## Year 6 SATs Maths: Arithmetic - Addition and Subtraction

Addition and subtraction are fundamental skills you'll use throughout the SATs Arithmetic paper. Being accurate and efficient with these operations is key to scoring well. You’ll need to work with whole numbers, including large numbers, and decimals.

## Column Addition

This is the most common method for adding larger numbers.

• Line up the numbers carefully according to their place value. Make sure the ones digits are aligned, tens digits are aligned, and so on. If you're adding decimals, line up the decimal points.
• Start adding from the rightmost column (the ones column or the smallest decimal place).
• If the sum of a column is 10 or more, write down the ones digit and carry over the tens digit to the next column on the left.
• Continue this process until all columns have been added.

## Column Subtraction

This method is used for subtracting larger numbers.

• Again, line up the numbers by their place value, ensuring ones are under ones, tens under tens, etc. For decimals, line up the decimal points.
• Start subtracting from the rightmost column.
• If the top digit in a column is smaller than the bottom digit, you need to borrow or exchange from the column to its immediate left. This means taking one from the digit on the left and adding ten to the current column.
• Remember to adjust the digit you borrowed from.
• Continue this process across all columns.

## Mental Methods

For simpler calculations, using mental strategies can be quicker.

• Partitioning: Breaking numbers into tens and ones (or hundreds, tens, ones) to add or subtract parts.
• Compensating: Rounding a number to a nearby multiple of 10 or 100, performing the calculation, then adjusting the answer.
• Using number bonds (e.g., to 10, 100, 1000) for quick recall.

## Checking Your Work

Always try to check your answers, especially in an exam.

• Inverse Operations: Addition and subtraction are inverse operations. You can check an addition sum by subtracting one of the addends from the total. Similarly, you can check a subtraction sum by adding the answer to the number you subtracted.
• Estimation: Before calculating, estimate the answer by rounding the numbers to the nearest 10, 100, or whole number (for decimals). This helps you see if your final answer is reasonable.

# Year 6 SATs Maths: Arithmetic - Multiplication & Division

Mastering multiplication and division is crucial for your Year 6 SATs Arithmetic paper. This section tests your ability to use formal methods, mental strategies, and understand key mathematical terms.

## Multiplication Strategies

Long Multiplication

This is for multiplying larger numbers, typically a 4-digit number by a 2-digit number. Remember these steps:

1. Multiply by the ones digit: Multiply the top number by the ones digit of the bottom number. Write down your answer, remembering to carry over any tens.

2. Multiply by the tens digit: Before you start, put a zero in the ones column of your next line. Then, multiply the top number by the tens digit of the bottom number. Write this answer below the first one, again carrying over if needed.

3. Add the results: Add the two rows of numbers you've calculated to get your final product.

*Example*: 2345 x 16

Mental Multiplication

• Multiplying by 10, 100, 1000: Digits move to the left. For whole numbers, add zeros. For decimals, the decimal point moves right (e.g., 3.4 x 100 = 340).
• Multiplying by multiples of 10: Multiply by the non-zero digit, then multiply by 10 (e.g., 15 x 20 = (15 x 2) x 10 = 30 x 10 = 300).

Factors and Multiples

• A factor is a number that divides another number exactly, with no remainder (e.g., factors of 12 are 1, 2, 3, 4, 6, 12).
• A multiple is a number in another number's times table (e.g., multiples of 5 are 5, 10, 15, 20...). The number itself is always its first multiple.

## Division Strategies

Long Division

This is for dividing larger numbers, typically a 4-digit number by a 2-digit number. Remember the 'Divide, Multiply, Subtract, Bring Down' (DMSB) process:

1. Divide: How many times does the divisor (the number you're dividing by) go into the first part of the dividend (the number being divided)?

2. Multiply: Multiply the answer from step 1 by the divisor.

3. Subtract: Subtract this result from the part of the dividend you were working with.

4. Bring Down: Bring down the next digit from the dividend.

5. Repeat: Continue the DMSB steps until no digits are left to bring down.

• Remainders: You might have a remainder. This can be written as 'rX', a fraction (remainder/divisor), or a decimal (by adding .0 and continuing the division).

Short Division (Bus Stop Method)

This is quicker for dividing by a 1-digit number. Write the dividend inside the 'bus stop' and the divisor outside. Work from left to right, carrying any remainders to the next digit.

Mental Division

• Dividing by 10, 100, 1000: Digits move to the right. For whole numbers, remove zeros. For decimals, the decimal point moves left (e.g., 450 ÷ 100 = 4.5).

## Key Terms

• Prime Number: A whole number greater than 1 with exactly two factors: 1 and itself (e.g., 2, 3, 5, 7, 11).
• Composite Number: A whole number greater than 1 that has more than two factors (e.g., 4, 6, 8, 9, 10).
• Product: The result of a multiplication.
• Quotient: The result of a division.
• Dividend: The number being divided.
• Divisor: The number you are dividing by.

Remember to check your answers where possible, especially for long calculations!

## Understanding Fractions

A fraction represents a part of a whole. The numerator (top number) tells you how many parts you have, and the denominator (bottom number) tells you how many equal parts the whole is divided into.

## Types of Fractions

• Proper Fractions: Numerator is smaller than the denominator (e.g., 3/4).
• Improper Fractions: Numerator is equal to or larger than the denominator (e.g., 7/5, 4/4).
• Mixed Numbers: A whole number combined with a proper fraction (e.g., 1 2/3).
• Converting Improper to Mixed: Divide the numerator by the denominator. The whole number is the quotient, the remainder is the new numerator, and the denominator stays the same. (e.g., 7/3 = 2 with remainder 1, so 2 1/3).
• Converting Mixed to Improper: Multiply the whole number by the denominator, then add the numerator. This becomes the new numerator, over the original denominator. (e.g., 2 1/3 = (2x3)+1 / 3 = 7/3).

## Equivalent Fractions and Simplifying

Equivalent fractions represent the same value even though they have different numerators and denominators (e.g., 1/2 = 2/4 = 3/6). To find equivalent fractions, multiply or divide both the numerator and denominator by the same number.

Simplifying fractions means finding an equivalent fraction where the numerator and denominator have no common factors other than 1. You divide both by their highest common factor (e.g., 6/9 simplifies to 2/3 by dividing by 3).

## Adding and Subtracting Fractions

To add or subtract fractions, they must have the same denominator. If they don't, find a common denominator (often the lowest common multiple) and convert them to equivalent fractions before adding or subtracting the numerators. The denominator stays the same. (e.g., 1/2 + 1/4 = 2/4 + 1/4 = 3/4).

## Multiplying Fractions

• Fraction by a whole number: Multiply the numerator by the whole number. The denominator stays the same. (e.g., 2/5 x 3 = 6/5).
• Fraction by a fraction: Multiply the numerators together and multiply the denominators together. (e.g., 1/2 x 3/4 = 3/8).

## Dividing Fractions

• Fraction by a whole number: Multiply the denominator by the whole number. The numerator stays the same. (e.g., 1/2 ÷ 3 = 1/ (2x3) = 1/6).

## Fractions of Amounts

To find a fraction of an amount, divide the amount by the denominator and then multiply by the numerator. (e.g., 2/3 of 12: (12 ÷ 3) x 2 = 4 x 2 = 8).

## Decimals: Understanding and Using Them

Decimals are a way to show parts of a whole number. The decimal point separates whole numbers from parts of numbers.

• The first digit after the decimal point is the tenths place (e.g., 0.1).
• The second digit is the hundredths place (e.g., 0.01).
• The third digit is the thousandths place (e.g., 0.001).

Adding and Subtracting Decimals

To add or subtract decimals, always line up the decimal points. You can add zeros at the end of a decimal to make them the same length, which helps keep columns tidy.

• Example: 3.45 + 1.2 = 3.45 + 1.20 = 4.65

Multiplying and Dividing by 10, 100, 1000

This is about moving the decimal point:

• Multiply by 10, 100, or 1000: Move the decimal point to the right by the number of zeros.
• Example: 2.3 x 100 = 230 (move 2 places right)
• Divide by 10, 100, or 1000: Move the decimal point to the left by the number of zeros.
• Example: 45.6 ÷ 10 = 4.56 (move 1 place left)

Multiplying Decimals by Whole Numbers

Ignore the decimal point at first and multiply as usual. Then, count how many decimal places were in the original decimal number. Put that many decimal places back into your answer.

• Example: 2.3 x 4
• 23 x 4 = 92
• 2.3 has one decimal place, so the answer is 9.2

Dividing Decimals by Whole Numbers

You can use the bus stop method. Keep the decimal point in the same position in your answer as it is in the number you are dividing.

• Example: 7.2 ÷ 3 = 2.4

## Percentages: Parts Out of 100

Percentage means "per hundred" or "out of 100". The symbol is %.

• 10% means 10 out of 100.

Converting Between Fractions, Decimals, and Percentages

• Fraction to Decimal: Divide the numerator by the denominator. (e.g., 1/2 = 1 ÷ 2 = 0.5)
• Decimal to Percentage: Multiply by 100. (e.g., 0.5 x 100 = 50%)
• Percentage to Decimal: Divide by 100. (e.g., 50% ÷ 100 = 0.5)
• Percentage to Fraction: Write it over 100 and simplify. (e.g., 50% = 50/100 = 1/2)

Finding Percentages of Amounts

Some common percentages are easy to find:

• 50% = divide by 2
• 25% = divide by 4
• 10% = divide by 10
• 1% = divide by 100
• You can combine these: to find 30%, find 10% then multiply by 3.

## Order of Operations (BODMAS/BIDMAS)

When you have a calculation with more than one operation (like addition, subtraction, multiplication, or division), you need to follow a specific order to get the correct answer. This order is called the order of operations, and we use the acronym BODMAS (or sometimes BIDMAS) to remember it.

• Brackets first: Always solve anything inside brackets `()` before anything else.
• Orders (or Indices): This refers to powers or square roots (e.g., 2²). While less common in Year 6 SATs, it's good to know it comes next.
• Division and Multiplication: These have equal priority. Work them out from left to right as they appear in the sum.
• Addition and Subtraction: These also have equal priority. Work them out from left to right as they appear in the sum.

Example: Solve 5 + 3 x 2

1. Multiplication first (M): 3 x 2 = 6

2. Then Addition (A): 5 + 6 = 11

So, 5 + 3 x 2 = 11

Example with Brackets: Solve (5 + 3) x 2

1. Brackets first (B): 5 + 3 = 8

2. Then Multiplication (M): 8 x 2 = 16

So, (5 + 3) x 2 = 16

Remember, if you have both division and multiplication, or both addition and subtraction, work from left to right!

## Missing Numbers (Inverse Operations)

Finding missing numbers in equations means figuring out the unknown value that makes the equation true. We use inverse operations to 'undo' what has been done and find the missing part.

• Addition is the inverse of subtraction.
• Multiplication is the inverse of division.

To find a missing number, you often need to rearrange the equation using the inverse operation.

Example 1: Missing number in addition

7 + ? = 15

To find '?', you can do the inverse of adding 7, which is subtracting 7 from 15.

15 - 7 = 8

So, ? = 8. (Check: 7 + 8 = 15)

Example 2: Missing number in subtraction

? - 5 = 12

To find '?', you can do the inverse of subtracting 5, which is adding 5 to 12.

12 + 5 = 17

So, ? = 17. (Check: 17 - 5 = 12)

Example 3: Missing number in multiplication

? x 4 = 28

To find '?', you can do the inverse of multiplying by 4, which is dividing 28 by 4.

28 ÷ 4 = 7

So, ? = 7. (Check: 7 x 4 = 28)

Example 4: Missing number in division

30 ÷ ? = 6

To find '?', you can think: 6 x ? = 30. The inverse of multiplying by 6 is dividing 30 by 6.

30 ÷ 6 = 5

So, ? = 5. (Check: 30 ÷ 5 = 6)

Always check your answer by putting the missing number back into the original equation to make sure it works!

## Long Multiplication and Division

This topic tests your ability to multiply and divide multi-digit numbers accurately. These methods are fundamental for the arithmetic paper.

## Long Multiplication

Long multiplication is used to multiply numbers with two or more digits. The standard method involves breaking down the multiplication into simpler steps:

• Set it up: Write the numbers vertically, aligning the ones digits.
• Multiply by the ones digit: Multiply the top number by the ones digit of the bottom number. Write your answer below the line, carrying over any tens to the next column. This is your first partial product.
• Add a placeholder zero: Before multiplying by the tens digit, place a zero in the ones column of the next row. This is crucial because you are now multiplying by a tens value.
• Multiply by the tens digit: Multiply the top number by the tens digit of the bottom number. Write this result in the row above the zero, carrying over as needed. This is your second partial product.
• Add the partial products: Add the two partial products together to get your final answer.

Example: To calculate 34 x 26:

1. 34 x 6 = 204 (first partial product)

2. 34 x 20 = 680 (second partial product, remember the placeholder zero)

3. 204 + 680 = 884

## Long Division

Long division is used to divide large numbers by two-digit numbers (or more). The 'bus stop' method is commonly used. A helpful acronym is DMSBR: Divide, Multiply, Subtract, Bring Down, Remainder/Repeat.

• Set it up: Write the dividend (the number being divided) inside the 'bus stop' and the divisor (the number you are dividing by) outside.
• Divide: Start from the leftmost digit(s) of the dividend. How many times does the divisor go into this part? Write the answer above as the first digit of your quotient.
• Multiply: Multiply the quotient digit you just wrote by the divisor.
• Subtract: Subtract this product from the part of the dividend you were working with.
• Bring Down: Bring down the next digit from the dividend to form a new number.
• Repeat: Go back to the 'Divide' step with your new number. Continue until all digits of the dividend have been used.
• Remainder: If there's a number left at the end, it's the remainder. You can write it as 'R' followed by the number, or as a fraction (remainder/divisor).

Example: To calculate 156 ÷ 12:

1. 12 goes into 15 once (1 above the 5).

2. 1 x 12 = 12.

3. 15 - 12 = 3.

4. Bring down the 6, making 36.

5. 12 goes into 36 three times (3 above the 6).

6. 3 x 12 = 36.

7. 36 - 36 = 0.

The answer is 13.

## Key Tips

• Times Tables: A strong knowledge of times tables is essential for both operations.
• Estimation: Always estimate your answer before calculating to check if your final answer is reasonable.
• Show your working: If you make a mistake, showing your steps can help you find it.