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Number (6Ni)

Mathematics Stage 6 Chapter Notes

What this chapter covers

Number — Integers, powers and roots
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Number (6Ni) notes

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In this chapter

  1. 1Understanding Place Value and Ordering Numbers
  2. 2Factors, Multiples, and Prime Numbers
  3. 3Written Methods for Addition and Subtraction
  4. 4Written Methods for Multiplication and Division

1. Understanding Place Value and Ordering Numbers

Our number system, the Hindu-Arabic system, is built on place value. This means the position of a digit in a number determines its value. For example, in the number 7,453, the '7' is worth 7 thousands, the '4' is worth 4 hundreds, the '5' is worth 5 tens, and the '3' is worth 3 units. This system extends to numbers after a decimal point, representing tenths, hundredths, and thousandths. To compare and order numbers, we can visualise them on a number line. Numbers to the right are always larger than numbers to the left. This is especially important for negative numbers, where -10 is smaller than -1 because it is further to the left of zero.

Key term

Place Value: The value of a digit based on its position within a number, such as units, tens, hundreds, or tenths.

Examiner insight

Examiners look for clear understanding of ordering, especially when a list includes a mix of positive numbers, negative numbers, and decimals.

Common pitfall

When ordering a mix of numbers, students often forget that a negative number like -10 is smaller than -1 because it is further to the left on a number line.

Fun fact

The idea of zero as a number and a placeholder was one of the most important inventions in the history of mathematics, allowing our modern place value system to work.

Worked example 12 marks

The population of a city is 2,407,158. Write this number in words and state the value of the digit '7'.

  1. 1

    Step 1: Break the number into groups of three from the right: 2, | 407, | 158.

  2. 2

    Step 2: Write each group in words, adding the period name (million, thousand). Two million, four hundred and seven thousand, one hundred and fifty-eight.

  3. 3

    Step 3: Identify the position of the digit '7'. It is in the thousands column.

  4. 4

    Step 4: The value of the digit '7' is seven thousand (7,000).

Worked example 22 marks

Arrange the following numbers in ascending order (smallest to largest): 3.5, -4, 2, -1.5, 0.

  1. 1

    Step 1: Identify the negative numbers: -4 and -1.5. The number furthest from zero on the negative side is the smallest. So, -4 is the smallest.

  2. 2

    Step 2: The next smallest is the other negative number, -1.5.

  3. 3

    Step 3: The next number is 0.

  4. 4

    Step 4: Identify the positive numbers: 3.5 and 2. The smaller of these is 2.

  5. 5

    Step 5: The largest number is 3.5.

  6. 6

    Step 6: Write the final ordered list: -4, -1.5, 0, 2, 3.5.

Recap

  • Each place value column is ten times greater than the column to its right.
  • When ordering numbers, a number line can help you visualise their positions.
  • Negative numbers get smaller as their numeral gets bigger (e.g., -5 is smaller than -2).
  • To read large numbers, group digits in threes from the right.
  • When ordering decimals, compare the digits in each place value column from left to right.

Quick check

  1. What is the value of the digit 6 in the number 3,562,817?1 mark
  2. Which is larger, -12 or -2?1 mark

2. Factors, Multiples, and Prime Numbers

Factors, multiples, and primes are special properties of whole numbers. A 'factor' is a number that divides into another number exactly, with no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. A 'multiple' of a number is the result of multiplying that number by another whole number (it's in the 'times table'). For example, multiples of 3 are 3, 6, 9, 12, etc. A 'prime number' is a special type of number that has exactly two factors: 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11. The number 1 is not a prime number.

Key term

Prime Number: A whole number greater than 1 that has exactly two distinct factors: 1 and itself.

Examiner insight

When asked for all factors, showing your method of finding them in pairs helps ensure you don't miss any and can gain partial marks.

Common pitfall

Confusing factors and multiples. Remember: factors are few (like the ingredients in a cake), while multiples are many (like the cakes you can make).

Fun fact

Prime numbers are used to create the strong encryption that keeps your online data, like bank details and passwords, safe.

Worked example 12 marks

Find all the factors of 48.

  1. 1

    Step 1: Start by finding factor pairs. Always begin with 1.

  2. 2

    1 x 48 = 48. So, 1 and 48 are factors.

  3. 3

    2 x 24 = 48. So, 2 and 24 are factors.

  4. 4

    3 x 16 = 48. So, 3 and 16 are factors.

  5. 5

    4 x 12 = 48. So, 4 and 12 are factors.

  6. 6

    5 does not divide 48 exactly.

  7. 7

    6 x 8 = 48. So, 6 and 8 are factors.

  8. 8

    7 does not divide 48 exactly. The next number is 8, which we already have.

  9. 9

    Step 2: List all the factors in ascending order: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.

Worked example 23 marks

Find the first three common multiples of 4 and 6.

  1. 1

    Step 1: List the first few multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36...

  2. 2

    Step 2: List the first few multiples of 6: 6, 12, 18, 24, 30, 36...

  3. 3

    Step 3: Identify the numbers that appear in both lists. The first is 12.

  4. 4

    Step 4: The second is 24.

  5. 5

    Step 5: The third is 36.

  6. 6

    The first three common multiples are 12, 24, and 36.

Recap

  • A factor divides a number exactly.
  • A multiple is found by multiplying a number by any whole number.
  • A prime number has only two factors: 1 and itself.
  • The number 1 is not a prime number.
  • The number 2 is the only even prime number.

Quick check

  1. List the first five multiples of 9.1 mark
  2. Is 17 a prime number? Explain why or why not.2 marks

3. Written Methods for Addition and Subtraction

For adding and subtracting larger numbers or decimals, we use written column methods. The key to success is carefully lining up the numbers according to their place value. For whole numbers, this means units under units, tens under tens, and so on. For decimals, you must line up the decimal points. In addition, if a column adds up to 10 or more, you 'carry' the digit to the next column on the left. In subtraction, if the top digit is smaller than the bottom digit, you must 'exchange' (or 'borrow') from the column to the left.

Key term

Exchanging: The process in column subtraction of regrouping from the column to the left, exchanging one of its value for ten in the current column.

Examiner insight

Clear, neat working in columns is essential. An examiner needs to be able to follow your steps, including any 'carrying' or 'exchanging' marks you make.

Common pitfall

When subtracting decimals, students often forget to add a placeholder zero, leading to errors like calculating 5.8 - 2.31 as 3.51 instead of the correct 3.49.

Worked example 12 marks

Calculate 5,847 + 672.

  1. 1

    Step 1: Write the numbers in columns, aligning the place values from the right.

  2. 2

    5847

  3. 3

    + 672

  4. 4

    ------

  5. 5

    Step 2: Add the units column: 7 + 2 = 9.

  6. 6

    Step 3: Add the tens column: 4 + 7 = 11. Write down 1 and carry 1 to the hundreds column.

  7. 7

    Step 4: Add the hundreds column: 8 + 6 + 1 (carried) = 15. Write down 5 and carry 1 to the thousands column.

  8. 8

    Step 5: Add the thousands column: 5 + 1 (carried) = 6.

  9. 9

    Final Answer: 6,519.

Worked example 23 marks

Calculate 72.5 - 8.63.

  1. 1

    Step 1: Write the numbers in columns, aligning the decimal points. Add a placeholder zero to 72.5 to make it 72.50.

  2. 2

    72.50

  3. 3
    • 8.63
  4. 4

    -------

  5. 5

    Step 2: Subtract the hundredths: 0 - 3. We need to exchange. Borrow from the 5 tenths, leaving 4. The 0 becomes 10. 10 - 3 = 7.

  6. 6

    Step 3: Subtract the tenths: 4 - 6. We need to exchange. Borrow from the 2 units, leaving 1. The 4 becomes 14. 14 - 6 = 8.

  7. 7

    Step 4: Subtract the units: 1 - 8. We need to exchange. Borrow from the 7 tens, leaving 6. The 1 becomes 11. 11 - 8 = 3.

  8. 8

    Step 5: Subtract the tens: 6 - 0 = 6. Remember to place the decimal point in the answer.

  9. 9

    Final Answer: 63.87.

Recap

  • Always line up numbers by their place value for column methods.
  • When working with decimals, always line up the decimal points.
  • In addition, 'carry' digits to the next column when a sum is 10 or more.
  • In subtraction, 'exchange' from the column to the left if the top digit is smaller.
  • Add placeholder zeros to make subtraction with decimals easier.

Quick check

  1. What is 999 + 102?1 mark
  2. What is 15.0 - 1.5?1 mark

4. Written Methods for Multiplication and Division

For multiplication, the column method is efficient. When multiplying by a two-digit number, you multiply by the units digit first, then multiply by the tens digit on a new line, remembering to put a zero as a placeholder in the units column. Finally, you add the two results together. For division, the 'bus stop' method is common. You work from left to right, dividing into each digit of the number. Any amount left over from one step is carried to the next digit. If there is an amount left over at the very end, it is called the 'remainder'.

Key term

Remainder: The amount 'left over' after a division when one number does not divide exactly into another.

Examiner insight

For division questions, read carefully whether the answer should be given with a remainder, as a fraction, or as a decimal. Giving the wrong format may lose a mark.

Common pitfall

In long multiplication, the most frequent error is forgetting to add a zero as a placeholder on the second line of working when multiplying by the tens digit.

Worked example 13 marks

Calculate 258 x 34.

  1. 1

    Step 1: Set up the column multiplication. Multiply 258 by 4 (the units digit).

  2. 2

    258 x 4 = 1032.

  3. 3

    Step 2: On the next line, multiply 258 by 30 (the tens digit). To do this, write a 0 in the units column, then multiply 258 by 3.

  4. 4

    258 x 3 = 774. So 258 x 30 = 7740.

  5. 5

    Step 3: Add the two results together.

  6. 6

    1032

  7. 7

    + 7740

  8. 8

    ------

  9. 9

    8772

  10. 10

    Final Answer: 8,772.

Worked example 22 marks

Calculate 615 ÷ 7.

  1. 1

    Step 1: Set up the division using the 'bus stop' method.

  2. 2

    Step 2: How many 7s in 6? 0. Carry the 6 over to the next digit, making 61.

  3. 3

    Step 3: How many 7s in 61? 8 (since 7 x 8 = 56). Write 8 above the 1.

  4. 4

    Step 4: The remainder is 61 - 56 = 5. Carry the 5 over to the next digit, making 55.

  5. 5

    Step 5: How many 7s in 55? 7 (since 7 x 7 = 49). Write 7 above the 5.

  6. 6

    Step 6: The final remainder is 55 - 49 = 6.

  7. 7

    Final Answer: 87 remainder 6 (or 87 r 6).

Recap

  • In long multiplication, remember the placeholder zero when multiplying by the tens digit.
  • The 'bus stop' method for division works from left to right.
  • A remainder is the whole number left over at the end of a division.
  • Multiplying by 10, 100, or 1000 moves the digits to the left.
  • Dividing by 10, 100, or 1000 moves the digits to the right.

Quick check

  1. What is 120 x 40?1 mark
  2. Calculate 98 ÷ 5 and state the remainder.2 marks

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. In the number 5,284,710, what is the difference between the value of the digit 8 and the value of the digit 1?2 marks
  2. List all the prime numbers between 20 and 35.2 marks
  3. A sequence starts at 3 and follows the rule 'multiply by 2 and add 1'. What are the first four terms of the sequence?3 marks
  4. Calculate 47.8 + 19.25.2 marks
  5. A library has 1,250 books. A school buys 388 of them. How many books does the library have left?2 marks
  6. A factory produces 275 cars every day. How many cars does it produce in 14 days?3 marks
  7. A school of 368 students is going on a trip. Each bus can hold a maximum of 45 students. How many buses are needed to take all the students?3 marks
  8. Arrange these values in order from smallest to largest: -5, 1.2, -5.5, 0, 1.02, -0.53 marks
  9. Nita is thinking of a number. It is a common multiple of 6 and 8. It is also a factor of 120. The number is greater than 40. What is the number?4 marks
  10. A shop sells apples for £0.35 each and bananas for £0.28 each. Mateo buys 4 apples and 3 bananas. How much change does he get from a £5 note?4 marks

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