M1 · Integers, decimals and calculations | Edexcel GCSE Maths

Revise the key ideas

An integer is a whole number: …, −2, −1, 0, 1, 2, …. In 307.46, the 4 means four tenths and the 6 means six hundredths. Zeros can be essential place holders.
Multiplying by 10, 100 or 1000 makes each digit worth 10, 100 or 1000 times as much. For example, 0.047 × 100 = 4.7. Dividing reverses this; do not describe the decimal point as changing the number's place-value system.
On a number line, values increase to the right: −7 < −2 < 0 < 0.4. The symbols ≤ and ≥ include equality; ≠ means not equal.
Signed numbers on a number line
Enlarge diagram: Signed numbers on a number line
Compare decimals by matching place values: 0.6 = 0.60 > 0.56. For fractions, use a common denominator or convert to decimals before ordering.
Adding a positive number moves right; subtracting a positive number moves left. −3 + 7 = 4 and −3 − 7 = −10. Subtracting a negative adds its opposite: 5 − (−2) = 7.
For multiplication and division, equal signs give a positive result and different signs give a negative result: (−4) × (−3) = 12; 12 ÷ (−3) = −4. This sign rule does not apply to addition.

For column addition and subtraction, line up decimal points and matching place values. 7.35 + 0.8 = 8.15. Exchange between neighbouring columns when required; use zeros to make places clear.
For long multiplication, multiply by each digit and account for its place value. 24 × 13 = 24 × 10 + 24 × 3 = 240 + 72 = 312.
For long division, divide, multiply, subtract and bring down the next digit. 156 ÷ 12 = 13 because 12 × 13 = 156. A remainder may need to become a fraction or decimal in the context.
To multiply decimals, first use whole numbers and then restore the total number of decimal places: 1.2 × 0.35 = 0.42 because 12 × 35 = 420 and there are three decimal places altogether.
To divide by a decimal, multiply both numbers by the same power of 10: 4.8 ÷ 0.06 = 480 ÷ 6 = 80. Changing only the divisor changes the question.
Use brackets first, then powers and roots, then multiplication/division from left to right, then addition/subtraction from left to right. 18 ÷ 3 × 2 = 12, not 3.
The reciprocal of a non-zero number is 1 divided by that number. The reciprocal of 5 is 15; the reciprocal of 23 is 32. A number times its reciprocal is 1.

Worked example: 4 + 3 × (7 − 5)² = 4 + 3 × 2² = 4 + 12 = 16. Show each stage so the order is clear.
Worked example: (−3)² = 9, but −3² = −9 because squaring happens before the minus sign outside the power. Brackets change which number is squared.
Check arithmetic using an inverse operation or an estimate. If 6.2 × 4.9 is reported as 303.8, estimating 6 × 5 ≈ 30 reveals the place-value error; the correct product is 30.38.

Practice questions with explained answers will be added here.

For now, cover the worked answers, try the calculations yourself, then compare each step. Include units and reasons where needed.