Reflecting
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Captions
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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:
- [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.
- [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
- 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.
- 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.