Number Sense in the Elementary School Mathematics Classroom

100

Number Sense in the Elementary SchoolMathematics ClassroomJohn W. McBrideCharles E. Lamb

Perhaps the greatest challenge to an ele-mentary mathematics teacher is moti-vating students to learn mathematics.Even though a teacher may have a verysound program, poorly motivated stu-dents frequently learn very little. All toooften, this results in a few successexperiences which, in turn, produces aless-than-positive attitude towardmathematics. Poor attitude underminesmotivation and the cycle continues. Thiscycle can be broken by engaging stu-dents in highly-motivational mathe-matics activities which result in imme-

diate success and recognition. One such activity is a classroom number sense pro-gram.

Background

A formal number sense program has been in existence in the state of Texas formany years now. It is administered by The University Interscholastic Leaguethrough the Continuing Education Division of The University of Texas atAustin. The league also organizes and governs competitive programs in areassuch as athletics, music, and forensics. In the secondary schools, competitionsare held at the district, regional and state levels. In the elementary schools, com-petition is concluded with district-level contests.Number sense competitions consist of paper/pencil tests that are administered

simultaneously to all students. Tests are provided by the UIL and consist of ap-proximately 70 to 80 items. Students are required to compute all problems men-tally and write only the answers on their papers. No erasing is permitted and"write overs" are counted wrong. Students receive five points for each correctanswer and lose four points for each incorrect answer or unanswered problem up

School Science and MathematicsVolume 86 (2) February 1986

Number Sense 101

to the last problem attempted. Problems on the test are representative of theschool mathematics curriculum for the age-level of students being tested and areorganized from simplest to most difficult.The number sense program in the State of Texas was developed to help mo-

tivate students to learn mathematics and to literally develop "number sense."Students develop number sense and become capable of computing rapidly asthey begin to grasp the structure, properties and operations of mathematics.These characteristics of mathematics can be taught to students by helping themlearn and understand numerous "shortcuts" which they can apply to a numbersense test. Many of these shortcuts are easy to learn and can be taught to studentsin a matter of minutes, thus enabling them to experience immediate success andrecognition. This can be highly motivating to students. For example, elementarystudents can be taught to multiply a whole number by 10 or a power of 10 (100,1,000, etc.) by annexing to the right of the number as many "O’s" as there are in10 or the power of 10 used as the multiplier. Teachers can develop number senseshort cuts from their curriculum or select them from resource books on numbersense (Lamb and Hutcherson, 1982; Skow, 1981).

A Classroom Number Sense Program

Teachers can easily organize a number sense program in their own classroom. Agood way to begin is to select a number shortcut relevant to the curriculum andteach it as a "trick" or "shortcut" to computation. It is imperative that theteacher be enthusiastic and that all students (within reason) experience successwith the shortcut and "have fun" with it. A good strategy is to let the students"push you" into showing them other shortcuts.After students have learned several shortcuts, introduce the idea of the compe-

titions. Show them a sample number sense test (Figure I) and let them try it. Ad-minister the test as you would for competitions. Allow students to score theirown tests and compute their own scores. Go over the test w^ith them and helpthem recognize problems that could have been solved w^ith the shortcuts theyknow. Display the test on a bulletin board and use it to lead students into newshortcuts. Challenge students to devise new short cuts and let them teach theseshortcuts to the class. For students with a fear of tests, it might be advisable totry a short or partial test at first.

Write your number sense tests from the curriculum you are covering in yourclassroom. Then, as you teach computational shortcuts, you can relate the short-cuts back to your mathematics curriculum and help students gain greater insightsand understandings. Challenge students to write test items and use them whenpractical.

Organize a regular schedule for teaching shortcuts and administering tests.Some teachers have found success in spending the first five or ten minutes of thedaily mathematics period teaching and practicing the shortcuts and the last


102 Numher Sense

NUMBER SENSEE-131

(1) 67 + 96 = ___

(2) 41 - 13 = ___

(3) 24 x 24 = ___(4) 42 + 16 + 17 =

(5) 440 - 40 == __(6) 2/2 x 30 = ___

(7) 3 X 9 + 3 X 13 = ____.

(8) 217 - 18 - 99 =____.(9) 1% - I2/ =____.

*(10) 27 x 63 == ____.(11)6 Multiplied by what number gives 90? ____.(12) 15%of40is____.(13) 3 meters = ____centimeters.(14) The next number in the sequence 1,2,1,2,1,2, . . .is.(15) What number subtracted from 44 gives 28? ____.(16) Which is smaller ’/or4/,?____.(17) 283 + 412 =____.(18) The area of a 3" x 4 "rectangle is____sq. in.(19) Vi of 6 feet 6 inches =____inches.

*(20) 20 + 21 + 22 + 23 + 24 + 25 = ____.

(21) The average of 24, 30,0 is____.(22) 43 = ____.(23) Write XLVIII in Arabic Numbers. ____.(24) Change 1 liwo, to base 10. ____ base 10.(25) Change 4 ion, to base two. ____base two.(26) 40 is ____% of 80.(27) The sum of two primes is 7. The smaller is ____.(28) The sum of two primes is 7. The larger is ____.(29) The sum of two primes is 7. Their product is ____.

*(30) 27 x 45 - 15 = ____.

(31) IfX + 23 = 40, X = ____.

(32) /,+/,+/,= ____.

(33) 3n = 126, n = ____.

(34) 232 = ____.

(35) -13+13 =____.

(36) (8 x 15) +(9x11) =____.

(37) The area of a square is 196 inches. Each side is ____ in.(38) 27 x 33 = ____.

(39) 22 x 44 = ____.

*(40) 1471 +6231 + 729 + 63 + 1000 = ____.

(41) 243 - 9 = ____.

(42) 1827 - 9 has a remainder of____.(43) % - % = ____.(44) 52 + 72 = ____.

(45) 62 + 82 + 102 - 42 = ____.

(46) 75% =____fraction.

FIGURE I


Number Sense 103

FIGURE I�Continued

(47) A cube has an edge 5 in. The volume is ____ cu. in.(48) Using 1983, write the maximum four-digit number. ____.(49) Using 1983, write the minimum four-digit number. ____.

*(50) (11 x 19)2 = ____.

(51) The reciprocal of % is____.(52) The negative reciprocal of - % is ____.(53) 17 x 23 ==____.

(54) 2/; x VA == ____.(55) 9 pounds =____ounces.(56) Express % as a decimal. ____.(57) The median of 3, 4, 5, is ____.(58) .35 = ____%.(59) The greatest common factor of 3 and 4 is ____.

*(60) 210 = ____.

(61) $2.86 + $.93 + $1.42 = $____.(62) At 150 pesos per dollar, 675 pesos = $___.(63) The largest prime divisor of 63 is ____.(64) Pencils are 3 for 50C, a dozen costs $____.(65) 7x7x7x7= ____.

(66) A man drove 600 miles at 15 miles per gallon. He used ____ gallons.(67) A coat is to sell for $10. Sales tax is 5^0. Total price is $____.(68) 534 - 6 = ____.

(69) February 1988 will have____days.*(70) 168 hours =____minutes.

^Success and recognition can be highly motivating and cancontribute to a positive attitude toward mathematics.^

20 minutes of the period on Fridays administering practice tests. Tests for com-petition are given on a monthly basis.Some schools have found success in holding competitions across the inter-

mediate (fourth, fifth, and sixth) grades. Classroom teachers coach their stu-dents and then send them to the all-school competitions. They invite the numbersense champions from each elementary school to compete in a district-sponsorednumber sense contest. A teacher from each elementary school is usually ap-pointed to coach the school’s team and take the team to the district meet.

Number Sense Shortcuts

Teachers wishing to begin a number sense program in their classroom or schoolshould compile a resource file or shortcuts that parallel their mathematics curric-ulum. Then, as they teach each area of the curriculum, they can introduce short-


104 Number Sense

FIGURE I�Continued

NUMBER SENSEANSWER KEY

E-131

(1) 163(2) 28(3) 576(4) 75(5) 11(6) 75(7) 66(8) 100(9) 4/,

*(10) 1615.95 - 1786.05(11) 15(12) 6(13) 300(14) 1(15) 16(16) 4/,(17) 695(18) 12(19) 39

*(20) 128.25 - 141.75(21) 18(22) 64(23) 48(24) 3(25) 100(26) 50(27) 2(28) 5(29) 10

*(30) 1140 - 1260(31) 17(32) %orl!^(33) 42(34) 529(35) 0

(36) 219(37) 14(38) 891(39) 968

*(40) 9019.3 - 9968.7(41) 27(42) 0(43) %(44) 74(45) 184(46) %(47) 125(48) 9831(49) 1389

*(50) 41496.95 - 45865.05(51) %(52) 4(53) 391(54) 6/.or 1.25 or25/(55) 144(56) .625(57) 4(58) 35(59) 1

*(60) 972.8 - 1075.2(61) $5.21(62) $4.50

(63) 7(64) $2.00

(65) 2401(66) 40(67) $10.50

(68) 89(69) 29

*(70) 9576 - 10584

cuts to motivate students to learn the content being presented. The shortcuts canalso be used to improve students’ understanding of the mathematics content. Forexample, a teacher can enhance students’ understanding of place value by teach-ing the following shortcut for addition, called "front end addition," and stimu-lating students to figure out why it works:

56 + 32 = ?1. Add the tens: 50 + 30 = 802. Add the ones: 6+2=83. Add the two sums: 80 + 8 = 88


Number Sense 105

Following are a few examples of the kinds of shortcuts elementary teachers canconstruct themselves or draw from selected references.Adding a single-digit number to a double-digit number when carrying is re-quired: Higher decade addition is a shortcut which can be used to improve stu-dents’ understanding of place value and the meaning of addition. It consists oftwo steps:

1. Sum the digits in the ones place of the two numbers. Write down the ones-placedigit in this sum. It will be the ones-place digit in the answer.

2. Add 1 to the tens-place digit in the double-digit number. Write it down as the tens-place digit in the answer.

The following example illustrates the shortcut:47 + 6 = ?1. 7 + 6 = 13. Write down the 3. It will be in the ones place of the answer.2. 4 + 1 = 5. Writedown the 5 in the tens place of the answer. Answer: 53.

Students learn to rapidly derive the answer mentally by thinking: 7 + 6 = 13, somy answer will be in the next decade (50’s) and end in 3, therefore my answer is53."Adding two double-digit numbers using partial addition is a shortcut which canbe used to improve students’ understanding of place value and addition. Themethod is as follows:

1. Add the first number and the tens-place digit of the second number.2. Add the ones-place digit of the second number to the sum.

The following example illustrates the method:27 + 48 = ?1. Think of 48 as 40 + 8 and add 27 to the 40: 27 + 40 = 67.2. Add the 8 to the 67 using higher decade addition: 67 + 8 = 75.

Students learn to rapidly derive the answer mentally by thinking:"27 + 40 = 67 + 8 = 75."

Adding three or more two-digit numbers using partial addition is an extension ofthe last shortcut and can be used to strengthen students’ understanding of placevalue. It consists of the two following steps:

1. Add the first two numbers using partial addition.2. Add the third number to the sum using partial addition. Continue adding in this

manner.

This method is illustrated by the following example:25 + 32 + 47 + 18 = ?1. Add 25 and 32 using partial addition: 25 + 30 = 55 and 55 + 2 = 57.2. Add the third number, 47, to the sum 57. Use partial addition: 57 + 40 = 97 and

97 + 7 = 104. Continuing on with partial addition, 104 + 10 = 114 and114 + 8 = 122.


106 Number Sense

Answer: 122

Students learn to mentally apply this short-cut by thinking as follows: "25 + 30= 55 and 2 is 57; 57 + 40 = 97 and 7 is 104; 104 +10=114 and 8 is 122."

^Many of these shortcuts are easy to learn and can be taughtto students in a matter of minutes, thus enabling them to ex-perience immediate success and recognition/’

Multiplying a number by 25 is a shortcut which can be used to improve students’understanding of the meaning behind multiplication and division as well as themeaning of remainders in division. It can also be used to help students grasp themeaning of the distributive property of multiplication over addition. This short-cut is done in two steps:

1. Divide the number by 4.2. Multiply by 100.

This method works because 25 equals 100/4. So, instead of multiplying by 25,you can multiply by 100 and then divide by four; or you could divide by 4 first,then multiply by 100. The latter is the best method for students to use if they arecomputing mentally without use of a pencil. The following example illustratesthis shortcut:

25 x 43 = ?1. Divide 43 by 4 and think of the answer as a mixed number: 43/4 = 103/4

Notice that when you divide a number by 4, your answer will either come outeven or have a remainder of 1, 2, or 3. If you think of division as repeated sub-traction, then dividing a number by four means * ’How many sets of 4 can be sub-tracted out of the number?" A remainder means that you have part of a set of 4.left over. For example, 43/4 ==� 10 3/4 means that you can subtract 10 sets of 4out of 43 and have 3 out of a set of 4, or 3/4, of a set of 4 left over.

2. Multiply by 100; 100 x 10 3/4 = 1075

Notice that 100 times 10 % means 100 times 10 plus 100 times 3/4 (distributiveproperty). Since your remainder will always be 0, 1, 2, or 3 when you divide by 4,you can have students multiply 1/4, 2/4, and 3/4 by 100 and learn that a re-mainder of 1/4 adds 25 to their answer, 2/4 adds 50 and 3/4 adds 75. Studentslearn to rapidly derive the answer to this multiplication problem by think-ing: "43 x 25 = 43/4 = 10 3/4; and 10 3/4 x 100 = 1000 + 75 which is 1075.Multiplying a single-digit and double-digit number using the distributive prop-erty is a shortcut which is particularly useful in helping students understand the


Number Sense 107

distributive property of multiplication over addition. It is illustrated as follows:1. Think of the double-digit number in expanded notation (e.g., 34 as 30 + 4)...2. Multiply the tens and ones in the double-digit number by the single-digit number

and sum the answer.The following example illustrates these two steps:7 x 34 = ?1. Think of the double-digit number, 34, in expanded notation: 30+42. Multiply the 10’s and 1’s in the double-digit number by the single-digit number

and sum the answer: (7 x 30) + (7 x 4) = 210 + 28 = 238

Students learn to mentally solve this problem by thinking: "7 x 34 = 7 x 30;which is 210 and 7x4 which is 28; 210 and 28 is 238.

Conclusion

A number sense program can be a valuable tool to an elementary teacher. Manynumber sense shortcuts can be readily taught to students and can provide themwith a way to obtain immediate success and recognition in mathematics. Successand recognition can be highly motivating and can contribute to a positive atti-tude toward mathematics. Number sense shortcuts can help students gain abetter understanding of mathematics concepts and thus truly develop "numbersense."

References

1. Lamb, C. E. and Hutcherson, L. R. Developing Number Sense. Austin, Texas: TheUniversity of Texas, 1982.

2. Skow, D. P. No Sense in Mathematics. Edinburg, Texas: D&R Enterprises, Inc., 1981.

John W, McBrideCharlesE. LambPan American UniversityThe University of Texas at AustinEdinburg, Texas78539Austin, Texas 78712-1294

MASS VS WEIGHTMass is the physical substance of a body. A body’s mass is the same every-

where. Weight is a product of mass times the force of gravity or acceleration.For example, a 61-Vi kilogram (150-pound) person on Earth would weigh about11 kilograms (25 pounds) on the Moon, where gravity is only about one-sixth ofEarth’s gravity, and virtually nothing in the orbiting Space Shuttle.


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Number Sense in the Elementary School Mathematics Classroom