Mathematics Operations and BODMAS Rules

Words and Figures

• Writing figures in words is done by writing down how you would say the number.
• For example, 6840 would be written as "six thousand, eight hundred and forty".

Operations

• Addition: Line up the numbers in columns and add from right to left, carrying over any tens.
• Subtraction: Line up the numbers in columns and subtract from right to left, borrowing when necessary.
• Multiplication: Use the column method. Multiply the first number by each digit of the second, working from right to left and adding the results.
• Division: Use the bus shelter method. Divide the first number by the second, working from left to right.

Order of Operations (BODMAS/BIDMAS)

• Brackets: Perform any operations inside brackets first.
• Orders/Indices: Then, perform exponents, square roots, etc.
• Division and Multiplication: Then, perform any divisions or multiplications, working from left to right.
• Addition and Subtraction: Finally, perform any additions or subtractions, working from left to right.

Rounding

• Rounding to the Nearest Hundred: If a number is between 200 and 300, round to either 200 or 300. Determine which by comparing the number to the midpoint of 200 and 300 (250).
• Rounding to the Nearest Thousand: If a number is between 7000 and 8000, round to either 7000 or 8000. Determine which by comparing the number to the midpoint of 7000 and 8000 (7500).
• Rounding to One Decimal Place: If a number is between 5.1 and 5.2, round to either 5.1 or 5.2. Determine which by comparing to the midpoint of 5.1 and 5.2 (5.15).
• Rounding to One Significant Figure: Use only one digit followed by zeros. Look at the second digit. If it's 5 or greater, round up the first digit. Otherwise, round down.
• Rounding to Two or More Significant Figures: Consider the digit after the desired number of significant figures. Round up if it is 5 or greater; otherwise, round down.

Rounding Numbers

• To round a number to a specific number of significant figures: count the digits from left to right, starting with the first non-zero digit.
• If the next digit is 5 or greater, round the last digit up.
• If the next digit is less than 5, keep the last digit as it is.
• Example: Rounding 1307 to two significant figures, use 13 and the third digit is 7, round up to 1300.
• When rounding decimal numbers, disregard any zeros before the decimal point.

Finding Highest and Lowest Possible Values

• When rounding to the nearest thousand, the lowest possible value is the number less than half the rounding unit (e.g., less than 500).
• The highest possible value is the number less than or equal to half of the rounding unit (e.g., less than or equal to 499).
• Example: If a population is 12,000 to the nearest thousand, the lowest possible value is 11,500 and the highest is 12,499.

Estimation with Rounding

• Estimation uses "nice numbers" by rounding to one significant figure.
• This simplifies calculations using whole numbers.
• Example: Estimate the cost of 78 magazines at £5.15 each. Round 78 to 80 and £5.15 to £5. The estimated cost is 80 x £5 = £400. Use the ≈ symbol to show an estimate.

Ordering Decimal Numbers

• Arrange decimal numbers in order by comparing digits in each place value column, starting from the leftmost column.
• If digits in a column are the same, compare the next column.
• Example: Arrange 7.81, 7.49, 7.3, 7.007, and 7.102 in ascending order: 7.007, 7.102, 7.3, 7.49, 7.81.

Arithmetic With Decimals

• When adding or subtracting, vertically align numbers so decimal points line up.

• Subtract or add each column from right to left, remembering to borrow or carry.

• Example: 4.2 - 1.79. Align the decimal points to get 2.41.

• Multiplication: count the decimal places in the numbers; the answer will have the same number of decimal places.

• Example: 0.8 x 0.3 = 0.24 (two decimal places). Alternatively, convert decimals to whole numbers (ex: multiply by 10 or 100) then multiply and divide by the same factor to get the answer.

• Division of decimals by whole numbers: Use the bus stop method; place the decimal point in the quotient in the correct position.

• Example: 11.4 ÷ 3 = 3.8.

• Division of a decimal by a decimal: multiply both numbers by a multiple of ten (e.g. 10, 100, etc) that makes the divisor a whole number. Then perform the division. Example: 15.7 ÷ 0.2 = 157 ÷ 2 = 78.5.

Real-Life Negatives and Ordering Negative Numbers

• Negative numbers represent values below zero.
• When ordering temperatures, smaller negative numbers are colder.
• Example: Temperatures from coldest to warmest (-8 °C, -7 °C, -6 °C, -4 °C, 0 °C, 2 °C).

Arithmetic Involving Negatives

• Adding a positive to a negative moves towards the right on the number line.
• Adding a negative to a positive moves towards the left on the number line.
• Subtracting a negative is the same as adding its positive.
• Subtracting a positive is the same as adding its negative.
• Example: 6 – 10 = -4, -7 + 12 = 5, -13 – 4 = -17, 5 + (-3) = 2, 8 – (-7) = 15, -10 + (-5) = -15.

Multiplication & Division With Negatives

• Positive x Positive = Positive
• Positive x Negative = Negative
• Negative x Positive = Negative
• Negative x Negative = Positive

Place Value

• Place value indicates the value of each digit in a number.
• The place value chart includes units, tens, hundreds, thousands, etc., and decimal place values (tenths, hundredths, thousandths).

Inequality Signs

• < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to)

Place Value Using Calculations

• Multiplying a number by 10 increases its value ten times.
• Dividing a number by 10 decreases its value ten times.

Multiples, Common Multiples, LCM

• Multiples: Results from multiplying a number by an integer.
• Common multiples: Shared multiples of two or more numbers.
• Lowest Common Multiple (LCM): The smallest common multiple of two or more numbers.

Factors, Common Factors, HCF

• Factor: A whole number that divides another without a remainder.
• Common factors: Shared factors of two or more numbers.
• Highest Common Factor (HCF): Largest common factor of two or more numbers.

Prime Numbers

• Prime numbers are whole numbers greater than 1 with only two factors: 1 and itself. (e.g., 2, 3, 5, 7)

Square Numbers, Squaring Numbers, Square Roots

• Square number: Result from multiplying a whole number by itself.
• Squaring: Multiplying a number by itself.
• Square root: The value that, when multiplied by itself, equals the original number.

Cube Numbers, Cubing Numbers, Cube Roots

• Cube number: Result from multiplying a number by itself three times.
• Cubing: Multiplying a number by itself three times.
• Cube root: The value that, when multiplied by itself three times, equals the original number.

Index Notation, Laws of Indices

• Index notation: Writing a number with a power.
• Multiplying with the same base: Add the exponents.
• Dividing with the same base: Subtract the exponents.
• Power over power: Multiply the exponents.

Negative Indices

• Negative index: The reciprocal of the base number raised to the positive value of the index. (e.g., x^-n = 1/x^n)

Product of Primes

• Every whole number greater than 1 is either a prime number or can be expressed as a product of prime numbers.
• Prime Factorization: Using a prime factor tree to write a number as a product of prime numbers.

Applying Product of Primes

• Use prime factorizations to find the least whole number to multiply by a given number to get a perfect square or cube. Factors must have even exponents for a perfect square, multiples of 3 for a cube.

Finding LCM and HCF

• Use prime factorizations to find LCM and HCF.
• LCM: Product of highest powers of all prime factors.
• HCF: Product of lowest powers of all common prime factors.
• Venn diagrams can be used to visualize common and unique prime factors.

This quiz covers essential math operations including addition, subtraction, multiplication, and division. It also explains the order of operations known as BODMAS/BIDMAS, which is vital for solving complex equations correctly. Test your understanding of writing numbers in words and applying these operations accurately.