Thursday, December 20, 2012

A Lesson Plan for the Teachable Moment: THE TWELVE DAYS OF CHRISTMAS AND PASCAL’S TRIANGLE

Created by: Judy Ann Brown LP '92 | Rutgers university Leadership Program: Exploring Discrete Mathematics in the Classroom | http://bit.ly/RI1P3Q

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.

                             

                            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.
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                            For more information, please send mail to Judy Ann Brown, judyann@ptdprolog.net
                            Web page originallycreated Wednesday, December 10, 1997
                            updated Friday, December 10, 1999

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