the twelve days of christmas
TRANSCRIPT
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
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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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