How long is a minute?

The question

How long is a minute?

Everyone knows the answer is sixty seconds. The exercise is about whether the room agrees on how sixty seconds feels, and about which number best describes the room.

Before you start

“Do you use Chromebooks in this class?” If the answer is yes and devices are in the room, run the Sheets version. Otherwise, run it on paper.

Open this exercise on your phone and tap Collect. Enter how many students are taking part.

Collect the data

This takes about three minutes. It is the only part of the exercise where timing matters, so read it through first.

1. Say: “Close your eyes or put your head down. When I say go, stay still. When you think a minute has passed, raise your hand and keep it up. Don’t say anything.”
2. Say “go” and tap Go at the same moment.
3. Tap the big button once for every hand that goes up. If three go up together, tap three times. If you mis-tap, use Undo.
4. At 120 seconds the app stops on its own and asks how many students are still waiting. It fills in the count you entered at the start, so you only need to change it if that count was wrong. Say “stop.”
5. Say “open your eyes.”

The app makes no sound and doesn’t vibrate, so it won’t give anyone a cue.

If a few students peek at the clock, let it go. It won’t change what the exercise teaches.

If there’s a ticking wall clock, students may count along with it. That’s fine too, and it can come up in the close.

If your phone dies or the page won’t load, write down the seconds on your stopwatch each time a hand goes up, and write 120 for anyone still waiting. Everything after this step works the same way.

Set up the grid

The app shows the list in the order the hands went up, with the count beside it. Copy it onto the board as it appears. Don’t sort it, since sorting is part of the students’ work.

Under the list, draw a number line from 0 to 120, marked every 10 seconds.

Sheets version: one column headed Seconds. Students type the list into A2 downward.

Build it

1. Dot plot. Put one dot on the number line for each number, stacking dots that land in the same place. On paper, students copy it. On the board, you can draw it yourself, or have a couple of students draw it while the rest start step 2.
2. Range. Longest minus shortest. Sheets: =MAX(A2:A30)-MIN(A2:A30)
3. Median. Sort the list and find the middle number. With 25 numbers, that’s the 13th. With an even count, take the two in the middle and go halfway. Sheets: =MEDIAN(A2:A30)
4. Mean. Add everything and divide by the count. On paper, pairs can split the list in half and add their halves together. Sheets: =AVERAGE(A2:A30)

Adjust A30 to wherever the last number actually is.

The answer

There’s no fixed answer, since the data is whatever the room produced. Once the class has worked out its range, median, and mean, tap Answer key in the app. It shows all three, calculated from the same list. That screen is for you, so don’t hold it up to the room.

If the class’s numbers don’t match, the usual reasons are an unsorted list for the median or a skipped number in the mean. Send them back to recount before telling them the answer.

Without the app, check these instead:

• The median should sit roughly in the middle of the dot plot.
• If a few students waited a long time, the mean will be higher than the median. If a few went very early, it’ll be lower.

Sanity check: the count of dots matches the count beside the list.

Where they get stuck

• Median from the unsorted list. The middle of the list as written isn’t the median. Sort it first.
• Sheets gives a mean or median that looks wrong. The range in the formula stops before the last row. Check that it covers every number.

Close

First: “Which number better describes this room: the mean or the median?” Let two or three students argue it. The dot plot usually settles it.

Then, if anyone was recorded at 120: “Those people hadn’t raised their hands yet. Their real number is bigger than 120. Does that change the mean? Does it change the median?” The mean is too low, and nobody can say by how much. The median doesn’t move, as long as fewer than half the room was still waiting. That’s why the median gets used when some of the data is cut off.

If there’s time left, run it again and ask whether the room got closer to sixty.

In the note