It was a pretty laid back day in my 6th grade math classes today. We spent some time developing an understanding of reciprocal unit rates (i.e. miles per hour and hours per mile) and worked with various strategies like ratio tables and double number lines. The format was pretty much guided examples, so it was no surprise to me when some students began to get restless during the second half of class. I decided at this point to break out a matching problem I thought up on the drive home last night.

On the board, I displayed three unit rates:

- 12 miles per hour
- 12 miles per minute
- 12 minutes per mile

I also displayed 3 situations:

- A person jogging
- A fighter jet flying
- A car driving out of a parking lot

After reading the rates carefully (just to make sure students did not confuse them), I asked what I thought would be a simple question:

**Which unit rate matches a situation best?**

I let the question hang in the silence for a moment, since I thought it would be obvious. After that ever important pregnant pause where every teacher is faced with the decision to scaffold or let students explore, I directed students to discuss matches with a partner. It was only 3 minutes, but it was a fantastic sharing time. As I circulated around, I heard students making arguments with their partners. The best part was the reasons students created. Some reasons were specific, while other students made reasons that are purely 6th grade.

“The car can’t be moving at 12 miles per minute. That’s too fast.”

“The plane has to moving faster than 12 miles per hour. That what’s it’s got to be.”

“The jogger is running at 12 miles per hour. I can run that fast.”

“The car is going 12 minutes per mile. What else could it be?”

When I called upon students to share their thinking with the class, the fighter jet paired with 12 miles per minute was the giveaway for most students. As one student in my second period said, “12 miles is a big distance and 1 minute isn’t really long, so the plane is the only thing that’s fast enough to have that rate.” It was a great moment in class. For some other students, it was like a switch had been flipped. Unit rates meant something for the first time. It wasn’t just an answer to a question anymore.

The discussion that followed was more debated. Some students argued that 12 miles per hour could describe a jogger, but other students dismissed the idea by saying that jogging is slower than 12 miles per hour. I asked if anyone knew the distance of a half marathon. One student shared about 13 miles, which allowed me to ask, “Will a jogger run almost a half marathon in one hour?” A couple students were silly and insisted it was possible, but you could see a new found confidence among students who were insisting, “NO WAY!”

It’s interesting how questions you think are simple can have more weight than you expect. Today also reminded me that I need to slow down to have students consider the meaning of units in a solution. How many times do we see students write erroneous units for answers, but fail to ask them, “What do the units of this answer tell us about the situation?”

This reminds me of a task from the Shell Centre in which a graph is presented, and students are asked something to the effect of: To what could this graph correspond?

I think there is a list of choices provided (parachuting, hitting a golf ball, etc) and I’m not even sure if the axes are labeled. (In my opinion, they needn’t be!)

Several similar tasks exist; the common-thread is the need for justification of whatever choice(s) students make — often this leads to lively discussion, as well!

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I love Shell Centre tasks! You got a great point about justification and I know my experience isn’t unique by any stretch of the imagination. It’s one thing for a student to get a solution, but it’s a totally different level of thinking that required to interpret information and solutions. I wonder if discussion with partners also lowers the perceived risk of these questions, too? Is the typical student more each to explain how they know to another student during a partner activity than answering in a whole group atmosphere?

Thanks for the comment!

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(For your last question: I am reading the word ‘each’ as ‘apt’; the inability to edit comments is slightly frustrating, I’m finding!)

It seems to me that small group formats (including partnered discussions and turns-and-talks) enable students to engage with material more actively and with a lower anxiety level. But I also want to return to something from your original post:

When I called upon students to share their thinking with the class, the fighter jet paired with 12 miles per minute was the giveaway for most students. As one student in my second period said, “12 miles is a big distance and 1 minute isn’t really long, so the plane is the only thing that’s fast enough to have that rate.” It was a great moment in class. For some other students, it was like a switch had been flipped. Unit rates meant something for the first time. It wasn’t just an answer to a question anymore.I think the real key is not only thinking about each of small group and whole class formats, but also striking the proper balance between them; as you said, the former was “fantastic” but “only 3 minutes.” And it’s in the whole class format that you have the insight described above.

In an ideal setting, the positive ramifications that you describe Carry Over from the small group to the whole class: students have a chance to think through material without feeling threatened, to voice their ideas, and to hear back from a peer — one who is attentive (but not silent) and constructive (if critical). Back in the whole class format, the students may now feel as if they have Something To Say; the hope is that this builds in a lower perception of risk and a greater willingness to voice thoughts to the entire class.

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