Problem (1) is presented as another opportunity for students to discuss the idea that the square root function has a restricted range because we only use “half” the outputs. The graphs shown in this picture also provide a reference when thinking about domains and ranges, which is the big idea of today’s lesson.
Problem (2) is essential to prepare students for the lesson, and the language used is deliberately un-intimidating. Ask students to justify each of their answers:
The quadratic function obviously has no “impossible inputs,” but it is presented this way so that students have to convince themselves that all inputs are allowed. Ask them:
The words “domain” and “range” are introduced in problem (3) and once students have a sense of what these words mean, I ask them:
Students will be provided with number lines to show domains and ranges during the lesson, which is one efficient way. They may also use inequalities, or just verbal descriptions for now. I ask them:
These three problems effectively give students the chance to make sense of most of the day’s lesson on their own. If you find some students floundering during this time, I reassign them a new partner or ask students who have some ideas to convince their peers. I have found that when a struggling student is put in the position of trying to decide whether or not they agree with a student who arrived at an answer more quickly, this helps change the “status” in the classroom, because now the struggling student has an important job. Again, this is only possible if you give no indication at all of whether or not you agree with a student’s answers. I say, “____________, can you convince ______________ that your answers to problem (2) make sense?” This brings in both MP1 and MP2.
The closing gives students another opportunity to articulate the same big ideas that we have been thinking about for a while. One interesting idea to highlight is related to questions (A) and (B).
You would think that the domain of the original function becomes the range of the inverse function, but this is not the case when it comes to quadratic and radical functions. It is true that the range of the quadratic function becomes the domain of its inverse, however. This is a deeper conversation to have with students who have already developed a deeper understanding.
Questions (C) and (D) are important to ensure that students don’t just start memorizing things. We want them not only to be able to find domains and ranges of radical functions concretely, but also to explain more abstractly the ideas behind what they are doing.
The restrictions on the domain come from the fact that there is no square root of a negative number. I like to push students further: Why can’t we? They should be able to say that the “square root” of a number is the number that we can square to get the original number, but any number that we square will result in a positive output. The restrictions on the range come from the fact that we need to restrict the outputs in order for the radical function to be a function. In other words, only one output is allowed for each input.
These conversations are what elevate the course to a higher level of thinking and help students develop deep and abstract understandings. Even if it is difficult to ensure that all students understand these big ideas each day, the more frequently that you have these conversations, the more this way of thinking will become part of the culture of the class and students will start to develop the idea that “doing math” does not mean simply working with numbers to get answers, but actually to make sense of what is going on and to create deeper generalizations.