James Bialasik SWEET HOME SENIOR HIGH SCHOOL, AMHERST, NY
Algebra I : Unit #8 - Extending Equations : Lesson #10

The Groupon Question: Solving Two Variable Equations

Objective: SWBAT solve an equation in two variables and use the equation to model a situation
Standards: HSA-CED.A.2 HSA-CED.A.3 MP1 MP2 MP3 MP4 MP7
Subject(s): Math
60 minutes
1 Launch - 15 minutes

At the start of this lesson, I have students work in pairs. I give them the opener and I project the question on the board.  I like my students to take time to think through the problem, before they solve it.  Once they begin their work, I encourage students to list as many combinations as they can. I'll observe and keep pushing students until they have all come up with a minimum of five combinations.  As they work, I will explain to the class that we are going to be posting the combinations on the graph projected on the board:

Solving_Two_Variable_Equations_Launch_Graph

Once this seed is planted, we will do a Think-Pair-Share about how to best display the combinations on the graph.  Afterward, I will survey the class to determine if any group is completely confused as to how they will display their information.  If so, I will try to find another group who can help them. With real world tasks like today's Groupon Question, I want my students to have confidence seeking assistance from peers.

Once all the groups are clear on what to do, I will ask one person from each pair put a possible combination on the graph. As the data is added, students will again turn-and-talk to drill a little deeper into the problem using the representation. This turn and talk is simply to give students a minute or two to process what they are seeing before the concepts are formalized through the discussion.

Launch-Reflection
Safety

I was very impressed by how smoothly the launch for this lesson went.  In my lesson narrative, I tried to have questions that would prompt students to think about various aspects of this task. This included items such as:

  1. the meaning of the intercepts
  2. rational values,
  3. connecting the points

Interestingly, my students came up with both the values of (0,25) and (25,0) as well as the value (12.5, 12.5) without any prompting.  This led to a more student led discussion (which is always better) where those students discussed why they chose those particular points.  

We were able to work off of the (12.5,12.5) value to determine that there would be many other combinations that we may have missed which would necessitate drawing a line through all of the points.  The students were also able to discuss how to come up with the equation x+y=25 and show that all of the values we graphed made this equation true.  

As you can see in the image, students were very precise in their point plotting. As more and more students added points you could hear them starting to discuss the fact that it was going to form a straight line.  The closing activity for this lesson also served to drive home the point that every point on the line was a solution to the equation given.

2 Discussion - 15 minutes

During this part of the lesson it is important to give students time to think about the process for and the concepts used when solving equations in two variables. We are going to focus on a series of points here:

  • What do the y-intercepts represent?
  • Is each graph continuous or discrete?
  • How many solutions are there? 
  • Is there a single equation that represents this situation?

First, I will pick three or four of the solutions that the students posted during the launch and list them next to the graph as ordered pairs.

Example: (15, 10), (10, 15), (20, 5)

Then, I will ask, "What do you notice about all of these values?" During the conversation that follows, I will guide students towards the understanding that all of the ordered pairs have a common sum of 25.  I will make explicit the fact that these ordered pairs are made up of two coordinates, usually x and y.  

Next, we will relate the Graph to the context. Students should understand that the graph represents an amount that you and your friend spend. When this is understood, I will ask, "Is there an equation that we can use to represent this situation?"  I will give students time to think-pair-share around this question even if a few groups have the answer.  I try to manage the conversation so that all of my students have time to think and process.  At the same time, I listen while students are sharing. I am trying to identify students that are constructing a solid (clear) rationale. I will call on those students to share with the whole class, to model and to help the class move forward quickly (MP3).  

Teacher's Note: If no students have put the points (0,25) or (25,0) on the graph, I will ask the question, "What if your friend doesn't want to eat?  How would we indicate that on the graph?"  If the above coordinates have been marked ask the class what those point represent.

To get to the idea of a discrete vs. a continuous graph, I may need to guide students towards some decimal values.  An easy way to do this is to ask, "What if you and your friend spent exactly the same amount on your meal, how much would this be?"  I will follow this easy question with a quick question, "What if your friend spends $10.63, how much can you spend?"  My hope is that students will jump to the conclusion that a continuous line can be used to represent all of the possible values. 

Teacher's Note: While this graph is not completely continuous (only goes to the hundredths place), we will call it continuous for the sake of this example.

In this portion of the questioning, we will conclude by considering why all of the coordinates are in the first quadrant.  You can show students that coordinates like (-5, 30) will certainly make the equation x+y=25 true.  However, solutions like this do not make sense in the context of this problem.

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3 Closure - 10 minutes

The Closing Activity for this lesson will assess students preliminary understanding of the day's concepts.  I will use it to prepare for differentiating instruction in future lessons.  The students investigate a situation where the value of the coefficient for x and y are different. Students will need to determine what these coefficients mean in the context of the problem and construct an argument to explain their thinking (MP4).