Numeracy

Addition and subtraction

​Integers

Make sure you use your knowledge of place value.

You can use partitioning or column method

Column addition is a written method we can use to add numbers.

Example 1:
5728 + 345

Step 1
Rewrite the question in columns (one number on top of the other): we need to line up the ones with the ones, the tens with the tens, the hundreds with the hundreds and the thousands with the thousands. It doesn't matter that 345 has no thousands, we can either leave a gap or write a zero.

​

Starting with the smallest column (on the right) we add down the columns.
The smallest column is the ones. 8 + 5 = 13
We write the 3 in the ones and carry the 1 (which is worth ten) over to the tens column..

​

​Step 2

As we work from right to left, the next column is the tens.
2 + 4 + 1 = 7

​

Step 3

The next column is the hundreds.
7 + 3 = 10
We write the zero in the hundreds column and carry the 1 over to the next column, the thousands column (The 1 is worth 1000 because we were adding hundreds, 700 + 300 = 1000)

​

​Step 4

The next column is the thousands. 5 + 1 = 6

5728 + 345 = 6073

​

Step 5

Multiple Choice

557 + 61

• 617

• 618

• 558

• 496

Draw

5273 + 261

5534

Draw

3459 + 3926

​7385

We can also subtract numbers using the column method.

Example 1:
536 - 287

Step 1
Rewrite the question in columns (one number on top of the other): we need to line up the ones with the ones, the tens with the tens and the hundreds with the hundreds. The first number has to be on top, and the number we are taking away must be on the bottom.

​

Starting with the smallest digits (the column on the right) we subtract the bottom number from the top number.
The smallest column is the ones, we have 6 - 7
6 - 7 would give us a negative answer, to avoid that we can take one of the tens (from the top number) and add it to the ones column (where it is worth 10). Instead of having 500 + 30 + 6 we now have 500 + 20 + 16

​

Step 2

We now have 16 - 7 in the ones column. 16 - 7 = 9

​

Step 3

The next column is the tens. We now have 2 - 8 which would be negative. Instead we can take one off of the hundreds and add it to the tens.
We can change 500 + 20 + 16 to 400 + 120 + 16

​

Step 4

We now have 12 - 8 in the tens column. 12 - 8 = 4

​

​Step 5

We now have 4 - 2 in the hundreds column. 4 - 2 = 2

536 + 287 = 249

​

Step 6

Multiple Choice

839 − 152

• 991

• 981

• 587

• 687

Draw

876 - 190

​686

Draw

1821 - 234

​1587

Addition and subtraction

Decimals

Use your place value knowledge and always ALWAYS line up the decimal points! IT MEANS PUT THEM OVER EACH OTHER.

24.52 + 19.7

It doesn't matter if we have decimals we are still following the same steps.

Step 1
Rewrite the question in columns (one number on top of the other). This time we are lining up the tens with tens, the ones with the ones, the decimal point with the decimal point, the tenths with the tenths and the hundredths with the hundredths. It doesn't matter that 19.7 has no hundredths, we can either leave a gap or write a zero.

​

Starting on the right, we add down the columns.
The smallest column is the hundredths.
2 + 0 = 2

​

STEP 2

As we work from right to left, the next column is the tenths.
5 + 7 = 12
We write the 2 in the tenths column and we carry the 1 over to the ones column. (The 1 is worth 1 because we actually added 0.5 and 0.7 which is 1.2)
The decimal point will stay in the same place.

​

STEP 3

The next column is the ones. 4 + 9 + 1 = 14
We write the 4 in the ones column and carry the 1 over to the next column, the tens column..

​

STEP 4

The next column is the tens. 2 + 1 + 1 = 4

24.52 + 19.7 = 44.22

​

STEP 5

Multiple Choice

8.7 + 2.35

• 10.05

• 11.5

• 11.05

• 10.5

Draw

2.08 + 5.7

​7.78

Draw

61.7 + 4.17

​65.87

54.7 - 28.36

Step 1
Rewrite the question in columns (the first number on top of the second number). This time we are lining up the tens with tens, the ones with the ones, the decimal point with the decimal point, the tenths with the tenths and the hundredths with the hundredths. 54.7 has no hundredths so we can write a zero there..

​

We start with the column on the right (the hundredths) and subtract the bottom number from the top number. In the hundredths column we have 0 - 6, we don't want to do this so we take one from the tenths column and add it to the hundredths column.

​

STEP 2

We now have 10 - 6 in the hundredths column. 10 - 6 = 4

​

STEP 3

The next column is the tenths. 6 - 3 = 3

​

STEP 4

The next column is the ones. We do not want to do 4 - 8. Instead we take one from the tens column and add it to the ones column.

​

STEP 5

We now have 14 - 8 in the ones column. 14 - 8 = 6

​

STEP 6

The next column is the tens. 4 - 2 = 2

​

54.7 + 28.36 = 26.34

STEP 7

Multiple Choice

15.23 - 6.25

• 8.89

• 8.98

• 9.89

• 9.98

Draw

60.52 - 34.72

​25.8

Draw

852.1 - 17.43

​834.67

To square a number we multiply the number by itself.
3 squared is 9 because 3 × 3 = 9
4 squared is 16 because 4 × 4 = 16

We can write squared using a small (superscript) 2.
5² means 5 squared
10² means 10 squared

​

SQUARES

Multiple Choice

Work out the value of 9²

• 18

• 81

​To work out the value of 9 squared we need to calculate 9 × 9
9 × 9 = 81
9² = 81

​To work out the value of -8 squared we need to calculate -8 × -8
-8 × -8 = 64 (A negative times a negative is a positive)
(-8)² = 64
If you are using a calculator you must put the -8 in brackets, otherwise the calculator will get the answer wrong!

Square rooting is the opposite of sqauring a number.
9² = 81, so the square root of 81 is 9

Square root 100 can be written as √100
√81 means square root 81

​

Square Roots

Multiple Select

Work out the value of √144

• 12

• -12

• 12 AND -12

​We are looking for the number that multiplies by itself to make 144
12 × 12 = 144 AND -12 x -12 = 144
√144 = 12 and -12

Multiple Select

Work out the value of √100

• 10 and -10

• 10

• -10

​We are looking for the number that multiplies by itself to make 100
10 × 10 = 100 and -10 x -10 =100
√100 = 10 and -10

Multiple Select

Work out the values of: √9

• 3

• -3

• 3 and -3

​There is a correct order to do calculations in, this is so everyone does calculations in the same way. When we do calculations we need to do them in the correct order.

The order of operations is:
1) Brackets
2) Indices
3) Dividing and Multiplying
4) Adding and Subtracting

Dividing and Multlipying are in the same line - they can be done in any order.
Adding and Subtracting are also on the same line - they can be done in any order but it is usually easier to do them from left to right.

Multiple Choice

3 + 5 × 4 =?

• 23

• 32

​We have no brackets and no indices.
We will start with the multiplication
5 × 4 = 20
So we now have 3 + 20
3 + 20 = 23

3 + 5 × 4 = 23

Draw

  Add brackets to make these expressions equal to each target

Rounding

To the nearest whole number

If the digit is 5 or above we will round up.

If it is 4 or below we will not.

Multiple Choice

Round 7.28 to the nearest whole number

• 7

• 8

​Rounding to the nearest whole number is the same as rounding to the nearest one.
7.28 is in between 7 and 8. We need to work out if it is closer to 7 or 8.
We can draw a line after the ones and look at the next digit, the tenths. If the digit is 5 or above we will round up (to 8), if it is 4 or below we will not (it will stay as 7).
We have a 2 in the tenths, so we will not round up. The answer is 7.

Multiple Choice

Round 12.74 to the nearest whole number.

• 12

• 13

Multiple Choice

Round 386 to the nearest ten.

• 390

• 380

​386 is in between 380 and 390. We need to work out if it is closer to 380 or 390.
We can draw a line after the tens and look at the next digit, the ones. If the digit is 5 or above we will round up (to 390), if it is 4 or below we will not (it will stay as 380).
We have a 6 in the ones, so we will round up. The answer is 390.

Multiple Choice

Round 1593 to the nearest ten

• 1590

• 1600

Multiple Choice

Round 1259 to the nearest hundred.

• 1300

• 1200

​1259 is in between 1200 and 1300. We need to work out if it is closer to 1200 or 1300.
We can draw a line after the hundreds and look at the next digit, the tens. If the digit is 5 or above we will round up (to 1300), if it is 4 or below we will not (it will stay as 1200).
We have a 5 in the tens, so we will round up. The answer is 1300.

Multiple Choice

Round 2.68 to one decimal place.

• 2.7

• 2.6

​When we round to one decimal place, we want just one digit after the decimal point.
2.68 is between 2.6 and 2.7. We need to work out if it closer to 2.6 or 2.7
We can draw a line after one decimal place and look at the next digit, the hundredths. If the digit is 5 or above we will round up (to 2.7), if it is 4 or below we will not (it will stay as 2.6).
We have a 8 in the hundredths, so we will round up. The answer is 2.7

Multiple Choice

Round 15.374 to two decimal places.

• 15.37

• 15.38

• 15.4

• 15.3

​When we round to two decimal places, we want two digits after the decimal point.
15.374 is between 15.37 and 15.38. We need to work out if it closer to 15.37 and 15.38
We can draw a line after two decimal places and look at the next digit, the thousandths. If the digit is 5 or above we will round up (to 15.38), if it is 4 or below we will not (it will stay as 15.37).
We have a 4 in the thousandths, so we will not round up. The answer is 15.37

factors and multiples

​Multiples

The multiples of a number are all the numbers in its times table.
The multiples of 5 are all the numbers in the 5 times table.

Draw

Write down all the factors of 15

​1, 3, 5, 15

Draw

Write down all the factors of 39.

​1, 3, 13, 39

​Factors

The factors of a number are the numbers that multiply together to make it.
The factors of 10 are the numbers that you can multiply together to make 10. 2 × 5 = 10, so 2 and 5 are both factors of 10 (so are 1 and 10).

A prime number is a number with exactly 2 factors.
5 is a prime number because its only factors are 1 and 5.
6 is not a prime number because its factors are 1, 2, 3 and 6.

Draw

Write down the first 5 multiples of 7

​7, 14, 21, 28, 35

HCF (Highest Common Factor) and LCM (lowest Common Multiple)

​Highest common factor

A common factor is a factor that is shared by two or more numbers. For example, a common factor of 8 and 10 is 2, as 2 is a factor of 8, and 2 is also a factor of 10. The highest common factor (HCF) is found by finding all common factors of two numbers and selecting the largest one.

For example, 8 and 12 have common factors of 1, 2 and 4. The highest common factor is 4.

Multiple Choice

Find the highest common factor of 9 and 21.

• 3

• 7

• 9

• 21

​Start by listing the factors of 9 and 21:

• factors of 9: 1, 3, 9

• factors of 21: 1, 3, 7, 21

The common factors of 9 and 21 are 1 and 3, so the highest common factor of 9 and 21 is 3.

The LCM

A common multiple is a number that is a shared multiple of two or more numbers. For example, 24 is a common multiple of 8 and 12, as 24 is in the 8 times tables (8×3=24) and 24 is in the 12 times tables (12×2=24).

The lowest common multiple (LCM) is found by listing the multiples of each number and circling any common multiples. The lowest one is the lowest common multiple.

Multiple Choice

Find the lowest common multiple of 5 and 6.

• 5

• 12

• 18

• 30

​List the multiples of 5 and 6:

• multiples of 5: 5, 10, 15, 20, 25, 30, 35...

• multiples of 6: 6, 12, 18, 24, 30...

The lowest common multiple of 5 and 6 is 30, as it is the first multiple found in both lists.

Multiple Choice

Find the Higest Common Factor: 8 and 16

• 1

• 4

• 8

• 2

Multiple Choice

Find the Lowest Common Multiple: 6 and 15

• 3

• 18

• 30

• 36

Multiple Choice

Find LCM; 10 and 15

• 10

• 15

• 20

• 30

FRACTIONS

​MIXED NUMBERS

2 1/2 is an example of a mixed number. This is when whole numbers and fractions are written together.

To turn mixed numbers into improper fractions, look at the denominator of the fraction first. This will be the denominator of the improper fraction.

Example

Turn 3 1/2 into an improper fraction.

The fraction in the mixed number has 2 as its denominator, so the improper fraction will also have 2 as its denominator. 3 whole ones are 6 halves. There is also another half left over. In total, this is 7 halves or 7/2..

​

​Converting mixed numbers to improper fractions

​To convert any mixed number to an improper fraction:

1. multiply the whole number by the denominator

2. add on the numerator

3 1/2 = 3×2 + 1 = 7 so, 7/2.

Denominator stays the same.

Draw

Change these mixed numbers into improper fractions

To convert an improper fraction to a mixed number, work out how many whole numbers there are by dividing the numerator by the denominator. Make the remainder the new numerator and leave the denominator as it was.

Example

Turn 7/5 into a mixed number.

7÷5 = 1 whole one, and 2 remaining.

Write 7/ 5 as 1 2/5.

​

Convert improper fractions to mixed numbers

Draw

Change these improper fractions into mixed numbers

Fraction arithmetic

Adding and subtracting fractions

Fractions with the same denominators can be added (or subtracted) by adding (or subtracting) the numerators.

For instance, 2/9+3/9=5/9 or 6/11−4/11=2/11.

If two fractions do not have the same denominator, then find a common denominator by making equivalent fractions.

Draw

Work out 4/7 + 1/3.

​Create a common denominator by looking for the lowest common multiple of 7 and 3. This is 21 (7×3=21).

Create equivalent fractions using 21 as the new common denominator.

4/7=12/21

1/3=7/21

So: 4/7+1/3=12/21+7/21=19/21

This is the final answer as the fraction cannot be simplified.

Draw

Work out 2 2/5 − 3/4.

​This sum contains a mixed number (2 2/5), which must be converted to an improper fraction.

This fraction cannot be simplified further, so this is the final answer.2 2/5=12/5 (2×5+2/5=12/5)

This gives: 2 2/5 − 3/4=12/5−3/4

Now create a common denominator by looking for the lowest common multiple of 5 and 4 which is 20.

Create equivalent fractions using 20 as the common denominator.

12/5=48/20

3/4=15/20

So: 12/5−3/4=48/20−15/20=33/20

The question was asked in mixed number format, so the answer should be given as a mixed number if possible.

Divide the numerator by the denominator:

33/20= 1 13/20

Draw

1 2/9 + 1/4

​53/36 or 1 17/36

Draw

2 1/3 + 1 1/4

​43/12 or 3 7/12

​Multiply fractions

To multiply fractions, you need to multiply the numerators together and multiply the denominators together.

Draw

3/4 × 2/5

​6/20 = 3/10

Draw

1 2/3 × 2/7

​1 2/3 = 3/3 + 2/3 = 5/3

5/3 × 2/7

5 × 2/3 × 7 = 10/21

Divide fractions

For dividing fractions, keep the first fraction as it is, change the divide sign to a multiply and flip the second fraction upside down.

One way to remember this is:

Keep it, change it, flip it.

Draw

2/5 ÷ 2/3

​2/5 × 3/2 = 2 × 3/5 × 2 = 6/10

6/10 = 3/5

Draw

3/4 ÷ 2 1/5

​We need to convert 2 1/5 to a top heavy fraction first.

3/4 ÷ 11/5

3/4 × 5/11 = 3 × 5/4 × 11 = 15/44

​Percentage of a quantity

Percent means out of 100

When we work out percentages without a calculator we usually start by working out 50%, 10% or 1%..

To find 50% of a number you divide by 2

To find 10% you divide by 10

To find 1% you divide by 100

Draw

Work out 10% of £320

​To work out 10% of an amount we divide it by 10.

£320 ÷ 10 = £32
10% of 320 = £32

​Sharing in a given ratio

A ratio can also be used to share a quantity into parts.

Example 1

Rebeckah and Amy share £280 in the ratio 5:2. How much money will they each receive?

1. Add up the ratio to find the total number of parts:

5 + 2 = 7 parts

2. Divide the total amount by the number of parts:

£280 ÷ 7 = £40

Each part is worth £40

3. Multiply by the ratio to find each person’s share:

5 × £40 = £200 (Rebeckah’s share)

2 × £40 = £80 (Amy’s share)

4. Check these add up to the original amount:

£200 + £80 = £280

Draw

Abbie and Ben share £120 in the ratio 2:1
Work out how much each of them get.

​The first step is to work out how many equal parts there are.
The ratio is 2:1 so there are 3 (2 + 1) equal parts

We now need to work out how much each of the parts is worth
We divide the £120 between the 3 parts
£120 ÷ 3 = £40
Each part is worth £40

Now we can work out how much Abbie and Ben get
Abbie has 2 parts so she gets £80 (2 × £40)
Ben has one part which is worth £40