Grade 3 · 4.MD.A.1 · 5.MD.A.1

Metric prefixes

Kilo, centi and milli turn up in front of meter, gram and liter. What does each little word part say about the size of a unit, and why does it say the same thing every time?

You need

The metric system and SI unitsGrade 3 Place valueGrades 1 to 5

Unlocks

Converting metric units by powers of tenGrade 4 Liquid volume: liters and millilitersGrade 3 Metric mass unitsGrade 3

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

Sanjar signs up for a bike race. The poster says the course is 4 km. Inaya reads the label on her cat’s eye drops: “1 mL”. Nilufar looks at the marks up the side of her baby brother’s bottle: they are in mL too. The letters m, L, k keep turning up in new mixes, and nobody memorizes every mix one at a time. They do not have to.

In each unit name, a short word part in front says how big, and the base unit after it says what is measured. There are only a few of these front parts, and each one always means the same size. Learn kilo-, centi- and milli- and you can read units you have never seen before.

The metric system and SI units set out the base units and the steps of ten between metric units. This lesson looks closely at the front parts themselves. It takes two days. Day 1 (Examples 1 to 3) takes kilo-, centi- and milli- one at a time and reads each on a place value chart. Day 2 (Examples 4 to 7) puts the same front part beside different base units, sorts out which units people really use, meets the rest of the family, and uses the prefixes inside a problem.

Definition

Day 1: three prefixes

A prefix is a word part put at the front of a word to change what it means, like re- in redo. In a metric unit the prefix names a size, and the base unit (meter, gram, liter) names what is measured.

kilo- means 1{,}000 of the base unit. A kilometer is 1{,}000 meters. Short form: k, as in km.
centi- means one of 100 equal parts of the base unit. Cut a meter into 100 equal parts and each part is a centimeter. Short form: c, as in cm.
milli- means one of 1{,}000 equal parts of the base unit. Cut a liter into 1{,}000 equal parts and each part is a milliliter. Short form: m, as in mL and mm.

The words are old. Kilo- comes from the Greek word for a thousand. Centi- comes from the Latin word for a hundred, the same word that gives a cent (100 cents make a dollar) and a century (100 years). Milli- comes from the Latin word for a thousand; the Roman mile was a thousand paces.

Kilo- makes a unit bigger than the base unit, by putting many base units together. Centi- and milli- make a unit smaller, by cutting one base unit into equal parts.

Day 2: the whole family

The same prefix can stand in front of any metric base unit, including the second. Between the three main prefixes sit three more, seldom used:

prefix	short form	size
kilo-	k	1{,}000 of the base unit
hecto-	h	100 of the base unit
deka-	da	10 of the base unit
(none)		the base unit itself
deci-	d	one of 10 equal parts
centi-	c	one of 100 equal parts
milli-	m	one of 1{,}000 equal parts

Past both ends the family keeps going, in bigger jumps. Mega- means a million of the base unit, and micro- means one of a million equal parts. Those are for science and computers, not for everyday measuring.

Intuition

Day 1: a prefix on a place value chart

Place value in one line (Place value): where a digit sits decides what it is worth, ones, tens, hundreds or thousands. A prefix tells you where the 1 lands when you count a unit in its base unit, or count a base unit in its parts.

Figure 1Kilo- and milli- put the first digit in the thousands place. Centi- puts it in the hundreds place.

Count one kilometer in meters and you get 1 thousand: the 1 sits in the thousands place. Four kilometers are 4 thousands of meters, 4{,}000 meters. Count one meter in centimeters and you get 1 hundred: the 1 sits in the hundreds place, because centi- cuts into 100 parts. Count one liter in milliliters and the 1 is back in the thousands place, because milli- cuts into 1{,}000 parts.

Figure 2Centi- means one of 100 equal parts. One part of the meter is a centimeter.

A centimeter is easy to see on a meter stick. The stick is cut by marks into 100 equal parts. One part is a centimeter, about the width of a small fingernail. The longer marks fall every 10 parts, so the meter is also 10 groups of 10 centimeters.

Day 2: a prefix works like the word dozen

A dozen eggs is 12 eggs. A dozen pencils is 12 pencils. The word dozen says 12 and nothing about eggs or pencils; the word after it says what you have 12 of. A prefix works the same way. Kilo- says 1{,}000 and nothing about length or mass; the base unit after it says what there are 1{,}000 of. So a kilogram is 1{,}000 grams and a kiloliter is 1{,}000 liters, without anyone having to tell you.

Cutting works the same way. Milli- says “cut the base unit into 1{,}000 equal parts”, whatever the base unit is. A millimeter is one of 1{,}000 equal parts of a meter, a milligram one of 1{,}000 equal parts of a gram, a millisecond one of 1{,}000 equal parts of a second.

Results

Day 1 and Day 2

> Kilo-: 1 kilo-unit is 1{,}000 of the base unit. > > Centi-: 100 centi-units make 1 base unit. > > Milli-: 1{,}000 milli-units make 1 base unit. > > Each prefix means the same size beside every base unit.

Every pair can be made, but only some are used. The everyday ones:

Beside the meter: kilometer (roads, races), centimeter (rulers, small lengths) and millimeter (very small lengths, such as a tire’s width) are all common.
Beside the gram: kilogram (people, bags of food) and milligram (the medicine in a pill) are common; the centigram is almost never used.
Beside the liter: milliliter (medicine, drink bottles) is common; the kiloliter is seldom used, and the centiliter is seldom seen in the United States.
Beside the second: the millisecond times a close race; kilo- and centi- are seldom used.

Why it is true

Why the same prefix means the same size. The prefixes were set up as number words, not as units. When the metric system was planned, one list of prefixes was made for every base unit at once, so kilo- meant 1{,}000 whatever unit it was put in front of, a meter or a gram alike. A word built from a prefix and a base unit means “the prefix’s size, of the base unit”. That is why the dozen picture works: the size word and the thing word do separate jobs.

Why a milli-unit is smaller than a centi-unit. Take one meter. Centi- cuts it into 100 equal parts; milli- cuts the same meter into 1{,}000 equal parts. The more equal parts one whole is cut into, the smaller each part has to be, the same way a pizza cut into 8 slices gives smaller slices than one cut into 4. So a millimeter is smaller than a centimeter, even though 1{,}000 is more than 100.

How many millimeters fit in a centimeter. The 100 centimeters in a meter and the 1{,}000 millimeters in a meter fill the same meter. Share the 1{,}000 millimeters equally among the 100 centimeters: since 100 \times 10 = 1{,}000, each centimeter holds 10 millimeters. On a ruler, the small marks between two centimeter marks cut each centimeter into 10 millimeters.

Why the chart works. Kilo- puts 1{,}000 base units together, and 1{,}000 is one thousand, a 1 in the thousands place. So any whole number of kilometers, written in meters, is that many thousands: the digit goes in the thousands place and zeros fill the hundreds, tens and ones.

Worked examples

Day 1 is Examples 1 to 3. Day 2 is Examples 4 to 7.

Example 1

A bike race in meters

Sanjar’s bike race is 4 km long. The signs along the course count the distance in meters. What does the sign at the finish line say? What does the sign say at the point where the riders have gone exactly 1 kilometer?

Solution

The prefix kilo- means 1{,}000 of the base unit, and the base unit here is the meter. So each kilometer is 1{,}000 meters, and the sign at the 1-kilometer point says 1{,}000 meters.

The whole race is 4 kilometers, which is 4 groups of 1{,}000 meters, or 4 thousands of meters. On a place value chart, 4 thousands put a 4 in the thousands place with zeros in the other places, as in the second row of Figure 1:

4 \text{ thousands} = 4{,}000

The finish sign says 4{,}000 meters. The number got bigger because the meter is the smaller unit: it takes many meters to cover the same road.

Example 2

A meter stick in a hundred parts

Inaya’s meter stick is cut by marks into 100 equal parts. What is one part called, and why? She lays two meter sticks end to end along the starting line of the bike race. How many of the small parts run along the two sticks?

Solution

Centi- means one of 100 equal parts of the base unit. The base unit is the meter, cut into 100 equal parts, so one part is a centimeter (Figure 2). It is the same idea as a cent: 100 equal cents make one dollar, and 100 equal centimeters make one meter.

Each stick has 100 parts, so the two sticks have

100 + 100 = 200

small parts. The line is 200 centimeters long. The number 100 is large, yet a centimeter is a small unit: the 100 counts how many parts the meter was cut into, so the bigger that count, the smaller each part.

Example 3

A full eye dropper

Inaya gives her cat eye drops from a small dropper. Squeezed full, the dropper holds about 1 milliliter. About how many full droppers would it take to fill a 1-liter bottle? Why is the milliliter a sensible unit for the dropper?

Solution

Milli- means one of 1{,}000 equal parts of the base unit, here the liter. So 1{,}000 milliliters make one liter, and it would take about 1{,}000 full droppers to fill the bottle. On the place value chart (Figure 1, last row) one liter counted in milliliters puts a 1 in the thousands place.

A full dropper is a tiny amount next to a liter. The liter is far too big a unit for it, and the milliliter, about the size of the dropper, gives the easy number 1.

The prefix matters. If the liter were cut into only 100 equal parts (centiliters), each part would be bigger than a milliliter: the 1{,}000 milliliters shared equally among 100 parts is 10 milliliters in each, since 100 \times 10 = 1{,}000. That is about 10 full droppers in one centiliter.

Example 4

Baby bottles in a liter

Figure 3Four bottles of 250 milliliters fill one liter.

Nilufar helps make her baby brother’s milk. Each bottle holds 250 milliliters. How many bottles can she fill from a 1-liter carton?

Solution

The carton is 1 liter, and milli- says 1{,}000 milliliters make a liter, so the carton holds 1{,}000 milliliters. Fill bottles one at a time and add up:

250 + 250 = 500, \qquad 500 + 500 = 1{,}000

Two bottles make 500 milliliters, and two more make 1{,}000. So 4 bottles fill one liter, as Figure 3 shows: the liter is cut into 4 equal parts of 250 milliliters each. The carton is used up exactly, with nothing left over.

The label on the carton could have said 1{,}000 mL instead of 1 L. Both name the same amount; the bigger unit simply needs the smaller number.

Example 5

Units you have never met

Sanjar finds an old science book that uses three units he has never seen: a hectometer, a dekaliter and a decimeter. His race results also list times in milliseconds. Use the prefix family to say what each unit is.

Solution

Split each name into a prefix and a base unit, then read the prefix’s size from the family table.

Hectometer: hecto- means 100 of, and the base unit is the meter, so a hectometer is 100 meters.
Dekaliter: deka- means 10 of, and the base unit is the liter, so a dekaliter is 10 liters.
Decimeter: deci- means one of 10 equal parts, and the base unit is the meter, so 10 decimeters make 1 meter.
Millisecond: milli- means one of 1{,}000 equal parts, and the base unit is the second, so 1{,}000 milliseconds make 1 second.

None of these needed to be learned on its own. The prefix gave the size and the base unit gave what is measured. A millisecond is very short; race timers use it when two riders cross the line almost together.

Example 6

Choosing the prefix

For race day, Sanjar wants a unit for each of these: (a) the length of the course, (b) the mass of his bike, (c) the width of his bike tire, (d) the water in his water bottle. Choose a prefix and a base unit for each.

Solution

First the base unit, from what is measured. Then the prefix that makes the unit close in size to the thing, so the number is neither huge nor less than 1.

(a) A length, so meters. A course is thousands of meters, so kilo- suits it: kilometers.
(b) A mass, so grams. A bike has the mass of thousands of grams, so kilo- suits it again: kilograms.
(c) A length, so meters, but a tire is only about as wide as a thumb, far less than a meter. Milli- cuts the meter into the smallest everyday parts: millimeters. (Centimeters would also work; tire makers often print the width in millimeters on the tire.)
(d) A liquid volume, so liters. A water bottle holds less than a large liter bottle, so milli- suits it: milliliters.

The same prefix kilo- served a length and a mass, and milli- served a length and a liquid volume. The prefix picks the size; the base unit picks what is measured.

Example 7

Laps of the race

The 4-kilometer race in Example 1 is ridden around a loop that is 500 meters long. How many laps does each rider ride?

Solution

Lap lengths are in meters, so first read the race in meters. Kilo- means 1{,}000 meters, so each kilometer is 1{,}000 meters. Two laps cover

500 + 500 = 1{,}000 \text{ meters}

which is exactly 1 kilometer. Every kilometer of the race takes 2 laps, and there are 4 kilometers, so the race is

4 \times 2 = 8 \text{ laps}

Check by adding up the meters: 8 laps of 500 meters are 4 pairs of laps, each pair 1{,}000 meters, and 4 thousands of meters are 4{,}000 meters, the full race.

Boundaries

A prefix needs a base unit. Kilo- alone is a number word, 1{,}000 of nothing. In everyday talk “a kilo” is a shortened name for a kilogram.
One prefix to a unit. A unit never has two prefixes; nobody writes a kilokilometer. A thousand kilometers is written 1{,}000 km.
Only metric units take these prefixes. There is no kilofoot or centipound. The second takes them (millisecond), but the minute and the hour do not.
The letter m has two jobs. Alone, m is the meter. In front of another letter, m is milli-: mm is a millimeter and mL a milliliter. Capital M is a different prefix, mega-.
The parts must be equal. Centi- and milli- cut the base unit into equal parts. A meter cut into 100 pieces of different lengths does not make centimeters.
Outside the metric system the word can drift. In computer memory, a kilobyte has sometimes meant 1{,}024 bytes instead of 1{,}000. In metric units kilo- is exactly 1{,}000.

Where this goes next

The three families come next one at a time: Metric length units for millimeters to kilometers, Metric mass units for milligrams to kilograms, and Liquid volume: liters and milliliters for milliliters and liters. Converting metric units by powers of ten uses the prefix sizes to change any measurement from one metric unit to another, including amounts that are not whole thousands or hundreds. The parts named by centi- and milli- return later as the hundredths and thousandths places of a decimal, where the place value chart grows to the right of the ones.

Exercises

Day 1: kilo-, centi- and milli-

Split each unit into a prefix and a base unit, and say what the prefix means: (a) kilogram; (b) centimeter; (c) milliliter; (d) millimeter.
Fill in each blank. (a) 1 kilometer is ___ meters. (b) ___ centimeters make 1 meter. (c) ___ milliliters make 1 liter. (d) 1 kilogram is ___ grams.
Sanjar rides a 6-kilometer practice ride. How many meters is that? In which place on a place value chart does the 6 sit?
A sign on the bike path says the next town is 5{,}000 meters away. How many whole kilometers is that?
Inaya has a ribbon 1 meter long, marked into centimeters. She cuts off 30 centimeters. How many centimeters are left?
Inaya squeezes 10 full droppers, about 1 milliliter each, into a measuring cup. About how many milliliters are in the cup? How many more full droppers would fill 1 liter?
Write 1 meter in centimeters and 1 liter in milliliters. In which place does the digit 1 sit each time, and why are the two places different?
A dollar is 100 cents. How many centimeters make a meter? What do the words cent and centimeter share?

Day 2: one prefix, many base units

Nilufar’s smaller baby bottles hold 200 milliliters each. How many of them fill 1 liter?
A jug holds 1 liter of water. Nilufar pours out 3 small bottles of 200 milliliters each. How many milliliters are left in the jug?
Use the prefix family to fill each blank. (a) 1 hectogram is ___ grams. (b) 1 dekameter is ___ meters. (c) ___ deciliters make 1 liter. (d) ___ centigrams make 1 gram.
1{,}000 milliseconds make 1 second. How many milliseconds are in 3 seconds?
Put these in order from biggest to smallest: centimeter, kilometer, meter, millimeter, hectometer, decimeter.
Choose a prefix and a base unit that suit each: (a) the medicine in one spoonful; (b) the mass of a baby; (c) the length of a baby’s foot; (d) the distance a plane flies between two cities; (e) the thickness of a coin.
Sort these into units people use often and units almost never seen: kilometer, kilogram, kiloliter, centimeter, centigram, milligram.
A 3-kilometer bike race has a sign every 100 meters. Into how many 100-meter pieces do the signs split the course?
A track lap is 200 meters. How many laps make 1 kilometer? How many make a 2-kilometer race?
Nilufar’s baby brother has a mass of 5 kilograms. How many grams is that?

Reasoning

Explain why centi- means the same size in centimeter and in centiliter.
Is a unit with milli- in front always smaller than its base unit? Is a unit with kilo- in front always bigger? Explain.
A millimeter is smaller than a centimeter, even though 1{,}000 is more than 100. Explain why.
Use a place value chart to explain why 8 kilometers, written in meters, is 8{,}000 meters.
Why does nobody give the length of a bike race in millimeters, or the medicine in an eye dropper in liters?

Answers

(a) Kilo- and gram: 1{,}000 grams. (b) Centi- and meter: one of 100 equal parts of a meter. (c) Milli- and liter: one of 1{,}000 equal parts of a liter. (d) Milli- and meter: one of 1{,}000 equal parts of a meter.
(a) 1{,}000. (b) 100. (c) 1{,}000. (d) 1{,}000. Kilo- is 1{,}000 of the base unit; centi- cuts it into 100 parts and milli- into 1{,}000.
6{,}000 meters. Each kilometer is 1{,}000 meters, so 6 kilometers are 6 thousands of meters; the 6 sits in the thousands place.
5 kilometers. 5{,}000 is 5 thousands, and each thousand meters is 1 kilometer.
70 centimeters. The meter is 100 centimeters, and 100 - 30 = 70.
About 10 milliliters, since 10 droppers of about 1 milliliter each make 10 \times 1 = 10. A liter is 1{,}000 milliliters, so about 1{,}000 - 10 = 990 more droppers.
1 meter is 100 centimeters, with the 1 in the hundreds place. 1 liter is 1{,}000 milliliters, with the 1 in the thousands place. Centi- cuts into 100 parts and milli- into 1{,}000, so the parts are counted into different places.
100 centimeters make a meter. Both words come from the Latin word for a hundred: a cent is one of 100 equal parts of a dollar, and a centimeter is one of 100 equal parts of a meter.
5 bottles. Adding 200 five times: 200, 400, 600, 800, 1{,}000, and a liter is 1{,}000 milliliters.
400 milliliters. Three bottles take 200 + 200 + 200 = 600 milliliters, and 1{,}000 - 600 = 400.
(a) 100. (b) 10. (c) 10. (d) 100. Hecto- is 100 of, deka- is 10 of, deci- cuts into 10 equal parts and centi- into 100.
3{,}000 milliseconds. Each second is 1{,}000 milliseconds, so 3 seconds are 3 thousands.
Kilometer, hectometer, meter, decimeter, centimeter, millimeter. The units with a size of 1{,}000 and 100 of the meter come first, then the meter, then the parts from biggest (one of 10) to smallest (one of 1{,}000).
(a) Milliliters. (b) Kilograms. (c) Centimeters. (d) Kilometers. (e) Millimeters.
Used often: kilometer, kilogram, centimeter, milligram. Almost never seen: kiloliter, centigram.
30 pieces. One kilometer is 1{,}000 meters, which is 10 hundreds, so 3 kilometers are 3 \times 10 = 30 hundreds of meters.
5 laps make 1 kilometer, since 200 + 200 + 200 + 200 + 200 = 1{,}000 meters. A 2-kilometer race is 2 \times 5 = 10 laps.
5{,}000 grams. Kilo- is 1{,}000 of the base unit, so 5 kilograms are 5 thousands of grams.
Centi- is a size word: it means “one of 100 equal parts”, whatever comes after it. In centimeter the meter is cut into 100 equal parts; in centiliter the liter is. The cutting is the same; only the thing being cut changes.
Yes to both. Milli- cuts the base unit into 1{,}000 equal parts, and a part is smaller than the whole. Kilo- puts 1{,}000 base units together, which is more than one base unit.
Both cut the same meter. Milli- cuts it into 1{,}000 equal parts and centi- into only 100. The more equal parts one whole is cut into, the smaller each part is.
Each kilometer is 1{,}000 meters, so 8 kilometers are 8 thousands of meters. On the chart, 8 thousands put an 8 in the thousands place and zeros in the hundreds, tens and ones, which reads 8{,}000.
A millimeter is tiny next to a race, so the race would be a huge number of millimeters, hard to picture. A liter is huge next to one dropper, so the dose would be far less than 1 liter. A sensible unit is close in size to the thing: kilometers for a race, milliliters for a dropper.