Number - Structure and calculation: apply the four operations, including formal written methods, to integers, decimals and simple fractions (proper and improper), and mixed numbers – all both positive and negative; understand and use place value (e.g. when working with very large or very small numbers, and when calculating with decimals)

The idea

Formal written methods include column addition/subtraction, long multiplication and short/long division. Place value is key: for 3,400,000 the '4' is 4 hundred thousand; for 0.0045 the '4' is 4 thousandths. When adding/subtracting decimals, align the decimal points. To multiply decimals (e.g., 0.5 × -0.03), multiply the integers (5 × 3 = 15), count the total decimal places (3), and apply sign rules to get -0.015. To divide by a decimal (e.g., 8.4 ÷ 0.2), scale both numbers to make the divisor whole: 84 ÷ 2 = 42. For fractions, convert mixed numbers to improper fractions first. To multiply, multiply numerators and denominators. To divide, invert the second fraction then multiply. To add or subtract, find a common denominator (e.g., 1/3 + 1/4 = 4/12 + 3/12 = 7/12). The rules for negative numbers (e.g., negative × negative = positive) apply to all number types.

What earns the marks

• Correct use of a formal written method, such as column addition or long division, with clear layout.
• Correctly handling decimal points in calculations, either by scaling for division or counting places for multiplication.
• Converting mixed numbers to improper fractions before multiplying or dividing.
• Finding a common denominator before adding or subtracting fractions.
• Applying the correct sign rules for calculations involving negative numbers across all number types.

Common mistakes and how to put them right

• Adding fraction numerators and denominators without finding a common denominator (e.g., 1/2 + 1/3 = 2/5).

  Put it right: Find a common denominator first: 3/6 + 2/6 = 5/6.

• Scaling only the divisor when dividing by a decimal (e.g., 7.5 ÷ 0.25 becomes 7.5 ÷ 25).

  Put it right: Scale both numbers by the same power of 10 to maintain equivalence: 750 ÷ 25.

• Incorrectly applying sign rules, especially with multiple negatives (e.g., 5 − (−2) = 3 or −4 × −3 = −12).

  Put it right: Two negatives make a positive in multiplication, division, and when subtracting a negative: 5 + 2 = 7 and −4 × −3 = 12.

Exam tips

• Estimate the answer first by rounding each number to one significant figure. This helps you check if your final answer is a reasonable magnitude.
• Show your full working for formal methods, as marks are awarded for the correct process even if you make a calculation error.
• On a calculator paper, always use brackets for negative numbers in operations like squaring or multiplication, e.g., enter `(-3.5)²` not `-3.5²`.