Systems of Equations (part 2)

None of what I’m sharing is new…but its me reflecting on the week…so I can reference back and make adjustments in building a better unit  of learning experiences for next time around.

To address some student questions, here are examples used in class to follow-up.

On white boards:

• y=x+1
• Pick a value for x.  Find y.  (Ex. (3, 4)
• Now, let’s double our equation.  What?!?  Yep, double it. 2(y=x+1)
• Okay. 2y=2x+2
• Use your same value for x from above and find y. (3, 4)
• What do you notice?
• Let’s multiply our first equation by 5. 5(y=x+1)
• 5y=5x+5.
• Use your same x value from above, find y.  (3,4).
• Did that happen for everyone?  Turn and talk…
• What if we took half of our equation?  .5y=.5x+.5
• for the same value of x, it works again (3,4)
• Then we go to Desmos to see the graph of our equation along with ALL of our versions of the equation.

Its a big idea that I don’t tell them.  They have observed why we can use this “magical” math thing is actually just a different version of the same equation…as one student put it “its the same equation, in disguise!”

But I also feel there is value in diverting from my original plan here to address the student’s struggle to figure out WHY? we do this in elimination, otherwise, it is literally, a “magical math thing” that just happens.

I need to do a better job of this – equivalent expressions / equations – earlier in the year, when we are looking at equations of lines…but also, how can I connect it with scale factors and similarity?  It all comes back to proportionality, but what strategies and tasks can I use to help my students make the connection and really develop a deep understanding?

Next on the list, we graphed our systems we’ve solved in Desmos.  Noticed and wonder…comparing our graphs to the work we’ve done algebraically.  Ohhhhh.  We found the intersection point!  Again, not me telling them, but they see it on their own.  I love that Desmos allows us to graph an equation in standard form.

Finally, I asked students to solve these equations and discuss their results in their groups:

•   4x -6y   = 12                 and             7x – 4y = -11
• -2x + 3y = -6                                     14x – 8y = 16

When does 0 = 0? ALWAYS!                   When does 0 = -6?  NEVER!

Again, we looked at the graphs in Desmos…

 

Same Line

Parallel Lines

Several quickly stated the first set was only a multiple of the first equation, so it would be the SAME line!  (yes. secret happy dance!)

And the parallel lines never intersect…the equations were multiples on one side, but NOT on the other, a student noticed.  Its a translation, just moving one line up or down – another student stated.  So, how can I use their intuitive thoughts to build a better lesson?

I found Racing Dots on teacher.desmos.com  –  based on an activity, Great Collide by Jon Orr – to bridge between special situations, graphing solutions, substitution and algebraic solutions – will share more on this task later!

 

Lesson Planner Resource #made4math

While trying to catch up on my reader – I ran across Simplifying Radicals post on using Google Docs to create lesson plans.  She had an update at the top and suggested reading a comment made by another reader.

I briefly went over to Common Curriculum to check it out.  I’ve created an account, watched a few of the videos, looked through suggestions by other users and played around with lesson plans.  I like it.  I think I will eventually like it a lot.

Basically, you set up your schedule – what you’re teaching; edit a template – creating category “planning boxes” that you will use often.  You can add / delete any given day.  Within the template – hover over the settings for the box and choose “Show on Class Website” for items you wish for students to have access to on the class website it automatically creates for you.  The standards box is automatically included – you can search either Math or ELA CCSS by keywords or standard #.

 

printview

 

Uploading files / linking to online resources / videos is VERY simple.  Currently files are quickly loaded from your computer, google drive, evernote, box and it seems dropbox will be added soon.

I created 2 separate resources boxes in my planner – Student – to include links to online resources / files for students to have access to on the class website; and another Teacher – to link resources I need.   This is one I just experimented – a drag and drop.  It displays the photo or file name.

 

I attempt to keep a class blog – but sometimes get behind keeping assignments / resources up to date.  What I think I’m going to LOVE about this site – as I’ve already mentioned, you can choose which planning boxes you want to show on the class website – then either give students site address OR post link on current blog, etc.

Website View

Excited about this new resource – Thanks to Debbie Hurtado for posting on Simplifying Radicals for the heads up!

 

PLN

While cleaning out a file cabinet this past week, I ran across a packet for a presentation I gave in 2002 at KCTM conference (the electronic file is long gone, but I’ll scan and post it soon).  As I read the description and looked through the activities – I asked myself, what was different about my teaching back then?  My explanations and reasoning in the packet were very thoughtful.  So what had changed?

In 2002, I had become a NBCT in Early Adolescence Mathematics.   I spent a lot of time talking with educators, sharing strategies, learning new things for my classroom. I was fortunate to attend several workshops focused on making Algebra more accessible for all students through the use of graphing calculators and hands-on learning activities.  The workshops provided great resources, but the conversations which took place with other educators is where the real growth began.  As part of the NBCT process, I was required to videotaped lessons and reflect on my own classroom…how what I was doing impacted student learning.  How could I make it better for my students?  When I felt I need to make a change, I could email someone with my network and ask for advice and ideas.

In recent years, I have gotten in to a bit of an educational rut.  I’m not blaming becoming a mom – but my child was/still is my priority, its just a little easier now to start venturing out to conferences and workshops again.  Relying soley on what my administration provides for my only source of professional growth, well, much like our students , one-size fits-all isn’t the best option for everyone. 

At times in recent years I have felt overrun by all I was being asked to do.  There were so many “things” I was told to implement into my classroom.  I hadn’t had time to process and study, to really buy in to what we were doing and why.  I attempted it all, checked it off the list, but I didn’t do any of it well.  I felt like a failure (hmmm…is this how some of my students feel at times???).  I don’t like being a failure.  I like to focus on one, maybe two things and put much effort in those and give good quality.  When I’ve mastered those, I can set new goals…isn’t that how it should be? 

This past semester, however, I began meeting with 2 colleagues weekly for some good wholesome talk about math class.  Our conversations ranged from things we were doing in our classrooms, lessons we were using, strategies for assessment and technology integration.  We shared videos of our classrooms and discussed what we saw, what worked well, and again, how we could make it better.  Reflection + Conversation = Growth for me.

Who knew what I was doing had a name.  This summer, I’ve learned it was my PLN – personal learning network…the people I have chosen to follow and connect with, the educators from around the world who seem to share the same philosophy and goals for education that I do.   As educators, we are much like our students in that we are all at different points.  My PLN is not required, but it allows me to focus on where I need to grow…I can focus on my own learning and teaching needs.

The one-size fits all required professional development can be so frustrating.  Yes, I suppose it’s needed for those who don’t want to take the initiative to find their own means for growing.  But by receiving constructive feedback from administration (& colleagues) – based on observations and other evidence, I should be able to determine my specific areas of growth (hmm…sounds a lot like self-assessment in my classroom for my students). 

Life-long learning is an essential characteristic found amongst effective educators and something that should be modeled for our students. With so many changes occurring in the field of educational technology, curriculum, pedagogy, and law, it is imperative that educators receive opportunities for growth in their school. Additionally, they should be provided with the knowledge and foundation to develop a Personal Learning Network. This will enable them to learn more according to their diverse interests and passions.  E.Sheninger

 

As a professional, I must hold my own practices up to scrutiny and then decide if those practices are worth keeping.  By expanding my PLN to include online resources like twitter and teacher blogs, I have opened an array of tools I had not even considered in the past.  I am inspired and encouraged by their tweets and blogs.  Whether in-person or online, by surrounding myself with like-minded educators, I can focus on my interests and passions – which will enhance the learning environment in my classroom and allow my students to focus on their own interest and passions.

Interested in PLN? @NMHS_Principal provides a wonderful blog on how to get started.  Not convinced you need a PLN? See skipvia’s fantastic video he provides at the end of his blog.

A couple of things from twitter I find interesting as a teacher and a mom:

FLAG – fix learning and grow via @jreulbach

Summer Fun: 12 Ways Parents Can Build a Mathematics Brain in Children

And I continue to learn about SBG…

Wordle Play

Technology in the Classroom Wordle