Representing Proportional Relationships in Different Ways (2 Day Lesson)

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Students will be able to represent a proportional relationship in multiple ways.

Big Idea

Presenting Proportions! Representing the same proportional relationship in different ways.


7 minutes

To start off this lesson, I will engage students in an opening exercise. In this exercise, a word problem will be presented to the class. As a group of 4 or 5 students, the students will solve this problem in a specific manner assigned by me. There will be six groups in the classroom set up according to ability to ensure that everyone participates. In other words, no one student should be relying on the ability of another, this should be a collaborative effort. The manner in which the groups will solve this problem is as follows:

  • Group 1: By creating a graph
  • Group 2: By creating a picture
  • Group 3: By creating a proportion using a flow map
  • Group 4: By creating a table
  • Group 5: By creating a double number line
  • Group 6: By creating an equation and solving


The groups will be given 4 minutes to solve the problem and present it on chart paper. Once a group is finished with their problem, they will post their solution on a designated wall in the classroom. Students will take 3 minutes to do a gallery walk to see the different ways the same problem was solved in different ways. During this gallery walk, students should be making notations of anything that they may have a question about, anything they notice, or any discussion points that they would like to bring up during the instructional piece. (MP1, MP2, MP3, MP4, and MP5)

Instruction & Teacher Modeling

15 minutes

During the instructional part of this lesson, I am looking for students to be able to demonstrate their understanding that there are many ways that a ratio relationship can be represented. During this time, the students should also realize the relationship between the different representations of the ratio presented. In addition, students should not only be able to understand the representations but also apply those representations to a real world scenario demonstrating fluency in using the multiple representations of ratio relationships. (MP1, MP2, and MP4)

For this reason, during our discussion, I am looking for students to articulate things such as, which representation makes the most sense to them, which representation does NOT make sense to them, as well as the correlations that they see between all or some of the representations. (MP1, MP3, and MP6)

To begin the instructional part of this lesson, I will have each of the groups present their solution to the problem presented in the opening exercise. Recall that each group had to solve their problem in a different manner from the other groups. As each group is presenting, I will facilitate discussion pertaining to how the solution is representative of the problem that was solved, as well as the concept of ratio and ratio relationships. It is at this time that the students who made notations about each type of solution will be able to voice what they have noted so that it can be addressed at this time. In reference to their notations, I am hoping to see where students may have caught mistakes, noticed relationships between the solutions, or have specific questions about how the solution method in question works. Once all groups have presented, I will then encourage students to discuss, as a class, the relationships they see between each solution method.

During this instructional piece, I will ensure to do the following:

  • Facilitate deeper understanding through probative questioning
  • Answer any lingering questions once the discussion has concluded
  • Address any misconceptions uncovered during this discussion. This will be done using peer interaction, probative questioning, prompting, or any other technique necessary to pull understanding from the student.


10 minutes

During the Try-It-Out portion of this lesson, we will focus on applying the concept of ratio to a real world satiation. To do this, I will present the students with a pattern of circles and triangles. Based upon what they know, the students will need to complete the following tasks:

  • Describe the relationship between the two elements of the pattern. In other words, describe the ratio relationship represented by the pattern.
  • Extend your pattern in a sequence of 10 terms
  • Choose one of the terms in your sequence.
  • Write a word problem using the ratio relationship you identified and the term in the sequence you chose.


I will guide the students through these tasks one by one. Upon completion, I will ask three students to share their words problems. While they are sharing, I will ask them questions to reveal their thinking as they completed the task.


The mathematical practice standards that are evident in this section of this lesson are as follows: MP1, MP2, MP4, MP7

Independent Practice

20 minutes

The students will explore the different ways to represent ratio relationships by creating a poster that displays all the different ways that they have learned to solve a problem that involves a ratio relationship. To do this, I will pass out pattern blocks to each of my students. The students will choose two colors out of the pattern blocks. Then, they will create a pattern using the blocks. Using their own pattern, they will create an eight pane poster that will contain the following elements.





Double Number Line

Table & Equation



Word Problem


My students will need to ensure that they have a proper representation of each of the elements indicated in the diagram above. The diagram will be graded based upon four elements using a rating. Those elements are as follows:

  • Accuracy
  • Visually appealing
  • Thoroughness of each representation
  • Ability to articulate their understanding of what their poster shows


This activity requires students to use the following mathematical practice standards: MP1, MP2, MP3, MP4, MP5, MP6, MP7, and MP8


20 minutes

To close out this lesson, selected students will share their patterns, word problems, their solution, and one of their representations of their ratio relationship. While they are presenting, I will ask strategic questions to ensure that the students presenting as well as those that are being presented to, understand the concept of today’s lesson. Some of the questions that I plan to ask are as follows:

  • What similarities did you see between the different representations?
  • Could you have started with any of the representations and still be able to derive all of the other representations?
  • Which representation do you like best? Why?
  • Which representation will you probably use more? Why?


Mathematical Practice Standards evident in this section: (MP3) 


TOTD: The students will place a sticky note on the parking lot containing any lingering questions they may have about the concept of ratio relationships. I will post the answers to their questions on my teacher website, so that all students can read the answers at their leisure. 

Homework: As an extension to today's independent exploration assignment, for homework, I will have my students continue with their posters by writing a brief essay. In this essay, the students will describe what each part of each of the representations of their solution means. In doing so, they must also demonstrate their understanding that there is a relationship present between each of the solution representations.