By this time, students should have a solid understanding of problem (1) and (3) in the warmup (Transformed Radical Functions Warm-Up.docx).
They should be starting to make the connection between the mathematical processes used to solve these two problems. When they work on these two problems I ask them:
Problem (2) is a challenge when it comes to think about how the two negative signs affect the graph. I ask students these questions:
Though it seems simple, asking students these questions helps them understand some of the complexity involved in these transformations. Eventually, they should be able to link these changes to the function to reflections over the x- and y-axes. Hopefully they can articulate this today.
These kinds of realizations happen most effectively through direct dialogue with individual students or pairs of students. When we ask students to write their answers to questions, this introduces a whole new road-block for many students who don’t love to write, and so they may miss the chance to develop that understanding if the question is only presented in that form. When we ask students a question as a whole class, it is unlikely that we are asking each student at the right time—if we ask the question before students have had the chance to think about it themselves, then they won’t develop the same depth of understanding. For this reason, I like to have the questions ready to ask students as I circulate, so that I can ask students these questions when they are most ready to answer them. The back-and-forth dialogue gives students the chance to think and sends the message to students that we value their individual learning. It takes a lot longer, but it is an important part of this way of teaching and learning.