Grade 4 · 4.MD.A.1

The customary system

Why are there 12 inches in a foot but 3 feet in a yard, and how do you know which of these units to use?

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Customary length unitsGrade 2

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Customary weight unitsGrade 4 Customary capacity unitsGrade 4

Where you are 6 in Systems of units

Measurement12

Systems of units6

The customary system

The metric system and SI units Metric prefixes Conversion factors and dimensional analysis Converting square and cubic units Multi-step problems with unit conversions

Why this exists

Gulsen works at a pet store. On one shelf she finds a leash marked 6 feet, a bag of dog food marked 30 pounds, and a fish tank marked 20 gallons. Three different words, because the three labels measure three different things: how long, how heavy, and how much a container holds.

A number alone tells you nothing. “The leash is 6” could mean 6 inches, far too short to walk a dog, or 6 miles, far too long to hold. A measurement is a number together with a unit, and the unit says what one of the counted things is. The units used every day in the United States form a family called the customary system. This lesson meets the whole family, gets a feel for how big each unit is, and asks why the numbers that tie the units together are so uneven.

This lesson takes two days. Day 1 (Examples 1 to 3) is the units, how big they are, and which one suits which job. Day 2 (Examples 4 to 7) is how the units compare in size, and why the steps between them are uneven.

Definition

Day 1: three families of units

The customary system is the set of units for length, weight and capacity used in everyday life in the United States. Inside each family, every unit is an exact number of the next smaller one. The first column is the short way to write each unit.

Length (how long, tall or far). You met these in Customary length units.

unit	short form	equals
inch	\text{in}	the smallest unit here
foot	\text{ft}	12 inches
yard	\text{yd}	3 feet
mile	\text{mi}	1{,}760 yards, which is 5{,}280 feet

Weight (how heavy).

unit	short form	equals
ounce	\text{oz}	the smallest unit here
pound	\text{lb}	16 ounces
ton	(none)	2{,}000 pounds

Capacity (how much a container holds, also called liquid volume).

unit	short form	equals
fluid ounce	\text{fl oz}	the smallest unit here
cup	\text{c}	8 fluid ounces
pint	\text{pt}	2 cups
quart	\text{qt}	2 pints
gallon	\text{gal}	4 quarts

These facts are exact, not rough guesses.

A unit only makes sense beside the units of its own family. You can ask how many feet are in a mile. You cannot ask how many pounds are in a gallon: one measures heaviness and the other measures room in a container.

Intuition

Day 1: a picture for each unit

To feel how big a unit is, keep an everyday object in mind that is about that size, called a benchmark. A benchmark is “about”, never exact, but it lets you judge whether a measurement makes sense.

unit	about the size of
inch	the width of a quarter coin, which is just under an inch across
foot	a school ruler, which is exactly one foot long
yard	a yardstick; a kitchen counter is about a yard high
mile	about four laps of a school running track, or about 20 minutes of steady walking
ounce	a slice of bread
pound	a loaf of bread
ton	a small car weighs between one and two tons
cup	a measuring cup, or a small glass of water
pint	a small carton of ice cream
quart	a carton of milk from the dairy aisle
gallon	a big plastic jug of milk

Choosing a unit. Each family has small units for small things and big units for big things. The best unit is the one closest in size to the thing being measured, so the number is easy to picture. A goldfish is a few inches long, a leash a few feet, a trip to the store a few miles.

Bigger unit, smaller number. Measure one leash in feet and again in inches. It takes more of the smaller unit to cover it, so the number in inches is bigger.

Results

Day 2: comparing the units in a family

Within a family the units line up from smallest to largest, and each step up is an exact whole number of the step below:

\begin{aligned} \text{in} &\xrightarrow{\times 12} \text{ft} \xrightarrow{\times 3} \text{yd} \\ \text{yd} &\xrightarrow{\times 1{,}760} \text{mi} \end{aligned}

\text{oz} \xrightarrow{\times 16} \text{lb} \xrightarrow{\times 2{,}000} \text{ton}

\begin{aligned} \text{fl oz} &\xrightarrow{\times 8} \text{c} \xrightarrow{\times 2} \text{pt} \\ \text{pt} &\xrightarrow{\times 2} \text{qt} \xrightarrow{\times 4} \text{gal} \end{aligned}

Read an arrow as “this many of the unit on the left make one of the unit on the right”. So the second arrow says 3 feet make a yard.

Skipping steps multiplies. To know how many of a small unit fit in a bigger one several steps up, multiply the numbers on the arrows between them. A yard is 3 feet and each foot is 12 inches, so a yard is 3 \times 12 = 36 inches.

Figure 1One yard is 3 feet, and each foot is 12 inches, so the yard is 3 \times 12 = 36 inches.

In the same way a quart is 2 \times 2 = 4 cups, and a pint is 2 \times 8 = 16 fluid ounces. Changing a whole amount from one unit to another, such as 7 yards into feet, is the next step, taught in Customary conversions.

The steps are uneven. In the metric system, every step is 10, 100 or 1{,}000: a meter is 100 centimeters, a kilometer is 1{,}000 meters (see The metric system and SI units). In the customary system the steps are 12, 3, 1{,}760, 16, 2{,}000, 8, 2 and 4. No single number runs through the family, so each fact has to be known on its own.

Why it is true

Why skipping steps multiplies. A yard is 3 feet, and each of those feet is 12 inches. So a yard is 3 groups of 12 inches, which is 3 \times 12 inches. Every longer chain works the same way: groups of groups is multiplication.

Why the steps are uneven: the units were not designed together. Most customary units are much older than the system they belong to now. They grew up in markets, farms and workshops long ago, in England and before that in Rome, and for centuries their sizes varied from town to town. Old stories tie them to everyday things. One story says the foot began as the length of a person’s foot. An old English tradition says an inch is three grains of barley laid end to end. The word mile comes from the Latin word for a thousand, and the usual story is that a Roman mile was a thousand double steps of a marching soldier. These are traditions about where the names and rough sizes came from, not how the units are fixed now.

Over time, laws fixed each unit to one exact size, and the units were tied together afterward, using whatever numbers the old sizes happened to give. The usual account of the mile is a good example. English farmers measured land in furlongs, said to be the length of one plowed furrow, 660 feet. A law set the mile at 8 furlongs, and 8 \times 660 = 5{,}280. Today every customary unit is defined exactly from a metric one: an inch is exactly 2.54 centimeters.

The metric system is the opposite case. It was planned on purpose, in France over two hundred years ago, so that every step would be a ten, a hundred or a thousand.

Why some uneven numbers are handy anyway. The odd-looking numbers are not random for everyday work.

12 can be split into equal whole parts in many ways: halves, thirds, quarters and sixths. A foot cut into 3 equal pieces gives 4 inches each. Ten splits evenly only into halves and fifths.
16 = 2 \times 2 \times 2 \times 2, so a pound can be halved four times and still come out in whole ounces. One reason often given for this: a market balance scale can split anything into two equal halves with no weights at all.
The capacity steps are mostly 2: a quart is two pints, a pint two cups. A liquid is easy to halve by pouring it into two matching containers. The name quart comes from quarter, since a quart is a quarter of a gallon.

Worked examples

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

Example 1

Choosing units for shelf tags

Gulsen writes tags for five things in the store: the length of a leash, the weight of a big bag of dog food, how much water a fish tank holds, the length of a goldfish, and the weight of a hamster. Which family and which unit suits each tag?

Solution

First decide what is measured, because that picks the family. Then pick the unit closest in size to the thing.

A leash’s length is a length. A leash is longer than a ruler but shorter than a car, so feet suit it: a tag of 6 feet. In inches the same leash is 6 \times 12 = 72 inches, which is true but harder to picture.
The dog food’s weight is a weight. A big bag weighs many loaves of bread, so pounds suit it: a tag of 30 pounds.
How much water the tank holds is a capacity. A tank holds many milk jugs, so gallons suit it: a tag of 20 gallons.
A goldfish’s length is a length, but a goldfish is only a few quarter coins long, so inches suit it: about 2 inches.
A hamster’s weight is a weight, but a hamster is lighter than a loaf of bread, so ounces suit it: about 5 ounces.

The family follows from what is measured; the unit follows from the size.

Example 2

Tags that cannot be right

Danylo finds three tags with the wrong unit. A puppy collar says it is 12 feet around. A bag of cat litter says it weighs 25 tons. A hamster’s water bottle says it holds 8 quarts. Fix each one by changing only the unit.

Solution

Compare each tag with a benchmark.

12 feet is 4 yards, longer than a couch. No puppy’s neck is that big. A puppy’s neck is about as long as a ruler, so the collar is 12 inches around.
A small car weighs between one and two tons, so 25 tons is heavier than ten small cars. A bag a person can carry is measured in pounds: 25 pounds.
8 quarts is 8 cartons of milk, which is 2 gallons, a bucketful. A hamster bottle holds about a cup, and a cup is 8 fluid ounces, so the tag means 8 fluid ounces.

Each time the number was fine and the unit was far off.

Example 3

One leash, three numbers

Themba unrolls a long training leash. It is 5 yards long. Write its length in feet and in inches. Which of the three numbers is largest, and why?

Solution

Each yard is 3 feet, so 5 yards is 5 groups of 3 feet:

5 \times 3 = 15 \text{ feet}

Each of those feet is 12 inches, so 15 feet is 15 groups of 12 inches:

15 \times 12 = 180 \text{ inches}

So the leash is 5 yards, 15 feet, or 180 inches. The number in inches is largest, because the inch is the smallest of the three units: it takes the most of them to cover the same leash. The number in yards is smallest, because the yard is the biggest unit. Check: a yard is 36 inches, and 5 \times 36 = 180.

Example 4

How many inches in a mile

How many inches are in one mile? Find it two ways, once going through feet and once going through yards.

Solution

Through feet. A mile is 5{,}280 feet, and each foot is 12 inches. Multiply the two steps:

5{,}280 \times 12 = 63{,}360 \text{ inches}

Through yards. A mile is 1{,}760 yards, and each yard is 3 \times 12 = 36 inches:

1{,}760 \times 36 = 63{,}360 \text{ inches}

Both ways give 63{,}360, as they must, since it is the same mile. A kilometer, by contrast, is 1{,}000 \times 100 = 100{,}000 centimeters: a 1 and zeros, because every metric step is a ten.

Example 5

Halving a pound of fish flakes

Gulsen has 1 pound of fish flakes and a balance scale. She pours the flakes into two bags until the scale is level, then halves each bag the same way, and keeps going until each bag holds 1 ounce. How many rounds of halving does she need, and how many bags does she end with?

Solution

A pound is 16 ounces. Each round of halving doubles the number of bags and halves what each bag holds:

after round	bags	each bag holds
0	1	16 ounces
1	2	8 ounces
2	4	4 ounces
3	8	2 ounces
4	16	1 ounce

Figure 2Halve a pound four times and each piece is one ounce. That is why 16 ounces make a pound.

She needs 4 rounds and ends with 16 bags of 1 ounce. This works out evenly because 16 = 2 \times 2 \times 2 \times 2. If a pound were 10 ounces, the second round would already ask her to split 5 ounces in half, which is \displaystyle 2\frac{1}{2} ounces, not a whole number.

Example 6

Cutting a foot of fleece

Danylo cuts a 1-foot strip of fleece into equal pieces to braid into cat toys. If he cuts it into 2, 3, 4, 5 or 6 equal pieces, how long is each piece? Which ways give whole inches?

Solution

A foot is 12 inches, so each piece is 12 divided by the number of pieces:

\begin{aligned} 12 \div 2 &= 6, & 12 \div 3 &= 4, \\ 12 \div 4 &= 3, & 12 \div 6 &= 2 \end{aligned}

These all come out whole, because 2, 3, 4 and 6 are factors of 12. Five pieces do not: 12 \div 5 = 2 remainder 2, so each piece is \displaystyle 2\frac{2}{5} inches.

Figure 3A foot splits evenly into halves, thirds, quarters and sixths. Only fifths miss the inch marks.

So halves, thirds, quarters and sixths of a foot are whole inches, and fifths are not. If a foot were 10 inches instead, a third of a foot would be \displaystyle 10 \div 3 = 3\frac{1}{3} inches. That is a real advantage of 12 for cutting and building: it has more factors than 10.

Example 7

Ordering four tank sizes

Themba compares four small fish tanks. They hold 3 gallons, 10 quarts, 20 pints and 50 cups. Put them in order from least to greatest.

Solution

The units are different, so compare them all in one unit. The cup is the smallest of the four, so every amount becomes a whole number of cups. Chain the steps: a pint is 2 cups, a quart is 2 \times 2 = 4 cups, and a gallon is 4 \times 4 = 16 cups.

3 gallons is 3 \times 16 = 48 cups.
10 quarts is 10 \times 4 = 40 cups.
20 pints is 20 \times 2 = 40 cups.
50 cups is 50 cups.

So 10 quarts and 20 pints hold exactly the same amount, 40 cups, and the order from least to greatest is

10 \text{ qt} = 20 \text{ pt} < 3 \text{ gal} < 50 \text{ c}

Boundaries

United States units only. Britain has units with the same names and different sizes. Its imperial gallon and pint are larger than the United States ones; an imperial pint is 20 imperial fluid ounces. The ton here is the United States ton of 2{,}000 pounds; the British long ton is 2{,}240 pounds, and the metric ton is 1{,}000 kilograms.
Ounce and fluid ounce are different units. The ounce measures weight and the fluid ounce measures capacity. They share a name, but 8 ounces of fish flakes and 8 fluid ounces of water are different kinds of amount.
Compare only inside one family. A gallon cannot be compared with a pound, since one is room in a container and the other is heaviness. How much a gallon weighs depends on what is in it.
Benchmarks are approximate; unit facts are exact. A slice of bread is about an ounce, and slices vary. A pound is exactly 16 ounces, always.
Time is uneven too, but belongs to both systems. An hour is 60 minutes and a day is 24 hours in the customary and the metric system alike; see Units of time.

Where this goes next

Next, Customary conversions changes a whole amount from one unit to another, such as 4 pounds into ounces or 54 inches into feet and inches, using the facts in this lesson. Customary weight units and Customary capacity units look more closely at the weight family and the capacity family. Later, Multi-step problems with unit conversions solves problems where a conversion is one step among several. The metric system and SI units is the other family of units, built on tens, and Conversion factors and dimensional analysis carries amounts between the two systems.

Exercises

Day 1: the units and how big they are

Is each unit a length, a weight or a capacity: quart, ton, yard, ounce, pint, mile?
Put these in order from shortest to longest: yard, inch, mile, foot.
Put these in order from least to most: gallon, cup, quart, pint.
Choose the best unit for each: (a) the length of a guinea pig; (b) the weight of a large bag of birdseed; (c) the water in a cat’s drinking bowl; (d) the distance Danylo drives to work; (e) the weight of one dog treat; (f) the water in a pond.
Each tag has the wrong unit. Fix it by changing only the unit: (a) a parakeet weighs 1 pound; (b) a dog collar is 16 feet around; (c) a bag of birdseed weighs 20 tons.
Choose the sensible measurement: (a) a leash: 6 inches, 6 feet or 6 miles; (b) a puppy: 8 ounces, 8 pounds or 8 tons; (c) a large fish tank: 55 cups, 55 pints or 55 gallons; (d) a guinea pig’s length: 10 inches, 10 feet or 10 yards.
Danylo walks his dog around a school track, and 4 laps is about a mile. About how many laps is 2 miles? About how many laps is half a mile?
Suppose each slice of a loaf of bread weighs 1 ounce. How many slices make 1 pound? How many make 2 pounds?
A hamster cage is 2 feet long. How many inches long is it? Which number is bigger, the one in feet or the one in inches, and why?
Themba says a bag of fish food marked 8 ounces and a bottle of tank cleaner marked 8 fluid ounces are the same kind of amount. Is he right?

Day 2: comparing units

How many feet are in a mile? How many yards? Use the two answers to find how many feet are in a yard, and check it against the table.
How many ounces are in one ton?
How many fluid ounces are in one quart? In one gallon?
How many cups are in 3 quarts? How many cups are in a half gallon?
(a) Start with 1 gallon and halve it 3 times. Which unit do you reach, and how many of them make a gallon? (b) Halve a pound 3 times. How much does each piece weigh, and how many pieces are there?
A yard of ribbon is cut into equal pieces. For 2, 3, 4, 5, 6, 8 and 9 pieces, say which give a whole number of inches per piece, and give each length.
Gulsen splits 1 pound of cat treats into 2, 4 or 8 equal bags. How much is in each bag? What happens with 3 bags?
Which is more? (a) 2 yards or 70 inches; (b) 1 mile or 5{,}000 feet; (c) 3 pounds or 50 ounces; (d) 5 pints or 3 quarts.
Put these bags of dog treats in order from lightest to heaviest: 2 pounds, 30 ounces, 36 ounces, \displaystyle 2\frac{1}{4} pounds.
Put these leashes in order from shortest to longest: 2 yards, 66 inches, 5 feet, 70 inches.
A meter is 100 centimeters and a centimeter is 10 millimeters. How many millimeters are in a meter? How many inches are in a yard? Which was quicker to work out, and why?

Reasoning

Use the furlong story to explain why a mile is 5{,}280 feet and not a round number such as 5{,}000.
Gulsen says the word quart hides a fraction. Which fraction, and a fraction of what?
Themba says metric is better because every step is a ten. Gulsen says 12 is better for cutting things up. Explain what is right in each view.
From cups up to gallons, every capacity step is \times 2 except one. Which step is it, and what unit could you put in the middle so that every step is \times 2?

Answers

Quart: capacity. Ton: weight. Yard: length. Ounce: weight. Pint: capacity. Mile: length.
Inch, foot, yard, mile. Each is a whole number of the one before: 12 inches make a foot, 3 feet a yard, 1{,}760 yards a mile.
Cup, pint, quart, gallon. A pint is 2 cups, a quart 2 pints and a gallon 4 quarts.
(a) Inches, since a guinea pig is about as long as a ruler or less. (b) Pounds. (c) Cups. (d) Miles. (e) Ounces, since a treat is far lighter than a loaf of bread. (f) Gallons.
(a) 1 ounce: a pound is a whole loaf of bread, far heavier than a small bird. (b) 16 inches: 16 feet is longer than a couch. (c) 20 pounds: 20 tons is heavier than ten small cars.
(a) 6 feet. (b) 8 pounds. (c) 55 gallons. (d) 10 inches. Each of the others is far too small or far too big next to its benchmark.
About 8 laps for 2 miles, since 2 \times 4 = 8. About 2 laps for half a mile, since half of 4 is 2.
16 slices make 1 pound, since a pound is 16 ounces. 2 pounds take 2 \times 16 = 32 slices.
24 inches, since 2 \times 12 = 24. The number in inches is bigger, because an inch is a smaller unit and more of them fit along the same cage.
No. The ounce measures weight and the fluid ounce measures capacity, how much room the liquid takes up. They share a name but are different kinds of amount.
5{,}280 feet and 1{,}760 yards. Then 5{,}280 \div 1{,}760 = 3, so a yard is 3 feet, which matches the table.
32{,}000 ounces. A ton is 2{,}000 pounds, each of 16 ounces, and 2{,}000 \times 16 = 32{,}000.
32 fluid ounces in a quart, since a quart is 4 cups and 4 \times 8 = 32. 128 fluid ounces in a gallon, since a gallon is 4 quarts and 4 \times 32 = 128.
A quart is 2 \times 2 = 4 cups, so 3 quarts is 3 \times 4 = 12 cups. A half gallon is 2 quarts, so 2 \times 4 = 8 cups.
(a) Half a gallon is 2 quarts, half again is a quart, half again is a pint. So you reach the pint, and 2 \times 2 \times 2 = 8 pints make a gallon. (b) Each piece is 2 ounces, since 16 \div 2 \div 2 \div 2 = 2, and there are 8 pieces.
A yard is 36 inches. Whole inches: 2 pieces of 18 inches, 3 of 12, 4 of 9, 6 of 6 and 9 of 4. Not whole: 5 pieces of \displaystyle 7\frac{1}{5} inches and 8 pieces of \displaystyle 4\frac{1}{2} inches, since 5 and 8 are not factors of 36.
8 ounces, 4 ounces and 2 ounces, since 16 \div 2 = 8, 16 \div 4 = 4 and 16 \div 8 = 2. With 3 bags, 16 \div 3 = 5 remainder 1, so each bag holds \displaystyle 5\frac{1}{3} ounces, not a whole number, because 3 is not a factor of 16.
(a) 2 yards, since 2 \times 36 = 72 inches, more than 70. (b) 1 mile, since it is 5{,}280 feet. (c) 50 ounces, since 3 pounds is 3 \times 16 = 48 ounces. (d) 3 quarts, since 3 quarts is 3 \times 2 = 6 pints, more than 5.
30 ounces, then 2 pounds, then 36 ounces and \displaystyle 2\frac{1}{4} pounds, which are equal. In ounces: 2 pounds is 32, and \displaystyle 2\frac{1}{4} pounds is 32 + 4 = 36.
5 feet, 66 inches, 70 inches, 2 yards. In inches: 5 feet is 60 and 2 yards is 72.
1{,}000 millimeters in a meter, since 100 \times 10 = 1{,}000. 36 inches in a yard, since 3 \times 12 = 36. The metric one is quicker: multiplying by 10 and 100 only adds zeros, while 3 \times 12 is a real multiplication.
In the usual account, a furlong was 660 feet, the length of a plowed furrow, and a law set the mile at 8 furlongs. So a mile is 8 \times 660 = 5{,}280 feet. The mile was built from an old farm unit that was not designed to make round numbers, so the result is not round.
A quarter. A quart is \displaystyle \frac{1}{4} of a gallon, since 4 quarts make a gallon.
Themba is right that tens make changing units easy: moving between metric units only shifts digits. Gulsen is right that 12 splits into equal whole parts in more ways: halves, thirds, quarters and sixths of a foot are all whole inches, while 10 splits evenly only into halves and fifths. Each system is good at a different job.
Quart to gallon, which is \times 4. Put the half gallon between them: 2 quarts make a half gallon and 2 half gallons make a gallon, so every step from cup to gallon is \times 2.