Sketching f' from f

My calculus class recently finished up sketching a derivative, given the graph of a function.

We began with using spaghetti and estimating the slope at each individual point. Among other things, I used this sheet from Math Teacher Mambo.

I told my students that the next step was to be able to sketch the basic shape of the derivative, sans spaghetti. No more estimating the slope at each individual point.

Students were having some trouble with this (they usually do, hmm). And then I thought about using color-coding, like so . . .

First we identified points where the slope was zero. We marked those in green on the original function and transferred the points to the x-axis of the derivative.

Next we identified regions where the function was increasing (positive slope). We shaded them in yellow and then shaded the corresponding region of the derivative ABOVE the x-axis only.

Then we identified regions where the function was decreasing (negative slope). We shaded those in blue and then shaded the corresponding region of the derivative BELOW the x-axis only.

Now, to sketch the basic shape of the derivative, you draw a graph that hits the green points and stays within the shaded regions.

It worked great! Students were all "Oh, now I get it!" Love those words!

We also needed a way to color-code places where the derivative was undefined, such as sharp points and discontinuities, so we added pink. A pink point translated to a vertical line on the graph of the derivative as a "don't touch this" signal.

Then we did a little more practice by matching some function/derivative cards. These are not my creation, but I am not sure where they came from to give credit.

Function and Derivative Cards by sqrt_1

There used to be lots of arguing and discussing during this activity, this time students breezed through it easy-peasy. I was just sitting there going ". . . but aren't you going to discuss . . . and argue and stuff?"

Next time I'll keep the spaghetti for sure. After that I'll have them try the card matching before the color-coding thing, and then back to the cards.

First on the Drop Pile

I wrote about this once before. Go back and read that if you want. Or don't. I'm about to summarize here anyway. :)

Yesterday the math PLC at our school did a "Keep. Drop. Create" activity. Among other things, "questions at the beginning of class" ended up in the drop pile. This is something I have felt strongly about for a while, so I am very happy that our whole department is in agreement.

I am probably preaching to the choir here, but if you give practice assignments on a regular basis and you're using the first ten or fifteen minutes of class to answer questions about yesterday's assignment, you should consider using that time for something else. Here's why:

Low student engagement. During question time, most of our students fit into one of three categories:

1. Students who didn't do much (or any) of the practice. They are now sitting there writing down the problems while you work them out. They're not thinking. They're transcribing.

2. Students who did most of the practice. When they got to a tough problem, they gave up because "I'll just ask that one in class". They learn to wait for help rather than persevering.

3. Students who did all of the practice before coming to class. They are now bored to tears while they watch your performance of math problems they already know how to do. Not learning.

(There may be a fourth category of students who legitimately have a question and are now eagerly anticipating your answer. But there are maybe two students in that category. Also, I would suggest that doing the problem for them while they watch is not the best kind of help.)

They're ready to learn. If there is a portion of the class period that students are most ready to do something, it is at the beginning. Don't lose them here.

Opportunity cost.  Instructional time is precious. Let's use those ten minutes for something else. Something that engages all students. We are working on a list.

In the mean time, we have decided to have little slips of scrap paper cut up and ready to go. At any moment we can do a quick formative assessment by posting a problem, collecting the slips, sorting them into piles, and identifying where students are having trouble.

What will you do with your extra ten minutes?

Or, an even better question, what other "math class traditions" need to go on the drop pile?

So Long, 2013

2013 brought a few firsts to my classroom (and life):

1.  I started SBG in my Calculus class. And I am wondering what took me so long. Students loved it. They liked that they could focus on learning without the stress. I loved that my valdectorian-competing students had complete control over their grades. Want a higher grade? All you need to do is simply demonstrate a higher level of knowledge. Boom. That's it. Next stop, my Algebra 2 classes. In order for SBG to be successful here, I must figure out how to be more efficient with all the paper work and re-assessing.

2.  Two Algebra 2s. This year my school decided to offer two levels of Algebra 2. I teach both. The basic level Algebra 2 was especially challenging. I think every student in that room hated math and everything associated with it. At least it felt that way some days. I put a lot of energy into managing behavior and felt like I didn't do justice to the math. It was just tough. Really tough. I have an opening in my schedule for second semester, so I will be able to split that class into two sections. I am very much looking forward to working with smaller groups. It will be better. I am feeling determined and hopeful.

3. I decided my (non-advanced) Algebra 2 class would be the best place to start Interactive Notebooks. I am pretty sure that what I am doing does not count as a true INB. There are no beautiful foldables or elaborate color-coded notes. (Even though I wish there were). But there is a lot of stuff glued into a notebook. I like that the constraints of the page size forced me to edit content and constantly ask what was really important for students to know/do. All but 1 or 2 students had perfectly completed/organized notebooks at the end of the semester. When it was time to review, everyone could easily locate what they were looking for. There is something about numbering pages and filling out tables of contents and gluing notes or practice into a composition notebook that equals organizational magic. I had my challenges with this group, but locating someone's missing assignment was not one of them.

4. On a personal note, and because I cannot resist writing about it, I ran my first EVER half marathon in 2013. This was a pretty big deal for me. I was the kid who dreaded, every year of my entire life, the day in PE class when we had to run a mile. As an adult I have loved what running has done for me . . . I am healthier, I have found friendships with running buddies, and I've figured out that I am capable of so much more than I ever imagined. I love sharing this story with my students, as many of them experience math the way that I experienced running. I like to think that I understand what they are feeling in some way.

I wish I could hug the guy who took this picture around mile 8 or so because . . . people behind me!

Lastly, because I hate to break tradition, my most-read (or least not-read?) posts from the last year:

How I Taught End Behavior:  This is my goal . . . more lessons like this where students are sorting and looking for patterns and figuring things out. More students doing, less teacher telling.

Trig Hand: Trick alert! My mistake was using this in my Algebra 2 classes. I won't do that again, even after focusing on the conceptual understanding. But I will use this with my Calculus students, and I have used it myself since I discovered it.

Plethora of Practice: I made two sets of cards for evaluating trig ratios, and found a lot of different ways to use them for a variety of practice sessions.

Desmos Test Question: I fell in love with Desmos this year, and I am still discovering all the many ways I can use this tool in my classroom. Here I used it for assessment.

Diving Into Programming: This year I dipped my toes into simple programming on a TI-83/84 calculator. After a few days, I am convinced that programming has a place in every math classroom. My dream is to have a project to go with each unit of the classes I teach. And the entry is so much lower than you think. You can do this, too.

This Lesson Cost $1: My intro to the zero product property.

Unit Circle: This post exemplifies what I love about blogging. I came asking for help, and I received some really helpful comments. I am thankful.

Happy 2014!!

Domain & Range from a Graph

My students have never had such accuracy in identifying domain and range from a graph.

Thanks to this little unsuspecting guy.

I blindfolded him and gave him eyes on the sides of his head.

And then he took a stroll across the x-axis to find to find domain.

Now the x-values that are included in the domain can be determined by stating the x-values where our little friend can "see" the graph.

Can he see it here? No.

What about here? No.

What is the smallest x-value where he can see the graph?  -9

Can he see it here? Yes

What is the largest x-value where he can see the graph? 9

So the domain is [-9, 9]

To find range, he strolls up the y-axis.

For a function that doesn't have endpoints, we talked about what the function looks like beyond the plane that is shown. Students really didn't have trouble with this.

All this silliness is the result of desperation.

For the first time ever, I am teaching two levels of Algebra 2. I am finding that some of my "standard" explanations aren't working with the "not advanced" group.

So then I'm all "Okay, that didn't work. Here, let me grab this toy bug from my closet." And stuff like this happens.

What if My Students Use Programming to CHEAT?

One question comes up as I've recently dipped my toes into programming on calculators with my students . . . You teach your students to program, but what if they start programming formulas into their calculators?

I've been thinking about this question, and playing out a few scenarios in my head . . .

1. Don't teach students to program. Maintaining ignorance is a good way to keep students from doing things you don't want them to do.

2. Teach them to program, but make sure they don't use it for cheating. Reset calculators before every test. This is a good way to make sure students don't spend too much time on any programs.

3. Let them program away. What might happen? Student who learns about the quadratic formula might think about programming it into her calculator. Later on, student encounters an equation, recognizes it is quadratic, correctly identifies a, b, and c and then executes a program she has coded herself to solve the quadratic.

I think I can live with option 3.

In fact, I'm over here smiling.

Why I Blog

Chiming in a little late, but I wanted to share my journey into blogging . . .

I still remember the day when I discovered math teacher blogs. I had been teaching for maybe eight years and I was in a rut. I had gotten to the point where I pulled out the folder of unit whatever from last year and wash, rinse, repeat. I was doing an adequate job, but I was bored.

On THE day I was sitting at my computer and I wondered if there were any math teachers out there blogging about teaching math. I started to search. I found Kate’s blog and Sam’s and a few others. Their blogrolls lead me to others. I started reading and I could not get enough. I tried log war and row games and other stuff. I got creative inspiration and I started to enjoy teaching again. My learning of new things was no longer limited to a rare conference here or there or from conversations with my two colleagues.

I started to recognize that I was a pretty mediocre teacher, and I stopped being content with mediocrity.

It was less than a year later that I started a blog of my own. Initially, I felt that I had gained so much from others that I wanted to contribute something to the community. It didn’t take me long to realize that what I had to share was just a drop in the pool of resources out there, but some people seemed to find what I had shared to be helpful and that felt great. I also wrote about what was on my mind and I was able to solidify many of my thoughts about teaching and learning through thinking them out in writing.

The real hook for me came when I started receiving feedback from others who were reading. I found that even when I posted something that I thought was super amazing, someone would give me an idea to make it even better. I asked questions when I wasn’t sure where to go with a topic, and I got answers. I am so very thankful to those commenters who have made me a better teacher.

I have subscribed to Sam’s philosophy about blogging – it shouldn’t be a chore. But while I refuse to let blogging be stressful, it has provided me with some motivation to produce share-worthy moments in my classroom.

I still mess things up all the time. I still spend too much time talking at the front of the room. I still have so much to learn . . . but there is no question that I am a better teacher than the blog-less version of myself.

And I’m definitely having more fun.

Programming Part 3

Day 3, for documentation purposes . . . I now have so many questions swimming around in my head related to how & when I would use programming in the future. But that's another post.

For Day 3, students were introduced to loops. This was a little more difficult for all of us, but the students were still super interested in the process.

Chris gave us these programs to try:

1. Use a loop to write a program that prints all multiples of three from 3-42.

2. Primality testing: Write a program that lets the user input a number and then checks if it is prime.

3. Number of numbers divisible by 3: Write a program that will let the user input a number, and then the program searches for and counts the number of numbers up to that number that are divisible by 3.

4. Product of first n numbers: Write a program that will let the user input a number, and then it will find the value of the product of all the integers up to that number.

5.  Averaging test scores:  Write a program that asks the user how many test scores they want to average, and then prompts them for each test score and finds the average.

Harder ones:

6.  Reducing Radicals:  Write a program that can reduce a radical. For example, if the user enters 20, it should return the numbers 2 and 5, representing 2 root 5.

7.  Finding perfect numbers:  A perfect number is a number that is equal to the sum of its divisors, besides itself. Example:  6 = 1 + 2 + 3. The first 4 perfect numbers were known in antiquity. Write a program that can find all perfect numbers up to a number inputted by the user. You will most likely need nested loops.

These were tough! I did not get through the list of programs today (or past #3).

The activity is still a hit with students. They are seriously begging for more!

Tomorrow they are each going to present one of their original programs.