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Choosing a sensible unit

A number needs a unit, and the unit should fit the size of the thing.

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 pick the unit that fits what is being measured, estimate a size in that unit, and say why a unit that is too big or too small is unusable.

2. What you already do

You already choose units without calling it that. You say a walk takes twenty minutes, not 1,200 seconds, and you say someone is nine years old, not 108 months. Both of the numbers you did not say are correct. This lesson is about why you left them alone.

You also know some sizes by heart. You know a door is taller than you, a car trip to another city takes hours, and a strawberry is light enough to hold on one finger. Those sizes are what you will use to choose.

3. Words for this lesson

4. A number needs a unit

A number on its own says nothing. 15 could be fifteen millimeters or fifteen kilometers, and those are not the same thing at all. Fifteen millimeters is about the width of your thumbnail. Fifteen kilometers is a long car ride. A measurement is always a number and a unit together.

Choosing the unit is choosing how big a step to count in. If you count a pencil in centimeter steps, you count about eighteen of them. If you count it in meter steps, you do not even finish one step: you get 0.18 m. If you count it in millimeter steps, you get 180 mm. All three are true. Only one is easy to say and easy to picture, and that is 18 cm.

So the rule for choosing a unit is short: pick the unit that gives a number you could say out loud without a string of zeros. A number between about one and a thousand is usually right. When the number comes out as 0.0002 or as 4,000,000, the unit is too big or too small for the thing.

5. Three kinds of units

Before you choose a unit, decide what kind of thing you are measuring. A unit for length cannot measure a mass, just as a clock cannot tell you how long a table is.

Each row goes from a small step to a big step. Ten millimeters make one centimeter, a hundred centimeters make one meter, and a thousand meters make one kilometer. A thousand grams make one kilogram. Sixty seconds make one minute, and sixty minutes make one hour. You will learn to change between these in later lessons. For now, what matters is that they go from small to big, so there is always a smaller unit and a bigger one to try.

6. Sizes you can carry with you

Scientists call a size you know by heart a benchmark. Benchmarks let you choose a unit, and make an estimate, without a ruler or a scale.

• A millimeter is about the thickness of a coin or a fingernail.
• A centimeter is about the width of your little finger.
• A meter is about the length of a big step for a grown-up, or the height of a kitchen counter.
• A kilometer is about ten or twelve minutes of walking.
• A gram is about the mass of a paper clip.
• A kilogram is the mass of a liter bottle of water.

When you meet something new, compare it with a benchmark. A crayon is about as long as nine little-finger widths, so it is about 9 cm. A school bus is about as long as twelve big steps, so it is about 12 m. The benchmark tells you the unit and the number at the same time.

7. Trying each unit out loud

When you are not sure which unit fits, try them all and listen. Here is a bag of apples with a mass of two kilograms.

• In grams, it is 2,000 g. That is a big number with a lot of zeros.
• In kilograms, it is 2 kg. That is easy to say.

Here is a staple, which is about 1 cm long.

• In meters, it is 0.01 m. That starts with zeros.
• In centimeters, it is 1 cm. That works.
• In millimeters, it is 10 mm. That works too.

Sometimes two units both work, like 1 cm and 10 mm. Then either is sensible. People who make small things, like jewelers and engineers, often use millimeters, because they want to be exact about small sizes. What you must avoid is the unit that gives a number no one could picture.

8. How to check your choice

A good choice passes three checks.

1. The kind matches. A time is in seconds, minutes or hours, never in meters.
2. The number is sayable. It has no long string of zeros before or after it.
3. The size makes sense. Picture the number of steps. If a dog were 30 g, it would weigh the same as a few paper clips. That cannot be right, so the unit must be kilograms.

The third check catches the most mistakes. It works in both directions: it tells you when a unit is too big, and when it is too small. Asking could a real one be that size? is how scientists check every measurement they make, even very clever ones.

9. Making an estimate

An estimate is a careful guess at a size. It is not a wild guess: you make it by comparing the thing with a benchmark and counting how many would fit.

Suppose you want to estimate the height of a kitchen table. First decide the kind: it is a length. Next, find a benchmark. A meter is about the height of a kitchen counter, and a table is a little lower than a counter. So the table is a bit less than one meter. Less than one of something is a sign to use a smaller unit, so try centimeters: a bit less than a hundred centimeters, about 75 cm.

An estimate does not have to be exact to be good. If the real table is 72 cm, then 75 cm was a fine estimate. If you had said 75 m, that would be a bad estimate, even though the number is the same, because a table seventy-five meters tall would be taller than a twenty-story building. The unit is part of the estimate, and choosing it well is most of the work.

10. Why scientists use the metric units

In the United States, you see two families of units. On road signs and in recipes you often see miles, feet, inches, pounds and ounces. These are called customary units. In science, and in most countries of the world, people use meters, grams and liters. These are the metric units, and this course uses them.

Scientists use metric units because every step is ten, a hundred or a thousand. That makes changing units easy: you move the decimal point. In the customary family, twelve inches make a foot and three feet make a yard, and those numbers have to be remembered one by one. The rule for choosing a unit is the same in both families, though: pick the one that gives a number you could say out loud.

11. In the world: reading a food label

Pick up a box of cereal and look at the side. The box says how much cereal is inside, for example 340 g. It does not say 0.34 kg, even though that is the same amount, because 340 is easier to read and compare on a shelf.

Now look at the nutrition label. A serving of cereal has about 3 g of protein and about 200 mg of salt. The makers switched to milligrams for the salt because in grams it would be 0.2 g, which is easy to misread. Each number on the box uses the unit that makes it a sensible size.

So a shopper comparing two boxes can do it in seconds. One box is 340 g and another is 510 g: the second has 170 g more, which is half as much again. That is only easy because both numbers are in the same, sensible unit.

12. In the world: a 5K race

Many towns in the United States hold a 5K fun run. The K stands for kilometers, so the course is 5 km long. Organizers could say 5,000 m or 500,000 cm, and both would be true, but runners want a number they can picture. At a steady jog of about 8 minutes for each kilometer, a runner can say the race will take about $5 \times 8 = 40$ minutes.

The same race uses other units too. The starting line is painted about 10 cm wide, the water cups hold about 200 mL, and the winner's time is given in minutes and seconds, such as 17 minutes 30 seconds. Every one of those units was chosen to fit the size of the thing it measures, which is exactly what you have practiced in this lesson.

13. A unit is never wrong, only unhelpful

The usual mistake is thinking a unit can be wrong. It cannot: an apple really is 150,000 milligrams. The unit can only be unhelpful, and it is unhelpful when the number comes out with a string of zeros on either end of it.

The second mistake is picking the unit you met most recently rather than the one that fits the thing. After a lesson about kilometers, it is tempting to measure everything in kilometers. Picture the object first; choose the unit second.

The third mistake is mixing up kinds: saying a bag of rice is 2 meters when you mean 2 kilograms. Meters measure how long a thing is, not how heavy. Deciding the kind first, before you think about size, stops this one.

14. The length of a pencil

1. Decide what kind of quantity it is.

   $\text{a length}$

   So the units to try are mm, cm, m and km.

2. Try it in kilometers.

   $0.00018\ \text{km}$

   Far too many zeros: a kilometer is far bigger than a pencil.

3. Try it in meters.

   $0.18\ \text{m}$

   Still less than one step, so still awkward.

4. Try it in centimeters.

   $18\ \text{cm}$

   Easy to say and easy to picture.

5. Choose and check the unit.

   $18\ \text{cm} \approx 18 \text{ finger widths}$

   A real pencil could be that long, so centimeters fit.

15. A trail sign that says 3000 m

1. Read the measurement as written.

   $3000\ \text{m}$

   It is true, but it has three zeros.

2. Name its kind of quantity.

   $\text{a length}$

   So the next bigger unit is the kilometer.

3. Recall the step between units.

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

   A kilometer is a thousand meters.

4. Count the kilometers in it.

   $3000 \div 1000 = 3$

   Three thousands make three thousand.

5. Write it in the better unit.

   $3000\ \text{m} = 3\ \text{km}$

   The same trail, said in bigger steps.

6. Check the size makes sense.

   $3\ \text{km} \approx 35 \text{ minutes of walking}$

   A real trail could be that long, so kilometers fit.

16. Packing a backpack for a hike

1. List the three things to measure.

   $\text{the backpack's mass, the water bottle's height, the hike's time}$

   Each one needs its own kind of unit.

2. Sort them into their kinds.

   $\text{mass, length, time}$

   A mass needs g or kg, a length mm to km, a time s to h.

3. Picture the backpack's mass.

   $\text{about } 4\ \text{kg} = 4000\ \text{g}$

   Four big water bottles: kilograms give the small number.

4. Picture the water bottle's height.

   $\text{about } 25\ \text{cm} = 0.25\ \text{m}$

   Less than one meter, so centimeters fit.

5. Picture the hike's length in time.

   $\text{about } 3\ \text{h} = 180\ \text{min}$

   Three hours is easier to say than 180 minutes.

6. Write all three with their units.

   $4\ \text{kg}, \quad 25\ \text{cm}, \quad 3\ \text{h}$

   Each is a number and a unit together.

7. Check each against a benchmark.

   $4 \text{ water bottles}, \ 25 \text{ finger widths}, \ \text{a long morning}$

   Every size could be real, so every unit fits.

17. Your turn: which unit for the thickness of a coin?

1. Decide what kind of quantity it is.

   $\text{a length}$

   So try mm, cm and m.

2. Try it in centimeters first.

   $0.2\ \text{cm}$

   Less than one: the unit is too big.

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

   Try it in millimeters instead.

18. Guided practice

Which unit would you use for the time to boil an egg?

• minutes
• hours
• days

19. Guided practice

Complete the worked solution: a stack of coins is $5$ cm tall. Write its height in millimeters and in meters, and say which units fit.

1. Write the height in millimeters.

   $5 \times 10 =$ _a mm

   Each centimeter is ten millimeters, so there are more steps.

2. Write the height in meters.

   $5 \div 100 =$ _b m

   A meter is a hundred centimeters, so there are fewer steps.

3. Keep the units that fit.

   $\text{centimeters or millimeters}$

   The meter answer starts with zeros, so meters are too big.

20. Guided practice

Match each everyday quantity to the unit that would make a sensible measurement.

21. Guided practice

Someone measured the height of a room in millimeters. Is that a sensible unit?

• No — meters would fit better
• Yes, that is the usual unit

22. Practice

Choose the unit you would measure each thing in.

23. Practice

Each measurement is written in a unit that is too small. Choose a better unit, and write the number in that unit.

24. Practice

Estimate the height of a kitchen table. Choose a sensible unit and give a number. You do not need to be exact: a sensible estimate is close enough.

Answer: unit: m / cm / km / mm

25. Somewhere new

A bag of flour is labeled 500 — but the unit has worn off. Which unit was it?

• grams
• kilograms

26. Lesson test

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

27. Test question

Choose the unit you would measure each thing in.

28. What you can do now

You can choose a unit that fits. Tell someone which unit you would use for the thickness of a coin, and why centimeters would be awkward.

Working for the steps left to you

17. Your turn: which unit for the thickness of a coin?, step 3