It has been a bit of a battle this year to convince our students that mathematics is not disconnected. They seemed to arrive in our classroom at 9 years old with the conviction that the discipline exists sequentially, layered based on varying degrees of difficulty, some of which will remain inaccessible to the more artistically minded for most of their lives. We have been working hard to share that math is in fact a wonderfully complex web of recurring concepts, ideas, and patterns accessible through many different points from a variety of perspectives, and consisting of infinite possibilities awaiting discovery.
Our year began again with conversations about what we call multiples, what it means to be a multiple, and what a multiple of the number one is. We wouldn’t let our students dismiss multiples of one as “obvious” or “easy,” insisting that they consider what it means to be the number one. For example, how the number one can be manipulated without losing its integrity and how it is a part of other larger numbers. Before the fall break, we had explored multiples of one to nine, discovered patterns, noticed which ones fall into columns on hundreds chart, and noticed which multiples connected to others and how.
On Monday of this week we shared Perry the Platypus’ birthday dilemma….
at this point in the year, students were quick to glue the problem in their journals and begin documenting their thinking as they worked through answering Perry’s question. Some flipped back through their journals to remind themselves of previous discoveries they had made about multiples. Some organized their work in charts and some in diagrams or bullet points. Each was now familiar with the idea of writing down every thought or “a-ha!” that resulted from their considering the problem.
Almost all of the students jumped right to identifying that in one year Perry would be a multiple of 2, 4 and 8 because every multiple of 8 is also a multiple of 2 and 4. Many wrote notes to remind themselves that the smallest multiple of any number is that number itself. One cool observation that resulted from this problem was that Perry would be a multiple of every single one of these numbers by the time he was 12 BUT that when he was 11, he would not be a multiple of any of these numbers! One student wrote… ‘It seemed important to notice that both 7 and 11 are only multiples of one and themselves…’ PRIME! Another student noticed that when Perry was an odd-numbered age, then he was only a multiple of odd numbers and he expanded to state that odd numbers can only have odd factors!
The coolest part of the week however, was the conversations that resulted at the tables who had begun working through an extension to the problem which asked..
How long will Perry have to wait to be a multiple of 2, 3, 4, 5 and 6 all at the same time?
At first the question seemed overwhelming and intimidating to many students. They had been preconditioned to focus on finding a solution. We suggested that they look instead at which numbers on a hundreds chart were definitely NOT a solution. For example, which numbers on a hundreds chart were NOT multiples of 5… Right away we were met with excitement..
“WE JUST ELIMINATED 80 NUMBERS! A multiple of 2, 3, 4, 5 and 6 HAS to end in 5 or 0.”
Students excitedly crossed 8 columns of numbers off their list when one double takes again…
“Wait… the number we’re looking for also has to be a multiple of 2.. it CAN’T be odd… FIVES ARE OUT!”
“So we’ve got ten numbers left and we’re looking to see which one of these ten is a multiple of 3, 4 and 6..”
“But if it’s a multiple of 3 it HAS to be a multiple of 6..”
“Right so just 4 and 6..”
Looking at multiples of 6 students immediately eliminated everything but 30, 60 and 90. One noticed that multiples of 6 only end in zero if they are multiplied by a multiple of 5. The last step was to eliminate numbers that were not multiples of 4. As 30 and 90 were eliminated, another student noticed that multiples of 4 which end in zero HAVE to have an even number in the tens spot.
As we wrapped things up for the day on Thursday, one student commented as he reluctantly closed his journal; “Mrs. Bailey, I never knew math was so exciting. It twists and turns and loops and connects all over the place and it just seems to go on forever!”
Math is beautiful.