Estimation and mental ComputationAuthor(s): Robert E. Reys, Barbara J. Reys and Harold L. SchoenSource: The Arithmetic Teacher, Vol. 34, No. 6, FOCUS ISSUE: CALCULATORS (February 1987),pp. 28-29Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41193092 .

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Estimation and mental Computation

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28 Arithmetic Teacher

I ■ t^ ^ x^ i ¥-^ ̂ • ^ • I / 4 x $2.00 is $8.00, and ) H I ■ Front-End t^ ^ x^ i Estimation ¥-^ ̂ • ^ • I ( 4x$0.i9isLsthan 4 x $2.00 is $8.00, and <■

I 1 1 ( $0.80. That's $8.80, 'M I '"fl^H|^^^H|H^^^^^^^HHP' ' so it >s less than $10.00. /H

I One reason for the current interest in estimation is the When a more accurate ^ ^^„jj^^-^ I I widespread use of the hand-held calculator. It is easy to estimate is required, students o j^^HUm I I press thè wrong key on a calculator and cause an unrea- can be taught to adjust their °

t^^^^^B H I sonable result to appear. A quick estimate can often ¡denti- initial front-end estimates. In ^^^^KÊ H I fy such an error so that it can be corrected. For example, is this example, the child cannot ^^^^^r I I the answer displayed reasonable? buy 5 note pads with $10.00, *^^r ■ I 6 918 + 5 721 +823 S^*^

^^^ ' but what about 4 pads? W ■ I r . ' This refined front-end estimation process has four H I Í I "I?5?TI a00) 6 918 + 5 721 + 823 1 steps: (1) identify the digits with the highest place values, ■ I '-11 -I.1 ^U ' ' (2) mentally compute with them using the appropriate place ■ I 225 / 1 1 is more than 6 + 5 thousand, ) values* fa) ̂ Ide how to adjust the estimate, and (4) men- ■ I UUUU / Ali ' ' or 1 1 000. f-J tally compute to get the adjusted estimate. These steps are ■ I DODD :B| - ^Xa^^'^

illustrated as follows: ^^~^m

Apemsalofth^t-enddigifeinthefirstt^ 4- 18 'Tan627J

A' 18V> I$k2^A7ÌS^ I and 5 thousand) is all that is required to see that the sum 2.37 ^XJ^--^ 2. 37'x ' jhafS $1 50 r^l I must be more than 1 1 000, so the displayed answer cannot + 33 /^-v^v-^%^ .83 ^v^>--^---^-^TB

jhafS $1 50

I be correct. 2. /$7 /^-v^v-^%^

+ $4 + $2' r*^™^^^^ I This simple front-end-estimation process is useful for ( is $13 ; 4. ( My estimate is ̂ m I making a "ball park" estimate of a ' computation. Consider ^-»S^s-S

; ( $13.00 + $1.50, m

this situation: ' >^-v--v-^.

'

^-v-v^v-v--^r-^ / ^or^a^t$14^50J| /Thave $10 00.^

/ ^ • ■• No- 5 x $2.00' |n addition to providing a reason for estimatifirthr^Tnl

v ? I1 b?o5 J > IS

$100?k so^$nn19/ viator can be used as a tool for teaching estimation, as ■

Wg^ v ?

KTTlS I1 b?o5 J

'^^?9w^- > $100?k so^$nn19/

*^*^ ;he viator followlng can be exampte used as ■*»*■ a tool for teaching estimation, as

I X^!&&&&' O ° i^

- *^*^ Think of a number to multiply by 8. If the answer is H |1^^^^5^d^>^.

° wiXS^K 00O between 610 and 630, then the ball goes in the H I v^dS^^^l fflNv mT^'Ä basket. Estimate the number, then check it with your H

M^^^^s^ V¿/ calculator- /^- your

-v I Edited by Robert Robert E. E. Keys Reys and and Barbara Barbara J. J. Reys Reys wOuOw M^^M ^^^n-^ I Robert Robert E. E. Reys Keys and and Barbara Barbara J. J. Reys Reys I University óf Missouri - Columbia j^ - >jKX)wW A^^S ^1 Columbia, AiQ 65211 /^ ¿jP$$jQ( ^-^B ■ Prepared by Harold Harold L. L. Schoen Schoen bp . y Enter rgn r^i ryi nn rg H Harold Harold L. L. Schoen Schoen

I /owfl City, IA 52242 '

- ' - - ' H

Robert Robert E. E. Reys Keys and and Barbara Barbara J. J. Reys Reys

Harold Harold L. L. Schoen Schoen

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February 1987 29

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| Tip Boardj

Teaching Tips

The four steps in the refined front-end-estimation process suggest an understanding of the skills that are prerequi- sites to prof idency with the process. These include identify- ing the digits that have highest place value, determining

I what that place value is, mentally computing with powers of ten, and adjusting the initial estimate. Your instruction and diagnostic efforts should be aimed at an understanding of these skills.

The first step in the long-division algorithm involves a front- end estimate.

53)42 583

( About 8, ( so 8 is our J I first digit )

I 8 53)42 583

Long division can be a way to get started with front-end es- timation, and learning, front-end estimation should improve your students' skill in long division.

Testing Tip

When you test your students' ability to estimate or when you examine commercial tests designed to do so, be care- ful that front-end estimates are counted as correct. Many people, including some test authors, teachers, and stu- dents, incorrectly equate estimation with rounding. If you wish to encourage and test front-end-estimation processes, an answer like 6000 must be treated as an acceptable esti- mate of 4705 + 2698. Multiple-choice items are especially unlikely to have 6000 as a choice or, worse, are likely to have it listed as an incorrect choice.

Research Report ■

In an interesting study by Robert E. Reys, Barbara J. Best- I gen, James F. Rybolt, and J. Wendell Wyatt (Identification ■ and Characterization of Computational Estimation Process- I es Used by In-School Pupils and Out-of -School Adults ■ [Wàshington, D.C.: National Institute of Education, 1980]), ■ a calculator was wired to give incorrect answers. Students ■ and adults who were good estimators were told to use the ■ defective calculator to check their estimates. Even these ■ good estimators, when confronted with the answer on the ■ calculator, usually accepted it as correct and assumed their ■ estimate was in error. This research suggests that students ■ often place too much trust in a calculator's answer. Teach- I ers should encourage their students to question the rea- ■ sonableness of all answers, including those produced by a I calculator ór a computer. ■

An Extension ■

Although we usually teach right-to-left algorithms for exact I addition and subtraction with paper and pencil, both can be ■ done using a front-end, çr left-to-right, algorithm. This front- I end approach has at least two advantages over the usual I right-to-left one: the first step gives a fair estimate of the fi- I nal answer immediately, and the process moves from left to I right, as does reading. I

Take a Look I

Here are some resources that will help students develop I the concept of estimation, as well as a variety of estimation I strategies: I A Reasonably Close Encounter was developed by the Min- I

neapolis General Math Project. It Is a rich source of I ideas for stimulating thinking about estimation. (Available ■ from Dale Seymour Publications) I

GUESS (A Guide to Using Estimation Skills and Strategies) I is a systematic development of thè front-end strategy, as I well as of other estimation strategies. (Available from I Dale Seymour Publications) m ■

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