Multiplication and Division Ideas
As with addition and subtraction, pupils need a sound understanding of multiplication and division concepts, and the links between them, in order to progress within many areas of maths.
There has to be an emphasis on pupils being able to efficiently use multiplication facts (or times tables) since they are essential for so many aspects of maths - pupils will not be able to divide or work with fractions accurately without having quick access to times table facts. However, we don't have to rely entirely on rote learning, which will not work for some pupils and for others it does not expose pupils to the underlying patterns within and between the multiple groups.
Make sure to emphasise the multiples of six, seven, eight and nine since pupils can struggle with these facts compared to the earlier multiples so provide lots of opportunities to work with these. Also, don't forget to include multiplying by zero and one which can sometimes catch pupils out because as teachers, we can assume they are obvious and therefore spend little time on them.
100 Grid Patterns
Exploring multiple patterns will help pupils to assimilate multiplication and division facts (which can be a struggle for some pupils) and this can be started on the hundred square, where multiples of numbers can be shaded and any patterns noticed can be gathered together. Allowing time to discuss the patterns and why they occur is time well spent. This can be as simple as looking at a set of multiples and looking for patterns in the digits within the multiples, or shading several multiple sets such as threes, sixes and nines on the same grid. It is often surprising how many older pupils - including some who are proficient at recalling multiplication facts - are surprised about insights such as the connections between the multiples of three, six and nine! When developing divisibility tests, the 100 grid provides a useful investigation tool.
Make sure that pupils are familiar with the multiplication grid and allow time to discover the symmetry within it (thanks to the commutative rule where 3 x 8 = 8 x 3) which cuts the number of facts to learn by half. Connections between different sets of multiples can also be explored such as doubling twos to get fours and doubling fours to get eights. It is also useful for exploring the links between multiplication and division, (for many pupils, 24 ÷ 3 is better as 3 x ? = 24) and for showing the shape of numbers. The array model of multiplication is also emphasised, which can be linked to area.
In its latest incarnation, the English National Curriculum has become more prescriptive regarding the written methods that pupils should have mastered by the end of their primary schooling. These are specified as: columnar addition and subtraction, short and long multiplication and short and long division.
Previously, pupils were allowed to use whatever method suited them best and were able to utilise a flexible approach which made sense to them. However, these approaches are still useful for helping pupils to make sense of multiplication and division calculations.
This method connects to arrays and by partitioning numbers before multiplying, it helps pupils to understand what is happening when we multiply larger numbers. If this is explained alongside the long multiplication algorithm it can help pupils to see why the algorithm works since some pupils struggle to remember the process for calculating using the algorithm. It is also an excellent approach for multiplying brackets when pupils meet algebra later on in their maths journey, rather than relying on remembering the process involved.
Based on repeated subtraction, this became a popular method for division with some pupils partly because they had control over the size of the chunks to choose each time and partly because they understood the process. In fact, when I showed this to a group of fourteen year olds, who had always struggled with division calculations, they were thrilled that they could now use a process that they understood. Of course, it does rely on the ability to subtract accurately and familiarity with the number line can help pupils to decide on the size of their chunks; multiples of ten can be used initially to reduce the likelihood of errors when subtracting.
Short and Long Division
These have been put together because I suggest that they are taught together. The reason for this is that many pupils (and adults) will attest that they can apply one method but are confused by the other, so they are not seeing the links between the two methods where we are simply more explicit in our writing when undertaking long division. I would also suggest that either method can be employed for any size divisor.
Of course, it is always worth writing the calculation as a fraction (if the dividend and divisor have common factors), returning to the method once the equivalent fraction with the smallest divisor has been found in order to work with an easier calculation.
See Multiplication Facts in Resources which will help pupils to explore
multiplication facts by focusing on patterns and connections.
An array is a useful image for multiplication and pupils can be provided with 2cm square paper, together with counters, to see how many rectangles they can create for a given number of counters.
1cm squared paper is also useful when introducing two-digit multiplication since it provides a visual grid where the side length and width of the original rectangle can be partitioned into multiples of ten and the remaining units, so that pupils can see how many squares there are in each section using multiplication. Make sure to point out the link between this and area since they are in fact finding the area of each section. Using base 10 materials on squared paper can also help pupils to understand the grid method and can illustrate the link between arrays and division.
It is worth displaying a 100 square and a multiplication grid in the classroom, as well as having individual smaller grids available for pupils to use as needed, alongside laminated L shapes to assist in reading the grid, as needed.
Pupils sometimes claim that they like multiplication but dislike division, even when working with numbers less than one hundred, which implies that they are not connecting multiplication and division. I have some quick starters where numbers appear and then the required operation and often faces drop when they see say, 72 ÷ 8, but when I ask the question, '8 x what results in 72?' then those who are secure with their multiplication facts can answer easily.
When dividing with larger numbers, using factors or writing the division as a fraction and cancelling, can result in a much easier calculation depending on the numbers involved. It is therefore worth exploring different approaches to calculations so that pupils can choose the most appropriate method or use an alternative to check a formal written calculation if the required method has been specified.
See Multiplication Facts in Resources, which contains images of multiplication grids that can be printed out and laminated.