Proportional relationships are ingrained in our everyday, they just make sense.  I think that is why I really enjoy proportions and seeing a students’ ability to find something that is “missing”.  Many students do well with proportions, there is a simple formula to teach, the steps remain the same, solve, label, etc.  A piece of cake. Other students look at a proportion and all they see two dreaded fractions. Que dark scary music.

Today, I am sharing how proportional relationships go far beyond just solving for a missing number and set a foundation for success in Algebra.

7.RP.2 Recognize and represent proportional relationships between quantities.

• 7.RP.2A  Decide whether two quantities are in a proportional relationship, e.g., 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.
• 7.RP.2B  Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
• 7.RP.2C  Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.
• 7.RP.2D  Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.

Visual Representations

These standards are heavily weighted with visual representations including tables, graphs on the coordinate plane, and diagrams.  Some students will have been exposed to these multiple representations in 6th grade with standard 6.RP.3 and equivalent ratios.  The goal is for students to fluently connect the table, equation, graph, verbal description, and even diagram together.   When given one piece of information, students should be able to represent that same information in multiple ways.  Students who can do this will have a firm foundation for 8th grade math and Algebra 1.

Ideas for teaching Proportional Relationships

There are tons of great ideas and activities out there, but below are a few of my favorite “tried and true”.  They are all easy to incorporate and provide scaffolding for students who struggle in the math classroom.

Multiple Representations Graphic Organizer

Some students need to see everything together and that is where this multiple representations graphic organizer comes in handy.  It is the perfect size for students to show their work and simultaneously is easy for students to draw on their own.  I really like how the components stay the same, but the given information changes.

These are perfect for introducing the various visual representations, but also can be laminated to use in tutoring or math intervention.  Once students are familiar with the graphic organizer, I would even give students butcher paper and various pieces of information.  Then, they would use markers to represent the remaining information.  A quick informal assessment idea or even partner work for a Friday.

Highlight the Unit Rate/Constant of Proportionality

A quick trick to help students see all the connections is to use a highlighter!  Model and have students highlight the unit rate/constant of proportionality within the different representations.  This is perfect for representing k in the equation and table, or meeting 7.RP.2d by labeling  (1, r) on the graph.

Emphasize Vocabulary

Hopefully, students are familiar with the term unit rate and how to find it.  Constant of proportionality sounds so cumbersome and challenging.  Unit rate is often emphasized when we are talking about unit price (cost per ounce, etc), so I think students have trouble seeing a graph or table and also using the phrase unit rate.  Either way, students need to be familiar with the verbiage and understand what a question is asking.  Here is a more detailed explanation for why they are the same.

Common Misconceptions

1. A relationship is not proportional unless (0,0) is visible in the table or graph
2. Dividing x/y to find k
3. General confusion about k
4. General disconnect between k and proportions
5. Proportional relationships only involve positive numbers
6. Mixing up x and y on the table, when it is not given

Anchor Chart Ideas

Anchor charts are fabulous ways to showcase the content in a visual manner for students to reference.  They can easily be created before the lesson or as you are teaching, depending on the content.  The visual representations of proportional relationships are perfect for anchor charts.

Ideas for Struggling Students

1. Practice finding different points in the relationships that aren’t listed on a table.
2. Begin with the equation and use and input-output table to create a table.
3. Use a four corners graphic organizer.
4. Match multiple representations.
5. Practice graphing with four quadrants (from 6th grade).

Hopefully, this gives you some ideas for teaching proportional relationships or even insight as to what knowledge your students are coming with.  I would love to hear other great activities or ideas you have used! Feel free to share in the comments.

Be sure to check out these different concepts and activities that are included in my Proportional Relationships Unit and Proportional Relationships Activity Bundle.