Rules ----> Tables ----> Graphs

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Objective

SWBAT find number pairs that fit a rule, plot the pairs, and sketch the graph.

Big Idea

Rules, tables, and graphs, oh my! Students work fluidly between multiple algebraic representations.

Opening

15 minutes

This activity asks students to begin with a rule, generate an In/Out table to represent the rule and then make the corresponding graph.  I introduce the activity by letting students know that in the past lesson they were working from graphs to tables to rules.  I tell them that today, they will be working in the opposite direction. Instead of starting with a graph, they will start with a rule.  They will then make a table, and from the table, make a graph.

I like to begin with some tables and rules they have generated in past lessons; I start with one of their examples.  I post just the rule on the board.  For example, a student generated rule might be: Out = 3(In) + 4

I ask students to come up with some number pairs that would fit that rule.  If students are stuck, I let them know they can choose any number for the In.

Together we generate a table.

Next, I ask students how they could represent this table on a graph.  I let them know that they will have to come up with the appropriate scales for the x and y axes.  I ask them for tips about how to create scales that will be represent their rule.  Together, we generate a class graph for the rule. I ask questions here about where the graph would end, and would it continue beyond the first quadrant.  I also ask about non integer values and negative numbers.


Investigation

25 minutes

Next, I let students get working on Rules, Tables, Graphs.  I think this is a nice activity for students to work on individually, especially because I do a lot of group work in my class.

This activity first gives students 3 equations to graph that are written with the words In and Out rather than using variables.  The second question is a quadratic.  The third equation will require students to scale the graph carefully.

The second activity, Graphing Linear Equations, gives students  equations that use x and y rather than In and Out.  This gives students the opportunity to transfer their learning to a more formal algebraic situation.  If it hasn't been discussed already, you can talk about x always being the independent variable and y as the dependent variable.  

Things to watch for as students work:

  • Order of Operations - I make sure students follow the order of operations as they populate their tables.  
  • I encourage students to examine negative values as well as non integer values for the In values.
  • Students will likely have trouble with the scales for their graphs! I always make sure to have lots of extra graph paper on hand. I also different sized graph paper available so students can decide what size suits their particular rule.
  • Students may be confused by what they see happening for the nonlinear examples. I remind them that not all algebraic relationships are linear. They might find the resulting parabolas to be interesting shapes if they haven't seen them before.

As students finish, I have them put up one of their graphs on the board in preparation for the whole group discussion.

 

 

Discussion + Closing

20 minutes

Next, I have different students present their graphs to the class.  I ask them to address specifically how they determined an appropriate scale.  I like to keep a list of strategies for this task as this is an ongoing issue for students.  As they share out, I encourage students to use the appropriate vocabulary when discussing their graphs (axes, ordered pairs, x and y coordinates, etc).

I place special emphasis on the relationships between tables and graphs. I might ask students how they see the graph as a reflection of the table. For example, for the linear equations, I might ask how they can see that the ordered pairs will graph to make a straight line in the table.  I extend this idea to ask about what's happening in the nonlinear graphs and how that relates to the corresponding tables.  Students should be able to see that though the In column of the quadratic function is going up by the same amount, the Out column is going up by increasing amounts.  I have found my students to be excited about the shape of the quadratic graphs as they may not have seen parabolas before.  

I make sure the discussion touches on the idea that the lines will extend beyond the first quadrant. I ask students if they sketched lines once they saw the shape of their graphs or if they left discrete points.  This is a good way to tie in non integer values as Inputs and to talk about discrete and continuous graphs if this terminology hasn't been covered already.

I leave time after the discussion for students to reflect on their learning.  Remember that the emphasis in today's lesson is on starting with a rule and then generating tables and graphs. I might focus in on the specific relationship between tables and graphs.  I have students complete an exit ticket based on the following prompt: How would you describe the relationship between tables and graphs?