Mental Calculations and Estimation of Sums and Differences

Mental Calculations

Mental calculations help us to solve or to find answers fast. In order to accomplish mental calculations we need to combine several abilities: the skills to understand and use algorithms, place values, and number properties. There are various procedures to perform mental computations.

Compatible Numbers

Find pair of numbers whose sum or difference is easy to compute.

Example:  

              17 - 12 + 43 = 17 + 43 = 60 - 12

Substitutions

A number is broken down into a convenient sum or difference of numbers. Here are some ways to compute the sum 127 + 38.

                              127 + (3 + 35) = (127 + 3) + 35 = 130 + 35 = 165

                              127 + (30 + 8) = (127 + 30) + 8 = 157 + 8 = 165

                              (125 + 2) + 38 = 125 + (2 + 38) = 125 + 40 = 165

Equal Differences

The difference between two numbers is unchanged when both numbers are increased or decreased by the same amount.

Example:  11 - 7 = 4

                                    11 + 9 = 20

                                      7 + 9 = 16   

                                                         20 - 16 = 4

Add-Up Method

Add up from the smaller to the larger number. 

         53 - 17

From 17 to 20 is 3, and from 20 to 53 is 33. S the difference is 3 + 33 = 36.

Estimations of Sums and Differences

The three most common techniques for estimating are rounding, using compatible numbers, and front-end estimation.

Rounding
An approximate sum or difference. The type of problem will determine to what place value the numbers will be rounded.
       1.    624 - 289 - 132 »
                                      600 - 300 - 100 = 200

       2.    4723 + 419 + 1040 »
                                         5000 + 400 + 1000 = 6400

       3.    812 - 245 »
                            800 - 200 = 600

Compatible Numbers
Sometimes a computation can be simplified by replacing one or more numbers by approximations in order to obtain compatible numbers. For example, to approximate 342 + 250, we might replace 342 by 350.

                                            342 + 250 » 350 + 250 = 600

Front-End Estimation
Is similar to left-to-right addition, but involves only the leading digit of each number.
Example:
                 433 + 684 + 288 » 400 + 600 + 200 = 1200

Basic Math Skills

Most of the information on this blog was taken from the text book: Mathematics for Elementary Teachers.