Grade 2 · 2.MD.A.2 · 4.MD.A.1

Smaller units give bigger counts

One bench, measured once in feet and once in inches, gets two different numbers. Why, and how can you tell before measuring which number will be bigger?

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Iterating a single unitGrade 1

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Smaller units give bigger counts

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Why this exists

Chukwuemeka and Aruzhan both measure the bench at the bus shelter. Chukwuemeka says it is 4. Aruzhan says it is 48. Neither made a slip. Chukwuemeka laid a foot ruler along the bench and counted feet. Aruzhan read an inch tape and counted inches. The bench is one bench; the two numbers belong to two different units.

Children already met this idea with home-made units. In Iterating a single unit a longer rain boot fitted fewer times along the same rug, and What a unit is, and why we agree on standard ones states the rule: a bigger unit gives a smaller count. This lesson carries that rule over to the standard units children now measure with every day, inches and feet and centimeters and meters. Those units sit on rulers and tapes, often two of them on one ruler, and a child who measures one object twice in two of them will get two different numbers. This lesson is about that moment: predicting which number will be bigger before measuring, explaining the difference by the size of the unit, and running the idea backward, so that two counts of one object tell which unit is the bigger one.

Every length here is a whole number of units, or the nearest whole number. Nothing is changed from one unit to another by a rule; each count is read off a tool.

The lesson runs over two days. Day 1 measures the same object in two standard units and compares the counts (Examples 1 to 3). Day 2 reasons from the counts alone: ordering units, sizing an unknown unit, and comparing lengths given in different units (Examples 4 to 6).

Definition

Day 1: one object, two units

To measure a length is to count how many copies of a unit fit along it, end to end, with no gaps and no overlaps. The answer is a number and a unit: 48 inches, 4 feet. The number alone is called the count here.

When one object is measured in two units of different sizes, the two counts are different. The bigger unit gives the smaller count, and the smaller unit gives the bigger count. Size and count move in opposite directions.

Day 2: reading the counts backward

The rule also runs backward. If one object has two counts in two units, the unit with the bigger count is the smaller unit. And if two objects have the same count in two different units, the object measured in the bigger unit is longer.

Intuition

Day 1: a ruler with two edges

Many rulers carry inches along one edge and centimeters along the other, as in Figure 1. Look at the spaces. An inch space on the top edge stretches across two centimeter spaces on the bottom edge and part of a third. Both edges run the same length of ruler, so the narrow centimeter spaces have to be packed in more tightly to cover it. An object laid along the ruler covers one stretch, and that stretch holds more centimeter spaces than inch spaces.

Figure 1One trowel, one ruler, two edges. It reaches 10 on the inch edge and just past 25 on the centimeter edge.

A child can feel this without any ruler. Cover a table edge with big books, then with small blocks. The table does not grow or shrink. The big books run out of table sooner, so fewer of them fit.

Day 2: three rows under one edge

Figure 2 lays three rows of units under one edge of a foam puzzle mat. Each row covers exactly the same edge. The top row is one big piece. The middle row is twelve middle-sized pieces. The bottom row is thirty small pieces and a sliver. Reading down, the pieces get smaller and the count gets bigger. Reading up, the pieces get bigger and the count gets smaller.

Figure 2One edge, three units. The bigger the pieces, the fewer it takes.

Once that picture is in mind, the counts can be read like a picture. A big count means the pieces were small. A small count means the pieces were big.

Results

> For one object: the bigger unit gives the smaller count, and the smaller unit gives the > bigger count. > > Backward: if one object has a bigger count in unit A than in unit B, then unit A is the > smaller unit. > > Same count: if two objects have the same count in two different units, the one measured in > the bigger unit is longer. > > Different counts, different units: the numbers alone do not say which object is longer.

The standard length units met so far, from smallest to biggest:

Unit	About the size of
centimeter	the width of a ones cube
inch	the width of a quarter coin
foot	a sheet of school paper, a little longer
yard	the width of a front door
meter	a little longer than a yardstick

So, for any object at least one centimeter long, its count in centimeters is bigger than its count in inches, its count in inches is bigger than its count in feet, and so on down the table.

Why it is true

Take an object and two units, one bigger than the other. Say the bigger unit fits along the object exactly 4 times, as the foot ruler does along the bench in Figure 3 (Example 2).

Now lay one small unit under each big one, so there are 4 small units too, one under each foot ruler. Each small unit is shorter than the big unit above it, so each one leaves a piece of its big unit uncovered. Four small units together reach less far than four big ones. They stop short of the end of the bench. More small units must be added to reach the end, so the count in the small unit is more than 4.

Nothing in that argument used the number 4 or the bench. Whatever the bigger unit’s count is, the same number of smaller units falls short, so the smaller unit needs more. That is the rule.

The backward rule follows from it. Suppose one object has 14 of one unit and 36 of another. If the unit with 36 were the bigger one, the rule would give it the smaller count, but 36 is not smaller than 14. So the unit with 36 cannot be the bigger one. It is the smaller one.

Worked examples

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

Example 1

The trowel on both edges

Afua lays a garden trowel from the flower box along a ruler with inches on the top edge and centimeters on the bottom edge, as in Figure 1. The end of the handle is at 0. Before she reads either edge, will the count in centimeters be bigger or smaller than the count in inches? Then read both edges.

Solution

Predict first. A centimeter is a smaller unit than an inch: on the ruler, each inch space is wider than two centimeter spaces together. The trowel covers the same stretch on both edges, and the smaller unit gives the bigger count, so the centimeter count will be bigger.

Now read. On the top edge the tip of the blade is at the 10 mark, so the trowel is 10 inches long. On the bottom edge the dashed line from the tip falls just past the 25 mark, much nearer 25 than 26. To the nearest centimeter the trowel is 25 centimeters long.

25 is bigger than 10, as predicted. Both measurements are right. “The trowel is 10” and “the trowel is 25” are both half an answer; with their units, 10 inches and 25 centimeters name the same trowel.

Example 2

The bench at the bus shelter

Chukwuemeka lays one foot ruler along the bench at the bus shelter, marks where it ends, and moves it along. The fourth placement ends exactly at the far end of the bench. Aruzhan is going to measure the same bench with an inch tape. Will she get more than 4, fewer than 4, or exactly 4? Then she reads the tape, as in Figure 3. What does she get?

Figure 3Four foot rulers fit along the bench. On the inch tape, the same bench ends at 48.

Solution

The placements touch with no gaps and no overlaps, so the bench is 4 feet long.

An inch is a much smaller unit than a foot: a foot ruler has 12 inch spaces on it (Customary length units). The smaller unit gives the bigger count, so Aruzhan will get more than 4, and a lot more, since even one foot ruler’s stretch already holds 12 inches.

On the tape, the far end of the bench is at the 48 mark. The bench is 48 inches long. The bench is 4 feet long and 48 inches long. The count grew because the unit shrank; the bench did not change.

Example 3

Which count is which?

Chukwuemeka measures the flower box twice, once with an inch ruler and once with a centimeter ruler. He writes down 14 and 36, and forgets to write the units. Which count is in inches and which is in centimeters?

Solution

Both counts are for one flower box, so the rule applies: the smaller unit gives the bigger count.

Of inches and centimeters, the centimeter is the smaller unit, as the two edges of a ruler show. So the bigger count, 36, belongs to centimeters, and the smaller count, 14, belongs to inches. The flower box is 14 inches long, and to the nearest centimeter it is 36 centimeters long.

Check the other way round. If it were 14 centimeters and 36 inches, the box would fit more inches than centimeters, which would make the inch the smaller unit. It is not, so that reading is impossible.

Example 4

Three units along a puzzle mat

Aruzhan measures one edge of a square foam puzzle mat three times, as in Figure 2. One foot ruler fits exactly. Twelve inch spaces fit exactly. Thirty centimeter spaces fit, with a sliver left over. Using only the three counts, put the foot, the inch and the centimeter in order from biggest to smallest.

Solution

All three counts are for the same edge. The mat edge is 1 foot, 12 inches, and to the nearest centimeter 30 centimeters. Put the counts in order from smallest to biggest:

1 < 12 < 30

The smallest count belongs to the biggest unit, and the biggest count to the smallest unit. So the units in order from biggest to smallest are the foot, the inch, the centimeter. The order of the counts and the order of the units run in opposite directions.

This matches the table in Results, and it matches Example 1, where the centimeter count was the bigger one again.

Example 5

A tape with no name

Aruzhan finds an old tape in a drawer. Its marks are evenly spaced, but she does not know what unit they are. She measures the bench from Example 2 with it and gets 16. What can she say about the size of the mystery unit compared with the inch and the foot?

Solution

The bench has three counts now: 4 feet, 48 inches, and 16 mystery units.

Compare the mystery unit with the foot. The mystery count, 16, is bigger than the foot count, 4. The bigger count belongs to the smaller unit, so the mystery unit is smaller than a foot.

Compare it with the inch. The mystery count, 16, is smaller than the inch count, 48. The smaller count belongs to the bigger unit, so the mystery unit is bigger than an inch.

4 < 16 < 48

The mystery count sits between the foot count and the inch count, so the mystery unit sits between the foot and the inch in size: bigger than an inch, smaller than a foot. The counts alone do not say exactly how big it is, but they pin it between two units she knows.

Example 6

Two ribbons for the flower box

Afua cuts a ribbon 20 inches long to tie around the flower box. Aruzhan cuts a ribbon 20 centimeters long. Whose ribbon is longer? Then Chukwuemeka brings a ribbon 30 centimeters long. Is it longer than Afua’s, since 30 is bigger than 20?

Solution

Afua and Aruzhan have the same count, 20, in different units. Twenty inches means twenty inch spaces end to end; twenty centimeters means twenty centimeter spaces end to end. Each inch space is longer than each centimeter space, so twenty of them reach further. Afua’s ribbon is longer.

Chukwuemeka’s 30 and Afua’s 20 are different counts in different units, so the numbers alone cannot be compared. Use something measured in both units. The puzzle mat edge in Example 4 is about 30 centimeters, and it is also 12 inches. So Chukwuemeka’s ribbon, laid along that edge, would reach about to the end of it: it is about 12 inches long.

Now both lengths are in inches. About 12 is less than 20, so Afua’s ribbon is longer. The bigger number, 30, belonged to the shorter ribbon, because its unit was the smaller one.

Boundaries

One object, or two equal lengths. The rule compares two counts of the same length. Counts of two different objects in two different units say nothing by themselves, as Example 6 shows: 30 centimeters is shorter than 20 inches.

Same count, different units. When the counts are equal, the bigger unit gives the longer object. That is the only comparison of two objects the counts settle on their own.

The rule gives a direction, not an amount. Knowing that a centimeter is smaller than an inch says the centimeter count is bigger. It does not say that 10 inches is 25 centimeters; that number had to be read off the ruler. Working out one count from the other is changing units, which belongs to Customary conversions and Converting metric units by powers of ten.

Both measurements must be done properly. Each count assumes the units were laid end to end, with no gaps and no overlaps, along a straight line, as in Iterating a single unit. A careless count in the small unit can come out smaller than a careful count in the big one.

Nearest whole units. A count that is “about 25” is still a count, and the rule holds for it when the two units are far apart in size, like the inch and the centimeter. Two units close in size can round to the same count: a stick one yard long is 1 yard, and to the nearest meter it is 1 meter too. Very short objects also need care. An object shorter than the bigger unit has a count of “less than 1” in it, which is still the smaller count.

Length with length. Both units must measure the same thing. A count in inches and a count in cups are not two measurements of one length, and nothing about them compares.

Where this goes next

Knowing which unit is bigger lets a child judge an answer before measuring, and that habit grows into benchmarks and estimating: picturing a doorway for a meter or a quarter for an inch. Reading a length off the two edges of a ruler leads into lengths on a number line, where a ruler’s marks become numbers.

Later the rule becomes exact. A foot is twelve inches, so a count in feet becomes a count in inches twelve times as big, which is Customary conversions and Converting metric units by powers of ten. And the smaller unit’s bigger count is also a finer one: measuring in smaller units gets closer to the true length, which is Measurement is approximate: precision of a tool.

Exercises

Day 1: one object, two units

Exercises 1 to 4 use Figure 4.

Figure 4Rulers for Exercises 1 to 4.

How long is the eraser on ruler A in inches? About how long is it in centimeters?
How long is the plant marker stick on ruler B in inches? About how long is it in centimeters?
How long is the pencil on ruler C in inches? About how long is it in centimeters?
For the pencil on ruler C, which count is bigger, the inches or the centimeters? Say why without looking at the numbers.
A fence board is 5 feet long. If you measure it with an inch tape, will you get more than 5, fewer than 5, or exactly 5?
A shelf is about 70 centimeters long. If you measure it with a meter stick, will the count be more than 70, fewer than 70, or exactly 70?
A garden hose is measured once in yards and once in feet. The two counts are 5 and 15. Which count is in yards?
A picture book is measured once in inches and once in centimeters. The two counts are 9 and 23. Which count is in inches?
Afua measures the hallway at school in meters and in centimeters. Without measuring, which count will be bigger?
A rope is measured in feet and gets the count 6. It is measured again in yards. Is the new count more than 6 or fewer than 6?

Day 2: reading the counts

A plank is 6 of unit P long and 18 of unit Q long. Which unit is bigger?
A long rope is 4 yards, 12 feet and 144 inches long. Order the yard, the foot and the inch from biggest to smallest using only the counts.
A flower box is 2 feet long and 24 inches long. A stick of unknown length fits along it 6 times. Is the stick longer or shorter than a foot? Longer or shorter than an inch?
One ribbon is 15 inches long and another is 15 centimeters long. Which is longer?
One path is 10 yards long and another is 10 feet long. Which is longer?
One string is 40 centimeters long and another is 40 meters long. Which is longer?
A strip of paper is 25 centimeters long. Another is 12 inches long. Use Example 1 to say which is longer.
A table is 7 of unit A long, and a desk is 7 of unit B long. Unit B is bigger than unit A. Which is longer, the table or the desk?
One wall is 9 of some unit long and another is 5 of a different unit long. Can you tell which wall is longer? What would you need to know?

Reasoning

Explain why any object at least one inch long has a bigger count in centimeters than in inches.
Chukwuemeka says, “My bench is 4.” Why is that not a full answer?
Can one object, measured with no part left over, have the same count in two units of different sizes? Explain.
One pencil is 15 long and another is 7 long, but the units are not given. Give units that make the 7 pencil the longer one, and explain.

Answers

2 inches, about 5 centimeters. The eraser ends at the 2 mark on the top edge. On the bottom edge it ends just past 5, much nearer 5 than 6.
8 inches, about 20 centimeters. The stick ends at the 8 inch mark, and on the bottom edge just past 20, nearest 20.
6 inches, about 15 centimeters. The pencil ends at the 6 inch mark, and on the bottom edge just past 15, nearest 15.
The centimeters. A centimeter is smaller than an inch, and the smaller unit gives the bigger count for the same pencil. The numbers agree: 15 is bigger than 6.
More than 5. An inch is smaller than a foot, so the same board takes more of them.
Fewer than 70. A meter is much bigger than a centimeter, so the count shrinks. The shelf is shorter than one meter stick, so its count in meters is less than 1.
5 is in yards. A yard is bigger than a foot, and the bigger unit gives the smaller count, so 5 yards and 15 feet.
9 is in inches. An inch is bigger than a centimeter, so it gets the smaller count: 9 inches and 23 centimeters.
The centimeters. A centimeter is much smaller than a meter, so the hallway holds many more of them.
Fewer than 6. A yard is bigger than a foot, so the rope holds fewer yards than feet.
Unit P. The plank has the smaller count, 6, in unit P. The smaller count belongs to the bigger unit.
Yard, foot, inch. The counts go 4 < 12 < 144, and the units run the other way: the smallest count, 4, belongs to the biggest unit.
Shorter than a foot, longer than an inch. The stick’s count, 6, is bigger than the foot count, 2, so the stick is the smaller unit of those two. It is smaller than the inch count, 24, so the stick is the bigger unit of those two.
The 15 inch ribbon. Same count, and an inch is bigger than a centimeter, so 15 inches reach further.
The 10 yard path. Same count, and a yard is bigger than a foot.
The 40 meter string. Same count, and a meter is much bigger than a centimeter.
The 12 inch strip. The trowel in Example 1 is 10 inches and about 25 centimeters, so the 25 centimeter strip is about as long as the trowel, about 10 inches. About 10 is less than 12, so the 12 inch strip is longer.
The desk. Same count, 7, and unit B is bigger, so seven B units reach further.
No. The counts are in different units. You need both walls measured in one unit, or at least to know which unit is bigger and by how much. If the 5 unit were much bigger, the 5 wall could be the longer one.
Lay the object along a ruler with both edges. It covers one stretch. Each inch space is wider than a centimeter space, so as many centimeter spaces as inch spaces fall short of the end, and more centimeters must be added. So the centimeter count is bigger.
It has no unit. 4 feet and 4 inches are very different benches. The bench is 4 feet, and it is also 48 inches, so “4” alone could mean the wrong thing.
No. If both counts were the same, say 8, then 8 big units and 8 small units would reach exactly as far. But each small unit is shorter than each big one, so 8 small units reach less far. The counts must differ.
For example, 15 centimeters and 7 inches. Line the two pencils up on a two-edged ruler. The first reaches the 15 centimeter mark, which is just short of the 6 inch mark, as on ruler C in Figure 4. The second reaches the 7 inch mark, further along. So the 7 inch pencil is longer. Any choice where the 7 is in a big enough unit works, such as 15 inches and 7 feet.