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Category Archives: Creating Equations & Inequalities

Systems of Equations – Take 3

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.

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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)?

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How can we use the graph to know whether it contains the point (2,3)?Screen Shot 2014-11-05 at 2.36.56 PM

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.

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And results from the second question:

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

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What’s significant about the green point?

What does it have to do with the given equations?

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

 

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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.

 

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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.

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And more:

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Maybe it’s time to generalize our results?

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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 …

 
 

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Systems of Linear Equations

We are working on Systems of Linear Equations in our Algebra 1 class.

The first day, our lesson goals were the following:

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

Level 3: I can solve a system of linear equations, determining whether the system has one solution, no solution, or many solutions.

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

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

We used the Mathematics Assessment Project formative assessment lesson Solving Linear Equations in Two Variables to introduce systems.

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Then we asked students to draw a sketch of a system of equations with one solution, a sketch of a system of equations with no solution, and a sketch of a system of equations with many solutions.

We used the Math Nspired activity How Many Solutions to a System to let students explore necessary and sufficient conditions for the equations in a system with one, no, or many solution(s). We used a question from the activity so that students could solidify their thinking about the relationship between the equations for different solutions.

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We sent the questions as Quick Polls so that we could formatively assess their thinking and correct misconceptions.

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We like using this type of Quick Poll (y=, Include a Graph Preview) while students are learning. The poll graphs what students type so that they can check their results and modify the equation when they do not get the expected result.

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(Note: on the second day, we asked again for students to create a second equation to make a system with one, no, and many solution(s), but the given equation is in standard form.)

 

On Day 2 of Systems, the learning goals are slightly different.

Level 4: I can solve a system of linear equations in more than one way to verify my solution.

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

Level 2: I can solve a system of linear equations, determining whether the system has one solution, no solution, or many solutions.

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

We started with a task from Dan Meyer’s Makeover series, Internet Plans. Part of this makeover was a complete overhaul of the context. Students write down a number between 1 and 25. They view this flyer (from Frank Nochese).

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And then they decide which gym membership to choose, using the number they wrote down as the number of months they plan to work out.

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When I saw the C’s and B’s on the number line on the blog post, I actually thought that the image was beautiful – a clear indication that one plan is better than the other two for a while, and then another plan is better than the other two. This image absolutely leads to the question of which one becomes better when – and thinking about why plan A is never the better deal.

Our students started calculating. Several asked what $1 down meant before they could get a calculation. Then they shared their work on our class number line.

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We were dumbfounded at the results. Really. It was hard to know where to start. We were prepared to help student develop the question with correct calculations. We hadn’t thought about what we would do when students couldn’t figure out which plan was the better deal for a specific, given number of months.

What would you do next if this is how your students answered?

 

We started with the lone C at 4 months, hoping someone would claim it and share his calculations. He did: I divided $199 by 12 months and then multiplied by 4 to get the cost for 4 months.

 

Another student said he couldn’t do that because you had to pay $199 for 12 months no matter how many months you actually used it.

 

Another student said that A was the better deal because they were planning to get every other month free by working out more than 24 days each month. We discussed how realistic it is to work out at a gym 24 days a month.

 

We did eventually look at a graphical representation of each plan. And talked about what assumptions had been made using these graphs.

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Can you tell which graph is which plan?

Can you write the equations of the lines?

In the first class we never made it to actually calculating the point(s) of intersection (class was shorter because of a pep rally), but other classes did calculate the point(s) of intersection.

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It’s not a waste of time to think about Plan C costing $199 for 12 months.

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But then we really should have created a piecewise function to represent the cost for 13-24 months as well.

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The number lines weren’t much better in subsequent classes.

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So maybe the students got to practice model with mathematics, just a little, even though we have a long way to go before students will be able to say, “I can create a system of linear equations to solve a problem”.

The teachers were definitely reminded of how much work we have to do this year, as the journey continues …

 

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