The students work to explain the relationship between the two algebraic expressions they have written.
Students’ Conceptual Challenges
In this episode, ET and Haleemah are asked to use part of each of their two equations, 3c + 2c + 4c = T and (3 + 2 + 4)c = T, to create a new equation. They are not sure what to do [2:37]. Eventually ET writes the equation 9c = T, which was not formed using the other two equations [3:05]. The teacher helps them by selecting 3c + 2c + 4c from one of their equations and 9c from ET’s new equation. She then asks Haleemah and ET whether or not these two expressions are equal. ET thinks they are equal, but his explanation relies only on symbolic reasoning (namely that you add the 3 + 2 + 4 to get 9, and then you put the c back in next to the 9 at the end). Later in the episode, they check the equation 3c + 2c + 4c = 9c by letting c = 5. In Episode 6, Haleemah and ET make sense of the meaning of each part of the equation in the game app context.
For use in a classroom, pause the video and ask these questions:
- [Pause the video at 2:11] Ask your students to consider the task in front of Haleemah and ET—how might they combine parts of the equations (3 + 2 + 4)c = T and 3c + 2c + 4c = T to form a new equation that is also true.
- [Pause the video at 3:41] ET has written a third equation, 9c = T. Ask your students if they can think of any new true equations they could make given this new equation.
- [Pause the video at 4:44] If you posed Focus Question #2 and your students did not discuss this new equation (3c + 2c + 4c = 9c), ask them if they think this equation is true. Encourage them to explain their reasoning to each other, and then ask for volunteers to explain their reasoning to the class.
- After watching the video, ask your students what they think it means for two equations to be equal. Highlight student ideas about equivalence. Perhaps students say that one side of one equation can be substituted for one side of the other equation and the other equation stays true. Perhaps they mention that both equations work the same way. Encourage them to explain their thinking.
- If your students have created their own equations that look different from the ones in the video, you can ask them to display them on whiteboards or through a collaborative app (e.g., see www.Padlet.com). Then ask pairs of students to pick two or three equations and debate whether or not the equations are equivalent. Ask for volunteers to share their partner’s thinking in their own words.