Noticing and Wondering

Tuesday evening, Max spoke at the Global Math Department meeting about Noticing and Wondering. He has spoken about this before at Twitter Math Camp and I was a bit intrigued about it then. When I saw he was speaking Tuesday night, I knew I had to be there.

Now, I'll be honest, this is NOT something I have done with my students. I have had the tendency to instruct without letting them do a whole lot of exploring. Part of it for me is that there is so much material to teach in Algebra 2 (and I am having to play catch up from Algebra 1) and part of it is my own comfort level. However, as I was thinking about my lesson on Friday about the Remainder and Factor Theorem, inspiration struck me on Wednesday. My Algebra 2 students were having an assessment on Thursday and there would be enough time afterwards for them to do a little noticing and wondering. After their assessment, I asked them to complete a paper with the following questions:

Do (x^3 + 6x ^2 - 3x + 7) / (x + 3) using synthetic division.
Find f(-3) if f(x) = x^3 + 6x ^2 - 3x + 7.
What do you notice?
What do you wonder?

Do (2x^4 + 6x^3 - 15x^2 + 15x - 50) / (x - 2) using synthetic division.
If f(x) = 2x^4 + 6x^3 - 15x^2 + 15x - 50, what is f(2)?
What do you notice?
What do you wonder?

Here are some of the responses I received:

Notice:

• I don't remember how to do (the f(__)) problem.
• same problem and you get the remainder
• the remainder is the same as the answer of the function and they have the same number of terms
• The remainder of the synthetic division is the same answer as the 2nd problem I worked out (the f(__) problem).
• I noticed that the numbers are the same in the problems and f(x) is the same number as it is in the box of in synthetic division.

(I had several answers that were similar to the 2nd, 3rd, and 4th notice bullets.)

Wonder:

• How to do it (the f(__) problem)
• Could you use synthetic division to find f(x)?
• (Written under the 1st wonder) I got the same remainder for the 2nd set of problems, but not the 1st set of problems. What did I do wrong on the top question?
• Can you use functions to solve synthetic division?
• What do they have in common? Why are they both remainders?
• How does that happen? Why are they the same?
• Are we going to do the same thing as before or is it different?
• I wonder if the two problems are related. I wonder if you can use the 2nd problem to figure out the synthetic division problem.
• Will this be the case every time? (Then the student added on the 2nd one when his answer did not match) Will the answer match the remainder if the number replacing x is negative?
• Why is the answer the same as the remainder?
• Are they connected? Is this another way?

My observations:

• Many (TOO MANY!) of my students did not recognize function notation or how to work with it. This is something we did review earlier in the year and it dismays me how many had no clue. Even several of my top kids came up to ask how to deal with f(-3) and once I told them, they remembered. However, that they even had to ask worries me.
• The students who didn't know how to do synthetic division (all the way) correctly obviously did not make the connection at all. Many of these students left the notice/wonder part blank.
• I half expected some smart-aleck responses from some of my students (especially some of my struggling ones). I didn't get any. However, I had a lot of blanks in the spaces asking for their noticings and wonderings.

I had chosen to do it this way, rather than orally, for two reasons:

1) I knew I was going to have extra time after their assessment and this would help keep them focused and quietly working on something.

2) I felt that this would give my students who do not catch on as quickly as my top students the opportunity to think about it and possibly make the connection on their own. In my Algebra 2 classes, I have a range of students from very bright students (who mostly took Algebra 1 in 8th grade) to students who struggle with math. Not all of my high-ability students caught it, and some of my middle ability students did put together the connection rather nicely. I hope that will help them tomorrow when we discuss the Remainder and Factor Theorems.

Originally, I didn't think I would give them their papers back. However, after reviewing them and reflecting some, I think I will give them back to them. Hopefully we'll have some nice discussion about it tomorrow.

Addendum:
On Friday, my principal came in to observe me during the 1st of my 4 Algebra 2 classes. (We are a Race to the Top school and we are doing the new Ohio Teacher Evaluation System this year.) Although I'm guessing I'll get dinged for this not being as much of a class discussion since they had completed the noticings and wonderings on paper the day before, I feel like it went pretty well. I would have liked some more discussion out of them. Part of it for me is that I haven't done this before and finding the right questions to elicit discussion out of them was a little bit of a challenge for me. I feel like we answered most of their wonderings, which is a good thing. :-)

Transformations Unit

Last year, I shared the transformations matching cards I used with my Advanced Algebra 2 students. With teaching transformations for the first time to all Algebra 2 students, I have revamped my lesson and cards. Thanks are due to @druinok for her help in hashing out what I was doing with this unit.

I'm working with F.BF.3:
Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

After discussions with both @druinok and our pre-calculus teacher, I decided to hold off on the f(kx) portion. @druinok shared that in her state, they don't do the horizontal stretches and compression in Algebra 2 and our pre-calculus teacher said that until you are working with a periodic function, the horizontal and vertical appear to be the same. So I will wait to bring in the f(kx) part until we get to graphing sine and cosine later this year.

I began very similarly to what Rebecka Peterson did by introducing parent functions to my students on day one. Here is what I gave my students:


(I don't know WHY the graphs keep showing up wrong, but they do. It looks right in Word but I can't get it to show correctly.)

It went way quicker than I anticipated - it only took about 20-25 minutes from start to finish. I haven't decided if in future years I will start into the notes following this or what to do to not leave so much open time on day one.

The second and third day, we worked through this packet:

 (Like the last one, still having issues with the graphs. Not sure why.)

The second day, we got through the first two pages of the packet. I had students work through the three graphs and descriptions and we did the summary piece together.

The fourth day, I had a meeting. I left an activity that their Algebra 1 teacher called "Around the World." I've done this as a scavenger hunt before. Here are the pages I used:

I had a brainstorm for my fifth day activity. I didn't feel real comfortable coming in after being out the day before and having them start into the assessment activity. So, after going over questions from the practice problems and the Around the World activity, I had students make "appointments" like in the Appointment Test Review activity that Mrs. H blogged about. Then I had students make up 1 or 2 equations for functions that they transformed (depending on how much time was left in class). The only guideline I gave them was that each equation had to have at least 2 transformations. Then, when they met with each appointment, they exchanged equations and had to find the transformations. This went pretty well for many students, although some still had some difficulty coming up with the transformations after they made up their equations. Most students went with two transformations. I collected their cards at the end of the period.

On the sixth day, I had students do the Transformations Matching Cards as their assessment for the activity. I had them work in pairs and allowed them to use the note pages. Rather than have them complete 5 sets like I had last year, I had them work through 3 sets. Students are matching pictures of the graph with the description of the transformations and the equation of the graph. I had students work with the five parent functions they graphed on the first day. Many of the equations came from what they generated on the 5th day. Here is the what I gave them:


They did very well with the assessment - I am sure part of that is that I allowed them to use their notes and their cards with the parent functions. Possibly next year I would allow them to work in partners but without notes, but still with the parent function cards. Overall, i am pleased how this unit went.

All of the files I used are shown through box.net - they are in docx format and you are welcome to download and adjust them as needed. If you are having trouble, feel free to email me at lmhenry9 at gmail dot com and I'll be happy to email you a copy directly. I hope this helps someone out.

My Version of Log Wars

Today in Algebra 2, we did my version of Logarithm War. For the uninitiated, here is the post from Kate Nowak with her version. With my Algebra 2 students, my goal was to have them practice solving for x in a logarithmic equation. If you've read my blog as of late, I have been very frustrated with their lack of arithmetic skills, especially with regards to powers lately.  What I did was have them get into pairs and grab whiteboards. There are 36 cards (so if I had an odd number of students, there could be a group of 3 and still divide the piles into equal numbers) and they were divide the cards into equal piles. Each student was to work out their problem and the person with the highest answer got the cards. If they were equal, they were to lay down another card face down and then a 3rd card face up, which they were to work out to determine the winner. Winner of each pair at the end of the period got a blow pop.

Here are the cards I used:



Thoughts from today -

• As usual, my Advanced Algebra 2 students were the most enthusiastic about this. They were certainly the loudest group of the 4. 
• Pairs in two of my classes thought the winner of the "hand" should be who got their answer correct first instead of who had the highest answer. Thoughts for a variation, maybe? I didn't want to emphasize speed today but accuracy instead.
• Most everyone participated. I had one pair in one class who didn't do more than maybe 4 pairs. "I already knew how to do it," was one of the students' comment in that pair. That student took away practice from his/her partner which wasn't helpful to him/her. :-( Even my lowest ability students did participate (although, again, maybe not as much as I would have liked, but at May 17th, I'll take whatever I can get).
• I was able to get around and help many students who needed help getting started. Once they got going, they were involved and it worked well.
• However, in all of my regular Algebra 2 classes, they stopped after going through the pack once. My last regular Algebra 2 class had a couple of pairs who kept going and in my Advanced Algebra 2 class, they kept going until I told them to quit. I could add cards and make a larger pack, but I'm not sure if I'm going to head there.
• Overall, it was a good activity and worth doing.

I should also add that I used Amy Gruen's Loop for Logs as a teaching aid for changing from logarithmic form to exponential form. It works very well and I am grateful for this great idea. If you haven't checked out her blog before, there are lots of good things there - stop and take a look!

Review Worksheet Twist

I decided to give my Algebra 2's one more review day. After some deliberation, I decided to give them a review worksheet - partially because they would have a copy of all the problems I wanted them to practice so they could practice on their own, partially because I didn't feel like trying to put together another review activity for them that they wouldn't put a whole lot of effort into. On the way in to work, I came up with a plan that I hoped would help get questions answered. I think part of this inspiration came from @druinok.

I handed out the worksheet and told the students I would project the answers on the SMARTBoard once I was done giving directions. They were to work on the worksheet problems and check their answers. If they were correct, they were to continue working on problems. If they were wrong, they were to try to figure out what they did wrong and fix it. If they couldn't figure it out, they could check their notes or with the neighbor, or they could find the card with the worked out problem that was posted on my wall:

If they still were stuck and none of the above helped them, then they were to sign up for help:

I had them do this so that when there were several of them who needed help, I could keep track of who I needed to get to. I also told them to ask about one problem at a time so that I could help as many as possible and not have someone monopolize my time.

Did it work? Well, not as well as I would have liked. I did see several students going and checking their work against mine on the wall. I had a couple of students who just went and flat out copied what I had on the wall onto their work paper. I did not have a rush of questions that I had anticipated, which I'm not sure if that's good or bad at the moment. I suspect that it's not good because I don't think they took the work time as seriously as I think they should have.

I do like that the onus was on the students to check their work against something that was correct before asking questions. I was trying to get them to not only work on the problems but to figure out what they did wrong before automatically running to me with questions and this did help the students who took advantage of it. Would I do it again? Maybe not this school year - but I think it was helpful for those students who did the activity in the right spirit.

Observations on Review Stations

I did review stations with my Algebra 2 classes the first two days back from Christmas Break. I had originally thought I would do them in one day, however we had a 2 hour delay and my normally 50 minute class period was trimmed to 30 minutes. Add to that I needed to pass back their quizzes from the last day before break and get them into the groups, and we were only able to do one activity. The second day, instead of having them do a warm up, I left directions on the SMART Board for students to get back into their groups from the day before and to get needed materials out and get ready to go.

The biggest question on my mind as I went to bed the night before we went back was whether all of the time I put into the activities would be worth it. I had probably spent 5-6 hours between deciding the activities and physically preparing all of them, including the answers. I had chosen activities and designed activities that would be mostly self checking. The dice activity in particular took a long time to find and type up the answers. I had invested a lot of energy in addition to the time and I hope my students got something out of it. Tomorrow for their warm up activity, I am having them complete a brief survey to get an idea of how they think it went.

My observations:

• In only one of my three classes did all groups appear to take all the activities seriously. In one class, one group didn't seem to be taking things very seriously, which was due to one student - I had hoped the other students in the group would pull this student in the right direction and unfortunately that didn't happen. In the other class, two groups kind of got off track - one due to a behavior issue, one due to poor pairing on my pair. I had to make some adjustments on the fly to groups due to absences and the one adjustment I had made ended up with a couple of struggling students as partners. Overall, most of the classes were engaged.
• Many of my students, even the "stronger" ones, did not remember what to do very well. Some of them got going quickly once they asked a question and we went back over how to do it. This was the most evident on the Row Game page for the Properties of Exponents.
• With the 3 x 3 match activities, several groups of students didn't fully work out the problems - especially the add/subtract polynomials. Maybe need to revise the problems so the answers aren't identifiable by the first term on that one? 
• I ended up with 5 groups of 4 (or 3 here or there). With 6 activities, this actually worked well, especially in my first class when I realized I screwed up the rotation and I needed to have 3 groups do the same activity with only 2 rotations left. I got much better with rotating activities in my other two classes.
• I originally was going to have students move, not the activities. However, having 6 physical spots to rotate through was going to be difficult without moving desks (and I have other classes between my 3 Algebra 2s and only 3 minutes between class periods). It worked out okay to rotate activities. However, this kept the partners the same. I think if I do this again, I would make them change partners every 2 to 3 activities.
• I originally had thought 5-6 minutes a station and get all 6 done in a 50 minute class period. Reality was 10 minutes a station and we got 5 done over approximately 70-75 minutes. If I want to do it in one class period, I'd have to shorten activities. However, I don't think they would have worked through as much or gotten as much. Definitely need to use my timer - not just half use it - to keep things moving.
• Overall, I think it was worth it. Would I do it again? Well... it was an awful lot of work. Still pondering that one.

One of my struggling students about halfway through today asked for a copy of the problems so he could take them to tutoring. At first, I had suggested to him to write down the problems he was working on to take to the tutor, but then I realized I had most of it in a printable form. A little later in the period, I think when he had the dice activity, he commented to me something along the lines of "this took you a lot of work." I replied that it did and that I appreciated he recognized that it did take a lot of work. Only kid who remarked that at all in three classes. I'll be interested to see their comments tomorrow on the survey.

Review Stations

My Algebra 2 students didn't do as well as I would have liked on their before break quiz. I had hoped that many would do well enough to master the exponent rules and adding and subtracting polynomials, but in reality, only a handful out of my 65 or so students did. When we return Tuesday, we need to do some review.

I had hoped to put together 6 stations - 3 exponent rules and 3 adding and subtracting polynomials, but at the moment I only have 4. I have spent several hours putting this together and at this late hour, I am questioning whether it will be worth it. My hope is that they will actually work through many of the problems and get much closer to mastering the skill. 

I intend to put the students in groups of 4 to rotate through the stations. I may have to adjust if I don't come up with two more stations. I am anticipating giving them 5-6 minutes per station. I know they won't necessarily get through all the problems, but they should get through most.

Right now the four stations I have are:

1) Match Puzzle for Exponent Rules. 

I printed this on cardstock and will have it cut out already to save time.  This is a 3 x 3 grid that has the questions and answers printed on the interior. They will have to match up the problems and answers.

2) Row Game for Exponent Rules.


In this case, I called it the partner game. Each partner works out their problem and their answers should match.

3) Match Puzzle for Adding and Subtracting Polynomials.



 Same directions as #1, but with adding and subtracting polynomials instead.

4) Cubes - Adding and Subtracting Polynomials.

I have constructed two cubes, one out of blue cardstock, the other of white cardstock. I have numbered the faces and written the following polynomials on the faces:

Blue Cube:
1: 3x^2 + 6x - 7
2: 8x - 2x^3 + 2x^2 - 1
3: 4x^3 - 3x^2 + x + 10
4: 5x^2 + 2x + 6
5: -5x^2 + 6x - 8
6: 6x^3 - 3x^2 + 7x + 10

White Cube:
1: 10x^3 - 7x^2 + 2
2: 3x^3 + 12x^2 - 6x - 9
3: 9x - 10x^2 + 3
4: 4x^3 + 6x^2 - 3x + 5
5: 5x^2 - 4x - 8
6: -6x^3 + 5x^2 + 3x - 8

Students will also have a wooden token that I will have put + on one side and - on the other side. The directions will tell them to roll the two cubes and flip the token to find out if they are adding or subtracting the two polynomials. I have specified that they are to either add blue + white or subtract blue - white. They are supposed to do this at least 12 times. Here are the answers I have left them so they can check:


At the moment (and it's pretty late), my back up for the last two stations is to create flashcards of 12 problems for both the exponent rules and adding and subtracting polynomials and have students work out the problem on the front and check their answers, which will be on the back. I'm not sure what else to do. I'm hoping that by having students move from station to station that they will work through the problems and hopefully get their questions straightened out. I'd like to think that this will be more effective than giving them a practice worksheet of x problems on exponent rules and adding and subtracting polynomials. We shall see.

Addendum:
Thanks to @pamjwilson - I am going to use Exponent Block for my 3rd exponent rules activity. This came via Sam Shah's wonderful virtual filing cabinet (check it out if you haven't been there!).

Addendum 2:
Thanks to @druinok - the final adding and subtracting polynomials activity will be something like I have... Who has....


Students are directed to deal out all of the cards. The oldest person starts and picks one of his or her cards. He or she reads the "Who has" part of their card. Students work out the problem. Whomever has the answer says "I have..." and then reads their "Who has..." at the bottom of that card. Students then work that problem and repeat until all problems are worked (or time is up).

Thanks again to everyone for your help!

Transformations Matching Cards

In my Advanced Algebra 2 class, we discussed the rules for transforming equations (shifts, compressing/stretching and reflections). Tomorrow is our last day before Winter Break (staff still has to go Friday), so rather than give practice problems that won't get done, I put together matching cards. I wrote 6 equations that are transformed from a base function, graphed them on the TI-84 and screen captured them, and wrote the description of how the new graph was transformed from the original. I did 5 sets. Then I put codes in the corner of each card (F#, G#, D#).

Students will work in pairs to match up the triples. I am going to provide them with a worksheet to record their information. Students will need to work through all 5 sets during the class period. I did make some sets easier than others. Cards are 3 1/2" wide by 3" tall and were originally done in Word. Enjoy!

Edit - I updated the box.net file on December 15th - found a typo when doing it in class.

My students got through 2 of the 5 sets in class today (over about 35 minutes or so). We will do the 3 sets they didn't get to the day they return from break. Actually, this will work out nicely, for it will get them back into the swing of things with what we left off with.
--Lisa