Those of you who are fans of the middle school curriculum Connected Math Project (CMP) will especially appreciate this string. I was preparing for a visit to MS 22, a middle school in the South Bronx —  my collaborator Erica Berger, a thoughtful and dedicated teacher, asked me to design a string to introduce linear relationships and to prepare students for a potentially messy investigation in CMP.

Some would say number strings are “curriculum neutral” or “curriculum impartial.”  That is, they are not tied to or loyal to any one curriculum — the routine of number strings can be a helpful and grounding experience for all kids. The challenge, for all of us, is locating and/or designing strings that will support the lesson or investigation that follows.

Before my visit to Erica’s class, I looked over the first investigation in CMP and used the values in the first investigation to guide me. I really wanted to link an imaginable or believable context (a car ride with dad) to a model (the ratio table) so that kids began to associate steady rate stories with the ratio table. I also wanted to avoid going straight for the unit rate (48 miles per hour), because that felt like a give-away and diminished kids’ opportunities to reason and make sense of rate. So we wrote this one and tried it with her kids.

String for Moving Straight Ahead, Investigation 1

Linear Relationships

Good morning, 7th graders.  I have a situation that I want us to think about today.  It might remind you of something that has happened to you, too.

Suppose you are on a trip outside of New York City, riding along in a car, going at a steady rate with no traffic whatsoever. Want to roll down the window and feel the wind in your hair?  You are cruising. After 45 minutes in the car, your dad says you’ve traveled 36 miles. I’m going to record the situation so far on this table.

Turn and talk to your partner about whether this table captures the situation so far or whether I should have written something different. Remember you’ve driven 36 miles in 45 minutes. Does this table capture what’s happened so far? Why or why not? Who could convince us?

[You may need to pose this question if kids are not convinced that 45 minutes is equivalent to .75 hour

So, what’s the same and what’s different about these tables?]

Sounds like we’re now convinced.  Cool, so let’s say you keep going at this rate. How far have you traveled now? How do you know?

Okay, and what about now? How far have you traveled? And how do you know?

This is a long car ride, huh?  Well, guess what?  It turns out that the whole trip took you 4.5 hours. How far have you traveled? And how do you know? I’ll give you a minute to think, and then you can talk to your partner and share your strategy. Maybe there are a few ways to think about this…..

[Expect that at least a few students will double the 144 because they will overgeneralize the pattern of doubling.  That’s great if it happens because it gets kids really defending their thinking.  When it happened with Erica’s class I just recorded the possible distances (288 miles, 216 miles), noted that we did not agree and then let kids describe how they got their answers.  It is so often the case that simply by talking through their thinking, a student will revise their ideas, “No, no, wait! I change my answer. I actually agree with Marissa now. I was doubling, but the 3.0 doesn’t double to make 4.5.”]

I forgot to mention an important detail  — you and your dad stopped for gas at the beginning of this journey, after only 15 minutes. We should probably put this on the table.  I’m going to add this line at the bottom. But first I’m wondering who can explain why I’m not writing “15” under time? And why .25 somehow represents 15 minutes? How could that be? Who feels like they could explain that?

Once the room is convinced by a student that 15 minutes is equivalent to .25 of an hour: So, how far had you traveled at that point?

We hope there are lots of different strategies here. All of these can be recorded on or near the table.

One thing we haven’t even talked about is this, our last question: How fast were you going on this trip?  In other words, what was your speed?  Take a moment to think about this and how you might explain it to all of us.

At this point there are several really nice ways to find the unit rate, and we tried to get kids thinking flexibly and creatively.  This could also lead to a discussion about speed in general, about how speed is a specific form of unit rate that is captured on a speedometer and is considered a “common measure” within the domain of ratio and proportion.