Repeating Your Reasoning
The students apply their method for finding the number of tiles in the border to swimming pools of different sizes.
Students’ Conceptual Challenges
- Haleemah and ET apply their method from Episode 1 to two new pools without any problems. However, if your students work on the tasks before viewing the video (see Focus Question 1), they may use a different method without realizing it. If that happens ask, “Did you use Haleemah and ET’s method for finding the number of tiles in the border of a pool with 10 tiles on a side, or did you use a different method?”
- If your students used a different method than Haleemah and ET, you can ask them to find the number of tiles in the border of the pool again, this time using the same method that ET and Haleemah used in Episode 1.
- Haleemah and ET continue to make sense of the structure of this problem. Their struggle in this video isn’t as obvious as in Episode 1, but they are still wrestling with how to name the two quantities they have identified. For example, Haleemah just refers to the number of tiles such as 15 and 13 [7:19] whereas ET uses both compass directions, such as 7 tiles on the “east” side of the border and 5 tiles on the “north” [1:40], and positional terms, such as 5 tiles on the top and bottom [2:10].
For use in a classroom, pause the video and ask these questions:
- [Pause the video at 1:04 and/or 4:20] Ask students to use ET and Haleemah’s method for counting the tiles in the border of a pool with 7 tiles on a side and/or 15 tiles on a side.
- [Pause the video at 6:40] Ask students if they can rephrase ET’s thinking about how he knew there were 13 tiles in the bottom of the border. Follow up by asking if that reasoning applies in earlier examples as well. Encourage students to be explicit about why subtracting 2 from the side length always works.
- [After viewing Episode 2] Ask students to use their own methods from before to find the number of tiles in the border of a pool with 7 (and/or 15) tiles on a side. Ask them discuss with their partner how their method compares to Haleemah and ET’s method.
- As students repeat their reasoning, ask them to take a moment to examine ET and Haleemah’s work on all three problems from Episodes 1 and 2. Encourage them to discuss with a partner what they notice stays the same in their expressions or equations throughout all three problems.
- Similarly, ask students what changes in their expressions or equations throughout all three problems.