THE CALCULATOR, MATH MAGIC, AND ALGEBRA

THE CALCULATOR, MATH MAGIC, AND ALGEBRAAuthor(s): James P. Herrmann and Brian GarmanSource: The Mathematics Teacher, Vol. 77, No. 6 (September 1984), pp. 448-450Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27964129 .

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THE CALCULATOR, MATH MAGIC, AND ALGEBRA

I have found that the calculator can be used both as a motivator and as a dynamic aid in developing insight into many common algebraic concepts, laws, and skills. I list specific benefits near the end of this article.

To get started, each student needs a cal culator that has an "exponential and nth root" button. Most students know how to use the buttons for the four basic oper ations, but few know how to use the many other buttons on the calculator. Conse

quently, a day or so should be devoted to

ways of using the memory to store partial results, the parentheses buttons to do multi

ple operations, and the exponential-and ttth-root key to do exponents and radicals.

With this done, the students are ready for a

bit of "math magic/' I describe a sequence of operations that

students can carry out on their calculators. For example, I might begin with the follow

ing math-magic trick :

1. Select any number.

2. Double it.

3. Add 4 to this product. 4. Multiply this sum by 5.


7. Subtract 320 from this product. 8. Divide this difference by 100.

9. The quotient should be your original number.

The slight smile that often appears when students "magically" get the original number is always exhilarating to see.

Next, I repeat the same instructions, but this time students select a letter instead of a number and write the response to the in structions using good mathematical nota tion. I remind them to use parentheses wisely. What results is an algebraic ex

pression not unlike those they have sim

plified in the past or will simplify in the near future. For the previous problem, a

typical response is

[(2x + 4)5 + 12] ? 10 - 320

100 "x'

Finally, I challenge them to discover the mathematical process and logic behind this

apparently magical result by simplifying the left side of the algebraic identity that

they have just created. The next example illustrates how the

exponential-and-ttth-root button can facili tate the solution of some powerful problems that will make the normally routine alge braic skills come alive.

The instructions are as follows :

1. Select a number.

2. Square it.

3. From this result subtract the original number.

4. Add 1 to this difference.

5. Multiply this sum by 1 more than the

original number.

6. Subtract 1 from this product. 7. Extract the cube root of this difference.

8. This number should be your original number.

"Sharing Teaching Ideas "

offers practical tips on the teaching of topics related to the secondary school cur

riculum. We hope to include classroom-tested approaches that offer new slants on familiar subjects for the

beginning and the experienced teacher. See the masthead page for details on submitting manuscripts for review.

448 Mathematics Teacher



Again each student follows the instruc tions using the calculator and then using pencil, paper, and a variable. Lastly, each student is invited to discover what is hap pening by cleverly manipulating the crea

ted algebraic expression. For this example, a likely response is

^(x2-x + l)(x + l)-l = x.

A final example has a slightly different twist. Have your students follow these in

structions :

1. Enter the number for your birth month.

2. Multiply it by 10.5.


5. From this product subtract the product of 5 and your birth month.

6. Add 165 to this difference.

7. Add your age to this sum.

At this point have one of your students

give you his or her calculator. Subtract 365 from the number shown in the screen and announce your student's birth month and

age. You know what these numbers are, since the number represented by the tens and units digit is the person's age and the

remaining digits (01, 02, 03, ..., 12) repre sent the birth month. After repeating this

procedure with several other students, you can share the impressive results by direc

ting the others to subtract 365 and watch

ing their eyes light up. Another variation on this theme that

many students enjoy is having them con

jure up their own set of "magic" instruc tions. These, in turn, can be given to the entire class.

I find that using the calculator in this

way benefits my students in many ways. Let me list several of these advantages :

They must know, use, and pay heed to the order of operations.

They must pay close attention to in

The preparation of this manuscript was supported in

part by the University of Tampa Faculty Development Grant.

structions, which improves their listening skills.

They hear mathematical language verbalized?such important, recurring words and phrases as sum, difference, product, quotient, factor, term, and extract the cube root.

They learn to translate verbal math ematical phrases into written phrases. A similar skill is needed to translate the writ ten language of many word problems.

They get first-hand experience in how a simple set of verbal instructions often leads to a complicated algebraic expression. This experience frequently helps relieve the initial anxiety they sometimes feel when en

countering complex expressions.

They have a chance to use their alge braic skills, often in many unexpected and clever ways.

They are more aware of the meaning of a written algebraic expression.

They are challenged to discover why an algorithm works, which may kindle the fire of excitement to pursue other math ematics.

They have fun.

Listed next are several algebraic identi ties that my students and I have developed. Each can be used to write a set of instruc tions for a new math-magic trick. Better

yet, create your own! (For simplicity, we assume is positive in each of the following identities.)

2. y/x2 + 2x + 1 - = 1.

3 (s/x-x)(y/x + *) + = L

3 ? 1 4. For 1,-

- 2 - + 9 = 10. ? 1

5. ( 3 + Sx2 + 3x + 1)1/3 -l

= x.

6. (x -

3)(x + 4) - 2 + 12 = .

7. (*/x - 3x2/3 + Sx113 - 1 + I)3

= -

9. (( 4)5)0 05 = .

September 1984 449



V 2 - + /

/ 4-1

11. For 1,V * ~

\ - 1 = .

+ 1

12. ypK3 + 2 -h + 1)(

- 1) + 1 = .

13. y [( + 1)( + 2)( - 1)( - 2) + 5

2 - 4]

2 = .

14. ( "1-h )"1-(l-f ) = .

15. (( "1

+ 1 1 - )-1?(2 + 1)-1 = .

Brian Garman

University of Tampa Tampa, FL 33606

RING-AROUND-A-TRAPEZOID A standard question in the geometry cur

riculum is to ask students to solve

questions like the following :

a) Determine the area of the trapezoid in

figure 1.

Fig. 1. Find the area of this trapezoid.

Fig. 2. Find the area of the shaded portion between the concentric circles.

6) Determine the area of the shaded por tion formed by the two concentric circles in figure 2.

With a little imagination, we transform

figure 1 into figure 2, as shown in figure 3. Let's begin our investigation by assuming that the area of the trapezoid in figure 1 is

where a is the length of the upper base, b is the length of the lower base, and h is the altitude. The area of the shaded portion of the two concentric circles in figure 2 is

(2) A = n(R2 -

r2),

where R is the length of the radius of the

large circle and r is the length of the radius of the small circle.

Looking at the transformation in figure 3, we suspect that h = R ? r. The circumfer ence of the large circle (C = 2nR) becomes b = 2nR, and the circumference of the small circle (c = 2 ) becomes a = 2nr. Substitut

ing these values in formula (1) yields

. h(a + 6) A =

A =

2 '

(R - r)(2nr + 2nR) 2

or A = n(R2

- r2).

Students can verify this transformation

by forming two concentric circles with

pieces of string, then "undoing" the string to form a trapezoid, making sure the differ ence in the radii becomes the altitude. "Before" and "after" areas are calculated

and then compared. Isosceles trapezoide

450 Mathematics Teacher



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THE CALCULATOR, MATH MAGIC, AND ALGEBRA