Systems of Equations – Take 3

05 Nov

What do you do with Systems of Equations in a high school Algebra 1 class?

Our students have some experience with systems from grade 8, but not as much as they eventually will with full implementation of our new standards.

NCTM’s Principles to Actions lists Establish mathematics goals to focus learning as one of the Mathematics Teaching Practices. We can tell that having a common language to talk about what we are doing is helping our students communicate to us about what they can and can’t (yet) do.

We started our unit on Creating Equations & Inequalities with the following leveled learning progression and questions:

Level 1: I can determine whether an ordered pair is a solution to a linear equation.

Level 2: I can graph a linear equation y=mx+b in the x-y coordinate plane.

Level 3: I can solve a system of linear equations.

Level 4: I can create a system of equations to solve a problem.

NCTM’s Principles to Actions lists Elicit and use evidence of student thinking as one of the Mathematics Teaching Practices. We need to know what students are thinking so that we can move their thinking forward. We created a leveled learning formative assessment so that we could see where students are.

Level 1: I can determine whether an ordered pair is a solution to a linear equation.

We have this y= Question configured to generate a graph preview. The equations students enter graph as they are typed, so the students are able to check their thinking as they go. (Note that we don’t have to configure the question to graph the equation as it is entered; we are choosing to do so while students are learning.)

The following are the results from one of our Algebra 1 classes. This is my favorite question.

How can we use the equation to know whether it contains the point (2,3)?

How can we use the graph to know whether it contains the point (2,3)?

What happens when we give students a point & ask them to create two equations that contain the point?

Do they know that they are creating a system?

Does it help them to know that eventually we will give them the equations and ask them for the point?

Level 2: I can graph a linear equation y=mx+b in the x-y coordinate plane.

And results from the second question:

Level 3: I can solve a system of linear equations.

What’s significant about the green point?

What does it have to do with the given equations?

At least half of our students don’t yet understand what the solution is.

And only a very few are able to solve the system from the question on 3.3, which is okay. This is the first day of the unit. We have the information we need to know how to proceed with the lesson.

I wrote about some of the tasks we used on the first two days of the unit here.

On the third day, we tried Dueling Discounts from Dan Meyer’s 101 Questions, which went better than Dan’s Internet Plans Makeover.

And more:

Maybe it’s time to generalize our results?

On the last day we tried Candy & Chips from 101 Questions, which also went well.

NCTM’s Principles to Actions lists Build procedural fluency from conceptual understanding as one of the Mathematics Teaching Practices. We know we aren’t there yet, but we are definitely making progress as the journey continues …

4 responses to “Systems of Equations – Take 3”

1. November 5, 2014 at 4:31 pm

My thinking is that “problems” should be attacked by ANY method first of all, trial and error, guesswork, stepping through a sequence of values, whatever. When there is some sort of agreement about the solution, then is the time to look at the formal approach. They would at least believe in it, even if they were still stubling over the details.

2. November 5, 2014 at 5:49 pm

This will be very helpful. Thanks.
Do you have a place where I can get that Bell 4-1 file? [Having them pre-made takes a lot of work, but pays huge dividends. I need to move beyond on-the-fly]
I would like to use it and share it.
I am teaching IM1 this year [no geometry], so this is germane.
You gave this as a 1st day. Do you also give it again near the end of the unit? I bet there is a NCTM about that–from informing our teaching to evaluating our teaching. I will await that post.

3. November 7, 2014 at 6:04 am

More ….
1. I would suggest that your leveled progressions need 2.5: I can see that if an ordered pair is a solution to an equation then the point represented by the ordered pair is a point on the graph of the equation. (there is probably a neater way of putting it)

2. Simple systems of linear equations can be solved by “observing and making use of structure”
For example x+y=10 and x+2y=13
It shouldn’t take them too long to see that when you add another y the total goes up by 3, so y=3.