Determine if Two Quantities are in a Proportional Relationship

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Objective

SWBAT determine if two quantities are in a proportional relationship using a table or a graph.

Big Idea

Students will practice MP1, MP2, MP3, MP4, and MP6 to determine which table demonstrates a proportional relationship between x and y values.

Bell Ringer

20 minutes

Hand students the bell ringer as they enter the room.  Students will sit in their I.T.T seats and begin right away using MP1, MP2, and MP6 to grapple through two problems. Allow students 10 minutes for I.T.T. Students will need to write their thinking strategies in their interactive notebooks. They will use this to share during pair up time.  Walk the room to check for understanding.  Students should show that they are able to test for equivalent ratios in a table.

Once students have worked individually for 10 minutes, have students discuss their work with their pair up partners.  Students should have 10 minutes to discuss their thinking and compare their responses.  Students should be able to guide one another through the process of solving each of these questions. This will put into practice MP3.  For struggling learners, you may want to check for vocabulary understanding.  During this time it will be important to check that students understand how to read an x and y table.  They may know the table as input/output, or function table. I ask students where they have seen a table like this before.  What were some of the uses of the table?  Students should use their prior knowledge of graphing on a coordinate grid to help them determine which table represents a proportional relationship between x and y.  Students may be tempted to choose a table that has a relationship between the x values separate from the y values.  It is important for the students to recognize the relationship must be between the x and y values. 

If students are able to determine the quantities that are in a proportional relationship by testing for equivalent ratios in a table, or graphing on a coordinate plane and observing whether the graph is a straight line through the origin they are meeting the expectation of the objective.  

Whole Group Discussion

10 minutes

During this time, students groups should have the opportunity to share out their pair up time discussions, and reveal each of their responses.  You may not have time to have each student share.  As you filter through the room during pair up time, attempt to identify a group who has understanding, some understanding and little understanding. 

During the whole group discussion have students debate their responses and defend their thinking.  This again will practice MP 3. As the facilitator of the discussion, you can head the discussion with open ended questions that will ask students to defend their answers.  For example:

Astudent may say I chose A for question number 1 because there is a constant change of (+2) between the x values, and a constant change of (+14) between the y values.  Now, plotting these points would not be a conducive strategy here. 

Ask students why?  Students should recognize that the ordered pairs have numbers in the 30’s and above.  This would be very time consuming to create a coordinate grid that would accommodate the amount of numbers needed on each axis. 

What would be another strategy that the students can use?  Students should feel comfortable writing the x and y values as ratios and reducing them to their lowest form.  Students who have been comfortable using the calculator to reduce fractions will need to practice divisibility rules, finding the GCF, and simple division.  This may done through homework assignments, bell ringers, exit tickets, or lunch bunch groups. 

Students will be tempted to choose the response in which they see a constant change between the x and y values.  Both A and C have a constant change between the values, where A has a constant change of (x7) and C (+9).  This will be a great time to discuss how a constant change of multiplication differs from a constant change of addition when determining a proportional relationship between quantities.  

Closing

10 minutes

It is important for students to know the correct process and correct answers in order for them to correct misguided thinking.  Go through the correct process in responding to each question.  Students will correct mistakes.  For example, for question number 1, I would go through eliminating each response that is incorrect first.  Response B is eliminated because when each ratio is reduced, none of the ratios are equivalent.  Response C is eliminated for the same reason.  However, some students may get as far as reducing each ratio and may choose C because they will see a pattern in the reduced ratios.  This will be good for the students to understand that the ratios must be equivalent.  

Exit Ticket

For the exit ticket, have students explain why graphing on a coordinate grid would not be a good strategy to use for this bell ringer.  Have students create a proportional table that would call for using the graphing strategy and explain the difference between their table and the correct table from question 1.