In Algebra 2 . . . I used these cards:
Hi Students! Here's a deck of cards. Each one has a polynomial function and its graph.
The degree of each polynomial is determined by its highest exponent. Talk with your group and agree on the degree of each one.
The leading coefficient is the number in front of that term with the highest exponent. Talk with your group and agree on the leading coefficient for each card.
Now we are going to focus on the ends of the graphs. Look at the left side. Does it point up or down? What about the right side? What you are looking at is called the "end behavior". Group the cards according to end behavior. Show me when you're done.
Next I want you to look for similarities in each group. What do the degrees have in common? What about the leading coefficients?
In 5 minutes or so, all of the groups had identified what the groups had in common according to even/odd degree and positive/negative leading coefficient.
We did some practice once they had this figured out.
Oh, and then I played some music and showed them some polynomials one at a time and they showed end behavior with their arms. And there was dancing.
This used to take me 4 minutes. I would write four rules on the board and they would copy them down.
Today it took 24.
But it was fun. And later I saw somebody end-behavior dancing in the hallway.
This approach is better than just telling them, right?
Showing posts sorted by relevance for query end behavior. Sort by date Show all posts
Showing posts sorted by relevance for query end behavior. Sort by date Show all posts
Thursday, January 10, 2013
Sunday, January 5, 2014
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!
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!!
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!!
Wednesday, August 28, 2013
Desmos Test Question
I am kind of excited about this (bonus) test question I used today . . .
My advanced algebra 2 students just finished studying a few of the basic parent functions and their transformations. Today, they took a test.
First there was a standard paper/pencil part of the test. Nothing unusual here.
Next, they picked up one of these cards containing the description of a parent function and a transformation.
The cards were color-coded according to difficulty level. Students were free to choose. Every card is different, so you won't be working on the same graph as your neighbor. Students were to pick up an iPad and use Desmos.com to create the function described.
Green Cards: Create a graph using a given parent function and animate one given transformation.
Purple cards: Animate two given transformations.
When finished, students bring me the iPad. I check that the graph matches the description, stamp the card, and clear out the graphs for the next person.
I was really pleased with the results. Almost everyone was successful in creating their graph. I don't feel like this needs to be a bonus question next time. It might even become a regular part of test-taking in my classroom.
Now I am thinking about the possibilities. My head is already spinning with ways I can use this question format for other topics:
Quadratics/Polynomials: Create a function with given vertex or given zeroes. Can you keep the vertex in place while animating the zeroes? Can you keep one zero in place and animate the other(s)? Create a function with given end behavior.
Systems: Create a system with one solution, no solution, or infinite solutions. Create a system with a given solution.
Rational function: Create a function with given vertical and/or horizontal asymptotes. Animate one or both asymptotes.
P.S. If you are wondering what we did in class BEFORE this assessment, here is a quick summary:
1. We spent several days sketching parent functions and transformations the old fashioned way, using paper and pencil. Students completed tables and plotted points, sketched graphs, looked for patterns, and generalized their discoveries.
2. Once students had mastered the basic functions and their transformations, I spent one class period introducing Desmos.com. I reserved our math department's shared iPad cart. I showed students how to enter equations, create sliders, and click play to ANIMATE (<3). They were as enamored as I was when I saw this a month ago at TMC13! Then I just let them play.
3. Finally, I started giving them a few challenges. Try to make an x^2 that moves vertically while stretching. Can you keep it from turning upside down? Can you restrict its movement to the second quadrant? Can you make it move horizontally along the line y = 2?
My advanced algebra 2 students just finished studying a few of the basic parent functions and their transformations. Today, they took a test.
First there was a standard paper/pencil part of the test. Nothing unusual here.
Next, they picked up one of these cards containing the description of a parent function and a transformation.
The cards were color-coded according to difficulty level. Students were free to choose. Every card is different, so you won't be working on the same graph as your neighbor. Students were to pick up an iPad and use Desmos.com to create the function described.
Green Cards: Create a graph using a given parent function and animate one given transformation.
Purple cards: Animate two given transformations.
When finished, students bring me the iPad. I check that the graph matches the description, stamp the card, and clear out the graphs for the next person.
I was really pleased with the results. Almost everyone was successful in creating their graph. I don't feel like this needs to be a bonus question next time. It might even become a regular part of test-taking in my classroom.
Now I am thinking about the possibilities. My head is already spinning with ways I can use this question format for other topics:
Quadratics/Polynomials: Create a function with given vertex or given zeroes. Can you keep the vertex in place while animating the zeroes? Can you keep one zero in place and animate the other(s)? Create a function with given end behavior.
Systems: Create a system with one solution, no solution, or infinite solutions. Create a system with a given solution.
Rational function: Create a function with given vertical and/or horizontal asymptotes. Animate one or both asymptotes.
P.S. If you are wondering what we did in class BEFORE this assessment, here is a quick summary:
1. We spent several days sketching parent functions and transformations the old fashioned way, using paper and pencil. Students completed tables and plotted points, sketched graphs, looked for patterns, and generalized their discoveries.
2. Once students had mastered the basic functions and their transformations, I spent one class period introducing Desmos.com. I reserved our math department's shared iPad cart. I showed students how to enter equations, create sliders, and click play to ANIMATE (<3). They were as enamored as I was when I saw this a month ago at TMC13! Then I just let them play.
3. Finally, I started giving them a few challenges. Try to make an x^2 that moves vertically while stretching. Can you keep it from turning upside down? Can you restrict its movement to the second quadrant? Can you make it move horizontally along the line y = 2?
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