Algebraic Expressions Lesson 4 Episode 6 (Teachers)

Reflecting

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ET and Haleemah reflect on their generalized algebraic expression and think about what would happen if their variable, v, was a negative number.

Episode Supports

Students’ Conceptual Challenges

Haleemah and ET use the equation L = 3 • v + 4 to determine that the end location of the scooter ride will be 19 meters, when the velocity is 5. They then examine a new equation, L = 3 • (−v) + 4. When asked to predict what L will be if v = 5, they quickly state −19 [3:57]. This suggests they are encountering a common challenge when working with negative numbers and algebraic expressions. While the negative symbol can be productively thought of as “the opposite of” (e.g., −7 is the opposite of 7 and −(−4) is the opposite of −4), the inclusion of a negative symbol does not necessarily mean the opposite of an entire expression or functional output.

Focus Questions

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

1. [Pause the video at 2:31] If you plan to use Supporting Dialogue Prompt #1 below, ask that before using this Focus Question. Haleemah says both equations are good, depending on “if you have a negative number or if you have a positive number.” ET says just one equation will work, since v is a variable and it can be any number and “it doesn’t matter if it’s positive or negative yet.” Ask your students who they agree with and why.
2. [Pause the video at 3:57] Remind your students that 19 was the end location that Haleemah and ET found when v = 5 and they used the equation L = 3 • v + 4. Ask your students to make their own prediction for what L will be, using the new equation L = 3 • (−v) + 4 when v is 5.

Supporting Dialogue

1. Early in the video, Haleemah seems to claim that you need both equations, L = 3 • v + 4 and L = 3 • (−v) + 4 depending on whether you use positive or negative values for v [1:47]. In contrast, ET claims that one equation is sufficient, namely L = 3 • v + 4 [2:19]. Ask your students to rephrase both Haleemah’s and ET’s arguments in their own words.
2. Ask your students to discuss with each other why they think ET and Haleemah initially guessed L = −19 when replacing v with 5 in the equation L = 3 • (−v) + 4. Encourage them to make an argument for why −19 is either a reasonable or an unreasonable guess.