Grade 2 · 2.MD.A.3 · 4.MD.A.1

Customary length units

A dog brush, a picnic table, a snow fort wall and the walk to the park are all lengths. Which named unit fits each one, how big is each unit, and how do the inch, the foot and the yard fit together?

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Measuring lengthGrade 1

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Why this exists

Amarachi and Hohepa help out at a pet shelter. Amarachi measures a shelf for new dog beds with her hand and gets 8 hand spans. Hohepa measures the same shelf with his hand and gets 10. Both counted carefully. The shelf did not change. Their hands are different sizes, so their units are different sizes, and the two numbers cannot be put side by side.

A unit only lets people share a length when everyone uses the same one. A standard unit is a unit whose size is fixed and agreed, so that an inch in one town is the same as an inch in any other town. In the United States the everyday standard units of length are the inch, the foot, the yard and the mile. They are called customary units. Most of the rest of the world uses centimeters, meters and kilometers instead.

This lesson gives each customary unit a size you can picture and a tool to measure with: an inch ruler, a foot ruler and a yardstick. It shows, by laying the tools end to end, that 12 inches make a foot and 3 feet make a yard. It ends with the mile, which is too long for any ruler, and with choosing the unit that fits the job. Reading a ruler is already known from Measuring length. Every length here is a whole number of units.

The lesson runs over four days, one unit a day. Day 1 is the inch (Examples 1 and 2). Day 2 is the foot (Examples 3 and 4). Day 3 is the yard (Examples 5 and 6). Day 4 is the mile and choosing a unit (Examples 7 and 8).

Definition

Day 1: the inch

The inch is the smallest of the four units. An inch ruler is a row of inch spaces stuck together, with a numbered mark at the end of each space. Short things are measured in inches: a dog brush, a pencil, a dog treat, a toy.

Day 2: the foot

The foot is the length of 12 inches laid end to end. Its plural is feet. A foot ruler is the common school ruler: one foot long, marked from 0 to 12 inches. Things about as long as a person or a piece of furniture are measured in feet: a dog bed, a bench, a picnic table.

Day 3: the yard

The yard is the length of 3 feet laid end to end. A yardstick is one yard long. Since each foot is 12 inches, a yardstick is marked from 0 to 36 inches, because

12 + 12 + 12 = 36

Longer things, indoors and out, are measured in yards or feet: a snow fort wall, a hallway, a garden row.

Day 4: the mile

The mile is a much longer unit, used for how far apart places are: from home to school, from the shelter to the park, from one town to the next. A mile is far too long to lay a ruler along.

Each unit has a short form, used on signs and labels: \text{in.} for inch, \text{ft} for foot, \text{yd} for yard and \text{mi} for mile. In this lesson the words are written out.

Intuition

Day 1: how big an inch is

Hold up one inch between a thumb and a finger. It is about as wide as a quarter coin. That one picture is enough to tell a child that a dog treat a few inches long is small enough to hold in a fist, and that a dog brush of 7 inches fits in one hand. (Building a set of these pictures and estimating with them is the work of Benchmarks, estimating and choosing a unit.)

Figure 1The brush starts at 0 and ends at 7. It is 7 inches long.

On an inch ruler, each space between neighboring marks is one inch. The brush lies from the 0 mark to the 7 mark, so it covers 7 inch spaces and it is 7 inches long. This is the same reading as in Measuring length; the only new thing is the name of the unit.

Day 2: twelve inches make a foot

Cut out twelve paper squares, each exactly one inch wide, and lay them end to end on a foot ruler. They cover it from the 0 mark to the 12 mark, with no gaps and nothing hanging over. That is the whole meaning of “a foot is 12 inches”: the foot ruler is twelve inches in a row.

Figure 2Twelve inch squares cover a foot ruler exactly. That is what a foot is.

A foot is about the length of a sheet of school paper, a little longer. A child can pick up a foot ruler and see that a dog bed is a few rulers long. To measure something longer than one foot ruler, lay the ruler down, mark where it ends, and move it along, as in Iterating a single unit. The count of rulers is the length in feet.

Day 3: three feet make a yard

Lay three foot rulers end to end beside a yardstick. The first one ends at the 12 mark on the yardstick, the second at the 24 mark, the third at the 36 mark, which is the end of the yardstick. Three feet are exactly one yard.

Figure 3Three foot rulers end to end are exactly as long as one yardstick.

A yard is about as wide as a front door. Because one yardstick takes the place of three foot rulers, a long length needs fewer moves with a yardstick, and the count comes out smaller. (Why a bigger unit always gives a smaller count is its own idea, Smaller units give bigger counts.)

Day 4: a mile is a walk

Nobody measures a mile with a ruler. The picture to carry is a walk: at an easy walking pace, a mile takes about 20 minutes. So a mile is the unit for trips, roads and trails, and the inch, foot and yard are the 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 and many chances to slip. 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
inch	inch ruler	the width of a quarter coin
foot	foot ruler	a sheet of school paper, a little longer
yard	yardstick	the width of a front door
mile	no ruler	a walk of about 20 minutes

12 \text{ inches make } 1 \text{ foot}

3 \text{ feet make } 1 \text{ yard}

A length is a number together with a unit. The same table can be 9 feet and 3 yards; the numbers differ because the units differ, and the length is the same.

Why it is true

The two facts in the box are not discovered; they are agreed. A foot was made to be twelve inches long and a yard was made to be three feet long, and every ruler maker follows the same agreement. That is exactly what makes a standard unit useful: any two people with any two foot rulers will match.

What a child can do is check the agreement with real tools. Lay twelve inch squares on a foot ruler, as in Figure 2, and they fill it exactly. Lay three foot rulers beside a yardstick, as in Figure 3, and they match it exactly. Looking at the numbers on a yardstick shows the same thing: the foot rulers end at 12, then 12 + 12 = 24, then 24 + 12 = 36.

Why can one thing be 9 feet and also 3 yards? Put nine foot rulers in a row along it. Every group of three rulers in that row is as long as one yardstick, and nine rulers make three groups of three, since 3 + 3 + 3 = 9. So three yardsticks cover exactly what the nine foot rulers covered.

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 dog brush on an inch ruler

Hohepa lays a dog brush on an inch ruler at the shelter, as in Figure 1. The end of the handle sits on the 0 mark, and the far edge of the bristles sits on the 7 mark. How long is the brush?

Solution

The brush starts at 0, so the mark at its other end counts the inch spaces it covers. Count them: from 0 to 1 is one inch, from 1 to 2 is another, and so on up to the space from 6 to 7. That is 7 spaces. The brush is 7 inches long.

Example 2

Hand spans and inches

Amarachi and Hohepa measured the shelf for the dog beds in hand spans and got 8 and 10. Then each of them measures it again with an inch ruler, moving the ruler along. Amarachi gets 30 inches. What does Hohepa get, and why did the hand spans disagree when the inches agree?

Solution

Hohepa gets 30 inches too. An inch is the same size on Amarachi’s ruler and on Hohepa’s ruler, so the same shelf holds the same number of inches, whoever lays them down.

The hand spans disagreed because the two hands are different sizes. Hohepa’s hand is smaller, so more of his hand spans fit along the shelf: 10 against 8. Neither count was wrong. Each was counted in a different unit, and a count only means something to another person when both use the same unit.

Example 3

Inch squares along two foot rulers

Aizada lays two foot rulers end to end along a dog bed, and they reach exactly from one end of the bed to the other. How long is the bed in feet? She then lays a long tape measure, marked in inches, along the same bed from its 0 mark. Which inch mark is at the far end of the bed?

Solution

Two foot rulers fit exactly, so the bed is 2 feet long.

Each foot ruler covers 12 inch spaces, as Figure 2 shows. The first ruler ends at the 12 mark on the tape. The second covers 12 more inch spaces, so it ends at

12 + 12 = 24

The far end of the bed is at the 24 mark. The bed is 2 feet long, and it is also 24 inches long: the same length named in two units.

Example 4

The picnic table in feet

At the park next to the shelter, Amarachi measures a long picnic table with one foot ruler. She lays it at the left end, marks where it ends with her finger, moves it up to her finger, and keeps going. The ruler’s end lands exactly on the right end of the table on the ninth placement. How long is the table?

Solution

Each placement covers one foot, and the placements touch with no gaps and no overlaps, so the count of placements is the length in feet. The table is 9 feet long. Figure 4 shows the nine rulers in a row above the table.

Figure 4The same table takes 9 foot rulers or 3 yardsticks. It is 9 feet long, and it is 3 yards long.

Example 5

The same picnic table in yards

Hohepa measures the same picnic table with a yardstick. How many times does he lay it down, and how long is the table in yards?

Solution

A yardstick is as long as 3 foot rulers. The nine foot rulers from Example 4 split into three groups of three:

3 + 3 + 3 = 9

Each group of three is as long as one yardstick, so three yardsticks cover the table exactly. Hohepa lays the yardstick down 3 times, and the table is 3 yards long.

Both measurements are right. The table is 9 feet long and it is 3 yards long. The number changed because the unit changed; the table did not.

Example 6

The snow fort wall

Aizada builds a snow fort with a long front wall. She lays a yardstick along the wall, marks its end in the snow, and moves it along. Five placements fit exactly, as in Figure 5. How long is the wall in yards, and how many foot rulers would fit along it?

Figure 5Five yardsticks fit along the wall with nothing left over. The wall is 5 yards long.

Solution

Five yardsticks fit with nothing left over, so the wall is 5 yards long.

Each yardstick is as long as 3 foot rulers. Five yardsticks are five groups of three foot rulers, so count by threes:

3 + 3 + 3 + 3 + 3 = 15

Fifteen foot rulers would fit along the wall. The wall is 5 yards long, which is 15 feet.

Example 7

Choosing a unit at the shelter

Amarachi is making labels for four things at the shelter. Which unit fits each one: inches, feet, yards or miles?

the length of a dog treat
the length of a dog bed
the length of the shelter’s long hallway
how far the shelter is from the park, a walk of about 20 minutes

Solution

A dog treat is shorter than a foot ruler; it fits in a hand. Inches give a small count, a few inches, so inches fit.

A dog bed is longer than a foot ruler, and a few foot rulers fit along it, so feet fit. (Inches would work too, but the count runs into the dozens.)

A long hallway takes many foot rulers. A yardstick laid down a number of times does it in fewer moves, so yards fit. Feet also work.

The walk to the park takes about 20 minutes, and that is the picture of one mile. No ruler is laid along a road, so miles fit, and the park is about 1 mile away.

Example 8

Two dog pens

The shelter has two outdoor dog pens. The first is 4 yards long. The second is 10 feet long. Which pen is longer, and by how much?

Solution

The two numbers are in different units, so they cannot be compared as they stand. Picture foot rulers along both pens. Along the first pen, 4 yardsticks fit, and each yardstick is as long as 3 foot rulers, so count the foot rulers by threes:

3 + 3 + 3 + 3 = 12

So 12 foot rulers fit along the first pen, and it is 12 feet long. The second is 10 feet long. Now both are in feet, and 12 is more than 10, so the first pen is longer. The difference is

12 - 10 = 2

The first pen is longer, by 2 feet. The bigger number, 10, belonged to the smaller unit.

Boundaries

A number alone is not a length. “The pen is 4” says nothing until the unit is named. 4 inches, 4 feet, 4 yards and 4 miles are four very different lengths.

Two lengths are compared in the same unit. 10 feet against 4 yards cannot be judged by the numbers, as Example 8 shows. Put both in one unit first, by laying out the tools in your head.

Whole units only, here. Every length in this lesson fits a whole number of units. A real dog bed may end between two inch marks. Then its length is “about” the nearer whole inch, as in Measuring length, or it is read in half inches and quarter inches, as in Measuring to the half and quarter inch.

A mile is known by a picture, not a ruler. “About 20 minutes of walking” depends on how fast someone walks, so it gives “about 1 mile”, never an exact length.

These units are for length. An inch, a foot, a yard and a mile say how long or how far. They do not say how heavy something is or how much it holds; those need other units, met in The customary system.

Where this goes next

Reading between the whole inch marks, in halves and quarters of an inch, comes in Measuring to the half and quarter inch. The metric units, Metric length units, work the same way with different sizes. The customary system puts the length units beside the customary units of weight and capacity. Later, the facts seen here by laying tools end to end, 12 inches to a foot and 3 feet to a yard, become a way to change any length from one unit to another, together with the mile as 5{,}280 feet, in Customary conversions.

Exercises

Day 1: the inch

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

Figure 6Rulers for Exercises 1 to 3.

How long is the dog treat in row A?
How long is the toy mouse in row B, tail included?
The pencil in row C starts at the 3 mark. How long is it?
Aizada measures a bench with her shoe and gets 9 shoe lengths. Hohepa measures it with his shoe and gets 7. Then Aizada measures it with an inch ruler and gets 40 inches. What will Hohepa get with an inch ruler?
One ribbon is 11 inches long and another is 8 inches long. Which is longer, and by how much?

Day 2: the foot

How many one-inch squares cover one foot ruler exactly?
Amarachi lays four foot rulers end to end along a bench, and they fit exactly. How long is the bench in feet? Which inch mark on a tape measure is at the far end?
Hohepa lays one foot ruler along a dog crate, then moves it along. It fits exactly 3 times. How long is the crate?
A doormat is as long as one foot ruler and 3 more inches. How many inches long is it?
Which is longer, 1 foot or 10 inches? By how much?

Day 3: the yard

How many foot rulers laid end to end match one yardstick? Which inch number is at the end of a yardstick?
A row of snow blocks fits 2 yardsticks end to end exactly. How long is it in yards? How many foot rulers would fit along it?
Aizada lays a yardstick along the fence of a dog pen 7 times, and it fits exactly. How long is the fence?
A path to the picnic tables is 6 yards long. How many foot rulers fit along it?
Which is longer, 2 yards or 7 feet? By how much?

Day 4: the mile and choosing a unit

Choose inches, feet, yards or miles for each: the length of a dog’s collar; the length of a sled; the distance from the shelter to the next town.
Amarachi’s walk to school takes about 20 minutes at an easy pace. About how far is it?
At the same pace, about how long does a walk of 2 miles take?
Would you measure the distance from the shelter to the park with a yardstick? Say why or why not.
A label at the park reads “Picnic table: 9”. What does the label leave out, and why does it matter?

Reasoning

Why do two people with two different inch rulers get the same length for a shelf, when two people using their hands do not?
The snow fort wall is 5 yards long and 15 feet long. Explain, with yardsticks and foot rulers, why both are right.
Why do the numbers on a yardstick run from 0 to 36?
“A foot is 12 inches.” How could a child check this with paper and a foot ruler?

Answers

4 inches. The treat starts at 0 and ends at the 4 mark, so it covers 4 inch spaces.
9 inches. The mouse starts at 0 and its nose is at the 9 mark.
7 inches. It covers the spaces from 3 to 10, and 10 - 3 = 7.
40 inches. An inch is the same size on every inch ruler, so the bench holds the same number of inches for both. The shoe counts differ because the shoes are different sizes.
The 11 inch ribbon, by 3 inches, since 11 - 8 = 3.
12 squares. A foot ruler is twelve inch spaces in a row, from the 0 mark to the 12 mark.
4 feet. Each foot ruler covers 12 inches, so the end is at the 48 mark: 12 + 12 + 12 + 12 = 48.
3 feet. Each placement covers one foot, with no gaps or overlaps.
15 inches. The foot ruler is 12 inches, and 12 + 3 = 15.
1 foot, by 2 inches. One foot is 12 inches, and 12 - 10 = 2.
3 foot rulers, and the end of a yardstick is the 36 mark, since 12 + 12 + 12 = 36.
2 yards. Each yardstick matches 3 foot rulers, so 6 foot rulers fit: 3 + 3 = 6.
7 yards. Each placement covers one yard.
18 foot rulers. Each yard holds 3 of them: 3 + 3 + 3 + 3 + 3 + 3 = 18.
7 feet, by 1 foot. Two yardsticks match 3 + 3 = 6 foot rulers, so 2 yards is 6 feet, and 7 - 6 = 1. Here the bigger number does name the longer length.
A collar in inches (it is shorter than a foot ruler). A sled in feet (a few foot rulers fit along it; inches also work, with a bigger count). The next town in miles (no ruler is laid along a road).
About 1 mile. A walk of about 20 minutes is the picture of one mile.
About 40 minutes. Each mile takes about 20 minutes, and 20 + 20 = 40.
No. The park is about a mile away, a walk of about 20 minutes, and laying a yardstick down along a whole road would take a huge number of placements. Distances between places are told in miles.
The label gives a number but no unit, and 6 inches, 6 feet and 6 yards are very different lengths. It should say “Picnic table: 9 feet” (or “3 yards”, the same length).
Every inch ruler has inches of the same agreed size, so the same shelf always holds the same number of them. Hands come in different sizes, so the count depends on whose hand it is.
Five yardsticks fit along the wall. Each yardstick is as long as three foot rulers, so 3 + 3 + 3 + 3 + 3 = 15 foot rulers fit along it too. It is one length, named in two units.
A yardstick is three foot rulers long, and each foot ruler is 12 inches, so the inch marks run 12, then 24, then 36 at the end: 12 + 12 + 12 = 36.
Cut paper squares exactly one inch wide and lay them end to end along the foot ruler. Exactly twelve of them cover it, from the 0 mark to the 12 mark, with no gap and nothing over.