Algorithms for Adding Whole Numbers

When you were young you were probably introduced to adding by using something concrete that you could see, touch and manipulate, called a manipulative.  One such manipulative is base-ten blocks consisting of units, strips and mats.

Lets try adding 125 and 137.  We can show that with the following mats, strips and units.

1 mat, 2 strips, 5 units





1 mat, 3 strips, 7 units



We could then show this as

 2 mats, 5 strips, 12 units

 

We can simplify the units

 2 mats, 6 strips, 2 units








Looking at the number of mats strips and units we can determine that our answer to 125 + 137 = 262.

Another approach is to use place-value cards.  Each card is labeled ones, tens, hundreds and so on placing the appropriate number of dots under each place value.  A mark in under ones is worth 1, a mark under tens is worth 10 ones, a mark under hundreds is worth 10 tens or 100 ones and so on.  Lets look at the same math problem we used previously.  125 + 137.  This can be shown using place-value cards like this...

Notice that there was a total of 12 dots in the ones column.  We can simplify by moving one dot into the 10s column leaving us with 2 dots in the hundreds, 6 dots in the tens and 2 dots in the ones for an answer of 262.

The instructional algorithm for adding looks like this

   125
+ 137
     12       5 + 7
     50       20 + 30
   200      100 + 100

   262

Which leads us to the final algorithm which we are used to using.

     1
   125
+ 137
   262   

All examples lead us to the same answer, 262.  Young learners start off using manipulative to gain understanding of adding and place-value and move onto the instructional algorithm and then the final algorithm.