# Hours of Sleep

At times, I fantasize about sleep. Oh how the thought of stretching out, falling asleep and staying that way for 8, 9, or 10 hours is so enticing. Instead, I can work the idea of sleep into a problem and hopefully encourage students to get to bed at a reasonable time.

My opening problem for my sixth graders this week was:

On average, I sleep about 8 hours 15 minutes each day. So, during September, I will likely sleep forÂ 8.15 x 31= 252.65 or about 250 hours.

I told students that my wife (their humanities teacher) disagreed with my calculations and hoped that they could settle our debate by agreeing with me or letting me know where I erred. (I came across a similar problem while looking over the Mathematics Assessment Project site which appears to have some good resources.)

To get started with class this day, I had students working individually. My plan was to have them work a bit independently and then collaborate their ideas once each student got into the problem. The 31 days was picked up on pretty quickly and for one student that was the only identified problem. A change to 30 days and then he was finished. The rest became involved in the problem and didn’t appear to want the collaboration time so I changed gears. As students wrapped up the problem, I asked them to make their thinking explicit as they each transferred to a whiteboard. This change moved the problem from a quick exercise to a focal point as a rich discussion ensued.

Writing 15 minutes as 0.15 was the key error for students to identify as we discuss ratios and the idea of “the whole”. Different pathways included:

I liked how this student laid out her thinking.

• putting 8 hours into minutes for the month, adding 15 and then converting back to hours and minutes
• moving from 8.15 to 8.25 hours as it was realized that 15 minutes is in fact a quarter of an hour

A round table whiteboard discussion was held to wrap up the problem. Each student brought his/her whiteboard to the circle and held it up for all others to see. A few solutions were essentially identical so we grouped those students. The task posed was for us to find where differences in problems remained. Students began to critically look at the work of others. The person whose work was on the whiteboard was in charge of the discussion and could accept suggestions and hints from others. Through this analysis, we found a student who had realized the errors but then worked the problem with 31 days while fixing the error of 0.15 minutes. Another student had good reasoning and it took the others some time to realize that a division error (decimal in the wrong place) was the culprit. Solutions that were similar in process but not in appearance were also flushed out. In the end, students were provided the opportunity to explain their thoughts, constructively critique others and revise solutions. I relished the opportunity to listen to the students talk math while only providing a rare nudge.

One of my goals this year is to create a community of mathematical discourse where it errors are fantastic and provide avenues of learning. By slowing the class down and opening up for the thoughts and discussions of students, I took a good step towards this community.