Grade 4 · 4.MD.A.1 · 4.MD.A.2

Units of time

The movie runs 3 hours, but the streaming app counts in minutes. How do you say the same amount of time in another unit, when the steps between the units are sixties and twenty-fours instead of tens?

You need

Multiplication factsGrade 3 Telling timeGrade 1

Unlocks

Calendars and datesGrade 4 Flow rateMath 7

Where you are 13 in Time and money

Measurement12

Time and money13

▴ 3 above Elapsed time Twenty four hour time and crossing midnight

Units of time

Calendars and dates Money ▾ Show all 13 (5 more)

Why this exists

Lowanna is planning a movie night, and Jorge and Mert are planning the sleepover that follows it. The popcorn needs a few minutes in the microwave, which counts in seconds. The movie runs a few hours, but the streaming app lists it in minutes. The sleepover lasts more than a day, and it is two weeks away.

Time comes in sizes from a blink to a lifetime, and no single unit fits them all. Counting a movie in seconds gives a number too big to picture; counting popcorn time in hours gives a tiny part of one. So time has a family of units, each one made of a set number of the next smaller one. To compare two amounts of time, or to give one in the unit someone asked for, you change it from one unit to another. The amount of time stays the same; only the unit you count it in changes.

Definition

The units of time are the second, the minute, the hour, the day, the week, the month and the year. Each one is made of whole groups of a smaller one:

Fact	Read it as
1 minute = 60 seconds	sixty seconds make a minute
1 hour = 60 minutes	sixty minutes make an hour
1 day = 24 hours	twenty-four hours make a day
1 week = 7 days	seven days make a week
1 year = 12 months	twelve months make a year
1 year = 365 days	except a leap year, which has 366

The units nest: a week holds days, a day holds hours, an hour holds minutes, and a minute holds seconds, like boxes inside boxes. A month is the one unit with no single size: it has 28, 29, 30 or 31 days, which is the business of Calendars and dates.

An amount of time can be written in two units at once, such as 2 hours 15 minutes: two whole hours and fifteen more minutes. This lesson is about amounts of time (how long), not about reading the clock, which is Telling time.

Intuition

Part 1: how the units nest, and changing a larger unit to a smaller one

Watch the minute hand on a clock. It sweeps once all the way around in 60 minutes, and in that time the hour hand moves from one number to the next: one hour. So an hour is 60 minutes. Two hours is two sixties, three hours is three sixties, and each hour you add brings 60 more minutes. A number of hours is that many equal groups of 60 minutes, and equal groups are multiplication:

\text{number of minutes} = \text{number of hours} \times 60

Lowanna’s movie night lasts 3 hours. In minutes it is 3 \times 60 = 180 minutes. A table lays the groups out, one more hour in each row:

Hours	Minutes
1	60
2	120
3	180
4	240
5	300
6	360

Read down, and the minutes climb by 60 each time the hours climb by 1. Read across, and every row says “hours times 60”. It is 6 \times 1, 6 \times 2, 6 \times 3 from the Multiplication facts, with a zero on the end, because 60 is 6 tens.

A double number line puts both units on one line: hours on top, minutes below, and each tick names one moment in both units.

Figure 1Hours and minutes on one line. Each hour along the top is 60 minutes along the bottom, so 3 hours sits at 180 minutes, and the half-hour tick between 1 and 2 hours sits at 90 minutes.

The same move works at every step of the nest. Minutes to seconds multiplies by 60, days to hours by 24, weeks to days by 7, years to months by 12. In each case the smaller unit is the one being counted, and it takes more of the smaller unit to fill the same time, so the number always goes up. That gives a quick check: if you change hours to minutes and get a smaller number than you started with, you went the wrong way.

When an amount is written in two units, change only the larger part and add the smaller part as it is. A short movie of 1 hour 40 minutes is

1 \times 60 + 40 = 60 + 40 = 100 \text{ minutes}

Look at that answer: 100 minutes, and it is not even close to “1 hour”. Part 2 is about why.

Part 2: why time is not counted in tens

Lengths in the metric system step by tens, and so do our numbers: ten ones make a ten, ten tens make a hundred. Time does not. Sixty seconds make a minute, sixty minutes make an hour, and twenty-four hours make a day. Where do those numbers come from?

The day and the year come from the sky. A day is about how long Earth takes to spin once so the Sun comes back to the same place, and a year is about how long Earth takes to travel once around the Sun. Nobody chose those lengths. The cutting up was chosen by people: the day into 24 hours, the hour into 60 minutes, the minute into 60 seconds. Counting in sixties is very old, and Numbers in other bases tells a little of where it comes from.

Sixty is a handy number to cut up. Half an hour is 30 minutes, a third is 20, a quarter is 15, a fifth is 12 and a sixth is 10, all whole numbers of minutes. A hundred cannot be cut into three equal whole parts.

The price is that place value does not carry across units. 100 minutes is not “1 hour”: it is 1 hour 40 minutes, because only 60 of those minutes make the hour. Changing from a smaller unit to a larger one means asking how many groups of 60 fit.

Lowanna lines up movies for the night, 200 minutes in all. How many hours is that? Three groups of 60 is 180, and four groups is 240, which is more than 200. So 3 whole hours fit, and 200 - 180 = 20 minutes are left over:

\begin{aligned} 200 \div 60 &= 3 \text{ R } 20 \\ 200 \text{ minutes} &= 3 \text{ hours } 20 \text{ minutes} \end{aligned}

Figure 2Lowanna’s movie list. Three whole hours fit in 200 minutes, and the shaded 20 minutes are too short to make a fourth hour.

The remainder is not thrown away. It is the minutes that are not enough to make another hour, and it stays in minutes, the way leftover inches stay inches in Customary conversions and as in Remainders and what they mean. It is always less than 60: if 60 or more were left, one more hour would fit.

Decimals need the same care. 1.5 hours means one hour and five tenths of another, and five tenths is one half. Half of an hour is half of 60 minutes, which is 30 minutes (taking a A fraction of a whole number). So

1.5 \text{ hours} = 60 + 30 = 90 \text{ minutes}

and the half-hour tick between 1 and 2 on the double number line sits at 90. Writing 150 minutes would treat an hour as 100 minutes, as if the 5 after the point meant 50 minutes. The .5 is a part of an hour, never a count of minutes.

Results

1 minute = 60 seconds, 1 hour = 60 minutes, 1 day = 24 hours, 1 week = 7 days, 1 year = 12 months.
Larger to smaller: multiply by the number in the fact. 3 hours = 3 \times 60 = 180 minutes.
Smaller to larger: divide by the number in the fact. The quotient counts the larger unit and the remainder stays in the smaller unit, always less than the number in the fact. 200 minutes = 3 hours 20 minutes.
Parts of an hour: half an hour is 30 minutes and a quarter of an hour is 15 minutes, so 1.5 hours is 90 minutes.

Why it is true

An hour is exactly the same stretch of time as 60 minutes, so any number of hours is that many groups of 60 minutes, laid end to end. Counting equal groups is multiplying, so hours times 60 gives the minutes. Nothing about the time changed; it was counted again in smaller pieces.

Going back, each hour uses up 60 of the minutes. Taking out as many whole groups of 60 as possible is dividing by 60: the quotient is the number of whole hours, and what is left is too small to make another, so it is less than 60. Putting the hours back as minutes and adding the leftover rebuilds the start: 3 \times 60 + 20 = 200. The same reasoning works with 24 for days, 7 for weeks and 12 for months in a year.

Worked examples

Examples 1, 2 and 3 go with Part 1, and Examples 4 to 7 with Part 2.

Example 1

Popcorn on a seconds timer

The microwave timer counts in seconds. The popcorn bag says 4 minutes. Make a table of minutes and seconds from 1 to 4 minutes, and say what to type in.

Solution

Each minute is 60 seconds, so a number of minutes is that many groups of 60 seconds. Each row multiplies the minutes by 60.

Minutes	Seconds
1	60
2	120
3	180
4	240

The last row is the one asked about: 4 \times 60 = 240, so Mert types 240 seconds. A second is smaller than a minute, so the number of seconds should be larger than the number of minutes, and 240 is larger than 4.

Example 2

Two weeks to the sleepover

The sleepover is 2 weeks away. How many days away is it? Jorge says that one week alone is more than 160 hours. Is he right?

Solution

Each week is 7 days, so 2 weeks is 2 \times 7 = 14 days.

For the hours in a week, each day is 24 hours, so 7 days is 7 \times 24. Split 24 into 20 and 4, multiply each part by 7, and add. Multiplying each part and adding is the distributive property:

7 \times 24 = 7 \times 20 + 7 \times 4 = 140 + 28 = 168

A week is 168 hours, and 168 > 160, so Jorge is right.

Example 3

A movie in the streaming app

The movie Lowanna picks runs 2 hours 17 minutes. The streaming app lists every movie in minutes. What number does it show?

Solution

The movie is 2 whole hours and 17 more minutes. Change the hours to minutes: 2 \times 60 = 120. The 17 minutes are already minutes, so add them as they are:

2 \times 60 + 17 = 120 + 17 = 137 \text{ minutes}

The app shows 137 minutes. Only the hours were multiplied.

Example 4

The whole movie list in hours

Lowanna’s list of movies adds up to 200 minutes. How long is that in hours and minutes? Check the answer.

Solution

Minutes are the smaller unit, so ask how many groups of 60 fit in 200. Three groups make 3 \times 60 = 180, and four make 240, which is too many. So 3 hours fit, with 200 - 180 = 20 minutes left:

200 \div 60 = 3 \text{ R } 20

The list runs 3 hours 20 minutes. The 20 minutes are fewer than 60, so they cannot make another hour, and the answer is finished. Check by going back: 3 \times 60 + 20 = 180 + 20 = 200. The figure in Part 2 shows these three hours and the shaded leftover.

Example 5

A sleepover in days and hours

Jorge’s sleepover starts on Friday evening and the last guest leaves 40 hours later. How long is that in days and hours?

Solution

Hours are the smaller unit and a day is 24 hours, so ask how many groups of 24 fit in 40. One group is 24; two groups are 48, which is more than 40. So 1 whole day fits, with 40 - 24 = 16 hours left:

40 \div 24 = 1 \text{ R } 16

The sleepover lasts 1 day 16 hours. The remainder 16 is less than 24, as it must be. Check: 1 \times 24 + 16 = 40.

Example 6

One and a half hours

The movie box says the film is 1.5 hours long. Mert says that is 150 minutes. How many minutes is it really?

Solution

The 1 is one whole hour, which is 60 minutes. The .5 is five tenths of an hour, and five tenths is one half. Half of 60 minutes is 60 \div 2 = 30 minutes. Together:

1.5 \text{ hours} = 60 + 30 = 90 \text{ minutes}

The film is 90 minutes long. Mert’s 150 would be right only if an hour had 100 minutes: he read the 5 as 50 minutes. A size check agrees: 150 minutes is more than 2 hours, since 2 \times 60 = 120, and 1.5 hours is less than 2 hours.

Example 7

Who slept longer

After the sleepover, Mert’s sleep tracker says he slept 500 minutes. Jorge says he himself slept 8.5 hours. Who slept longer, and by how much?

Solution

The two amounts are in different units, so put them in one unit first. Minutes are the smaller unit, so change Jorge’s hours to minutes. The 8 is 8 \times 60 = 480 minutes, and the .5 is half an hour, 30 minutes:

8.5 \text{ hours} = 480 + 30 = 510 \text{ minutes}

Now compare: 510 > 500, so Jorge slept longer, by 510 - 500 = 10 minutes.

Check the other way, in hours and minutes. 500 \div 60 = 8 \text{ R } 20, since 8 \times 60 = 480 and 500 - 480 = 20, so Mert slept 8 hours 20 minutes. Jorge’s 8.5 hours is 8 hours 30 minutes, and 30 - 20 = 10, the same 10 minutes.

Boundaries

Months have no single size. A year is always 12 months, but a month is 28 to 31 days and a year is 365 or 366 days, so there is no one fact for changing months to days or weeks. Counting days on a calendar is Calendars and dates.
Amounts of time, not clock times. These facts change how long something lasts. Finding how long it is from one clock time to another is Elapsed time, and times that run past midnight are in Twenty four hour time and crossing midnight.
Not base ten. Place value does not carry from one unit to the next. 100 minutes is 1 hour 40 minutes, and in a decimal number of hours the digits after the point are parts of an hour, not minutes: 1.5 hours is 90 minutes.
Leftovers stay in the smaller unit. When dividing leaves a remainder, it is kept in the smaller unit and is less than the number in the fact (60, 24, 7 or 12); otherwise a whole larger unit is still hiding in it.

Where this goes next

Measured time feeds almost every other measurement. Story problems with several conversions, or with hours and minutes added and regrouped, are in Multi-step problems with unit conversions. How long it is from the start of the movie to lights out is Elapsed time. Clocks in different places on Earth that read different hours at the same moment are Time zones. And once time is a unit you can divide by, you can measure how fast something happens in each minute or each hour, as in Flow rate.

Exercises

Part 1: larger to smaller

Change 7 minutes to seconds.
Change 9 hours to minutes.
Change 4 days to hours.
Change 6 weeks to days.
Make a two-column table of hours and minutes for 10, 11 and 12 hours.
A long movie runs 3 hours 5 minutes. How many minutes is that?
The opening credits run 2 minutes 40 seconds. How many seconds is that?
A sleepover lasts 1 day 9 hours. How many hours is that?
Lowanna’s cat is 3 years old. How many months is that?
Jorge’s summer vacation lasts 5 weeks 3 days. How many days is that?

Part 2: smaller to larger, decimals and comparing

Change 170 minutes to hours and minutes.
Change 95 seconds to minutes and seconds.
Change 50 hours to days and hours.
Change 30 days to weeks and days.
Change 270 minutes to hours and minutes.
At the sleepover, Mert’s clock app says the party has lasted 1{,}000 minutes. How long is that in hours and minutes?
How many minutes is 3.5 hours?
A weekend sleepover lasts 1.5 days. How many hours is that?
How many minutes is a quarter of an hour? How many hours is half a day?
One movie runs 1 hour 55 minutes. Another runs 118 minutes. Which is longer, and by how much?
Which is longer, 1{,}200 seconds or a quarter of an hour? By how much?

Reasoning

Explain, without a table, why changing hours to minutes multiplies by 60, and why the number of minutes is always larger than the number of hours.
Jorge says 300 minutes is 3 hours, “because the 3 is in the hundreds place”. Explain what is wrong, and give the right number of hours.
Mert says 2.5 hours is 250 minutes. Lowanna says it is 150 minutes. Who is right? Explain.
Show that half, a third, a quarter, a fifth and a sixth of an hour are each a whole number of minutes. If an hour had 100 minutes, which of those five would not be a whole number of minutes?

Answers

420 seconds. Each minute is 60 seconds, so 7 \times 60 = 420.
540 minutes. 9 \times 60 = 540.
96 hours. 4 \times 24 = 4 \times 20 + 4 \times 4 = 80 + 16 = 96.
42 days. Each week is 7 days, so 6 \times 7 = 42.
600, 660 and 720 minutes: 10 \times 60 = 600, 11 \times 60 = 660 and 12 \times 60 = 720. Each row is 60 minutes more than the one before, because it is one hour more.
185 minutes. 3 \times 60 + 5 = 180 + 5 = 185.
160 seconds. 2 \times 60 + 40 = 120 + 40 = 160.
33 hours. 1 \times 24 + 9 = 24 + 9 = 33.
36 months. Each year is 12 months, so 3 \times 12 = 36.
38 days. 5 \times 7 + 3 = 35 + 3 = 38.
2 hours 50 minutes. 170 \div 60 = 2 \text{ R } 50, since 2 \times 60 = 120 and 170 - 120 = 50.
1 minute 35 seconds. 95 \div 60 = 1 \text{ R } 35, since 95 - 60 = 35.
2 days 2 hours. 50 \div 24 = 2 \text{ R } 2, since 2 \times 24 = 48 and 50 - 48 = 2.
4 weeks 2 days. 30 \div 7 = 4 \text{ R } 2, since 4 \times 7 = 28 and 30 - 28 = 2.
4 hours 30 minutes. 270 \div 60 = 4 \text{ R } 30, since 4 \times 60 = 240 and 270 - 240 = 30.
16 hours 40 minutes. 1{,}000 \div 60 = 16 \text{ R } 40, since 16 \times 60 = 960 and 1{,}000 - 960 = 40. Check: 960 + 40 = 1{,}000.
210 minutes. The 3 hours are 3 \times 60 = 180 minutes, and the .5 is half an hour, 30 minutes, so 180 + 30 = 210.
36 hours. One day is 24 hours, and half a day is 24 \div 2 = 12 hours, so 24 + 12 = 36.
15 minutes and 12 hours. A quarter of an hour is 60 \div 4 = 15 minutes, and half a day is 24 \div 2 = 12 hours.
The 118 minute movie is longer, by 3 minutes. 1 hour 55 minutes is 60 + 55 = 115 minutes, and 118 - 115 = 3.
1{,}200 seconds is longer, by 300 seconds, which is 5 minutes. A quarter of an hour is 15 minutes, which is 15 \times 60 = 900 seconds, and 1{,}200 - 900 = 300. Or in minutes: 1{,}200 \div 60 = 20 minutes, and 20 - 15 = 5.
Each hour is the same amount of time as 60 minutes. So a number of hours is that many groups of 60 minutes, and counting equal groups is multiplying by 60. A minute is much shorter than an hour, so it takes more minutes than hours to fill the same time: every hour turns into 60 minutes, so the count of minutes is 60 times the count of hours, which is larger for any number of hours above 0.
5 hours. Place value counts in tens, but an hour is 60 minutes, not 100, so the hundreds digit does not count hours. Dividing by 60: 300 \div 60 = 5, with no remainder. Check: 5 \times 60 = 300.
Lowanna is right: 2.5 hours is 150 minutes. The 2 is 2 \times 60 = 120 minutes, and the .5 is half an hour, 30 minutes, so 120 + 30 = 150. Mert read the .5 as 50 minutes, which would only work if an hour had 100 minutes.
With 60 minutes: half is 60 \div 2 = 30, a third is 60 \div 3 = 20, a quarter is 60 \div 4 = 15, a fifth is 60 \div 5 = 12 and a sixth is 60 \div 6 = 10, all whole. With 100 minutes, half (50), a quarter (25) and a fifth (20) are whole, but a third and a sixth are not: 100 \div 3 = 33 \text{ R } 1 and 100 \div 6 = 16 \text{ R } 4.