A clock face only has the numbers to , but a day is much longer than twelve hours. When a clock says , how do you know whether it means the morning or the evening?

Grade 2 · 2.MD.C.7

Morning and afternoon on a twelve hour clock

A clock face only has the numbers 1 to 12, but a day is much longer than twelve hours. When a clock says 7{:}00, how do you know whether it means the morning or the evening?

You need

Telling timeGrade 1

Where you are 13 in Time and money

Measurement12

Time and money13

Telling time

Morning and afternoon on a twelve hour clock

Telling time to the nearest five minutes Elapsed time Twenty four hour time and crossing midnight ▾ Show all 13 (8 more)

Why this exists

A child who can tell time can look at a clock and say “seven o’clock”. But seven o’clock comes twice every day: once in the morning, around the time many children wake up, and once in the evening, around the time many families finish dinner. The clock face looks exactly the same both times. If a friend says “come over at seven”, the clock alone cannot say which seven is meant.

The reason is that a clock face has only twelve hours on it, and a day has twenty-four. The hour hand goes all the way around the face once, and the day is only half over, so it goes around a second time. Every time a clock can show turns up twice.

People solve this with two short labels. a.m. marks a time in the first half of the day and p.m. marks a time in the second half. With the label, “7{:}00 a.m.” and “7{:}00 p.m.” are two different times, and nobody has to guess.

This lesson runs over two days:

Part 1: why each clock time happens twice, what a.m. and p.m. mean, and the two special times, noon and midnight, where one half of the day ends and the other begins.
Part 2: choosing a.m. or p.m. from what is happening at that time, and putting the events of one day in order when some are a.m. and some are p.m.

A real clock helps, and so does a long strip of paper that the adult turns into a day line (the first picture below), with the child’s own day marked on it: getting up, school, lunch, home, dinner, bed.

Definition

Part 1: Two halves of one day

A day runs from one midnight to the next. Midnight is the middle of the night, when most people are asleep and the date changes. Halfway through the day is noon, the middle of the day, around lunchtime.

a.m. times are the times from midnight up to noon. These are the night after midnight and the morning.
p.m. times are the times from noon up to midnight. These are the afternoon, the evening and the night before midnight.

The letters come from two old Latin phrases. a.m. is short for ante meridiem, “before midday”, and p.m. is short for post meridiem, “after midday”. A child does not need the Latin. “a.m. is before noon, p.m. is after noon” is enough.

The label is written after the time, with a space: 9{:}15 a.m., 3{:}40 p.m. Some clocks and signs write AM and PM in capital letters or without the dots. They mean the same thing.

Noon and midnight themselves. The two times where the halves meet need a rule of their own.

Noon is written 12{:}00 p.m.
Midnight is written 12{:}00 a.m.

The reason for this choice is the minute that follows. One minute after noon is already in the afternoon, 12{:}01 p.m., so noon is given the same label as the hour it starts. One minute after midnight is the very start of a new day, 12{:}01 a.m., so midnight takes the a.m. label of the hour it starts. Both are still easy to mix up, so people very often just say “noon” and “midnight” instead of “12{:}00 p.m.” and “12{:}00 a.m.”. That is a perfectly good way to write them, and it is what many schedules and tickets do.

Intuition

Part 1: The day line

Lay the whole day out as one long line, from midnight on the left to the next midnight on the right. Each tick is one hour.

Figure 1One whole day, midnight to midnight. The hour numbers run 12, 1, 2, \ldots, 11 in each half.

Read the numbers under the ticks from left to right: 12, 1, 2, 3 and on to 11, then 12 again in the middle, then 1, 2, 3 and on to 11, then 12 at the far end. The same twelve numbers run twice, once in each half of the line. The left half is a.m. and the right half is p.m.

Now picture the hour hand of a clock while a finger moves along the line. At the left end, at midnight, the hour hand points at 12. As the finger moves right, the hand moves around the face: 1, 2, 3 and on. When the finger reaches noon, in the middle of the line, the hand is back at 12, having gone exactly once around. Then it does the whole trip again for the p.m. half. That is why each position of the hands belongs to two places on the line, one in each half.

Figure 2The same clock face at two different times of the same day.

So a clock tells you where in its half of the day you are, and the label tells you which half. You need both.

Two things about the day line are worth pointing out to a child, because they are not what the numbers suggest.

The hour 12 comes first in each half. In each half the hours run 12, 1, 2, \ldots, 11. So 12{:}30 p.m. is only half an hour after noon, early in the afternoon, and 12{:}30 a.m. is only half an hour after midnight, in the middle of the night. A 12 on the clock does not mean “late”; it means the half of the day has just begun.

The label changes at 12, not at 1. At 11{:}55 a.m. it is still morning. Five minutes later the hour hand reaches 12 and the label changes to p.m. The hour number changes to 1 an hour after that. The same happens at night: 11{:}55 p.m. is the end of one day, and five minutes later, at midnight, a new day starts with a.m.

Part 2: Clues about which half

When a time is said or written without a label, the clue is what is happening. Some things happen in the a.m. half for nearly everyone: the middle of the night, getting up, breakfast, the start of school. Some happen in the p.m. half: lunch just after noon, the end of the school day, dinner, the evening. Light helps as well: the sun comes up in the a.m. half and goes down in the p.m. half.

These clues are about everyday life, not mathematics, so a clue can be weak. A person can eat lunch at 11{:}45 a.m., and in summer it can still be light at 8{:}00 p.m. When the clue is not enough, the honest answer is “it could be either”, and the label is what settles it.

Results

Three facts follow from splitting the day into two halves at noon.

Every clock time happens twice a day. Each position of the hands is reached once in the a.m. half and once in the p.m. half. So “7{:}00” alone names two times, and “7{:}00 a.m.” or “7{:}00 p.m.” names one.

Noon and midnight. Noon is 12{:}00 p.m. and midnight is 12{:}00 a.m.; saying “noon” and “midnight” avoids any doubt.

Putting one day in order. Within one day, every a.m. time comes before every p.m. time. Inside one half, the times go in clock order with 12 as the first hour: 12, 1, 2, \ldots, 11.

Why it is true

The face of a clock has 12 numbers, and the hour hand moves from one number to the next in one hour. So it goes once around the face in 12 hours. A day has 24 hours, and

12 + 12 = 24

so in one day the hour hand goes around the face exactly twice. The first trip starts at midnight and ends at noon; the second starts at noon and ends at the next midnight. During each trip the minute hand goes around once an hour, so the two hands together pass through every position they can make, once per trip. Two trips means every position twice, once in each half.

That is also why the order rule holds. Every a.m. time is on the first trip and every p.m. time is on the second, so every a.m. time comes first. Inside one trip the hour hand starts at 12 and passes 1, 2, \ldots, 11 in turn, so 12 is the first hour of each trip, not the last.

Worked examples

Example 1

Part 1: The same clock, twice a day

This is the kitchen clock at Chibueze’s house. His grandmother gets up to bake bread while it is still dark outside, and she sees this clock as she turns on the oven. Later that day Chibueze comes home from school and sees the clock showing the very same thing. Write each time with a.m. or p.m.

Figure 3The kitchen clock at Chibueze’s house.

Solution

First read the clock. The long minute hand points at 6, so it is half past the hour. The short hour hand is halfway between 4 and 5, so the hour is 4. The clock says 4{:}30.

The grandmother is up before the sun, in the early morning. That is after midnight and before noon, so it is the a.m. half: 4{:}30 a.m.

Chibueze comes home after school, in the afternoon. That is after noon and before midnight, so it is the p.m. half: 4{:}30 p.m.

The clock gave the same answer both times. Only the label, which came from what was happening, tells the two times apart. Between them the hour hand went once all the way around the face.

Example 2

Part 1: Noon and midnight on the houseboat

Miroslava is staying with her grandmother on a houseboat. The ship’s bell rings at 12{:}00 in the middle of the day for lunch, and the clock shows 12{:}00 again in the middle of the night. Write both times two ways. Then write the time half an hour after each one.

Solution

The middle of the day is noon, where the a.m. half ends and the p.m. half begins. Noon is written 12{:}00 p.m., or simply “noon”.

The middle of the night is midnight, where one day ends and the next begins. Midnight is written 12{:}00 a.m., or simply “midnight”.

Half an hour after noon the minute hand has moved from 12 to 6, and the hour hand is halfway between 12 and 1. The clock says 12{:}30. It is just after lunch, in the afternoon, so it is 12{:}30 p.m.

Half an hour after midnight the clock also says 12{:}30. It is the middle of the night, the very start of a new day, so it is 12{:}30 a.m.

Both 12{:}30 times are early in their half. On the day line each sits just to the right of a 12 tick. That is easy to forget, because 12 is the biggest number on the face.

Example 3

Part 1: Where the label changes

Chibueze watches the classroom clock on the hour from 10{:}00 a.m. Name the next three times on the hour, with labels.

Solution

Each hour the hour hand moves on to the next number and the minute hand comes back to 12.

After 10{:}00 a.m. the hour hand reaches 11. It is still before noon, so the next time on the hour is 11{:}00 a.m.

Next the hour hand reaches 12. This is the middle of the day, noon, so the label changes here: 12{:}00 p.m.

Next the hour hand reaches 1. Noon is behind us, so this is 1{:}00 p.m.

So the three times are 11{:}00 a.m., 12{:}00 p.m. (noon) and 1{:}00 p.m. Notice the order of the changes. The label changes from a.m. to p.m. when the hour hand reaches 12; the hour number starts again at 1 one hour later. On the day line, the 12 tick in the middle is the first tick of the p.m. half.

Example 4

Part 2: Which half of the day?

Kalea’s family spends a day at the beach. They leave home at 7{:}15, just as the sun comes up. They eat their sandwiches at 12{:}20, a little after noon. They watch the sun go down over the water at 7{:}40. Write each time with a.m. or p.m.

Solution

The sun comes up in the morning, which is after midnight and before noon. So they leave home at 7{:}15 a.m.

The sandwiches are eaten a little after noon. Every time from noon until midnight is p.m., so lunch is at 12{:}20 p.m. The 12 does not make it a.m.: it means this is early in the p.m. half.

The sun goes down in the evening, after noon. So they watch the sunset at 7{:}40 p.m.

The hour 7 came up twice in one day, once in each half, just as the day line says it must.

Example 5

Part 2: When the clue is missing

Miroslava feeds the ducks from the houseboat at 9{:}10. Is that a.m. or p.m.? Then answer the same question for “Miroslava feeds the ducks at 9{:}10, before breakfast.”

Solution

The first sentence gives no clue about the half of the day. People do feed ducks in the morning, and at 9{:}10 p.m. on a summer evening it could still be light enough to do it. So the honest answer is that it could be either: 9{:}10 a.m. or 9{:}10 p.m.

The second sentence adds “before breakfast”. Breakfast is the first meal of the day, in the morning, so this is the a.m. half: 9{:}10 a.m.

This is exactly the job of the label. When the sentence says enough, the label can be worked out; when it does not, only the label settles it, which is why tickets, schedules and invitations write it down.

Example 6

Part 2: Putting a beach day in order

Here is more of Kalea’s beach day, mixed up. Put the events in order from earliest to latest.

Swimming at 2{:}50 p.m.
Building a sandcastle at 10{:}35 a.m.
Ice cream at 12{:}45 p.m.
Looking for shells at 8{:}25 a.m.
Packing up to go home at 5{:}10 p.m.
Flying a kite at 11{:}40 a.m.

Solution

Sort by half of the day first. Every a.m. time is before noon and every p.m. time is after noon, so all the a.m. events come first.

The a.m. events are the shells at 8{:}25, the sandcastle at 10{:}35 and the kite at 11{:}40. In clock order that is 8, then 10, then 11.

The p.m. events are the ice cream at 12{:}45, the swimming at 2{:}50 and packing up at 5{:}10. In the p.m. half the hour 12 comes first, right after noon, so the ice cream is the earliest of these. Then 2{:}50, then 5{:}10.

In order: shells 8{:}25 a.m., sandcastle 10{:}35 a.m., kite 11{:}40 a.m., ice cream 12{:}45 p.m., swimming 2{:}50 p.m., packing up 5{:}10 p.m. On the day line they sit left to right as the dots A to F.

Figure 4Kalea’s beach day on the day line. Left to right is earliest to latest.

Example 7

Part 2: Which comes first?

In one day, which comes first, 11{:}20 a.m. or 12{:}40 a.m.? And which comes first, 11{:}20 p.m. or 12{:}40 p.m.?

Solution

Both pairs are in the same half of the day, so the half does not decide it. What decides it is that 12 is the first hour of each half.

12{:}40 a.m. is 40 minutes after midnight, near the very start of the day. 11{:}20 a.m. is late in the morning, close to noon. So 12{:}40 a.m. comes first.

12{:}40 p.m. is 40 minutes after noon, just after lunchtime. 11{:}20 p.m. is late at night, close to the next midnight. So 12{:}40 p.m. comes first.

In both halves the time with the 12 came first, even though 12 is the bigger number. On a clock face the hour hand reaches 12 at the start of each half and 11 near the end.

Boundaries

a.m. and p.m. only belong with a twelve hour clock. They say which trip around the face the hour hand is on. A clock that numbers the hours of the day 0 to 23 gives every hour its own number and needs no label; that comes in a later grade.
“Earlier” and “later” here are inside one day. The order rule, a.m. before p.m., compares times from the same midnight to the next. 11{:}00 p.m. tonight comes before 6{:}00 a.m. tomorrow, because the second one is in the next day.
Clues from daily life can fail. Sunrise, meals and school are good guides, but they are habits, not rules. People who work at night eat breakfast in the p.m. half, and the time of sunrise changes through the year. When a clue is weak, the time stays “a.m. or p.m.” until the label is given.
12{:}00 needs care. By the usual rule noon is 12{:}00 p.m. and midnight is 12{:}00 a.m., but because the two are so easily confused, “noon” and “midnight” are the clearest way to say them.
A label says when, not how long. Knowing that one event is at 10{:}35 a.m. and another at 2{:}50 p.m. puts them in order. Finding how much time passed between them is a later step, not part of this lesson.

Where this goes next

The two halves of the day are the base for working with time. Every time read on the face, to the nearest five minutes as in telling time to five minutes, still needs its a.m. or p.m. Counting how long something takes, especially when it starts in the morning and ends in the afternoon, is Elapsed time: going from 11{:}40 a.m. to 12{:}45 p.m. means passing noon, where the label changes. Later, Twenty four hour time and crossing midnight numbers the afternoon hours 13 to 23, so the second trip around the face gets its own numbers and no label is needed. The day itself sits inside the week, the month and the year, which Units of time and Calendars and dates take up.

Exercises

Part 1: a.m. or p.m.?

Write each time with a.m. or p.m.

3{:}25 in the afternoon
10{:}50 in the morning
4{:}20 in the night, before the sun comes up
9{:}05 at night, an hour or two before midnight
What is another way to write 12{:}00 p.m.?
What is another way to write 12{:}00 a.m.?
A clock says 12{:}35, and everyone is eating lunch. a.m. or p.m.?
A clock says 12{:}35, and everyone is asleep. a.m. or p.m.?
The time is 9{:}00 p.m. Name the next three times on the hour, with labels.
Which time on the hour comes just before 1{:}00 p.m.? Give it two ways.

Part 2: Read the clock, then choose

Use the clocks for Exercises 11 to 14. Read each clock, then write the time with a.m. or p.m.

Figure 5Clocks for Exercises 11 to 14.

Clock A shows when Kalea’s family eats dinner on a picnic blanket.
Clock B shows when Kalea builds a second sandcastle, an hour before lunch.
Clock C shows when a thunderstorm wakes Miroslava in the middle of the night.
Clock D shows when Chibueze’s class starts its first lesson of the school day.

Part 2: Which half of the day?

An owl hoots outside Miroslava’s window at 1{:}10 in the dark of night. a.m. or p.m.?
Miroslava eats lunch on the houseboat at 12{:}25. a.m. or p.m.?
Chibueze’s soccer practice ends at 5{:}55, just before dinner. a.m. or p.m.?
Miroslava watches the sun come up over the lake at 6{:}20. a.m. or p.m.?
A store sign says “Open 9{:}00 to 9{:}00”. The store is open all day and closed all night. Which 9{:}00 is a.m. and which is p.m.?

Part 2: Put the day in order

Put in order from earliest to latest: 4{:}15 p.m., 9{:}50 a.m., 12{:}30 p.m., 6{:}05 a.m.
Put in order from earliest to latest: 1{:}45 p.m., 12{:}55 p.m., 11{:}10 a.m.
In one day, which comes first: 12{:}50 a.m. or 5{:}00 a.m.?
In one day, which comes first: 7{:}30 p.m. or 8{:}10 a.m.?

Reasoning

Why does every time on a clock face happen twice in one day?
Is every a.m. time earlier than every p.m. time in the same day? Explain how you know.

Answers

3{:}25 p.m. The afternoon is after noon.
10{:}50 a.m. The morning is before noon.
4{:}20 a.m. Before sunrise is after midnight and before noon.
9{:}05 p.m. Before midnight and after noon is the p.m. half.
Noon. 12{:}00 p.m. is the middle of the day, where the p.m. half starts.
Midnight. 12{:}00 a.m. is the middle of the night, where a new day starts.
12{:}35 p.m. Lunch is just after noon, so it is early in the p.m. half.
12{:}35 a.m. Everyone asleep at 12{:}35 means just after midnight, early in the a.m. half.
10{:}00 p.m., 11{:}00 p.m. and 12{:}00 a.m. (midnight). The label changes when the hour hand reaches 12 in the middle of the night.
12{:}00 p.m., which is noon. The hour hand is on 12 one hour before it reaches 1.
6{:}30 p.m. The minute hand is on 6 and the hour hand halfway between 6 and 7. Dinner is in the evening, after noon.
11{:}00 a.m. The hour hand is on 11 and the minute hand on 12. An hour before lunch is still morning.
2{:}30 a.m. The minute hand is on 6 and the hour hand halfway between 2 and 3. The middle of the night after midnight is a.m.
8{:}00 a.m. The hour hand is on 8 and the minute hand on 12. School starts in the morning.
1{:}10 a.m. At 1{:}10 p.m. it is the middle of the day; in the dark of night it is just after midnight.
12{:}25 p.m. Lunch is just after noon, and the hour 12 is the first hour of the p.m. half.
5{:}55 p.m. Just before dinner is late afternoon.
6{:}20 a.m. The sun comes up in the morning.
The store opens at 9{:}00 a.m. and closes at 9{:}00 p.m. It is open all day, so it opens in the morning and closes in the evening.
6{:}05 a.m., 9{:}50 a.m., 12{:}30 p.m., 4{:}15 p.m. The a.m. times come first; in the p.m. half, 12 is the first hour.
11{:}10 a.m., 12{:}55 p.m., 1{:}45 p.m. The a.m. time comes first, then 12 before 1 in the p.m. half.
12{:}50 a.m. It is less than an hour after midnight, and 5{:}00 a.m. is hours later.
8{:}10 a.m. Every a.m. time comes before every p.m. time in the same day.
The hour hand goes once around the face in 12 hours, and a day has 12 + 12 = 24 hours. So it goes around twice, once from midnight to noon and once from noon to midnight, and every position of the hands is reached once on each trip.
Yes. The a.m. times are all from midnight up to noon and the p.m. times are all from noon up to the next midnight. Every a.m. time is before noon and every p.m. time is at noon or after, so every a.m. time comes first.