Algebraic Expressions Lesson 3 Episode 6 (Teachers)

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In this episode, we see evidence of a conceptual challenge that ET has worked to overcome. Extending one’s understanding of arithmetic to algebraic expressions is challenging, and ET gives voice to this challenge when he says “When you think of addition, you think of a total sum; you don’t think of a total sum with a variable with all possible outcomes” [7:48]. This is similar to when Haleemah was working to see (3 • 2) simultaneously as something to compute and as an entity in Episode 1 of this lesson.

For use in a classroom, pause the video and ask these questions:

1. [Pause the video at 1:15 and again at 4:36] Haleemah and ET will describe what various parts of their equations represent in the context of the apps. This is the work of reasoning abstractly and quantitatively—they are contextualizing the symbols in the equation by describing the quantities those symbols represent and the relationships between them. Before your students watch these clips, ask them to contextualize the same symbols and then compare their answers with ET and Haleemah’s. 
2. [Pause the video at 3:47] ET and Haleemah have shown that when the cost of an app is $5, both expressions, (3 • 5) + (2 • 5) + (4 • 5) and 9 • 5, yield the same total cost, $45. Ask students to discuss how that might help them see why the two expressions are equal, that is (3 • 5) + (2 • 5) + (4 • 5) = 9 • 5. Follow up by asking them if the expressions are equal even if they don’t calculate each of the expressions. Push them to explain their reasoning. Highlight any comments that include ideas about the distributive property or that utilize groups of language (e.g., 3 groups of 5 and 2 groups of 5 and 4 groups of 5 is really just 9 groups of 5).

1. To interpret 3 • 5 as an entity, Haleemah says “15 is the amount of money spent” and asks ET if he agrees. He says he does, but then pushes for specificity by suggesting that they add “it’s the amount spent by Danyal” [1:35]. Ask your students to review each other’s descriptions of the symbols in their equation and give suggestions for making them more precise.
2. After ET explains why contextualizing each equation helps him understand why they are equivalent, pause the video [8:25]. Ask your students to consider what ET said and try to put it into their own words. Ask them what else they would like to add, or if they are thinking about it differently from ET.