Cambridge Lower Secondary CheckpointStage 6

Number (6Np)

Mathematics Stage 6 Chapter Notes

What this chapter covers

Number — Place value, ordering and rounding
ShareWhatsAppPost
Number (6Np) notes

Unable to load PDF

The notes viewer could not load. Please refresh the page.

Read online free. Download a watermarked copy with a free account.

Read the notes

The full Number (6Np) notes as text: skim, search, and jump between subtopics.

~13 min read

In this chapter

  1. 1Understanding Place Value
  2. 2Positive and Negative Numbers
  3. 3Factors, Multiples and Primes
  4. 4Rounding and Estimation
  5. 5Addition and Subtraction Methods
  6. 6Multiplication and Division Methods

1. Understanding Place Value

Place value is the value of each digit in a number. Our number system, the Hindu-Arabic system, is a base-10 system. This means the value of a digit is determined by its position, with each place being ten times larger than the place to its right. For example, in the number 345.67, the '4' is not just 4, it's 4 tens (40), and the '7' is not 7, it's 7 hundredths (0.07). Understanding this is crucial for ordering numbers and performing calculations.

Key term

Place Value: The numerical value that a digit has by virtue of its position in a number.

Examiner insight

Examiners often test place value by asking students to order lists of decimals with varying numbers of decimal places. Ensure you can confidently compare numbers like 2.3, 2.301, and 2.29.

Common pitfall

When comparing decimals, students often think 0.45 is larger than 0.5 because 45 is larger than 5. Always compare digit by digit from the decimal point outwards.

Fun fact

The concept of zero as a placeholder, which is essential for our place value system, was a revolutionary mathematical idea developed in ancient India.

Worked example 13 marks

Consider the number 6,734,502.89. Write down the value of the digit 7, the digit 5, and the digit 9.

  1. 1

    Identify the position of the digit 7. It is in the hundred thousands column. So, its value is 700,000.

  2. 2

    Identify the position of the digit 5. It is in the hundreds column. So, its value is 500.

  3. 3

    Identify the position of the digit 9. It is in the hundredths column (two places after the decimal point). So, its value is 9/100 or 0.09.

Worked example 22 marks

Use the symbols <, >, or = to compare the following pair of numbers: 45.06 and 45.6

  1. 1

    Start by comparing the digits from the left. The tens digit (4) and the units digit (5) are the same in both numbers.

  2. 2

    Next, compare the tenths digit (the first digit after the decimal point). In 45.06, the tenths digit is 0. In 45.6, the tenths digit is 6.

  3. 3

    Since 0 is less than 6, the number 45.06 is less than 45.6.

  4. 4

    The correct statement is 45.06 < 45.6.

Recap

  • Each place in a number is ten times the value of the place to its right.
  • The decimal point separates the whole number part from the fractional part.
  • To compare numbers, look at the digits from left to right in the highest place value column.
  • Zeros used as placeholders are important, for example in 50.4 versus 5.04.

Quick check

  1. What is the value of the digit 3 in the number 12.035?1 mark
  2. Put these numbers in order from smallest to largest: 2.7, 2.07, 2.77, 2.707.2 marks

2. Positive and Negative Numbers

Negative numbers are numbers less than zero. They are written with a minus sign (-) in front. A number line is a great way to visualise them. Zero is the central point. Positive numbers are to the right of zero, and their value increases as you move further right. Negative numbers are to the left of zero, and their value decreases as you move further left. This means -10 is smaller than -2.

Key term

Negative Number: A number that is less than zero, representing an opposite value or a deficit.

Examiner insight

Questions involving temperature, bank balances, or elevation above/below sea level are common contexts for negative numbers. Be ready to apply your knowledge to these real-world scenarios.

Common pitfall

A common mistake is thinking that because 10 is bigger than 5, -10 must be bigger than -5. Remember that on the number line, -10 is further to the left and is therefore smaller.

Worked example 13 marks

Place the following numbers on a number line and then write them in ascending order (smallest to largest): 3, -4, 0, -1, 5, -6

  1. 1

    Draw a number line from at least -6 to 5.

  2. 2

    Mark the position of each number on the line. -6 will be furthest to the left.

  3. 3

    Next will be -4, then -1, then 0, then 3, and finally 5 will be furthest to the right.

  4. 4

    Ascending order means reading the numbers from left to right on the number line.

  5. 5

    The correct order is: -6, -4, -1, 0, 3, 5.

Worked example 22 marks

The temperature in Moscow is -8°C. In London, it is 5°C. Which city is colder, and by how much?

  1. 1

    Compare the two temperatures, -8°C and 5°C. Negative numbers are colder (smaller) than positive numbers.

  2. 2

    Moscow is colder because -8 is less than 5.

  3. 3

    To find the difference, you can count on a number line from -8 up to 5. From -8 to 0 is 8 degrees. From 0 to 5 is 5 degrees.

  4. 4

    The total difference is 8 + 5 = 13 degrees. So, Moscow is 13°C colder than London.

Recap

  • Negative numbers are to the left of zero on a number line.
  • The further left a number is on the number line, the smaller its value.
  • A larger negative number (like -20) is smaller in value than a smaller negative number (like -2).
  • The difference between a positive and a negative number is found by adding their distances from zero.

Quick check

  1. Which is smaller, -15 or -12?1 mark
  2. What is the difference in value between 6 and -7?1 mark

3. Factors, Multiples and Primes

Factors are numbers that divide exactly into another number with no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. Multiples are the result of multiplying a number by an integer (a whole number). For example, multiples of 3 are 3, 6, 9, 12... A Prime Number is a whole number greater than 1 that has exactly two factors: 1 and itself. Examples include 2, 3, 5, 7, 11.

Key term

Prime Number: A whole number greater than 1 with only two factors: 1 and itself.

Examiner insight

Be prepared for questions that combine concepts, such as 'Find a prime number that is a factor of 30'. This requires you to know both definitions and apply them together.

Common pitfall

Students frequently mix up factors and multiples. Remember: Factors are Few (there's a limited number of them for any given integer), but Multiples are Many (they go on forever).

Worked example 13 marks

Find all the factors of 48.

  1. 1

    Start by finding pairs of numbers that multiply to make 48.

  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

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

Worked example 22 marks

Find the first three common multiples of 4 and 6.

  1. 1

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

  2. 2

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

  3. 3

    Look for the numbers that appear in both lists. These are the common multiples.

  4. 4

    The first common multiple is 12.

  5. 5

    The second common multiple is 24.

  6. 6

    The third common multiple is 36.

Recap

  • Factors divide into a number exactly.
  • Multiples are the 'times table' of a 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 7.1 mark
  2. Is 51 a prime number? Explain your answer.2 marks

4. Rounding and Estimation

Rounding makes numbers easier to work with. To round a number, you look at the digit to the right of the place you are rounding to. If this digit is 5 or more, you round up. If it's 4 or less, you round down. Estimation involves rounding numbers in a calculation before you work it out to get a rough answer. This is a brilliant way to check if your final calculated answer is sensible.

Key term

Estimation: The process of finding an approximate answer to a calculation by rounding the numbers first.

Examiner insight

Examiners look for you to show the rounded numbers you used for your estimation. Just writing down an estimated answer without showing your working (e.g., '60 x 8') may not get you full marks.

Common pitfall

When rounding a number like 9.96 to one decimal place, many students incorrectly write 9.9. The '6' rounds the '9' up to 10, which means you must carry over, resulting in the correct answer of 10.0.

Worked example 13 marks

Round the number 47,839 to the nearesta) ten,b) hundred, andc) thousand.

  1. 1

    a) To round to the nearest ten, look at the units digit (9). It's 5 or more, so round the tens digit up. 47,839 becomes 47,840.

  2. 2

    b) To round to the nearest hundred, look at the tens digit (3). It's 4 or less, so the hundreds digit stays the same. 47,839 becomes 47,800.

  3. 3

    c) To round to the nearest thousand, look at the hundreds digit (8). It's 5 or more, so round the thousands digit up. 47,839 becomes 48,000.

Worked example 22 marks

Estimate the answer to 58.7 x 8.21. Show your rounding.

  1. 1

    To estimate, we round each number to one significant figure (its most important digit).

  2. 2

    Round 58.7 to the nearest ten. It becomes 60.

  3. 3

    Round 8.21 to the nearest whole number. It becomes 8.

  4. 4

    Now perform the estimated calculation: 60 x 8 = 480.

  5. 5

    So, the estimated answer is 480.

Recap

  • To round, look at the digit to the right of the target place value.
  • If the digit is 5 or more, round up.
  • If the digit is 4 or less, round down (keep the target digit the same).
  • Estimation is rounding before calculating to get a rough answer.
  • Always state the numbers you have rounded to in an estimation question.

Quick check

  1. Round 3.14159 to two decimal places.1 mark
  2. Estimate the answer to 402 ÷ 19.8.2 marks

5. Addition and Subtraction Methods

For adding and subtracting numbers, especially large numbers or decimals, the column method is the most reliable. The key is to align the numbers correctly according to their place value. For decimal numbers, this means lining up the decimal points directly underneath each other. You can add placeholder zeros to make the numbers the same length after the decimal point, which helps with alignment.

Key term

Inverse Operation: An operation that reverses the effect of another operation; addition is the inverse of subtraction, and subtraction is the inverse of addition.

Examiner insight

In multi-step problems, show your working for each calculation clearly. Even if you make a mistake in one part, you can still earn marks for correctly setting up the problem and for subsequent correct calculations based on your initial error (known as 'error carried forward').

Common pitfall

The most frequent error in decimal addition and subtraction is misaligning the decimal points, for example placing 45.67 under 378.5 as if they were whole numbers.

Worked example 12 marks

Calculate 378.5 + 45.67

  1. 1

    Write the numbers in columns, aligning the decimal points.

  2. 2

    378.50

  3. 3

    + 45.67

  4. 4

    ---------

  5. 5

    Add a placeholder zero to 378.5 to make it 378.50. This helps with alignment.

  6. 6

    Starting from the right (hundredths column): 0 + 7 = 7.

  7. 7

    Tenths column: 5 + 6 = 11. Write down 1 and carry over 1 to the units column.

  8. 8

    Units column: 8 + 5 + 1 (carried) = 14. Write down 4 and carry over 1 to the tens column.

  9. 9

    Tens column: 7 + 4 + 1 (carried) = 12. Write down 2 and carry over 1 to the hundreds column.

  10. 10

    Hundreds column: 3 + 1 (carried) = 4.

  11. 11

    The final answer is 424.17.

Worked example 22 marks

A shopkeeper has a 10kg bag of flour. He sells 2.75kg. How much flour is left?

  1. 1

    This is a subtraction problem: 10 - 2.75.

  2. 2

    Write the numbers in columns, aligning the decimal points. Remember that 10 can be written as 10.00.

  3. 3

    10.00

  4. 4
    • 2.75
  5. 5

    ---------

  6. 6

    Starting from the right (hundredths column): you can't do 0 - 5. You need to borrow.

  7. 7

    Borrow from the units (0), which needs to borrow from the tens (1). The 1 becomes 0, the units become 10, then 9, and the tenths become 10, then 9, and the hundredths become 10.

  8. 8

    Hundredths: 10 - 5 = 5.

  9. 9

    Tenths: 9 - 7 = 2.

  10. 10

    Units: 9 - 2 = 7.

  11. 11

    The final answer is 7.25kg.

Recap

  • Always align numbers by their place value when using column methods.
  • For decimals, this means lining up the decimal points.
  • Use placeholder zeros to avoid errors, especially in subtraction.
  • Remember to 'carry' in addition and 'borrow' (or exchange) in subtraction.
  • You can check a subtraction answer by using the inverse operation: addition.

Quick check

  1. Calculate 5.8 + 12.341 mark
  2. Calculate 20 - 5.672 marks

6. Multiplication and Division Methods

For multiplication, the grid method or long multiplication are common formal methods. For division, short division (the 'bus stop' method) is used when dividing by a single-digit number, while long division is used for dividing by two-digit numbers or more. Knowing your times tables is essential for both operations.

Key term

Product: The result obtained by multiplying two or more numbers together.

Examiner insight

Show your working clearly, especially for long multiplication and division. A clear layout helps you avoid errors and allows the examiner to award partial marks if you make a small slip.

Common pitfall

In short division, a common error is forgetting to place a zero in the answer when a number doesn't divide. For example, in 612 ÷ 6, the answer is 102, not 12. After dividing 6 by 6, you must check how many 6s go into 1 (which is 0) before carrying the 1 over.

Worked example 13 marks

Calculate 256 x 24

  1. 1

    Use the long multiplication method.

  2. 2

    First, multiply 256 by 4 (the units digit of 24). 256 x 4 = 1024.

  3. 3

    Next, multiply 256 by 20 (the tens digit of 24). A quick way is to put a zero in the units column and then multiply 256 by 2. 256 x 2 = 512, so 256 x 20 = 5120.

  4. 4

    256

  5. 5

    x 24

  6. 6

    ------

  7. 7

    1024 (256 x 4)

  8. 8

    5120 (256 x 20)

  9. 9

    ------

  10. 10

    Add the two results together: 1024 + 5120 = 6144.

  11. 11

    The final answer is 6144.

Worked example 22 marks

Calculate 471 ÷ 3

  1. 1

    Use the short division ('bus stop') method.

  2. 2

    Write the calculation as 3 | 471.

  3. 3

    How many 3s go into 4? 1, with a remainder of 1. Write 1 above the 4 and carry the remainder 1 next to the 7, making it 17.

  4. 4

    How many 3s go into 17? 5 (since 3x5=15), with a remainder of 2. Write 5 above the 7 and carry the remainder 2 next to the 1, making it 21.

  5. 5

    How many 3s go into 21? 7 exactly. Write 7 above the 1.

  6. 6

    The digits above the line give the answer. The final answer is 157.

Recap

  • Long multiplication involves multiplying by the units, then the tens, then the hundreds, and adding the results.
  • Remember to add a zero as a placeholder when multiplying by the tens digit in long multiplication.
  • Short division is a quick method for dividing by a single-digit number.
  • In division, remainders are carried over to the next digit to the right.
  • Knowing multiplication facts makes division much easier.

Quick check

  1. Calculate 15 x 100.1 mark
  2. Calculate 84 ÷ 4.1 mark
  3. What is the remainder when 55 is divided by 8?1 mark

End-of-chapter exercise

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

  1. A number is made from the digits 3, 4, 5, 6, 7, 8. The number is a multiple of 10. The digit in the ten thousands place is a prime number. The digit in the hundredths place is worth 0.04. The digit in the hundreds place is double the digit in the tenths place. What is the number?4 marks
  2. The temperature at midday is 7°C. By midnight, the temperature has dropped by 11°C. What is the temperature at midnight?2 marks
  3. Find the lowest common multiple (LCM) of 8 and 12.2 marks
  4. A baker makes 240 cakes. He packs them into boxes of 15. How many full boxes can he pack?3 marks
  5. A plank of wood is 3.5 metres long. A builder cuts off a piece that is 1.82 metres long. What is the length of the remaining piece of wood?2 marks
  6. Estimate the answer to (198 x 51.3) / 9.9. Show the rounded values you use for your estimation.3 marks
  7. Tickets for a concert cost £38.50 each. A school buys 25 tickets. What is the total cost?3 marks
  8. Identify all the prime numbers between 20 and 30.2 marks
  9. I am thinking of a number. It is a multiple of 7. It is also a factor of 42. It is not a prime number. What is my number?3 marks
  10. Put these numbers in order from smallest to largest: -5, 0.5, -5.5, 5.05, -0.52 marks

Go deeper

Practise and revise with member-only material for this chapter.

Workbook

Exam-style practice questions with full worked solutions.

Unlock with a plan

Chapter at a Glance

A single-page summary of every key idea for fast revision.

Unlock with a plan

Free notes are just the start.

Unlock every Workbook and Chapter at a Glance, and generate your own worksheets and predicted papers.

Explore plans

Related chapters

Number (6Ni)Number (6Nf)Algebra (6Ae)Algebra (6As)Geometry and Measure (6Gg)Geometry and Measure (6Gp)