Linear vs Non-Linear Functions

In Algebra I, we are working on determining whether a function is linear or non-linear.  We have been using visualpatterns.org as Thursday bellringer problems each week, so I started this lesson by showing the students 3 patterns (2 linear, 1 non-linear) and asked them to write an equation for each pattern and graph the points using my version of Sheri Walker's handout from visual patterns.org.

The first two were pretty easy for them since they looked like the bellringer patterns (nice and linear).  The third one was a little tricky.  With some assistance the students were able to figure out the pattern.  We then talked about what made the third pattern different than the first two.  A lot of students picked up on the fact that the rate of change was different, some even mentioned that the equation had an exponent.  We compared the graphs and discovered that we couldn't connect the points with a line.  This opened up the discussion about how a constant rate of change (here's the INB page for rate of change from @mathequalslove) is what determines a linear pattern.  We practiced with a few examples and that was a wrap for day 1.

Today we started by reviewing with Plickers.  First time I've used them...LOVE!!!  I gave them four patterns and they had to determine if they were linear or non-linear.  Plickers are seriously awesome.  I will definitely be using them more often.  Great formative assessment.

Then we built our own non-linear and linear patterns with pattern blocks.  This was a great opportunity for my co-teacher and I to walk around, get a feel for who was still struggling and offer more one-on-one instruction.  I asked students to write their names on post-its and put them on their desks.  Then each student figured out the equation for their own linear pattern.  I had the students switch a few times, record whose pattern they were working on and figure out the equation.  I made sure I got a picture of each student's desk with name so that I could do some error checking.

 

 

 




Overall, this was a great activity.  I feel like students are really developing an understanding of rate of change and the fact that constant rate of change determines a linear function.

Next we will be delving deeper into linear functions and start making the connection between rate of change and slope.  I'm planning on doing a lot of experiments and investigations to help develop really solid understanding of linear functions.  I LOVE Andrew Busch's resources and will probably be using several of them.

Algebra Party!!

We are moving at a snail's pace in Honors Geometry this year... New 8 period, shorter class, day (up from 7), new textbook & curriculum, and ever changing daily schedules have made it hard to find my grove to get through material!  That being said, we are working on solving problems involving segments and angles using algebra.  I didn't want this week to just be a boring week where I show lots of examples in class and students do lots of examples for homework, so I decided to throw a big ole Algebra Party.

It all started with some angle bisector algebra problems.  I used a carousel approach where I gave each group of 4 students 4 different problems and a big whiteboard.  There were 4 steps to each problem:  Draw and label a picture, write an equation, solve the equation, and solve for the requested segment.  Each student did step 1 for their own problem, then rotated to the next problem to check and do step 2, etc. until all four problems were complete.  The students said that it really helped them to understand the different types of problems better.

Then we had a half day with 15 minute classes on Friday.  To ramp up the excitement for a week of lots of Algebra, I decided to have the students make party hats.  They are really very simple hats, but they loved it!  Our school motto this year is "212 it - go the extra degree" and one student made a 212 hat:  It says:  STEAMY MATH 44,944 / 212 = 212 

The second party game was clock partners.  I had the students sign up for partners for each hour of the clock and just called out random hours for three problems.  They got to work with different partners and do some problems with vertical and supplementary angles. 

To conclude the week of party games I will be doing two more party games.  The first will be another carousel activity, but randomly assigning students to groups to practice complementary and supplementary angles.  The second will be a scavenger hunt with acute and obtuse inequalities.  

Who said math class had to be boring?

Average Velocity vs Instantaneous Velocity

In Calculus, I wanted to start my unit on Derivatives by getting the students to conceptualize the limit definition of the derivative.  I decided to use an idea from @ThinkThankThunk that he blogged about here.  Now, I didn't start by doing this on day 1, but I really liked the idea of somehow demonstrating the difference between average and instantaneous velocity.

Day 1 & 2:  I went outside and we had one of the students run/walk/jog at varying speeds for 200 ft.  I timed her first for the 200 ft, then every, 50 ft, then every 25 ft.  We discussed the difference between average and instantaneous velocity and how we could go about determining instantaneous velocity.

Day 3, 4, & 5:  We used the CBR2 to do a ball bounce activity to help develop the limit definition of the derivative.  Handouts are here, here (Activities 1.3, 1.8, and 1.9), and here. And a little Desmos demo thrown in.

Overall, I felt like the lesson went well.  Yes, it took quite a few days, but the understanding of what a derivative is, what it is measuring, and how it measures that will be invaluable as we move on to more difficult topics.  

Barbie Zipline!

Well, I took a risk with this, and it kinda tanked, but we learned a lot in the process...  I tried using Barbie Zipline in Geometry to reinforce and add jazz to the distance formula.

For the lesson on distance formula, I start out by using Dan Meyer's Taco Cart to get students thinking about the Pythagorean Theorem.  I always like to talk to them about why the Pythagorean Theorem works and show them at least one proof, because they rarely get that experience.  My handout is here.

After we have some good discussion about the Pythagorean Theorem and they are amazed at the proof, I proceeded to show them the connection between the pythagorean theorem and the distance formula.  We practiced a little bit and then we were ready for some fun.

This year, I decided to give them a taste of a STEM application and do the Barbie Zipline activity that @JStevens009 described in his blog.  I gave the students the initial information that we would be zip-lining from the top of the bleachers (30 ft high).  This was their starting point (0, 30).  Our zipline would attach at the top of the bleachers and then down to the top of a yard stick.  They had to then determine the ending point (?, 3) that would provide Barbie with a safe and fun journey.  I required them to show their work using the distance formula.

Initial whiteboard brainstorming:

I had them finalize their work and hand it in.  Then we went out to do the zipline.  Unfortunately, only 4 groups out of 11 had a successful zipline experience.  It was so cool for those groups.  The other groups had designed ziplines that were too long and their Barbies got stuck in the middle.  I think this had to do with the fact that we didn't spend enough time in class discussing what type of zipline might have the best design.  I will definitely incorporate that into my initial discussions for next year and lead them into discussing the fact that they really need to pay attention to the steepness (or slope) of their zipline.

This prompted me to change the way that I wrapped up the lesson.  Instead of calculating each group's speed and having a discussion about which group provided the best zipline experience as I had originally planned, we had a great discussion in class about what went wrong in the designs that didn't work and how to make this a better project for next year.  I LOVED the insights that the class gave.  They picked up very quickly on the fact that the zipline design needed to include a limit on the length of zipline (possible less than 100 ft), that the slope needed to be a certain "steepness", and that perhaps the location could have been a bit different (we had one barbie run into the rail at the bottom of the bleachers during her descent).  We also talked a little about the physics application of G-Force that we could possibly calculate.

So, this didn't go as planned, but I think I met my goal overall of giving them a fun and exciting application of the distance formula while getting them to think about design flaws in the experiment.

2015-2016 Goals

• Day one ideas - I want to be very purposeful for day 1.  I DO NOT want to go over rules or read my syllabus.  I want students to do meaningful math and work cooperatively.  So here's what I came up with...
• Bellringer sheet - students will pick up a bellringer sheet when they walk in the door and work on their first Monday bellringer.  This will teach them where to find the bellringer sheets and what they can expect every day in my room.  My music timer will be playing and they will get a taste of the routine of my classroom without me really saying a word.
• Algebra 1 - four fours activity a la @nhighstein to review order of operations without "reviewing" order of operations.  I will have them work in partners or groups.  What will be great about this activity is that my co-teacher and I will have the opportunity to walk around the room and get an idea of how the students work together, who is struggling, etc. on DAY 1!!!  
• Calculus - A Taste of Calculus activity followed by a student questionnaire (here is the teacher reference for the questionnaire) for homework.  I don't remember where I got these, but I definitely didn't make them myself.  If I can figure out where I found them, I'll come back and give credit where credit is due.
• Probability and Statistics - I haven't figured this one out yet, but my husband is teaching this course as well (we both teach math in the same school....two doors away from each other) so I'm hoping we can brainstorm something amazing in the next two weeks.
• Honors Geometry - Partner or Group (haven't decided) Geometry vocabulary activity.  You can read about it here.  Students take turns describing a diagram and the partners try to draw it only using the description.  The skeleton sheet is here.  I have to add points and labels.  
• Updated weekly Bellringer sheets.  I've used weekly bellringer sheets for as long as I'm teaching.  At the beginning, they would just be composed of problems that were warm-ups for the daily lesson.  Usually those "Check Skills" at the beginning of each section or the spiral review problems at the end of each problem set in my textbook.  A few years ago, I decided to rethink my bellringers.  I LOVED what @algebrainiac1 was doing with her weekly bellringer sheets and modeled mine after hers.  You can read about what I did here (in my first ever blog post....5 posts ago).  Then I read about how she was going to change hers to a two-week rotation and thought it would be a great idea to do that for my Algebra 1 class that I is a two-period block class for the first half of the year.  I'm not sure if I'm going to do it as a two-week rotation or have them do the week 1 bellringer for 1st period and the week 2 for 2nd period.  Here's my finished product.
• High Fives every day - What a GREAT idea!!!  I start school in two weeks and I am beyond excited to start this simple way to make students feel welcome in my classroom.  You can read about it here.
• New lesson plan format - I totally stole this idea from @algebrainiac1.  You can read her post about it here.  Totally LOVE the Table of Contents add-on for Google Docs.  I've used a Google Spreadsheet for my lesson plans for the past few years and I loved that I could embed it on my Homework Helpline for my students so that posting homework daily isn't something I can ever forget about.  I have my lesson plans all set up for the year, so it is a go, but I'm not sure I am going to love it as much as my spreadsheet.  We'll see. 
• Music cues and music timer chrome extension - LOVE this from @chrisrime (http://t.co/Vk0Mh3tNR4).  He created a Musical Timer Chrome Extension that I'm going to use to automatically play music every day except Wednesdays.  We have a different schedule on Wednesdays, so I need to figure out a plan B.
• #teach180 - because I have always wanted to join #180blog and blog every day, but I'm just don't think it will work, but taking a picture a day is definitely doable.
• BLOG!!!!  I really want to....  I took tons of pictures in my classroom over the last few years with every intention of blogging about what I was doing, but somehow they never made it here (obviously since this is only post 2 after a two year break!).  I need to think of blogging as my own personal reflection tool.  I can't help feeling somehow inadequate in the blogging department because I don't feel like I have a whole lot of original ideas to share, just reflections of ideas that I stole from somewhere else (actually, MOST of them come from the other amazing math teachers I follow in the #MTBoS).  

Trying my hand at blogging...again.

With 2 of 3 completely inspiring and motivating (seriously) In-service days under my belt, I've decided it's time to get into this blogging thing for real.....


Ok, so that is the blog post I started LAST YEAR!!!  I never finished it.  So, I guess it's always a good thing to try and try again, right?

The start of the school year for me is about two weeks away with student day one in three.  I have so many ideas swimming around in my head that I thought blogging about at least one of them might get me on the right track.  So, here goes.

I've been teaching Honors Geometry for quite a few years and have been thinking about how to go about the vocabulary I teach during the first week of school.  I've tried several ways to go about it, but I really hate that it becomes a lot of teacher-led lecture at the beginning of a year that I really want to be focused on collaboration and student-centered instruction.  So, I had an idea.  I have a list of terms from the first chapter that I think are important for students to know, especially the notation.  I created this form for student notes.  I want the students to focus on coming up with an understandable definition and the notation for each term.  After doing the first one or two with them to show them what I have in mind, I had the idea to divide the terms up and assign each group of students a few of them.  Then, using their book, the internet, etc, each group will come up with a way to fill in the information in the chart for each term.  They will put their ideas on post-it chart paper (LOVE this stuff!).  Then, I want to have a gallery walk where each group will have an opportunity to see what has been done for each term.  They will have an opportunity to add or comment on each term via post-it.  After all is said and done, the students will have an opportunity to fill in each of their charts with the information.

I don't know how it will go, but I think that giving the students this voice and ownership at the beginning of the course will set the tone that I'm looking for for the rest of the year.  Does anyone out there have any other creative ideas for vocabulary terms?  I'd love to hear from you!

Exploring the #MTBoS: Mission #1

It has taken me a while to decide what to write about for this blog post. The mission is to write about "one thing that happens in your classroom that makes it distinctly yours. It can be something you do that is unique in your school… It can be something more amorphous… However you want to interpret the question! Whatever!" I'm not all that sure what makes my classroom distinctly mine, so I've decided to write about some things that have become routines and habits in my classroom. 

Before I get to that, I'd like to just reflect on some of the things that have occurred in my teaching career that have ultimately led to me finding this amazing community of math teachers in the "Mathtwitterblogosphere".  I have been teaching for 13 years at the high school level (grades 7-12).  I have taught all courses except Algebra II and Calculus.  I absolutely LOVE my job most days and wouldn't trade it for any other.  But, I started off (as most teachers do) by teaching the way I was taught in a mostly lecture, notes, examples, homework format.  I started becoming increasingly dissatisfied with my teaching style and started to question the status quo.  I started looking for ways to incorporate cooperative learning, hands-on, and discovery activities in my classroom.  I wanted to find ways that I could structure my classroom so that my students were doing more "work" than I was.  I wasn't trying to find a way to be lazy about teaching, but I already know how to do this stuff and I wanted my students to have as much experience with math as possible.  I wanted to find ways to become a facilitator instead of a deliverer.  So, as I hunted for resources to fit this teaching style eventually stumbled upon twitter and I can't believe how amazing it is.  It seems like I've finally found some other math teachers who are like minded and willing to share ideas and resources.

Before I go any further, let me just say that I really don't think that I'm all that great of a teacher.  I know that there are so many things that I need to improve on.  I guess that's why I'm out there pretty much begging, borrowing, and stealing resources a lot more that I'm sharing my own.

Well anyway, onto some of the things that have become routines and habits in my classroom:

• I always have my students working in some sort of groups.  At the very least in pairs, but often in groups of 4 or 5.  I encourage my students to talk to each other all the time.  I don't want to be the only person that they can ask questions of.  I LOVE when my students start to discuss things with each other and debate about math problems without any prompting from me.
• I try to incorporate discovery into as many lessons as I possibly can.  I really want students to see things for themselves instead of just believing what the teacher tells them.  I want them to understand WHY things are true and behave the way they do.  I want them to see connections between mathematical ideas, like how the pythagorean theorem is related to the distance formula and the equation of a circle.  I LOVE incorporating technology like Geometer's Sketchpad and Desmos to facilitate student discovery.
• I do bellringers (warm-ups) with my students almost every day.  I've done this for quite a few years usually just pulling problems right from the textbook.  Every year, though, I have found myself having to remind students to work on their bellringers at the beginning of almost every class.  This year, I changed the format of my bellringers to daily themes.  Using resources such as Daily Desmos, Estimation 180, Would You Rather, Visual Patterns, and Graphing Stories, to give my students yearlong practice with skills that are central to my courses.  I can't believe how much this has changed how my students approach bellringers and the rich discussions that unfold while they are trying to figure out a pattern or how to graph a situation.  Here are my bellringer sheets for Honors Geometry and Precalculus.
• I like to encourage "good mathematical practice" in my classroom (hence the title of my blog) and get students to think about what that means.  This means addressing things like giving an exact answer unless asked for an estimate (students like to give rounded decimal answers for EVERYTHING!!), expressing answers in the same form as the problem (decimals vs fractions), precisely graphing points and lines on a coordinate plane, drawing and labeling diagrams carefully, etc.  I actually considered making a  "ten commandments of good mathematical practice," but haven't gotten around to it, yet.  
• I really, really LOVE it when students come up with different ways of thinking through a problem or find shortcuts (I LOVE SHORTCUTS!!) to solve a problem.  I encourage students to do math in their head and show as much work as they need to in order to solve problems.  I will often ask a student to explain their reasoning to me if they can do a math problem in their head and then encourage them to try something a little bit harder.

My ultimate goal is for my classroom to become as student centered as possible.  I have a long way to go, but I think that I've come a long way from where I started.  

Thanks for visiting =)