Bridge over troubled water
A. Partitioning and "bridging through 10" are examples of methods suggested in the National Numeracy Strategy to help pupils become proficient in mental calculations and to lay foundations for later, more formal, pencil-and-paper methods. In the Strategy document you will find a description of the stages that help one understand this process. Page 32 states that, as outcomes, Year 1 pupils should: "Begin to partition and recombine by breaking units of 6, 7, 8 or 9 into '5 and a bit'. For example, work out mentally that: 5 + 8 = 5 plus (5 and 35 + 5 + 3 10 + 3 = 13."
The numbers are partitioned into 5s and hence easily into 10s.
Page 40 of the document says pupils should be taught to:
"Add or subtract a pair of numbers mentally I by bridging through 10 or 100, or a multiple of 10 or 100, and adjusting. As outcomes, Year 1 pupils should: Begin to add a pair of single-digit numbers, crossing 10.
* Use two steps and cross 10 as a middle stage. For example, work out mentally that: 6 + 7 = 13 and explain that: 6 + 7 = 6 + 4 + 3 10 + 3 = 13" The use of tallying into groups of five, using matchsticks or straws, helps pupils to develop a feel for breaking the number into "five and a bit".
Consider the strategy's example of 6 + 7; using the tally of matchsticks this becomes 5 + 1 + 5 + 2 = 10 + 3 = 13.
Helping pupils to grasp the numbers through this tallying process is extremely useful. You can then introduce the number line, so they have a visual representation of the process. In Cambridge-Hitachi's Mult-e-Maths, lesson 7 from Year 3 of the Addition and Subtraction strand contains a useful tool for showing a number line representation of bridging through 10.
Add any two-digit and single-digit numbers and they are represented by a "loop" on the number line, eg 16 + 7 becomes 16 + 4 + 3 = 20 + 3 = 23.
Dragging the right-hand end of the loop splits it into two parts whose sizes can be adjusted to demonstrate the usefulness of adding 7 to 16 by adding 4 and then 3. Changes to the number line are reflected in the lower addition statement on the screen.
The total can then be discussed and revealed as answers in the written statements and as a label on the number line. Partitioning is also used with larger numbers: bridging through multiples of 10.
The addition can also take place through compensation. This is quite an advanced approach and not necessarily one that pupils would naturally attempt. This is where the numbers are "made up" to the next easiest number and then that amount is subtracted as compensation. eg 17 + 9 = 17 + 10 - 1 26 26 + 11 = 26 + 10 + 1 = 37 You may find the following worth reading: * Ian Thompson's paper on mental strategieswww.m-a.org.ukdocslibrary 2076.pdf
* How Do We Want Children to be Numerate by Mike Askew and Margaret Brown of King's College London www.bera.ac.ukpublicationspdfs520668_Num.pdf
* For more information on Mult-e-Maths (the bridging through 10 tool is included as part of the demo): www.cambridge-hitachi.com productsprimarymultemathsstrands.htmNoNo
* In the online game 'Add it up', you select numbers to clear the board, providing fun and mental practice: www.flasharcade.comgame.php?additup2
* The following could be used for practice or to set problems for the class to discuss in a lesson starter: www.amblesideprimary.comamblewebmentalmathsnumberbond.html
* A selection of activities isavailable at: www.coxhoe.durham.sch.ukCurriculumNumeracy.htm
You need to visit the site first: I couldn't get all of the activities to work, but there are plenty to choose from.