Identifying the Constant of Proportionality
Lesson 12 of 21
Objective: SWBAT identify the constant of proportionality in a table, graph, or word problem and interpret its meaning in the context of the problem.
Students enter silently according to the Daily Entrance Routine. The Do Now assignment spirals through some concepts and skills students need to master as evidenced by recent Exit Tickets and graded assignments. The first question includes a problem that will test out students’ knowledge of rational number operations. This one may stump some of my students, but I want to see who can make progress with it.
Often, when students sit staring blankly at a problem, stumped, I offer possible problem solving strategies. In order to recall such strategies students are encouraged to review their notes. In some cases students may not be able to do so because of a lack of notes or an organizational issue. In these cases my last resort is to use a whiteboard to get students started on a strategy. My targeted strategy for solving each question on today's Do Now is listed below:
- Bar models to understand a combination of fractions involves adding and the connection to the LCD, pieces that fit in both the fraction 1/6 and ¾.
- A picture to show the relationship between number of pizza’s sold (n) and amount per pizza $8
This picture will resemble the pictures drawn in 5th grade when my students learned how to identify operations in one step problems
Recall of the ordered pairs in a coordinate plane (x, y) and recall of the constant of proportionality as k = y/x
(this question is the best example to use as an opportunity to celebrate students with neat, organized notes)
After giving students 6 – 7 minutes to work independently and write their solutions on the board, we will review their answers. I prefer for students to explain their own work, but also keep in mind the allotted 10 minutes for this section. If we are running out of time, I review only the answers and focus primarily on a student explaining the solution for #3 which has the most bearing in today’s lesson.
After reviewing the answers to today's Do Now problems, I will give my students our Guided Practice for this lesson. As we begin I ask all students to copy the aim for the lesson from the white board, writing it at the top of this handout. I restate it by saying that today we will be identifying the constant of proportionality in tables, graphs, and word problems.
Next, I will ask all students to read Problem #1 independently. After a 30 – 40 second wait, I will ask another student to identify the units for the constant of proportionality in this problem. I say, "What will the units be and where should they be placed within a fraction?" After successfully identifying hours for the denominator, and, the parts of the fence painted for the numerator, I will ask a student to come to the front and write a proportion that uses this initial ratio to answer the question:
What rate can be used to describe the fence painting in this problem?
I will prompt the student to describe the steps they are taking. I will ask him/her to justify the numbers and the placement of the units in the second ratio in the proportion.
For the second problem students must identify the constant of proportionality within a table. Again, this is a great opportunity to ask students to recall calculating the constant of proportionality from their notes. I plan to celebrate those students who are organized enough to turn to this information easily (see my Celebrating Organizational Experts reflection).
For the last problem I expect that my students may use as many as three different way to identify the constant of proportionality. For this final example, I ask students to work with neighbors to identify the constant of proportionality in at least two different ways. I budget about 3-4 minutes for this activity as I walk around to collect as many different strategies as possible. Each time a new strategy surfaces, I ask that a student show it on the board. The three strategies I expect to see are:
- Identifying k from the equation y=4x (y=kx; k=4)
- Locating the point (1,4) on the graph
- Using any of the other points on the line and reducing to find the unit rate (constant of proportionality)
After we complete our work with the Guided Practice problems, my students are ready to begin today's primary Class Work. As usual, a timer will be set up on the board. Today I ask my students to work independently and silently for the first 10 minutes. Any students who finish the assignment within that time frame will be able to go to Booths to check their answers with another student. Then, they can work together to display one of their solutions on chart paper. This task rewards them for their expertise, while helping then learn that experts are people who are recognized for sharing their knowledge.
The 10 problems included in today's classwork require students to refer back to the notes taken during the Guided Practice (or a prior lesson). With this lesson, I know in advance that I will be working with a small group in a designated part of the room. Some of these join me at their discretion, others are chosen by me based on my observations during the Guided Practice. Any student who did not "pull enough weight” during the Guided Practice may need to work with me without the opportunity to earn Booths.
As more and more students complete the Practice, I will begin to distribute chart paper to pairs of students. My goal is to have as many problems as possible displayed on the board (and for as many students as possible to practice writing up their responses). On the chart paper, students need to justify their answer using a definition or explanation that reflects formal mathematical knowledge communicated clearly. For example, if the constant of proportionality was identified from an equation, students will need to refer back to the basic function y = kx to defend their answer:
For example, in question #3 B = 0.4w; y = B and x = w, which means 0.4 is k; for these problems MP7 is also useful as students make use of the structure of an equation to identify the constant of proportionality.
If the constant of proportionality was identified from a table of values or word problem (not including an equation), an argument will need to be presented about the dependent and independent variables:
For example, in question #4 the price depends on the number of comics bought, thus k = dependent/independent = 6/2 = $3/comic
During the last 10 minutes of class I will be selecting students to explain their work on some of the questions that proved most difficult today. Generally, I either have a list of questions prepared before the lesson or I have made an effort to identify the questions to focus on during Class Work.
I want my students to explain their solutions to motivate Mathematical Practice 3. This strategu also frees me up to walk around the room to ensure students are on task, actively listening, asking questions, and copying down the work.
My expectations for this practice worksheet are that any problems not completed during class must be completed for homework.