Introduction to Linear Equations in Two Variables

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Objective

SWBAT identify the solution to a linear equation in two variables and explain what the graph represents.

Big Idea

Interactive lesson moving between small groups discussions and whole group sharing full of thought provoking questions.

Bellringer: Warm-up

10 minutes

Ask students to get a sheet of paper or at least a half sheet.  Have them divide the paper vertically down the middle and label the left side Pre-Lesson and the right side Post Lesson.  Ask them to paraphrase the following questions on the left-hand side of the page under "pre" lesson and then answer each question.  Write the following questions on the board or use your projector to project the questions.

         1. What is an example of a linear equation in two variables?

         2. What does it mean to solve a linear equation in two variables?

         3. How many solutions are there to a linear equation in two variables?

         4. What does it mean to graph a linear equation?  What is the importance of the graph?

 

It is important to see this pre lesson knowledge and then compare it to the post lesson knowledge to gauge how effective this lesson is at addressing each of these questions.  Answering these questions is the goal of this lesson.  The big idea is that students will easily be able to answer each question with depth and conceptual understanding.  Convey the importance of these questions to the students and that these are the focus concepts of the lesson.  Using this writing strategy is actually also applying a literacy strategy.

Clarifying and Sharing Learning Intentions and Criteria for Success


 

Beginning the New Activity

30 minutes

This lesson does not have a handout, as it is more interactive between you and the students in small groups and you and the class as a whole group.  You are working with student to directly answer the questions in the given warm-up by completing this interactive lesson together. 

 

Put the following equation up on the board:  x + y = 10.  Bring student groups (homogeneous groupings of two or three at most students) an index card and ask them to write two solutions to the equation x + y = 10. Allow students about two minutes to complete this task then collect the index cards and record the answers on the board for everyone to see.  You expect to have whole number solutions and what is great is when students in one group write the solution x = 4 and y = 6 then anther group writes the solution x = 6 and y = 4.  Discuss as a group, are these two solutions the same solution or different. Also important to look for is HOW students record their solutions.  Most students will probably write x = 4 and y = 6 but if anyone writes the solution as (4, 6) bring this up for discussion about why it would be written in this format.   Ask your students during this whole class time if they can come up with any more solutions that are not on the board yet and continue to write these solutions (this is when students should begin to think of negative integers and even rational numbers).  

Put another linear equation on the board:  2x - y = 12 and ask student groups to take back their cards and put two new solutions to this equation on the cards.  Repeat the process of collecting cards and writing solutions on the board.  Ask students if they can add any more solutions to the ones listed so far.  (Hopefully you will begin to get negative numbers and fractions as options)

Ask students to discuss the following questions within their groups:

1. What does it mean to solve a linear equation in two variables?

2. How many solutions are there to a linear equation in two variables?

Walk about the room listening to discussions and selecting students to present their answers during the whole class mini-wrap session.  After about 3 – 5 minutes of small group discussion, pull the class together for a whole group mini wrap-up session lead by the student groups you just selected.  A great point to make here is that when x and y are both unknown, the options are limitless, but as soon as one variable (either x or y) have a value the other unknow has limited choices in order to make the equation true. (Good knowledge to begin building for solving systems of equations in the next unit).

Pass out graph paper or ask students to get their own out.  Ask students to graph two solutions from the board for the equation x + y = 10 (pick the two from your previous discussion of x=4 and y=6 vs. x=6 and y=4 if this was a good conversation among your students).  Discuss now if students believe these solutions are the same solution or different (obviously different as they graph in different locations).  Next, ask students if they notice anything unique about their graph?  Script a few responses or guesses on the board and have them graph any third solution from the board they want.  Again, is there anything unique about the graph?  Script any new information or guesses on the board.  Ask students to graph any three additional solutions they chose.  Pick two to four example graphs from your students who chose different solutions to graph.  Look at these example graphs under the document camera.  Put the following questions on the board for students to discuss in their small groups.

  1. Do these appear to be correct graphs? 
  2. How do you know if they are correct or not? 
  3. What are you looking for in these graphs compared to your own?

Move about the room talking to groups and asking for their answers.  Choose groups to present during the mini wrap-up session as you move about the room.  After about five to ten minutes hold a mini-wrap-up over these questions. 

Next, post the question, “What is unique about the solutions to a linear equation in two variables when you graph them?”  Allow groups one minute to talk and then share answers as a whole group – make sure you script these great responses on the board (you are looking for solutions lie in a line or along the same line which should lead to the discussion that a linear equation has infinite solutions). 

The final question for whole class discussion is, “What is the minimum number of points you must graph in order to find the line which contains all solutions?”  Allow time for students to discuss and then ask them to hold up their hands.  Let them know on the count of three they should show their group answer using their fingers.  Of course you are looking for students to show two fingers but ask students with other numbers to come to the board and demonstrate.  Especially interesting would be to have a student demonstrate at the board (with a smart board you can have a blank graph paper as a background) how to graph x + y = 12 using only one point (you know someone will try this).  Then let someone else who held up three fingers try to “fix” the x + y = 12 graph and so on until you arrive at two points as a class decision. Allowing students to expirment with graphing the minimum number of points is bringing in math practice standard six (MP6), attending to precision.

 

Wrapping Up the Lesson

10 minutes

Ask students to return to the bellringer paper and answer the same questions from the warm-up using the post lesson side.  Collect these papers as a formative assessment (give them back to students as notes on the following day).  Assign the graphing page as homework for tonight.