Grade 2 · 2.MD.A.3 · 4.MD.A.1

Metric length units

A stuffed turtle, a bunk bed, an ant and the walk to a friend’s house are all lengths. Which metric unit fits each one, how big is each unit, and how do the centimeter, the meter, the millimeter and the kilometer fit together?

You need

Measuring lengthGrade 1 Decimal place valueGrades 4 to 5

Unlocks

The metric system and SI unitsGrade 3

Where you are 7 in Length and units

Measurement12

Length and units7

▴ 2 above Customary length units Length on a number line

Metric length units

Measuring to the half and quarter inch Converting metric units by powers of ten

Why this exists

Dzifa and Tawhiri both measure the same bunk bed. Dzifa says it is 2. Tawhiri says it is 200. Both are right, and both are useless to anyone else until they say what they counted: Dzifa counted meters, Tawhiri counted centimeters. A length is a number of units, and a unit only helps people share a length when its size is fixed and agreed.

In the United States the everyday units are the inch, the foot, the yard and the mile (Customary length units). Almost everywhere else, and in science everywhere, people use the metric units: the centimeter, the meter, the millimeter and the kilometer. They are worth learning for a second reason too. They fit together by tens, hundreds and a thousand, the same way ones, tens and hundreds fit together in a written number.

This lesson gives each metric unit a size you can picture and a tool to measure with. The main units are the centimeter and the meter, measured with a centimeter ruler and a meter stick. That 100 centimeters make a meter is seen, by laying ten-rods from a set of base ten blocks along a meter stick, not learned as a rule. The millimeter and the kilometer are met by their size. The lesson ends with choosing the unit that fits the job. Reading a ruler from 0 is already known from Measuring length, and every length here is a whole number of units.

The lesson runs over four days. Day 1 is the centimeter (Examples 1 and 2). Day 2 is the meter (Examples 3 and 4). Day 3 is the millimeter (Examples 5 and 6). Day 4 is the kilometer and choosing a unit (Examples 7 and 8).

Definition

Day 1: the centimeter

The centimeter is a small unit. A centimeter ruler is a row of centimeter spaces stuck together, with a numbered mark at the end of each space. Most school rulers have centimeters on one edge and inches on the other; the centimeter edge is the one with more, closer marks. Short things are measured in centimeters: a crayon, a stuffed animal, a book.

Day 2: the meter

The meter is the length of 100 centimeters laid end to end. A meter stick is one meter long, marked from 0 to 100 centimeters. Things about as long as a person or a piece of furniture are measured in meters: a bunk bed, a rug, the side of a ball pit, a hallway.

Day 3: the millimeter

The millimeter is the tiniest of the four units. Look closely at a centimeter ruler: between two numbered marks there are small marks that cut each centimeter into 10 tiny equal spaces. Each tiny space is one millimeter, so 10 millimeters make a centimeter. Very small things are measured in millimeters: an ant, a seed, a bead.

Day 4: the kilometer

The kilometer is a long unit, used for how far apart places are. It is the length of 1{,}000 meters laid end to end, far too long to lay a meter stick along.

Each unit has a short form, used on rulers, signs and labels: \text{cm} for centimeter, \text{m} for meter, \text{mm} for millimeter and \text{km} for kilometer. In this lesson the words are written out.

Intuition

Day 1: how big a centimeter is

In most classroom sets of base ten blocks, the little ones cube is one centimeter along each edge. That is the picture to carry: a centimeter is about as wide as a ones cube, a little narrower than a fingertip. (Building such pictures and estimating with them is the work of Benchmarks, estimating and choosing a unit.)

Figure 1The turtle starts at 0 and ends at 12. It is 12 centimeters long.

On a centimeter ruler each space between neighboring numbered marks is one centimeter. The turtle lies from the 0 mark to the 12 mark, so it covers 12 centimeter spaces and it is 12 centimeters long. This is the same reading as in Measuring length; the new thing is the name of the unit.

A ten-rod from the same set of blocks is ten ones cubes in a row, so it is 10 centimeters long. That gives a second way to measure: lay rods and cubes along a thing and read the length as tens and ones.

Day 2: one hundred centimeters make a meter

Lay ten-rods end to end along a meter stick. The first rod ends at the 10 mark, the second at 20, the third at 30, and the count goes up by ten each time, as in skip counting by tens. The tenth rod ends at the 100 mark, which is the end of the meter stick.

Figure 2Ten rods of 10 centimeters cover a meter stick exactly. A meter is 100 centimeters.

Ten rods of ten cubes are one hundred cubes, the same as ten tens make a hundred in a written number. So a meter is 100 centimeters, and the picture shows it, with no rule to remember. A meter is a little longer than a yardstick, which is the one picture of its size to carry.

To measure something longer than one meter stick, lay the stick down, mark where it ends, and move it along, as in Iterating a single unit. The count of sticks is the length in meters. Because each meter stick holds 100 centimeters, the same length in centimeters counts by hundreds: 100, 200, 300 and so on.

Day 3: ten millimeters make a centimeter

Look at the ruler under a magnifying glass. Between the 0 mark and the 1 mark the small marks make 10 tiny spaces; the middle mark, after 5 of them, is often drawn a little longer to help the eye. Each tiny space is one millimeter.

Figure 3Each tiny space is one millimeter. The ant covers 6 of them, so it is 6 millimeters long.

A millimeter is about as thick as the wire of a paper clip. An ant is a few millimeters long, so an ant is measured by counting tiny spaces from the 0 mark, the same way a turtle is measured by counting centimeter spaces.

Since every centimeter holds 10 millimeters, a length in centimeters is turned into millimeters by counting by tens: 1 centimeter is 10 millimeters, 2 centimeters are 20, 3 are 30.

Day 4: a kilometer is a walk

Nobody measures a kilometer with a meter stick. The picture to carry is a walk: at an easy pace, a kilometer takes about 10 to 15 minutes. So the kilometer is the unit for trips, roads and trails, and the other three are units for things you can lay a tool along.

Choosing a unit comes down to the size of the thing. A unit far smaller than the thing gives a huge count. A unit bigger than the thing does not fit along it even once. The good unit is the one a few, or a few dozen, of which fit along the thing.

Results

Days 1 to 4

Unit	Tool	About the size of
millimeter	the tiny spaces on a centimeter ruler	the thickness of a paper clip wire
centimeter	centimeter ruler, ones cube	the width of a ones cube
meter	meter stick	a little longer than a yardstick
kilometer	no ruler	a walk of about 10 to 15 minutes

10 \text{ millimeters make } 1 \text{ centimeter}

100 \text{ centimeters make } 1 \text{ meter}

1{,}000 \text{ meters make } 1 \text{ kilometer}

A length is a number together with a unit. The same bunk bed is 2 meters and 200 centimeters; the numbers differ because the units differ, and the bed is the same.

Why it is true

The facts in the box are agreed, not discovered. The meter was chosen first, and the other units were made from it by tens. The names say so. “Centi” is the same start as in “cent”: 100 cents make a dollar, and 100 centimeters make a meter. “Kilo” means a thousand: 1{,}000 meters make a kilometer. “Milli” means a thousandth part, and the box shows why: a meter is 100 centimeters, each of 10 millimeters, and 100 tens are 1{,}000, so a meter is 1{,}000 millimeters. The same starts appear on other units, such as grams and liters (Metric prefixes).

What a child can do is check the agreement with real tools. Lay ten-rods along a meter stick, as in Figure 2, and ten of them fill it exactly. Count the tiny spaces between 0 and 1 on a ruler, as in Figure 3, and there are ten.

Why can one bunk bed be 2 meters and also 200 centimeters? Each meter stick along it is 100 centimeters, so two of them cover 100 + 100 = 200 centimeters. It is one length, named in two units. The bigger unit gives the smaller count, which is its own idea, Smaller units give bigger counts.

Worked examples

The examples follow the days: Examples 1 and 2 are Day 1, Examples 3 and 4 Day 2, Examples 5 and 6 Day 3, and Examples 7 and 8 Day 4.

Example 1

A stuffed turtle on a centimeter ruler

Altynai lays her stuffed turtle along a centimeter ruler, as in Figure 1. The tip of its tail is at the 0 mark and the front of its head is at the 12 mark. How long is the turtle?

Solution

The turtle starts at 0, so the mark at its other end counts the centimeter spaces it covers. From 0 to 1 is one centimeter, from 1 to 2 is another, and so on up to the space from 11 to 12. That is 12 spaces. The turtle is 12 centimeters long.

Example 2

Rods and cubes along a stuffed snake

Tawhiri’s stuffed snake is longer than his ruler. He lays ten-rods and ones cubes from a set of base ten blocks along it instead. Four rods and then three cubes reach exactly from the tip of the tail to the tip of the head, as in Figure 4. How long is the snake?

Figure 4Four ten-rods and three ones cubes fit along the snake exactly. It is 43 centimeters long.

Solution

Each ten-rod is 10 centimeters long, so the four rods cover 10, 20, 30, 40 centimeters, counting by tens. Each ones cube is 1 centimeter, so the three cubes add 3 more:

40 + 3 = 43

The snake is 43 centimeters long: 4 tens and 3 ones, written just as the number 43 is written. The rods and cubes are a centimeter ruler cut into pieces.

Example 3

The bunk bed in meters and in centimeters

Dzifa lays a meter stick along the floor beside the bunk bed, marks where it ends, and moves it along. The second placement ends exactly at the far end of the bed, as in Figure 5. How long is the bed in meters? How long is it in centimeters?

Figure 5Two meter sticks fit along the bunk bed. It is 2 meters long.

Solution

Each placement covers one meter, and the placements touch with no gaps and no overlaps, so the count of placements is the length in meters. The bed is 2 meters long.

Each meter stick is 100 centimeters, as Figure 2 shows. The first stick covers 100 centimeters and the second covers 100 more:

100 + 100 = 200

The bed is 200 centimeters long. That is why Dzifa said 2 and Tawhiri said 200: the same bed, counted in two units.

Example 4

A mat for the ball pit

The long side of a ball pit is 3 meters. Altynai has a foam mat 250 centimeters long to lay along that side. Does it fit? If it does, how much of the side is left uncovered?

Solution

The two numbers are in different units, so they cannot be compared as they stand. Put both in centimeters. Each meter is 100 centimeters, so 3 meters is three hundreds:

100 + 100 + 100 = 300

The side of the ball pit is 300 centimeters. The mat is 250 centimeters, and 250 is less than 300, so the mat fits. The part left uncovered is

300 - 250 = 50

The mat fits, with 50 centimeters of the side left over. Notice that the bigger number, 250, belonged to the shorter thing: centimeters are a much smaller unit than meters.

Example 5

An ant on the ruler

Altynai’s class keeps an ant farm. An ant walks onto a ruler, and for a moment its back end is at the 0 mark, as in Figure 3. Its head reaches the sixth tiny mark, before the 1 mark. How long is the ant?

Solution

The ant starts at 0, so count the tiny spaces it covers: from the 0 mark to the first tiny mark is one, and so on up to the sixth. Each tiny space is one millimeter. The ant is 6 millimeters long.

It is shorter than one centimeter, because it does not reach the 1 mark. That is why the millimeter is the unit for it: in centimeters, all one can say is “less than 1”.

Example 6

A tunnel in the ant farm

Along the glass of the ant farm, one tunnel runs from the 0 mark of a ruler exactly to the 3 mark. How long is the tunnel in centimeters? How many millimeters is that?

Solution

The tunnel starts at 0 and ends at the 3 mark, so it is 3 centimeters long.

Each centimeter holds 10 tiny spaces. Count them by tens, one centimeter at a time:

10 + 10 + 10 = 30

The tunnel is 3 centimeters long, and it is 30 millimeters long.

Example 7

Choosing a unit

Dzifa is writing labels. Which unit fits each one: millimeters, centimeters, meters or kilometers?

the length of an ant in the ant farm
the length of a stuffed bear
the long side of the ball pit
how far it is to Tawhiri’s house, a walk of about 12 minutes

Solution

An ant does not reach even one centimeter mark. Only millimeters give it a whole number, a few millimeters, so millimeters fit.

A stuffed bear is shorter than a meter stick but longer than a few ones cubes. A few dozen centimeters fit along it, so centimeters fit.

The side of a ball pit takes a meter stick laid down a few times, so meters fit. (Centimeters work too, but the count runs into the hundreds.)

A walk of about 12 minutes is within the 10 to 15 minutes that picture one kilometer. No stick is laid along a road, so kilometers fit, and Tawhiri lives about 1 kilometer away.

Example 8

The rest of a kilometer

Altynai’s school holds a walk-a-thon around a path that is 1 kilometer long. Altynai has walked 600 meters so far. How many more meters does she have to walk?

Solution

The two lengths are in different units, so put the kilometer in meters. Picture meter sticks laid along the path. Every hundred sticks make 100 meters, and counting by hundreds:

100, 200, 300, 400, 500, 600, 700, 800, 900, 1{,}000

Ten hundreds make 1{,}000, so the path is 1{,}000 meters long. Altynai has walked 600 of them. The rest is

1{,}000 - 600 = 400

She has 400 meters more to walk. Check: 600 + 400 = 1{,}000, the whole path.

Boundaries

A number alone is not a length. “The bed is 2” says nothing until the unit is named. 2 millimeters, 2 centimeters, 2 meters and 2 kilometers are four very different lengths.

Two lengths are compared in the same unit. 3 meters against 250 centimeters cannot be judged by the numbers, as Example 4 shows. Put both in one unit first.

Whole units only, here. Every length in this lesson is a whole number of units. A real thing may end between two marks. Then its length is “about” the nearer whole unit, as in Measuring length, or it is read in a smaller unit, as the ant was. Reading unlabeled marks on a meter stick is the work of Measuring tools and reading their scales.

A kilometer is known by a picture, not a ruler. “About 10 to 15 minutes of walking” depends on how fast someone walks, so it gives “about 1 kilometer”, never an exact length.

These units are for length. Centimeters, meters, millimeters and kilometers say how long or how far. They do not say how heavy something is or how much it holds; those need grams and liters, met in The metric system and SI units.

Where this goes next

The metric units are built so that every step between them is ten, a hundred or a thousand, and The metric system and SI units shows the same plan for weight and capacity, with the same name starts (Metric prefixes). Later grades measure to the millimeter as a matter of course and write lengths between whole units with a decimal point, such as 1.25 meters, which needs Decimal place value. Changing any metric length from one unit to another by tens and hundreds becomes Converting metric units by powers of ten. The customary units, Customary length units, do the same job with sizes that do not go by tens.

Exercises

Day 1: the centimeter

Use the rulers in Figure 6 for Exercises 1 to 3.

Figure 6Rulers for Exercises 1 to 3.

How long is the toy whistle in row A?
How long is the stuffed fish in row B?
The toy block in row C starts at the 5 mark. How long is it?
Tawhiri lays 3 ten-rods and then 6 ones cubes along a stuffed dog, and they fit exactly. How long is the dog?
A stuffed cat is 27 centimeters long and a stuffed rabbit is 32 centimeters long. Which is longer, and by how much?
A stuffed bear on the top bunk is 50 centimeters tall. How many ten-rods stacked end to end match its height?

Day 2: the meter

How many ten-rods laid end to end match one meter stick? Which number is at the end of a meter stick?
Dzifa lays a meter stick along a playground fence 4 times, and it fits exactly. How long is the fence in meters? In centimeters?
A rug is as long as one meter stick and 20 more centimeters. How many centimeters long is it?
Which is longer, 1 meter or 90 centimeters? By how much?
The rail of the top bunk is 180 centimeters long. Is it longer or shorter than 2 meters, and by how much?
A hallway is 500 centimeters long. How many meter sticks fit along it end to end?

Day 3: the millimeter

Use the ants in Figure 7 for Exercises 13 to 15. Each ant starts at the 0 mark.

Figure 7Ants for Exercises 13 to 15.

How long is ant A?
How long is ant B?
How long is ant C, in millimeters?
How many millimeters are in 2 centimeters?
A seed in the ant farm is 5 millimeters long. Is it longer or shorter than 1 centimeter, and by how much?
Which is longer, 8 millimeters or 1 centimeter? By how much?

Day 4: the kilometer and choosing a unit

Choose millimeters, centimeters, meters or kilometers for each: a bead on a bracelet; a pillow on the bunk bed; a soccer field; the drive to the next town.
Dzifa’s walk to the library takes about 14 minutes at an easy pace. About how far is it?
A trail is 1 kilometer long. Tawhiri has walked 700 meters of it. How much farther is it to the end?
Would you measure the distance from school to the zoo with a meter stick? Say why or why not.

Reasoning

Dzifa says the classroom wall is 6 and Tawhiri says it is 600. Can both be right? Explain with meter sticks and centimeters.
How could a child check, with base ten blocks and a meter stick, that a meter is 100 centimeters?
Why does the same length always give a bigger number in centimeters than in meters?

Answers

6 centimeters. The whistle starts at 0 and ends at the 6 mark, so it covers 6 centimeter spaces.
14 centimeters. The fish starts at 0 and its nose is at the 14 mark.
4 centimeters. It covers the spaces from 5 to 9, and 9 - 5 = 4.
36 centimeters. Three rods are 10, 20, 30 centimeters, and six cubes add 6: 30 + 6 = 36.
The rabbit, by 5 centimeters, since 32 - 27 = 5.
5 ten-rods. Counting by tens, 10, 20, 30, 40, 50 takes five rods.
10 ten-rods, and the end of a meter stick is the 100 mark: 10, 20, \ldots, 100 is ten tens.
4 meters, which is 400 centimeters: 100 + 100 + 100 + 100 = 400.
120 centimeters. The meter stick is 100 centimeters, and 100 + 20 = 120.
1 meter, by 10 centimeters. One meter is 100 centimeters, and 100 - 90 = 10.
Shorter, by 20 centimeters. Two meters are 100 + 100 = 200 centimeters, and 200 - 180 = 20.
5 meter sticks. Counting by hundreds, 100, 200, 300, 400, 500 takes five sticks.
4 millimeters. Ant A covers 4 tiny spaces from the 0 mark.
9 millimeters. Ant B covers 9 tiny spaces; it ends just before the 1 mark.
13 millimeters. Ant C passes the 1 mark, which is 10 millimeters, and covers 3 more tiny spaces: 10 + 3 = 13.
20 millimeters. Each centimeter is 10 millimeters, and 10 + 10 = 20.
Shorter, by 5 millimeters. One centimeter is 10 millimeters, and 10 - 5 = 5.
1 centimeter, by 2 millimeters. One centimeter is 10 millimeters, and 10 - 8 = 2.
A bead in millimeters (it is shorter than one centimeter). A pillow in centimeters (a few dozen fit along it). A soccer field in meters (a meter stick laid down many times). The next town in kilometers (no stick is laid along a road).
About 1 kilometer. A walk of about 14 minutes is within the 10 to 15 minutes that picture one kilometer.
300 meters. A kilometer is 1{,}000 meters, and 1{,}000 - 700 = 300. Check: 700 + 300 = 1{,}000.
No. The zoo is a trip, not a thing to lay a stick along; a meter stick would have to be laid down a huge number of times. Distances between places are told in kilometers.
Yes. Six meter sticks fit along the wall, so it is 6 meters. Each meter stick is 100 centimeters, so counting by hundreds, 100, 200, 300, 400, 500, 600, the same wall is 600 centimeters. Dzifa counted meters and Tawhiri counted centimeters. Each should say the unit.
Lay ten-rods end to end along the meter stick. Ten of them cover it exactly, ending at the 100 mark. Each rod is 10 ones cubes, so the meter stick is as long as 100 ones cubes, which is 100 centimeters.
A centimeter is much smaller than a meter: 100 of them fit in each meter. So each meter stick that fits along a thing adds 1 to the count in meters but 100 to the count in centimeters. One stick gives 1 and 100, two give 2 and 200. Smaller units are counted more times.