Estimation and Computation

Mental Computation is the process of finding an exact answer to a computation mentally, without pencil, paper, calculator, or any other computational aid.

Understanding some specific mental computation techniques can help you efficiently and accurately carry them out.The count on technique is an efficient method for adding when one of the addends is 1,2, or 3; 10,20, or 30; 100, 200, or 300; and so on.

Understanding some specific mental computation techniques can help you efficiently and accurately carry them out.The count back technique is an efficient method when subtracting 1,2, or 3; 10, 20, or 30; and so on.

Numbers that are easy to compute mentally are called compatible numbers.The choose compatible numbers technique involves selecting pairs of compatible numbers to perform the computation, usually involving a basic fact.

The break apart numbers technique involves breaking the numbers in a computation into manageable parts to permit the use of the basic commutative, associative, and distributive properties. This technique also draws on a firm understanding of numeration.

Working toward a solution1. Understand the problem.2. Develop a plan.3. Implement the plan.4. Look back.5. Is the answer reasonable?6. Is there another way to solve the problem?

Estimation techniques draw on an understanding of numeration and knowledge of basic facts. Estimation also involves decisions about whether an estimated answer is acceptable for a given situation and which technique should be used to arrive at the estimate.

Finding how many students, days, lunches, classes, and so on.

Finding how much length, area, volume, time, and so on.

Finding a sum, difference, product, or quotient.

A process for finding a number reasonably close to the exact answer for a calculation.

Rounding is a process of replacing a number or numbers in a calculation with the closest multiple of 10, 100, 1,000, and so on. Rounding is based on locating the point halfway between consecutive multiples of 10, 100, 1,000, and so on.

The simplest way to use front-end estimation technique involves calculating with the leftmost, or front-end, digit of each number as if the remaining digits were all zeros.Use this technique when an estimate is needed quickly and a rough estimate is acceptable.

The clustering technique involves looking for the number about which the addends cluster and then multiplying by the number of addends. For example:A salesperson recorded the number of customers that visited the store between 9 and 11 A.M. on the weekdays. The numbers - 48, 55, 47, 52, and 53 - all cluster around the same number, 50. The estimated total number of customers that came into the store between 9 and 11 A.M. during the week is 5 x 50 = 250.Use this technique to estimate sums when the addends in the calculation cluster around the same number. You can also use it in a similar manner for some products.

An algorithm is a step-by-step procedure for adding and subtracting whole numbers. There is more than one algorithm for adding whole numbers. The most commom algorithms for addition of whole numbers use notions of place value, properties, and equivalence to break calculations into simpler ones.

Base-ten blocks can be used to find a sum and thus provide models that help explain the addition algorithms.

Expanded algorithm for addition:369 +244 613Think:Add hundreds: 300+200=500Add tens: 60+40=100Add ones: 9+4=13Add the hundreds, 613tens, and ones

Models can be used to explain algorithms for subtraction in much the same way as they are used to explain addition algorithms.There is more than one algorithm for subtracting whole numbers. Most common algorithms for subtraction of whole numbers use notations of place value, properties, and equivalence to break calculations into simpler ones. The simpler ones are then used to give the final difference.

Using base-ten blocks to find differences demonstrates that many different procedures for modeling subtraction are also available. Using base-ten blocks models the take-away interpretation of subtraction.

Use the expanded algorithm for subtraction:245-18227Think:There are no hundredsto subtractSubtract tens: 245-10=235Take away 5 ones: 235-5= 230Take away 3 moreones: 230-3=227 (the difference)