Grade 3 · 3.MD.A.2

Metric mass units

A paper clip, a hamster and a melon are each heavy by a different amount. Which units say how much matter each one holds, and how do you get a number for it, first by feel and then with a balance or a scale?

You need

Metric prefixesGrade 3 Benchmarks, estimating and choosing a unitGrade 2

Unlocks

Story problems with grams, kilograms and litersGrade 3 DensityMath 6

Where you are 3 in Mass and weight

Measurement12

Mass and weight3

Mass is not the same as weight Customary weight units

Metric mass units

Why this exists

Coumba runs a produce stand. A customer picks up a melon and asks, “How heavy is it?” Coumba could say “pretty heavy” or “about as heavy as a big book”, but the customer’s big book is not her big book. To answer in a way that means the same thing to everyone, she needs a unit: an amount of heaviness that everyone has agreed on, so that “two of them” says exactly how much.

The metric units for how heavy a thing is are the gram and the kilogram. They measure mass, which is how much matter (how much stuff) an object holds. On Earth, an object with more mass is heavier to lift, so in this lesson “mass” and “how heavy” go together, and both are said plainly. Scientists keep mass and weight apart for a good reason that shows up off the Earth; that story is in Mass is not the same as weight. Everything in this lesson happens on the ground, at a produce stand and a pet shop.

Knowing the units is only half of it. A person also needs a feel for them, so that “a 2 gram bag of apples” sounds as wrong as it is, and a way to get a number: first an estimate by hand, then a measurement with a pan balance or a digital scale.

This lesson takes three days. Day 1 (Examples 1 and 2) is the gram and the kilogram, a picture of each, and choosing the unit that suits an object. Day 2 (Examples 3 and 4) is estimating a mass by comparing it with an object whose mass you already know. Day 3 (Examples 5 to 7) is reading a digital scale and a pan balance.

Definition

Day 1: the gram and the kilogram

Mass is how much matter an object holds. A melon holds more matter than a grape, so it has more mass.

A gram is a small unit of mass, about the mass of one paper clip. Its symbol is \text{g}, so 40\text{ g} is read “forty grams”.

A kilogram is 1{,}000 grams. The word part kilo- means a thousand, as it does in kilometer (Metric prefixes). A kilogram is about the mass of a full 1-liter bottle of water. Its symbol is \text{kg}, so 3\text{ kg} is read “three kilograms”.

1\text{ kg} = 1{,}000\text{ g}

Two more units sit at the far ends and are only met now and then.

A milligram (\text{mg}) is one of 1{,}000 equal parts of a gram, so 1{,}000 milligrams make 1 gram. A single grain of rice is a few dozen milligrams. Milligrams turn up on medicine and vitamin labels.
A metric ton (\text{t}) is 1{,}000 kilograms. Outside the United States it is spelled tonne. A small car has a mass of about 1 metric ton.

unit	size	about as heavy as
\text{mg}	1{,}000 make 1\text{ g}	a mosquito is a few
\text{g}	base unit	a paper clip
\text{kg}	1{,}000\text{ g}	a 1-liter water bottle
\text{t}	1{,}000\text{ kg}	a small car

The two units used every day are the gram and the kilogram. Grams suit things you could hold between your fingers or in one cupped hand: a strawberry, a pencil, a hamster. Kilograms suit things you lift with your arms: a bag of potatoes, a sack of dog food, a person.

Intuition

Day 1: a picture of a kilogram made of grams

A single paper clip is about 1 gram, which is too light to feel much in the hand. A box of 100 paper clips is about 100 grams, and that you can feel: it sits in the palm with a little weight. Ten such boxes together are about 1{,}000 grams, which is 1 kilogram, about the same as a full 1-liter bottle of water.

Figure 1Ten boxes of 100 paper clips make about 1 kilogram, about the mass of a full 1-liter bottle of water.

Those two pictures, the box of clips and the bottle of water, are the two benchmarks the whole lesson leans on. A benchmark (also called a referent) is a familiar object whose size you already know and can bring to mind, as in Benchmarks, estimating and choosing a unit.

The same amount can be named in either unit. A melon of 2 kilograms is also 2{,}000 grams, since each kilogram is 1{,}000 grams and 2 thousands is 2{,}000. The bigger unit gives the smaller number, because fewer big units are needed to make up the same mass. A sensible unit is one that gives a number that is easy to say and picture: “2 kilograms” rather than “2{,}000 grams” for the melon, “20 grams” rather than a tiny part of a kilogram for a mouse.

Day 2: estimating with a referent

To estimate a mass is to give a sensible “about” number without measuring. The method is to compare by hand. Hold the object in one hand and a referent in the other, and feel which one pushes down harder. Then ask how many referents the object feels like.

If it feels lighter than the box of 100 paper clips, it is less than about 100 grams.
If it feels heavier than the box but lighter than the water bottle, it is somewhere between about 100 grams and 1{,}000 grams. Comparing with two or three boxes held together narrows it down.
If it feels heavier than the water bottle, it is more than about 1 kilogram, and comparing with two or three bottles (or 1 kilogram bags of rice) together narrows it down.

An estimate is honest when it is given as “about” or as a range, such as “between 100 and 200 grams”. Hands cannot feel the difference between 170 grams and 175 grams, so naming one exact number would claim more than the hands know.

Size can fool the eye. A pillow is big and light; a rock the size of a fist is small and heavy. That is why a mass is estimated by holding, not by looking.

Results

Day 3: two ways to measure a mass

A pan balance has a beam that rests on a center post, with a pan at each end. The object goes in one pan and known masses (metal blocks marked 100\text{ g}, 50\text{ g}, 20\text{ g} and so on) go in the other.

> Reading a pan balance. The side that goes down holds more mass. When the beam is level, the > object has the same mass as all the known masses together, so its mass is their total.

A digital scale has a flat top and a screen. With nothing on it, the screen shows 0. Put an object on top and the screen shows its mass as a number and a unit.

> Reading a digital scale. Check that it shows 0 when empty, then read the number and the > unit. The same mass reads as a smaller number in kilograms than in grams: > 1\text{ kg} = 1{,}000\text{ g}.

Why it is true

Why a level beam means equal masses. The beam is like a seesaw balanced on its middle. Each pan is pulled down by how heavy its load is. If one side is heavier, it wins and goes down. The beam can only sit level when neither side wins, which is when the two loads are equally heavy. Both pans are side by side in the same place, so equally heavy means equal mass. (Why the balance still agrees with itself even on the Moon is the subject of Mass is not the same as weight.)

Why the known masses are added. Mass is an amount of matter. Put a 100 gram block and a 50 gram block in the same pan and that pan holds all the matter of both, 100 + 50 = 150 grams. Nothing is lost or gained by setting them side by side, so the masses in one pan add.

Why a bigger unit gives a smaller number. Measuring is counting how many of a unit fit into the mass. A kilogram is 1{,}000 times as much as a gram, so it takes only 1 kilogram where it takes 1{,}000 grams. For any mass, the count in grams is 1{,}000 times the count in kilograms.

Worked examples

Day 1 is Examples 1 and 2. Day 2 is Examples 3 and 4. Day 3 is Examples 5 to 7.

Example 1

A sensible mass for a bag of apples

The sign on a bag of apples at Coumba’s produce stand says 2, but the unit has been smudged off. Which is the sensible mass for the bag: 2 grams or 2 kilograms?

Solution

Test each choice against a benchmark.

2 grams is about the mass of 2 paper clips. A single apple is far heavier than two paper clips; it pushes down on the hand like a box of many clips. A 2 gram bag could not hold even one apple.
2 kilograms is about the mass of 2 full 1-liter bottles of water. A bag of apples lifted by its handle feels like that: heavy enough to notice, light enough to carry with one hand.

The sensible mass is 2 kilograms. In grams, that bag is 2{,}000 grams, since 2 thousands of grams is 2{,}000 grams.

Example 2

Grams or kilograms at the pet shop

Bintou is writing labels at the pet shop, but she has only written the numbers so far. Choose grams or kilograms for each.

item	number
a parakeet	30
a sack of dog food	15
a goldfish	3
a guinea pig	1

Solution

The number alone cannot decide: a small number may go with the big unit and a large number with the small one. For each item, picture both choices with the benchmarks.

The parakeet, 30. Thirty kilograms is about thirty water bottles, as heavy as a large dog. Thirty grams is about thirty paper clips, a small handful, which suits a little bird that perches on a finger. Grams.
The sack of dog food, 15. Fifteen grams is a pinch of clips; the empty sack alone is heavier. Fifteen kilograms is fifteen water bottles, a big sack you hug to lift. Kilograms.
The goldfish, 3. Three kilograms is three water bottles, far too heavy for a small fish. Three grams is three paper clips. Grams.
The guinea pig, 1. One gram is a single paper clip, much too light for an animal you hold with both hands. One kilogram is a water bottle, about right. Kilograms.

item	mass
a parakeet	30\text{ g}
a sack of dog food	15\text{ kg}
a goldfish	3\text{ g}
a guinea pig	1\text{ kg}

The largest number in the list, 30, belongs to one of the lightest animals, because it is counted in the small unit.

Example 3

A cabbage against a bag of rice

Ibrahima keeps a 1 kilogram bag of rice under the counter at the produce stand as his referent. A customer asks how heavy a large cabbage is. Ibrahima holds the cabbage in one hand and the bag of rice in the other. The cabbage pushes down harder. Then he holds two 1 kilogram bags of rice together in one hand. Now the bags push down a little harder than the cabbage. Estimate the mass of the cabbage.

Solution

Each comparison puts the cabbage on one side of a known amount.

Heavier than one bag: the cabbage is more than about 1 kilogram.
Lighter than two bags: the cabbage is less than about 2 kilograms, which is 2{,}000 grams.

So the cabbage is between about 1 kilogram and 2 kilograms. The two bags felt only a little heavier, so the cabbage is nearer 2 kilograms than 1. A fair estimate is “a bit less than 2 kilograms”. Kilograms are the right unit, since the cabbage is heavier than the water bottle benchmark.

When Ibrahima then sets the cabbage on his scale, the screen reads 1{,}800 grams. That is between 1{,}000 grams and 2{,}000 grams, and much nearer 2{,}000: the estimate was a good one.

Example 4

How heavy is a hamster?

At the pet shop, Bintou cups a hamster gently in one hand and holds a box of 100 paper clips in the other. The hamster pushes down harder than the box. Then she holds two such boxes together: they push down harder than the hamster. Estimate the hamster’s mass, and say which unit suits it.

Solution

One box of clips is about 100 grams, and two boxes are about 100 + 100 = 200 grams. The hamster is heavier than one box and lighter than two, so its mass is between about 100 grams and 200 grams. A fair estimate is “about 150 grams”, halfway between.

The hamster is far lighter than a 1-liter water bottle, so it is well under a kilogram. Grams give a number that is easy to say; in kilograms the hamster would be only a small part of 1. Grams suit it.

Example 5

A melon on a digital scale

Coumba sets a melon on the digital scale at her produce stand. The empty scale showed 0. With the melon on it, the screen shows the left reading in the figure. She presses the unit button and the screen changes to the right reading. What is the melon’s mass, and why are there two numbers?

Figure 2One melon, one mass, two readings. The screen shows the number and the unit; both must be read.

Solution

Read the number and the unit together. The left screen says 2\text{ kg}: the melon has a mass of 2 kilograms. The empty scale read 0, so the whole reading belongs to the melon.

The right screen says 2000\text{ g}, which is 2{,}000 grams (screens leave out the comma). The melon did not change. Each kilogram is 1{,}000 grams, so 2 kilograms is 2 thousands of grams:

2\text{ kg} = 2{,}000\text{ g}

Both readings name the same mass. The kilogram reading has the smaller number because the kilogram is the bigger unit. A check with the benchmark: 2 kilograms is about two full water bottles, which is a fair mass for a melon.

Example 6

Balancing the hamster

Bintou now measures the hamster from Example 4 on the pet shop’s pan balance. She sets the hamster in the left pan. She puts a 100 gram mass in the right pan, but the hamster’s side stays down. She adds 50 grams: still down. She adds 20 grams: still down, only just. She adds 5 grams, and the beam is level, as in the figure. What is the hamster’s mass?

Figure 3The beam is level, so the hamster has the same mass as the four masses together, 175 grams.

Solution

Each time the hamster’s side stayed down, the hamster had more mass than the blocks so far, so Bintou added more. When the beam is level, the hamster has the same mass as all the blocks in the right pan together. Add them, one block at a time:

\begin{aligned} 100 + 50 &= 150\\ 150 + 20 &= 170\\ 170 + 5 &= 175 \end{aligned}

The hamster has a mass of 175 grams. That lies between 100 grams and 200 grams, inside the range Bintou estimated by hand in Example 4.

Example 7

Carrots that tip the beam

Ibrahima puts a bunch of carrots in the left pan of a balance. His masses are 500 grams, 200 grams, 100 grams and 50 grams. He tries three ways, shown in the figure from top to bottom.

With 500 grams in the right pan, the carrot side goes down.
With 500 grams and 200 grams, the masses side goes down.
With 500 grams, 100 grams and 50 grams, the beam is level.

What does each try say about the carrots?

Figure 4The side that goes down holds more mass. The carrots are more than 500 grams, less than 700 grams, and exactly balance 650 grams.

Solution

The side that goes down holds more mass.

First try. The carrots went down, so the carrots have more than 500 grams.
Second try. The masses went down, so the masses have more than the carrots. They total 500 + 200 = 700 grams, so the carrots have less than 700 grams. After two tries the carrots are known to be between 500 grams and 700 grams, though neither try was level.
Third try. The beam is level, so the carrots have exactly the mass of the blocks:

500 + 100 + 50 = 650

The carrots have a mass of 650 grams. That fits both earlier tries: 650 is more than 500 and less than 700. Since a kilogram is 1{,}000 grams, the carrots are a bit more than half a kilogram.

Boundaries

Units must match before masses are put together. 2\text{ kg} and 300\text{ g} are not 302 of anything. Write the kilograms as grams first: 2{,}000 + 300 = 2{,}300 grams.
Mass is not size. A large object can have little mass and a small one a lot. Benchmarks are compared by holding, never by looking.
An estimate gives a range, not an exact number. Two careful people can give different estimates, and both can be fair.
A balance reads only as finely as its smallest mass. With blocks no smaller than 1 gram, an object that falls between two whole numbers of grams leaves the beam slightly tipped every time; the closest you can say is “between” those two numbers. How finely a tool can measure is the subject of Measurement is approximate: precision of a tool.
A digital scale must read 0 when empty, or every reading is off by what the screen showed first. A scale also has a top limit: a kitchen scale cannot weigh a person.
Mass and weight are the same story only on Earth. Everything here is measured on the ground. Off the Earth an object keeps its mass but its weight changes; see Mass is not the same as weight.

Where this goes next

Once masses can be measured, they can be put to work. Story problems with grams, kilograms and liters solves story problems with grams, kilograms and liters, adding, subtracting, multiplying and dividing masses given in the same unit. The partner units for how much a container holds, the liter and the milliliter, are in Liquid volume: liters and milliliters, and the United States units for how heavy a thing is, ounces and pounds, are in Customary weight units. How all the metric units share one pattern of tens is in The metric system and SI units. Much later, mass is divided by volume to say how tightly matter is packed, which is density.

Exercises

Day 1: grams, kilograms and sensible sizes

Write with a symbol: (a) seven kilograms; (b) forty grams. Read aloud: (c) 15\text{ kg}; (d) 300\text{ g}.
Choose grams or kilograms for the mass of (a) a watermelon; (b) a single strawberry; (c) a sack of potatoes; (d) a pencil.
At the produce stand, which is the sensible mass for one pear: 200 grams or 200 kilograms?
At the pet shop, a bag of cat litter is labeled 10. Is that 10 grams or 10 kilograms?
Complete each label at the pet shop with \text{g} or \text{kg}: (a) a kitten’s collar, 20; (b) a large dog, 35; (c) an empty hamster food bowl, 90; (d) a sack of rabbit food, 4.
Use 1\text{ kg} = 1{,}000\text{ g}. How many grams are (a) 4 kilograms; (b) 9 kilograms?
A box holds 100 paper clips. About how many grams is the box of clips? About how many such boxes together make about 1 kilogram?
Choose milligrams, grams, kilograms or metric tons for the mass of (a) a dump truck; (b) a single sesame seed; (c) a school backpack full of books; (d) an orange.

Day 2: estimating with a referent

Coumba holds a bunch of grapes in one hand and a 1 kilogram bag of rice in the other. The bag pushes down harder. Then she compares the grapes with a box of 100 paper clips, and the grapes push down harder. Between which two masses is the bunch of grapes? Is “about 500 grams” a fair estimate? Is “about 5 kilograms”?
Ibrahima’s sack of onions feels about as heavy as 3 bags of rice held together, each bag 1 kilogram. Estimate the mass of the onions in kilograms and in grams.
At the pet shop, a small bag of birdseed feels about as heavy as 4 boxes of 100 paper clips held together. Estimate its mass.
Choose the best estimate for the mass of a pumpkin: 4 grams, 40 grams or 4 kilograms.
A pillow is much bigger than a brick. Which do you expect to have more mass, and how could you check with your hands alone?
A puppy at the pet shop feels heavier than 2 bags of rice held together, but lighter than 3 bags. Each bag is 1 kilogram. Between how many grams is the puppy’s mass?

Day 3: reading a balance and a scale

A pan balance is level with a bag of nuts in the left pan and masses of 200 grams, 100 grams and 20 grams in the right pan. What is the mass of the nuts?
A pear balances these masses: 100 grams, 50 grams, 20 grams, 10 grams and 1 gram. What is the mass of the pear?
A bag of rabbit food is in the left pan. With 500 grams in the right pan, the rabbit food side goes down. With two 500 gram masses in the right pan, the masses side goes down. What can you say about the mass of the rabbit food?
The digital scale at the produce stand reads 3\text{ kg} for a bag of oranges. What is the mass of the oranges in grams?
One scale shows 900\text{ g} for a bag of carrots. Another shows 1\text{ kg} for a bag of beets. Which bag has more mass?
Coumba’s masses are 100 grams, 50 grams, 20 grams, 10 grams, 5 grams, 2 grams and 1 gram, one of each. Which of them would she put in the pan to balance a lemon of 128 grams?

Reasoning

A pan balance is level with a bag of feathers in one pan and a small metal block in the other. Which has more mass? Explain.
Is it always true that a mass has a bigger number in grams than in kilograms? Explain.
A digital scale shows 3\text{ kg} for a box of squash, and another shows 400\text{ g} for a bag of peppers. Explain why the two together are not 403 of anything, and find their total mass in grams.
Explain why an estimate of a mass is often given as a range, such as “between 100 and 200 grams”, rather than as one exact number.

Answers

(a) 7\text{ kg}. (b) 40\text{ g}. (c) Fifteen kilograms. (d) Three hundred grams.
(a) Kilograms: a watermelon is heavier than a water bottle. (b) Grams: a strawberry is about as heavy as a few paper clips. (c) Kilograms: a sack of potatoes is lifted with the arms. (d) Grams: a pencil is a few paper clips.
200 grams. Two hundred kilograms is heavier than two grown-ups; 200 grams is about two boxes of paper clips, a fair mass for a pear.
10 kilograms. Ten grams is ten paper clips, lighter than the empty bag; ten kilograms is ten water bottles, a heavy bag.
(a) 20\text{ g}. (b) 35\text{ kg}. (c) 90\text{ g}. (d) 4\text{ kg}. Picture each number of paper clips or of water bottles and keep the one that fits.
(a) 4{,}000 grams: 4 thousands of grams. (b) 9{,}000 grams: 9 thousands of grams.
About 100 grams, since each clip is about 1 gram. About 10 boxes, since 10 \times 100 = 1{,}000 grams, which is 1 kilogram.
(a) Metric tons. (b) Milligrams. (c) Kilograms. (d) Grams.
Between about 100 grams and 1{,}000 grams. “About 500 grams” is fair, since it is inside that range. “About 5 kilograms” is not: that is five bags of rice, and the grapes are lighter than one.
About 3 kilograms, which is about 3{,}000 grams, since 3 thousands of grams is 3{,}000.
About 400 grams, since 4 \times 100 = 400.
4 kilograms. Four grams and forty grams are a few clips or a small handful of clips, far too light for a pumpkin; four kilograms is four water bottles.
The brick. Hold the pillow in one hand and the brick in the other and feel which pushes down harder. Size does not show mass.
Between 2{,}000 grams and 3{,}000 grams, since 2 kilograms is 2{,}000 grams and 3 kilograms is 3{,}000 grams.
320 grams, since 200 + 100 + 20 = 320.
181 grams, since 100 + 50 + 20 + 10 + 1 = 181.
More than 500 grams and less than 1{,}000 grams. The first try shows the food is more than 500; the two masses total 500 + 500 = 1{,}000 grams and went down, so the food is less.
3{,}000 grams, since each kilogram is 1{,}000 grams and 3 thousands is 3{,}000.
The beets. 1\text{ kg} is 1{,}000 grams, and 1{,}000 is more than 900.
100, 20, 5, 2 and 1 grams, since 100 + 20 + 5 + 2 + 1 = 128. This is the only way: without the 100 the rest total only 88, the 50 with the 100 is already 150, and after 100 + 20 the last 8 grams can only be 5 + 2 + 1.
Neither: they have the same mass. A level beam means the two loads are equally heavy, whatever their sizes.
Yes, for any mass that is not zero. Each kilogram is 1{,}000 grams, so the count in grams is 1{,}000 times the count in kilograms, which is bigger.
The 3 counts kilograms and the 400 counts grams, so they are counts of different units and cannot simply be added. Writing 3 kilograms as 3{,}000 grams makes the units match: 3{,}000 + 400 = 3{,}400 grams.
Hands can tell heavier from lighter but cannot feel small differences, such as 170 grams against 175 grams. A range says what the hands really know; one exact number would claim more.