Hand students the Bell Ringer as they enter the room. For this bell ringer, students will work on problems, 1 and 2.
Before starting this bell ringer, go over the homework from the lesson on identifying the constant of proportionality from a table. Review the big points of the lesson when going over the homework. This will transition students into identifying the constant of proportionality from a graph. Some important key points to help with transition are:
1. Key vocabulary
2. How to identify x and y values and how to use that information to identify these values plotted on a coordinate grid.
3. How to solve for unit rates.
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 plots on a coordinate grid. This lesson may pose difficulty in reading the plotted points because the y values are not on a whole number. It will be important for students to identify when solving for unit rates they will end up with a repeating decimal or complex fraction and relate that to how it looks on the grid.
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 rate represented on the grid. Some students may still struggle with converting the x and y values into rates. The order of the rate 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. Students may also struggle with reducing the rate into a unit rate because the division will not compute evenly. 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 grid 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, but knowing that the first number in the rate 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. They will need to be comfortable with seeing a decimal or fraction as the numerator. Students should understand that dividing the numerator and denominator by the denominator will convert the rate to a unit rate. Some students will quickly relate the fact that the plotted points are not exactly on a whole number for both the x and y axis and understand that the unit rate will not be a whole number out of 1. This should lead students to think in terms of fractions or decimals when computing for the unit rate.
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 1 with the response of 3/5 if they have little understanding. Students will see that there is a plotted point of (3,5). They will understand how to write the ordered pair as a rate, but not know how to write the rate with the relationship of y to x.
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 1, I would start with identifying one point is not exactly on a whole number for the y value. In question 1, the lower plotted point has a variable that represents the y value. This may be difficult for some students. Estimation will be great when finding the value of this variable. This would lead into a discussion on finding the unit rate and how to write the rate with the correct relationship of y to x. I would show them what this rate looks like. This will eliminate the response of 3/5. Once that response is eliminated I would reduce the rate to a unit rate and talk about what 1.6666666666 looks like on the grid. This should lead into the conversation of equivalent rates. Students should know that both plotted points are equivalent. Identifying equivalent rates on a grid or table is a way to identify constant of proportionality in a grid or table. This is an opportunity to tie in previous conversations from prior lessons.
This resource can be used as a guide in creating questions for a homework assignment for this lesson. The resource was found on Math Worksheet Land.