Number and Place Value

Number and Place Value

• Place value
• Comparing large numbers
• Rounding numbers
• Negative numbers
• Factors and multiples
• Highest common factor (HCF) and lowest common factor (LCM)
• Prime numbers
• Square and cube numbers

Place value

Place Value

Place Value

Place value is a way of understanding how much each digit in a number is worth, based on where it is in the number.

Each place in a number has a specific role.  We use a decimal system, which means that the value of each number increases 10 times for each place:

Place Value

Place value is a way of understanding how much each digit in a number is worth, based on where it is in the number.

Each place in a number has a specific role.  We use a decimal system, which means that the value of each number increases 10 times for each place:

Place Value

Place value is a way of understanding how much each digit in a number is worth, based on where it is in the number.

Each place in a number has a specific role.  We use a decimal system, which means that the value of each number increases 10 times for each place:

Place Value

Place value is a way of understanding how much each digit in a number is worth, based on where it is in the number.

Each place in a number has a specific role.  We use a decimal system, which means that the value of each number increases 10 times for each place:

Place Value

Place value is a way of understanding how much each digit in a number is worth, based on where it is in the number.

Each place in a number has a specific role.  We use a decimal system, which means that the value of each number increases 10 times for each place:

Place Value

Place value is a way of understanding how much each digit in a number is worth, based on where it is in the number.

Each place in a number has a specific role.  We use a decimal system, which means that the value of each number increases 10 times for each place:

This number is just a simple way of writing

Place Value

We call the maximum value of each number column a "base value".  Because we use a decimal system, our number system is a base-10 system.

There are number systems that are not base-10:

Computers are built on a binary system (base-2)

Place Value

We call the maximum value of each number column a "base value".  Because we use a decimal system, our number system is a base-10 system.

There are number systems that are not base-10:

Computers are built on a binary system (base-2)

Place Value

We call the maximum value of each number column a "base value".  Because we use a decimal system, our number system is a base-10 system.

There are number systems that are not base-10:

Computers are built on a binary system (base-2)

Place Value

We call the maximum value of each number column a "base value".  Because we use a decimal system, our number system is a base-10 system.

There are number systems that are not base-10:

Computers are built on a binary system (base-2)

Place Value

We call the maximum value of each number column a "base value".  Because we use a decimal system, our number system is a base-10 system.

There are number systems that are not base-10:

Computers are built on a binary system (base-2)

Place Value

We call the maximum value of each number column a "base value".  Because we use a decimal system, our number system is a base-10 system.

There are number systems that are not base-10:

Computers are built on a binary system (base-2)

Place Value

We call the maximum value of each number column a "base value".  Because we use a decimal system, our number system is a base-10 system.

There are number systems that are not base-10:

Computers are built on a binary system (base-2)

Place Value

We call the maximum value of each number column a "base value".  Because we use a decimal system, our number system is a base-10 system.

There are number systems that are not base-10:

Computers are built on a binary system (base-2)

Place Value

We call the maximum value of each number column a "base value".  Because we use a decimal system, our number system is a base-10 system.

There are number systems that are not base-10:

Computers are built on a binary system (base-2)

Place Value

We call the maximum value of each number column a "base value".  Because we use a decimal system, our number system is a base-10 system.

There are number systems that are not base-10:

Computers are built on a binary system (base-2)

This row represents the 1s.  There can be a one or there can be zero ones.

Place Value

We call the maximum value of each number column a "base value".  Because we use a decimal system, our number system is a base-10 system.

There are number systems that are not base-10:

Computers are built on a binary system (base-2)

Because this is a binary system, the this row represents not the 10s but the 2s

Place Value

We call the maximum value of each number column a "base value".  Because we use a decimal system, our number system is a base-10 system.

There are number systems that are not base-10:

Computers are built on a binary system (base-2)

And this row represents the 2 x 2, so we can say it represents the 4s

Place Value

We call the maximum value of each number column a "base value".  Because we use a decimal system, our number system is a base-10 system.

There are number systems that are not base-10:

Computers are built on a binary system (base-2)

Finally, this number represents the 2 x 2 x 2 = 8s.

Place Value

We call the maximum value of each number column a "base value".  Because we use a decimal system, our number system is a base-10 system.

There are number systems that are not base-10:

Computers are built on a binary system (base-2)

This number is made up of:

Place Value

We call the maximum value of each number column a "base value".  Because we use a decimal system, our number system is a base-10 system.

Computers are built on a binary system (base-2)

Place Value

And the Roman's didn't use a real base system at all.  In Roman numerals the row doesn't have a specific meaning.  Instead each number has a symbol.

2026 is MMXXVI in Roman numerals

Place Value

And the Roman's didn't use a real base system at all.  In Roman numerals the row doesn't have a specific meaning.  Instead each number has a symbol.

2026 is MMXXVI in Roman numerals

The symbol M = 1000.  There's two of them, so 2 x 1000 = 2000

The symbol X means 10.  Again, there are two of them, so 2 x 10 = 20

V means 5.  There's one of these, so 1 x 5 = 5

I means 1.  There's one I, so we have 1 x 1 = 1

2000

20

5

1

Place Value

And the Roman's didn't use a real base system at all.  In Roman numerals the row doesn't have a specific meaning.  Instead each number has a symbol.

2026 is MMXXVI in Roman numerals

The symbol M = 1000.  There's two of them, so 2 x 1000 = 2000

The symbol X means 10.  Again, there are two of them, so 2 x 10 = 20

V means 5.  There's one of these, so 1 x 5 = 5

I means 1.  There's one I, so we have 1 x 1 = 1

2000

20

5

1

2026

Place Value

In our own decimal system, however, the position of a number determines it's value.  Let's analyse a larger number:

4  2  0  5  3  1  7

Place Value

In our own decimal system, however, the position of a number determines it's value.  Let's analyse a larger number:

4  2  0  5  3  1  7

7 is in the ones place, so it's worth 7

4

2

0

5

3

1

7

Place Value

In our own decimal system, however, the position of a number determines it's value.  Let's analyse a larger number:

4  2  0  5  3  1  7

Place Value

Understanding place value helps you with everything in maths - reading, writing, rounding and even multiplying big numbers!

Practice: For each digit in the number 8,506, write its value.

8,506

Ordering and Comparing Numbers

Ordering numbers means arranging numbers either from smallest to biggest (ascending order) or biggest to smallest (descending order).

Ordering and Comparing Numbers

Ordering numbers means arranging numbers either from smallest to biggest (ascending order) or biggest to smallest (descending order).

Let's organise these numbers in descending order

Ordering and Comparing Numbers

Let's organise these numbers in descending order

Ordering and Comparing Numbers

Let's start by sorting by the left-most digits (the largest value) first.

Ordering and Comparing Numbers

Now, let's sort by the next digit too.  As you can see they are already in the right order: 20, 20, 18, 12 and 12

Ordering and Comparing Numbers

Now, include the three first digits from the left. The order is still correct: 202, 201, 184, 125 and 125.

Ordering and Comparing Numbers

We can now compare by the last digit.  As you see, the two last two numbers must switch place.

Ordering and Comparing Numbers

That's it.  We have sorted the numbers in descending order.

Ordering and Comparing Numbers

Comparing numbers helps us find out if one number is equal to, smaller than or bigger than another number.

Ordering and Comparing Numbers

Practice: Compare these numbers using <, >, =

342    456

1230    1230

9876    7654

4321678    4329876

Ordering and Comparing Numbers

Practice: Compare these numbers using <, >, =

342    456

1230    1230

9876    7654

4321678    4329876

Again we use the method of scanning left to right

Ordering and Comparing Numbers

Practice: Compare these numbers using <, >, =

342    456

1230    1230

9876    7654

4321678    4329876

Again we use the method of scanning left to right

Ordering and Comparing Numbers

Practice: Compare these numbers using <, >, =

342    456

1230    1230

9876    7654

4321678    4329876

Again we use the method of scanning left to right

Ordering and Comparing Numbers

Practice: Compare these numbers using <, >, =

342    456

1230    1230

9876    7654

4321678    4329876

Again we use the method of scanning left to right

Ordering and Comparing Numbers

Practice: Compare these numbers using <, >, =

342    456

1230    1230

9876    7654

4321678    4329876

From the fourth digit, we can conclude that the number on the right is the largest.

Rounding Numbers

Rounding makes a number simpler while keeping its value close to the original.

It can help us estimate quantities quickly and it can be a good way of double checking answers.

To round a decimal to a given place:

Rounding Numbers

Example: Round 763 to the nearest ten.

Rounding Numbers

Example: Round 763 to the nearest ten.

As we are rounding to tens, we look first at the units (to the right of the tens)

The value is 3, which is less than 5

We keep the value 6 and we change the following values to 0

Rounding Numbers

Example: Round 763 to the nearest ten.

As we are rounding to tens, we look first at the units (to the right of the tens)

The value is 3, which is less than 5

We keep the value 6 and we change the following values to 0

Rounding Decimals

Let's try one more, step-by-step

Rounding Decimals

Round to nearest thousandth

Rounding Decimals

First, identify the thousands place

Rounding Decimals

Look at the value to the right.

It's more than 5.

Rounding Decimals

We change the thousands up one from 7 to 8

Rounding Decimals

We set all the following values to 0

Rounding Decimals

Negative Numbers

Negative Numbers

Negative numbers are numbers that are less than 0. They are written with a minus sign in front.

Examples: -3, -7, -15.

Where do we use negative numbers?

• Temperature: -5°C is colder than 0°C

• Money: If you owe £10, it's written as -£10

• Depth: Below sea level is negative, around -50 metres

Negative Numbers

How to compare negative numbers:

0

1

2

3

4

-4

-3

-2

-1

Negative Numbers

Practice: Find the largest number from the following list.

-1342     -1508     -1511     -1430

Negative Numbers

Adding negative numbers

Because negative numbers represent less than 0, when you add them to a positive number, they will make the total lower, not higher.

Negative Numbers

Adding negative numbers

Because negative numbers represent less than 0, when you add them to a positive number, they will make the total lower, not higher.

Negative Numbers

Adding negative numbers

Because negative numbers represent less than 0, when you add them to a positive number, they will make the total lower, not higher.

Negative Numbers

Adding negative numbers

You can imagine this on a number line where positive numbers increase to the right and negative numbers to the left:

Negative Numbers

Adding negative numbers

The -3 represents three jumps to the left.

Negative Numbers

Adding negative numbers

The +5 represents five jumps to the right.

Factors and Multiples

Factors and Multiples

A factor of a number is a number that divides into it exactly, without leaving a remainder.

Factors and Multiples

12

1 x 12

2 x 6

3 x 4

Example:

The factors of 12 are 1, 2, 3, 4, 6 and 12.

1

12

2

6

3

4

A factor of a number is a number that divides into it exactly, without leaving a remainder.

Factors and Multiples

A factor of a number is a number that divides into it exactly, without leaving a remainder.

12

1 x 12

2 x 6

3 x 4

Example:

The factors of 12 are 1, 2, 3, 4, 6 and 12.

1

12

2

6

3

4

The highest common factor (HCF) is the largest factor that two or more numbers have in common.

Example: Find the HCF of 12 and 16.

                              Factors of 12: 1, 2, 3, 4, 6, 12.

                              Factors of 18: 1, 2, 3, 6, 9, 18.

                              Which factors do 12 and 18 have in common?

                              1, 2, 3, 6 are common factors.

Factors and Multiples

The largest common factor is 6

Factors and Multiples

HCF are important because we can use them to simplify and compare fractions.

Factors and Multiples

HCF are important because we can use them to simplify and compare fractions.

The HCF for 4 and 20 is 4

Factors and Multiples

HCF are important because we can use them to simplify and compare fractions.

The HCF for 4 and 20 is 4

Factors and Multiples

HCF can also help you dividing other things into equal parts.

Factors and Multiples

HCF can also help you dividing other things into equal parts.

Let’s say you have 48 crayons and 60 coloured pencils. You want to put them into identical art kits for friends. You want each friend to get the same number of crayons and the same number of coloured pencils, with nothing left over.

Factors and Multiples

You can often use your times table knowledge to quickly identify common factors.

We can immediately see that 6 is the highest factor shared by both numbers.  But with larger numbers this may not be quite as easy.  Let's look at something more complicated.

What is the Highest Common Factor of these numbers?

First let's write each number as a product of its prime factors.

What is the Highest Common Factor of these numbers?

First let's write each number as a product of its prime factors.

What is the Highest Common Factor of these numbers?

First let's write each number as a product of its prime factors.

Factors and Multiples

A multiple of a number is the result of multiplying it by another whole number.

Example: Some multiples of 5 are 5, 10, 15, 20, 25, 30, ...

Notice that multiples are what you see in a number's times table!

Factors and Multiples

HCF and LCM

The lowest common multiple (LCM) is the smallest multiple that two or more numbers share.

Example: Find the LCM of 4 and 6.

                                       Multiples of 4: 4, 8, 12, 16, 20, 24, ...

                                       Multiples of 6: 6, 12, 18, 24, 30, ...

                                       The common multiples are 12, 24, ...

                                       The lowest common multiple is 12.

HCF and LCM

There are two strategies we can use to find LCM:

HCF and LCM

2. Find LCM by by Prime Factorization.

HCF and LCM

2. Find LCM by by Prime Factorization.

You can also find LCM by listing all the prime factors of the numbers.

Now pick each number as it appears the most.

The longest entry for 3 is [3 x 3 x 3]

HCF and LCM

2. Find LCM by by Prime Factorization.

Now pick each number as it appears the most.

The longest entry for 3 is [3 x 3 x 3]

The longest entry for 2 is [2 x 2]

HCF and LCM

LCM are important because we can use them to compare, add and subtract fractions.

HCF and LCM

LCM are important because we can use them to compare, add and subtract fractions.

HCF and LCM

LCM are important because we can use them to compare, add and subtract fractions.

The LCM for 9 and 4 is 36, so we multiply 9 x 4 = 36

The LCM for 9 and 4 is 36, so we multiply 12 x 3 = 36

HCF and LCM

LCM are important because we can use them to compare, add and subtract fractions.

HCF and LCM

LCM are important because we can use them to compare, add and subtract fractions.

Now that both fractions have the same denominator, we can add the numerators.

You can learn more about this in our tutorial dedicated to fractions.  For now it's just important to remember that finding the LCM is essential for these types of calculations.

HCF and LCM

LCM are also useful for word problems and real-life situations.

Let's say you're looking at two routes:

"

Bus A & B both arrives at 10am.  After that, Bus A arrives every 15 minutes and Bus B arrives every 22 minutes.

Anna arrives on Bus A and John arrives on bus B.  They missed the 10am busses. What is the earlies they can meet up if they both want to arrive at exactly the same time?

Instead we can look for the lowest common multiple of 15 and 22:

Instead we can look for the lowest common multiple of 15 and 22:

So, the two bus routes will align every 330 minutes. Thus the time will be

Prime Numbers

What are prime numbers?

A prime number is a number that only has 2 factors. This means that is cannot be divided evenly by any other number.

Examples:

Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19...

Non-prime numbers: Non-prime numbers: 4, 6, 8, 9, 10...

Interesting facts! The number 1 is not prime because it has only one factor. The number 2 is the only even prime number.

Prime Numbers

How to check if a number is prime: 

• Start dividing the number by smaller numbers (starting from 2).
• If it can be divided evenly by any number other than 1 and itself, it’s not prime.
• If no division works, it’s prime!

Prime Numbers

Practice questions:

Is 13 a prime number?

List all prime numbers between 1 and 20.

Which of these is not prime?

A) 7           B) 9          C) 11         D) 13

Square Numbers

A square number is the result of multiplying a number by itself.

Example: 4 x 4 = 16, so 16 is a square number.

Common square numbers:

1² = 1

2² = 4

3² = 9

4² = 16

5² = 25, and so on!

So, 1, 4, 9, 16, 25, ... are all square numbers.

Square Numbers

A square number is the result of multiplying a number by itself.

Square numbers can be visualised as a two dimensional square.

Cube Numbers

A cube number is the result of multiplying a number by itself a total of three times.

Example: 3 x 3 x 3 = 27, so 27 is a cube number.

Common cube numbers:

1³ = 1

2³ = 8

3³ = 27

4³ = 64, and so on!

So, 1, 8, 27, 64, 125 ... are all cube numbers.

Cube Numbers

A cube number is the result of multiplying a number by itself a total of three times.

Cube numbers can be visualised as a three dimensional cube.

Square and Cube Numbers

Practice:

What is 7²?

What is 5³?

Is 36 a square number?

List all the square numbers between 1 and 50.

What is the cube of 3?