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

4546₂4

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)