We return to our home learning series with a third post from the Atrium School, an independent school in the Boston area. During the past school year, Atrium participated in the PoP: USA research along with 5 other schools. As the next year may include online or blended teaching for many, how the Atrium teachers maintained their commitment to playful learning in math is instructive.

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Multiples of nines, square numbers, and the Fibonacci sequence: 5th graders playing with patterns

by Ben Mardell

Since the start of this school year, fifth graders have been discussing the emerging pattern on their monthly calendar, put there by co-teachers Sam Bloch and Diane Foster. As each day brought more data, the children worked to discern what specific symbols meant and speculated on what the next day might bring. By February, the class found the icons purchased by their teachers to be a bit boring, so for March there was a competition to select and use a student-made pattern. This continued for April and May.

Meeting virtually at 2:30 pm for an end-of-school day zoom call, the class continued to discuss the calendar pattern. It is May 18th and a pizza slice icon appears. It joins another pizza slice on the 9th, a variety of face symbols, and some smaller emojis (pine trees, lightbulbs and a face with sunglasses).

As the fifth graders join the zoom call, Sam asks in the chat box what they think the pizza slice is about. There is consensus that it involves multiples of nines, and will next appear on the 27th.

Sam then asks about the small emoji with sunglasses that appears on the 1st, 4th, 9th, and 16th. The group quickly concludes that this involves square numbers. With 18 days of data, the class has figured out most of what the monthly pattern involves. But one puzzle remains: the pine trees on the 1st, 2nd, 3rd, 5th, 8th, and 13th.

Diane notes, “this is a really hard one that we haven’t seen before.”

Elyse observes that the pine trees appear on both even and odd days. Then Olivia begins an explanation:

Well, one plus two, one and two, they both have the pine tree. And one plus two is three. Three has another pine tree. And two plus three is five, and five has another pine tree. Five plus three is eight. Eight has another pine tree. And eight plus five is thirteen, and thirteen has another pine tree.

Sam: "So when do we see the next pine tree?" Olivia responds:

Well thirteen plus eight is twenty, right? So, on the 21st.  

Sam: Yeah on Thursday. Thank you Olivia. Now does anyone know what that pattern is famously called? Alright birthday boy, do it." Nate answers:

It’s called a Fibonacci sequence. My dad talks about it all the time. He’s like, ‘this is a Fibonacci sequence’ and I’m like ‘okay, cool’.  

Sam: (laughs) Yeah, it is cool. Fibonacci sequences are awesome. They appear a lot of times in nature and in repeating patterns that you see without even knowing it. So thank you to the person that made this calendar for including the Fibonacci sequence.

Calendar math mysteries solved, the class moves on to talk about a project they are working on and the tasks for tomorrow.

The calendar activity continued in June with a new pattern created by another student. Like past months, the June pattern inspired lively conversations—both verbally during the afternoon meetings, and in the chat area of Zoom. In fact, while Sam clearly prefers in-person teaching, he has been struck by how the chat feature has enabled shier students to share their ideas more frequently. He also notes how continuing the calendar pattern activity helped maintain continuity in the transition to distance learning, and supported the feeling of classroom community as he, Diane, and the students discussed current patterns and reminisced about past patterns as well.

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