Leadership Program Exploring Discrete Mathematics in the Classroom

# The Twelve Days of Christmas and Pascal's Triangle

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 a-laying   7 swans a-swimming   8 maids a-milking   9 ladies dancing 10 lords a-leapin' 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 twelve-day 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: [K-4] and [5-8]
• Standard 3: Mathematics as Reasoning: [K-4] and [5-8]
• Standard 4: Mathematical Connections: [K-4] and [5-8]
• Standard 6: Number Sense and Numeration: [K-4]
• Standard 7: Computation and Estimation: [5-8]
• Standard 13: Patterns and Relationships: [K-4]

Other Resources:

• Lancaster, Ron, "Media Clips," The Mathematics Teacher, 88 (9), (Dec. 1995), pp. 748-749.
• Raphel, Annette, "Critical Thinking During the December Holidays," The Arithemetic Teacher, 41 (4), (Dec. 1993), pp. 216-219.