Two Engaging Proportional Reasoning Tasks

I hope that there are other 6th (and 7th) grade teachers out there that might find this analysis useful if they are looking for ways to increase student engagement, thinking, and discourse around percents, fractions, and proportional reasoning standards.  This engaging learning opportunity can be used at the beginning of a unit as an inquiry-based exploration and pre-assessment.  It can also be used as a way of assessing student learning in the middle or the end of a unit.  It’s a low-floor opportunity that allows for students at all levels to participate.  It also allows for rich discussion and sense-making because solutions can be reached via multiple strategies.

Please feel free to plagiarize and make it your own as you see fit.  I’d love to hear from you if you use it so we can get better together.   You can find our images and handouts here.  (Special thanks to Graham Fletcher for providing such an amazing 3-Act Lesson and also to Andrew Stadel for his sensational Estimation180 activities.)

Before reading further, it may be useful to watch the Act 1 video on Graham’s website and also Andrew’s Estimation180 image.  Go ahead and check them out here and here.  I’ll wait.

And…welcome back!  Let’s dive in.

Some quick background:

I conducted a lesson inquiry using these visuals with a team of 6th grade teachers.  Both lessons were unique experiences with their own twists and turns and choices.  The lesson pathway below shows the main flow of the learning experience and took about 60 minutes.  Further analysis and takeaways can be found at the end of this post, but I’ve tried to embed a lot of our thinking within the flow of the lesson.

This lesson was taught to 2 classrooms of 6th grade students.  Students had completed the units on ratios and proportional reasoning from their GoMath curriculum.  We used this lesson as a way to assess student conceptual understanding of proportional reasoning and their problems solving skills.  We were curious to see how much knowledge students had retained.

Our objectives (as teachers):

  • We wanted all students hooked, engaged, motivated, positive, energetic….and to persevere in solving proportional reasoning problems.
  • Specifically, we were curious about:  How do students respond to the visual hooks?  Does this approach increase engagement and learning for students at all levels?
  • Bigger questions we were wondering about:  How do my students talk about math?  As an instructor, how do I navigate openness in a lesson in a way that is structured enough to promote student learning for all?

Learning objectives (for students):

  • You will use strategies to solve some math problems.
  • You will communicate your reasoning to each other.


We displayed the image below.

How much pie did I eat last night for dessert?  Go ahead.  Make an estimate.  Now share your answer with your elbow partner.  If they’re the same, explain why you agree.  If they’re different, try to convince your partner why you’re right.

Andrew’s image allows for a low-floor entry point in to the lesson.  We wanted to activate and engage student thinking about part-to-whole fractions and percents.  We handed out strips of paper that had the image on it so that students could draw on the paper.  We also had the image loaded up on Google classroom so students could access it on their iPads.

In the image below, you can see the strips of paper with the image.  A few students (like this one) divided the pie into fourths.  Students were able to identify this error quickly because they could visually see that the “point” of the missing piece wasn’t a square corner.

Many students (as we had hoped) drew directly on the paper image or on their iPads.  Here is an example of what we often saw:

This student sought to attend to more precision by making measurements on the image.

Most students arrived at “1/5” as an answer in about 5 minutes.  We anticipated that “1/5” would be a more common answer than “20%,” but we wanted to have both fractions and percents to be a part of the lesson vernacular.  Fortunately, Andrew’s answer forces the issue to talk about percents.

We told students to ignore the 72˚ if they didn’t know what that meant.  Instead we focused on the 20%.

You all seem to agree that I ate one-fifth of the pie because we can divide the pie evenly into five pieces and it’s clear that I ate one of them.  Therefore, I ate one out of five pieces or one-fifth of the total pie.

But the answer says 20%.  What does this mean about our answer?  Are we correct?  

We let students discuss and unpack this moment briefly.  Both classes had several students that recognized that one-fifth was equivalent to 20/100 and therefore 20%.  We showed this strategy on the board showing 1/5 x 20/20 = 20/100 or 20%.

This was the key point of the warm-up and why Andrew’s image is perfect:  We were able to introduce the grade-level content (percents) by using a very low-floor strategy (area model).  This approach allowed for students of all levels to organically make meaning of the problem that was going to come next.  Furthermore, both strategies (area model and proportional reasoning/percents) were reviewed in the context of a problem.  Students of all levels can get on board and experience some positive momentum.

(An interesting enrichment question for early finishers of this problem: “If I ate a piece that size every night, how long would it last?”)

Through (Part 1):

We then used a modified version of Graham’s Kool-Aid Kid task as a performance task for students to practice their proportional reasoning skills and to reinforce the relationship between fractions and percents.

Great job.  Take a look at this short video for our next problem.

We played the Act 1 video.  We paused the video on this image.  Students also had access to the image on paper and on their iPads.

What do you notice?

“He drank most of it!”  “He drank more than half!”  We scribed useful sentences on the board.

What percentage of the Kool-Aid did I drink?  Make an estimate.

Notice that we were very explicit about saying percentage.  We wanted to keep the focus on grade-level thinking while also providing a tool (an image they could draw on) that allowed them to use a prior grade-level strategy (area model).

Many partitioned right on the paper or on their iPad image.  Students seem to think it was either 66.6% or 75%.  There were some rich math fights going on as students tried to convince each other.  We had anticipated this debate when we were planning the lesson and were excited to see that the debate seemed to fuel interest.

Here samples of some student work.  You’ll see that students were going back and forth about thirds and fourths.  (I’m a math poet!)

This student (below) made an interesting (and useful) error in converting two-thirds in to a percent.

He knew he had made a mistake because 15% didn’t make sense because he knew that 2/3 was more than 50%.  He wasn’t the only one to make this mistake so we were able to unpack the misunderstanding.  We copied some of his work on the board.

I’ve seen several students do this.  Please look at the work on the board.  What mistake is being made here?  And what could we say to someone making this mistake to help their thinking?

Whenever possible, let student work drive the discourse, especially when it comes to clarifying misconceptions.

Through (Part 2):

Here’s my favorite pivot question!  I find it really useful in having students think contextually about numerical information.

If you wanted to make your estimate more accurate, what information might be useful to you?

We let students discuss in groups.  Most groups in every class realized that they needed to know how much Kool-Aid was in there at the beginning and at the end.

We modified Graham’s Act 2 images and displayed these instead:

From here, students set out to do some work.  We were curious to see how students made sense of the problem and progressed to the solution.  Would they find the difference of 500 and 125 first and then divide 375 by 500?  Would they divide 125 by 500 first and then do subtraction (from 1 or 100%) later in the process?  Would they simplify fractions?  Would they set up a proportion?

Here’s a snapshot of some of the student work we saw as we monitored the room.

The work sample above shows how this problem is accessible to students with conceptual understanding of fractions.  She’s able to create a tape diagram (that looks like the mug) to help here see that she needs four “125s” to make 500.  Therefore, I drank three-fourths and left one-fourth in the glass.  After sharing her work with us and her neighbors, we encouraged her (and others with a similar approach) to prove/show that three-fourths was 75%.

This student offered us an opportunity to explore an error that connected to an error from the warm-up (finding the value whole to part).  She got stuck because 4 or 4% didn’t make sense.  She couldn’t contextualize the meaning of the numbers well enough to see her error.

We copied her work on the front board.

What did these students do?  What did they find out?

The student’s bar model visual from above provided a useful instructional tool to help this student see that 4 was a useful number in the solution.

We saw some students simplifying fractions to find the percent.

I scribed this student’s work (from above) so it was clearer to follow (see below).

This student (below) shows how he used multiplication to change the simplified fraction into a percent.

Not many, but a few students set up a proportion and used some reasoning skills to find the missing variable.

 This student said:  “I know that 500 is 5 times a 100.  So I know I need to find out what times 5 is 125.  And that number is 25.  Therefore the answer is 25%.”

Great reasoning.  Right answer to a different question.  A lot of students forgot to think about their answer in context, choosing 25% instead of 75% as the correct answer.

When productive struggle had been maximized (after about 2o minutes), we orchestrated a math discussion around their strategies and tried to help them make sense of and make connections between the math strategies.  The discussion would usually go well for about 5-10 minutes.  Some questions we asked:

What did this student do?  Who else did a similar strategy?  Who can explain it in a different way?  Of all the strategies we discussed, which strategy seemed the most efficient for you?  Which one seemed the most concrete?  


We then showed the answer.  Again, we modified Graham’s Act 3 video/image to reflect choices we had made earlier in the lesson.

We then asked students to reflect on the lesson experience.  Did they enjoy math more today than they usually do?  Why?  Or why not?

There were several responses like this one from students:

(Note:  I don’t think the goal is to make math easier, just more accessible.)

We also modified this Illustrative Mathematics task as an exit ticket. 

Reflections, Takeaways, Analysis:

At the end of the day, we reflected on our objectives and student learning.  Some things we noticed and wondered and a few professional goals moving forward.

  • It’s important to let kids explore.  This tasks was appropriately challenging; it hit the sweet spot between accessibility and challenge for almost all students.
  • Showing the visuals increased student interest and engagement compared to working out of the textbook.
  • It’s important to let students formulate the problem for themselves.  We can do this by creating a headache for them that requires mathematical skills/reasoning to solve.
  • The numberless introduction (Andrew’s pie image) was low-floor so all students were on board right from the start.
  • Furthermore, the introduction allowed us to review skills in context using student work to facilitate discussion.  This flow was more engaging than reviewing skills out of context using explicit direct instruction.
  • It’s important to do tasks like these.  The textbook leads student thinking too much; it limits their choice and voice in the learning process.  These tasks allow for an exchange of ideas.

An invitation:

As always, feedback and comments are welcome.  What inspired you?  What opportunities did we miss?

Help us get better together.

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