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# Boost From Level 4 to Level 5

To give you a better idea of what is on the digital download below is a list of the main categories and further information of what is expected:

## Calculations (Non-calculator)

The questions on the National Tests often test the children's understanding of the number system as well as their ability to perform calculations using the four basic operations.

It is therefore worth spending some time discussing the structure of the number system and the fact that it is one of the greatest inventions of all time.

Place value is an extremely important idea embedded in the number system and children should be developing a good understanding of how each digit takes its value and how this value changes as the digits move to the left or right when multiplying or dividing by powers of 10.

They should also understand that concepts may be combined. For example, to multiply by 500, multiply by 5 and then by 100; to test to see if a number is divisible by 15, test to see if it is divisible by 3 and divisible by 5.

## Calculations (Calculator)

After analysing all the past NCT papers, we have found that questions in this section fall mainly into the following types which are practised on the pages in this module shown:

1. Simple calculations where the only skill required is proficiency in the application of basic operations on the calculator.
2. Simple calculations involving diagrams or stories or where the results of calculations must be put onto a diagram or into a table.
3. Inverse operations (these come up frequently) in which a missing number must be found to make a calculation true.
4. What could the missing number(s) be? In these problems, there are a number of possible answers, any one (or set) of which is acceptable.
5. Rounding numbers to the nearest 100 etc or saying which of a group of numbers is nearest to a given number.
6. Problems involving algebraic concepts, but without the letters. (Eg the sum and difference of two numbers is given. What are the two numbers?)
7. Negative numbers which so far have mainly been about differences between two negative numbers or a positive and a negative number.
8. Explaining why certain answers are possible or impossible for certain problems. We expect to see an increase in the number of this type of problem on the NCT papers in the future.

Plenty of practice is therefore given in these areas. Some of the questions are simple enough to be done without a calculator, but calculators should be used to practice the operations and reading the displays so children feel confident when the more difficult problems are tackled.

## Properties of Numbers

Here we are looking at odd, even, square numbers, triangle numbers, prime numbers, multiples and factors

Technique for finding factors:

There is a very quick and logical method of finding factors of numbers apart from just guessing and children should be taught this method.

Say we want to find the factors of 60.

1 is obviously a factor and it divides into 60 sixty times, so write 1 on the far left and 60 on the far right…

`1                                                          60`

Try the next number that seems likely to be a factor, in this case 2.

This goes into 60 thirty times, so write 2 after the 1 and 30 before the 60…

`1,  2                                                  30, 60`

Try the next numbers in the same way.

`1,  2,  3,  4,  5,  6,            10,  12,  15,  20,  30,  60`

Next we try 7, but this is not a factor. You can stop when you get to the square root of 60 (about 7.7). You now have all the factors.

It is easy to see that there are no factors between 30 and 60, for example, because if there were, they would divide into 60 between 1 and 2 times which is clearly impossible.

This method gives you “two for the price of one” because each time you find a factor you divide it into 60 and immediately obtain another.

We also have work on how to find a remainder with a calculator.

## Sequences

This is quite a straight forward module looking at sequences in different ways, except that the arithmetic is more difficult than in the Level 3/4 module and the required understanding of number greater.

There is always a starting number and a rule for continuing the sequence. Sometimes children are given the sequence and need to understand the rule. Sometimes they are given the rule and need to complete the sequence.

The rules that are usually used in NCT papers are of the following forms:

Adding a constant number to the previous term.

Subtracting a constant number from the previous term.

Multiply the previous term by a fixed number, especially doubling and trebling.

Multiplying by a fixed number and adding or subtracting a constant.

(Eg ×2 and +5).

Other types occasionally appear and we have included some in this module.

## Fractions and Percentages

(Calculators may be used for the more difficult questions in this module)

A good understanding of equivalent fractions is absolutely necessary for competent work in fractions. Make sure your children are able to change a fraction into any number of equivalent fractions and that they are able to cancel and lecnac (opposite of cancelling) fractions at will.

For example, in questions where children have to spot two fractions which have the same value, one method is to cancel all the fractions to their lowest terms and same value fractions can then be easily spotted. Children should also be able to convert fractions to decimals by dividing the numerator by the denominator with a calculator (fractions are, after all, division sums). This will enable them to easily find fractions that are less than or more than another fraction.

Children should be able to quickly make all possible proper** fractions from a group of numbers.

E.g. the numbers 2, 5, 6 and 7 give the proper fractions:

`     2/5,  2/6,  2/7,  5/6,  5/7 and 6/7`

Children should be able to find simple fractions of quantities and shapes and should know the percentage equivalents of simple fractions.

Lastly, they should be able to calculate percentages of quantities by either recognising the equivalent fraction of the percentage or by dividing by 100 and multiplying by the percentage.

** A fraction of value less than 1.

## Probability

Moving from level 4 to level 5 on probability demands much more clarity in answering. Usually the questions ask that the answer is put in the form of a fraction.

Make sure that children understand that when asking for an answer as a fraction 1/8 is correct, but 1 in 8 will not gain a mark. Decimal fractions eg 0.125 are usually accepted as correct.

When explaining comparisons between spinners it is important to state the probability ie 2 in 5 or 2/5 compared to 3 in 6 etc

Vague answers such as, “because it is bigger,” or, “it is the same space for both,” will not gain a mark at this higher level.

Again the terms more likely, less likely and equally likely keep re-occurring in the tests.

To work out the probability of an event, first count how many possibilities there are altogether ‐ this will give you the bottom number (denominator) of your fraction. Then work out how many possibilities for the event you are considering ‐ this will give you the top number (numerator).

An understanding of simple equivalent fractions is also needed at this stage.

Children need to know that ? is equivalent to 2/8 etc.

## Algebra

Teacher/Parent Health Warning

These worksheets are difficult for children in year 6. They should only be used with children who are well advanced at level 4 or are at level 5 already. They should certainly not be used with children who are working at level 3. These are intended as practice pages for children who already have a good understanding of the basic concepts of algebra.

Virtually no questions on algebra appear at level 3 or 4. They are all either upper level 4 or level 5.

Children should be able to spot patterns and describe these in words and algebraic terms.

They should be able to give the next terms in sequences, having spotted the patterns.

Spelling note The correct plural of ‘formula’, i.e. ‘formulae’ seems to have been dropped in all but the most academic of circles in favour of ‘formulas’. The latter spelling is the one that children come across in mathematics text books and with their ICT work. We have, therefore, used this spelling in our modules.

## Interpreting Graphs, Tables and Charts

Tables of prices or distances are popular questions and can prove to be quite tricky to answer, even with a calculator. At level 4/5 the questions usually involve multiple operations to reach the answer (eg two of one thing plus one of another). Encourage very careful reading of the table to ensure that the correct numbers are chosen.

Pupils need to be acquainted with Venn diagrams and be able to interpret what they show.

Not only are pupils expected to know the terms mean and mode but they are also expected to be able to explain the use of them on graphs.

Line graphs are introduced on the 4/5 questions. It is important for children to realise that a line graph shows continuous data ie every point on the line has a value. Block graphs or bar charts can not be converted into line graphs as the values between the bars have no value.

Scales also differ at this level. Encourage children to look carefully to see exactly how the numbers are going up or down on an axis and what each square or interval represents.

Reading information from pie charts is also introduced. Again these questions can be harder than they first appear as similar pie charts can have different total values.

## Scale, Proportion, Conversion of Units

Nearly all the work on this topic occurs at levels 4 and 5. Consequently, many of the concepts covered are quite difficult.

Some of the calculations are also tricky, so a calculator may be needed for the harder examples.

The work falls into the following categories:

Number sequences. This is allied to the work covered in the algebra module, but is a good introduction to ratios and proportion.

Proportion: Examples in this topic are normally of the recipe ingredients type. Children are given the quantities of materials needed to make a cake, concrete path etc and are asked to find how these quantities vary as the number of cakes/total amount changes.

Conversion. This involves conversion of currencies, litres to gallons, miles to kilometres etc. Children may be required to perform a simple calculation or use a chart, table or graph.

Reading scales. Questions in this section normally involve an object placed against a ruler drawn on the page. The idea is to use the scale to give the length of the object, which may involve a direct reading or the difference between two readings.

## Scale Drawing

There have only been a few questions on scale drawing in the test papers at KS2 since their inception and these seem to have been aimed at children working at levels 4 and 5.

Nevertheless, every mark counts and children should have good proficiency with scale drawing. They should be able to draw and measure lines to the nearest millimetre and draw and measure angles to the nearest degree. A sharp pencil always helps!

Any form of angle measurer is permitted (there are several on the market), but many will still use the traditional 180 or 360 degree protractor.

## Co-ordinates

All the questions in this topic so far set concern only co-ordinates in the first quadrant (i.e. with positive x and y co-ordinates). However, problems involving negative co-ordinates are covered in the year 6 syllabus of the Primary Framework Document and may therefore pop up at any time. With this in mind, most of the problems in this module are set with co-ordinates in the first quadrant, but there is a section involving negative co-ordinates at the end.

The whole module involves a complete understanding of how position is described using co-ordinates. Children should not only be able to describe the position of a point using co-ordinates, but should also be able to say how far one point is from another in both the x and y directions. They should then be able to find the co-ordinates of intermediate points and notice relationships such as “To get from one dot to another on this line you go along two squares and up one”. (No algebra is needed to describe these relationships at this stage apart from knowing that the number on the horizontal axis is the x co-ordinate and the one on the vertical axis is the y co-ordinate.

When a shape is reflected on a grid (whether the grid is shown or not), children should be taught to find the distance from the mirror line to the corners of the shape and use this information to find the co-ordinates of the reflected shape.

## Properties of Shapes

This module gives practice in sorting shapes according to their properties. Children should be familiar with the material in the level 3/4 module including a good knowledge of triangles, different types of quadrilaterals, pentagons, hexagons and octagons and should be able to find the perimeter and area of simple shapes.

In addition, they should know that the sum of angles around a point is 360°, the sum of the angles of a triangle is 180°, angles in an equilateral triangle are each 60° and the sum of the angles in a quadrilateral is 360°.

They should be able to calculate angles using any of the above, e.g. one of the equal angles in an isosceles triangle is 35°, what are the other two angles?

They should know and recognise acute, right, and obtuse angles.

## Measures, Perimeter and Area

A good knowledge of the metric system of measuring length is needed to succeed with this section, including converting millimetres to centimetres and centimetres to metres. Children may well be asked to compare measurements written in mm, cm or m.

One of the best techniques to use with these comparisons is to convert all the measurements to centimetres and then compare ‐ but don't forget to convert back to the originals when writing in the answers!

Perimeter questions assume knowledge of properties of shapes eg an isosceles triangle has two sides of equal length, an equilateral triangle has 3 equal sides of equal length and a regular polygon has all its sides of equal length.

Popular questions also involve working out perimeters, but firstly having to work out lengths of sides from the given information. Many children find these two and three part operations to get an answer very tricky and need plenty of experience of how to attack them.

A further complication is that it is often stated that the drawings are not to scale ‐ a clear warning not to try and work the answer out by using a ruler!

Area questions are seldom straightforward. A popular type is to have two rectangles together forming a third shape. Again, length of sides has to be worked out before the area can be calculated.