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5th & 6th Grade Math – Multiple Representations of Numeric Patterning

Clip 7/14: Problem 2


The next step of the lesson is to hand out the “Digging Deeper” worksheets. The first prompt on the worksheet asks learners to sketch and color code their drawing to show what is staying the same and what is changing. Notice how Maddy and Kelsey’s model doesn’t actually represent learner B’s solution to the task.

Blue Ones show the Old Ones: Here two learners are puzzling about learner B. David is explaining to Adrian how blue tiles represent the “old tiles,” or the tiles from the previous stage.

Teacher Commentary

Where does the 4 come from?

Kelsey asks this question first of her class mates and then of her partner Maddy. Maddy has a clear understanding of how learner B sees the pattern growing – “4 + 3 + 3….” Take a look at their worksheet as you watch Maddy explain learner B to Kelsey. This is exactly why discourse is so important.. There are two amazing things happening in this clip. First, Kelsey has had sufficient time to grapple with the task at hand and is clearly able to articulate where she is stuck. Second, there is so much learning happening as Maddy is forced to think of the best way to show Kelsey what she sees. She even invents this rainbow/shell system of separating the “+3”s. The mathematical authority in this interaction resides with both girls. Kelsey is empowered to make sense of learner B’s work while Maddy is challenged to explain what she knows. At the end of this interaction I was wondering if Kelsey really “got it,” but when Maddy and Kelsey present their thinking to the class, watch who does the talking!

Blue Ones show the Old Ones: Adrian and David illustrate a general theme I noticed in my class during this lesson. Using color coding to show what changes and what stays the same is a convention that was not commonly understand in our class. Here David explains that the blue tiles represent the old tiles, but this representation doesn’t match numerically with Learner B’s solution, which was “4+3+3+3…” for stage 11. David and Adrian’s model shows the following numerically:
Stage 1: 1+3
Stage 2: 4 +3
Stage 3: 7+3

The learners struggled with the idea of how to show what stayed the same and what changed using only two colors of tiles.