Lessons Learned from a Review Lesson on Proportional Relationships In Graphs

Sometimes, I learn more from my lessons than my students do.   Of course, just as is true form my students, I have to be paying attention.   This week, as we were reviewing proportional relationships, I noticed a few things that I didn’t expect.

Setting the Stage – The Prequel

First, this is a review.   My students learned about proportional relationships last year and this was intended as a refresher leading into work with linear relationships a little later in the  year.

This lesson was the second in a sequence of four on proportional relationships.   The previous day, we had worked with proportional relationships in tables.   It had gone about as I had expected.   After a few minutes and a little bit of discussion, students recalled the criteria as being a constant rate of change and containing the point (0,0) (even if it is only implied).   They had also recognized that one could get the y value by multiplying the x value by a constant.   I was pretty happy.

The Lesson

Students were going to build on their experience from the previous lesson using three different representations.   Each table group was given a set of two word problems.    For each word problem, they were to create a table and a graph.   Finally, they were to determine whether the relationship was proportional and explain why or why not.

Problem 1:  Evie built a robot that can put ping pong balls in a cup.    It took it three minutes to travel the course, put the ball in the cup and return to the start point.  Evie’s robot was going to compete in a 12 minute race.   Create a table and a graph showing the robot’s performance.   Determine if the relationship is proportional.

Problem 2: Sofia was really excited about all the things she was seeing in Italy.   She called Natalia to tell her all about the volcano she had visited.   It cost her four dollars for the first minute and one dollar for every minute after that.   Create a table and a graph showing the cost of the phone call.

I chose to give them two problems with a constant rate of change but with different y-intercepts because this seemed to be their sticking point in determining whether something was proportional.    They were successfully determining whether the rate of change was constant but were sometimes hitting a wall on the y-intercept.   Hence, I wanted to focus there.

As each group worked, they were to document their result for each problem on a poster.   The groups would then do a gallery walk to see each other’s work.   This is some of what they saw.


Clearly, there are some errors in thought in this work.   However, the gallery walk served it’s purpose.   Students noticed the differences in the different solutions and they started talking.

“Time is the independent variable.   It should be in the left column.”

“What is it when you connect the points? or don’t connect them?”  “Continuous or discrete.”

“Should the points be connected for the ping pong balls?   You can’t have part of a ping pong ball.”   “We were weren’t counting ping pong balls, we were counting laps and you can have part of a lap.”  “Should it be time or should it be laps?”

The conversations were not about the question that I posed.   Everyone got which graphs were proportional.   They did so without relying on the table (which I had made part of the task as a scaffold in case anyone needed it and to also have them make connections across the two representations).   They were important conversation reviewing the finer points of making graphs that I hadn’t expected to be necessary.   They were though, and using student work to drive the conversation was the best thing that could have happened.

After a class debrief, students worked with table groups to complete a proportional relationship card sort.   (You can get the card sort here.)


What I Learned

  • Most of my students easily translated the word problems into graphs.
  • Most of my students had a good grasp of proportional relationships in graphs.   I should have added more layers of difficulty in my problem set.   I focused primarily on the y-intercept as the tell-tale indicator of their mastery because I expected they would get the “constant rate of change” very readily in a graph.   Perhaps I should have added a scenario with exponential growth and perhaps I should have added a scenario in which part of the scenario is linear and part of it is not.
  • My students are rusty with regard to the finer points of making tables and graphs (where does the independent variable belong, what does it mean to be continuous or discrete, what exactly should the variables be).   This proved to be an important lesson for my students, but not for the reasons that I expected.   I’m thankful that I took the time to review a little bit rather than just rushing forward with new material, which is a somewhat natural inclination for me.   Taking a little bit of time to review important ideas was time well spent.

Coming Attractions

Lesson three in this sequence will explore proportional relationships in equations.   Lesson four will look at proportional relationships in all four representations.

Proportional Relationship Anecdotal Records

You just can’t hurry some things.   They take time.    In middle school, grasping the big ideas of proportional relationships seems to be one of those things.    Students need to build an understanding of a constant rate of change and how it presents in tables, graphs, equations, and verbal representations.   They need to build an understanding of what it means to have no “start-up” value in each of these representations as well.   They need to explore these ideas in a little of different ways in order to make sense of it.

Because there are so many pieces to this idea, there are a lot of different places that understanding can break down.   In order to know where that breakdown is happening for each student, I like to take anecdotal records.   At the end of the day, I can go back and see who knows what and make decisions about what I need to do the next day to help each of them move forward.   The thing is, I don’t have time to write a paragraph about each kid as I walk around the class looking at their work and listening to their discussion.    I need something that just takes a second or two for each kid at any given time.    To that end, I decided to create an anecdotal record form specific to proportional relationships.

I started with what I want to know my students can do.

  • Recognize a proportional relationship in a table
  • Recognize a proportional relationship in a graph
  • Recognize a proportional relationship in an equation
  • Recognize a proportional relationship in a verbal representation (word problem)
  • Be able to connect proportional relationships represented in a table and a graph.
  • Be able to connect proportional relationships represented in a table and equation.
  • Be able to connect proportional relationships represented in a table and word problem
  • Be able to connect proportional relationships represented in a graph and equation
  • Be able to connect proportional relationships represented in a graph and a word problem

If students can’t identify a proportional relationship in a given representation, I also want to know if the break down is the rate of change or the y-intercept.

I came up with this Proportional Relationships Anecdotal Records form.    I have one box for each student (I can use more than one page for a given period).   I put the student’s initials in the box and then circle the place where a breakdown is happening.   If the student is not recognizing a proportional relationship, I can write in k if the breakdown is the rate of change or the letter b if the breakdown is at the y-intercept.


I will probably use this to drive a brief review activity the following day.    I will pair students to do a card sort or to use the cards to play rummy.   Initially, I will pair them so that a student who has the idea down is working with a student who doesn’t.  After working with the idea for several days, I may place them so that kids who are struggling with the same issue are working together.     At that point, I can work with that small group to address whatever disconnect is still in place.   You can get the card sort here.

Accommodating Imperfection – Proportional Relationships Cards with Multiple Variations For Play

Successful design is not the achievement of perfection but the minimization and accommodation of imperfection. – Henry Petroski

I know that I will not ever design the perfect lesson any more than I will ever create the perfect design as an engineer.  I’m not sure that perfect exists in this world.   While Dr Petroski (a civil engineering and history professor at Duke University) focused his work on failure analysis in an engineering context, the underlying principle he espouses applies to my work as an educator as well.    On any given day, I know that some students will walk away  not having fully mastered the concepts addressed in class.   So, I plan ways to revisit concepts in small chunks of time until everyone does “get it.”  When I plan ways to revisit concepts, I try to create activities that I can use a lot of different ways because I want to be able to use them more than once rather than having to create an endless array of materials.

Proportional Relationships – What I Want Them To Know

I want students to know that proportional relationships are linear and go through the origin.

Proportional Relationships – What I Want Them To Be Able To Do

I want to ensure that students see proportional relationships in tables, graphs, equations, and word problems.   In each representation, I want them to see the constant rate of change and that there is no “start-up” value (the y-intercept is 0).     I want to incorporate Trail’s work (Twice-Exceptional Gifted Children) to support conceptual learners, so I envision a whole-part-whole instructional sequence:    what is a proportional relationship; how do you see it in a table, how do you see it in a graph, how do you see it in an equation, how do you see it in a word problem;   how is the constant rate of change shown in each representation, how is the “no-start up” shown in each representation (compare and contrast these in the different representations).

Proportional Relationship Cards

I created a set of 48 cards.   There are twelve cards for each of the four representations.   The cards can be used separately or together.   That is, I can use just the cards relating to a single representation if I want to focus on that representation.   Alternatively, I can use cards from all four representations if I want students to make connections across representations.   You can download the cards and game instructions by clicking on the link below the photos.

Proportional Relationship Card Sort and Game

Activity 1 – Quiz-Quiz-Trade

Quiz-Quiz-Trade is a Kagan Cooperative Learning Structure.   In this structure, students partner and quiz each other.   Then, they find a new partner and repeat the process.   Marzano’s research (Classroom Instruction That Works) shows that using cooperative learning structures produces gains of 27%    It also shows that incorporating movement increases levels of engagement (The Highly Engaged Classroom).


Activity Two – Give One, Get One

Like Quiz-Quiz-Trade, students work with a partner and quiz each other.   I will use this activity when I want students to make connections across representations.   It incorporates movement and a cooperative learning structure.   It is outlined in Marzano’s The Highly Engaged Classroom.   I have students form two lines facing each other with about three feet in between the lines (the structure does not specify this, but I find it works well this way).   Each student is given a card.   I will give one line a single representation and the other line a different representation.   The cards for partners will be different representations of the same problem.   Each partner will have to find the rate of change and the y-intercept using their card.   When a partner has found them, he or she steps forward.   When both partners are in the middle, they quiz each other on what the rate of change is/how it is shown on their card and on the y-intercept/how it is shown in their card.  When they are done, they step back into the line.   When all the pairs are done,  have one line pass their card to the next person and the other line shift (line one passes the card down one, line two shifts up one).

This is an activity that I do for 5 minutes at the end or start of class.   I end it based on time.   I don’t try to have every student do every problem.

Activity Three – Rummy

Students play in groups of 2 to 4 players.   They use the entire set of 48 cards to match the table, graph, equation, and word problem.


This is an activity I will use so that students compare and contrast the different representations.   Marzano’s research (Classroom Instruction That Works) shows that finding similarities and differences can produce gains of 45%.   His research (The Highly Engaged Classroom) also shows that using a game increases levels of engagement.

I may have students play the game in heterogenous groups as a general review activity.

I may use this as a differentiated instruction activity.  I will have students play the game in groups according to their level of mastery.   Students who have not attained mastery play the game.   Students who have attained mastery play a different game reflective of their own skill gaps.

I will have students play the game for 5-10 minutes at the end of class.   If the game is not over, the player with the most sets wins the game (and a piece of candy)

Activity Four – Card Sort with Three Variations

Students work individually, in pairs, or in table groups to sort the cards.   In the first variation, they work with a single representation and sort them into proportional/not proportional categories.   In the second variation, they work with a mixture of representations to sort them into proportional/not proportional categories.   In the third variation, they work with a mixed set of representations and find the matching cards (same situation represented in a table, graph, equation, word problem).

I may have students work in table groups as a general review activity.

I may have students work with intentional pairing.   In this scenario, I pair a student who is struggling with the concept with a student who has mastered the concept.    As they sort the cards, the discussion is scaffolded for the student who has not yet attained mastery.

I may have students work individually and use this as a formative assessment.





Proportional Relationships Auction Activity

I read Sarah Carter’s post this morning on her revised Function Auction Activity here.   As I read it, I thought it might be nice to do something similar for proportional relationships. Here is my version of a proportional relationship activity based on Sarah’s work.

Proportional Relationship Auction

You can download the file by clicking on the text below the photo.

I haven’t tried the activity yet, but am looking forward to using it as a review of proportional relationships leading into work with linear equations in the fall.


Necessity Is The Mother Of Invention – #ILookLikeAnEngineer

Necessity is the mother of invention.   Unfortunately, the “necessity” can be all too easily forgotten as an essential component in education.    I teach what I teach, in part, out of necessity but it is my necessity not that of my students.   I need to teach the curriculum that I teach because it aligns with the standards set forth by the state but it is not a burning necessity for my students no matter how many times I tell them the essential questions and how they will use it in the future.   Knowing something only becomes a burning necessity in the mind of an eleven year old when they see a need to know it so they can do something they want right now.

So how do we create that need to know?   I think we give kids real problems that they really want to solve.   It’s not something that I can do every day, but I try really hard to find time and space to do it every year.   To do this, I  compact lessons and I accelerate where I can.   This year, I managed to squeeze out almost a month at the end of the year to do an engineering project with my students.

Request For Proposal

Students were presented with a Request For Proposal (RFP) from a fake toy company.    The proposal indicated that this fake toy company was seeking to expand market share to include more girls in their customer base for motorized toys.    The toy company wanted those bidding on the contract to conduct market research and build a toy to meet that need.    The toy company indicated that the toy must meet one of three different criteria:  travel 3 m in 3 s, climb 1 m at a 15 degree slope in 2 s, or climb 1 m at a 30 degree slope.

Creating a Team and Conducting Market Research

Students were assigned teams and formed mini-companies that would bid on the RFP.   They created a team name, logo, and slogan.   Then, they conducted customer surveys with both adults and children in the target age range.    They analyzed the data and determined the type of toy the customer was seeking.

Building Technical Knowledge

ChJmkKSUkAEA9Q-During the same time-frame, students built knowledge of how gear trains work.   They began by building gears on a frame and exploring relationships between the rotations of the gears and the number of teeth on the gears (gear ratios, teeth ratios).   Next, they added a motor and wheels so that they could calculate the rate on a 3 m course and measure the rim force on the wheel.   They repeated this process with gear ratios ranging from 1:3 up to 225:1.   As they did this, they were building important skill in construction as well as an understanding of the different kinds of performance they might expect from different kinds of gear ratios.    From there, they measured rim force on the tooth of a gear connected to the motor.    They did so for different sized gears and then learned how to calculate torque.    With this knowledge, they could explain why certain gear ratios would not move and why certain gear ratios would be well-suited to climbing.   At this point, they had built sufficient knowledge to answer the first stages of that burning question of how to build a toy that would meet each of the criteria.

Making a Prototype

Each team began construction of a basic prototype to meet their desired criteria.   This amounted to attaching the motor and the desired gear train along with the wheels on the frame structure.   Students then tested their motorized frame to see if it met the criteria.   Once they had a basic working prototype, they started constructing a body to give the toy the desired aesthetics.    As they constructed the body, they continued to test the toy to make sure the additional weight did not place them out of compliance with the criteria in the RFP.    They repeated tests multiple times and used median values in order to eliminate outlier trials resulting from poor testing technique.

Sealing the Deal – Writing a Written Proposal and Giving an Oral Presentation


When the toy was completed, each team wrote a written report in response to the RFP and prepared an oral presentation.   The final stage of the project required each team to present their toy to a panel of judges representing the fake toy company.   I recruited 3 engineers and a soon-to-be lawyer to represent both the technical and business interests of the company for the panel of judges.   (I am lucky enough to have Sandia National Laboratories nearby and willing to provide this kind of support to encourage excellence in math and science.)   The judges selected a winning team based on the presentation and a demonstration of the toy.   (The winning team members each got a gift card to Cold Stone Creamery).

While this last stage is not “math”, it is very much a part of what engineers do and I wanted my students to appreciate the importance of being able to communicate effectively as an engineer.  Reading, writing, and speaking are just as much essential skills for an engineer as are math and science mastery

Why It Mattered

  • Students got to experience the engineering process, which is so much more powerful than hearing about it.
  • Girls had to learn how to make something and how to make it work.   It’s not that they are any less adept, but many of them are much less experienced.   This results in a certain amount of hesitancy, initially,   Having to make it work pushes them past this hesitancy and they discover just how good they are at it.    Giving girls this experience and confidence is important in leveling the playing field when it comes to engineering.
  • Students used the math that they have learned this year to do something real that mattered to them (finding unit rates, conducting surveys, making data representations, analyzing data to make decisions, finding medians, using equations to calculate torque, measuring radii).
  • Students had to find ways to work together – teams could not shift part way through the month long project.
  • Students who lacked confidence as speakers learned that public speaking is a learned skill and that you get better at it with practice.   (I made each team do a dry run of their presentation in front of their classmates and get feedback the day before the final presentations.  They took the feedback and were so much better the second day.)

Gallery of Toys


Which Bicycle Shop Should I Use –A Lesson Introducing Proportional Relationships in Tables, Graphs, and Equations

“I’m thinking about purchasing a custom-made bicycle. Bicycle City charges $160 plus $80 per day that it takes to make the bicycle. Bike Town charges $120 per day to make the bicycle. For what number of days will the charge be the same at both bicycle shops?”

I introduced the lesson with this question (taken from the NY Institute for Learning). We began with our usual process of making sense of the problem (What do you notice? What do you wonder? What is the question? What do you know?). Then, students set to work on the task individually.


Once everyone had solved the problem, I had three different students present their solutions (one who had used a table, one who had used a graph, and one who had used equations) to the class using the document camera. This gave us the chance to talk about how you find the solution using different solution paths.


At that point, I told students that a proportional relationship was one in which you could multiply the x value by a constant and get the y value. I asked them to figure out whether either bike shop’s pricing was proportional and how they saw that in their particular solution. This gave us the opportunity to discuss how they see the constant rate of change in each representation. It also gave students the chance to discover that a proportional relationship goes through the origin for themselves.

We wrapped up the lesson by completing a foldable summarizing the characteristics of a proportional relationship and how to see them in a table, graph, and equation. You can download the foldable by clicking on the link.

Proportional Relationships

The lesson is designed to give a whole-part-whole understanding of the concept. This is particularly important with students who are at the extremes on the conceptual/sequential dimension of cognitive style. I have several twice-exceptional students who are at one end of this spectrum or the other. I struggled with how to help them succeed for the entire first semester. I started building in the use of a whole-part-whole structure and the use of graphic organizers for note-taking more after reading Twice-Exceptional Gifted Children by Beverly Trail, EdD. It has had a pretty big impact for these students (test scores moved from the 50-60% range to the 98% range on the last test). I don’t know if that will continue to hold, but with that kind of improvement, I am going to keep incorporating this technique and find out.

The way that the foldable is designed, students can compare and contrast how one sees a proportional relationship in a graph, table, and equation by looking across the foldable.   The can also compare and contrast an example of relationship that is proportional and a relationship that is not proportional by looking down a column.   Marzano’s research shows that the use of comparing and contrasting can result in significant gains in student learning.

I also incorporated the use of color in the completion of the foldable to highlight the important points. This was not as intentional as it should have been. Early in the day, we worked through the foldable summarizing the ideas in the lesson. As the day progressed, I thought it would be a good idea to have kids highlight the key points in each representation (constant rate of change and going through the origin). By fourth period, I finally refined it so that kids used one color to highlight the rate of change and a different color to highlight that the relation goes through the origin. This seemed to really bring out the idea across representations, so I will do it this way from the start the next time I teach this lesson.