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Mathematical Algorithms

We use algorithms to solve basic addition, subtraction, multiplication, and division problems. Most adults learned to solve each one of these mathematical operations one way. The procedure was memorized and you never really understood why it worked, you just know that it did.  If you made a mistake doing it, you had no sense of your mistake because you had no basis for knowing that your answer didn’t make any sense conceptually. The focus was on the steps of the algorithm and not on conceptual understanding of the process. What we know is that some children never mastered an algorithm no matter how many times it was taught.  The procedures of the algorithm never made sense to them conceptually or the steps to solve it were just too confusing.  Other countries have used different algorithms to solve these same basic operations.  Some of these algorithms are more conceptually based and seem to work better for some children.  It doesn’t matter which algorithm a child uses as long as they are successful, efficient, and accurate using it.  A child should know that there is more than one algorithm that can be used to solve a problem. Children should develop a variety of computational methods and the flexibility to pick the best one for a given situation. There are 3 basic algorithms for addition: Partial-Sums, Column Addition, and Opposite–Change Rule.

Partial Sums You can add two numbers by adding partial sums, working one place-value column at a time, and then adding all the sums to find the total.

Example: Partial Sums                                                                  475                                                                 + 338 Add the hundreds (400 + 300).                 700 Add the tens (70 + 30).                         +   100 Add the ones (5 + 8).                             +     13 Add the partial sums (700 + 100+ 13).        813

Column Addition To use the column addition algorithm,                       draw lines to separate the ones, tens, hundreds.  Add the digits in each column and then adjust the results.

Example: Column Addition                                                hundreds    tens    ones                                                         2        9        4 Add the digits                                  4        2        8 in each column                                  6       11       12   Since 11 tens is 1 hundred          hundreds   tens     ones and 1 ten, add 1 to the                        2       9      4 hundreds column                       +         4       2      8 and change the                                    7       1      12 number in the tens column                   to 1.                                               hundreds    tens    ones Since 12 ones is 1 ten and                    2       9       4 1 one, add one to the tens          +        4       2       8 column, and change the number            7       2       2

Opposite Change Rule If you add a number to one part of the sum you must subtract the same number from the other part.  The results stay the same. 7 + 9 = 16 (7+3) + (9-3) = 10 + 6 = 16 This idea can be used to rename the number being added so that one of them ends in a zero, making it easier to add.

Example: Opposite-Change Rule   Rename the first number and then the second.                   add 3    add 40       300 +444 subtract 3 441 subtract 40 + 401   701