6000 + 600 + 40 + 1500 + 150 = 40 = 8,330

Multiple

A multiple is a number that can be divided by another number a given number of times without leaving a remainder. For example, multiples of 3 include: 3, 6, 9, 12, 15.

Prime Numbers

Prime numbers can only be divided by themselves or one to leave answer that is a whole number. 1 is not considered to be a prime number. Ask your child if they can see any patterns that identify what numbers are prime numbers, and what type of numbers cannot be prime. Hopefully they will identify that even numbers cannot be prime because they can always be divided by themselves, 1 and 2.  

Composite Numbers

A composite number is a number that is not a prime number. In other words, it is any number that can be divided by itself, 1 or another number, and the result would be a whole number.

Square Numbers

A square number is the answer to one number being multiplied by itself. For example: 

3² 6 because 2 x 2 =4. 4² is 16, because 4 x 4 = 16. 4³

Sum: Sums are addition questions. If you want to give a person a series of questions that involve multiplying, dividing, subtracting or adding, these are best described as being calculations.

Square Root

A square root of a number is found by dividing a given number by that same number. For example, the square root of 16 is 4 because 16 divided by 4 is 4.

Triangle (triangular) Numbers

A number that can make a triangular dot pattern. Example: 1, 3, 6, 10 and 15 are triangular numbers.

Addition

Practise being able to add single and two-digit numbers together at speed.

Subtraction

Practise being able to subtract single digit numbers from numbers ranging from 1 to 19. A fun way to do this is to take a pack of cards. All face cards (cards with a Jack, Queen or King on them are worth have a value of 10. Ace is worth 11. Take two cards at random. Subtract the smaller number from the bigger number. Tie how quickly you can get through the whole pack. There is a 10 second time penalty for each mistake. 

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Column Subtraction

Make it very clear that it is the digit in the bottom row that is subtracted from the number directly above it. This is often explained as 'top number take away bottom number.'

Encourage the learner to think of column subtraction that includes hundreds, tens and units as using fictitious (made up) currency. There are pounds coins, ten pound notes and hundred pound notes, ten pound notes and pound coins with pound coins representing units. Monopoly money can be useful for representing £100 notes.

Each time you will need to ask yourself: knowing how many pound coins (or whatever the currency unit is) I have, can I take away the number of pound coins (or whatever currency unit is being used.) If not because there are too few, you will need to exchange (swap) something. If your cross out the number of ten pound notes and replace it with one number smaller, you can add 10 to the number of pound coins.

Rounding

Rounding is rarely given enough attention. Children need not only to be able to round quickly, but be in the habit of rounding. It is part of another key tool: estimation.

Rounding to the nearest 10

If the units digit is 5, 6, 7, 8 or 9, round up to the next 10 by increasing the tens digit by 1 and changing the units to 0.

If the units digit is 0, 1, 2, 3 or 4, round down by changing the units digit to 0.

Rounding to the nearest 100

Answering questions written in sentences

RUCESAC is an excellent acronperym that you can use to help you answer questions written using a combination of words and numbers.

R = Read the question. You might need to do this two or three times.

U = Understand and underline. Check that you understand the question and underline key words and numbers.

C = Choose which operations (add, subtract, multiply or divide) you will need to perform. Do you need to add, subtract, multiply or divide. To answer some questions you will need to perform more than one operation.

E = Estimate. Make a rough guess as to what the answer will be so that you can compare your answer to it later on to make sure it looks right.

S = Solve the question by using the underlined words and numbers and performing the operations you chose earlier.

A = Answer the question. Write down your answer using neat handwriting.

C = Check your work. Does the answer make sense? Can you do the opposite to check that your answer is right?

2-Dimensional Shapes - Think of these as flat shapes. They have only a length and a height.

• 3-sided shapes are called triangles.
• 4-sided shapes are called quadrilaterals.
• 5-sided shapes are called pentagons.
• 6-sided shapes are called hexagons.
• 7-sided shapes are called heptagons.
• 8-sided shapes are called octagons.
• 9-sided shapes are called nonagons.
• 10-sided shapes are called decagons.

Types of Triangle

All the internal angles (angles inside a triangle) add up to 180 degrees. This is important because if you know the size of two angles, you can work out the size of the third angle by adding together the two angles you do know and subtracting that answer from 180.

There are four types of triangle. The can be remembered as the RISE triangles. Each letter of RISE stands for a different type of triangle: right angle triangles; isosceles triangles, scalene triangles; equilateral triangles. Their properties are detailed below:

Equilateral triangles:

All equilateral triangles have:

• three lines of symmetry
• three angles that are the same size
• three lengths which are the same size

Isosceles triangles

All isosceles triangles have:

• one line of symmetry
• two angles which are the same size
• two lines which are the same length

Scalene triangles:

All scalene triangles have:

• no lines of symmetry
• no angles which are the same size
• no angles which are the same length

The one key fact about a right angle triangle is that it contains a right-angle, A right angle is 90ᴼ

3-dimensional shapes

Children should learn to recognise the following shapes:

sphere; cube; cuboid (3-D rectangle); cube; triangle-based pyramid; square-based pyramid

Angles

• An acute angle is less than 90 degrees.
• A right angle is 90 degrees.
• An obtuse angle is greater than 90 degrees but less than 180 degrees.
• A straight angle is 180 degrees.
• A reflex angle is greater than 180 degrees but less than 360 degrees.

3-D Shapes

Be able to identify: cubes, cuboids, spheres, triangular prisms, cylinders, hexagon-based prisms. triangle-based pyramids and square-based pyramids.

Telling the Time

There are two types of clock: analogue and digital. Have your child wear a watch and frequently ask them what time it is.

Your child will need to know: 1) How many seconds there are in a minute (60); 2) How many minutes are in an hour (60) and 3) How many hours are in a day (24). The latter point might not be obvious to your child. You might need to talk about people who work at night, people who travel at night-time. Not everyone starts work in the morning and finishes in the evening.

It can be useful to talk about what time frequent key activities happen. For example: What time does school start; What time does school finish; What time does a particular club start and end.

Talk about why it is useful to tell the time and to be able to this accurately.

When learning the differences between a.m. and p.m. time, you might want to describe a situation where a school letter explains that the coach leaves at 7 on the first of December. If one person assumed 7 to mean 7:00 in the morning, then they could be too early. Equally, if a person thought it meant 7:00 in the evening, they could easily miss their trip if it actually left in the morning. The a.m./p.m. notation is very important. 

Area

The area of a rectangle can be found by multiplying its length by the height. The area is always written in units squared.

A square is a type of rectangle. The area of a square can be found by multiplying the length of one side by itself.

The area of a triangle can be found by multiplying the length of the triangle by the height of a triangle. Then, dividing the whole answer by 2.

Finding the Area of Compound Shapes

You may find a shape such as an L shape, which could be split up into different rectangles. This is an easy way to work out the area.

Some shapes could be split up into both triangles and rectangles. This is another easy way to work out the area.

Derived Information and Area

Some questions involving area may require you to use mathematical facts to work them out. This is called derived information. For example, if two the horizontal and vertical lengths of a rectangle have their lengths labelled, you know that the opposite sides are the same length.

Volume 

To work out the volume of a cube, use the formula: length x height x depth. The area is in units (e.g. centimetres) cubed.

4 x 5 x 6 = 120cm³

 The 3 means cubed. The 3 shows how many dimensions the shape has. 

The Bus Stop Method

The Bus Stop method is a tool used to answer division questions.

The largest number goes inside the bus stop. You could think of this as being the number of seats available.

The number you are dividing the large number by is the number that is written on the left-hand side of the bus stop. Think of this as being the number of people in a group. The group of people want to stay together at all times.

Step 1: Work out how many groups of the people staying together, can fit into the first digit. Note: I think of this as being how many groups of the given number of people can sit on the number of seats shown by the first the number inside the bus-stop. Each seat can have just one person sitting in it. Write the answer on top of the bus stop, directly above the first digit in the bus-stop.

Step 2: If you find that there were some 'seats' left over, in small writing, write the number of seats left over, directly in front of the next digit in the bus-stop. This will form a 2-digit number.

Step 3: See how many groups of the given number will fit into the next digit (or two-digits if you just wrote one number in front of another because you had seats left unfilled). Write the answer on top of the bus stop, directly above the digit in the bus stop showing the number of seat(s) that you were just working with.

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Symmetry

A square has four lines of symmetry.

A rectangle that is not a square has two lines of symmetry.

An equilateral triangle has three lines of symmetry.

An isosceles triangle has one line of symmetry.

Missing Number Questions

Nearly all major tests include questions where a number is missing, For example: 4 + BLANK = 10. The secret to these is to do the inverse, for example, expect to have to subtract if the question says to add. Remember that all the information you are given in a question has been put there for a reason. It is all useful. Expect to have to use all of it.

Reading a Scale

On almost every maths reasoning paper there are questions that involve reading a scale. It is very rare for each line on a scale to represent 1. Usually you have to count in twos, fives, 10s or 20s.

Factors

Factors are numbers that can be multiplied together to get a certain number. So 2 and 9, and 3 and 6 and 1 and 18 are all factors of 18. A key difference between factors and multiples is that factors are the same number as the start number or smaller than the number you started with. Multiples, on the other hand, are the same size or bigger than the number you started with.

Fun Fact: if the digits making up a number add up to a multiple of 9, 3 is also a factor.

Volume

To work out the volume of a cuboid, multiply the length by the height by the length.

Roman Numerals

I = 1

II = 2

II = 3

IV = 4 (1 less than 5)

V = 5

VI = 6 (1 more than 5)

IX = 9 (1 fewer than 10)

X = 10

XI = 11

XIV = 14 (10, then 1 fewer than 5)

XX = 20

XXX = 30

XL = 40

L = 50

XC = 90 (10 fewer than 90)

C = 100

CD = 400 (100 fewer than 500)

D = 500

M = 1,000

Rules:

1) If a symbols is repeated, all of the values represented together. E.g. XXX = 300 (100, + 100 +100).

2) If a smaller number is on the left-hand side of a larger number, subtract the smaller number from the bigger number. E.g. IX = 9 (subtract 1 from 10)

3) If a smaller number is on the right-hand side of a larger number, add together the smaller number and the larger number. E.g. VI = 6 (add 1 to 5) 

General: Think carefully about place value. You need to represent the column (ones/tens/units/hundreds etc. before the next type of number is represented. E.g. XCV = 95 because it's 10 fewer than 100, then add 5.

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Fractions

Key points to remember: Fractions almost always involves either multiplication or division.

The numerator is the name of the top number of the fraction.

The line between the top number and the bottom number in a fraction really means 'out of.'

The denominator is the name of the bottom number of a fraction.

Often you will need to make denominators the same. For example: 2/3 + 1/4 =. In this question the denominators are different. Multiply the two denominators (e.g. 3 x 4). This gives a common denominator (12). A common denominator is the name for denominators which are the same.

Whatever you multiplied a denominator by, you must multiply the numerator above it by the exact same amount. In 2/3, the 3 was multiplied by 4 so, so the numerator above it must also be multiplied by 4. 

In 1/4, the 4 was multiplied by 3 so the numerator (1,) musty also be multiplied by 3.

Finally, add the numerators (just the numerators) together. Keep the denominators the same.

Multiplying Fractions

Multiply the numerators (top numbers) together. Leave the denominators. I.e. only the numerator will change.

Diving Fractions

Swap the position of the second numerator with the second denominator, then multiply the two numerators together to make the new numerator. Then, multiply the two denominators together to produce the new denominator.

Fractions of an Amount

To find a fraction of an amount e.g. 3/5 of 20, divide the amount by the denominator (bottom number,) then multiply that answer by the numerator.

Using the example 3/5 of 20: 

20 divided by 5 = 4. 4 x 3 = 12. The answer is 12.

Simplifying Fractions

Find a number that both the numerator and the denominator can be divided by, that will give an answer that is even. For example: 16/24

Since both 16 and 24 are even, they can both be divided by 2. This would give us 8/12. Again, both numbers can be divided by 2. This time the fraction would be 4/6. Yet again, both numbers can be divided by 2. This time the answer would be 2/3. Since 2/3 cannot be divided by a whole number and given an answer that is an integer (whole number,) 2/3 is the answer in it's simplest form.

Remember, all even numbers can be divided by 2. 

Top Tip: Use 2, 3, 5 and 10 as starting points to see if the numerator and denominator can be divided by them to produce a whole number. 

Fractions, Decimals and Percentages

Children who recognise the equivalents between some of the key percentages, decimals and fractions are at a huge advantage compared with those who do not.

Decimals

Remember that 1 = whole. A decimal is a proportion of one whole. 0.1 is a tenth (1/10) of a whole.

100% = 1 =1/1

50% = 0.5 = 1/2

25% = 0.25 = 1/4

33% = 0.33 = 1/3

75% = 0.75 = 3/4

12.5% = 0.125 = 1/8

Converting a Decimal into a Percentage

To convert a decimal into a percentage, move the number two places to the left so 0.27 becomes 27 per cent.

Percentages

Per cent means out of 100. 100% means the whole of something. 50% means 50 out of 100 which is the same as a half.

To work out a percentage, divide a numerator out of the denominator below it; then multiply the answer by 100. For example, a score of 28 out of 32 on a test is converted to a percentage by dividing 24 by 32, then multiplying the answer by 100. 

Co-ordinates

To read co-ordinates, remember that the first number refers to the number on the horizontal (across) axis. The second number is the number on the vertical (up and down) axis. If a question involves negative numbers, the negative numbers on the horizotnal axis are on the left-hand side of 0. The negative numbers on the vertical axis appear below the 0. 

Word Problems

A typical word problem involves noticing what's the same and what's different. For example:

Alfie bough 3 pens and 2 bookmarks. He paid £7.00.

Ben bought 3 pens and 1 bookmark. He paid £5.00.

What was the cost of a) a book mark and b) a pen?

What's the same: Both Alfie and Ben bought at least two pens and two bookmarks.

What's different: Ben bought one more bookmark than Alfie. The difference in what they paid (£2.00) must be the cost of a bookmark. Knowing that each bookmark costs £2.00, I can see that the remaining money must have been spent on the pens. Ben spent £3.00 on 3 pens so each pen must have cost him £1.00.

Missing Information Questions

Some questions include tables with missing amounts. The missing amounts can all be derived (worked out) from the other information in the table. Make sure you look at any columns or rows that include totals. If you know a total from a column and there is one number that is missing, you can add all the numbers in that column together and subtract that amount from the total. This will give you the missing amount.

Example:

For the question above, add together all the choices the boys made and subtract that amount from 22. The answer will allow you to complete the Mint column. Then, add the boys and girls' Mint totals to complete that row from the Total column. Continue to use this process to complete the table.

Months of the Year

Children are expected to know the months of the year in order. Furthermore, they need to know how many days are in each month. The following rhyme might help so children should learn it off by heart:

30 days has September,

April, June and November,

All the rest have 31,

Except for February alone,

Which has 28 days each year

but 29 days each leap year.

Children are expected to know that a leap year occurs once every four years. 

Conversions (Equal amounts) 

Children must learn the following facts that they can use tor conversions. Note 'convert' means 'to change.'

There are 100 cm (centimetres) in one metre, so 200 centimetres in 2 metres and 300 centimetres in 3 metres.

There are 10 mm (millimetres) in one centimetre, so 20 mm in 2 centimetres and 30 mm in 3 centimetres.

There are 1,000 metres in 1 Km (one kilometres) so 2,000 m in 2 Km and 3,000m in 3 Km.

There are 100 pennies in £1.00 (one pound) so 200 pennies in £2.00 and 300 pennies in £3.00.

There are 1,000 millilitres in 1L (one litre) so 2,000 millilitres in two litres and 3,000 millilitres in three litres.  

There are 500 ml in half a litre.

Ratios

A ratio can be shown like this:   3 red marbles: 5 blue marbles.

The example above shows that in every 8 marbles, 3 of them are red and five them are blue.

Once you know a ratio, you can divide a whole amount by the total of items in the ratio. For example: If there were 40 marbles and you needed to know how many were blue, and you knew that 5 in every 8 were blue, you could divide the total number of marbles (40) by 8 (giving you 5) and then multiply the 5 by the number of marbles that were blue (5) producing the answer of 25.

Step 1: Add together all the numbers that make up the ratio.

Step 2: Divide the total number of things by the answer to Step 1.

Step 3: Multiply the answer to Step 2 by the number in the ratio of things you wanted to know e.g. number of marbles that were a given colour.

Money Conversions and Ratios

Money conversion is a type of ratio. Most questions concern exchanging pounds for either euros or dollars. £1.00 is never the same value as either one euro or one dollar. There is roughly 1.2 dollars to one pound and 1.4 dollars to a pound. Always use the exchange rate you are given in the question.

Multiply the number of pounds by the amount in euros to convert pounds into euros.

Divide the number of pounds by the number of euros to convert into pounds.