##
* *Reflection: Intrinsic Motivation
Investigating a Radical Function - Section 1: Lesson Beginning (Opener)

When I taught this lesson today, the opening dialogue didn't go as well as I had hoped in some ways.

As usual, I began class by reviewing one problem from the Weekly Workout. Conveniently (and a little bit by design) the chosen problem involved an inequality of the form . This led to a standard quadratic equation in terms of *x*, and I made sure to "accidentally" leave this problem on the board off to one side.

At this point, I chose to proceed with my plan of questioning the class about radical numerical expressions and irrational numbers. The response from the students was weak, they were quiet, and they didn't seem to understand the point of the questions. I would have dropped the whole thing and cut to the chase, but it was clear that they weren't finding the questions easy to answer. Many had apparently never heard the term "radical" in mathematics, and there was confusion about the relationship between radicals and irrationals. In other words, I felt trapped; the conversation was slow, but I couldn't simply drop it in good conscience.

As soon as I moved on to a radical function, the class became more engaged and lively, which left me wondering how I could have gotten to that point more quickly.

After some reflection, I think that if I were to do it again I'd use that inequality from the Weekly Workout as my starting point. Here we have the variable *y* contained in a radical, and the outcome is a parabola. What if the variable *x* were contained in the radical? What would happen then? In my minds eye, I can imagine that the students would be more immediately engaged in this line of investigation.

In the end, this isn't the sort of thing you can plan completely. Different classes and different students will be engaged by different kinds of questions. You need to study your students and their habits of thought, and you need to be able to move flexibly one one path to another, always keeping the same goal in mind. The goal is to inspire the students to *think* about the topic at hand.

*Engaging Students in Discussion*

*Intrinsic Motivation: Engaging Students in Discussion*

# Investigating a Radical Function

Lesson 1 of 6

## Objective: Students will be able to investigate a single radical function, construct its graph, and have their first encounter with an extraneous solution to a radical equation.

#### The Investigation

*25 min*

First, I will hand out Investigating Radicals 1 and ask the students to begin working individually. This initial investigation is fairly mechanical, with student evaluating the given function and producing a graph. At this stage, I think it's very important that they make the graph by hand, so I will not allow graphing calculators yet.

As students finish the graph, and I've seen that it's correct, I'll hand out Investigating Radicals 2. This really brings us to the heart of the lesson since these questions will focus the students' attention on the limited domain and range of the function. As they begin solving radical equations, they will see extraneous solutions arise and they will be able to use their graph to identify them. In the next section, the students themselves will explain the how and why of these features of radical functions.

Please see the video narrative for more details.

*expand content*

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- UNIT 1: Modeling with Algebra
- UNIT 2: The Complex Number System
- UNIT 3: Cubic Functions
- UNIT 4: Higher-Degree Polynomials
- UNIT 5: Quarter 1 Review & Exam
- UNIT 6: Exponents & Logarithms
- UNIT 7: Rational Functions
- UNIT 8: Radical Functions - It's a sideways Parabola!
- UNIT 9: Trigonometric Functions
- UNIT 10: End of the Year