** Consider the following problem**:

**If -5 ≤ x ≤ 4, and f(x) = 2x**

^{2}- 3, how many integer values are possible for f(x)?One can simply view this as a more challenging question to pose to your honors/accelerated students, but, for me, it's an opportunity for all your students to think more deeply about important concepts. I feel strongly that our role here is to ask the key questions which will guide them toward understanding the "big ideas" underlying this problem. In fact, we can turn this question into an extended activity: 15-20 minutes).

Here is one idea for creating the environment currently being recommended. Please keep an open mind before concluding that there is simply not enough time for these explorations...

**WITH YOUR LEARNING PARTNER(S):**

1. Sketch the graph of the function on the given domain from recognition of quadratic functions and by making an x-y table with 4-5 points. WRITE YOUR INFERENCES FROM THIS. For example, from the sketch we believe that the greatest y-value on this domain is ___.

WRITE your conjecture for the answer to the problem: ____

2. Using the TABLE feature of your graphing calculator, with TblStart = -5 and ΔTbl = 1, display the Table. Now turn TRACE on and analyze the graph on this domain. Does this alter or confirm your conjecture from Step 1? YES NO

3.

**The following statement is plausible bu**

**t**

**FALSE.**

**The domain consists of 10 integer values. Therefore there are also 10 integer values for f(x), so the answer is 10.**

**Explain why this is wrong. There is more than one error!**

4.

**The correct answer is 51.**Depending on the class, a few, if not several, students should be able to come up with the correct answer and provide a thorough explanation.

5. Group Discussion:

- Ask students how they might have approached this question if it appeared on a standardized test? Plug in x-values? Use the graphing calculator? Guess? Skip it?
- Ask the group what made this questionable formidable for some students? How important was understanding what was asked for?
- Review one successful approach to solving the problem by calling on individual students to give the "next" step.

NOTE: This problem also presents a highly teachable moment for students to see an application of the

*in Precalculus (or more intuitively in Algebra 2). Help them make the connection! Is this easy for us to do?*

**Intermediate Value Theorem****Your thoughts?**

"All Truth passes through Three Stages: First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur Schopenhauer (1778-1860)

You've got to be taught To hate and fear, You've got to be taught From year to year, It's got to be drummed In your dear little ear You've got to be carefully taught. --from South Pacific

## 3 comments:

A good follow-up question is for how many x in that range the value of f(x) is integer. Can't answer that question without fully understanding the original one.

Thank you, Alexander. Excellent extension that adds another level. This is why I depend on my readers for inspiration. Are you a math educator?

This type of question should be part of a textbook problem set for students. Let me know if you know of such a text! This is the reason I wrote my book of challenge problems. Much more than an SAT prep book, it is a resource for teachers who believe that student understanding will only grow when they are presented with nonriutine problems that are not superficial.

I might use this for IVT (coming up in a few days). I very much like the misleading effect the table in the calculator will have... Hmm. Maybe even use this to motivate IVT?

I'll let you know. Thanks.

Jonathan

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