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Primary Magazine - Issue 60: Maths to share - CPD for your school


Created on 23 January 2014 by ncetm_administrator
Updated on 05 February 2014 by ncetm_administrator

 

Primary Magazine Issue 60Multiplication sign courtesy of Wikimedia Commons, in the public domain
 

Maths to share - CPD for your school  

This is the third of our series of four explorations of ways in which you can help your children to develop their conceptual understanding of the four operations. In Issue 58 we looked at addition, in Issue 59 we looked at subtraction, and in this issue we explore multiplication. For more about this operation see Issue 25, which explores the basics of this concept. In this issue we will focus on the development of the written columnar method.

The National Curriculum requires teachers to teach multiplication of two-digit numbers by a single digit number, using mental methods and also progressing to formal written methods in Year 3. In Year 4 the children should be taught to multiply 2 and 3 digit numbers by a single digit using formal written layout. In Year 5 they should be taught to multiply numbers up to 4 digits by a one- or two-digit numbers using a formal written method, including long multiplication for two-digit numbers. In Year 6 the children should be taught to multiply multi-digit numbers up to 4 digits by a two-digit whole number using the formal written method of long multiplication. However, as mentioned in Issue 57, so long as the formal methods are taught, it is up to colleagues in individual schools to decide when these are taught.

For the meeting you will need copies of the multiplication and division sections of the National Curriculum and equipment such as straws and base ten equipment.

If you have them, place value counters would be helpful, or simply sets of three different coloured counters to represent hundreds, tens and ones.

You will also find it helpful to provide squared paper or wrapping paper that shows arrays, counters, paper and straws.

Begin your staff meeting by writing these calculations on the board:

  • 24 x 50
  • 24 x 4
  • 24 x 15
  • 136 x 9
  • 245 x 1.6.

Give colleagues a few minutes to discuss ways to solve each calculation. Take feedback, discussing the different strategies they have used.

There are several ways to answer these calculations, including some efficient mental calculation strategies. It might be worth highlighting the more obvious methods, such as:

  • 24 x 50: multiply by 100 and halve
  • 24 x 4: double and double again, or partitioning
  • 24 x 15: multiply by 10, halve that and add the two together
  • 136 x 9: multiply by 10 and subtract 136, grid method, formal method
  • 245 x 1.6: multiply by 16 by doubling four times and then dividing by 10, grid method, long multiplication.

Ideally, we would all want our children to develop the ability to look at a calculation and decide which method is the appropriate one to use for multiplying numbers. Sometimes it might be that a mental calculation strategy is the most efficient, sometimes it might be the column method. Remind colleagues that this means teaching mental calculation strategies remains important. In the notes and guidance section of the National Curriculum for each year group there is an expectation that children use mental calculation strategies; for example, in Year 6 it states that ‘they undertake mental calculations with increasingly large numbers and more complex calculations’.

As with addition and subtraction, it is probably wise to begin teaching the column method with a simple calculation, such as 18 x 3. Begin by exploring this using base 10 equipment and place value counters, and model the approach in a similar way to this…

Ask colleagues to make 18 using the equipment that you have available. They could do this three times, placing them underneath one another:

three rows of 18 blocks

Ask them what they notice about their arrangement. They should notice that it forms the basis for the grid method. Give out straws and paper (for labels) and ask them to create a grid:

three rows of 18 blocks broken down into grid

Next, ask them to re-group the ones into tens and ones and to show the answer linking it to the way in which the children would solve the calculation

three rows of 18 blocks broken down into grid sowing 5x10s and a 4

30 + 24 = 54

You could point out that before adding the two numbers together their answer was 30 24.

Now use counters or squared or wrapping paper and ask colleagues to make an array for 18 x 3:

three rows of 18 counters

Repeat the process from above:

three rows of 18 counters broken down into a grid

Next, ask what is the same and what is different about the two. Agree that both show the commutative element of multiplication and both show the inverse between multiplication and division. Base 10 clearly shows the 10s and ones but the counters don’t until they are marked. The base 10 loses the array element when the ones are grouped into tens. The counters remain ungrouped so don’t lose the array.

Using these models, demonstrate how the array can be transferred to the partitioning method for multiplication and then progress to the formal method using columns:

1 8   1 8  
x 3   x 3  
2 4   5 4  
3 0    2  
5 4      

Ask colleagues to tell you what is the same and what is different about these methods. This is a great question to ask the children, helping to develop their reasoning skills. It is important to stress that the children’s recording needs to be developed alongside the kinaesthetic manipulation of whatever resources they use.

Finish your meeting by leading a discussion on when the children in your school should be taught to move from partitioning or the grid method of multiplication to the written method of column multiplication. Compare colleagues’ thoughts with the expectations from the National Curriculum. As a group make a decision to write into your school calculation policy.

You might like to show one or more of these video clips that have been produced by the NCETM; the titles are self-explanatory:

Multiple representations of multiplication shows a Year 2 class exploring different representations for multiplication statements in order to help them develop an understanding of their multiplication tables.

The commutative law for multiplication shows a Year 2 class using manipulatives to describe a multiplication statement and then moving on to explore the commutative law using arrays.

Grid method as an interim step shows a Year 4 class using manipulatives to deepen their understanding of the grid method.

Moving from grid to column method shows a Year 6 class developing their understanding of the column method by exploring the similarities and differences between this and the grid method.

We hope that you have found this article helpful. If you decide to use it for staff professional development, please let us know - we'd love to hear what you did.

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Explore further!
If you've enjoyed this article, don't forget you can find all previous Maths to share features in our archive, sorted into categories, including Calculation, Exploring reports and research, and Pedagogy.

Image credit
Page header courtesy of Wikimedia Commons, in the public domain

 
 

 

 
 
 
 
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