Systems of Equations (part 2)

Standard

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…

 

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?

racing dots

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!

 

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