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

Clip 2/14: Numeric Pattern Planning - Part B


In this clip, Margie Trainer, Stacy Emory, and Francis Dickinson share their take on re-engagement as Fran introduces us to the original task and the lesson at hand. Fran shares what he saw as he reviewed student work on the "Buttons" task. In this preparation, he identified two learners' descriptions of the 11th stage to use during the re-engagement lesson. Fran describes the general flow of the lesson as planned. You’ll also hear a description of the “what’s my rule?” number talk. Margie and Fran then discuss why this number talk was chosen. Margie asks Fran about his use of the think-pair-share strategy in his efforts to foster discourse in the classroom. Fran, Stacy, and Margie discuss the ways in which learners may struggle with different representations of the pattern in the "Buttons" task. Also, they discuss how the learners are developing their ability to generalize the pattern with algebraic representation.

Teacher Commentary

This lesson is a reengagement lesson designed for learners to revisit a problem-solving task they have already experienced. My colleague, Stacy Emory, best describes reengagement by comparing it to re-teaching. Re-teaching is a teacher directed activity where we plan a different lesson to address something that is perceived to be a misconception with our students. Reengagement is a learner-centered activity wherein the original task is posed in such a way that we may expose learners to different strategies, alternate solutions, or even misconceptions. Think of the original task as a formative assessment that helps you shape the lessons that follow.

All of my 6th grade learners were able to successfully draw the 4th stage of this pattern and mostly all of them were able to correctly identify the number of white buttons in stages 5 and 6. The two exceptions to stages 5 and 6 were learners who scored the special case points for counting the black button along with the white buttons.

13 out of 28 learners incorrectly identified 33 as the number of total buttons in stage 11. These learners also went on to incorrectly list 72 as the number of total buttons in stage 24. This error, I believe, solely to be a critical reading miscue. These learners were leaving out the black button and missing the language “total number of buttons.” I base this inference on the abundance of learner work which successfully describes the growth of the white buttons.

The learners were mostly successful at generalizing the pattern in their own words or through the use of number sentences.

Some learners were describing the pattern as adding on:

“I added 4 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 [ to get stage 11.]”

Other learners were moving towards thinking multiplicatively about the +3-ness of the growth:

“ I counted by 3’s for each pattern...”

Less than half of the class chose to represent the growth with a number sentence such as:

“I [did] (11 x 3) +1 = 34 buttons [stage 11.] I added one for the black button.”

Overall, my 6th grade class performed very well on this task, which is why we can use this reengagement lesson to begin looking at multiple representations of functions.

All of the teacher tools you see at work in my classroom have accumulated over the years of observing other math teachers in video and in person. It is easy to become isolated in our profession, but there is a lot to be gained from observing others at work in their craft.

Materials & Artifacts