TEACHING MATHEMATICS WITH TECHNOLOGY: Teaching Mathematics Using Calculators

TEACHING MATHEMATICS WITH TECHNOLOGY: Teaching Mathematics Using CalculatorsAuthor(s): David L. Pagni and George W. BrightSource: The Arithmetic Teacher, Vol. 38, No. 5 (JANUARY 1991), pp. 58-60Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41194756 .

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TEACHING MATHEMATICS WITH TECHNOLOGY

^^^^^^^™ Teaching Mathematics Using Calculators

last three years has seen a marvelous resur- gence of interest in the use of calculators for

teaching mathematics. Much of the credit goes to professional organizations like the National Coun- cil of Teachers of Mathematics that have promoted the use of calculators. The renewed interest in the use of calculators in schools coupled with the sale of over 250 million electronic hand-held calcula- tors in the last ten years in the United States (about three for each household) suggests that an oppor- tunity exists for a revolution in mathematics educa- tion.

When I began conducting calculator workshops three years ago, few of the elementary school teachers who participated used calculators with their students. This situation has changed dramati- cally each year since then, as higher percentages of the participants say they are using calculators as part of their mathematics program. However, the same participants ask for model lessons of how to integrate calculators into the mathematics curricu- lum. Many textbooks pose problems that suggest the use of a calculator, but teachers seem to want more. In addition to using the calculator to perform arithmetic operations, teachers want to know how to teach mathematical concepts and problem-solv- ing techniques with a calculator.

Take, for example, the so-called "constant fea- ture" of the four-function calculator. For addition it works like this:

Keystrokes Display

The logic of the calculator allows it to "remember" the "add three." In mathematics we call this exam- ple of remembering the operator +3. As long as

no other keys are pushed, if we enter any other number and push '=', the calculator will add three to the number.

Keystrokes Display

This capability, called the constant feature of add- ing three, is equivalent to having a "function ma- chine" that always adds three to the entered num- ber (see fig. 1).

If we do not enter a number but just push 0/ the result is to add three to the number shown on the calculator display.

Keystrokes Display

Consider the power of this feature to help stu- dents understand the concept of multiples of a number. If we clear the calculator, thus beginning with zero on the display, then push '±' Гз1Т=1 0 0E]we get a one-to-one correspondence between the number of times we press '=' and the number of threes we have added (see fig. 2). Eventually we can make the connection to multipli-

Prepared by David L. Pagni, California State University, Fullerton, Fullerton, С А 92634 Edited by George W. Bright, University of North Carolina at Greensboro, Greensboro, NC 27412-5001

53 ARITHMETIC TEACHER

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^^^^^^QQH Keystrokes Display Addition Multiplication

ION/CI 0 0 0x3

[+] Щ] Ξ зо + з ΐχ3 '=' 6 0 + 3 + 3 2x3

|3 9 0 + 3 + 3 + 3 3x3

0 12 0 + 3 + 3 + 3 + 3 4x3

^^^^^^^Я Number of =s Equivalent

Keystrokes Display pressed power

0 4 2 22 В 8 3 23 В 16 4 24 0 32 5 25

cation. Students who use this feature to explore multiples of various numbers, including common multiples of pairs of numbers, gain an intuitive grasp of products long before that concept is intro- duced.

A bit of investigation reveals that constant fea- tures are available for subtraction, multiplication, and division. The use of the constant subtraction feature leads to an intuitive concept of division as repeated subtraction:

Number of 19s

Keystrokes Display subtracted

В 47 2 0 28 3 0 9 4

The use of the constant multiplication feature leads to the concept of repeatedly multiplying by a number, later formalized as raising to a power (see fig. 3). Notice that the constant operator for multiplication is the first number keyed in followed by the multiplication symbol, in this instance, 2x. Also notice the one-to-one correspondence be- tween the number of times the equals key is pushed and the power to which the constant oper- ator is raised.

Worksheet 1

Name

Find the missing numbers by using the calculator to find mul- tiples of the first number.

1. 2, 4, 6, 10 14 2. 7,14 ,28, ,42 , ,11 3. 12, 24 , 60 96 132 ,

,168 4. 15, , ,120 э. ι, , / , / , / 6. 78, , , ,

Use multiples to solve these problems. 7. Sam and his brother are both between the ages of 35 and

50. Sam is older than his brother. Both their ages are multiples of 8. How old is Sam? How old is his brother?

8. Jeff is between 1 1 5 and 1 20 centimeters tall. His height is a multiple of 9. How tall is Jeff?

9. Misha has between 50 and 65 books in her room. The number of books is a multiple of both 4 and 5. How many books does Misha have?

10. Julie and Stan collected between 30 and 60 pounds of aluminum cans for the school's can drive. The number of pounds of cans is a multiple of 4, 6, and 8. How many pounds of cans did they collect?

From the Arithmetic Teacher, January 1991

^^^^^^^M Number of times divided Equivalent

Keystrokes Display by 2 power

H 32 2 22 В 16 3 23 В 8 4 24 0 4 5 25

The value to elementary-level students of the constant feature for division may not be as obvi- ous. One advantage is the time saved when divid- ing a collection of numbers by the same number, for instance, when testing for divisibility by a cer- tain number.

Continued on next page

JANUARY 1991 39

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Worksheet 2 Continued from previous page

Keystrokes Display

Repeated division by 2 can also be equated to raising to a power, in this instance, the power of the divisor rather than the power of the multiplier, as in the example of multiplication (see fig. 4). Thus, dividing 128 by 2^_or 32, is equivalent to

five times). Using the constant feature to investigate mathe-

matical concepts is one way that the calculator can be used to learn mathematics rather than simply to do mathematical computations. The calculator be- comes a manipulative in the sense that the student can manipulate numbers so as to formulate mathe- matical connections. A foregoing example con- nected the idea of multiples with repeated addition of a constant and eventually with multiplication. Another example made the connection between repeated multiplication by a constant and raising to a power (the use of exponents). Students thus gain mathematical power by adding this instrument to their arsenal for exploring and understanding numbers and systems of numbers.

The problems for grades 3-4 in worksheet 1 use the constant feature for addition to explore

Name

1 . Use your calculator to find the following values. Begin with

(σ) 36 (Ò) 37

(с) 73 (c/)1512 (e)353 (f) 854

2. List the calculator keys you would press to get the follow- ing values. Begin with [c]. (a) 25 (Ò) 52 (c) 84 (d) 4s

3. List the calculator keys you would press to get the follow- ing results. Begin with 'C'. Use the calculator's constant function.

(σ) 729 (b) 15 625 4. Which number is larger - 67 or 76? 5. If you were lucky enough to triple your money each time

you bet, how many times would you have to bet before you would have 177 147 times the amount you started with?

From the Arithmetic Teacher, January 1991

multiples. In worksheet 2, the problems for grades 5-6 explore exponents (raising to a power) through the use of the constant feature for multiplication. ·

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The Math Department seeks applications for two tenure track positions at the

assist, or assoc. professor rank for Aug. 1991. Applicants should have a Ph.D. or Ed.D. by Aug. 1991 in Math Education with a research interest in elementary

education; the ability to teach a range of courses including content and methods courses; an active research program; a commitment to teacher education and

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ARITHMETIC TEACHER

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