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Converting lengths

Millimeters, centimeters, meters and kilometers, and which way the number moves.

Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.

1. What you will learn

You will convert between mm, cm, m and km, and predict whether the number should get bigger or smaller first.

2. What you already do

You can already multiply and divide by $10$, $100$ and $1000$, and you already know that a meter is longer than a centimeter. Converting is those two things used together, and nothing else.

You also met the rule that a unit should fit the thing, and you have seen that the same pencil can be 18 cm or 0.18 m. In this lesson you learn to get from one of those numbers to the other, and to be sure which way to go.

3. Words for this lesson

4. Bigger unit, smaller number

The metric length units go up in steps: $10$ mm make a cm, $100$ cm make a m, $1000$ m make a km. Converting is multiplying or dividing by one of those numbers, and the only thing to get right is which.

The check that always works: a bigger unit needs a smaller number. A door is $210$ cm or $2.1$ m — the same door, and the meter answer is the smaller number because a meter is a bigger step. Think of counting a hallway in footsteps and then in giant leaps: you need fewer leaps than footsteps, because each leap covers more ground.

So before you calculate anything, ask: is the new unit bigger or smaller? If it is bigger, the number must go down, so divide. If it is smaller, the number must go up, so multiply. Then use the step between the two units. Deciding the direction first means you can never be tricked into going the wrong way.

5. The ladder of length units

It helps to picture the units as a ladder, with the biggest at the top.

Going down the ladder, to a smaller unit, you multiply: each big unit breaks into many small ones. Going up the ladder, to a bigger unit, you divide: many small ones join to make fewer big ones.

To go more than one rung, take one rung at a time. From millimeters to meters is two rungs up: divide by 10 to reach centimeters, then by 100 to reach meters. That is the same as dividing by 1000, because $10 \times 100 = 1000$.

6. Why the decimal point moves

Multiplying by 10 moves every digit one place to the left, so the number gets ten times bigger. Dividing by 10 moves every digit one place to the right. Multiplying or dividing by 100 moves them two places, and by 1000, three places.

So converting a metric length is just moving the decimal point. $2.4$ km to meters is three places to the left: $2400$ m. $350$ cm to meters is two places to the right: $3.50$ m, which is $3.5$ m. When there are not enough digits, add zeros to fill the places: $7$ km is $7000$ m.

This is the reason scientists like metric units. In the customary units used on many American signs, 12 inches make a foot and 5,280 feet make a mile, and converting those takes real multiplication. In metric units it takes only a decimal point.

7. Two ways to check

After every conversion, check your answer in two ways.

1. The direction check. Did the number move the way you predicted? Going to a bigger unit, it must get smaller. If $250$ cm came out as $25000$ m, it went the wrong way.
2. The picture check. Does the answer describe a real thing? A table that is $250$ cm long is $2.5$ m, about the length of a car's hood. If you had got $0.025$ m, the table would be as short as your thumb.

The direction check catches multiplying when you should divide. The picture check catches using the wrong step, like 10 instead of 100. Together they catch almost every mistake.

8. Adding lengths in different units

You cannot add $1$ m and $30$ cm to get $31$ of anything. The two numbers count different-sized steps. First put both lengths in the same unit, then add.

For example: a rope is $1.2$ m and another is $45$ cm. In centimeters, the first is $1.2 \times 100 = 120$ cm, so together they are $120 + 45 = 165$ cm. In meters, the second is $45 \div 100 = 0.45$ m, so together they are $1.2 + 0.45 = 1.65$ m. Both answers describe the same total length. Use the unit the question asks for, and check that the total is longer than either piece alone.

9. Where the numbers 10, 100 and 1000 come from

The names of the metric units tell you their steps, once you know three little word parts.

You have met centi- before in cent: a cent is a hundredth of a dollar, just as a centimeter is a hundredth of a meter. So 100 cents make a dollar, and 100 centimeters make a meter. The same word part means the same step.

The same parts are used for mass and volume too. A kilogram is a thousand grams, and a milliliter is a thousandth of a liter. Learn the three word parts once, and you know the steps for every metric unit you will ever meet, including ones you have never seen before.

10. Converting a length that has two parts

Sometimes a length is written with two units at once, like 1 m 35 cm. That means one meter and thirty-five centimeters more. To write it in one unit, convert the big part and add the small part.

In centimeters: $1$ m is $100$ cm, and $100 + 35 = 135$ cm. In meters: $35$ cm is $35 \div 100 = 0.35$ m, and $1 + 0.35 = 1.35$ m. Look at the two answers: the digits 1, 3 and 5 appear in both, and only the decimal point has moved. That is a good check that you did it right.

Going the other way, a length like $2.08$ m splits into $2$ m and $0.08$ m, and $0.08 \times 100 = 8$ cm, so it is 2 m 8 cm. Watch the zero: $2.08$ m is not 2 m 80 cm. The zero holds the tens place empty, and 8 cm is much shorter than 80 cm. A doctor measuring your height might write 1 m 32 cm on your chart, and the computer stores it as 1.32 m or 132 cm. All three say the same height.

11. In the world: a track meet

At a school track meet, the long track is $400$ m around. A mile race is about four laps, and a runner who finishes $4$ laps has gone $4 \times 400 = 1600$ m. The coach writes it on the board in kilometers: $1600 \div 1000 = 1.6$ km. Kilometers are the bigger unit, so the number got smaller.

In the long jump, the tape measure is marked in centimeters. A jump of $285$ cm is written in the results as $2.85$ m, because the official records use meters. The judge divides by $100$ and checks the answer: nearly three meters is a good jump for a third grader's older brother, and a sensible size.

If the judge had multiplied instead, the result sheet would say $28500$ m, a jump of more than twenty-eight kilometers. Nobody would believe it, which is exactly the picture check at work.

12. In the world: building a birdhouse

A plan for a birdhouse says each side is $15$ cm wide and $20$ cm tall, and the entrance hole is $32$ mm across. The builder buys a board $1.2$ m long.

To see if the board is long enough, the builder converts it to centimeters: $1.2 \times 100 = 120$ cm. The four sides need $4 \times 20 = 80$ cm, the roof and floor need about $30$ cm more, so the total is $110$ cm. That fits in $120$ cm, with $10$ cm to spare.

The drill bit for the hole is sold in millimeters, and $32$ mm is $3.2$ cm, about the size of a bluebird. Builders, carpenters and engineers convert lengths all day long, and they always predict the direction first, because a cut in the wrong place wastes wood that cannot be put back.

13. Learning the operation instead of the reason

The mistake is learning the operation instead of the reason: centimeters to meters means divide by a hundred, remembered without the why, comes out as multiply about half the time.

So do not remember the operation. Decide which unit is bigger, say the bigger unit takes the smaller number, and let that tell you whether to multiply or divide. If your answer went the wrong way, you will see it straight away.

A second mistake is using the wrong step: dividing by 10 to get from centimeters to meters, because 10 is the step you used last. The ladder table fixes this. Before you calculate, find the two units on the ladder and read the step between them.

A third mistake is losing places when the decimal point moves. Converting $7$ km to meters, some learners write $7.000$ and stop, because they moved the point but did not fill the empty places. Write the zeros: $7 \times 1000 = 7000$ m. Then do the picture check: seven kilometers is a long walk, and seven meters is across a room.

14. 210 cm to meters

1. Name the two units.

   $\text{have: cm} \qquad \text{want: m}$

   Know where you are and where you are going.

2. Predict the direction first.

   $\text{m is bigger, so the number gets smaller}$

   A bigger unit needs a smaller number.

3. Find the step between them.

   $100\ \text{cm} = 1\ \text{m}$

   One rung on the ladder.

4. Divide by the step.

   $210 \div 100 = 2.1$

   Dividing makes it smaller, as predicted.

5. Write the answer and check it.

   $210\ \text{cm} = 2.1\ \text{m}$

   A door just over two meters tall is a real door.

15. 2.4 km to meters

1. Name the two units.

   $\text{have: km} \qquad \text{want: m}$

   This time you are going down the ladder.

2. Predict the direction first.

   $\text{m is smaller, so the number gets bigger}$

   It takes many meters to make a kilometer.

3. Find the step between them.

   $1\ \text{km} = 1000\ \text{m}$

   One rung, but a big one.

4. Multiply by the step.

   $2.4 \times 1000 = 2400$

   The decimal point moves three places to the right.

5. Write the answer with its unit.

   $2.4\ \text{km} = 2400\ \text{m}$

   Bigger, as predicted.

6. Check it against a picture.

   $2400\ \text{m} \approx 30 \text{ minutes of walking}$

   A real path could be that long.

16. A 1.5 m plank cut into pieces of 25 cm

1. Notice the two different units.

   $1.5\ \text{m} \text{ and } 25\ \text{cm}$

   You cannot compare them until they match.

2. Choose one unit for both.

   $\text{centimeters}$

   Then the piece size stays a whole number.

3. Predict the direction first.

   $\text{m} \to \text{cm: bigger number}$

   Centimeters are the smaller unit.

4. Convert the plank's length.

   $1.5 \times 100 = 150\ \text{cm}$

   One hundred centimeters in each meter.

5. Count the pieces.

   $150 \div 25 = 6$

   How many 25 cm pieces fit in 150 cm.

6. Check by adding them back.

   $6 \times 25 = 150\ \text{cm} = 1.5\ \text{m}$

   The pieces make the whole plank again.

7. State the answer in words.

   $6 \text{ pieces of } 25\ \text{cm}$

   The number of pieces has no unit: it is a count.

17. Your turn: 6 mm in centimeters.

1. Predict the direction first.

   $\text{cm is bigger, so the number gets smaller}$

   Decide before calculating.

2. Your turn: work this step out. Its working is at the end of the packet.

   Find the step between them.

3. Your turn: work this step out. Its working is at the end of the packet.

   Divide by the step.

18. Guided practice

A table is $7020$ cm long. Give its length in meters.

Answer: unit: m

19. Guided practice

Complete the worked solution: a door is $2.53$ m tall. How tall is it in centimeters, and in millimeters?

1. Predict the direction first.

   $\text{m} \to \text{cm: bigger number}$

   A centimeter is smaller than a meter.

2. Convert meters to centimeters.

   $2.53 \times 100 =$ _p cm

   There are a hundred centimeters in each meter.

3. Convert centimeters to millimeters.

   $\text{that} \times 10 =$ _q mm

   There are ten millimeters in each centimeter.

20. Guided practice

Match each pair of units to the number of smaller units in one larger unit.

21. Guided practice

An ant is $19$ mm long. Give its length in centimeters.

Answer: unit: cm

22. Practice

Fill in each length in the unit the table asks for.

23. Practice

A path is $6.8$ km long. How long is it in meters?

Answer: unit: m

24. Practice

A crafter has two pieces of ribbon. One is $3.2$ m long and the other is $35$ cm long. How much ribbon is there altogether, in centimeters?

Answer: unit: cm

25. Somewhere new

A carpenter cuts a plank $5100$ mm long for a shelf. The store sells wall brackets by the meter. How long is the plank in meters?

Answer: unit: m

26. Lesson test

Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.

27. Test question

Fill in each length in the unit the table asks for.

28. What you can do now

You can convert lengths. Tell someone why the answer in meters is a smaller number than the answer in centimeters.

Working for the steps left to you

17. Your turn: 6 mm in centimeters., step 2

17. Your turn: 6 mm in centimeters., step 3