NOTICE Lab - Video Club 3: Flag Area

Video Club 3: Flag Area

In this video, two 5th grade students work on creating a flag with different cut out pieces. The students reason about what fraction of the whole different pieces are. They solve this task and express their thinking by comparing different cut out pieces, gesturing, and talking with one another and the interviewer.

Video Club Overview

Video Club Goals:

- Teachers will attend to and reason about students’ ideas about proportional reasoning, particularly related to how different sizes and kinds of shapes can make equivalent fractions.
- Teachers will attend to and reason about students’ ideas related to the substitution, particularly related to relationships between smaller and larger shapes.
- Teachers will attend to and reason about students’ multiple modes of expressing their ideas.

Materials:

- Flag Area task and diagram of flag cutouts (last page of the task)
- One full sized piece of construction paper per group
- Video loaded on Anotemos and shared with all participants
- If watching in person, each participant should have a device on which they can view the video and headphones.

Get Ready (20 mins, could be done on a different day than the video club): To prepare participants for viewing the video, review the problem. Give participants a few minutes to work on the problem themselves. Note that there are a cutouts and construction paper needed for this problem (see materials).

Once teachers have worked on the task, have a discussion about the seeds that came up for teachers.

Exploring the Math

A number of algebraic ideas can arise as students determine how to fill in the area of the flag. First, students use proportional reasoning and equivalence as they determine what fraction the different cutouts are of the whole piece of construction paper, and whether or not some cutouts are equivalent fractions. They also substitute some shapes for others as they combine smaller shapes to make larger shapes. These substitutions are also related to compositions of function as students begin to understand that they can “plug” smaller shapes into bigger shapes.

Several seeds of algebraic thinking could be activated by the during the task:

Belong or Not?- While doing the task, students might experiment with whether certain shapes and sizes fit into their flag configuration, whether smaller shapes fit into larger shapes, and/or whether different shapes are equivalent to one another. As students make these determinations, they group shapes into particular sets (shapes that are the same size, shapes that fit into other shapes, etc.) On a more abstract level, they also create sets of equivalent quantities. For example, students might identify sets of equivalent fractions like {1/1,4/4, 8/8} or {½, 2/4}.

Replacement- As students determine what shapes they can use for the flag, they might experiment with combining shapes to create equivalent areas. For example, two triangles might be put together to create the equivalent area of a larger rectangle. This seed might also be activated during more formal algebra tasks, like substituting values for function inputs.

Noticing Structure- Students will notice that the bigger shapes and the whole flag itself can be made from smaller shapes. The idea that a bigger shape can be made out of smaller shapes connects to the idea of composition. Although the context here feels more geometric- shapes fitting into other shapes- it might help students visualize functions fitting into other functions in more traditional function problems.

Note: This is not an exhaustive list of seeds! Encourage teachers to identify other potential seeds of algebra, even if they have not been cataloged

Viewing and Discussing the Video

Have participants watch the approx. 6 and a half minute video clip individually on the Anotemos platform. Participants should be encouraged to draw and comment on the video when they see interesting moments of student thinking, particularly those when seeds of algebra seem to be activated.

In most of the video, the students and interviewer speak Spanish. It is likely that a number of teachers in the group do not speak Spanish. Encourage teachers to attend to the other modes by which students express thinking, particularly their gestures and how they manipulate the cutouts. The draw tool is particularly useful for annotating these moments.

Discussion: In discussing the video, focus on the seeds of algebra that get activated as the students complete the task. Talk about how those seeds might connect to more formal algebraic thinking and concepts. In addition, emphasize how the students’ non-verbal modes of thinking convey important algebraic thinking related to proportional reasoning and substitution.

Description of Goal:

Teachers will attend to and reason about students’ ideas about proportional reasoning, particularly related to how different sizes and kinds of shapes can make equivalent fractions.
Teachers will attend to and reason about students’ ideas related to the substitution, particularly related to relationships between smaller and larger shapes.
Teachers will attend to and reason about students’ multiple modes of expressing their ideas.

Table - Video Key Moments

VC3 table

Potential Discussion Questions:

The students often look to fit smaller into bigger shapes. How algebraic thinking might be involved there?
- What questions could you ask the students to connect their ideas of “shapes fitting into other shapes” with fractions?

How do the students connect equivalent shapes? How does that help them solve the problem?
- Are there questions you could ask about equivalence that might help the students connect to the written form of the fraction?

How do the students use the structure of the problem to help them solve?

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