Subtracting Mentally (The Unit-Heavy Method)Author(s): Darrell MorganSource: Mathematics in School, Vol. 29, No. 4 (Sep., 2000), p. 8Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30212356 .

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SUBTRACTING MENTALLY

the unit-heavy method

by Darrell Morgan

A keen darts-playing colleague impressed me by stating that he could subtract any number from 301 instantly. I was struck be the speed of delivery of the answer and when I enquired as to how he did the subtraction, he said that he had learnt all the answers!

In trying to impersonate this impressive stunt I have been experimenting with mental methods for subtraction. Rather than subtracting from 301 it is far easier mentally to split 301 into 290 and 11 (as the sum is done using traditional methods). This means mentally that any number subtracted from the units need to add to 11 whilst any number subtracted from the tens and hundreds must add to 29.

For example,

301-85 becomes 2911-85=216 (8+21= 29 and 5+6=11) 301-163 becomes 2911-163= 138 (16 +13= 29 and 3+8= 11)

The (in)significance of this second subtraction will not be lost on any darts aficionados.

Assuming that any scores ending in 0 or 1 could be dealt with quite simply then it is possible that my colleague may well have learnt all the composite pairs that add to 11 and 29.

The unit-heavy method of subtraction has its uses in other areas of the curriculum and can be clearly demonstrated with reference to A level statistics, where candidates are frequently required to calculate probabilities under the normal curve, which then require subtraction from 1. Using traditional methods for subtraction 1-0.8732 becomes

1.0000 borrowing as required gives 0.99910 -0.8732 -0.873 2

Inverting this sum and treating it as an addition gives

O.abcd -0.8732 0.99910

i.e. every column must add to nine except the last column which adds to ten.

Here we need a=l1, b=2 and c=6 to make the required nines and d=8 to make the ten. Hence, 1-0.8732 can be calculated immediately (as 0.1268) without pen and paper by making each column add to 9 except the last which adds to 10.

By simply pairing composite numbers to 9, and last column 10, in this manner the subtraction becomes quicker and easier mentally than it does by hand (or even by calculator as is the usual procedure for many of my sixth form students).

This method of unit-heavy (addition to cope with) subtraction mentally is very useful for pupils preparing for Key Stage 3 mental tests.

Hence, problems such as

x10-x4.57 can be solved instantly as x5.43 (by adding to x9.910)

100-83 can be solved instantly as 17 (by adding to 910)

1-0.65 can be solved instantly as 0.35 (by adding to 0.910).

As well as probability and basic number work the unit- heavy method for subtraction is particularly useful in angle work.

Angles on a straight line problems need to add to 180, i.e. 1710,

e.g. 180-63=117 (6+11=17 and 3+7=10) 180-124=56 (12+5= 17 and 4+6= 10).

Angles about a point problems need to add to 360, i.e. 3510,

e.g. 360-148=212 (14+21=35 and 8+2= 10) 360-79=281 (7+28=35 and 9+1= 10).

Once confident in the use of the unit-heavy method pupils quickly develop their own strategies for KS3 questions such as:

(1998 higher test B, question 5)

(1998 lower test C, question 15)

(1999 higher test B, question 12)

I spend two pounds twenty. How much change will I get from five pounds? x4.'00-x2.20=x2.80.

A jacket costs fifty-two pounds. In a sale the price is nineteen pounds less. What is the sale price? x412-x19=x33.

What is two thousand minus fifty-seven? 19100-57= 1943

I have found that pupils respond superbly to this method of teaching mental subtraction and love to impress their friends/parents with instant answers to difficult problems such as x1000 000-x364 972 and 1-0.864712395 by using the simple rule that each column adds to nine except the last which adds to ten. '

Reference Key Stage Three Mental Mathematics Tests 1998-9, ACAC.

Keywords: Subtraction; Mental Methods.

Author Darrell Morgan, Porthcawl Comprehensive School, Porthcawl, Bridgend, Glamorgan CF36 3ES.

8 Mathematics in School, September 2000

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