chapter 11 number sense and estimation with whole numbers · 2019-02-20 · 196 chapter 11 number...

194

All of the topics related to whole numbers are intimatelyrelated. Number meanings and relationships (Chapter 6)are connected to the way students understand the oper-

ations (Chapter 7), and both of these collections of ideas con-tribute to basic fact mastery (Chapter 8). As children solvestory problems, the numbers that are used begin to get larger,and they use the ideas of place value (Chapter 9) to invent avariety of computational strategies (Chapter 10).

The ideas discussed in this chapter are part of the fulldevelopment of flexible and fluent thinking with whole num-bers. The number sense activities will help children buildrelationships specifically related to computation. The skills ofestimation can begin as early as third grade and later extendalready developed mental strategies to deal with real-worldsituations that do not demand exact answers.

BIG Ideas1. Multidigit numbers can be taken apart in a wide variety

of ways,including by decades (36 is 30 and 6),with place-value concepts (36 is 2 tens and 16 ones),and with othermultiples (480 is 500 less 20 or 475 and 5). Numbers caneasily be recombined when these parts are recognized.

2. Nearly all computational estimations involve replacingor substituting difficult-to-handle numbers with close“nice” numbers so that the resulting computation canbe done mentally.

NUMBER SENSE FOR WHOLE-NUMBER COMPUTATIONNumber sense has been a part of every topic discussed so farin this text. It refers to flexible thinking and intuitive ideas

Chapter 11Number Sense

and Estimation with Whole Numbers

about number. Here we examine only a small slice of num-ber sense, the ability to take numbers apart and combinethem in flexible ways. This flexibility with multidigit num-bers is extremely important for students to develop if they areto improve their use of invented strategies and algorithms forwhole numbers.

Working with Tens and HundredsMost invented strategies for whole-number computationgrow out of children’s work with direct modeling. Initially,students require support for their thinking by writing thesteps they use on paper. Because they are in the process ofcreating the strategies they are using, it is not reasonable toexpect them to operate completely mentally. However, manyinvented strategies can become automatic, and when thathappens, students can begin to use them without resortingto pencil and paper. This is somewhat analogous to master-ing the basic facts. Efficient strategies for the facts are devel-oped using assorted models, and children appear to be veryslow and deliberate about something as simple as adding6+8. But soon these strategies become so ingrained that stu-dents can quickly respond to basic facts. The activities herewill help students become more efficient with their inventedcomputational strategies by providing them with greater flex-ibility with numbers.

ACTIVITY

11.1 Arrow MathWith a 0–99 chart in view, write a number on thechart followed by one or more arrows. The arrow canpoint up, down, left, right, or diagonally. Each arrowrepresents a move of one square on the board.

After students become adept at “Arrow Math,”talk about what each arrow means. The left and rightarrows are one-less and one-more arrows. The up anddown arrows are the same as 10 less and 10 more.What do each of the four possible diagonal arrowsrepresent? (adapted from Hope, Leutzinger, Reys, &Reys, 1988).

The possibility of a 0–999 chart can be discussed withyour class, and you may even want to make one. Show a0–99 chart, and draw a second 10!10 grid below it. “Whatnumber would go in these squares?” (100 to 199) “And if wehad some more charts below this one, what numbers wouldbe in each of them?” Each new grid is the next 100 numbers:200 to 299, 300 to 399, and so on.

With an extended chart idea, the “Arrow Math” activitycan be extended to include “super arrows.”

In the next example, note how three super up arrows anda regular down arrow are actually a good way to subtract 290.

“Arrow Math” is only mildly problematic, and there issome evidence that the activity does not directly contributeto computation strategies (Cobb, 1995). However, the fol-lowing activity connects arrows on the chart more directlyto computation.

ACTIVITY

11.2 More Arrow MathBegin with an arrow math task as before. For example:

After getting a result, ask, “How much did we add?”(21) “Now write an equation to go with this arrowproblem.” (56+21=77)

This works well in a similar manner for up andthen left arrows for subtraction:

81 81 – 43 = 38

56

“674 minus 300 is 374, plus 10 is 384”674

Read: “426, 526, 626, 636, 637, 638”426

Read: “63, 73, 83, 84”63

Read: “45, 35, 25, 14, 13, 12”45

That is, one way to subtract 43 from 81 is first to sub-tract 40 and then to subtract 3 more. The next twoexamples offer another twist.

In the first example, you add 20 and then subtract 1,a nice way of adding 19. In the second example, 30is subtracted and then 1 added—a good way to sub-tract 29.

“More Arrow Math” certainly requires more than fourexamples. After the more straightforward tasks like the firsttwo, begin with the equations, and have students write andcomplete the arrow problem: For 27+34, you first go down3 (add 30) and then right 4 (add 4). How would you do457-238? When adding or subtracting a number ending in8 or 9, a good strategy, as indicated in “More Arrow Math,” isto add or subtract a close multiple of 10 and then compensate.

TECHNOLOGY NOTEMath and More 3: Patterns on Lattices (EduQuest, 1994) allowsstudents to create a wide variety of arrow paths on a numberchart that scrolls as high as 999. The software is a tool only,providing the number chart and arrows. Unfortunately, thestart and end numbers are always visible. However, the soft-ware is well designed to explore this idea, and the teachersupport material provides good explanations. It is importantto ask, however, does using the computer add something ofvalue to the task of doing and engaging in mathematics thatyou wouldn’t have without it? ■

ACTIVITY

11.3 Calculator Tens and HundredsTwo students work together with one calculator.They press 10 to make a “;10 machine.” Onechild enters any number. The other student says orwrites the number that is 10 more. The is pressedfor confirmation. The roles are then reversed. Thesame activity can be done with plus any multiple of10 or multiple of 100. On their turn to challenge, stu-dents can select what kind of machine to use. Forexample, they may first press 0 300 and thenpress 572. The other student would say or write 872and then press for confirmation. This game canalso be played using instead of to practicementally subtracting tens or hundreds.

As described, “Calculator Tens and Hundreds” is verysimilar to “Arrow Math” without the 0–99 chart. Students

+-

=

=+

=

=+

64 and 72

CHAPTER 11 Number Sense and Estimation with Whole Numbers 195

196 CHAPTER 11 Number Sense and Estimation with Whole Numbers

30

85

Takeaway

Takeaway

232Takeaway

673 Takeaway

Figure 11.2 Counting back or subtraction using both models

and numerals.

3056

45

470

745

Figure 11.1 Flexible counting on or addition using both

models and numerals.

doing these activities will soon discover that it is fairly easyto add and subtract hundreds and tens mentally. In “Cal-culator Tens and Hundreds,” students could easily beginto add or subtract numbers such as 240 or 73. Every stu-dent who has not been taught to add and subtract in thetraditional pencil-and-paper manner will begin these tasksmentally starting with the largest amounts, just as in“Arrow Math.”

Many of the skills and concepts developed so far alsoappear in the next activity, which combines symbolism withbase-ten representations. In all of these activities, class dis-cussion should have children explain how they did the exer-cises or tell how they thought about them.

ACTIVITY

11.4 Numbers, Squares,Sticks, and DotsAs illustrated in Figure 11.1, prepare a worksheetor overhead transparency on which a numeral andsome base-ten pieces are shown. Use small squares,sticks, and dots for base-ten pieces to keep draw-ing simple. Students write the totals that they com-pute mentally.

Figure 11.2 is a take-away version of the same activ-ity. As shown, the amount removed can be either thenumeral or the squares and sticks. Try it both ways.

Notice how many ideas related to invented strategies andcounting money are included in Activity 11.4. The next twoactivities push for the same thinking done mentally.

ACTIVITY

11.5 Moving Models MentallyDisplay some tens and hundreds base-ten models onthe overhead. Students say the number shown. Nexttell how much will be added or subtracted, as in“minus 300,” and have students say the result. Buildanother number and repeat. Finally, move to exam-ples with trades. Show 820. “Minus 50.” (Think:“Minus 20 is 800, and 30 more is 770.”)

ACTIVITY

11.6 The Chain GameEach student has one or more cards with a numberwritten on it. After naming a start number, theteacher calls out, for example, “Plus 300” or “Minus70.” The student holding the result card calls out theresult. Then the teacher reads the next change, andthe chain continues. If the cards are prepared so thatsome numbers occur more than once, all students


630510

480

500Start with –20

+60

+120

–50

+40

. . .

+200

+120

–300

–160

550

430 390

330

Figure 11.3 A chain game for adding and subtracting tens

and hundreds.

Two more makes 30 and70 more is 100, so 72.

28 and 70 is 98 and 2makes 100, 72.

Has to be 70-somethingbecause 80 more is too much.70 and 2 goes with the 8, 72.

Figure 11.4 Using little ten-frames to help think about the

“other part of 100.”

must remain alert even after their numbers have beencalled. Figure 11.3 illustrates the ideas with onlyseven cards.

with a partner using their calculator. They make thecalculator into a Parts of 100 Machine by pressing100 100 . Now when they press any numberless than 100 followed by , the calculator will givethe other part of 100 (with a minus sign).

=

=-

Thinking About Parts of NumbersThe activities in the previous section had children countingon or counting back in various ways. The next few activitiesfocus on taking numbers apart.

ACTIVITY

11.7 Break It in Two PartsPick any two-digit or three-digit number. The task isto find ways to make the number in two parts. Forexample, 375 can be 300 and 75 or 215 and 160.Children should be permitted to use base-ten piecesor other models if they wish.

Several potential problems may arise from the lastactivity. Treat them as opportunities for learning. Childrenwho do not have good place-value concepts may write 3 and 75 for 375. This may be inadvertent, or it may be asignal of a larger problem. Suggest modeling the numberwith base-ten pieces, and see how the child connects thepieces and the digits.

Some children may see an easy pattern: 374 and 1, 373and 2, 372 and 3, and so on. Compliment the nice pattern.Then ask, “What would the other part be if one part is 120?Would you have to continue your pattern all the way to 120?”

The next activity is similar but focuses on parts of impor-tant nice numbers such as 50, 100, or 300.

ACTIVITY

11.8 The Other Part of 100Give students a number, say, 28, and havethem determine “the other part of 100” (72). Dis-cuss strategies. If students are having difficulty withthis, try using the little ten-frame cards (BlacklineMasters). Show the given part of 100 with the littleten-frame cards as in Figure 11.4. When studentsare more adept, they can practice by themselves or

Students can play “The Other Part” of 50 or 200 or 450or any round number; 50 is probably the best number tobegin with. In the following activity, the nice number is seenas a part of another number. Often in computations, it is use-ful to recognize a nice part of a number, operate on that part,and then deal with the rest.

ACTIVITY

11.9 50 and Some MoreSay a number between 50 and 100. Students respondwith “50 and ___.” For 63, the response is “50 and13.” Use other numbers that end in 50, such as “450and Some More.”

Compatible numbers for addition and subtraction arenumbers that go together easily to make nice numbers.Numbers that make tens or hundreds are the most commonexamples. Compatible sums also include numbers that endin 5, 25, 50, or 75, since these numbers are easy to work withas well. The teaching task is to get students accustomed tolooking for combinations that work together and then look-ing for these combinations in computational situations.

ACTIVITY

11.10 Compatible PairsSearching for compatible pairs can be done as aworksheet activity or with the full class using theoverhead projector. Prepare a transparency, or


Make 106

51

3

2

4

97

85

3712

31

41 13

3819 22

928 240

165150 375

815

635

435760

565

550

365

240

450280 185

720

25

3545

5

95 15 55756585

85415 350

335260

125Make 50

Using fives tomake 100

Make 500

Make 1000

Figure 11.5 Compatible-pair searches.

duplicate a page with a search task. Five possibili-ties of different difficulty levels are shown in Figure11.5. Students call out or connect the compatiblepairs as they see them.

. Then they continue to add 8 mentally, challeng-ing themselves to say the number before they press

. They should see how far they can go before mak-ing a mistake.

The constant addend in “Challenge Counting” can beany number, even a two- or three-digit number. Try 20 or 25.Try 40 and then 48. As an added challenge, after a studenthas progressed eight or ten counts, have the student reversethe process by pressing followed by the same number andthen , , . . . Discuss patterns that appear.

ACTIVITY

11.13 Little Ten-Frame Additionand SubtractionProvide a set of little ten-frame cards for each oftwo students. Each student makes a number withhis or her cards. When both have their numberready, they place it out so both can see. Then theytry to be the first to tell the total. For the subtrac-tion version, one student makes a number greaterthan 50 and the other writes a number on paperthat is less than 50. The written number is to besubtracted from the modeled number. Studentsshould be encouraged to share strategies to see howfast they can get.

ASSESSMENT NOTESThe activities presented so far in this chapter not onlydevelop appropriate number sense skills but also provideclues when students are having difficulty going beyondcounting by ones when solving multidigit story problems.Students who exhibit difficulty with any of these activitieswill also have difficulty with almost any type of inventedcomputation and certainly with the estimation activities tocome. For example, how do students go about the exercisesin Activity 11.4, “Numbers, Squares, Sticks, and Dots” (p. 196)?This activity demands that children have sufficient under-standing of base-ten concepts that they can use them inmeaningful counts. If students are counting by ones, perhapson their fingers, then more practice with these activities maybe misplaced. Consider additional counting and groupingactivities (Chapter 9) where students have the opportunityto see the value of groups of ten. In Activity 11.7, “Break Itin Two Parts” (p. 197), look for the types of numbers chil-dren use. Can they be encouraged to use parts that are niceor round numbers? When they select one part, how do theygo about finding the other part? Perhaps the task should bemodified to use numbers such as 20 or 30 so that you cansee what number concepts are being employed.

==

-

=

=

ACTIVITY

11.11 Compatible CalculationsPresent strings of numbers that use compatibles:

30+80+40+50+10

25+125+75+250+50

95+15+35+5+65

Strings such as these can be approached in two ways,and each should be practiced. One way is to searchout compatible combinations such as 5 and 95 in thethird example. The other way is simply to add oneaddend at a time, saying the result as you go. For thefirst example, that would be “30, 110, 150, 200, 210.”

Change some of the operations to subtraction fora good variation.

Flexible ThinkingStudents may approach the following activities in a variety ofways. While they are good for workstation activities, it isoften useful to have a class discussion of the strategies stu-dents are using.

ACTIVITY

11.12 Calculator ChallengeCountingStudents press any number on the calculator (e.g.,17), then 8. They say the sum before they press+


These are just two examples of using the activities in thissection in a diagnostic manner. Consider thinking aboutother activities in this way. Build them into a brief interviewwith students you need more information about.

INTRODUCING COMPUTATIONALESTIMATIONThe long-term goal of computational estimation is to be ablequickly to produce an approximate result for a computationthat will be adequate for the situation. In everyday life, esti-mation skills are valuable time savers. Many situations do notrequire an exact answer, so reaching for a calculator or a pen-cil is not necessary if one has good estimation skills.

Good estimators tend to employ a variety of computa-tional strategies they have developed over time. Teachingthese strategies to children has become a regular part of thecurriculum. Beginning about grade 3, we can help childrendevelop an understanding of what it means to estimate acomputation and start to develop some early strategies thatmay be useful. From then on through middle school, chil-dren should continue to develop and add to their estimationstrategies and skills.

Understanding the Concept of an EstimateEstimation of measures and quantities is a significant part ofthe measurement strand. Students estimate lengths, weights,and other measures as both a practical skill and a means todevelop unit familiarity. Students also estimate quantities:How many jelly beans in the jar?

Computational estimation is a significantly different skill.There are no physical materials; it is not a visual task. Con-sider that computation has always meant “get the exact cor-rect answer.” Computational estimation involves somecomputation; it is not a guess at all. Since it is based on com-putation, how can there be different answers to estimates?The answer, of course, is that any particular estimate dependson the strategy used and the kinds of adjustments in thenumbers that might be made. Estimates also tend to vary withthe need for the estimate. Estimating your gas mileage is quitedifferent from trying to decide if your last $5 will cover thethree items you need at the Fast Mart. These are new and dif-ficult ideas for young students.

Suggestions for EstimationInstructionHere are some general principles that are worth keeping inmind as you help your students develop estimation skills.

Find Real Examples of EstimationDiscuss situations where computational estimations areused in real life. Some simple examples include figuringgas mileage, dealing with grocery store situations (doingcomparative shopping, determining if there is enough topay the bill), adding up distances in planning a trip, deter-mining approximate yearly or monthly totals of all sorts ofthings (school supplies, haircuts, lawn-mowing income,time watching TV), and figuring the cost of going to asporting event or show including transportation, tickets,and snacks. Help children see how each of these involvesa computation (in contrast to measurement estimates). Youcould sit down with a calculator or pencil and paper andcompute an exact answer. Discuss why exact answers arenot necessary in some instances and why they are neces-sary in others. Look in a newspaper or magazine to findwhere numbers are the result of estimation and where theyare the result of exact computations.

Use the Language of EstimationWords and phrases such as about, close, just about, a littlemore (or less) than, and between are part of the language ofestimation. Students should understand that they are tryingto get as close as possible using quick and easy methods,but there is no correct estimate. Language can help conveythat idea.

Build on Related Skills and ConceptsEstimation skills are related to mental computation skills andnumber sense. Most estimation strategies are based on theidea of using nice numbers that are close to the numbers inthe computation. The nice-number substitutes lend them-selves to a mental computation that was not possible with theactual numbers.

A real-world number sense also plays a role. Is $2.10,$21, or $210 most reasonable for thirty 69-cent soft drinks?It is much easier to focus on 7!3 and use a result thatmakes sense than to compute 0.69!30 and try to place thedecimal correctly. Similar assists come from knowing if thecost of a car would likely be $950 or $9500. Could atten-dance at the school play be 30 or 300 or 3000? A simplecomputation can provide the important digits, with numbersense providing the rest.

Accept a Range of EstimatesWhat estimate would you give for 27!325? If you use20!300, you might say 6000. Or you might use 25 for the27, noting that four 25s make 100. Since 325∏4 is about81, that would make 8100. If you use 30!300, your esti-mate is 9000, and 30!320 gives an estimate of 9600. Is oneof these “right”?

By listing the estimates of many students and letting stu-dents discuss how and why different estimates resulted, theycan begin to see that estimates generally fall in a range around


$1.50Over

orunder?

5 candy bars

17¢

89¢1 ofeach

39¢

$0.23

0.05

0.490.35

0.15

Candy Bar

43¢

Figure 11.6 “Over or Under?” is a good beginning estimation

activity.

1362 + 583 409 + 186 391 – 156 503 – 388

38 × 26 173 × 52

Which of these is closest to 600? To 1000? To 100?

7214 ÷ 70 913 ÷ 18

2 3 4

5 6 7 8

Figure 11.7 A textbook drill page can be a good source of

estimation exercises.

the exact answer. Different approaches provide differentresults. And don’t forget the context. Some situations call formore careful estimates than others.

Beginning Formats for EstimationConsider the threat a third-grade student perceives when youask for an estimate of the sum $349.29+$85.99+$175.25.The requirement is to come up with a number. Even text-books, in an attempt to teach a particular estimation strat-egy, will lead students through a series of rounding andadding and come up with a singular answer. This is con-trary to the idea of what an estimate really is: producinganswers that are “good enough” for the purposes. The pur-poses often determine what we need to know. For the threeprices, the question “About how much?” is quite differentfrom “Is it more than $600?” How would you answer eachof those questions?

Each activity that follows suggests a format for estima-tion where a specific numeric response is not required.

ACTIVITY

11.14 Over or Under?Prepare several estimation exercises on a transparency.With each, provide an “over or under number.” InFigure 11.6, each is either over or under $1.50, but thenumber need not be the same for each task.

ACTIVITY

11.15 Which One’s Closest?Make a transparency of a page of drill-and-practicecomputations (see Figure 11.7). Have students focuson a single row or other collection of five to eightproblems. Ask them to find the one with an answerthat is closest to some round number that you pro-vide. One transparency could provide a week’s worthof 5-minute drills.

ACTIVITY

11.16 Best ChoiceFor any single estimation task, offer three or fourpossible estimates.

How close the choices are will determine the diffi-culty of the task. Sometimes it is a good idea to usemultiples of ten, such as $21, $210, and $2100.

Even with these multiple-choice tasks, the three-part les-son format remains useful. Present the exercise, have studentsquickly write their choice on paper (this commits them to ananswer), and then discuss why the choice was made. All threeparts may take only 5 or 10 minutes. In the discussion, awide variety of estimates and estimation methods will beshared. This will help students see that estimates fall in arange and that there is no single correct estimate or method.

COMPUTATIONAL ESTIMATIONSTRATEGIESEstimation strategies are specific algorithms that produceapproximate results rather than exact results. The strategiesdiscussed in this section are the ones good estimators use.

65¢79¢

39¢

About how much in all?


(a) Front-end addition, column form

(b) Front-end addition, numbers not in columns

4 8 93 7

6 5 1+ 2 0 8

4 + 6 + 2 = 12about 1200

8 + 3 + 5 = 16about 160 more about 1360Adjust:

Adjust:

0 + 4 + 2 = 6about $60

3 + 2 + 7 = 12$12 or $13 more

about $73

$ 27.25$ 42.50

$ 3.98

Figure 11.8 Front-end estimation in addition.

It is always a good idea simply to present an estimationtask and see what strategies and ideas students will use, inmuch the same spirit as developing invented strategies. How-ever, it is most likely that you will need to be more directivein suggesting at least some of these strategies. After intro-ducing a strategy, you can have students practice it so that itis clearly understood.

When students have seen a variety of strategies, stopspecifying which strategy to use. Rather, provide a task, andlet students come up with an estimate using whatever strat-egy makes sense to them. Even in groups of three or four, stu-dents can each make an estimate and then compare bothestimates and strategies. This helps in the selection of a strat-egy as well as in allowing for a range of answers.

Front-End MethodsFront-end methods focus on the leading or leftmost digits innumbers, ignoring the rest. After an estimate is made on thebasis of only these front-end digits, an adjustment can bemade by noticing how much has been ignored.

Front-End Addition and SubtractionFront end is a good beginning strategy for addition or sub-traction. This approach is reasonable when all or most of thenumbers have the same number of digits. Figure 11.8 illus-trates the idea. Notice that when a number has fewer digitsthan the rest, that number is ignored completely.

After adding or subtracting the front digits, an adjust-ment is made to correct for the digits or numbers that wereignored. Making an adjustment is actually a separate skill. Foryoung children, practice first just using the front digits. Payspecial attention to numbers of uneven length when not in acolumn format.

When teaching this strategy, present additions or subtrac-tions in column form, and cover all but the leading digits. Dis-cuss the sum or difference estimate using these digits. Is it moreor less than the actual amount? Is the estimate off by a little ora lot? Later, show numbers in horizontal form or on price tagsthat are not lined up. What numbers should be added?

The leading-digit strategy is easy to use and easy to teachbecause it does not require rounding or changing numbers.The numbers used are there and visible, so children can seewhat they are working with. It is a good first strategy for chil-dren as early as the third grade.

Front-End Multiplication and DivisionFor multiplication and division, the front-end method usesthe first digit in each factor. The computation is then doneusing zeros in the other positions. For example, a front-endestimation of 48!7 is 40 times 7, or 280. When both num-bers have more than one digit, the front ends of both areused. For 452!23, consider 400!20, or 8000.

Division with pencil and paper is almost a front-endstrategy already. First determine in which column the firstdigit of the quotient belongs. For 7)———

3482, the first digit is a4 and belongs in the hundreds column, over the 4. There-fore, the front-end estimate is 400. This method always pro-duces a low estimate, as students will quickly figure out. Inthis particular example, the answer is clearly much closer to500, so 480 or 490 is a good adjustment.

Rounding MethodsRounding is the most familiar form of estimation. Estimationbased on rounding is a way of changing the problem to onethat is easier to work with mentally. Good estimators followtheir mental computation with an adjustment to compensatefor the rounding. To be useful in estimation, rounding shouldbe flexible and well understood conceptually.

Rounding in Addition and SubtractionWhen a lot of numbers are to be added, it is usually a goodidea to round them to the same place value. Keep a runningsum as you round each number. In Figure 11.9 (p. 202), thesame total is estimated two ways using rounding. A combi-nation of the two is also possible.

In subtraction situations, there are only two numbers todeal with. For subtraction and for additions involving onlytwo addends, it is generally necessary to round only one ofthe two numbers. For subtraction, round only the subtractednumber. In 6724-1863, round 1863 to 2000. Then it iseasy: 6724-2000 is 4724. Now adjust. You took away a


6 of these

×

Adjust:

Adjust:

about $25.00

Used 500—toohigh—each day.

About 7300

in one day.Travel 15 days.

6 × $4.00is $24.00

485 → about 5005 × 15 is 75,

so 7500

$4.17

485 miles

46 × 83 =

5 tens × 8 tens40 hundred

or4 thousand

roundup

50

rounddown

80

Figure 11.10 Rounding in multiplication.

I’ll roundto tens:

ignore thenext two

and $10 is $60.Then $20 more—about $80.$50

I’ll use theclosest dollar:

48 and 2 is 50, then 51, 58 and25 more—78 and 5—$83.

$24.95$7.1085¢

$1.89

$48.27

Figure 11.9 Rounding in addition.

bigger number, so the result must be too small. Adjust toabout 4800. For 627+385, you might round 627 to 625because multiples of 25 are almost as easy to work with asmultiples of 10 or 100. After substituting 625 for 627, youmay or may not want to round 385 to 375 or 400. The pointis that there are no rigid rules. Choices depend on the rela-tionships held by the estimator, on how quickly the estimateis needed, and on how accurate an estimate is required. Themore adjusting and “playing around” with the numbers, themore accurate you are likely to be. After a while, however,you should reach for the calculator.

Rounding in Multiplication and DivisionThe rounding strategy for multiplication is no different fromthat for other operations. However, the error involved can besignificant, especially when both factors are rounded. In Fig-ure 11.10, several multiplication situations are illustrated,and rounding is used to estimate each.

If one number can be rounded to 10, 100, or 1000, theresulting product is easy to determine without adjusting theother factor.

When one factor is a single digit, examine the other fac-tor. Consider the product 7!836. If 836 is rounded to 800,the estimate is relatively easy and is low by 7!36. If a moreaccurate result is required, round 836 to 840, and add twopartial products. Then the estimate is 5600 plus 280, or 5880(7!800 and 7!40).

If possible, round only one factor—select the larger oneif it is significantly larger. (Why?) For example, in 47!7821,47!8000 is 376,000, but 50!8000 is 400,000.

Another good rule of thumb with multiplication is toround one factor up and the other down (even if that is notthe closest round number). When estimating 86!28, 86 isbetween 80 and 90, but 28 is very close to 30. Try rounding86 down to 80 and 28 up to 30. The actual product is 2408,only 8 off from the 80!30 estimate. If both numbers were

rounded to the nearest 10, the estimate would be based on90!30, with an error of nearly 300.

With one-digit divisors, it is almost always best to searchfor a compatible dividend rather than to round off. Forexample, 7)———

4325 is best estimated by using the close com-patible number, 4200, to yield an estimate of 600. Roundingwould suggest a dividend of 4000 or 4300, neither of whichis very helpful.

When the divisor is a two-digit or three-digit number,rounding it to tens or hundreds makes looking for a missingfactor much easier. For example, for 425)———

3890, round thedivisor to 400. Then think, 400 times what is close to 3890?

Using Compatible NumbersWhen adding a long list of numbers, it is sometimes usefulto look for two or three numbers that can be grouped to make10 or 100. If numbers in the list can be adjusted slightly toproduce these groups, that will make finding an estimate eas-ier. This approach is illustrated in Figure 11.11.

In subtraction, it is often possible to adjust only onenumber to produce an easily observed difference, as illus-trated in Figure 11.12.


3 hundreds—I didn’t use the 48

—about 350

14 and 6 is 20.The 8 and 11 (and change)

is another 20—that’s 40

and 5 more—about $45.

Bridget’sSteakLasagnaWine

Caesar SaladPiePudding

Restaurant$14.1011.50

8.796.15

2.752.00

Scout Troops,

2000

Eagles

Explorers

Wolfpack

Grizzlies

Cougars

Braves

Mountaineers

41296317654885

# of

Merit

Badges

100

100

100

Figure 11.11 Compatibles used in addition.

$67.9543 to 63 is 20.

About $24 savings.Sale Price$43.50

Figure 11.12 Compatibles can mean an adjustment that produces

an easy difference.

Cookies SoldTroop 124

Looks like about 60 each—8 × 60 is 480

1. Marcie

2. Sally

3. Chris

4. Yvonne

5. Yolanda

6. Andrea

7. Meggan

8. Jo Ann

68

42

81

35

57

60

71

63

Figure 11.13 Estimating sums using averaging.

Frequently in the real world, an estimate is needed for alarge list of addends that are relatively close. This might hap-pen with a series of prices of similar items, attendance at aseries of events in the same arena, cars passing a point on suc-cessive days, or other similar data. In these cases, as illus-trated in Figure 11.13, a nice number can be selected asrepresentative of each, and multiplication can be used todetermine the total. This is more of an averaging techniquethan a compatible-numbers strategy.

One of the best uses of the compatible-numbers strategyis in division. The two exercises shown in Figure 11.14 illus-trate adjusting the divisor or dividend (or both) to create a

Figure 11.14 Adjusting to simplify division.Source: From GUESS (Guide to Using Estimation Skills and Strategies) (box II,cards 2 and 3), by B. J. Reys and R. E. Reys, 1983, White Plains, NY: DaleSeymour Publications. Copyright 1983 by Dale Seymour Publications.Reprinted by permission of Dale Seymour Publications.

division that comes out even and is therefore easy to do men-tally. Many percent, fraction, and rate situations involve divi-sion, and the compatible-numbers strategy is quite useful, asshown in Figure 11.15 (p. 204).


After entering the setup with the start # as shown, players taketurns pressing a number, then to try to get a result in thetarget range.

Addition:Press:

START15321653

790400215

→→→

800410220

TARGET

0 (start #)

Subtraction:Press:

START1841

129

25630475

→→→

30635485

TARGET

0 (start #)

Multiplication:Press:

START67

14339

110035001600

→→→

120036001700

TARGET

(start #) 0

Division:Press:

START2039

123

255015

→→→

306020

TARGET

0 (start #)

=

=

=

=

=

+

!

-

÷

Figure 11.16 “The Range Game”—a calculator game.

73% of our 132 eighth graders watch more than 2 hoursof TV per day.

If I use 120, one-fourth is 30and three-fourths is 90.(That’s probably a little low.I used a bigger percentagebut fewer students.)About 90 to 95.

73% → about 75% or .34

The chances of getting a winning lottery ticket are about 1 in 8. Zeke bought 60 tickets. About how many “winners”is reasonable?

A box of 36 thank-you cards is $6.95. How much is thatper card?

About 7 or 8 winners.

of 60 → of 64 is 8.18

18

$6.95 is close to $7.20. So thesecards cost a little less than 20¢ each.

36 × 2 is 72 → or 36 × 20 is 720.

Figure 11.15 Using compatible numbers in division.

ESTIMATION EXERCISESThe ideas presented so far illustrate some of the types of esti-mation and thought patterns you want to suggest to your stu-dents. But making up examples and putting them into a realisticcontext is not easy and is very time-consuming. Fortunately,good estimation material is now being included in virtually alltextbooks. This exposure, however, will probably not be suffi-cient to provide an ongoing, intensive program for developingestimation skills. Try adding to the text using one or more of themany good teacher resource materials that are available com-mercially. Some of these are listed at the end of the chapter.

The examples presented here are not designed to teachestimation strategies but offer useful formats to provide yourstudents with practice using skills as they are being developed.These will be a good addition to any estimation program.

Calculator ActivitiesThe calculator is not only a good source of estimation activitiesbut also one of the reasons estimation is so important. In the

real world, we frequently hit a wrong key, leave off a zero or adecimal, or simply enter numbers incorrectly. An estimate of theexpected result alerts us to these errors. The calculator as anestimation-teaching tool lets students work independently or inpairs in a challenging, fun way without fear of embarrassment.With a calculator for the overhead projector, some of the activ-ities described here work very well with a full class.

ACTIVITY

11.17 The Range GameThis is an estimation game for any of the fouroperations. First pick a start number and an opera-tion. The start number and operation are stored inthe calculator. Students then take turns entering anumber and pressing to try to make the resultland in the target range. The following example formultiplication illustrates the activity: Suppose that astart number of 17 and a range of 800 to 830 are cho-sen. Press 17 to store 17 as a factor. Press anumber and then . Perhaps you try 25. (Press 25

.) The result is 425. That is about half the target.Try 50. The result is 850—maybe 2 or 3 too high. Try48. The result is 816—in the target range! Figure11.16 gives examples for all four operations. Preparea list of start numbers and target ranges. Let studentsplay in pairs to see who can hit the most targets onthe list (Wheatley & Hersberger, 1986).

=

=

=!

=


“The Range Game” can be played with an overhead cal-culator with the whole class, by an individual, or by two orthree children with calculators who can race one another. Thespeed element is important. The width of the range and thetype of numbers used can all be adjusted to suit the level ofthe class.

ACTIVITY

11.18 Secret SumThis calculator activity uses the memory fea-ture. A target number is selected—for example, 100.Students take turns entering a number and pressingthe key. Each of the numbers is accumulated inthe memory, but the sum is never displayed on thescreen. If one player thinks that the other playerhas made the sum go beyond the target, he or sheannounces “over,” and the (memory return) keyis pressed to check. If a player is able to hit the tar-get exactly, bonus points can be awarded. Interestingstrategies quickly develop.

“Secret Sum” can also be played with the key. Firstenter a total amount in the memory. Each player’s number isfollowed by a press of and is subtracted from the mem-ory. Here the first to announce correctly that the other playerhas made the memory go negative is the winner.

ACTIVITY

11.19 The Range Game:Sequential VersionsSelect a target range as before. Next enter the start-ing number in the calculator, and hand it to the firstplayer. For addition and subtraction, the first playerthen presses either or , followed by a number,and then . The next player begins his or her turnby entering or and an appropriate number,operating on the previous result. If the target is 423to 425, a sequence of turns might go like this:

Start with 119.

350 D 469 (too high)

42 D 427 (a little over)

3 D 424 (success)

For multiplication or division, only one operationis used through the whole game. After the first or sec-ond turn, decimal factors are usually required. Thisvariation provides excellent understanding of multi-plication or division by decimals. A sequence for atarget of 262 to 265 might be like this:

Start with 63.

5 D 315 (too high)

0.7 D 220.5 (too low)=!

=!

=-

=-

=+

-+

=

-+

M–

M–

MRC

M+

1.3 D 286.65 (too high)

0.9 D 257.985 (too low)

1.03 D 265.72455 (very close!)

(What would you press next?)

Try a target of 76 to 80, begin with 495, and useonly division.

TECHNOLOGY NOTEEstimation skills are often embedded in software drill-and-practice packages, although very few are designed forestimation skills alone. Mega Math Blaster (Davidson), PowerRangers ZEO: Power Active Math (MacWarehouse, 1996), andCornerStone Mathematics, Grades 3–8 (SkillsBank, 1996) arethree examples. Power Rangers has six levels for each of itsskills, including one called “estimation,” which proves to beexercises in rounding. The estimation activities in Mega MathBlaster are in an arcade format. This adds the useful elementof speed but may be a turn-off for students who do not carefor the format. CornerStone Math has test materials both inthe software and in paper form that are useful for evaluationand practice in test taking.

The Blue Falls Elementary title in Fizz & Martina’s MathAdventures (Tom Snyder Productions, 1998) includes threeestimation problems in each of the four episodes. (The Fizz& Martina programs are each built around adventure storieswith group problem solving in a realistic context.) Studentsare encouraged to estimate and then explain the strategy theyhave used. At least one strategy is provided. ■

Activities for the Overhead ProjectorThe overhead projector offers several advantages. You canprepare the computational exercises ahead of time. You cancontrol how long a particular computation is viewed by thestudents. There is no need to prepare handouts. Commercialmaterials such as GUESS cards (Figure 11.14, p. 203) can becopied onto transparencies for instructional purposes.

ACTIVITY

11.20 What Was Your Method?Select any single computational estimation problem,and put it on the board or overhead. Allow 10 sec-onds for each class member to get an estimate. Dis-cuss briefly the various estimation techniques thatwere used. As a variation, prepare a problem with anestimation illustrated. For example, 139!43 mightbe estimated as 6000. Ask questions concerning thisestimate: “How do you think that estimate wasarrived at? Was that a good approach? How should it

=!

=!

=!


be adjusted? Why might someone select 150 insteadof 140 as a substitute for 139?” Almost every esti-mate can involve different choices and methods.Alternatives make good discussions, helping stu-dents see different methods and learn that there is nosingle correct estimate.

ACTIVITY

11.21 In the BallparkUsing a page of problems from a workbook, write inanswers to six or seven problems. On one or twoproblems, make a significant error that can be caughtby estimation. For example, write 5408∏26=28(instead of 208) or 36!17=342 (instead of 612).Other answers should be correct. Encourage the classto estimate each problem to find errors “not even inthe ballpark.”

ESTIMATING WITH FRACTIONS,DECIMALS, AND PERCENTSFractions, decimals, and percents are three different notationsfor rational numbers. Many real-world situations that call forcomputational estimation involve the part-to-whole relation-ships of rational numbers. A few examples are suggested here:

SALE! $51.99. Marked one-fourth off. What was theoriginal price?

About 62 percent of the 834 students bought their lunchlast Wednesday. How many bought lunch?

Tickets sold for $1.25. If attendance was 3124, abouthow much was the total gate?

I drove 337 miles on 12.35 gallons of gas. How manymiles per gallon did my car get?

The examples in this chapter have not used fractions, dec-imals, or percents. To estimate with such numbers first involvesan ability to estimate with whole numbers. Beyond this, itinvolves an understanding of fractions and decimals and whatthese two types of numbers mean. Calculations with percentsare always done as fractions or as decimals. The key is to beable to use an appropriate fraction or decimal equivalent.

In the first of the examples just presented, one approach isbased on the realization that if $51.99 (or $52) is the result ofone-fourth off, that means $52 is three-fourths of the total. Soone-fourth is a third of $52, or a little less than $18. Thus about$52+$18=$70, or about $69, seems a fair estimate of theoriginal cost. Notice that the conceptualization of the probleminvolves an understanding of fractions, but the estimation skill

involves only whole numbers. This is also the case for almostall problems involving fractions, decimals, or percents.

From a developmental perspective, it is important to seethat the skills of estimation are separate from the conceptualknowledge of rational numbers. It would be a mistake towork on the difficult processes of estimation using fractionsor percents if concepts of those numbers are poorly devel-oped. In later chapters, where rational number numerationis discussed, it is shown that an ability to estimate can con-tribute to increased flexibility or number sense with fractions,decimals, and percents.

LITERATURE CONNECTIONSInteresting literature often provides a diversionary context inwhich the mathematics may be a bit more fun and possiblymore realistic. Even if realism in these two books is a bitstretched, the context makes a welcome change of pace.

The Twelve Circus Rings(Chwast, 1993)Based on the same pattern as “The Twelve Days of Christmas,”the 12 circus rings each contain more animals or more acts:“six acrobats, five dogs a-barking, four aerialists zooming,three monkeys playing, two elephants, and a daredevil onhigh wire.” The colorful illustrations add to the excitementof the circus and the growing number of performers. After12 days, how many aerialists are zooming? Remember thatthere are four aerialists not only in ring 4 but also in rings5 through 12. The story lends itself to some early estima-tion but can easily be extended to create some fun ques-tions. “About how much do the bows cost for all the dogsa-barking if each bow costs $1.29?” Both you and your stu-dents can make up estimation questions that challenge theskills you’ve been developing. In addition to the potential forboth estimation and mental computation, the author alsoconsiders patterns as another extension.

The 329th Friend(Sharmat, 1979)This book not only offers the opportunity to examine men-tal math strategies and practice in the context of the story butis also about friendship and the need to be liked. The storyis about Emery Raccoon, who has no friends. Because he islonely, Emery invites 328 strangers to lunch to make newfriends. There are considerations about the number of dishes,knives, forks, and spoons that suggest multiplying 329 bysmall numbers and different ways of doing this. The story iseasily expanded into other similar number questions. At theend, Emery discovers that although his guests ignored him,there was one friend that was there all along—himself.


ASSESSMENT NOTESSince estimation involves a certain element of speed, teach-ers often wonder how they can test it so that students are notcomputing on paper and then rounding the answer to looklike an estimate. One method is to prepare a short list ofabout three estimation exercises on a transparency. The cardsin the GUESS boxes (Reys & Reys, 1983) are a ready sourcefor these, or you can simply write bare computations. Stu-dents have their paper ready as you very briefly show oneexercise at a time on the overhead, perhaps for 20 seconds,depending on the task. Students write their estimate imme-diately and indicate if they think their estimate is “low” or“high”—that is, lower or higher than the exact computation.

They are not to do any written computation. Continue untilyou are finished. Then show all the exercises, and have stu-dents write down how they did each estimate. They shouldalso indicate if they think the estimate was a good estimateor not so good and why. By only doing a few estimates buthaving the students reflect on them in this way, you actuallyreceive more information than you would with just theanswers to a longer list.

A theme of these assessment notes is that your studentsgive you a lot of information each day, and all you have todo is record it. Information can be gathered from your dailyestimation activities in much the same way as for inventedstrategy computations, by using a recording sheet for yourobservations (see Chapter 10).

REFLECTIONS ON CHAPTER 11Writing to Learn1. Describe specific examples where number sense plays a direct

role in the development of invented computational strategies,mental computation, and estimation.

2. Why might the idea of an estimate for a computation be a dif-ficult concept?

3. Describe each of the general strategies that were offered for con-ducting estimation activities.

4. Young children often find it difficult to come up with an esti-mate. What are the suggested techniques where students arerequired to estimate but not to actually produce an estimate?

5. Describe each of these estimation strategies. Be able to make upa good example and use it in your explanation.

a. Front-end addition and subtraction.

b. Front-end multiplication.

c. Rounding in addition.

d. Rounding in subtraction.

e. Rounding in multiplication.

f. Compatible numbers in addition and subtraction.

g. Compatible numbers in division.

6. How can the calculator be used to practice estimation strategies?

For Discussion and Exploration1. In the past, the computation curriculum focused only on the

traditional algorithms. There was only one way to compute. Inthe K–8 curriculum today, how much emphasis should be givento each of the forms of computation that have been discussed(invented strategies both with pencil and paper and mental, tra-ditional algorithms and variations, and finally estimation)? Tryto support your argument on the basis of real-world use.

2. Do the curriculum standards for your state require computa-tional estimation? Mental computation? For “straightforwardcomputation,” is there any requirement that a particular algo-rithm or method of computation be used? How does your state’scomputational guidelines influence what computation methodsare taught in school? Do you agree with what is required?

Recommendations for Further ReadingHighly RecommendedBresser, R., & Holtzman, C. (1999). Developing number sense: Grades

3–6. Sausalito, CA: Math Solutions Publications.Bresser and Holtzman are classroom teachers who have compiled13 worthwhile number sense activities covering a range of top-ics including mental computation, number meanings, andestimation. With each activity, the authors provide pages ofextensions, practical suggestions, and answers to questions con-cerning using the activity in the classroom. Most activitiesinclude examples of students’ work.

Reys, R. E., & Nohda, N. (Eds.). (1994). Computational alternativesfor the twenty-first century: Cross-cultural perspectives from Japanand the United States. Reston, VA: National Council of Teachersof Mathematics.This book addresses similarities and differences in computationin the United States and Japan. Divided into three sections, thearticles discuss mental computation, estimation, and the use ofcalculators in the two countries. Respected researchers from bothcountries have contributed to this work. Although not designedas a resource book, the thoughtful teacher will find practicalideas as well as interesting perspectives.

Reys, R. E., & Reys, B. J. (1983). Guide to using estimation skills andstrategies (GUESS) (Boxes I and II). White Plains, NY: Cuisenaire–Dale Seymour.Still among the best materials for teaching computational esti-mation strategies, these two hefty boxes of cards provide the


Reys, R. E. (1998). Computation versus number sense. Mathemat-ics Teaching in the Middle School, 4, 110–112.

Reys, R. E., Trafton, P. R., Reys, B. J., & Zawojewski, J. S. (1987).Computational estimation (grades 6, 7, 8). White Plains, NY: Cuise-naire–Dale Seymour.

Reys, R. E., & Yang, D. (1998). Relationship between computationalperformance and number sense among sixth- and eighth-gradestudents in Taiwan. Journal for Research in Mathematics Education,29, 225–237.

Sowder, J. T., & Kelin, J. (1993). Number sense and related topics.In D. T. Owens (Ed.), Research ideas for the classroom: Middlegrades mathematics (pp. 41–57). Old Tappan, NJ: Macmillan.

Sowder, J. T., & Wheeler, M. M. (1989). The development of con-cepts and strategies used in computational estimation. Journal forResearch in Mathematics Education, 20, 130–146.

Thornton, C. A., Jones, G. A., & Neal, J. L. (1995). The 100s chart:A stepping stone to mental mathematics. Teaching Children Math-ematics, 1, 480–483.

Trafton, P. R. (1986). Teaching computational estimation: Estab-lishing an estimation mind-set. In H. L. Schoen (Ed.), Estimationand mental computation (pp. 16–30). Reston, VA: National Coun-cil of Teachers of Mathematics.

Van de Walle, J. A., & Watkins, K. B. (1993). Early development ofnumber sense. In R. J. Jensen (Ed.), Research ideas for the class-room: Early childhood mathematics (pp. 127–150). Old Tappan,NJ: Macmillan.

teacher of grades 4–8 with a complete program. Two of the cardsare shown in Figure 11.14. Use them as transparencies, in smallgroups, or as independent work.

Other SuggestionsAuriemma, S. H. (1999). How huge is a hundred? Teaching Children

Mathematics, 6, 154–159.Hunting, R. P. (1999). Rational-number learning in the early years:

What is possible? In J. V. Copley (Ed.), Mathematics in the earlyyears (pp. 80–87). Reston, VA: National Council of Teachers ofMathematics.

Leutzinger, L. P., Rathmell, E. C., & Urbatsch, T. D. (1986). Devel-oping estimation skills in the primary grades. In H. L. Schoen(Ed.), Estimation and mental computation (pp. 82–92). Reston, VA:National Council of Teachers of Mathematics.

McIntosh, A., Reys, B. J., & Reys, R. E. (1997). Number SENSE: Sim-ple effective number sense experiences, grades 3–4. White Plains,NY: Cuisenaire–Dale Seymour.



Reys, B. J. (1986). Teaching computational estimation: Conceptsand strategies. In H. L. Schoen (Ed.), Estimation and mental com-putation (pp. 31–44). Reston, VA: National Council of Teachersof Mathematics.

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