What are Solutions?

So before I get into all the mechanics of equation solving I like to talk about what a solution is. I used this really great resource to guide my lesson - Mathematics Assessment Project CLASSROOM CHALLENGES Sorting Equations and Identities.  And we followed up by completing this worksheet. 

Then we filled in our ISN pages...

and here is the document above.

Expressions, Equations & Inequalities

This is a really starting unit in Algebra One and often is what people think of when they think of Algebra. An internet search of “what is algebra” gave this as a response

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols.

This is pretty fundamental to the study of Algebra – it gives us a symbolic way to work with numbers and to generalize. So this unit gets us started in how to do that. Students really need to understand what they are doing and not just memorize.

Sadly the symbolic nature of Algebra gets a bad rep and people can get very frustrated and joke about their frustration.

We are just going to move on from this and try to shed some light on how to teach this unit using ISNs.

Just to give some context, where is what we are working with from our curriculum:

Essential Questions:

·         How does one create an equation/inequality using symbolic algebra, specifically how are operations used in building? 

·         How are inverse operations used to deconstruct an equation/inequality by creating a series of equivalent equations/inequalities ultimately resulting in a solution?

·         How and in what ways do you use symbolic algebra to represent and explain mathematical relationships and real world situations? 

·         What is a solution set for a linear equation or linear inequality?

·         How do we utilize technology to justify a solution method?

Corresponding Big Ideas:

·        Obtaining the solution to an equation or inequality, no matter how complex, always involves the process of undoing the operations.

This is a long unit, so here is my TOC listing - I'm going to have to break down my foldables into a few blog posts (thus this is part 1). just remember my handwriting disclaimer...

I started the unit with the students being given a variety of mathematical items - numerical expressions, algebraic expressions, equations and inequalities. Check out my sorting cards HERE.  They had to sort them into 4 groups any way they saw fit (they weren't given those category names) - then we discussed and played around with what they came up with (at one point I gave them the category names and asked them to adjust their sorting into those groups). This was to warm them up to the vocabulary of these symbolic representations. From their we did little frayer diagrams of each in our notebooks:

For a structured right hand side sample, check this out. 

equation solving "practice" on the right side (my students should know these without instruction from last year's class). You can also find more equation solving ISN samples here. 

I added the "backwards inequalities" from the foldables I found on line. Here is what I started with...

and corresponding practice on the right hand side opposite each frayer diagram. Some of the "practice" was somewhat informational as you can see (I included those that were).  What's a frayer diagram? Learn more HERE.

Sequences Algebra One

I am used to teaching sequences and series in Algebra 2 & Pre-Calculus. But four years ago when we were given the task of aligning our Algebra 1 curriculum with the Common Core - lo & behold the first unit was "Patterns" which really was all about sequences - arithmetic, geometric and other sequences (Fibonacci, etc). And they wanted us to use function notation rather than sequence notation and students were supposed to naturally discover the formulas for a term in an arithmetic sequence and geometric sequence, ha! We gave it our best shot. After teaching this unit for three years I see now that the Common Core no longer has this as their first unit and that sequences are blended into other units. We have fine-tuned it over the last 3 years and aren't going to toss it just yet. Connecticut is making their own "Algebra for All Common Core" curriculum, so we'll just wait and see what they want us to do (it will include an end of year exam for all students in the state taking Algebra 1).

It's actually kind of fun to start with patterns, it gets students thinking mathematically. I have students do some exploring with patterns in worksheet they get for homework on the first day of school. You can find that here.

Then we develop the concepts & vocabulary. And "come up with" the arithmetic formula using what they already know about the slope intercept form of linear equations (y = mx + b) from there they see they have to find the "0 term" for the equation/formula. Geometric is harder to develop, so we kind of just give it to them. We revisit both sequences when we do linear and exponential functions. Overall it's a short unit with a cool little performance task at the end.

Foldables for Sequences

I use three sets of foldables for my ISN. Remember my page numbers are all wrong, I'm going to be better about the left side even & right side odd this coming school year. Also these are my first attempts at foldables and I'm not crazy about how everything lined up. I got better at that as the year progressed. The heading for each is a link to access that foldable via box.com.

INTRODUCTION TO SEQUENCES

ARITHMETIC SEQUENCES

GEOMETRIC SEQUENCES

Those are all Left Side Learning Pages (3 in all for this unit). The Right Side Reflect pages for opposite arithmetic & geometric are:

I don't have the actual examples - just notes to myself as to what to have students do.

And we do some cool stuff with the recursive rule and our calculators (TI-83 or TI-84). You can set them to generate a sequence recursively just by kind of following the rule. Type in the first term, [enter] then type in [+] "the common difference" [enter], You will see the first & second terms on the calculator screen. Then just keep typing in [enter] to generate more terms (the 3rd, 4th, etc). Not especially efficient if you want the 50th term but it's cool to do. And we talk about this is how computers generate terms. But they can do it super fast.

And if you are interested in my performance task, it's kind of cool. I made a honeycomb model out of cardboard for students to see what the core of the table looks like. Of course that is at school so I can't post a picture yet! Honeycomb Performance Task with the exploratory task we do the day before the PT and honeycomb graph paper.

Things I Learned From My First Year Doing Interactive Student Notebooks

Planning and organization The summer before I started doing the ISN I did a lot of reading on the Internet. I searched "interactive student notebooks"  and read many blogs to see how different people did them. One thing I found is that everyone said do not do not try to create your ISN pages beforehand. Before reading this, I thought I would use the summer before to create all the pages for my algebra 1 class. No no no no. I heeded the advice and throughout the year I created the pages as we went. This was very important, Because even when I planned a little bit ahead and perhaps made a few pages I invariably made changes on what I had done. So if you are an organized advanced planner like I am, you need to let this go and just go with the flow through the year and create things as you need them. But there was some very useful advanced preparation that I could do. I spent some time last summer searching for examples of different foldables for the various topics I teach. I searched on the internet and I searched in Pinterest. As I found things that would be useful I saved them in folders for each unit in my algebra 1 class. This way throughout the year when I was creating foldables I could open a folder for a unit and find things and look at things that I found over the summer. Very useful to do that advanced searching and saving. But I still did some searching as I went during the year, because you can't always anticipate what you might want a foldable for.   Supplies and Such As I said I am a very organized person, so I wanted to be organized in my approach to having students make their interactive notebook. So again from the reading I did over the summer I found different ways to organize. One important thing is the arrangement of my classroom. I have my student desks in groups of 5 but try to have only four students sit in a group so the fifth desk is open to me to sit and help the students. My classroom is all closed up for the summer. So I can not show pictures of my classroom and how it is organized. Later in August I will take pictures and post a blog about my classroom set up. But here is a template for my seating chart.  I change my seats every month. I have a large white board in both the front and back of my classroom. I have an LCD player affixed to the ceiling through which I can project images using my document camera. The document camera has been crucial for smooth completion of foldables and notes. As we create the foldable and notes pages I write along with the students and they can see what I am doing with the document camera projecting the image. I only started using the camera in January and just love it and wish I had started sooner. I may have to buy one if I can't borrow one from the school library again. I have files on the counters in my room organizing any foldables that I've created and extra foldables for students who need one because they were absent or some other reason. I also have drawers where I keep colored paper and my original templates in files. I will have to take pictures of all these and include them in the blog post I will do in August that shows my classroom set up. Training My Students As student walk into my classroom they see the student objectives for the block posted on my front whiteboard. They also see what pages need to be added to their TOC or table of contents. I tried to have any foldables we are going to use on each grouping of desks so there is no time spent in class handing them out. Also each group of desks has a basket of supplies. In there they will find scissors, glue sticks, tape, and colored markers and pencils. If the basket of supplies are still on the side counter the students know to retrieve a basket for their group if they see a TOC addition on the whiteboard. We don't do the interactive student notebook every class but we do it most classes. How I use the ISN in my Lessons Generally I like to develop new concept by having students do some kind of exploration, usually in groups. Sometimes this will be by following instructions on my whiteboard and sometimes it will be using worksheets or a packet of information or problems. I use the ISN to summarize concepts. I don't use them to drive the lesson. It is a way to pull things together and solidify concepts and write things out for the student reference. Then the right-hand reflection page is used for students to do practice or to come up with examples or even the next class block do some warm up problems that are related to the foldable concept on the left hand page. This approach has worked very well for me. I get lots of positive feedback from students. There is another teacher in our department who also used these notebooks the first time this past school year. She actually got a lot of complaints. But I think she was using them to drive her instruction. All her notes were done in the notebook. I like to have some time beforehand where students explore and do problems and figure things out and we have discussion. Then we used to know books to summarize. Also I try not to get too fancy with the foldables. I like to be a little creative but some kids just are not into that, so I don't want to get so complicated that it distracts them from the concepts. There are always colored pencils and markers in their baskets so students can get individually creative if they want.

Beginning The Algebra ISN

How do we begin?

Well I started off one way last year and this year I am going to start off completely differently.

Last year we did a bunch of things I will call "Me at a Glance" Activities. these took "part" of three class blocks as I interspersed them within the whole first day class expectation talk, and the first unit (Patterns). I basically used Sarah's set up and introductory materials  and they were fun but I'd rather try a different approach this year and make the ISN all about Algebra.

The notebooks

My plan is to hand out the composition notebooks on the second day of class (our district purchases the notebooks for the students - since we do not have textbooks we are able to use textbook funds for this. Doing this we know every student has the same notebook and everyone has one the day we start using them). We use MEAD 100 sheet composition notebooks, college ruled. We'll be doing this after the kids work in groups processing the patterns they worked with for HW (will cover in later blog post). I haven't decided yet if we are going to decorate the cover this year. This past year we just pasted on a title (which I printed out in a number of different colors and fun type faces for students to choose from).

Cover Page

We'll fill in the cover page. Here's my "generic one" - more a reminder for me of what students should have on their page.

QUICK DISCLAIMER - I do not have the best handwriting. I do lots more typing these days so my handwriting has deteriorated. But my students get used to it. My student teacher this year had BEAUTIFUL handwriting - it looked like a font it was so perfect and lovely. But my students know better than to expect that from me. 

Inside Front Cover

 I am going to do a foldable of Mathematical Practice Standards. I refer to these many times during the year and have posters around the room describing them. I was thinking of putting an "Algebra Reference Sheet" on the inside front cover but there are lots of reference pages throughout the notebook so nevermind. Anyhow - found this on line on the "i is a number" blog and it's great!

Then we'll talk a bit about why these ISNs are useful and how we will be using them in class. We don't have textbooks so these essentially will be personal textbooks for Algebra One. All students are responsible for having all the content up to date in their notebooks even if they are absent. So if they miss a class they need to come see me before or after school to get what they missed. Or have a classmate pick it up for them. I could even email them what they miss (with a pdf image of the filled out page).

Table of Contents

Next we'll set up out TABLE OF CONTENTS (TOC). This will become our regular daily routine. On my front board students will see as they come into class SWBAT (with the objectives of the class listed, just an FYI to them, they don't need to copy these) and TOC with the pages listed that we will be using in class that day and that they should set up as soon as they sit down. So here are my table of contents pages for the whole year for Algebra One (2014-2015 school year - CAUTION - do not try to fill out a whole year's worth in advance. You'll be creating this notebook day by day. I'll post soon a blog of "things I learned my first year with ISNs").

The unknown for me was, how many pages was this going to take? Our notebook has 200 in all (but one of those is the cover page and one is page zero). So I got close to using the whole thing!

Our First pages

Finally we'll fill in the first two pages. These are informational but also have a little content for our first unit. I really want to follow the format this year of "Left Side Learning" and "Right Side Reflection". I was pretty good at it this past year but was afraid I'd run out of room and sometimes things got rushed. Also I completely messed up the numbering this past year. If you look inside a book - right hand pages are odd numbered and left hand pages are even numbered. But that means if you start with a left page like I do you have to start with page 0 (zero). I didn't think of that and all year my page numbering completely bothered me as it didn't match the formatting of books. Sigh, this year I'll do it right.

So here are my first two pages:

The foldable and the examples complement what the students had for homework the previous night and worked on sorting in groups at the start of class. So I should get some pretty creative examples on those right hand pages!

And I am just starting to use box.com to file documents that I share on my blog. So here is my first try at sharing documents for the math practice standards foldable and the what is a sequence foldable.

Honors PreCalculus Pacing

I'm not going to do much posting on PreCalculus yet. But I also teach the honors level PreCalculus in our school. It's in quite a state of flux as we see what Algebra 2 will now be covering and what will be left for us to do. We are making some adjustments for the 2015-2016 school year with some topics already covered thoroughly in Algebra 2 which used to be mainstays of our PreCalculus curriculum. We are skipping exponential & logarithmic functions completely now and also skipping Right Triangle Trig (covered in Geometry & Algebra 2) and Law of Sines & Cosines (Algebra 2).

I did an ISN for this class this past school year (14-15) and will do one this coming year. But I will be deleting and adding things. So I'll post those pages as I go this coming school year (although I may start this summer if I get Algebra 1 done....). But for now, FYI teachers - here is our pacing guide for PreCalculus.

North Haven Public SchoolsScope and SequenceMathematics – PreCalculus L3

Our Algebra One Pacing Guide

So this is not very exciting but might be useful to my fellow math teachers. Here is the pacing guide we used this past school year for Algebra One. I teach the L3 (our advanced class). We actually were able to follow the guide pretty closely.

PACING GUIDE 2014 - 2015

Algebra 1 Level 2

Unit Title	Pacing	Dates*	Common Core Standards
1: Patterns (Algebraic & Geometric Sequences, Fibonacci, Pascal’s Triangle – honeycomb problem performance task)	3 weeks	Aug 27 – Sept 19	SMP 4,8 F.IF.3 F.BF.1, F.BF.2
2: Linear Equation and Inequalities (Solving, Justifying steps, applications) – relay for life performance task	5 weeks	Sept 22 – Oct 28	SMP 1,6,7 N.Q.1, N.Q.2, N.Q.3 A.SSE.1 A.CED.1, A.CED.4 A.REI.1, A.REI.3
3: Functions (definition, domain, range, function notation, linear vs non-linear in general way)	3 weeks	Oct 29 – Nov 21	SMP 4,5 A.CED,2 A.REI.10 F.IF.1, F.IF.2, F.IF.4, F.IF.5, F.IF.7b, F.IF.9
4: Linear Functions (rate of change, slope, linear equations in slope intercept, point slope & standard form – graph, write, use for applications, linear inequalities in two variables) homecoming balloons performance task	5 weeks	Nov 24 – Jan 13	SMP 1,2 F.IF.6, F.IF.7, F.IF.8 F.LE.1, F.LE.2, F.LE.5
Midterm Exams		Jan 15- 16, 20 -21
5: Scatter Plots and Trend Lines (graph, correlation, find equation for line of best fit by hand & with graphing calculator – snack food performance task)	4 weeks	Jan 22 – Feb 23	SMP 3,4,5 F.IF.7 S.ID.1, S.ID.2, S.ID.3, S.ID.6, S.ID.7, S.ID.8, S.ID.9
6: Systems of Linear Equations (examine two conditions with two unknowns with a picture, with a table, graphically, algebraically, also solve by graphing, substitution, elimination, applications) business letter performance task	5 weeks	Feb 24 – March 30	SMP 1,5 A.CED.3 A.REI.5, A.REI.6, A.REI.11
7: An Introduction to Exponential Functions (properties of exponents, rational exponents, simplifying radicals, exponential functions) – car comparison performance task	4 weeks	March 31 – May 5	SMP 1,4,6 N.RN.1, N.RN.2 A.SSE.1b, A.SSE.3c F.IF.7e, F.IF.8b F.BF.2 F.LE.1a, F.LE.1c, F.LE.2, F.LE.3, F.LE.5
8: Quadratic Functions and Equations ( start with “mini” polynomial unit – add, subtract, multiply & factoring for skill development. Then do patterns – first & second differences, key features of quadratic functions – AOS, vertex, intercepts, symmetry and the different forms – intercept, standard & vertex. Also solving quadratic equations. And quadratic regression on calculator)	4 weeks	May 6 – June 4	SMP 1,2 N.RN.3 A.SSE.3a, A.SSE.3b A.APR.1 A.CED.1, A.CED.2 A.REI.4a F.IF.4, F.IF.7a, F.IF.8a F.BF.3
Final Exams		June 8 - 11

*Tentative changes may occur due to snow days and other scheduling issues.

**Algebra I L3 covers the same materials but at a faster pace.  Some units may take the same amount of time but may be covered in more depth.

Starting Out

I've been teaching since January 1987 and have taught both middle school and high school mathematics over the years. 10 years at the middle school level (7th grade, so fun, but the math got a bit boring) and the last 15 years at the high school level. I've taught almost every math subject and level in high school (not calculus or statistics). I used to say Geometry was my favorite math subject but haven't taught that in years. Right now I teach Algebra One and Pre-Calculus. I am always open to trying new things and being creative in my teaching. I do love to teach, but it also consumes me and I have to remember to slow down and have some "me" time during the school year.

One part of being a good teacher is being a good member of the team - that is working together with other math teachers -supporting each other and sharing materials. It's very easy to just shut your classroom door and be a hermit but it can be very rejuvenating to work with others. Sometimes I have to force myself to do this when I get very busy but it's very important to ones sanity. So over the years I've worked with many different teachers with different levels of collaboration - I love where I am now with a bunch of younger new teachers who are excited about teaching and trying new things. I like to mentor new teachers and to have student teachers. I'm always sharing my materials with everyone. I strive to have creative, thought provoking, and rigorous materials.

I feel compelled to start (and hopefully keep up with) this blog because of all the wonderful materials I have found on the internet from other teachers. I feel that I should share my materials for any interested teachers who could use them. And I am not a fan of "paid" teacher sights. I know our salaries are not wonderful but I don't want to be charging other teachers to supplement my income knowing they are also struggling. So I'll do what I can to upload and provide *free* materials for anyone who can use them.

This is a work in progress.

Another impetus to starting this - I just finished my first year with "Interactive Student Notebooks" (ISNs) and love love love them!  I am a creative crafty person so they really appeal to that part of my personality. Also we are in a state of flux right now with our curriculum as we adopt the common core. This is our third year in our algebra one classes, so I'm feeling pretty solid with the sequence of what we are doing (even though the standards have changed and now our state, Connecticut, is modifying them for our use - so more changes coming...). The biggest problem is, no textbooks. Any textbooks that claim to be "common core compatible" are just the same old thing with new labels on them. So creating ISNs is a great way to have "student made textbooks".

And do I start a new thing slowly and cautiously? No, I do not. I used ISNs in all my classes - Advanced Algebra One, Applied Algebra One (lower level) and Honors PreCalculus. Phew, a lot of work creating materials (and borrowing/modifying found materials). But so so worth it. I got tons of very positive feedback throughout the year from almost all my classes (not my lower level class, but that's another post) and their parents. So it's full steam ahead to using these in the 2015-2016 school year.

So what I plan to do over the summer is to post my Advanced Algebra One ISN materials. It's still a work in progress but that curriculum is more or less set and I'll be following the same format in the coming school year. Pre-Calculus is in a greater state of flux so I'll start that after completing Algebra One but that will be something that I'll have to continue over the school year as we following our new "scope & sequence". All stuff I'll share when I get to it.

Advanced Algebra One - this class is an advanced level class for 9th graders. Traditionally advanced students take Algebra One in 8th grade but our school system identifies some students who were not ready for Algebra One in 8th grade but are stronger than average and need to be challenged in 9th grade. It's an interesting class - great kids, very motivated to work hard and they (mostly) enjoy math. The only problem is that some are very hard workers but don't have that strong aptitude and intuitive sense needed for a real advanced class. But most are quite successful in this course, it just catches up with them in Algebra 2 if their cognitive development doesn't speed up to accommodate the much more abstract concepts in that course (and the precalculus course the year after). That's another whole different conversation!

I'm very lucky with this class as we have block scheduling (82 minutes blocks) and most classes meet every other day but this Algebra class meets EVERY DAY. So we have lots of time. But we also have lots of content to cover. That's one of the "crazy producing" facts of teaching - gotta finish the curriculum! And the common core was supposed to get away from the "mile wide, inch deep" curriculum with fewer topics and greater depth. Sure we now have 8 units when we used to have 13. But there is actually MORE material than we had before, they just combined some of the units. Think we math teachers can't count? So anyhow, I'm very lucky to meet with them every day. I would never never get through everything if we met every other day.

One of the most important things to acknowledge is that none of what I create is truly original - everything has some seed of inspiration. I can't thank everyone but over time I will reference where I have found things. The important thing is to use what's out there and if you can share what you have.

My main reference consistently through the year has been the Math=Love blog. She is a prolific blogger who has great materials! But there are others too. I used Everybody is a Genius when I was first starting out (sadly she has not updated her blog since summer 2014).  Also check out Mrs. Hester's Classroom. and I do a lot of searching on pinterest for ideas. Very little of my material is original. Just modified and tweaked borrowings from other generous teachers.

So next post - I'll start off with my ISN pages and materials. This was just some background and some resources I used.