Number
Integers, Decimals and Order of Operations
Year 7 · Year 8 · Year 9
- ✓By the end of this lesson students will be able to perform the four basic operations (addition, subtraction, multiplication, division) with integers and decimals.
- ✓By the end of this lesson students will be able to apply the rules for operations involving negative numbers.
- ✓By the end of this lesson students will be able to correctly apply the order of operations (BIDMAS) to solve multi-step calculations.
- ✓By the end of this lesson students will be able to solve problems involving a combination of integers, decimals, negative numbers, and multiple operations.
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.
Decimals are numbers that include a fractional part, represented by digits after a decimal point. For example, 3.5, 0.25, -1.7.
The four basic mathematical operations are addition (+), subtraction (-), multiplication (× or *), and division (÷ or /). These operations can be performed on integers and decimals.
When adding or subtracting negative numbers: - Adding a negative number is the same as subtracting: e.g., 5 + (-3) = 5 - 3 = 2. - Subtracting a negative number is the same as adding: e.g., 5 - (-3) = 5 + 3 = 8. - When adding numbers with different signs, subtract the smaller absolute value from the larger absolute value and keep the sign of the number with the larger absolute value: e.g., -7 + 3 = -4. - When adding numbers with the same sign, add their absolute values and keep the common sign: e.g., -7 + (-3) = -10.
When multiplying or dividing negative numbers: - If the signs are the same (both positive or both negative), the answer is positive: e.g., (-3) × (-4) = 12, 10 ÷ 2 = 5. - If the signs are different (one positive and one negative), the answer is negative: e.g., (-3) × 4 = -12, 10 ÷ (-2) = -5.
BIDMAS is an acronym that dictates the order in which operations should be performed in a mathematical expression to ensure a consistent result. The order is: B - Brackets first I - Indices (powers and roots) D - Division M - Multiplication A - Addition S - Subtraction Division and Multiplication have equal priority and should be performed from left to right. Similarly, Addition and Subtraction have equal priority and should be performed from left to right.
Key facts to remember
- 1Integers are whole numbers, including positive, negative, and zero.
- 2Decimals represent numbers with fractional parts.
- 3BIDMAS stands for Brackets, Indices, Division, Multiplication, Addition, Subtraction.
- 4Division and Multiplication have equal priority; Addition and Subtraction have equal priority. Work from left to right for operations of equal priority.
- 5Adding a negative number is the same as subtracting (e.g., 5 + (-3) = 5 - 3).
- 6Subtracting a negative number is the same as adding (e.g., 5 - (-3) = 5 + 3).
- 7When multiplying or dividing, if signs are the same, the result is positive. If signs are different, the result is negative.
Worked examples
Example 1
Calculate: 15 - 3 × (8 + 2)
Answer
-15
Remember to follow BIDMAS: Brackets, then Multiplication, then Subtraction.
Example 2
Evaluate: -6 × 2.5 + (12 ÷ -3)
Answer
-19
Pay close attention to the signs when multiplying and dividing negative numbers.
Example 3
Calculate: (4.2 - 1.8) × 5 + 20 ÷ (-4)
Answer
7
Remember that multiplication and division have equal priority, as do addition and subtraction. Work from left to right for operations of equal priority.
Common mistakes
- ✗Incorrectly applying the order of operations, especially confusing the priority of multiplication/division or addition/subtraction.
- ✗Making sign errors when adding or subtracting negative numbers (e.g., 5 - (-3) becoming 2 instead of 8).
- ✗Making sign errors when multiplying or dividing negative numbers (e.g., -4 × -2 becoming -8 instead of 8).
- ✗Ignoring brackets or performing operations outside brackets before those inside.
- ✗Not working from left to right for operations of equal priority (e.g., doing addition before subtraction if addition appears first).
Exam tips
- ★Always write down each step of your calculation clearly. This helps you track your progress and makes it easier to spot errors.
- ★Use BIDMAS as a checklist for every multi-step calculation. Mentally (or physically) tick off each stage as you complete it.
- ★Be extra careful with negative numbers. Circle or highlight negative signs to ensure you don't miss them.
- ★If working with decimals, ensure your column alignment is correct for addition/subtraction, and count decimal places carefully for multiplication.
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