Grade 3 · 4.MD.A.1 · 6.RP.A.3D

The metric system and SI units

Why does the same word, kilo, show up on a scale at the veterinarian’s and on a road sign, and what makes the metric units fit together so neatly?

You need

Metric length unitsGrade 2

Unlocks

Metric prefixesGrade 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

Lenka works at a veterinarian’s office. A beagle steps onto the floor scale and the screen reads 11 kg. On her drive home, a road sign says the next town is 11 km away. The two signs share a number and a letter k, and the k means the same thing on both: a thousand. Eleven kilograms is eleven thousands of grams. Eleven kilometers is eleven thousands of meters.

That shared k is the whole idea of the metric system. The customary units grew up one at a time, each with its own odd number to the next: 12 inches make a foot, 3 feet make a yard, 16 ounces make a pound. Each one has to be remembered on its own. The metric system was planned instead, as one system: one main unit for each thing you measure, every other unit made from it by tens, and the same short word naming the same size every time. Learn the pattern once and it works for length, mass and liquid volume alike.

This lesson takes two days. Day 1 (Examples 1 to 3) is the base units, which attribute each one measures, and how to choose a unit of a sensible size. Day 2 (Examples 4 to 7) is the pattern of tens that ties the units together, and how it compares with the customary units.

Definition

Day 1: one base unit for each attribute

An attribute is the feature of an object or event that you measure: how long it is, how much matter it has, how much liquid it holds, how long it takes. A unit is the amount you count in. The metric system is a system of units in which

each attribute has one base unit, and
every other unit for that attribute is the base unit made 10, 100 or 1{,}000 times bigger or smaller (or by more tens still), and is named by putting a prefix, a short word part, in front of the base unit’s name.

attribute	what it tells you	base unit	short form
length	how long, tall or far	meter	\text{m}
mass	how much matter, which is what makes a thing heavy	gram	\text{g}
liquid volume	how much a container holds	liter	\text{L}
time	how long something takes	second	\text{s}

Mass is what a balance or a scale measures. Everyday talk calls it weight, and Mass and weight explains how the two words differ.

The second is shared: the customary system and the metric system use the very same second, minute and hour. Time is the one attribute here whose bigger units do not go up by tens (see Boundaries).

The units met most often:

attribute	smaller units	base unit	bigger unit
length	millimeter (\text{mm}), centimeter (\text{cm})	meter (\text{m})	kilometer (\text{km})
mass	milligram (\text{mg})	gram (\text{g})	kilogram (\text{kg})
liquid volume	milliliter (\text{mL})	liter (\text{L})	kiloliter, rarely used

The length units are in Metric length units, mass in Metric mass units, and liquid volume in Liquid volume: liters and milliliters. The prefixes kilo-, centi- and milli- each mean one fixed size, the same beside every base unit; the next lesson, Metric prefixes, takes them one by one.

Intuition

Day 1: a size you can picture for each unit

To judge whether a measurement makes sense, keep an everyday object in mind that is about the size of each unit, called a benchmark. A benchmark is “about”, never exact.

unit	about the size of
millimeter	the thickness of a dime
centimeter	the width of your smallest fingernail
meter	a meter stick; one long step by an adult
kilometer	the distance you walk in 10 to 15 minutes
gram	a paper clip
kilogram	a full 1-liter bottle of water
milliliter	about twenty drops from a medicine dropper
liter	a large bottle of water

Choosing a unit takes two decisions, in this order. First, name the attribute. That picks the family: a length gets a unit made from the meter, a mass one made from the gram, a liquid volume one made from the liter. Second, inside that family, pick the unit closest in size to the thing. A unit far too small for the thing gives a huge number that is hard to picture; a unit far too big gives a number less than 1.

Results

Day 2: the pattern of tens

> One base unit for each attribute. Meter for length, gram for mass, liter for liquid volume, > and the second, shared with the customary system, for time. > > Every other unit is the base unit times or divided by tens. 10, 100 or 1{,}000, where > 100 = 10 \times 10 and 1{,}000 = 10 \times 10 \times 10. > > A prefix means the same size beside every base unit.

prefix	means	example
kilo-	1{,}000 of the base unit	a kilometer is 1{,}000 meters
centi-	one hundredth of the base unit: 100 of them make one base unit	100 centimeters make a meter
milli-	one thousandth of the base unit: 1{,}000 of them make one base unit	1{,}000 milliliters make a liter

The customary units do not share steps. Each step is its own number:

family	steps in the customary system
length	12 inches in a foot, 3 feet in a yard, 1{,}760 yards in a mile
weight	16 ounces in a pound, 2{,}000 pounds in a ton
liquid volume	2 cups in a pint, 2 pints in a quart, 4 quarts in a gallon

The customary lesson, The customary system, meets those units in full.

Figure 1The customary steps are 12, 3 and 1{,}760, three numbers with no pattern. The metric steps are 10, 100 and 1{,}000, all made of tens.

Units made from other units. Some units are built out of others. A square centimeter, the area of a square 1 centimeter on each side, is made from the centimeter; counting these squares to find an area is Measuring area with unit squares. The liter is made from length too: it is the space inside a box 10 centimeters on each side (Example 7). A speed in kilometers per hour is made from a length unit and a time unit. Scientists keep a short official list of base units (the meter, the kilogram, the second, and a few more for things such as electric current and temperature) and build every other unit from that list.

Why it is true

Why tens? Our way of writing numbers is built on tens: ten ones make a ten, ten tens make a hundred, ten hundreds make a thousand. Units that step by ten line up with those places (The place value chart). The same length can be written as

1 \text{ m} = 10 \text{ dm} = 100 \text{ cm} = 1{,}000 \text{ mm}

where \text{dm} is the decimeter, 10 centimeters, a unit that sits between the centimeter and the meter but is seldom used. Each step down, the digit 1 moves one place to the left. Compare 1 \text{ yd} = 3 \text{ ft} = 36 \text{ in}: the numbers 3 and 36 follow no place pattern, so each has to be learned on its own.

Why the same words? Because the system was designed all at once. It was planned in France in the 1790s. The meter was chosen so that 10{,}000{,}000 meters would reach from the North Pole to the equator. The gram was set as the mass of the water that fills a cube 1 centimeter on each side. The prefixes were chosen once, for every unit together, so kilo- had to mean the same thing in kilogram as in kilometer. Nobody chose the customary steps that way; 16 ounces in a pound and 3 feet in a yard come from separate old trade measures.

Because the gram was tied to water and a cube, a liter of water has a mass of about a kilogram (Example 7), which is why a full 1-liter water bottle is the benchmark for a kilogram.

Worked examples

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

Example 1

Three measurements at the veterinarian’s office

Lenka wants to record three things at work: the mass of a dog in for a checkup, how much water her large water bottle holds, and the length of the hallway from the waiting room to the exam rooms. For each, name the attribute, the metric base unit for it, and the unit that suits the size.

Solution

The attribute comes first, because it decides which family of units is even possible.

The dog. Its mass is the attribute, so the base unit is the gram. A gram is about a paper clip, and a dog has the mass of thousands of paper clips, so grams would give a huge number. The kilogram, a thousand grams, suits the size: a medium dog is somewhere around 10 to 25 kilograms.
The water bottle. How much it holds is a liquid volume, so the base unit is the liter. A large water bottle holds about 1 liter, which is just right, so the liter itself suits it.
The hallway. Its length is the attribute, so the base unit is the meter. One long step is about a meter, and a hallway is some number of long steps, so meters suit it. Centimeters would give a number in the hundreds or thousands; kilometers would give a number much less than 1.

thing	attribute	base unit	unit that suits it
dog	mass	gram	kilogram
water bottle	liquid volume	liter	liter
hallway	length	meter	meter

Example 2

Which number fits the chart

Lenka’s chart for one afternoon shows three numbers whose units were left off: a puppy’s mass of 2, a dose of medicine of 5, and a kitten’s mass of 600. Choose a sensible metric unit for each.

Solution

Name the attribute, then test each possible unit against a benchmark.

The puppy, 2. A mass, so grams or kilograms. Two grams is two paper clips, far too light for any puppy. Two kilograms is about two full 1-liter water bottles, a fair size for a small puppy. The unit is kilograms.
The medicine, 5. A liquid volume, so milliliters or liters. Five liters is five large water bottles, far more than a pet could swallow. Five milliliters is one teaspoon. The unit is milliliters.
The kitten, 600. A mass. Six hundred kilograms is about the mass of a cow, so that cannot be right. Six hundred grams is six hundred paper clips, less than one water bottle, a fair mass for a young kitten. The unit is grams.

The number 600 looks big, yet it goes with the smaller unit. A big number in a small unit and a small number in a big unit can describe the same amount.

Example 3

Planning a science fair project

Ondrej and Marketa are testing whether bean plants grow taller when given more water. For their science fair poster they will record: (a) the height of each plant every few days, (b) the water poured into each pot daily, (c) the mass of a handful of bean seeds before planting, and (d) how long it takes the water to soak into the soil each time. Choose an attribute and a metric unit for each.

Solution

(a) Height is a length, so a unit made from the meter. A young bean plant is a few fingernail widths to a hand tall, so centimeters suit it.
(b) Water poured is a liquid volume, so a unit made from the liter. A small pot takes much less than a large water bottle, so milliliters suit it.
(c) The seeds’ mass is a mass, so a unit made from the gram. A handful of seeds has the mass of a few paper clips to a few dozen, so grams suit it.
(d) Soaking time is a time, so the second. Water soaks into a pot in seconds, not hours.

A millimeter and a milliliter share a prefix but measure different attributes: the prefix tells the size, the base word tells the attribute.

Example 4

The same word kilo, twice

On Lenka’s scale the beagle has a mass of 11 kilograms; on the road sign the town is 11 kilometers away. Why does the same word kilo appear in both? And without being told, what should a kiloliter be?

Solution

Split each unit name into its two parts.

unit	prefix	base unit	meaning
kilogram	kilo-	gram	1{,}000 grams
kilometer	kilo-	meter	1{,}000 meters

The prefix kilo- means “1{,}000 of”. The base word says 1{,}000 of what. So kilo- appears in both because both units are a thousand of their base unit; the base words differ because one measures mass and the other length.

A kiloliter must therefore be 1{,}000 liters: the prefix supplies the 1{,}000 and the base word supplies the liter. A kiloliter is a real unit, though a rare one. Nothing in the customary word “pound” tells you it is 16 ounces.

Example 5

A meter against a yard

Figure 2A meter is a little longer than a yard, by about 3 inches.

At the science fair, Marketa lays a meter stick next to a yardstick. Which is longer, and by about how much? Then: Lenka runs a 100-meter race, and a football field is 100 yards between the goal lines. Which is longer?

Solution

A yard is 36 inches. A meter is about 39 inches, so the meter stick is longer, by about 3 inches. The figure draws both to the same scale; the shaded piece is the part of the meter past the end of the yard.

The race and the field both have 100 of their unit, so compare the units. Each meter is longer than each yard, and both have the same count of 100, so 100 meters is longer than 100 yards. Every meter adds its extra 3 inches or so, a hundred times over.

Example 6

Two ladders of length units

Marketa lists four length units in each system from smallest to largest, with the number of each unit that makes the next, as in Figure 1. Compare the steps. How many steps of ten lead from a millimeter to a meter, and what does that say about how many millimeters make a meter?

Solution

The customary ladder climbs by \times 12, then \times 3, then \times 1{,}760: three unrelated numbers. The metric ladder climbs by \times 10, then \times 100, then \times 1{,}000, and every one of these is made of tens:

100 = 10 \times 10, \qquad 1{,}000 = 10 \times 10 \times 10

Between the named units are units seldom used, so the ladder really has a rung at every ten. From the millimeter to the meter the rungs are millimeter, centimeter, decimeter, meter: three steps of ten. Three steps of ten make

10 \times 10 \times 10 = 1{,}000

so 1{,}000 millimeters make a meter. That matches the prefix milli-, one thousandth, exactly as the system promises.

Example 7

A box that holds a liter

Figure 3A box 10 centimeters on each side holds 10 layers of 100 centimeter cubes, 1{,}000 cubes in all. That space is exactly one liter.

For the science fair, Ondrej builds an open box 10 centimeters long, 10 centimeters wide and 10 centimeters tall, and fills it with centimeter cubes, each 1 centimeter on every side. How many cubes fit? What does this box have to do with a liter?

Solution

Fill the bottom layer first. It has 10 rows of 10 cubes, the same count as squares covering a 10 by 10 rectangle (Area of rectangles and squares):

10 \times 10 = 100 \text{ cubes in one layer}

The box is 10 centimeters tall, so 10 layers stack up:

100 \times 10 = 1{,}000 \text{ cubes}

The space inside a box 10 centimeters on each side is exactly 1 liter. The space of one centimeter cube is exactly 1 milliliter. So the 1{,}000 cubes show 1{,}000 milliliters filling 1 liter, the same “one thousandth” that milli- means everywhere. Filled with water, the box holds about 1 kilogram of water, since each centimeter cube of water has a mass of about 1 gram. Counting cubes in a box is the start of Volume with unit cubes.

Boundaries

Time does not go by tens. The second is the base unit of time in the metric system, but the minute is 60 seconds, the hour 60 minutes and the day 24 hours. These are older than the metric system and are used the same way everywhere, so they were kept.
The official base unit of mass is the kilogram. The prefixes attach to the word gram, so this lesson treats the gram as the base of the family. On the scientists’ official list, though, the base unit of mass is the kilogram, the only base unit whose name already has a prefix.
The liter is not on the official list. It is the space of a box 10 centimeters on each side, a unit made from the meter, kept because it is handy for liquids.
Units of different attributes never compare. A kilogram is not bigger or smaller than a kilometer. They measure different things.
Facts across the two systems are about, not exact steps. Inside the metric system, 1{,}000 grams make exactly a kilogram. A meter is about 39 inches, a kilogram is a little more than 2 pounds, and a liter is a little more than a quart: measured facts, not steps of tens.
A liter has a mass of about a kilogram only when it is water. A liter of honey has more mass; a liter of cooking oil has less.

Where this goes next

Metric prefixes takes kilo-, centi- and milli- one at a time, along with the less common prefixes that fill the gaps. Converting metric units by powers of ten uses the pattern of tens to change a measurement from one metric unit to another, and Story problems with grams, kilograms and liters solves story problems in grams, kilograms and liters. Units made from two others return as Rates from measurements, such as kilometers per hour, and as density, grams in each milliliter. Temperature has its own metric scale, the degree Celsius, met in Converting between Celsius and Fahrenheit.

Exercises

Day 1: attributes, base units and sensible sizes

Name the attribute (length, mass, liquid volume or time) and the metric base unit for each: (a) how much juice a pitcher holds; (b) how heavy a backpack is; (c) how far a paper airplane flies; (d) how long a song plays.
Choose the more sensible unit: (a) the mass of a cat, grams or kilograms; (b) the mass of a strawberry, grams or kilograms; (c) the water in a bathtub, milliliters or liters; (d) a dose of cough medicine, milliliters or liters.
Choose the more sensible unit: (a) the width of a postage stamp, centimeters or meters; (b) the distance between two towns, meters or kilometers; (c) the length of a swimming pool, meters or kilometers.
A pet rabbit is weighed at the veterinarian’s office. Is its mass 2 grams or 2 kilograms? Use a benchmark to explain.
A fish tank at the science fair holds 40 of some unit. Is it 40 liters or 40 milliliters?
Fill in a sensible metric unit: (a) a school gym is 30 ___ long; (b) a crayon is 9 ___ long; (c) a bag of flour has a mass of 2 ___; (d) a medicine cup holds 10 ___.
A paper clip has a mass of about 1 gram. Is a box of 100 paper clips closer to 100 grams or 100 kilograms?
At the science fair, a toy car rolls down a ramp in 4 ___. Fill in the unit, and name two other units of time that the metric and customary systems share.

Day 2: the pattern of tens

A centimeter is one hundredth of a meter, so 100 centimeters make a meter. Using the same prefixes, how many centiliters make a liter? How many milligrams make a gram?
Order from smallest to largest: (a) kilogram, milligram, gram; (b) meter, millimeter, kilometer, centimeter.
Sort these facts into metric and customary, and say which steps are made only of tens: 12 inches in a foot; 100 centimeters in a meter; 16 ounces in a pound; 1{,}000 grams in a kilogram; 4 quarts in a gallon; 1{,}000 milliliters in a liter.
Lenka runs 50 meters. Ondrej runs 50 yards. Who runs farther, and why?
A kilogram is a little more than 2 pounds. A dog has a mass of 10 kilograms. Is that more or less than 20 pounds?
A liter is a little more than a quart. Which holds more, a 2-liter bottle or a 2-quart jug?
How many steps of ten lead from a centimeter up to a meter? Use them to say how many centimeters make a meter.
Marketa fills a box 10 centimeters long and 10 centimeters wide with centimeter cubes, 5 layers high. How many cubes is that? How many milliliters of space do they fill, and what part of a liter is that?
A square centimeter is a square 1 centimeter on each side. Which attribute does it measure, and which unit is it made from? How many square centimeters cover a rectangle 3 centimeters by 4 centimeters?

Reasoning

Explain, using place value, why “10 millimeters in a centimeter” is easier to work with than “12 inches in a foot”.
People sometimes say “a kilo” when they mean a kilogram. Explain why kilo- on its own does not name a unit.
Is it always true that measuring something in a metric unit gives a smaller number than measuring it in a customary unit? Give an example to support your answer.
The metric system makes almost everything go by tens. Why does a minute still have 60 seconds?
Which is bigger, 1 liter or 1 meter? Explain.

Answers

(a) Liquid volume, liter. (b) Mass, gram (kilograms suit a backpack). (c) Length, meter. (d) Time, second (a song lasts a few minutes, and a minute is 60 seconds).
(a) Kilograms: a cat has the mass of several full water bottles. (b) Grams: a strawberry is about as heavy as a handful of paper clips. (c) Liters: a bathtub holds many large water bottles. (d) Milliliters: a dose is a spoonful, far less than a bottle.
(a) Centimeters: a stamp is a few fingernails wide. (b) Kilometers: towns are many minutes of walking apart. (c) Meters: a pool is some number of long steps long.
2 kilograms. Two grams is two paper clips, far too light for a rabbit. Two kilograms is about two full 1-liter bottles of water, a fair mass for a rabbit.
40 liters. Forty milliliters is a few spoonfuls, too little for a fish; forty liters is forty large water bottles, the size of a fish tank.
(a) Meters. (b) Centimeters. (c) Kilograms. (d) Milliliters.
About 100 grams, since 100 clips of about 1 gram each is 100 \times 1 = 100 grams. A hundred kilograms is heavier than most adults.
Seconds. The metric and customary systems share the minute and the hour (also the day).
100 centiliters make a liter, since centi- means one hundredth beside any base unit. 1{,}000 milligrams make a gram, since milli- means one thousandth.
(a) Milligram, gram, kilogram. (b) Millimeter, centimeter, meter, kilometer.
Metric: 100 centimeters in a meter, 1{,}000 grams in a kilogram, 1{,}000 milliliters in a liter; each is made only of tens, since 100 = 10 \times 10 and 1{,}000 = 10 \times 10 \times 10. Customary: 12 inches in a foot, 16 ounces in a pound, 4 quarts in a gallon; none of 12, 16, 4 is made of tens.
Lenka. A meter is longer than a yard, and both ran 50 of their unit, so 50 meters is farther than 50 yards.
More. Each kilogram is a little more than 2 pounds, so 10 kilograms is a little more than 10 \times 2 = 20 pounds.
The 2-liter bottle. Each liter is a little more than a quart, and both have 2 of their unit.
Two steps: centimeter to decimeter, decimeter to meter. So 10 \times 10 = 100 centimeters make a meter.
One layer is 10 \times 10 = 100 cubes, so 5 layers are 100 \times 5 = 500 cubes. They fill 500 milliliters, which is half a liter, since 500 + 500 = 1{,}000 milliliters make a liter.
Area, and it is made from the centimeter. The rectangle is 3 rows of 4 squares, so 3 \times 4 = 12 square centimeters.
Ten millimeters fill one centimeter, so going from millimeters to centimeters is the same as going from ones to tens on a place value chart: the digits shift one place. Twelve does not match any place, so 12 inches in a foot has to be remembered and worked out separately.
Kilo- means “1{,}000 of” and needs a base word to say 1{,}000 of what. A kilogram is 1{,}000 grams, a kilometer 1{,}000 meters and a kiloliter 1{,}000 liters. In everyday talk “a kilo” is just a shortened name for a kilogram.
No. A meter is longer than a yard, so a length in meters is a smaller number than in yards. But a centimeter is shorter than an inch: a strip 4 inches long is a little more than 10 centimeters, a bigger number. Which number is smaller depends on which unit is bigger.
Minutes, hours and days were in use worldwide long before the metric system, and changing them would have changed every clock and calendar. A ten-hour day was tried for a short time in France in the 1790s and dropped. So the metric system took the second as its unit of time and kept the minute of 60 seconds.
Neither. A liter measures liquid volume and a meter measures length, so they cannot be compared. Only units of the same attribute can be bigger or smaller than each other.