Once Upon a Time: the 12-Hour Clock Face1
Early ES / Mathematics
Mathematics, Number, Quantity, Time
During the week prior to the seminar, display in the classroom as many different types
of clocks and watches as you can. Discuss during “calendar time” and during math
class what they all have in common (the use of Numbers!) and how they are also
different.
Have students practice telling time with the classroom clock or other 12-hour analog
clock, so that they get used to the representation of hours and minutes, including the
standard of 12 hours in a “day” and 60 minutes in an hour.
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The source for part of the title and some of the core questions in this plan is one of the Problems of the Month from
the wonderful web site: http://www.insidemathematics.org/assets/problems-of-themonth/once%20upon%20a%20time.pdf .
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Distribute the text and ask students to examine it closely in pairs and with pencil in hand
(note that there will be a lot of marking / writing on the text during this seminar cycle so
it’s fine to use multiple copies of the text for various stages of the work).
Have students label the hour hand and the minute hand on their clocks. Discuss what
each shows.
Discuss with students what this version of a clock tells us and what it doesn’t tell us
(whether it’s 10:08 at night or in the morning?).
Share as appropriate the following facts about clocks leading up to the seminar:
1. Clocks existed before clock faces;
2. The original clocks were striking clocks—their purpose was to ring bells to mark
the passage of a certain amount of time (and people told time with their ears, not
their eyes);
3. Early clocks were erected as tower clocks in public places, to make sure the bells
could be heard—so that everyone knew what time it was;
4. Eventually, it was discovered that the same mechanism that rang the bells could
drive the arms on a visual clock face as well;
5. Originally, an “hour” meant 1/12th of the period of time between sunrise and
sunset (and so hours got longer and shorter during the year);
6. But as clocks themselves became more dependable, the length of an “hour” was
standardized to mean 1/24th of the average solar day.
Provide age-appropriate definitions for the following terms—by referring specifically to
the text and having students refer to their own copies to find numbers or other markings
on the text that refer to each element of time:
Day: the time it takes the earth to rotate once relative to the sun
Hour: 1/24th of a (solar) day
Minute: 1/60th of an hour (also consisting of 60 seconds)
Addition: the process of calculating the total of two (or more) numbers
Subtraction: the process of taking one number or amount away from another
As well as any other math vocabulary you would like to explore with your students as
part of this particular seminar cycle.
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Have students sit in their original (Inspectional Read) pairs with their copies of the text
and distribute a 12-inch ruler to each pair of students. Using a piece of ribbon or other
flexible material, demonstrate how a circle that is broken into 12 equal units can be cut
just before the number “1” and straightened out into a straight line of 12 equal units.
Discuss with students how a clock is like a ruler (for measuring time) and different from
a ruler (it measures something you can’t see). Make a list on the board of all the ways
that a clock is both like and different from a ruler, including the names of the units of
measurement (hour vs. inch, etc.).
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 What would be another, more descriptive name for the clock face? (roundrobin response)
 What about the clock face made you think of that name? (spontaneous
discussion)
 Why do you think the clock face is round? Would another shape be better
for measuring time? Why or why not?
 Why do you think the “12” is at the top of the clock and the “6” at the
bottom?
 When it is ten minutes past ten, how many minutes must pass before the
big hand (the minute hand) gets to the 6? How did you figure it out?
 When it is seven o’clock, how many minutes must pass before the big
hand (minute hand) gets to where the little hand (hour hand) was at seven
o’clock? Explain how you got your answer.
 How many different ways can we use a clock to solve a problem like this?
 If you were going to design your own clock, what picture would you put on
the clock face to represent what a clock does? Explain your choice.
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Have students work in pairs to say how many different ways we can use clocks to solve
math problems. Encourage them to say everything they said, heard, or thought during
the seminar.
How can you use a clock face to solve a math problem? After “reading” and discussing
the 12-hour clock face, about hours and minutes, write a math problem (similar to the
ones in the seminar) and then explain how you would use a clock face to solve it. Use
evidence from the clock face to support your response. (Informational or Explanatory /
Explain)
(LDC Task#: 14 )
Post the core seminar questions on the board for students to use as examples. Then
have them work with their partners (from the Transition to Writing activity) to write a
similar math problem based on the examples. Check to be sure that the problems they
have composed will work for this exercise.
Remind students of their answers during the seminar discussion and ask them to list the
steps they would use (as individuals—not pairs) to solve the problem that each
composed with his or her partner.
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Have students expand the lists from the previous step (as individuals, not pairs) to write
either a paragraph or a series of short sentences in which each student describes in
detail how s/he would go about solving the problem.
Have the students work with their partners to do a collaborative revision of their
paragraphs by taking turns reading their papers aloud to each other so that the listener
can then ask one clarifying question and make one clarifying suggestion. Allow time for
students to write a second draft based on these questions and suggestions.
Work with each pair in turn to proof the two paragraphs with the students—marking and
discussing any spelling, punctuation, or grammar errors with the two students. Allow
time for all students to prepare a third draft (in which they correct any mistakes) suitable
for publication!
Publish the problems and solutions by:
1. Displaying a large copy of the original text (a 12-hour clock face) on a bulletin
board either in the classroom or in the hallway outside;
2. Displaying the problems each pair wrote in a larger circle around the clock face;
and
3. Displaying the student process solutions in yet a larger circle adjacent to the
appropriate problems.
4. Invite other classes from the same grade level to view the problems and critique
the solutions.
Terry Roberts
National Paideia Center
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