Section A — Chapters 1–22
Integers: Operations and Properties
Junior Cycle — 1st Year
- ✓Define integers and represent them on a number line.
- ✓Perform addition and subtraction of integers accurately.
- ✓Perform multiplication and division of integers accurately.
- ✓Apply the commutative, associative, and distributive properties to integer calculations.
- ✓Apply the correct order of operations (BODMAS) when solving problems involving integers.
Key concepts
Integers are whole numbers, including positive numbers (1, 2, 3, ...), negative numbers (-1, -2, -3, ...), and zero (0). They do not include fractions or decimals. We can represent integers on a number line, with zero at the centre, positive numbers to the right, and negative numbers to the left.
When adding integers:\n* If the signs are the same, add the numbers and keep the common sign. (e.g., 3 + 5 = 8; -3 + (-5) = -8)\n* If the signs are different, subtract the smaller absolute value from the larger absolute value, and keep the sign of the number with the larger absolute value. (e.g., -3 + 5 = 2; 3 + (-5) = -2)
To subtract an integer, add its opposite. This means change the subtraction sign to an addition sign and change the sign of the number being subtracted. (e.g., 5 - (-3) becomes 5 + 3 = 8; -5 - 3 becomes -5 + (-3) = -8)
When multiplying integers:\n* If the signs are the same, the answer is positive. (e.g., 3 × 5 = 15; -3 × -5 = 15)\n* If the signs are different, the answer is negative. (e.g., -3 × 5 = -15; 3 × -5 = -15)
When dividing integers:\n* If the signs are the same, the answer is positive. (e.g., 15 ÷ 3 = 5; -15 ÷ -3 = 5)\n* If the signs are different, the answer is negative. (e.g., -15 ÷ 3 = -5; 15 ÷ -3 = -5)
The order in which you add or multiply integers does not change the result. This property applies to addition and multiplication, but not subtraction or division.
The way in which integers are grouped when adding or multiplying does not change the result. This property applies to addition and multiplication, but not subtraction or division.
Multiplication distributes over addition or subtraction. This means you can multiply a number by a sum (or difference) or multiply it by each part of the sum (or difference) separately and then add (or subtract) the results.
BODMAS is an acronym to remember the correct order for performing operations in a mathematical expression.\n* **B**rackets first\n* **O**rders (powers and square roots)\n* **D**ivision and **M**ultiplication (from left to right)\n* **A**ddition and **S**ubtraction (from left to right)
Key facts to remember
- 1Integers include positive whole numbers, negative whole numbers, and zero.
- 2Adding a negative number is the same as subtracting a positive number (e.g., 5 + (-3) = 5 - 3).
- 3Subtracting a negative number is the same as adding a positive number (e.g., 5 - (-3) = 5 + 3).
- 4When multiplying or dividing, if the signs are the same, the answer is positive. If the signs are different, the answer is negative.
- 5BODMAS (Brackets, Orders, Division/Multiplication, Addition/Subtraction) dictates the order of operations.
- 6The Commutative Property allows changing the order of numbers in addition and multiplication.
- 7The Associative Property allows changing the grouping of numbers in addition and multiplication.
- 8The Distributive Property links multiplication with addition/subtraction: a × (b + c) = (a × b) + (a × c).
Worked examples
Example 1
Evaluate: -12 + 7 - (-3) - 10
Answer
-12
Remember to change 'subtracting a negative' to 'adding a positive'.
Example 2
Calculate: (-8) × 4 ÷ (-2)
Answer
16
Multiplication and division have equal priority; perform them from left to right.
Example 3
Evaluate: 5 × (8 - 12) + 20 ÷ (-4)
Answer
-25
Always follow BODMAS carefully, step by step.
Common mistakes
- ✗Incorrectly handling double negative signs in subtraction (e.g., 5 - (-3) often mistaken as 5 - 3).
- ✗Making sign errors in multiplication and division, especially when there are multiple negative numbers.
- ✗Not following the correct order of operations (BODMAS), leading to incorrect results.
- ✗Confusing the properties of integers, particularly applying commutative/associative to subtraction/division.
- ✗Treating 'minus' as always meaning subtraction, rather than also indicating a negative number.
Exam tips
- ★When dealing with addition and subtraction of integers, visualise a number line if you are unsure of the direction.
- ★Always show your working step-by-step, especially when applying BODMAS, to avoid errors and gain partial marks.
- ★Pay close attention to the signs of numbers throughout your calculations; a single sign error can lead to a completely wrong answer.
- ★Practice questions involving all four operations and brackets to become proficient with the order of operations.
Ready to practise?
Try a problem on this topic
Snap a photo or type a question — get step-by-step working instantly.