The Rich Mathematics of a Popular Holiday Song
One of the questions that always stumped the students was the one that had to do with the popular song “12 Days of Christmas”. The question was simple:
How Many Gifts Do You Get In Total In The Song?
Everyone, and I mean everyone, would offer the answer of 78, by simply adding up the numbers 1 to 12. There was a strong collective agreement from the students that this had to be the answer! So, you can imagine their “confusion” when I would point to a student–with all the other students frantically waving their hands to be picked as well–and tell them that their confidently reasoned answer of 78 was incorrect.
A sense of disbelief–and of curiosity as to why this answer was incorrect–completely fill the room. As a teacher, I would let this self-doubt simmer for a while before interjecting any kind of guidance on how to look at this seemingly innocent question in more detail.
And then, much to their dismay, I would start singing the actual song!
“On the first day of Christmas my true love gave to me, a partridge in a pear tree. On the second day of Christmas my true love gave to me, two turtle doves AND a partridge in a pear tree…”
I always emphasized the “and” when I sang that verse, and it quickly clued the class in that the number of gifts on the second day is three NOT two. And so the mathematical hunt began to find the correct number of gifts over the twelve days.
A great visual that shows where students went wrong and, ironically, that reveals part of the solution, is this one:
Students think that this visual represents the total number of gifts, when in fact it only represents the total number of gifts on the last day–the twelfth day! Sing the song for yourself and you’ll see.
When students actually start counting the gifts on each day, they start to generate the numbers 1, 3, 6, 10, etc. These are triangle numbers, and the number of gifts on the final day is also a triangle number, as visually depicted above. At this point, students realize they have to add up all these triangle numbers to get the correct total sum of gifts in the song.
However, what is interesting is that the detailed numbers of the song–days, gifts on each day, and total–is wonderfully embedded in Pascal’s Triangle.
The yellow line represents each day. The blue line represents the total number of gifts after each day. This is sometimes referred to as the “hockey stick” pattern. You go down as far as you want and then make a “blade” with one number. In the case of 12, the “blade” is 78, the number of gifts on that day. The pink circled number is the summation of all the blue triangle numbers. And this is the correct answer to the question of how many gifts in total one receives over 12 days–364!
This is a great question to introduce students to the magic and patterns of Pascal’s triangle while celebrating the holidays!