No captions Captions Stop the video above first if it is playing.
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ET and Haleemah make sense of Nicole’s algebraic and arithmetic equations and create a drawing that they believe describes how she was seeing the pattern.
Episode Supports
Students’ Conceptual Challenges
When analyzing a rectangular array of rows and columns of stones, the number of stones in one row also determines the number of columns in the array. ET and Haleemah seem to struggle with this idea, notably when ET describes one value in Amir’s equation as representing the number of columns as being different from the same value in Nicole’s equation, which he sees as representing the number of stones in a row [3:20].
Focus Questions
For use in a classroom, pause the video and ask these questions:
[Pause the video at 1:52] Ask your students to examine Nicole’s equation, x2 + x • (x + 1) + x2 = T. Ask them what they think about this equation and how Nicole might have been viewing the picture of the cobblestone pattern when she created this equation.
[Pause the video at 3:50] ET describes the 32 as meaning “three times three” and says the first three represents the three stones in one column while the second three represents the three stones in one row. Ask your students how they see the 32 in the diagram, and if they agree or disagree with how ET is seeing each 3.
[Pause the video at 5:28] Haleemah says that she sees 3 • 4 by thinking about base times height. Ask your students what they think Haleemah means. Follow up this question by asking how thinking about base times height might help ET better see how the two methods (Amir’s and Nicole’s) are similar.
Supporting Dialogue
After watching the episode, ask your students to use the “groups of” language (featured at the end, starting at 8:11), to describe the number of gray stones in the second section. You can anticipate that some students might see four groups of three by attending to the number of columns and stones per column, or they might see three groups of 4 by attending to the number of rows and stones per row. Encourage discussion by asking if these two ways of seeing the second section yield the same number of stones, which is a concrete way to discuss the commutative property of multiplication.
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