# Making Sense of Quantities.

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?”

[cross posted on the betterQs blog]