##
* *Reflection: Connection to Prior Knowledge
More with Solving Two Variable Equations - Section 2: Investigation

My students did well with this investigation. This lesson was really a benchmark lesson for me because it represents a shift in my thinking about mathematics. I have always been a "y=mx+b" person when it comes to graphing lines. Then I would worry when students were faced with difficult equations where getting y alone was challenging. Now students are understanding that a line is made up of the infinite number of solution points to its equation. Because of this, all they have to do is find coordinates that make the equation true. I feel that this is a much more rich understanding than simply finding y-intercept, plotting slope, etc. Students are really beginning to understand what a linear equation (or any equation for that matter) in two variables represents.

I did notice that students are still struggling with part "e" on question 1. They are able to plot the points that make an equation true but are still unsure why the points should be connected with a line. In my observations and discussions with students this is due to their lack of understanding of rational numbers in general. The real reason the points are connected with a line is because there are an infinite number of inputs and outputs (coordinates) in between each set of integer coordinates. Students were having trouble grasping this because they are not comfortable "playing around" with x-values such as 0.1 or 0.25 or 1.375 or 4.3245. They were great with finding integer values that work but their lack of comfort with decimal values made it difficult to put down a solid explanation for question "e".

I was pleasantly surprised how well students did on the last question of the investigation (4x+5y=0). I thought that the values would give the students trouble but they were very strategic about picking x and y-values that would yield integer results. Once again, students intentionally stayed away from any decimal values that would make the equation true. In the past, this would be one of those equations that students would have difficulty graphing because they were locked into the y=mx+b mode. Letting students find values that make the equation true offers them some flexibility when they have equations in this form.

*Connection to Prior Knowledge: Deciding on a new approach*

# More with Solving Two Variable Equations

Lesson 11 of 13

## Objective: SWBAT solve two variable equations by graphing the equations and using ordered pairs.

#### Launch

*10 min*

This Opening Activity will require that students reason quantitatively about the number of students that could attend a party. I structure the beginning of class in way very similar to the previous lesson.

- I give each pair of students some time to determine possible solutions.
- I post a graph on the board.
- I have one student from each pair come up and post their solutions to the question.

This time, however, when students mark an ordered pair on the graph, I will ask them to label the coordinates as well. Here are some questions that I will use to help guide the discussion and help students to notice the structure in this linear equation:

**Is there an equation we can use to model this situation?**I will Guide students towards seeing that x + 2y = 30 would model the situation appropriately since each freshman (x) is only bringing $1 and each sophomore (y) is bringing $2.**So if x + 2y = 30 is our equation, why didn't anyone mark the coordinate (5, 12.5)?**This is an important point, so I will give students time to process at this point. When students share out, point the conversation towards the fact that the equation is a model, but in the context of the problem we cannot have decimal values because our variables stand for number of students.**Yesterday all of our coordinates added up to the same number. But look at our coordinates today (2, 14), (4, 13), (6, 12)...Why is the sum not consistent?**It may take some thought but eventually students will come to the understanding that in the Groupon question, both parts were equal (x + y =25). In this question the sum of*x*and twice*y*will give the same sum of 30 because that is the equation. I plan to show students using the coordinates (that is why we had students list coordinates) that this is the pattern. Explain to them that all points on the line will have the same relationship (even ones like (5, 12.5) which are not in the context of this problem).

#### Resources

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#### Investigation

*20 min*

As we begin to work on the investigation I will ask students to do a Think-Pair-Share and list as many important ideas as they can think of dealing with solving two variable equations based on what they have seen so far (see Collaboration: Building Communication and Student Ownership for an example of how this works in my classroom). Students will probably come up with many ideas, but they will fall within the following categories:

- Two variable equations have many solutions.
- The solutions can be represented by coordinates on a line (or curve) in the coordinate plane.
- All of the ordered pairs that make up the line (or curve) must be solutions to the equation.

I explain to my students that they are going to continue to investigate these ideas by exploring some more linear functions that don't necessarily have a context (that is, their domain is all the real numbers). I give students about 10-15 minutes to complete the investigation. We may ither summarize the results as a whole class, or I may let groups share what they discovered. In either case, here are some points of emphasis for this investigation:

1c) I will look to see if students are finding an organized way to list ordered pairs. For example, start with a y-value and then find the corresponding x value. Choose y-values in some logical order (y=1, y=2, y=3...). Students will then realize that they can choose any y-value and find the corresponding x value.

1e) A line drawn (with arrows on both ends) illustrates the fact that we don't have to (nor can we) list all of the solutions we can use a line to show that there are an infinite number and the follow the pattern of coordinate pairs on the line.

2b) There is only 1 x-value that will make this equation true, namely, x= -2. Since we are graphing the solution in the coordinate plane, we can use any y-value as long as the x-value remains x=-2.

2a/2c) Have students discuss the values they chose for x and y. Students will notice that some choices lead to easier calculations and there is nothing wrong with making your life easier from time to time! For example, in 2a the choice of y=1 or y=-1 leads to integer answers for x while y=0 does not. Since students are trying to graph their solutions (1, -2) is easier to graph than (0, -3/2).

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#### Closure

*10 min*

This Ticket out the Door requires students to match equations to graphs. I allow students to work on this activity in pairs due to the understanding that can be developed through their conversations. Students have not yet been exposed to graphing lines by slope-intercept or another method. So, they will rely on finding solutions to each equation in order to match the equation to the graph (MP8).

I encourage my students to choose x and y values that fall in the 10 x 10 grid for efficiency. Also, students will be developing a great deal of conceptual understanding from this activity. I will listen to and make note of student conversations during this closing activity. This will help me to get a sense for how you to formalize the concepts of rate of change, intercepts, and linear functions in future lessons.

*expand content*

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- LESSON 1: Creating Equations to Solve Problems
- LESSON 2: More Solving Problems with Equations
- LESSON 3: Creating Inequalities to Solve Problems
- LESSON 4: The Exercise Plan (Day 1 of 2)
- LESSON 5: The Exercise Plan (Day 2 of 2)
- LESSON 6: Solving Literal Equations
- LESSON 7: Literal Inequalities
- LESSON 8: Rewriting Formulas
- LESSON 9: Printing Presses: Solving more difficult Linear Equations
- LESSON 10: The Groupon Question: Solving Two Variable Equations
- LESSON 11: More with Solving Two Variable Equations
- LESSON 12: Solving Two Variable Inequalities
- LESSON 13: Modeling with Two Variable Equations and Inequalties