The Twelve Days of ChristmasAuthor(s): Charlene OliverSource: The Mathematics Teacher, Vol. 70, No. 9 (DECEMBER 1977), pp. 752-754Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27961076 .

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sharing teaching ideas

The Twelve Days of Christmas

Since students are especially tense and

restless the day before the Christmas holi

days, I was delighted to find Robert A.

NewelPs article, "The Twelve Days of

Christmas," in the December 1973 issue of

the Mathematics Teacher. As a result, I

gave my second-year algebra class the fol

lowing problem:

On the first day of Christmas, my true

love sent to me a partridge in a pear tree.

On the second day of Christmas, my true

love sent to me two turtle doves and a

partridge in a pear tree. The pattern con

tinues until, on the twelfth day of Christ

mas, my true love sent to me twelve

drummers drumming, eleven pipers pip

ing, ten lords a-leaping, nine ladies danc

ing, eight maids a-milking, seven swans

a-swimming, six geese a-laying, five gold

rings, four calling birds, three French

hens, two turtle doves, and a partridge in

a pear tree.

Problem: Find the total number of gifts

given in "The Twelve Days of

Christmas." Show your work.

Use a formula or shortcut if

you can find one.

After handing out the problem, I sug

gested that one approach was to make a

table (table 1). Students supplied entries for

the first three days. We then briefly dis

cussed patterns and tried a few suggested ?th terms, none of which worked in the

table.

TABLE l

Day Number of Gifts Total Number of Gifts

~1 i i 2 3 (1 + 2) 4 (1 + 3) 3 6 (1 + 2 + 3) 10 (1 + 3 + 6)

12

Less than thirty minutes after the assign ment was made, 15-year-old Jackie Malone

appeared at my desk with the correct for

mula and the correct numerical solution.

To understand Jackie's position, try finding the formula without using any mathematics

beyond the first semester of second-year

algebra. Limit yourself to thirty minutes

before reading further. Jackie began by using three variables, y

for the number of days, for the number of

gifts given on the >>th day, and for the

total number of gifts given during y days

(fig. 1). After constructing the first three

columns of his table, z, x, and>>, Jackie had

the insight to look at the ratios x/y and z/x and added these columns to his table. He

then concluded that

x_ _ y + 1

y ~

2

and that

? = y + 2 " 3

Sharing Teaching Ideas offers practical tips on the teaching of topics related to the secondary

school curriculum. We hope to include classroom-tested approaches that offer new slants on

familiar subjects for the beginning and the experienced teacher. Please send an original and four

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752 Mathematics Teacher

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s.

y if) ? "t6!ftf a & , / (4) y I * * I I I ( )

i 3 2 /i (f) 10 (3 + 1+0 c ? * '7 if)

20 [ ? 2

5<o Pi d> f JL% if) #4 ;?? 7 y 3 (j)

J< y i 3* If)

w 7^ /2 ^

above fads ) SL> ~ >$

Fig. 1

Jackie knew that

\xJ\yJ A student with a background in ad

By substitution, he derived his general vanced mathematics can use summations to

equation: solve the problem. S, the number of gifts

December 1977 753

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given on the nth day of Christmas, where is an integer between 1 and 12 inclusive, is

(2) s=?/=M^?i). Therefore the formula for , the total num

ber of gifts given during days of Christ

mas, is

i = l ? ? / = 1 ? =1

By other methods, we know

(3) t l-l Thus, from (2) and (3),

= I + 1)(2? + 1) J_ + 1) 2

" 6 2

' 2

.,_ ( + 1)(2?+ 1)

-r "* + 3?2 + 2? (4) r=?_?.

To find the total number of gifts given

during the twelve days of Christmas, let =

12. Then

T = 123 + 3(122) + 2(12) = 364

6

Equation (4) is equivalent to Jackie's

equation (1). Jackie's solution to the "Twelve Days of

Christmas" clearly illustrates the capabili ties of at least a few high school students

and the need for assigning some problems

requiring a creative solution.

Charlene Oliver Stinnett High School

Stinnett, TX 79083

Solving Algebraic Inequalities

The following strategies form the foun dation for solving more complicated in

equalities. The strategy chosen depends on

the student's prerequisite skills, whether or

not the strategy can be easily generalized, and the efficacy of the strategy in actual classroom use. In my experience, strategy 1 causes students at the precalculus and cal culus level many difficulties. Strategies 2 and 3 seem to be very effective. Strategy 4 is a good picture of the solution set, but it is difficult to generalize to inequalities in

volving three terms of the type A/(ax + b). In describing the strategies for finding

the solution set of inequalities of the form

ax -f b cx + d

where A, B, a, b, c, d, are nonzero real

numbers, the example

?L_<-i_ 3x + 2 Ix - 3

will be used.

Strategy 1?Case Method

The procedure is to multiply both sides

by the quantity (3x + 2) (Tx ?

3). The cases

arise because the inequality is reversed if

(3x + 2)(7x -

3) < 0. Thus, the cases are as

follows:

1. 3x + 2 > 0 and Ix - 3 > 0

2. 3x + 2 > 0 and Ix - 3 < 0

3. 3x + 2 < 0 and Ix - 3 > 0

4. 3x + 2 < 0 and Ix - 3 < 0

For example, in case 3,

754 Mathematics Teacher

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