Proportional Reasoning by Jumping Rope

It’s my hope that there are other 6th (or 7th) grade teachers out there that might find this analysis useful if they want to use this lesson in their classrooms to engage their students in exploring proportional reasoning (6.RP.1, 6.RP.2, 6.RP.3).  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.  (Special thanks to Graham Fletcher for providing such an array of amazing 3-Act Lessons!)

Before reading further, it may be useful to watch the lesson resources on Graham’s website.  Go ahead and check them out here.  I’ll wait.

And…welcome back!  Let’s dive in.

Some quick background:

I conducted a lesson inquiry using Rope Jumper with a team of three 6th grade teachers.  Every lesson was a unique experience with it’s own twists and turns and choices.  The lesson pathway below shows the main flow of the learning experience and took about 50 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 three classrooms of 6th grade students.  One class had not yet explored ratio language or proportional reasoning.  The other two classes had spent about a week and a half in that chapter of their curriculum.  One critical thing we noticed, the students that had been learning about double-number lines, tape diagrams, and ratio tables never used those tools and structures.  It was a wake-up call to us that students need to constantly be learning tools and strategies in problems solving contexts that are accessible to all learners.

Our objectives (as teachers):

• Students will collaborate with each other to solve a problem in a classroom that has an atmosphere that is positive and engaging.
• Students will show what skills they have using models or symbolic operations to solve a problem.
• Teacher will focus on making sure there is a healthy balance of teacher talk and student talk.
• Teacher will monitor student learning, look for anticipated students responses, select and sequence student work to share in discussion, and make connections between their models and symbolic notation.

Specifically, we wanted to see what sense making strategies they choose.  Would they draw models?  Which models?  Would they do repeated addition or subtraction?  Trial and error with multiplication?  How would they make sense of those last 2 seconds?

Learning objectives (for students):

• You will be engaged and collaborate with each other to solve a problem.

Some quick notes here…  We wanted to keep the language very simple and focus on their thinking and collaborating skills.  While we were looking for evidence of proportional reasoning, we did not want to burden the front of the lesson with too much vocabulary and we also wanted to keep the lesson open for multiple pathways.

This objective was posted clearly on the board and referenced throughout the lesson at transition points.

Into (Act 1):

We began with this simple introduction.

Today, you’re going to collaborate with each other and share your thinking in order to solve a problem.  To help jumpstart your thinking this morning, let me ask you:  How many times do you think you could jump rope in 30 seconds?

Students share in groups and then as a whole class.  Keep it brief.

Hmmmm.  Well what do you think the world record might be?

Students make some guesses.

I have a video I’d like you to watch of someone trying to break the world record.

Watch Graham’s Act 1 video.  

Based on what you see, how many jumps do you think she was able to make in 30 seconds?  Make an estimate.

We had them share their estimates at their group tables.  Then asked each group to decide on a group estimate.  Teacher recorded groups estimates on the board.

Wow!  Those are a wide range of estimates from as low as ___ to as high as ___.  If you wanted to make your estimate more accurate, what information would you like to know?  

I’ve used this phrasing a lot in lessons like these.  It works effectively to justify the need for the math they are about to do.

Once students realize they need to know how many seconds had passed and the number of jumps, they are ready for the next video.  Some students wanted to know the rate at which she was jumping or asked “How many jumps is she doing in a second?”  We unpacked those questions by asking students to think about the two variables that are going on in the rate:  number of jumps and time.

At this point, play the second video.  Be ready to pause the video near the end so that it freezes with 7 seconds showing.  It will most likely show 39, 40, or 41 jumps.  It came up each of these in the three lessons we taught and the lesson flowed smoothly despite the numbers being different.  For this write-up, I’m using 40 jumps in 7 seconds.

What does this information tell you?

We wanted to make sure students understood what the numbers meant in context.  Some students immediately saw that their estimate couldn’t be right.  In some classes, we encouraged students to discuss and unpack that reaction at their tables and talk about why some estimates (like 100) are clearly wrong.

Only after we felt that all students had a solid understanding of the problem, we moved forward.

Through (Act 2):

Use some math reasoning to see how many jumps she would do in 30 seconds.  Let’s see how close we can get!

Teacher monitored the room looking for different strategies kids would use.  For students that didn’t show much initiative out of the gate, we made sure they were clear on the problem and tried to give them a motivational boost.  We encouraged them to talk to others and listen to their approaches.

If students are still really struggling, we asked them how many jumps might she have made in 14 seconds?  And how could you use that information?  

If students succeeded quickly, we asked them to show their thinking in different ways and/or encourage them to get more accurate.

I did not do a good job catching samples of student work, but here’s the scoop on what you should see from the most concrete to the most complex.  We selected student work samples of each approach and sequenced them in this order (concrete to complex).  These works samples became the backbone of the discussion at the end of Act 2.

Most students can find out how many jumps she does it 28 seconds.  It’s how many jumps she does in the last two seconds that allows for rich discussion.

Almost all students could reason that if she made 40 jumps in 7 seconds, she can jump 80 jumps in 14 seconds, 120 jumps in 21 seconds, and 160 jumps in 28 seconds.  We asked students:  How could you organize your work so it was clearer for someone to read?  Look for tape diagrams and ratio tables.  We didn’t see students using double number lines, but if you do, share that structure too!

By sharing these models, the issue of the last 2 seconds becomes clearer to see visually.  Some students used estimations like “Well, if she did 40 in 7 seconds, she probably did about 10 in 2 seconds.”  Most students didn’t show the math skills to get more accurate than that.  However, student work around the issue was really interesting.  We shared the more abstract examples near the end of the discussion.

Some students were interested in finding the unit rate.  If she jumps 40 times in 7 seconds, her rate is about 5.7 jumps a second.  Many students rounded this to 5 or 6 and went from there.  We encouraged these students to think about if these were over- or underestimates and how they could use those numbers to get a more accurate answer.  For example, if her rate is 5 jumps per second, she’ll jump 150 times in 30 seconds.  If her rate is 6 jumps per second, 180 times in 30 seconds.  5.7 is closer to to 6 than it is to 5, therefore her jumps are closer to 180 than 150.  Students weren’t this articulate obviously, but the reasoning is there in most classes and should be discussed.

Some students knew they could use a calculator and multiply 5.7 x 30 to get 171 jumps in 30 seconds.

Some students wanted to find the unit rate by replaying the video and pausing it after 1 second.  It was an interesting approach, but they realized that it was hard to determine exactly how many jumps she made in that first second.

This lesson creates the opportunity for students to want to geek out with each other and learn from each other.  They collectively want to get a more accurate answer and they’re inspired to learn more.

Look at some of this work I’ve collected.  It seems that we are getting some similar answers.

This was our prompt to begin closure to the student work time.  We presented visual models of the relationship (up to 28 seconds) first.  This allowed some struggling students to share their work and participate in the discussion.  Then we explored the “extra 2 seconds” issue.

We then shared work about the unit rate and also asked students who used a calculator to find the unit rate and then multiplied that answer by 30 to share their reasoning.  How did they know that the calculator was telling them the answer they wanted?

To bring closure to this discussion, we asked Which of these strategies makes the most sense to you and why?  We had them pair-share before using equity sticks to call on pairs to share with the rest of the class.

Once we had maxed out student learning in this moment, we moved forward.  In some lessons, we didn’t share all the strategies in the interest of saving enough time for closure or because student focus was fading.

Beyond (Act 3):

So, the world record is 160.  Does your math suggest she might break it?  Why might the math not accurately match reality?  What assumptions are we making?

Discuss.  Most classes realized that she may get tired and that her rate at the beginning will not be constant.  This point is a great time to talk about how math can model (predict) certain real-life situations, but sometimes real-life situations are complicated to model.  

Ready to see the answer?  YES!

Play the video and pause it at 10 seconds.  How do we feel about our answer?

This allows for a formative assessment.  Are more students able to reason proportionally than at the beginning of the lesson?  Consider pausing the video again (maybe at 15 seconds) and having students discuss again before playing the video all the way through.

After playing the video, we returned to our original estimates to see who got the closest.

For closure, we had them think and journal silently for 5 minutes with the following prompt:   What is something new you learned about math or about yourself as a math thinker today?

We then asked for volunteers who wanted to share.

Reflections, Takeaways, Analysis:

• This activity engages all learners and is open enough that all learners can find success.
• It was amazing (terrifying?) that students who had been studying tape diagrams, ratio tables, and double numbers lines never used those structures on their own volition.  Some only used them once we asked them to think of ways they could organize their work.
• As a result, we realized the need to do more learning activities like these so students can practice making connections between the skills they are learning and how they can be used to solve real-world, interesting and compelling problems.
• This lesson hinges on unpacking the last 2 seconds.  Go slowly through this moment.
• We are wondering about more effective ways for students to make connections between each other’s work.

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