No answer booksAuthor(s): Wendell HallSource: The Mathematics Teacher, Vol. 75, No. 1 (January 1982), pp. 13, 15Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27962736 .

Accessed: 12/09/2014 17:18

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to The Mathematics Teacher.

http://www.jstor.org

This content downloaded from 74.64.97.109 on Fri, 12 Sep 2014 17:18:43 PMAll use subject to JSTOR Terms and Conditions

Logarithms as areas Andrew Linn (April 1981) derived the identity

that

In ab = In a + In b,

where the In u is defined in terms of the area under

the curve y = 1/jc. The same result can be obtained

by using a linear transformation that preserves area and maps the curve y

= 1/jc onto itself.

Let be the linear transformation represented by the matrix

where is a nonzero real number. Then each point (a, b) maps into the point (ka, b/k),

LO Mk\ Vb\ V\lkb\ Let (a, Ma) be a point on the curve y

= l/x. Then the image of (a, Ma) is the point (ka, Mka), which is also on the curve y

= l/x. Thus, the curve y = Mx

maps onto itself under the transformation . Also since the determinant of Tis 1, area is preserved.

Assuming a, b > 1, the areas of the shaded re

gions in figureis 1 and 2 are In a and In b, respec

Fig. 1

Fig. 2

tively. Then the region in figure 2 maps onto the region in figure 3 under the transformation

where

[Ma 0

L 0 Ma

Ma 0 ] LO a?

} [i] -

] and

il Kl -

[ti Since preserves area, the area of the shaded

region in figure 3 is also In b. Thus, In ab - In a +

In b (fig. 4).

Fig. 4

I have used this transformation approach with

geometry, second-year algebra, and calculus stu dents in high school and have found it very suc

cessful.

John C. Huber Pan American University Edinburg, TX 78539

No answer books It was quiet after school. The young man paused

outside the room, glanced inside, smiled, and

started in. His hair was neat. His lips pushed through a medium beard and full, bushy mous tache.

"Hi, Mr. Hall! I'm Eddie G-, thought I'd stop in!" (Now I remembered through the growth, hair, and years: Eddie?back row, often absent,

usually sleepy, friendly, a close 'pass.') "Mr. Hall, just wanted to let you know I'm do

ing OK. I'm a department manager. Went to night school and took accounting. I've got a raise and now maybe a promotion soon. That 'making change' test you gave each of us really paid off. Really helped that you had us do those record keeping problems." (They had repeatedly kept as signment records on monthly income/expense/bal ance, inventories, stock cost/value/gain, and similar

work.) "But I finally figured out why math is so hard outside school: they just don't have answer books out there!"

Right! We teachers read answers, confirm students' so

lutions on the chalkboard, grade tests, and select

January 1982 13

This content downloaded from 74.64.97.109 on Fri, 12 Sep 2014 17:18:43 PMAll use subject to JSTOR Terms and Conditions

texts that have answers to odd exercises. We really want the students to know if their answers are cor rect!

But there are really no answer books outside the classroom. What can we do to give our trainees a more realistic situation? I propose the following techniques, but others can be added: (1) check all solutions to equations; (2) verify "hard solutions"

with a calculator; (3) solve arithmetic problems in two different ways; (4) make a pattern, diagram, or

model; (5) write work in a neat, orderly way; and

(6) have students work in teams. Each member does the same problem. Then answers are com

pared. All members rework any problem on which there is disagreement.

Now, it is essential that such activities be truly valued by the teacher. This must be reflected by comments, the length of assignments, the evalua tion system, and the teacher's own solution-check

methods. Good luck to Eddies and Edwinas everywhere!

Let's give them the skills and the confidence to make their own answer books.

Wendell Hall North Eugene High School Eugene, OR 97404

Subtraction skills I have developed an exercise for strengthening

subtraction skills that require borrowing. Begin by reversing the digits in any nonpalindro

mic number between 10 and 100. Subtract the smaller number from the larger one. Continue this

process with the difference until the answer is 9. For example, if we begin with 63, reverse it to ob tain 36; then 63

- 36 = 27, 72 -

27 = 45, and 54 -

45 = 9. It turns out that, at most, five subtractions are needed to reduce any nonpalindromic number between 10 and 100 to 9. Students can categorize the numbers in this range into five classes.

Henry Lulli John Fremont High School Los Angeles, CA 90003

Students construct tests After ten years of teaching, I think I have finally

found the definitive way of preparing students for mathematics exams, and I have entitled it "make

up." Before each exam, my students prepare an exam

that they believe I could have constructed. The en

tire exam, with point values for each problem, is

neatly prepared on one paper, and all the solutions are worked out on separate scrap paper. The entire

package is presented to me on the day of my exam.

While the students are taking the exams presented to them, I am analyzing the exams given to me.

To help students prepare their exams, I suggest that they ask themselves the following questions:

1. What were the most important concepts ex

pressed in the material?

2. How are the topics related? Can one question test more than one concept?

3. Can you think of any "twists" or "trick prob lems" to include?

As an additional incentive, a bonus of ten points is added to the grade of the student whose exam on the same material is selected as a make-up exam. In case of a tie, the bonus points are split. (It has

happened that six students received \2 points each.)

Students whose exams are selected generally score the highest on my exam. There is a great deal of debate over why this is true. Did those students understand the material so well that they were able to put together a high quality exam, or was it that

by making up the exam they were able to under stand the material?

For my part, I could not be happier. Besides per

forming better, my students are less anxious about

taking exams, and I now have an excellent file of

make-up tests for future examinations.

Marilyn K. Simon San Piego State University San Diego, CA 92182

Different forms To minimize the time devoted to making more

than one form of a test, I leave spaces on the spirit master around some arithmetic or algebraic expres sions. After making some copies of the original form, I make changes and run another batch. If af ter writing or typing the test, a slip sheet is placed between the master and the carbon sheets, the

changes can be written in lightly with a red pencil so that they may be seen readily.

Here are some examples to illustrate how

changes can be made.

Form A: _ 9 is a prime number.

Form B: _ 29 is a prime number.

Form A: _ - = ?

2 -2

Form B: _-= ?

2 -2

Form A: _The graph of the equation 3jc +

y = 6 will intersect the y-axis at the

point whose coordinates are

A (0, 6) (6, 0) C (0, 3) D (3, 0) E (2, 0)

Form B: _The graph of the equation 3jc + 2y = 6 will intersect the y-axis at the point whose coordinates are

A (0, 6) (6, 0) C (0, 3) D (3, 0) E (2, 0)

Form A: Simplify: ^jc ? Vx.

Form B: Simplify: ? Vx.

If copies are made on a mimeograph machine, the longer version can be typed first. After some

copies have been run, correction fluid can be used to block out the numbers or signs.

Jacqueline N. Pay ton

Virginia State University Petersburg, VA 23803

January 1982 15

This content downloaded from 74.64.97.109 on Fri, 12 Sep 2014 17:18:43 PMAll use subject to JSTOR Terms and Conditions