Identify the Constant of Proportionality from a Table

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Objective

SWBAT identify the constant of proportionality from a table.

Big Idea

Students will use MP1, 2, 3, 4, 6, and 8 to solve problems involving identifying the constant of proportionality from a table.

Bell Ringer

20 minutes

Hand students the Bell Ringer as they enter the room. For this bell ringer, students will work on problems, 1 and 2.  Problems 3 and 4 will be their homework.

Students will sit in their Individual Think Time 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? 

For this bell ringer, as you walk the room you are checking for students to understand what constant of proportionality is.  Students should understand they are looking to find the unit rates of each ratio represented in each of the tables.  Some students may still struggle with converting the x and y values into ratios.  The order of the ratio is crucial to how the question reads.  For example, in question 1, the order of the rate is y to x. 

One common mistake students will make is compare the x value to the y value.  If they do this the order of the rate will not compute to the correct unit rate.  Prior knowledge students will need in order to be successful in this task: Students will need a strategy in finding unit rates, they will need to know how to read a table in order to put the x and y values in the correct order, and students will need to know the three ways to write a rate. 

I prefer my students writing the ratio in fraction form, by knowing that the first number in the ratio is the numerator and the second number in the rate is the denominator. Students will need to be able to reduce the rate into a unit rate. Students should understand that dividing the numerator and denominator by the denominator will convert the rate to a unit rate.

One other helpful hint that I find works for my students is to change the horizontal table to a vertical table.  

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 evoke students to defend.  For example a student may respond to question 2 with the answer of 1/11.  This mistake is caused by not relating the y value to the x value.  Although each rate will reduce to 1/11, this will be the time to discuss how to identify a unit rate.  Students must remember that all unit rates are out of 1.  For question 1, students may see the relationship of x to y as being (x18) this will be a great time to discuss how to use that relationship to talk about constant proportionality in regard to unit rates. If students recognize multiplying consistently by 18 is the constant proportionality this will be a great alternative strategy for students to use.  You may want to tie in the prior discussion over constant change of addition vs. constant change of multiplication between x and y values.  

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 solve for the constant proportionality by setting up the rates with the comparison of y values to x values, then solve for the unit rate.  This will eliminate all other responses.  I would also show the students the relationship between the x and y values and discuss how this relates to constant proportionality.  The same process will work for question 2. 

Homework

Have students complete question 3 and 4 from the handout given.  Go over the responses during the bell ringer time for the next day's lessons.