Graphing Systems of Equations

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SWBAT graph systems of linear functions to find common points/solutions. SWBAT define variables and create equations to model a system of equations.

Big Idea

Students use technology to create and interpret systems of equations graphically!

Entry Ticket

15 minutes

For today's Entry Ticket: Graphing Systems I have students create and graph three linear equations on the same coordinate plane. 

The intent of this entry ticket is to practice and review prerequisite skills. Namely, I want students to be able to create, manipulate and interpret equations to create graphical representations of linear functions. There is no scale given on the graph, which is a strategic decision. I want students to think about the details of a graph (MP.6) and think about a scale that would be reasonable and give them information for this problem. 

Students work quietly or in small groups on the entry ticket. I find that assigning an effort grade to entry tickets helps students stay engaged, motivated and on task. I check in after 5-10 minutes and assign an effort grade based on observations of student conversations and work produced. I then review the entry ticket as a class. I try to act more as a facilitator during the review as I want students to lead and manage the conversation and process. 



Explicit Instruction and Active Note-taking: Graphing Systems

20 minutes

During this section, I utilize the Class Notes: Graphing Systems of Equations to provide explicit instruction around how to graph systems of equations. Students take Two-Column Notes for the next section of class on graphing systems. I introduce this presentation as a review on graphing linear functions. The main difference is now we are looking at two functions on a graph simultaneously, but the underlying concepts and skills to graph linear functions are the same as in previous lessons. 

During this time students are actively engaged in the conversation through Turn and Talks, writing notes and asking higher level questions of myself and their classmates. 

Guided Practice: Solving Systems Graphically

20 minutes

During this Guided Practice: Graphing Systems of Linear Functions section, I have students work in small groups on graphing and solving systems. I include an example where both equations are in slope-intercept form, and another example where students need to rearrange equations, create a table or use another strategy to graph the functions. 

I have every student write out the work for these examples and include them in the class notes for the day so they can use the examples as a reference for additional practice, homework, etc. 

These Guided Practice: Graphing Systems of Linear Functions problems provide students with an opportunity to engage in MP.6. Specifically, students have to attend to multiple details when graphing systems of equations. I like leaving out the scale on graphs because it encourages students to really think about an appropriate scale for the problem, and for word problems thinking about the scale also encourages students to think about the context and meaning behind the problem.


Collaborative Work: Graphing Systems

20 minutes

I then want to peel back the scaffolding for the lesson and have students work together to construct their own understanding of the day's concepts. 

For group work I provide students with practice problems to solve together. If there is time, students present their work to the class. I like to use a variety of instructional materials for practice. Some examples include textbooks examples, Kuta Worksheets, Khan Academy and/or my own resources/examples.

In this particular class I like using the Babysitting and Band Problems for group work. Both problems require students to integrate their thinking and solve systems by applying all of the tools learned in the unit (creating equations, graphing, algebraic manipulation, interpreting solutions based on the scenario, etc.) 

In this section I also included example word problems used in a previous lesson on creating equations. Since students already created the equations for the situation on the Collaborative Practice worksheet, in today's lesson I have them focus on representing the system of linear functions graphically. 

Exit Ticket

15 minutes

For the Exit Ticket students work independently on graphing a system of equations. I purposively give only one equation in slope-intercept form. The reason for this is I want to see if students can graph and also see if they can rearrange equations or use different strategies to graph the equations. 

I like to have students complete an Idea Organizer as part of this closing activity. Having students prove their solution in more than one way, helps students to make connections between different ways of describing functions. 


Teacher Note: The graphing can be given as the Exit Ticket and the Idea Organizer can be left for the night's homework assignment to provide students with more time to process and organize their thinking into a written response. 

Homework: Khan Academy Practice Graphing to Solve Linear Systems

For homework students complete at leat 5 practice problems on Graphing systems of linear equations on Khan Academy. I like this interactive applet because it provides a lot of nice support for students to practice the concepts.

For example, the site provides a coordinate plane and makes it easy for students to move and manipulate the graphs. This is an excellent example of Universal Design for Learning that not only supports students with fine motor difficulties, but a wide range of students who struggle setting up graphs. 

In addition the website provides hints to help students initiate the process of graphing and solving systems of linear equations. And as always, Khan Academy has an excellent repertoire of instructional videos for students to review when/if they get stuck on a particular problem.