 = 8
Example 2: (-4) + (-2) = -6
Example 3: 7 + (-9) = -2
Example 4: (-3) + 5 = 2

Visual example:

Consider two buckets where adding integers represents the number of stones in the buckets. Adding negative numbers means removing stones.

+--------+    +--------+    +--------+ 
|    +3   | +  |   +4    | =  |   +7    |  Both are positive 
+--------+    +--------+    +--------+

+--------+    +--------+    +--------+
|   -4    | +  |   -3    | =  |   -7   |  Both are negative 
+--------+    +--------+    +--------+

+--------+    +--------+    +--------+
|   +5    | +  |   -6    | =  |   -1   |  Mixed signs 
+--------+    +--------+    +--------+

Subtraction of integers

Subtraction of integers is closely related to addition. In fact, subtraction can be understood as adding opposites. For example, subtracting a number is the same as adding its negative counterpart.

  • Positive - Positive: Subtract the numbers; if the first number is smaller, the result will be negative.
  • Negative - Negative: Subtract the numbers; if the first number is more negative (or less negative but with a larger absolute value), the result shows the larger absolute value.
  • Positive - Negative: This operation turns into addition. For example, (a - (-b)) becomes (a + b).
  • Negative - Positive: The operation usually results in a more negative outcome.

Subtraction examples

Example 1: 9 - 5 = 4
Example 2: (-6) - (-4) = -2
Example 3: 7 - (-3) = 10
Example 4: (-8) - 5 = -13

Visual example:

If putting marbles into buckets represents addition, then taking marbles out in subtraction can be viewed as follows:

+--------+  -  +--------+  =  +--------+
|   +5    |    |    +3   |    |    +2   |
+--------+     +--------+     +--------+

+--------+  -  +--------+  =  +--------+
|   -4    |    |   -2    |    |   -2   |
+--------+     +--------+     +--------+

+--------+  -  +--------+  =  +--------+
|   +9    |    |  -(-4) |    |   +13   |
+--------+     +--------+     +--------+

Multiplication of integers

Multiplication of integers follows straightforward rules regarding the sign of the product:

  • Positive × Positive: The product is positive.
  • Negative × Negative: The product is positive.
  • Positive × Negative (or Negative × Positive): The product is negative.

Examples of multiplication

Example 1: 4 × 3 = 12
Example 2: (-3) × (-2) = 6
Example 3: 5 × (-4) = -20
Example 4: (-6) × 7 = -42

Visual example:

Think of multiplication as repeated addition or expansion:

4 * 3
+--------+  *  +--------+    |   +12   |
|   +12   |    |   +12   |
+--------+     +--------+

-3 * -2
+--------+  *  +--------+    |   + 6    |
|   +6    |    |   +6    |
+--------+     +--------+

5 * -4
+--------+  *  +--------+    |  -20    |
|  -20    |    |  -20    |
+--------+     +--------+

Division of integers

Division of integers also follows special rules regarding the sign of the quotient:

  • Positive ÷ Positive: The quotient is positive.
  • Negative ÷ Negative: The quotient is positive.
  • Positive ÷ Negative (or Negative ÷ Positive): The quotient is negative.

Partition examples

Example 1: 8 ÷ 2 = 4
Example 2: (-12) ÷ (-3) = 4
Example 3: 18 ÷ (-2) = -9
Example 4: (-15) ÷ 5 = -3

Visual example:

Think of division as the opposite of multiplication, or dividing objects into equal groups:

8 ÷ 2 = 4
+--------+  ÷  +-------+ = +--------+
|   +8    |   |   +2   |   |   +4   |
+--------+     +-------+   +--------+

-12 ÷ -3 = 4
+--------+  ÷  +-------+ = +--------+
|  -12    |   |   -3   |   |   +4   |
+--------+     +-------+   +--------+

18 ÷ -2 = -9
+--------+  ÷  +-------+ = +--------+
|   18    |   |   -2   |   |   -9   |
+--------+     +-------+   +--------+

Conclusion

Mastering operations on integers is fundamental in mathematics and is necessary to solve more complex equations and problems. Please remember the rules of signs when performing these operations.


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Operations on Integers

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