Category Archives: Systems of Equations

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…


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  –  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!


Systems of Equations Unit (part 1)


So many thoughts this past week as we began Solving Systems in Algebra I which will likely lead to multiple posts…

Here is the Systems Organizer Student Assessment Tracker.  I’m not satisfied with it yet, I’ve adjusted an old Algebra 2 unit, but I know by next year, this will be one of our strongest, most purposeful units.

I’ve been using the Candy Store problems since Mary and Alex shared them at TMC-Jenks.  A great problem solving task with manipulatives to introduce systems of equations.  My only change is to adjust for the U.S. candies, Solving systems CB S (Thanks for sharing your file, Mary!).  I plan to bring in a candy treat to students to celebrate their journey when we end the unit.

Based on prior assessments, in class observations, I purposefully separated students on skill level for this unit.  I intended it to be for me to have time to focus on groups with weaker algebra skills, while letting the others move on at their own pace.  I pulled those few who tend to “do the work” into groups together which would allow for those who follow along in tasks or let someone else do the thinking, then they copy it down-be required to do their own thinking.

Here’s what I notice – my “algebra” kids struggle, my “struggling” kids soar – with the hands on task!  It just goes to show, students do have good, strong reasoning skills when allowed to think on their own.  Each group gets a cup of pennies and two different types of pattern blocks with a white board and marker – although I think next year, I will hold off on the white board and marker until AFTER they solve the first one – I really want them to rely on reasoning and number sense before trying to jump in and create equations…although that is the end goal.

The beginning was often guess and check, but I loved hearing their number reasoning as they progressed through the problems.  Let me say, I have about 15 students spread between the 5 class who are still mad at me because they did not like the struggle.  I just kept patting them on the back, asking questions and when they began to engage, I’d walk off and let them continue.

After most students experienced some level of success with the Candy Store problems, we reviewed/introduced linear combination (elimination) to solve systems when presented in Standard Form.  I had examples ready, but as we practiced them on white boards as whole class, students were asking the questions:

  • what if the terms match and aren’t opposite?
  • what if nothing eliminates?
  • why does multiplying the equation by a number work?

It was great when it was students asking and not me leading.

In their groups, they received 11 cards – with solutions on one side and system equations on the other.  The cards were placed so all solutions were facing up and a start card.  When everyone in the group had solved/verified, they located the solution and flipped the card to find a new system to solve.


Nothing more than a glorified worksheet – handwritten while waiting at my daughter’s piano lessons.  But the discussions they were having as they solved on the whiteboards were so valuable…immediate feedback, peer assessment.

It was a good day.  The first time since Christmas Break that I felt confident we were moving forward.  (I know… its March.)

I’ve been trying to be more purposeful in ending class and allowing time for reflection. Students were asked to copy 2 of the problems into their INBs, solve and verify – basically creating their own notes / examples to refer back to.

Each student received a sticky note and was asked to complete the sentences:

  • I used to think…
  • Now, I know…
  • Caution…watch out for…

And they placed them according to their level of confidence as they exited the room:


This was on Wednesday.  I felt that they had built confidence, addressed common errors and misconceptions and had seen how the algebra could offer an efficient model in problem solving.  Yet, I still had a few groups who were strong/quicker with number reasoning when solving them.


Systems Linear Programming


As an intro to this lesson, I shared this scenario…

You are bidding a contract for Company ABC.  The order is for 12,000 dozen of a product and needs to be completed within 3 months.

First student question, why would anyone need 12,000 dozen of anything?  They felt this amount was a ludicrous number.  (After many summers working at Fruit of the Loom, I knew this was within reason, but a nice discussion anyhow.)

Well, is it?
According to Apple Press Info, if it’s as popular as  iPhone6…no. 

First Weekend iPhone Sales Top 10 Million, Set New Record

We figured if we had equal distribution among all 50 states, this was quite doable.

Do we have the man hours to fulfill this order in 3 months?

There are…21 (bc there were 21 students in class today) workers in this particular unit…who work 8 hours per day, each of you can complete 10 products in 1 hour.  Yes, I just made these up, but that’s what we worked with.

After a few minutes, we started sharing processes, quickly a bit of an argument – why did you do it this way? Should you have….?  Others arrived at the same solution, but with varying approaches.

I could kick myself for not taking a picture of their suggestions.  Some nice verifying one another going on.  However, they were not sure what those values represented…they could get the “right” values but lost when I asked for a label.

Watching students grapple with the numbers, made me realize how far out of reality we’ve taken students math skills.  I just want to do a better job of letting them make sense of problems themselves.

We determined it would cut it close, but we could likely finish this job, maybe requiring a bit of overtime to meet the deadline.

Now, as we make an offer for the contract, what are costs west consider? This leading to an idea of our linear programming. 
Wages, materials, utilities, insurance, packaging,  shipping, etc.  One student even said, there’s a lot to consider. Me, knodding, yes.

Is this a great example intro. Nah. But I feel it’s a nice way to show students there are many options a company must consider prior to the contract, production, sale.

Now, to the hard part.  A variety of students, some with adequate graphing skills, others struggling to find the line x> 3.

Modeling Systems


Sort of a rambling post. But trying to make some sense of my thinking…

I always appreciate posts from @emergentmath.  This particular post made me pause, I had just completed the MARS task, Boomerangs, he references.  We are in the midst of our systems unit.

I used Mary & Alex ‘ s suggestions with beginning systems without the algebra.  Conversations were great, students’ strength in reasoning was evident.

I plan to use Geoff’s suggestion for a matching/sorting activity this werk for students to see the benefits of each type of tool to solve systems.

But where I struggle is with this standard:


I am experiencing some pushback from a handful of students who are able to reason and solve a system without actually modeling it algebraically.

Their reasoning is correct.  They verify their solutions and interpret them correctly.  They can sketch a graph yet “refuse” to model as a system of equations.  I struggle because “their math” is right on.  I realize places where algebraic models can help but I honestly can’t tell them my way is better…yet the standard says…

It feels almost like I am punishing them if I make them model it algebraically.

Then I have others who are not sure where to start.  The equations model provides them a tool, yet they will not embrace it.

How do others handle this situation in your classrooms?

I use graphical, alongside a numerical table of values, with solving/verifying with the equations, letting them see their own connections eventually.

My biggest goal for systems is to provide enough modeling for students to actually “see a context” to connect/make sense of a naked system of equations.

This is where I believe skill/drill has ruined the power and beauty of math.  Finding an intersection point but what in the world does in mean?  It’s a point on a graph. Whoopee.  Why isn’t it all taught in context as a model?

I listen, I Learn #MTBoS30 Post 12


A student had a question about a system of equations problem in their review packet this past week.  It seemed like a straight forward problem, but they were struggling with it. Another realization I must be very intentional to spiral early topics throughout the year.

Another student quickly blurted out the answer, so I asked them to explain their approach.  Here is a look at their thinking…


Again, I don’t think this way.  My brain has been trained to use the traditional multiplication/linear combination.  The student’s idea is exactly the same, yet it makes so much more sense, not so procedural, but making sense of the numbers.  I even used this approach with a couple of students who were needing some help…they quickly picked it up and was able to adjust their thinking and apply to other problems.

Yes, when I give them an opportunity to share…I listen and learn.

Battleships and Mines


Battleships and Mines

I cannot remember the exact place I aquired this activity – I’m thinking its CORD related from years back.

Students create a 20 x 20 Quadrant I coordinate system using string and the tiles on the floor.

In small groups, they are given their shipping lane equation and 3 enemy lane equations.  When each system is solved, they place a “mine” on the coordinate solution.  When all groups have plotted their mines, we graph the enemy lanes – using 2 points from the line.

Happy to say all groups earned at least 3’s on the completion of their task.  It was simple implement, a quick review and students said it was fun.

It was an opportunity for students to share different methods for solving systems, ranging from putting in slope-intercept form and using graphing calculator, substitution and elimination – they verified their work by comparing with others within the group.  The group had to come to concensus on where to place the mine (solution).  Conversations were great – when there was a disagreement, I observed them looking for mistakes, explaining to their teammate how to correct.  Almost everyone was fully engaged in the activity

Self-assessment was simple – if it wasn’t a direct hit, the group needed to revisit their work.  During our debriefing, group mistakes ranged from a simple mis-plot on the grid; using the y-coordinate for the x when finding the second coordinate; simple arithmetic – forgot to divide by the coefficient;  The groups who had to revisit their work – were able to explain how to correct their work.

Scoring Guide:

4 – All 3 systems are solved correctly, mines are plotted correctly, to result in direct hits.

3 – All systems are solved correctly, but a misplot results in an almost hit.

2 – one of the systems contains errors in solution.

1 – 2 of the systems contain errors in solution.