Algebra 1 Unit 4 Interactive Notebooks [Revised]: Exponential Functions

This year I’ve committed to posting each unit of both my Algebra 1 and Algebra 2 INBs.

My district is moving to a standards based curriculum, and has identified priority standards for every course. These are the standards we are required to address and assess our students over, so they pretty much form our units.

I normally teach year-long courses (47 minute classes per day). However, due to a lot of district specific things involving SPED students, a new course introduction, and graduation requirements, this year I taught a block course (94 minute classes per day) of Foundations of Algebra first semester, which covered skill gaps students would need to find success in Algebra 1. This semester, I am teaching (most of) those same students Algebra 1 on a block schedule. This means I get to start Algebra 1 from the beginning in the same year and revise my activities and INB pages!

You can find my Algebra 1 (year long class) INB posts here:

Unit 1 | Unit 2 | Unit 3 | Unit 4 | Unit 5 | Unit 6

And my Algebra 2 INB posts here:

Unit 1 | Unit 2 | Unit 3 | Unit 4 | Unit 5 | Unit 6

And finally, my posts from this second round of Algebra 1 here:

Unit 1 | Unit 2 | Unit 3

 

The fourth standard we cover in Algebra 1 is A.CED.2 (again, but the first time we cover it with linear functions and this time we discuss exponential functions):

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

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Skill 1: I can identify when a relationship is exponential

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As mentioned in my post from my first time teaching this unit this year, my students put everything in the wrong place with a blank inside to this foldable, so I added some typed pieces on the inside of “ex” and “look for” so they would know where to put information and examples. We used the same card sort as I did the last time, but I had them work with a partner. After they finished the sort for the second time after we discussed the definitions, they split up the cards so that each of them had 1 graph, 1 equation, and 1 table for both linear and exponential functions, which they glued in for examples. This meant I needed less copies, and also the examples fit on their page more nicely.

Skill 2: I can graph an exponential function

I edited the instructional page on this to include tips for some of the things my yearlong Algebra 1 students constantly forgot, and I think it was a nice reference for this group of students to look at. I also changed one of the original functions I had used, because I didn’t realize it had a negative multiplier when I was using my snipping tool to get examples. We just moved the parentheses when we did it last time so that it would work out, but I just chose a different problem this time. I still think the linear example on here is the most important, because my students tend to just do the thing we just learned by rote and not actually think about it, and this is a good reminder that you NEED to think about it, in case something doesn’t fit the thing you JUST learned.

Skill 3: I can write an equation to represent an exponential function

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I didn’t change anything on this page from the last edition – my students really excelled at this skill and I think there is an example for most problems they would encounter, like when the pattern is dividing or when the table is actually linear. This group of students struggles a lot more with their math facts, so we had to discuss a lot more how to figure out what the y-values were being multiplied by, since these students couldn’t just look at it and see the number.

Skill 4: I can write an equation to represent exponential growth or decay

I realized the last time I did this unit that this skill needed a lot more scaffolding. I put more examples in and put typed instructions for finding the rate/multiplier, since many of my students forgot to write the instructions last time and then were stuck. On the graph page, I included an x/y table, which was a small step that went a long way in my students remembering how to graph once they had written an equation. I like this iteration of these notes.

We also did the World Population Project again, but this time I did it before they took their test, and I think it helped them get a lot more practice before taking the test. Most of my students wrote all correct equations for the exponential growth and decay test questions!

 

You can find all of the files for these notes here, in PDF and Publisher form.

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Algebra 1 Unit 6 Interactive Notebooks: Working with Polynomials

This year I’ve committed to posting each unit of both my Algebra 1 and Algebra 2 INBs.

My district is moving to a standards based curriculum, and has identified priority standards for every course. These are the standards we are required to address and assess our students over, so they pretty much form our units.

You can find my Algebra 1 (year long class) INB posts here:

Unit 1 | Unit 2 | Unit 3 | Unit 4 | Unit 5

And my Algebra 2 INB posts here:

Unit 1 | Unit 2 | Unit 3 | Unit 4 | Unit 5 | Unit 6

And finally, my posts from a 2nd go around I’m teaching of Algebra 1 here:

Unit 1 | Unit 2 | Unit 3

 

I am behind, friends. I have two new posts due for the 2nd go around of Algebra 1, this one and the other unit that goes with this standard, and one for Algebra 2 that need to be written. Fourth quarter is happening, everyone.

The 6th standard we teach in Algebra 1 I split into two units. It is A.REI.4:

Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions.

I split it into a unit where I teach the background skills needed for solving quadratics – basically working with polynomials and being able to use vocabulary, add, subtract, and multiply them. Then the second unit contains all the actual methods for solving quadratic equations.

So, the working with polynomials part:

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Skill 1: I can classify polynomials by their degree and number of terms

 

This should look familiar from our Algebra 2 polynomial unit. The terms page comes from Sarah Carter, and she also uses a Frayer model. You can find her post here. Students are always annoyed by all the writing here, but then they appreciate it later when they have this reference to use when they don’t understand the words in a question or need to classify their answer. One of my class periods really enjoyed trying to guess the terms before we wrote them using their knowledge of language, which was really fun! I realized there are a lot of different prefixes that mean “two” and “one”. We also had fun brainstorming other words that start with “mono”, “bi” or “tri”.

Skill 2: I can add and subtract polynomials

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This page is also the same one I use in Algebra 2. I find it works out pretty well for both groups, working more as a review for Algebra 2 and an introduction for Algebra 1.

Skill 3: I can multiply polynomials

 

This is the first time my students have worked with algebra tiles, and this exploration becomes pretty important because we then do explorations with algebra tiles for both factoring and completing the square. It’s good to let the students struggle with this for a bit and to go around and individually discuss with them whether their answers make sense or not. Even though many of my students profess to hate the tiles, they also admit that it makes the algebraic method seem much easier and that it makes more sense when they’ve done the tiles first.

I also really liked the side by side comparison I did with the tiles vs the box method in the second page, because it helped the students who DID really like using the tiles see why the tiles would not always be helpful and that there were faster, more efficient ways to do it. I think that this year I have really gotten better in relating our work with algebra tiles to the actual algebraic method, and it’s actually improved my own understanding of the algebra, so I think that’s a win!

 

The only thing I want to change about this unit in the future is in the practice, not the INB pages. When we took a quiz over this, my students pretty much across the board either added/subtracted every problem, or multiplied every problem, without paying attention to what they were actually being asked to do. I think adding more practice where addition, subtraction, and multiplication were mixed would clear this up pretty quickly.

 

You can find the files for these pages here, in PDF and Publisher forms.

Focus (Lent 2018)

Lent and the Easter season are my favorite times of the year. I love the preparation time of Lent and participating in the prayer, fasting and almsgiving to prepare ourselves to rejoice at the resurrection of Jesus on Easter. It’s like New Year’s, only more meaningful for me. A chance to reform yourself, to start something new.

However, every year when Lent rolls around, I struggle with what to “give up”. I really dislike the superficiality of the “sacrifices” many people around me make for Lent, and I want my Lenten sacrifice to be really meaningful for me. When I was younger, I gave up things like chewing gum, chocolate, or drinking pop. I gave up pop for several years, which actually kickstarted my vow to not keep any pop in my home anymore, which has transformed into me currently being over a year pop-free entirely! So these more superficial things definitely can have an effect on your life and making it richer and healthier and better. But this year, I couldn’t think of anything to give up that would really affect me that much.

My first thought was to give up superfluous spending. Then I realized how stressed out about money I’ve already been this year and realized that putting more monetary restrictions on myself would not be beneficial to my mental health. I thought of fast food, which I’ve also done in the past, then realized that the only times I eat fast food anymore are when I’m traveling or occasionally when I have a really awful day. I just wasn’t coming up with anything.

Then I thought of the years where I’ve ADDED something, and immediately knew I was on the right track. My facebook “on this day” had started giving me memories of the year I posted a Bible verse as my status every day of Lent, and I remembered the time I did Liturgy of the Hours every day as well. My church going habits were quite strong when I was in college, but dropped off a lot when I started teaching for many reasons. I want to get back into being involved with my church, which is something I realized during the meditation and mindfulness course my staff did for 8 weeks recently. When I am more involved in Church, I feel calmer, more peaceful, and get all those health benefits you get from meditation or mindfulness practices.

I have had a habit for a few years now of listening to Christian music every Sunday on my drive to and from church, to set the tone. I decided that for Lent this year, I would extend that. Any time I’m in my car, only Christian music. Nothing else. Can’t change the station.

I was not expecting it to be as impactful as it was. Forcing myself to listen to the talking parts and not station flip constantly, listening to the positive messaging of the songs, I found myself with so much more focus as I was driving throughout Lent. It started to spread beyond the car, too, as I found myself humming the songs when I was preparing lessons, listening to them while I worked on grad school assignments, singing them to myself as I got ready for bed. I could focus on other things because this one decision was always made for me, and I didn’t have to form opinions on every song that came on or decide if I wanted to hear it. This definitely became a meditation practice for me, and changed my temperament drastically through the season of Lent. In general, I’ve felt calmer, and even when I’ve gotten stressed, I’ve been able to come  back to my normal mood more quickly.

Easter came and went last weekend, and I got in my car on Monday…and I didn’t want to change the radio station! It felt weird. I’ve changed it a few times since, but I keep gravitating back towards the Christian station.

I was surprised that such a small change had such a big impact on my life. I’ve been able to focus on thinking about other things in my life while I’m in the car, and like I said, the meditative/mindful health benefits have extended into the rest of my days. I think considering our actions more mindfully and choosing to make one small change could really reinvigorate us for the rest of the school year!

Is there a location or a part of your day that you could make a small change in? Maybe you’ll decide to end or start your day with a different routine, or pick the car as your place for change also…leave me your ideas in a comment!

I’ll leave you with my favorite song that I’ve kept coming back to again and again…several times during Lent I found myself sitting in my car in my garage or in a parking lot, listening to the end of this one and taking the time to be thankful and peaceful – at least for the duration of the song.

Book Recommendations (Vol. 05)

The first quarter of 2018 is almost over! What?!

That means it’s time for the next volume of my book recommendation posts. Each quarter, I post briefly about the top 5 books I’ve read in that quarter.

I’ve also recently started a mini project with Megan Morgan in my district to get teachers to share what they’re reading each Wednesday – you can follow the #dcsdpln hashtag or either of our twitter accounts to catch these posts! It’s super fun to see what my colleagues are reading, either for fun or for professional development opportunities, or for grad school.

Anyways, on to my recommendations for this first part of 2018!

Read previous editions:

Vol. 01 | Vol. 02 | Vol. 03 | Vol. 04

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I wrote my last post of 2017 when I had read 80 books. I ended up reading 88 books in 2017, and I have read 13 books in 2018 so far! Some of this top 5 is crossover from those last 8 books.

In Defense of the Princess: How Plastic Tiaras and Fairytale Dreams Can Inspire Smart, Strong Women – Jerramy Fine

This is an exploration of the attributes that make a princess – from our Disney heroes to the real life princesses still ruling across the world today. Fine explores character traits that they all have in common and how we can embody them in our daily lives, and explores her personal journey to becoming a princess.

This was a great (and fast) read – a defense of the idea that women can be strong, smart, and accomplished without giving up the desire to be feminine and to wear a tiara. It’s about how we can embrace the qualities that make a true princess – strength, kindness, diplomacy, serenity, rationality, and a giving spirit. I loved it. Perhaps a particularly relevant read as we all join in the fervor of welcoming a new princess, Meghan Markle!

Stamped from the Beginning: The Definitive History of Racist Ideas in America – Ibram Kendi

This book uses five major figures from American history to lend context to a comprehensive description of the development of racist ideas in our history. How did they begin, how did they spread, why are they still supported? Assimilationists, segregationists, antiracists, racists. Activists and artists. Every single historical figure you have heard of and many that you have not.

This is what it’s billed as – a comprehensive history of racism in our country. It was highly recommended by a professional development presenter that has spoken to our staff several times whom I intensely respect, and so I did not shy away from getting my hands on this 500 page work. I found myself pretty much every other page going, “I did not know that. I DID NOT KNOW THAT. Oh my god, I did not know that.” Lots of hard truths. Well worth my time reading the 500 pages. It’s not a light or fun or easy read, but if you want to see a better America I think reading this would be a good starting point on that journey.

Tempests and Slaughter – Tamora Pierce

This is the first book in a series following the mage Numair, first seen in the Immortals Quartet also by Pierce. We backtrack to when he was a boy, in mage school, meeting some of the other infamous characters from that quartet and discovering the roots of their relationships and powers. You see Numair (called Arram still here) discovering the vast potential of his power, his struggles with the idea of slavery and nobility, and the beginnings of some very dark goings on.

I grew up reading Pierce’s works in the Tortall universe – The Immortals, The Lioness, Protector of the Small, Daughter of the Lioness. I met her when I was probably 13 years old and have a signed book from her that is one of my most treasured books. Her works were a strong influence in developing my understanding of feminism and what it means to be a strong woman. With that in mind, the entire experience of reading this book I felt like I was coming home to Tortall (well, Carthak). Everything felt familiar and lovely and I was immediately enveloped in the Numair backstory. I am so deeply in love with this world and the other Tortall books formed so much of myself that I really couldn’t do anything but love this. It is very much a background building book for the rest of this series and someone not already in love with this world may complain that not enough happens in this one plot wise, but I cannot cannot cannot wait for more of this.

Truly Devious – Maureen Johnson

Stevie is obsessed with the Ellingham Academy mystery – many years ago, after the unique private school on top of a Vermont mountain opened, the founder’s wife and daughter went missing and are believed to have been murdered by someone using the name Truly Devious. Stevie gets accepted to the Academy with the goal of solving the case, but does not expect to get sucked into one of her own. Death is back at Ellingham Academy, and Truly Devious is, too. Maybe? Murder, or an accident? Related to the first, or a copycat?

I am genuinely atrocious at figuring out clues in mystery novels, and so I literally have NO IDEA what is going on with this case and cannot wait to get my hands on the next one to get some more information. This was gripping and mysterious from the start, and I love dual timeline novels as well. Who are these people Stevie is going to school with????? Is one of them a murderer???? What even? Maureen is an expert mystery builder, and all of her characters seem so incredibly real.

The Lost Colony (Artemis Fowl #5) – Eoin Colfer

10,000 years ago, after the battle for Irleand between human and fairy, all of the fairies moved underground. The demons, however, refused to surrender. Their warlocks used their magic to take the island they lived on out of time and into Limbo. Now, the spell is breaking down, and demons are starting to appear in the human world unexpectedly. Artemis discovers this, and so does another young human girl – could this be his intellectual equal? Of course Artemis and Holly have to try to figure out how to save the day.

The Artemis Fowl books are a series I began reading when I was in the preteen target audience, but only made it through book 3. I didn’t even realize there were more books until I saw an announcement about a film being made and looked into re-reading them. These books are still a delight to read as an adult – fairies, legends, technology, all of the dwarf potty humor – and I am almost finished with the series. This installment is my favorite so far because time travel always blows my mind to think about and it’s fascinating to explore the different structures authors use to set it up. The climax of this one had me on the edge of my seat, wondering how they would be able to get out of this situation because there are 4 more books so they have to survive! I highly recommend this series, especially if you are looking for books to suggest for your middle school aged kids, or one to read with your own kids or out loud in class….or just for you! There’s also a silly code that runs along the bottom edge of the pages in each book, which 10 year old me was STOKED to decode for herself but which adult me just looks up on the internet to read the translation 😉

 

Happy reading! Share your recommendations in the comments!

Algebra 1 Unit 3 Interactive Notebooks [Revised]: Systems of Equations

This year I’ve committed to posting each unit of both my Algebra 1 and Algebra 2 INBs.

My district is moving to a standards based curriculum, and has identified priority standards for every course. These are the standards we are required to address and assess our students over, so they pretty much form our units.

I normally teach year-long courses (47 minute classes per day). However, due to a lot of district specific things involving SPED students, a new course introduction, and graduation requirements, this year I taught a block course (94 minute classes per day) of Foundations of Algebra first semester, which covered skill gaps students would need to find success in Algebra 1. This semester, I am teaching (most of) those same students Algebra 1 on a block schedule. This means I get to start Algebra 1 from the beginning in the same year and revise my activities and INB pages!

You can find my Algebra 1 (year long class) INB posts here:

Unit 1 | Unit 2 | Unit 3 | Unit 4 | Unit 5

And my Algebra 2 INB posts here:

Unit 1 | Unit 2 | Unit 3 | Unit 4 | Unit 5 | Unit 6

And finally, my posts from this second round of Algebra 1 here:

Unit 1 | Unit 2

 

The third standard we cover in Algebra 1 is A.REI.6:

Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

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I finally wrote a post about my journey through the other side of this index page – the evolution of my student proficiency log! You’ll notice if you’re a regular reader of my INB posts that I have been avoiding talking about them for awhile and that I only show the index side in these posts, and I am finally ready to share, but they deserve their own post 🙂

In my first post about this unit earlier this year, I said:

I pretty much like the way I organized this, except for the way I presented the elimination method

This means many of the pages I used are unchanged – but there are a few  besides just the elimination that I made some minor changes to. However, I probably won’t have as much to say about each skill and you can read my original post about this unit to get more insight into the other pages.

Skill 1: I can determine if a given point is a solution to a system of equations

 

Getting down those definitions and learning how to check possible solutions. None of my students use the “storing values in the calculator” method to check their solutions, but our curriculum coordinators in the district specifically told us that we need to show them this, so it’s there.

Skill 2: I can graph a system of equations to find the solution

First we talk about the possible outcomes when solving a system of equations. This was more fluid for these students than my first Algebra 1 group this year, because I added a page that is essentially just like this into their original solving equations unit. The only different thing about solving equations really is that you also have to know what the graphs look like, and that there is an x AND y value.

I changed the graphing pages to be cleaner and reduced the number of examples. Really, I have discovered that my students just need more practice graphing – by themselves, without me. So these notes have one example of each possible outcome, and then they practiced. I like these notes, but I really struggled to get the x and y axis to show up in Publisher the way I wanted to, so I ended up just drawing them in with a permanent marker before I made copies. I then tried to scan this copy so that I could just print it in the future, but the office copier hates me recently and it did not turn out well. It’s included in the files anyway, so you can see how poorly this ended up. Drawing in the axes each time you make copies is really not that bad, compared to a long and drawn out fight with the scanner!

Skill 3: I can solve a system of equations when given the value of one variable

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The only thing I changed here from before is that we still did a notice/wonder with the systems included in these notes, but I did not have them attach that notice/wonder to their notes, we just did it on the board. I still really like this as an intermediary step, since it’s essentially a review of solving equations once they substitute in that known variable.

Skill 4: I can solve a system of equations using substitution

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I kept this one exactly the same. Upon using it with this group of students, I think it may be beneficial to cut the number of examples here as well, like I did with the graphing. We hand wrote an example with no solutions in the center fold of this stapled notes section, since there ends up being two blank half pages in the middle, but I plan to just change it to be two one solution examples and one special case example.

Skill 5: I can solve a system of equations using elimination

I am moderately happy with how I set up the elimination method this time. We started with looking at what happens when you multiply entire equations by a constant and discussing what elimination means as a word. We looked at the equation x+2=4 and solved it, and then I asked what would happen if we multiplied the entire equation by a number of their choice. They chose 4. They were adamant during the entire process that the solution was now going to be x=8, even down to the moment where we were ready to divide 8 by 4. Their minds were blown. I think they’re still mad at me about it. But it stuck in their heads that multiplying these equations by some constant doesn’t affect their answers!

We then began with systems that are already set up for elimination – where either the x coefficients or y coefficients are already opposites. This was the step I missed last time. This was easy for my students.

Then we moved into how you can GENERATE this opposite effect if it isn’t there to begin with. I think one of the issues is that my students just aren’t fluent enough in multiples to think through this well. “What do 8 and 12 both go into?” is the hardest question in the world for them. I think I need to bring multiplication tables into this the next time I teach it. Anyways, we practiced just getting these opposites a few times, and then went into the last page of notes which is three examples of completing the elimination method all the way through to a solution.

I think part of my issue with teaching elimination is that it is my personal preference for a solution method when I solve systems myself, and so I do it so automatically that I think I have trouble thinking through all of the intricacies that my students are going to struggle with. Would love to accept suggestions from people on this one!

Skill 6: I can write and solve a system of equations to represent a situation

I didn’t change these last two pages at all from the last time I taught this. I did, however, struggle to get students to actually consider the most efficient method when they were solving systems. I tell them that they can choose whatever method they want to use, but I kind of screwed myself over with that statement when several of my students decided they were going to solve every single system by graphing it – and they were not quite proficient in graphing. I ended up encouraging them to look back at these notes and having them think about what they system was “set up for” and I won some of them over. The main issue was that they wanted to use graphing every time (or some of them were on team substitution or team elimination), but they really only knew how to use that method if the system was already set up for it. So they would go, “I want to graph this system but what is the y intercept” and I would respond “that equation is not in slope intercept form” and then they would give up. I’ll have to work on that.

 

You can find the files for these pages here, including a subfolder with the previous versions of the pages.

 

 

 

Student Proficiency Logs: A Journey

If you have been following my Interactive Notebook posts closely, you will have caught a hint of this journey that I’ve been on. It really begins last year, before I was using INBs in all of my classes, and it did not involve INBs at the time.

One of my big goals even in my first year teaching was to try to help my students process where they are at in the course. I want them to be able to answer questions like:

  1. How did I do on this assignment?
  2. What does the score I got mean?
  3. What skills do I need more work on?
  4. What am I missing that I need to make up?
  5. What do I understand well / what am I proficient at?
  6. Do I need to schedule myself for Miss Mastalio during intervention period?
  7. What is the end goal of what we’re doing in class?
  8. What are we actually working on in class right now?

My first two years teaching, I printed out student grade reports at the start of each week, passed them out, and answered any questions students had about them. That pretty much only answers questions 1 and 4 on the list above. (Maybe question 6 but we did not have an intervention period then so it didn’t apply.)

During the 2016-2017 and 2017-2018 school years, I created a grade log form to go with this Monday activity. I still printed out grade reports for each student and handed them out on Mondays, but alongside this log page.

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In addition to questions 1 and 4, this now started to address question 5, and 6 came into play as we added a form of our intervention period.

This year, our district is starting the transition to standards based grading. With this change, I have thought a lot more about the meaning of scores and how to effectively communicate goals and proficiency to students.

The past several years, I also had students filling out an opener sheet daily. They were tasked with copying down the opener problem and the lesson objective for the day, 4 out of 5 days every week. This kind of addressed question 8 from the list.

This year, I changed my grading policies so that student practice work does not affect their final grade anymore – only assessment materials do. I knew that this meant it would be hard to convince them of the usefulness of writing down openers and objectives, so I decided to do away with that sheet. We still do an opener and read the objective daily, but we just do them as a class and don’t record them individually. That’s another story as well (perhaps another blog post?) but I think it’s been pretty equal in effectiveness to what I was doing before, plus I don’t have to grade 60 opener sheets every Friday.

I also started using INBs with all of my classes this year, so my thought was that I could incorporate the grade log and the opener sheet components into something in their INB. The start of this school year got away from me, so I didn’t quite get this done. My index pages for my INBs do address questions 7 and 8 though – they clearly show all the skills that we are working on day to day, and also describe the big standard that is the end goal of each unit. We talk through how these all fit together each time we add to our INBs. The objectives I post in my classroom are exactly the same wording as our skills in our notebooks. This part of my question list started to fit together nicely.

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On the back page of the indexes, where it wrapped around the other side of a page in the notebook, I originally just had a large heading with the name of the unit. I realized that something similar to my old grade logs could go in that space, and began to work on this proficiency tracking log.

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Wow, is this way too complicated or what? Can any of you even figure out how to fill it out? What was I thinking??????

The thought was that students would check their grade on a chromebook or on their phone (we are being limited more severely on printing/copies this year so I no longer feel like I can print a grade report for each student each week) and then fill this out with the results. At this point, we’re really getting to all the questions on my list. They know that scores of 3 or 4 are proficient, and 1 and 2 are not. A 0 means they are missing that assignment/assessment. Question 1,2, and 4, check. Questions 3 and 5 are answered by them identifying which skills they are proficient in. Question 6 can be answered by them considering their grade and identifying if there’s an assessment they’re missing. Questions 7 and 8 are answered by the index side of this page and our discussions in class.

But this form is just way too much for a quick check and is harder to figure out than it is helpful. I immediately realized this when I tried to have students fill it out for the first time. It was bad.

 

Back to the drawing board!

I’ve finally come across one I like:

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They check Campus and their grades, and then use those scores plus their own reflection on their learning to give themselves a proficiency mark from 0-4. I have signs posted in my room that describe how each of those numbers would feel, and I give them a brief reminder on Tuesdays when we mark this: “0 means you literally have no idea at all what the skill even is, 1 is you just starting, 4 is give me any test right now with this and I will ace it”

I write on the board which skills they should be marking – the ones we’ve covered in the last week – and remind them that they can adjust old markings if they feel they’ve improved.

Questions 1,2, and 4 are addressed by their quick check of the gradebook. Questions 3 and 5 are addressed by their reflection on the skill statements and their scores. Question 6 is easy if they missed an assignment, or if they seem to be marking all 1’s and 2’s I suggest they schedule themselves for me during intervention time. Questions 7 and 8 are essentially the questions they should be answering to themselves as they reflect.

I’m happy with this. I feel like it could benefit from some written record of their scores, but I don’t want to veer into making it too complicated again. I may add some boxes in the empty space at the bottom for them to record assessment scores for the unit. The goal is for this to be a quick and automatic process at the start of class each Tuesday, and I think we’re getting closer to that.

I also like the way this looks because I think it’s easy to explain to a parent, and so I could use this at conferences to communicate strengths and weaknesses of each student without relying so heavily on the student’s grade for that explanation.

 

Do you use any sort of grade/proficiency tracking with your students? How do you run it and do you like it? Leave a comment and let me know, I’m definitely looking for better ideas here!

You can download an example of this index/proficiency log page here for PDF or here for Publisher.

Algebra 2 Unit 6 Interactive Notebook: Solving Polynomial Functions

This year I’ve committed to posting each unit of both my Algebra 1 and Algebra 2 INBs.

My district is moving to a standards based curriculum, and has identified priority standards for every course. These are the standards we are required to address and assess our students over, so they pretty much form our units.

You can find my Algebra 1 (year long class) INB posts here:

Unit 1 | Unit 2 | Unit 3 | Unit 4 | Unit 5

And my Algebra 2 INB posts here:

Unit 1 | Unit 2 | Unit 3 | Unit 4 | Unit 5

And finally, my posts from a 2nd go around I’m teaching of Algebra 1 here:

Unit 1 | Unit 2

Our 6th prioritized standard in Algebra 2 is A.APR.2:

Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).

Really, the goal is to use the remainder theorem to solve polynomial functions.

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I am extremely happy with the way this unit turned out – I think the sequencing was good, I think the notebook pages I made are effective, I think we did good activities along the way, and of course the most important, my students demonstrated really strong proficiency on this standard. I think it’s my best-taught unit so far this year, so I’m taking a moment to acknowledge myself.

Skill 1: I can multiply polynomials

This is a review – we touched on multiplying in their second unit when they worked on quadratics, but that was around October and I don’t expect them to automatically remember it in February. I also gave them two different methods to use – the distributive property and the box method. The instructions for the box method one come from Sarah Carter, but I didn’t keep all of the same examples. Every single one of my students used the box method after we did these notes, but I think it’s worth them knowing that that isn’t THE ONLY WAY to multiply polynomials. We ended up practicing this skill on a day we had a late start from snow (and we watched Olympic highlight videos for the first 20 minutes of class…) but they still got enough practice to feel confident about it again.

Skill 2: I can divide polynomials

We learned three different methods to divide. We did the same examples with all three methods, so that they could start to see the relationships between the methods. I then let my students choose their preferred method, with the caveat that synthetic division only works in certain cases. Pretty much all of them settled on synthetic division when it works and the box method when it didn’t. I think they had traumatic memories from long division with integers that made them not want to do long division with polynomials. We learned a lot of vocabulary that they did not remember, also: dividend, divisor, vinculum….

I need to add some more words to the box method notes because later, my students kept forgetting where the dividend showed up – did it go on top or in the boxes? A sentence or two in their notes would help with this.

Skill 3: I can use the Remainder Theorem to identify factors and zeros of polynomials

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Before we did these notes, we did an exploration where my students basically figured out what the Remainder Theorem was on their own. True, the statement of it is what the standard says so it was already on their index page, but the way it is stated in the standard is extremely mathy and not in student friendly language.

Once they figured it out, we put it into notes with this one example. We used this to practice identifying values of f(x) and to find factors.

Skill 4: I can solve a polynomial function

I broke this down further into several pieces – we started with just factoring the polynomials. This began with a piece of reasoning about how the Remainder Theorem would be useful to us that made my students incredibly frustrated because I would not tell them the answer until they figured it out themselves.

We had already figured out that if we could find factors, that would be nice, because they go into the polynomial evenly without those pesky remainders! So I wrote the stuff in black up on the SMART board and posed the question, “where on the graph would I find P(a)=0?” They were stumped. We had to break it down. What does it mean that P(a)=0? We eventually figured out that we were looking for where y=0, and then we were stumped for a bit longer, and then we figured out that we needed x values, and eventually one of my students yelled out X-INTERCEPTS! and they were all very excited.

Then I handed them the factoring notes page and they were like, “Miss Mastalio, it’s right on here, why couldn’t you just give us this???” Oops, made them discover math on their own. Anyways, once we figured out we needed the x-intercepts, it was pretty straightforward from there. They graphed, looked for integer x-intercepts, and then used synthetic division. I actually had the realization for the first time ever that you don’t even need to have them look for a GCF first – if you use the synthetic division, you’ll end up with the GCF as a result at the end! UNLESS there is an irreducible quadratic in the factorization, but I decided I was ok with not having a GCF factored out in that circumstance since our priority standard is focused more on using the Remainder Theorem to find factors and our assessment is mostly focused on using this to solve the polynomials, and you’ll still end up with the same solutions whether or not you factor out the GCF.

ANYWAYS, we factored polynomials to an end result of only linear and quadratic factors. Then, we reviewed solving quadratics with complex solutions. This is one of the units that I was not entirely happy with when I taught it at the start of the year. I did not focus enough on the complex solutions and focused more on solving quadratics with real solutions, which is an Algebra 1 standard. When I change this next year, I may not have to do as intense of a review during this unit. We used the Quadratic Formula until we hit the “taking the square root” point of simplification, and then we looked separately at simplifying those square roots. I like the way I explained this at this point in the year, and with a few adjustments to make it an introduction instead of a review, I think I will use these notes in our Unit 2 next year to teach it the first time.

Once we had practiced factoring and finding those quadratic solutions, we finally focused in on solving. I broke it into 4 steps: factor, find the linear solutions, find the quadratic solutions, write your solutions as a list (making sure to check that you have the same number of solutions as the degree of the polynomial). The students responded really well to writing the synthetic division as a continuous chain, once they realized that they needed to divide their answer by the next factor and that they could get away with not rewriting it.

 

You can find these files here, in PDF and editable forms!