Graphing Ratios

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Objective

SWBAT graph the pairs of values displayed in a ratio table.

Big Idea

A graphical representation of an equivalent ratio will be linear because every time you increase one value by a given amount, you increase the other value in proportion to the first.

Think About It

5 minutes

Students work independently on the Think About It problem.  After 2 minutes of work time to graph the ordered pairs, I ask for a student sample for the document camera.  I ask students what they notice about the relationship between the numbers as we move to the right.  They'll notice that we increase the x-coordinate by 4 between each point, and, that we increase the y-coordinate by 6 between each point.  I model this on the student sample, by annotating the graph.

Instructional Note: In a previous unit, students learned to graph in all quadrants in the coordinate plane.  This is their first exposure to graphing linear functions.  The key idea for this problem, which leads us into the intro to new material, is that plotting points from a ratio will form a straight line because all of the pairs have a constant relationship.  

Intro to New Material

15 minutes

This lesson combines creating ratio tables with graphing on the coordinate plane.  To start the Intro to New Material section, I have students create a ratio table to represent the situation. Students may ask if they can create a tape diagram as the visual.  Because we're working with multiple equivalent ratios, I want them to use a table.  We then graph the ordered pairs and discuss the line. 

I think that students may ask why they need to graph the ordered pair in order to be able to answer the question about kids and adults at the beach.  In response, I plan to tell them that we don't need to graph in order to answer the questions! Instead, I will explain that graphs help us to organize the problem, to see trends, and to make predictions.  

The key takeaway from this section is that whenever you plot equivalent ratios, the graph will always form a straight line.  Students can check for the reasonableness of their answer (in this case, the accuracy of the graph) by seeing if the equivalent points form a straight line.  If they don't, then one of the coordinate pairs does not represent an equivalent fraction.

Partner Practice and Discussion

15 minutes

Students work in pairs on the Partner Practice.  As they work, I circulate around the room.  I am looking for: 

  • Are students correctly labeling the axes?
  • Are students using their ratio tables to plot the points?
  • Are students showing clear, logical work?
  • Are students providing an answer to the specific question?
  • Are students checking for the reasonableness of their answer?
  • Are students correctly plotting their coordinates?

I am asking:

  • What does the ratio mean in this problem?
  • What is the value of each part?  How do you know?
  • Why does it make sense that graph you created is a straight line?
  • How do you know that you’ve graphed the equivalent ratios correctly?
  • What is the question asking you to find?

 

After 10 minutes of partner work time, I bring the class back together.  In the previous lesson, students learned to work with scale factors.  Some pairs may have used scale factor to create the ordered pairs, and I show a scale factor sample. If no pair used scale factor, then I create my own example for students on the document camera.  We discuss why this method works when creating graphs.  My preference is for students to plot at least 3 ordered pairs when creating a line.     

 

Independent Practice

15 minutes

Students work on the Independent Practice.  I circulate around the room, ensuring that student work  is meeting all the criteria for success.  

Note that students do not yet know the language of independent and dependent variables.  There may be students who put what would be considered to be the independent variable along the y axis.  For this lesson, I allow that.  The aim of this lesson is for students to plot equivalent ratios.  We will learn about independent and dependent variables later in the school year.  

An extension for students could be to have students create the ratio table from a given graph. 

Closing and Exit Ticket

7 minutes

After independent work time, I ask for a student sample for problem 6.  I ask students how they can use only the graph to determine whether or not the graph represents equivalent ratios.  I have students turn and talk with their partners. and then have 2 students share out their responses.  

Students work on the Exit Ticket to end the lesson.