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## Intervals of Increase and Decrease – Put to Bed!

Ever since I wrote Common Core Algebra I, I’ve had numerous discussions with math teachers around the country who weighed in on the subject of intervals of increase and decrease of functions. In my answer key, I often include the x-coordinates of the turning points in the intervals, although I’m not entirely consistent on it.

Now, I’ve had folks email me clearly upset that I’ve included the x-values of turning points in the intervals. Their logic is sound. If an x-value of a turning point is included in an interval of increase, then it would also be included in an interval of decrease. Thus making the function both increase and decrease there.

So, their point was not lost on me. In fact, my inconsistency on the convention is actually consistent with the fact that the turning point x-values can either be included or excluded. I wrote a really lengthy post on why and won’t go into it at depth. If you want to read it, here’s the link:

Unit #3.Lesson #4.Intervals of Increase and Decrease

Ultimately, my belief has always been that the Regents would take either convention. Of course, they never saw fit to clarify this point in their CC Algebra I standards clarifications. More on that later.

For all intents and purposes, though, the debate is now over because the January 2016 Common Core Algebra I exam had a free response question on it that included the question of a decreasing interval. Up to this point, we had only seen multiple choice questions and this issue didn’t arise in those.

Here’s Question #33 from this January. It’s a standard projectile motion question that we love so much. I like the question quite a bit because students can explore it on their calculator. Click on the image to see it better.

It’s the second part of the problem, where students are asked for the interval of time over which the height is decreasing. By the way, using whatever technique, the time to the peak height is 2 seconds.

Now, the first thing I looked at was the standard state rubric. My heart was beating. Would it state that we had to take off one point for the student including the 2 in the decreasing interval? What about 5 seconds when it hits the ground? Is the height decreasing there? Here’s the state rubric. Again, click to see clearer.

Wow! They didn’t even state the interval. Notice, they are quite clear on the time to the maximum being 2. But, they aren’t even willing to state what the “correct” interval is. Just that the student has given a “correct” interval with an explanation written.

O.k. Well, those of you who have graded these tests before now know where we turn to – the sample responses. For those of you not in the business, these are copious amounts of sample responses with the appropriate points given for the work shown. Here’s the first full credit response (4 points) they gave us:

Oh, snap! The first full credit solution includes the turning point (and the endpoint). Most importantly, it states unequivocally that the student has a complete and correct response. Thus, that settles it for good. The turning points must be included for full credit.

Oh, wait. Here’s the second full credit solution they provide:

Wait, now they’re accepting 2<t<5 for full credit. Huh?!? I guess the turning point x-coordinate doesn’t have to be included.

O.k. Enough with the sarcasm. What these two responses now give us is clarity! A student can include the x-coordinate of the turning point in an interval of decrease or increase (as seen from our first sample response) or they can leave it off (as seen from our second sample response).

So, I’m glad I won’t have to address that ever again. I’ll just refer to this post.

On a much larger picture, though, this illustrates something I have been very troubled by in the current transition to Common Core in New York – the lack of specificity in the standards. Seriously! There are so many issues out there that are completely predictable that need clarity like this one. The good and overworked folks up in Albany could just make a list to give us the clarity we need to create robust and consistent curriculum. Instead they try to get the word out in random forums, emails, and footnotes. I don’t know if this is some sort of philosophical change up there, but it makes is harder to do our jobs well.

I hope that with a new head of the Regents and change in the wind, they will consider making their expectations clear to us in very fine grained detail. This will allow us to craft our own stories of rich mathematics and not get bogged down in trying to answer questions about what will be on the exam and what will not.

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## CC Alg I Units 3 and 4 Review SMART Notebook Files – by Julie Merana-Spanarelli

Our good friend and frequent contributor Julie, from Central Islip, is back again with a great gift heading into the review portion of the year. She has created SMART Notebook files for Units 3 and 4 Reviews that we put out recently. You can access the reviews at:

Comprehensive Unit Reviews – by Kirk

Julie does a lot of great things with animations within these files, so experiment with all of the pieces of the files if you are an experienced SMART Board user. Here are her files:

Unit-3.Functions Review Smart

Unit-4.Linear-Functions-and-Arithmetic-Sequences.Review

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## Unit #3 Smart Notebook Files – by Ann Murray

So, our good friend, Ann Murray, has contributed another round of Smart Notebook files just in time for the holiday break. She has given us Unit 3 files. So, those of you with SMART Boards be sure to give these a look over your time off.

Thanks Ann for your continued contributions to teachers around New York state and the country.

U3 Lesson 1 Intro to Functions

U3 Lesson 2 Function Notation

U3 Lesson 3 Graphs of Functions

U3 Lesson 4 Graphical Features & Terminology

U3 Lesson 5 Exploring Fns w Calculator

U3 Lesson 6 Average Rate of Change

U3 Lesson 7 Domain and Range

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## Unit #3.Lesson #4.Intervals of Increase and Decrease

So, I’ve had a number of people ask about whether the x-coordinate of a turning point should be included in an interval of increase or decrease or whether the intervals should always be exclusive.

Believe it or not, I’ve been tossing this question around for awhile with my good friend Brian Battistoni at Arlington High. This is a great question and it can be argued both ways. What we really need is for NYSED to just issue a clarification, so we can all be consistent. Maybe they will, maybe they won’t. I think it is a cool question because it can be argued both ways. Here’s how…

The Exclusive Case (x-coordinates of turning points not included):

This is the convention that most people will first think of because if the turning points were included, then they would have to be included in both intervals of increase and intervals of decrease, thus seemingly making the function both increasing and decreasing at the same point. This doesn’t sit well with people. On top of that, if you look at it from a calculus perspective and define the function as increasing when its slope is positive and decreasing when its slope is negative, then turning points, where the slope is either zero (smooth) or undefined (corners), should be excluded.

The Inclusive Case (x-coordinates of turning points are included):

To make and understand this argument, we need to agree on what the question actually is. Remember, the question is always phrased like “give the interval over which f(x) is increasing.” So, let’s examine the function:

Most calculus books will define an interval of increase as follows:

In other words, bigger x’s give bigger y’s. But, in the case of the quadratic above, wouldn’t the interval:

fit this definition? Remember, the question is not whether the function is increasing at x=4. It is whether x=4 is part of an interval where bigger x’s give bigger y’s. The x-coordinate of the turning point would certainly be included in this definition and is certainly included on the Advanced Placement Calculus exam. This inclusion really gets at a fundamental geometric idea regarding intervals of increase and decrease:

An interval of increase is a stretch of the function that can be considered going uphill and an interval of decrease is a stretch of the function going downhill. Aren’t the bottom and tops of hills part of these stretches?

What’s the correct definition? Well, that’s up to the bureaucrats to decide, which is weird to say. The one sample problem Albany gave us was the exclusive use. I’m amazingly inconsistent on this because of my own struggles with two equally valid interpretations.

Anyone want to weigh in?

## The Domain and Range of a Function

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## Average Rate of Change

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## Exploring Functions Using the Graphing Calculator

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## Graphical Features

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## Graphs of Functions

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