Algebra+2

= = = = Upload Algebra 2 lesson files here. Along with your file, give a brief description for your choosing it to be the lesson of the week.

Lesson of the Week

10/5/11

Radical Showdown Comments: Ashley, Using a game format as a teaching/learning is very powerful. Your reflection pointed to that too. I hope you continue to explore this option as a tool for many types of lesson formats including review for testing. Dr. C.



//Lesson of the Week//

//9/29/11//

//Simplifying Radicals//



Comments, Ashley, this is indeed a difficult topic. One major misconception students have is that sqrt(x^2) = x which is not the case; the sqrt(x^2) = abs(x). A great way to demonstrate this is on the graphing calculator with y1=sqrt(x^2); y2=x; and y3= abs(x). Then a rich discussion can take place about why the graph of y1 matches the graph of y3 and not y2. Dr. C.

= = =Week 1 Lessons - Due by Jan. 30=

Radicals Lesson = = by Mike Refici

Since this was the only real lesson I taught this week, it was an easy choice for lesson of the week. It seems like everyone is doing radical these days. I tried to jazz it up and promote critical thinking with some key questions. At the same time, some of my students get anxiety when I assign homework problems that I have not modeled exactly in class.

__Comments__ //Hey Mike, I think it's a great idea to have the students brainstorm "opposites" at the beginning to build a relationship with squares and square roots. Maybe just be careful to come back to them being "inverses" so they don't get confused on vocabulary. -Sean//

Mike, Sean makes a good point. We usually think of "opposites" in connection with signed numbers...perhaps that is why the students were a bit slow to catch what you were going for. Did anyone get the perfect square & perfect cube number? Dr. S. = =

= = Operations on Radicals Brian Resendes

So, as you can read in my reflection, this lesson was kind of a dud. Up until today I felt that I was doing well until I graded a quiz. I feel like I need to make my lessons less traditional, but I'm not too sure how to do that for everything, or how to use discovery learning, scaffolding, etc in such a short time restraint.

Comments //Hey Brian, I don't know if your school is using the Glencoe texts like ours is...but if you are, ask the teacher about the box of supplemental resources that come with each text. I've been looking through some to brainstorm ideas. For example, in this chapter, they do an exercise on isometric dot paper to explore if sqrt of 2 + sqrt of 2 is the same as sqrt of (2+2).//

Brian, is it commonly understood that "radicals" mean "square root"? When is the index that could indicate cube root introduced? Dr. S.

= =

Roots and Radicals = = by Mike Iavarone

My apologies in advance if I messed up. I'm uploading this as my "Lesson of the Week" because it's the first one I taught, and for a first lesson, I thought it went pretty well. It was about roots and radicals, which isn't the most exciting thing in the world, but we've got another few days and then we move //into quadratics//!! I'm also linking to a website I found which includes an awesome "Power Point Jeopardy" game involving radicals! (among other things) [|I know it says 8th grade but there's so much stuff!]

Comments I like the idea of getting them to work in groups on a discovery learning activity. You have definitely found a more interesting way of looking at a potenially really dry topic. Do you have a copy of the graphic organizer that you could upload? -Jon Mike, your approach was solid. Seems as if your class has had lots of experience with skills before. Yes, names help! Dr. S.

Mike, I like how you set up your lesson, including what you anticipate to get for answers to questions. I've found this to be useful, especially to anticipate what wrong answers or misconceptions students may have in order to be prepared to ask the appropriate questions or give the appropriate explanation to help clear up confusion. Also, thanks for the link! -Ben

**Week 2 Lessons - Due by Feb. 6**

**Radical Equations and Inequalities (Mike Iavarone)**



This is my lesson of the week because it actually went pretty well! It's a lesson on solving equations and inequalities - it includes a technology component that I actually haven't been able to try yet, but I still plan to. Boring stuff, but the launch actually got them involved - it's a problem involving roller coasters that I've been building up for about a week! When was the last time you rode a roller coaster and said, "You know, I can probably build a lesson on radicals around this?"

Mike, I really like how you've been able to make a lesson on a complicated topic that is fun and meaningful to the students. They were probably so interested in finding the solution that they were more motivated to work through any struggles that they may have had. I'm impressed that you planned well enough to think to ask the question about the roller coaster in a previous class. If only I had that kind of foresight! -Ben

Mike, nice opening question! Have a question for you (and others): do most texts emphasize checking for a 'no solution' situation after the algebra calculations have been performed? That seems a bit opposite to the emphasis in pre-calculus, which is to check the domain where the function is defined. Am wondering whether checking assumptions before doing the work would be a worthwhile mathematical habit of mind to cultivate. Thoughts anyone??? Dr. S.

We actually did both; they'd set everything under the radical greater than or equal to zero to see where the function is defined. THEN solve the inequality much like you would a radical equation. This would give something like 2 < x < 5. Then they'd test a point less than 2, a point between 2 and 5, and a point greater than 5. Lots of checking. - Mike

**Mike - This is a very well thought out lesson. I always learn alot from your ideas. The rollercoaster launch is a sure winner. I think a good key question may have been "Why does a rollercoaster have a velocity of zero at the top." It's kind of counter-intuitive to think of a roller coaster standing still. Then, the examples you chose were scaffoled very well to support the students learning. Nice job picking an example that had no solution. You really set yourself up nicely for a good teachable moment about checking your work. I am wondering if you every brought closure to the rollercoaster problem during the launch? It may have been advantageous to now give away the solution right away, so you could return to it at the end to wrap things up. - Mike Refici**

Rational Exponents Mike Refici


 * This lesson was observed by Dr. Sullivan. Luckily it went very well. I actually had to make some changes to this lesson at the last minute because class was canceled the day before due to an assembly. It's more of a lead in to rational exponents. My main exploration was a nice discovery of why negative exponents produce the recipricol of a positive exponent. The whole section on radicals is much more interesting than I had ever thought. I wish we could spend more time on it, but the curriculum calls for full speed ahead.**

Mike, I like the idea of turning calling on all of the students into a game. I have a horrible time memorizing names, so I think I could use this to both make sure that everyone is participating and to help me learn names. -Ben

Mike, I totally understand what you're going through with the launch questions. My students seem completely unmotivated to do it at all because it's something they are not used to doing. But it's definately something I'm not giving up on! Sara

One way to be sure all students are called on during a class is to have names on a deck of index cards or popsicle sticks. They can be drawn randomly. When you have gone through the entire deck you know you have called on everyone. It can be hard for students to adjust to this at first, so you might consider giving them a 'bye' once during a class, but make every effort to call on them again before the class ends. You can also consider having a student ask another for help on a question....supporting student-to-student discourse is a worthwhile goal. Dr. S. 

Simplifying Radicals and Trig Functions Kevin Simpson

This lesson went well even though it progressed a little slower than I had originally though it would. I didn't quite get to finish everything that I had planned for the day. It turned out to be a mostly skills based lesson. Any ideas of how to liven it up would be appreciated. -Kevin

Kevin, Planning is solid. I could not help but wonder what the reaction might have been had you asked students to conjecture about the flagpole height and asked them to offer a reason for their conjecture...you could have written the conjectures down and said something like, "We will be able to see whose conjecture is closest after we add to our math repertoire." What do others think? Dr. S.


 * Kevin- First, I thought it was awesome that you noted the literal meaning of trigonometry as triangle measure. The vocabulary in Math is often rich with these literal connections, and it can really help students understanding when they learn to discover meaning in the actual words. Regarding your lesson, I agree wit Dr. S. that you could have expanded on your launch. Is the ultimate solution related to shadows and similar triangles? If so, you certainly have a ways to go before arriving at that. So it may be helpful to return to that question at the end of the period as well, even if only breifly, to chart your progress so it does not get forgotten. Also, it seems like this lesson had a lot going on, although I do realize you have block classes. So it was good that you tried to incorporate review at the beginning, but maybe unfortunate that it overshadowed getting to the trig functions. Would it have been better to just introduce sin, cos, and tan first, and then move into sec, cosine, cotan? Also, oppposite/adjacent is a concept students often get mixed up. It may have been an opportunity to get some students out of their seat to demonstrate this concept. Finally, perhaps a good key question would have been "Can we use sin cos, and tan, if we do not have a right triangle? Why or why not?" - Mike Refici**

Rational Exponents **Brian Resendes**



I liked this lesson because every step of it I introduced new topics by how they relate to older topics. It allowed the students to memorize the new information easier, and also show connections between different topics.

__Comments:__ Brian, I am glad you saw the value in stepping back with this lesson! Dr. S.


 * Brian, Nice Lesson. I too have noticed that going slow but steady is the most efficient way to progress through the curriculum at a good pace. And I noticed all the spots where you scafoolded the material. I thought it was great that you had the class develop the rule for converting between rational exponents and radicals. Did you at all discuss the meaning of why they are called rational exponents? I remember my class had some difficulty with this section, and some students got hung up on the idea of an alternative representation as opposed to an algorithmic process for finding a value. I am wondering how we could incoporate critical thinking into this section. Perhaps, "What values of n will make 3^(n/2) an integer?" -Mike Refici**

** Week 3 Lessons - due by February 13 **

Review of Linear Equations Mike Iavarone

This is my lesson of the week because it drove what happened for the rest of the week. We're going to be moving into quadratics soon, and my class groans every time they hear the word "graph." I decided to have them work in groups on graphing several lines, just to see how bad they are at it. It turns out they don't give themselves enough credit! The idea is pretty simple: set up stations and have them move around. It makes a very boring 50 minutes of material mildly interesting and is a great way to assess prior knowledge and all that stuff.

__Comments__

Good management skills are critical to the success of a class like this. They were evident throughout. Dr. S.

**Complex Numbers**
 * Mike Refici**



This lesson did not go well. My students got confused while I was deriving the i chart. It was a bad feeling and most of the class was not pleased with me. Upon reflection, I came up with a nice interactive excercise that gets them out of their seats. It should help to clear up some misunderstanding and also get them motivated with some hands on learning.

__Comments__

Mike, I found myself wondering whether this was the initial introduction to 'i' and complex numbers. <span style="font-family: 'Comic Sans MS',cursive;">It has been my experience that students are quite content knowing that there is no solution to a problem in the reals; many see no need to look for one in another number system! Coaxing them to see that there is value to having the complex number system won't come from needing to solve x^2 + 1 = 0, I'm afraid. I wonder whether your students might be wowed by pictures of the Mandlebrot set or some other fractals, for which the complex numbers are essential. <span style="font-family: 'Comic Sans MS',cursive;">Dr. S.

**UNIT CIRCLE**
 * Kevin Simpson**

This lesson required the students to develop the unit circle. This required them to use a lot of the concepts that we have been talking about in class over the past couple of weeks. This is the beginning of the semester and this subject requires quite a bit of work on behalf of the the students, but they have been working very hard to understand the concepts. I didn't get to finish the entire lesson in one day, but they did make progress during the class period. I'll just need to continue for the next day.

__Comments__ <span style="color: #00ff31; font-family: 'Comic Sans MS',cursive;">Kevin- Good idea not to force finishing the plan. I'm finding that a plan is what it is...just a plan. I'm finding it hard to know just how much is enough...sounds like you did a good job adapting your plan. -Sean

<span style="font-family: 'Comic Sans MS',cursive;">Sean, nice insight! A plan is a plan, a roadmap.....sometimes detours and slowdowns happen. Kevin, you will be happy later that you took extra time here. Dr. S.

**Week 4 Lessons - Due by Feb 28**

<span style="display: block; font-family: 'comic sans ms',cursive; text-align: center;">** Radical Expressions not so "radical, dude"- ** This lesson was a rough one. For some reason, I tried to cram the whole section into one lesson...bad idea. I ended up reteaching the next day and it went much better.
 * Sean Caffrey **

**Comments** Sean, your reflection said it all....once the notion of combining the powers in the radicand and the index is clear, the rest is connection. Glad you retaught. Dr. S.

Using Color Coding to Develop a Linear Programming Model By: Benjamin Robertson This lesson was taught in a "Topics in College Math" class. This class is a class that seniors who do not take precalculus, but need four years of college prep. math take. It gives an overview of topics that students may come across in their college courses. This lesson was designed to help students break down written scenarios when writing them as linear programming models. Often, these problems are very wordy and can be intimidating to students. This lesson was designed to help students to break long word problems into smaller, more workable pieces of information.



Ben, how many prior lessons did students have in this area? Yes, there is a lot of information in these situations....do students perceive the situation as plausible or a manufactured one? Will you incorporate graphing calculator technology into future lessons? Dr. S.
 * Comments:**

Finding Real Roots of Quadratics ** Brian Resendes **

This was the first lesson I've ever taught using graphing calculators. The students were surprisingly excited to use the calculators, and they enhanced the lesson much more than I expected.

<span style="display: block; font-family: 'Comic Sans MS',cursive; text-align: left;">**Comments:** This plan seems rather incomplete. I did not see how the calculators were integrated. Dr. S.

** The Real Number System ** This lesson was a bridge in between chapters. I wanted to make sure everyone knew what I meant when I used words like rational, integer, real number, on a daily basis. I also include some fun things at the beginning and end that I find pretty neat. Mike- I really like your launch. This is a problem that can be understood by students of a variety of levels. It is definitely the type of problem that will start some good discourse in the room amonst students. -Ben
 * Mike Refici **
 * Comments: **

<span style="font-family: 'Comic Sans MS',cursive;">Mike, Nice lesson. What answer did you expect to the question about "more numbers - whole or integers"? You might want to read the article in the current issue of the //Mathematics Teacher// entitled, "The Infinite Hotel." Also, Where did you place the Real numbers in your list? Dr. S.

Arc Length & Area of a Sector **By Kevin Simpson** With this lesson I tried to get the students to make a connection between the circumference and area of a circle to the formulas for finding the arc length and area of a sector. I think a couple of the students in the class understood where it was going, but it seems that most just wanted to plug numbers into a formula.

Comments: <span style="font-family: 'Comic Sans MS',cursive;">Kevin, Is the fundamental idea of a ratio clear to your students? If not, that would definitely create confusion. Dr. S. = =

**Week 5 Lessons - Due by March 6**

<span style="color: #a21010; display: block; font-family: 'comic sans ms',cursive; font-size: 120%; text-align: center;">Solving Quadratic Equations by Factoring (A follow-up lesson in pairs for deeper understanding) Sean Caffrey

Comments: The lesson materials are well thought out, and they were effectively used. Dr. S.

Quadratics Review **Brian Resendes**





This lesson was done because almost all my students seemed completely lost before a quiz (partially due to the unexpected snow day). Afterwards they had a better understanding, although not complete.

Comments:

<span style="font-family: 'Comic Sans MS',cursive;">Were the notes planned to be a review? It is not quite clear. Dr. S.

<span style="color: #ff0600; display: block; font-family: 'comic sans ms',cursive; font-size: 110%; text-align: center;">**Selected Topics** Mike Refici This just in... Probablitiy is awesome! It certainly has some appeal to students that Algegra 2 does not provide. This lesson revolved around us discussing some interesting questions from the pre-assessment for our unit. I am now looking forward demonstrating why the probablity of rolling a 12 at any given time is __not__ // 1/12. //
 * Intro to Probability**

Comments:

This promises to be a nice unit for these seniors. Your first lesson shows you are off to a good start. Keep us posted! Dr. S.

Solving a Linear Programming Model by Graphing Benjamin Robertson

I chose this lesson because it was a lesson which was observed and while it is not the most innovative lesson, it definitely served its purpose of giving the class one last step by step overview of how to solve a linear programming model by graphing. One of the best parts of this lesson was that I was able to get about 80% of the class to answer at least one question in the class.



Comments:

<span style="font-family: 'Comic Sans MS',cursive;">The LP could be used easily by a substitute teacher if s/he had the examples you planned to use. That is a nice help! Dr. S.

Discovering the Law of Sines Kevin Simpson

This is a lesson that I found on the Illuminations website and adapted it to use with my class. It went pretty well overall. The students are still having some difficulties solving algebraic expressions of one variable when they also involve a trig function, but they are improving. This was only the first day of a two day lesson on the Law of Sines. I continued the next day and we did quite a few examples of using the law of sines. The quiz I gave on this Friday went very well.

Comments: <span style="font-family: 'Comic Sans MS',cursive;">Good work on adapting other materials. Realizing that new tools are needed for non-right triangles can be a rather large leap for students, even though it is a logical progression for us. Dr. S.

**Week 6 Lessons - Due by March 13**

Quadratic Formula Brian Resendes This was the lesson where we sang the quadratic formula. It was not only really entertaining, but it was really helpful and I could hear the students singing while doing problems.

Comments: <span style="font-family: 'Comic Sans MS',cursive;">This is a nice illustration of Gardner's Multiple Intelligences! Dr. S.

<span style="color: #000000; display: block; font-family: 'comic sans ms',cursive; text-align: left;"> ** Review of Solving Quadratic Equations ** Sean Caffrey
 * Jeopardy! **

Nicely planned. I can see why the class went well for you. Dr. S.
 * Comments: **

** Linear Programming Review Stations Part 1 **
 * Benjamin Robertson **


 * This was the first half of a test review for a unit on Linear Programming. This portion of the review focused on the skills needed to complete a linear programming problem. The next portion will focus on the conceptual understanding of the problems. This lesson went really well. It allowed the students to work together to get their questions answered and I was able to help out when a group needed assistance. **

Comments: Nice planning on this one, Ben. Dr. S.

<span style="color: #3d085e; display: block; font-family: 'arial black',gadget,sans-serif; text-align: center;"> Real Number, Operations and Simplifying and Evaluation Expressions Kevin Simpson

Now that we have finished with trig, it's back to the beginning of the book and a review of topics covered in Algebra I. This lesson went pretty well. It was the first time I actually had a planned lesson fit perfectly into the class time and I was able to accomplish everything that I wanted to.

Comments: Nice job with the timing on this one! Dr. S.

=Lessons due by 3/20=

Vertex Form Activity Brian Resendes This was an activity we did to introduce the students to vertex form. It was fairly successful and pretty interesting to read the students' responses.

Comments: <span style="color: #000000; font-family: 'Comic Sans MS',cursive;">Nice work on making a student-centered learning lesson, Brian. Dr. S.

**Solving a Linear Programming Problem Using the TI-83**
 * Benjamin Robertson**

This is actually a lesson from last week, but it is one that I thought went really well. Students in this class do not have a lot of experience with the graphing calculator, so the step by step directions worked well in walking them through each step in the process. Students were very impressed with the calculators abillty to show the feasible region.



Comments: The worksheet that guided the use of the calculator to graph the feasible region and determine the optimal solution was nicely done. I imagine that other students will want to copy this one. Dr. S.

Lesson Due 4/3

<span style="color: #1db918; display: block; font-family: 'comic sans ms',cursive; font-size: 120%; text-align: center;">**Graphing Quadratic Equations Follow-Up Lesson**
 * (lesson via Smart Board)**
 * Sean Caffrey** <span style="color: #000000; display: block; font-family: 'comic sans ms',cursive; font-size: 120%; text-align: left;">

Comments Sean - I think it is very cool that you are using a SmartBoard. I have only seen them breifly at a substiute teaching assignment once, and it certainly looked like a lot fun. Maybe you can share your experiences with the Smart Board at Seminar. Thanks. Mike Refici

Sean, nice use of the technology. How did students react to your using it? Dr. S.

<span style="display: block; font-family: 'arial black',gadget,sans-serif; text-align: center;"> Graphing Square root functions/inequalities **Brian Resendes**

This was the first lesson in my unit plan, and was done in a different order than the book. It was one of the first lessons where I had pretty much no problems, and finished right on time. I think the decision to change the order from the book helped the students make connections to this section.

Comments: The ideas in this plan extend those of quadratic functions nicely; however, terminology in the lesson plan is not as precise as is needed for students! Dr. S.

=Lessons Due 4/27= = =

Inverses Brian Resendes

= = This lesson had to do with finding and confirming inverses of functions both in function notation, and those written as sets of ordered pairs. I think it worked out nicely because I was able to carry over specific ideas, like flipping x-y, and previous knowledge, operational inverses, to help the students understand this. Comments: Am curious about the vocabulary in your text here...does it say 'flip'? I found your use of the triangle unusual - can you explain why you started with it? Dr. S.

Mike Refici Point Slope Form Equations

This was a lesson in which we continued to graph lines using point slope form as well as slope intercept form. Also, I tried to have students see the connection between the two forms by manipulating one to look like the other.The lesson went well, although there is a lot of material within the lesson and some students struggled.

Comments: I presume that this lesson was for Algebra 1. Even though students prefer step by step (they don't have to think as much), keep prompting them for conjectures and reasons. Dr. S.

=Lessons Due 5/1=

= = "Extreme Form" Brian Resendes = = This lesson was a recovery type of lesson. The Extreme Form Project (also uploaded here) was not implemented correctly. I did not offer enough support or in class gauging of how my students were progressing. I spent an entire class going over how to start the project and allowing the students to work together on it to try to support them more. Comments: Brian, you made the best of a not-the-best situation. I imagine that you learned a lot from this! Dr. S.

Transformations of Parent Graphs Calculator Investigation  Benjamin Robertson

This lesson had students investigate the effect of different operatons on a parent function using the graphing calculator. The one change that I would make to this lesson if I taught it again would be to have students conjecture what they think would happen to the graph prior to graphing it, then test their conjecture using the graphing calculator.



Comments: Ben, the only suggestion I have for this lesson is to ask students to conjecture what the graphs will look like before they use the graphing calculator. In this way, the process is an active one for students and not a button-pushing exercise. Dr. S.