Whole Number Arithmetic Online Documentation

Use Estimating Skills


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Learners can estimate the answer to a problem when they do not need an exact number answer. They need to know which situations require exact answers. Estimation arrives at a sometimes accurate, but usually only approximate, answer to computation problems.

Learners can learn to estimate an answer to be more, or to be less, than the exact total. You can present examples of this in many situations. For example, when shopping in a store, pose problems such as:

You know that pencils cost 24 cents and you have 60 cents. Do you have enough to buy one; two; three? If you estimate by rounding 24 to 20, you might estimate that you have enough money to buy three pencils (3 * 20 = 60). If you change 24 to 30, you would determine that you only have enough to buy two pencils. In this case it is all right to estimate, but you have to be careful not to make the error of estimating too small a number, which might leave you not able to buy what you expected. So in this case, it might be good to use the rule when shopping, if you only have limited amount, to always estimate high. Then add what you have for a total, and make sure it is less than what you have to spend, and you will have enough.

Learners can also estimate to check answers to problems. For example if the problem is :

204 X 10,369

You know that your answer has to be more than : 200 x 10,000, or 2,000,000 If you answered with a smaller number, there is an error.

There are many ways to estimate numbers. Each learner may develop individualized methods for estimating. These methods include:

    rounding
    using front-end numbers
    clustering numbers

Rounding

If you do not need specific numbers, try rounding. The usual rule for rounding is that if digit is 5 or more, round up. If the digit is less than 5, round down (98 becomes 100, 12 becomes 10, etc.).

Follow these steps to round a number:

  1. Decide to what place you will be rounding a number -- the nearest tens, hundreds, thousands, or ten-thousands.
  2. Find the place in the number you are rounding to. For example 23,456 rounded to the nearest thousands, the place is at the three: 23, 456. (Rounded to the tens, the place would be at the 5.)
  3. Decide whether to change the digit in the rounding place. Look at the digit to the right of the place you are rounding to, (in this case 4). If the digit to the right of the rounding digit is less than 5, do not change the digit in the rounding place. If the digit to the right of the rounding place is more than 5, or equal to 5, increase the digit in the rounding place by 1. In this case since the number 4 is less than 5, you do not change the digit in the rounding place (leave it 3).
  4. Change all the digits to the right of the rounding place to 0. So, 23,456 rounded to the nearest thousands is 23,000. Rounded to the nearest tens it would be 23,460.
To estimate a sum in addition, round each number to the nearest whole number, then add. To estimate a difference in subtraction, round each number to the nearest whole number, then subtract. To estimate a product in multiplication, round each factor in the problem, then multiply the rounded factors. To estimate a quotient in division, it may help to replace numbers used with other numbers so that you can use basic facts. For example, 4,809/59 could be changed to be 4800/60 = 80 so that the division can be done mentally.

Front End Estimation

You usually work addition, subtraction, and multiplication problems from right to left, starting with numbers in the ones place (for more definition, see place value in the Glossary). However, when estimating, it is sometimes useful to start with numbers at the front end, or in the leftmost position.

For example, to estimate the sum of these numbers:

    492
    507
    654
    849
First, add the front end digits. Add 4 + 5 + 6 + 8 (hundreds) to total 2300. Then add the rest. You could group the remaining numbers:
    92 + 7 is approximately 100
    54 + 49 is approximately 100
So you can see that the rest of the numbers total around 200.
Add the 200 to 2300.
The answer is approximately 2500.

By adding just the leftmost number, you at least know a number that your total must be larger than. For example, to add the numbers: 856 + 234 + 456, add 8+2+4. You know that the final sum will be more than 1400.

You can also estimate differences in subtraction using front end estimation. For example in this problem: 9436 - 8654

First, estimate 9,000 - 8,000 = 1,000 (9 - 8 = 1)

Then, compare the rest. Since 436 is less than 654, the first estimate (1,000) is about 200 too much. So, adjust the first estimate by subtracting 200 from 1000, and the final estimate is 800.

Clustering

Look at numbers to find a number that they cluster around, or are close to. For example, to use clustering to add these numbers:

    594
    603
    622
    586
You note that these 4 addends cluster around 600.
You can estimate by figuring 4 X 600, which makes the estimate 2400.

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Last Updated on October 6, 2000 by Chuck at SSSoftware.com