Leadership Program
Exploring Discrete Mathematics

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Lesson Ideas
Created by: Judy Ann Brown LP '92
Materials: A copy of Pascal's triangle, colored pencils
Object of the lesson:
Using Pascal's triangle, find the number of items given each day in the song, "The 12 Days of Christmas."
Content/Procedure:
The first (red) diagonal of Pascal's triangle indicates the numbers of new gifts given on the consecutive days. 1 partridge in a pear tree
2 turtle doves
3 French hens
4 calling birds
5 gold rings
6 geese alaying
7 swans aswimming
8 maids amilking
9 ladies dancing
10 lords aleapin'
11 pipers piping
12 drummers drumming
The second (green) diagonal of Pascal's triangle indicates the combined gifts given on the consecutive days. 1 = 1 partridge in a pear tree
3 = 2 turtle doves + 1 partridge in a pear tree
6 = 3 French hens + 2 turtle doves + 1 partridge
in a pear tree
10 = 4 calling birds + 3 French hens + 2 turtle doves
+ 1 partridge in a pear tree
15 = 5 gold rings + 4 calling birds + 3 French hens
+ 2 turtle doves + 1 partridge in a pear tree
...The third (yellow) diagonal of Pascal's triangle indicates the total number of gifts given. 1 = 1 partridge in a pear tree
4 = 2 turtle doves + 1 partridge in a pear tree
+ 1 partridge in a pear tree
10 = 3 French hens + 2 turtle doves + 1 partridge
in a pear tree + 2 turtle doves + 1 partridge
in a pear tree + 1 partridge in a pear tree
20 = 4 calling birds + 3 French hens + 2 turtle doves
+ 1 partridge in a pear tree + 3 French hens
+ 2 turtle doves + 1 partridge in a pear tree
+ 2 turtle doves + 1 partridge in a pear tree
+ 1 partridge in a pear tree
...By the fourth day of Christmas you will have received a total of 20 gifts. Look closely at Pascal's triangle  do you see a Christmas stocking? Continue the pattern and see if you can find the cell in Pascal's triangle that shows the total number of gifts given over the twelveday celebration.
You might want to try out Ken Williams' interactive version of Pascal's triangle on the Web.
Mathematical Content:
The material in this lesson addresses the following 1989 NCTM Standards:
 Standard 1: Mathematics as Problem Solving: [K4] and [58]
 Standard 3: Mathematics as Reasoning: [K4] and [58]
 Standard 4: Mathematical Connections: [K4] and [58]
 Standard 6: Number Sense and Numeration: [K4]
 Standard 7: Computation and Estimation: [58]
 Standard 13: Patterns and Relationships: [K4]
Other Resources:
 Lancaster, Ron, "Media Clips," The Mathematics Teacher, 88 (9), (Dec. 1995), pp. 748749.
 Raphel, Annette, "Critical Thinking During the December Holidays," The Arithemetic Teacher, 41 (4), (Dec. 1993), pp. 216219.
For more information, please send mail to Judy Ann Brown, judyann@ptdprolog.net
Web page created Wednesday, December 10, 1997
updated Friday, December 10, 1999