**Section 8**

**Using Place Value to Operate on Numbers**

It is possible to solve almost every problem we
encounter by
counting. This strategy however becomes tedious when we deal with
large numbers
and many groups. Everyone uses counting for small situations and
simple
problems.

*Task 1*

*How many dots are there altogether in this
image? *

Did you count? Did you group and count? Think
about the
different ways this could be done.

Onwards

Once there are larger numbers involved such as
345 + 67
another method is needed. Efficiency is probably the best approach
in deciding
upon a strategy.

Taking the 67 apart and writing it as 50 + 12 +
5 and adding
5 to 345 to make 350, adding 50 to 350 to make 400 and then adding
12 to make a
final sum of 412 is fairly efficient. This strategy is called
“tidy numbers”
and works best when one number is near a ten or hundred. In this
case 345 is
near 350.

Another “tidy” way would be 345 + 67 = 340 + 5
+ 60 + 7 =
(340 + 60) + 6 + 7 = 412

The brackets here means that happened first.

This is an example of mental arithmetic and
most people in
the pre-computer and calculator days had quite well developed
skills for
computing answers. All that has been lost!

Algorithms

Combing larger and more numbers such as 2345 +
4278 + 765 +
321 + 45464 needs a more organised and
efficient
approach.

2345

4278

765

321

__45464__

What has been
created here is a
position based table so that the columns can be totalled. The
smallest or units
first building to tens and then tens building to hundreds and so
on.

2345

4278

765

321

__4546 _{2}4__

__3__

The units column
adds to 23
which is 20 + 3. The 3 is written in the units column and the 2 is
“carried
into the “tens” column. Now the tens are added 4 + 7 + 6 + 2 + 6
and the extra
2. This totals 27 which is really 270 because it is counting tens.
The 7 is
recorded and the 2 is carried to the hundreds column and so on.

*Task 2*

*Complete this
addition using
the traditional algorithm for addition as explained above. *

2345

4278

765

321

__45464__

Show all the
“carries”. The
position of the carries is best in the column it applies to but
could be at the
top or bottom of the table. I was taught to write a small version
of the number
at the bottom of the column but soon changed to using the room at
the top above
the appropriate column an d crossing it out after being used. If I
remember
correctly I also gained “a licking of the strap” for being
disobedient.
Creative thought was not encouraged in the 1960’s.

The other
significant method to
find a total involves using a calculator or a spreadsheet. This is
now probably
the best and most efficient method in the modern world.

*Task 3*

*Use a calculator
and then a
spreadsheet find the total to the problem in Task 2. *

**Reminder**

**It is better to
solve a
problem in 5 ways than to solve 5 problems. (Polya 1945)**