What a unit is, and why we agree on standard ones
Anyone can see that a pencil is longer than a crayon. How can we say how long the pencil is, with a number that someone else, even someone far away, could check?
Where you are 14 in What measurement is
Why this exists
A child can already tell which of two things is longer by standing them side by side (comparing two objects directly). But that only answers “which one?”. It cannot answer “how long?”, and it cannot help when one of the things is a table that will not move.
A number can travel where the table cannot. To get one, we pick one small thing, a cube, a paper clip, a hand, and count how many copies of it fit along the table. That small thing is a unit, and counting copies of it is measuring.
A count of hand-spans works well for the child who made it. It stops working as soon as the number has to reach someone else: a grandparent’s hand is a different size, and a shop in another town has never seen the child’s hand at all. So after saying what a unit is, this lesson has three more parts. It looks at the everyday things children measure with first (non standard units), at the one rule that makes any of them fair (why units must all be the same size), and at why people agreed long ago on units that are the same for everyone (why standard units exist).
Definition
What a unit is
A unit is one fixed amount that we choose to measure with. For length it might be the length of one cube, one paper clip or one hand-span. “Fixed” means every copy of the unit is the same size: every cube in the box, every paper clip from the same packet.
To measure the length of an object with a unit, lay copies of the unit along the object, end to end:
- the first copy starts exactly at one end of the object,
- each copy touches the one before it, with no gaps and no overlaps,
- the copies keep going until they reach the other end.
Then count the copies. The count, said together with the unit, is the measurement. In Figure 1, six cubes fit along the pencil and the last one finishes exactly at its point, so the pencil is 6 cubes long. (When the object ends partway along a copy, Example 5 shows what to say.)
The unit is part of the answer. “The pencil is 6” does not say how long it is; “the pencil is 6 cubes long” does.
Non standard units
A non standard unit is an everyday thing used as a unit: a paper clip, a cube from a box, a crayon, a straw, a shoe. The oldest are parts of the body, because a body is always there to measure with:
- a hand-span: the hand spread flat, from the tip of the thumb to the tip of the little finger;
- a footstep: one foot, heel to toe;
- a pace: one ordinary walking step;
- an arm span: both arms stretched out sideways, from fingertip to fingertip;
- a cubit: the forearm, from the elbow to the tip of the middle finger.
Measuring with a non standard unit follows exactly the rules above: start at one end, no gaps, no overlaps, count. There are two ways to lay the copies. With many copies, such as a box of paper clips, lay a row of them along the object. With only one copy, such as a hand, put it down, note where it ends, move it so that it starts exactly there, and keep count. That second way is iterating a single unit.
A sensible unit is a good deal smaller than the object, so that it fits several times, but not so small that the counting takes all day. Paper clips suit a book and footsteps suit a room. Paper clips across a playground would mean hundreds of copies, and a footstep along a crayon fits less than once.
Why units must all be the same size
A count of units is a measurement only when every copy of the unit is the same size: for length, the same length. Color and shape do not matter. A red cube and a blue cube from the same box are copies of one unit, because each reaches exactly as far along the object as the other.
In Figure 2 each ribbon is “5 clips long”, but the clips come in two sizes. Ribbon A has 5 small clips, ribbon B has 5 big clips, and ribbon C has some of each. The three ribbons are plainly different lengths, yet the count is the same for all three. “5 clips” says how long something is only when every clip is the same size.
Why standard units exist
A standard unit is a unit that has been fixed once, agreed on by many people, and copied carefully onto rulers and tapes, so that it is the same size for everyone, in every place, year after year. The centimeter, the meter, the inch and the foot are standard units of length. A hand-span is not: it changes from person to person, and even for one child it grows.
The point of a standard unit is that a measurement can travel. “The table is 120 centimeters long” means the same length to a carpenter in another town, who has never seen the table, as it does to the person who measured it, because both of their tapes carry the same centimeter.
A short history. People measured with their bodies for thousands of years, and the names of old units show it. The cubit of ancient Egypt began as the length of a forearm, and the foot began as about the length of a foot. Forearms differ, so for large buildings Egyptian builders used a royal cubit of fixed length, about 52 centimeters, and made measuring rods to match it; several of these cubit rods survive in museums today. Even so, for most of history a “foot” in one town was a different length from a “foot” in the next, which made buying and selling between towns a muddle.
In the 1790s scientists in France proposed one unit for everyone, the meter, and tied it to the size of the Earth: one ten-millionth of the distance from the North Pole to the equator. In 1799 a metal bar of that length, as near as their survey of the Earth could make it, was placed in the national archives in Paris. In 1875 seventeen countries signed the Meter Convention, an agreement to share the meter, and from 1889 the official meter was a new metal bar kept near Paris, with carefully measured copies sent to the countries that had signed. In 1960 the bar gave way to a length measured with light, and since 1983 the meter has been the distance light travels in empty space in \displaystyle \frac{1}{299\,792\,458} of a second, which a laboratory anywhere can reproduce without any bar. The inch has been fixed from the meter since 1959: one inch is exactly 2.54 centimeters.
Intuition
What a unit is
Each cube covers one cube’s worth of the pencil. With no gaps and no overlaps, every bit of the pencil is covered once and only once, so counting the cubes counts how many cube-lengths the pencil has in it.
Try it with real cubes. Stand the flat end of the pencil against the side of a book, push the first cube up to the same side, and push each next cube tight against the last. Say one counting word for each cube. The last word said is the length, for the same reason the last word in any count tells how many there are (How counting tells how many).
Non standard units
Children are measuring with non standard units long before anyone gives them a ruler: “the hall is 20 steps long”, “my book is 3 hands long”. Walk heel to toe across a room with the child, each of you counting your own footsteps. You will get different counts for the same room. Neither of you is wrong; you used different units. That small surprise is where the rest of this lesson starts.
Why units must all be the same size
A train of 5 carriages tells you how long the train is only if every carriage is the same length. Swap one carriage for a longer one and the train grows, but it is still “5 carriages”. A count of units is just like that: it holds on to a length only while every unit is the same size.
Why standard units exist
Picture a grandmother far away knitting a scarf as long as her grandchild’s arm span. “It is 8 of my hand-spans” is no help to her: her hand is bigger, so 8 of her spans would make a longer scarf. “It is 100 centimeters” works, because the centimeters on her tape are the same as the centimeters on the child’s tape. A standard unit lets two people who never meet share a length.
Results
What a unit is
A measurement is a number and a unit together. 6 cubes and 6 paper clips are different lengths.
With the same unit, the longer object gets the bigger count, and taking the smaller count from the bigger one says how much longer it is.
With a different unit, the same object gets a different count. A bigger unit gives a smaller count; a smaller unit gives a bigger count.
Lengths in the same unit, laid end to end, add.
Non standard units
Anything can serve as a unit of length, as long as every copy of it is the same size.
A body unit stays about the same for one person, and differs from person to person.
Why units must all be the same size
With every unit the same size, the same count always means the same length. With units of different sizes, one count can stand for many lengths, and one object can get many counts.
Why standard units exist
A measurement in a standard unit means the same length to everyone, so measurements made by different people, in different places, can be compared and added as if one person had made them all.
Why it is true
Everything here comes from the rules for laying the units.
What a unit is
Why no gaps and no overlaps. In Figure 3 the same ribbon is measured three times. In row A the cubes touch, and 6 fit. In row B there are gaps: the cubes still reach from end to end, but the parts of the ribbon in the gaps have no cube on them and are never counted, so the count is 4, too small. In row C the cubes lie over each other: some parts of the ribbon have two cubes on them and are counted twice, so the count is 8, too big. Only when every bit is covered exactly once does the count say how long the ribbon is.
Why start at the end. A first cube placed a little way along leaves the part before it uncovered, which is a gap at the start.
Why every copy must be the same size. If some cubes were big and some small, “6 cubes” could mean a short length or a long one, and nobody could check it. The part below, on why units must all be the same size, shows this in full.
Why a bigger unit gives a smaller count. Each copy of a bigger unit covers more of the object, so fewer copies reach the end. If a rod is exactly as long as 2 cubes, each rod can be swapped for 2 cubes, so every rod in the count becomes two cubes. Example 3 does this for one strip.
Why lengths add. Two ribbons laid end to end, with no gap, make one long ribbon. The clips that covered each ribbon together cover the long one, end to end, once each. So the long ribbon’s count is the two counts put together.
Non standard units
Nothing in the rules for laying units says what the unit has to be. The rules ask only for copies that start where the last one stopped and are all the same size. A paper clip from one packet, a cube from one box, one hand spread the same way each time: each of these meets that, so each is a real unit, and every result above holds for it. A body unit meets it only for one person. A second person brings a different-sized copy, which is a different unit, and that is why the two counts in Example 2 differ.
Why units must all be the same size
Same size, same count, same length. Take two rows of 5 clips, every clip the same size, and lay the rows side by side, starting together. The first clip of one row matches the first clip of the other exactly, so the second clips start together too, and match; and so on to the fifth. The two rows end together, so they are the same length. That is why “5 clips” names one length, and why counts in one unit can be compared and added.
Mixed sizes, same count, different lengths. Now swap one clip in a row for a bigger one. The count is still 5, but the row is longer by however much the big clip is longer. So with mixed sizes, “5 clips” can be many different lengths, as Figure 2 shows.
Mixed sizes, one object, different counts. Go the other way. Take a ribbon exactly as long as 10 small clips, and a box holding small clips and big clips, each big clip as long as 2 small ones. The ribbon could be covered by 10 small clips, by 5 big clips, or by 2 big and 6 small, which is 8 clips. One ribbon, three counts, and which one you get depends on which clips happened to come out of the box. A count that changes with the luck of the draw is not a measurement.
Why standard units exist
Two people’s counts can be compared only if their units are the same size: that is the part just shown. To know that two hand-spans are the same size, the two hands would have to be put side by side, and two people far apart cannot do that. So a unit made from bodies cannot be checked between people who are apart, and their counts cannot be compared.
A standard unit removes the problem once and for all. Every ruler marked in centimeters is made to match the one agreed centimeter. If my centimeter matches the agreed one, and yours matches the agreed one, then mine matches yours, since two lengths that each match a third length match each other. So our two counts are in the same unit after all, and everything this lesson says about one unit (compare, take away, add) works between us, even though we never meet.
Worked examples
Do each one with real objects where you can.
Which is longer, and by how much?
A pencil is 6 cubes long. A crayon is 4 cubes long. Which is longer, and by how many cubes?
SolutionBoth were measured with the same cubes, so the counts can be compared straight away. Six is more than four, so the pencil is longer.
To see by how much, picture the two rows of cubes side by side, starting together at the left. The crayon’s 4 cubes match the pencil’s first 4 cubes, and the pencil has cubes left over:
The pencil is 2 cubes longer than the crayon. The numbers compared the two without putting them side by side, which works only because the unit was the same.
Hand-spans across a table
A child spreads her hand flat on a table, from the tip of her thumb to the tip of her little finger. That stretch is her hand-span. She measures the table: it is 7 of her hand-spans long. Her grown-up measures the same table with his own hand-span and gets 4. Who is right?
SolutionBoth are right. They used different units.
To measure with a hand-span, put the thumb at one edge of the table and spread the hand flat. Then move the thumb to exactly where the little finger was, spread again, and count each spread. That is one hand used over and over, the way iterating a single unit works. If the hand is spread the same way each time, it is the same size each time, so it is a fair unit for that one person.
The grown-up needed fewer spreads, 4 against 7, to cross the same table. So each of his spreads must cover more of the table: his hand-span is the bigger unit, as a grown-up’s usually is. The table did not change; only the unit did.
So “the table is 7 hand-spans long” is only a full answer when we also know whose hand. This is why people came to agree on units that are the same for everyone (see the part on why standard units exist, and Example 8).
One strip, three units
A strip is measured three times, with small cubes, with rods and with long rods, as in Figure 4. Each rod is as long as 2 cubes, and each long rod is as long as 4 cubes. How long is the strip in each unit?
Count each row. 12 cubes fit along the strip, 6 rods fit, and 3 long rods fit. So the strip is 12 cubes long, 6 rods long, and 3 long rods long.
The three counts agree, and we can check. Each rod is 2 cubes, so 6 rods are
cubes. Each long rod is 4 cubes, so 3 long rods are
cubes. Same strip, three different counts: the bigger the unit, the smaller the count.
Two ribbons end to end
A red ribbon is 4 paper clips long. A blue ribbon is 3 paper clips long. They are laid end to end, with no gap, to make one long ribbon. How long is it?
SolutionThe 4 clips along the red ribbon and the 3 clips along the blue ribbon together cover the long ribbon from end to end, with no gaps and no overlaps. So count them all:
The long ribbon is 7 paper clips long. Check by laying 7 clips along it.
The two lengths could be added only because they were in the same unit. 4 clips and 3 cubes do not make 7 of anything.
When the units do not come out even
Figure 5 shows a stick with cubes laid along it. How long is the stick?
The first cube starts at the left end of the stick. Five whole cubes fit and the stick keeps going, but it ends partway along the sixth cube. So the stick is not 5 cubes long and not 6 cubes long. It is between 5 and 6 cubes long: longer than 5 cubes, shorter than 6. That is the honest answer. “A little more than 5 cubes long” is also true.
If one whole number is wanted, pick the nearer one and say “about”. The stick’s end is nearer the end of the fifth cube than the end of the sixth, so it is about 5 cubes long. The word “about” is what tells the listener the length is not exactly 5 cubes.
Choosing a sensible unit
A child measures a bookshelf three ways. It is 20 paper clips long, 5 crayons long, and less than 1 of her arm spans long. Each crayon is as long as 4 paper clips. Do the counts agree, and which unit suits a bookshelf best?
SolutionThe counts agree. Each crayon could be swapped for 4 paper clips, so 5 crayons cover as much as
paper clips, which is what the clips gave. Fewer crayons than clips, because the crayon is the bigger unit.
All three answers are true, but they do not say the same amount. “Less than 1 arm span” tells us only that the shelf is shorter than her outstretched arms; it cannot tell this shelf from a much shorter one. “20 paper clips” says how long the shelf is, but it took twenty careful placements, and every placement was a chance to leave a gap. “5 crayons” also says how long the shelf is, with only five placements. A unit a good deal smaller than the object, but not tiny, is the sensible choice: crayons for this shelf, footsteps for a room.
Five clips, two lengths
A box holds small paper clips and big paper clips, and each big clip is as long as 2 small clips. Lena measures a ribbon with small clips only and gets 5 clips (row A in Figure 2). Sam measures another ribbon with clips from the same box: small, big, small, big, small, also 5 clips (row C). Are the two ribbons the same length?
SolutionNo. Two counts can be compared only when every clip is the same size, and Sam’s were not. To compare fairly, change everything into small clips.
Each big clip can be swapped for 2 small ones. Sam’s row has 3 small clips and 2 big ones, which is
small clips. Lena’s is 5 small clips. So Sam’s ribbon is longer, by
small clips. Ribbon B in Figure 2, “5 clips” as well, is 2 + 2 + 2 + 2 + 2 = 10 small clips: the same count, twice Lena’s length. The cure is to measure with one size of clip only.
Fish tanks in two towns
Maya says her fish tank is 6 hand-spans long. Her cousin Leo, who lives in another town, says his is 5 hand-spans long. Whose tank is longer? Later each of them measures again with a centimeter tape: Maya’s tank is 60 centimeters long and Leo’s is 50 centimeters long. Now whose is longer, and by how much?
SolutionFrom the hand-spans we cannot tell. Maya used her hand and Leo used his, so the two counts are in different units, even though both are called hand-spans. If Leo’s hand is the bigger one, his 5 spans could reach further than Maya’s 6; if it is smaller, they reach less far. Nobody knows without putting the two hands side by side, and they live in different towns.
With the tapes, both counts are in the same standard unit, so they can be compared like any two counts in one unit. 60 is more than 50, so Maya’s tank is longer, by
centimeters. Nobody had to carry a fish tank to another town.
One strip, inches and centimeters
A strip is exactly 4 inches long. Laid along a row of centimeters, as in Figure 6, it reaches a little past 10 centimeters. Why does the strip get more centimeters than inches, and are both measurements right?
Both are right. The inch and the centimeter are both standard units, but they are different sizes: one inch is exactly 2.54 centimeters, a little more than two and a half. The bigger unit gives the smaller count, so the strip is fewer inches than centimeters.
We can check the centimeter count. Each inch is 2.54 centimeters, so 4 inches are
centimeters: a little more than 10, and much nearer 10 than 11, so about 10 centimeters, as the figure shows.
Both counts can travel: anyone with an inch ruler gets 4 for this strip, and anyone with a centimeter ruler gets a little more than 10. What cannot be done is to compare “4 inches” with “10 centimeters” by their numbers alone. 4 is less than 10, yet the two are the same strip. Changing one standard unit into another belongs to systems of units.
A class makes its own standard
A class wants every table to measure the classroom rug and get the same answer, but there are no rulers. They cut one stick and call its length a class stick. Then, for each of the five tables, they cut a new stick, laying it against the first stick and trimming it until the two ends match exactly. Table 1 finds the rug is 5 class sticks long. Why should every other table also find 5? And why should every copy be cut against the first stick, rather than each copy against the one cut just before it?
SolutionEach table’s stick matches the first stick, so any two tables’ sticks match each other: two lengths that each match a third length match each other. So every table is measuring with the same unit, laid by the same rules along the same rug, and gets the same count, 5 class sticks. The class has made a standard unit of its own: one kept original, and copies made to match it. That is what the Egyptian builders did with their cubit rods, and what the meter bar in Paris was for.
Copying from copies is riskier. Each trim might be a hair too long or too short. Cut every stick against the original and each one is only one small slip away from it. Cut each stick against the last copy and the slips can pile up down the line, so the fifth stick may be off by five slips. That is why a standard is kept as one official length, and copies are checked against it, not against each other.
Boundaries
The unit must be the same kind of thing as what is measured. Cubes laid end to end measure length. They do not say how heavy something is, or how much water a cup holds. Those need their own kinds of unit: a scoop for how much a jug holds, a weight for how heavy. Attributes you can measure sorts out which is which.
Units in a straight row measure along a straight line. A wiggly worm has to be straightened first, or measured along its wiggles, or the count is not its length.
Not every length is a whole number of units. Objects often end partway along a unit, as in Figure 5. Then we say “between 5 and 6 cubes”, or “about 5 cubes”, or use a smaller unit to get closer. No choice of unit makes every measurement come out exact (measurement is always approximate).
A body unit belongs to one person. A hand-span or a heel-to-toe footstep stays about the same for one person, if made the same way each time. Two people’s counts for the same table differ, as in Example 2, and neither is wrong.
A body unit drifts even for one person. A hand-span is only about the same each time: the fingers spread a little more or a little less, and over a year the hand grows. A measurement in hand-spans from last year and one from today are not quite in the same unit.
A non standard unit is fine while the measurement stays at home. One child, or one class sharing one box of identical cubes, can compare and add its own measurements. Only when the number has to reach someone else does the unit need to be standard.
Standard does not mean only one. Centimeters and inches are both standard. Two measurements in different standard units cannot be compared by their numbers alone (4 inches and about 10 centimeters are the same strip, Example 9) until one is changed into the other.
A standard unit is only as good as its copy. A badly printed ruler, or a tape measure stretched with use, gives the wrong count, however standard the unit written on it.
Where this goes next
The ideas here are used again straight away. Iterating a single unit measures with just one cube or one hand, moved along and counted, and looks closely at gaps and overlaps between units. It also turns the three-unit strip of Example 3 into a rule: smaller units give bigger counts.
The standard units come next: centimeters and meters in metric length units, inches, feet and yards in customary length units, and how a whole family of units fits together in systems of units.
Later, a ruler replaces the row of cubes: each space between its marks is one unit, already laid end to end with no gaps (measuring tools, measuring length). Area works the same way with square tiles (area as covering), and so does volume, with cubes filling a box.
Exercises
Count the units
Use Figure 7. In each row the units start at one end of the object and finish at the other, with no gaps and no overlaps.
- How many cubes long is ribbon A?
- How many paper clips long is pencil B?
- How many rods long is stick C?
- Each rod in row C is as long as 3 cubes. How many cubes long is stick C?
Which is longer?
- A book is 9 cubes long. A box is 12 cubes long. Which is longer, and by how many cubes?
- A shoe is 8 cubes long. A sock is 6 cubes long. Which is shorter, and by how many cubes?
- A spoon is 5 paper clips long. A fork is 5 paper clips long, measured with the same clips. What can you say about the spoon and the fork?
One thing, different units
- A strip is 10 cubes long. A rod is as long as 2 cubes. How many rods long is the strip?
- A rope is 4 rods long, and each rod is as long as 3 cubes. How many cubes long is the rope?
- A shelf is 8 cubes long. It is also 4 rods long. How many cubes long is one rod?
- A grown-up walks heel to toe across a mat and counts 6 footsteps. A child, whose foot is shorter than the grown-up’s, walks heel to toe across the same mat. Will the child count more than 6, fewer than 6, or exactly 6?
- Two children measure the same rug with their own hand-spans. One counts 9 and the other counts 6. Whose hand-span is bigger?
Putting lengths together
- A red ribbon is 4 cubes long and a blue ribbon is 5 cubes long. They are laid end to end with no gap. How long are they together?
- A bench is 9 of Ava’s hand-spans long. Ava has measured 6 hand-spans from one end. How many more hand-spans does she need to reach the other end?
- A cube train is 7 cubes long. Two more cubes are snapped onto the end. How long is the train now?
Non standard units
- Which of these would be the most sensible unit for measuring the length of a classroom: paper clips, crayons or footsteps? Say why.
- Ella measures a bench by moving one hand along it and counts 6 hand-spans. Once, though, she put her thumb down a little past where her little finger had been. Is the bench longer or shorter than 6 of Ella’s hand-spans?
- Jo measures a crayon with her footstep and says it is “less than 1 footstep long”. That is true. Suggest a better unit for the crayon, and say why it is better.
Same size or not?
Use Figure 8. In each row the units start at one end of the object and finish at the other, touching, with no overlaps.
- The cubes under ribbon D are two different colors, but all the same size. How many cubes long is ribbon D?
- Each big clip in row E is as long as 2 small clips. How many small clips long is pencil E?
- Someone counts the clips in row E and says “pencil E is 5 clips long”. Why does that not say how long the pencil is?
- One of the 4 straws in row F was cut short by mistake. Is stick F longer or shorter than 4 full straws?
- A box holds big clips and small clips, and each big clip is as long as 2 small clips. Pia’s book is 4 big clips long. Raj’s book is 6 small clips long. Whose book is longer, and by how many small clips?
Standard units
- Aiden says his toy snake is 9 hand-spans long. His cousin, in another town, says her toy snake is 7 hand-spans long. Can you tell whose snake is longer? Why or why not?
- Which of these are standard units: centimeter, hand-span, inch, paper clip, meter, footstep?
- A book is 9 inches long. Measured in centimeters, will the number be bigger than 9, smaller than 9, or exactly 9?
- A ribbon is exactly 2 inches long, and one inch is exactly 2.54 centimeters. About how many centimeters long is the ribbon? (Figure 6 may help: 2 inches is half of that strip.)
- A class makes a class stick and cuts one copy of it for each table. One table’s copy is cut a little too short. Will that table’s count for the rug be bigger or smaller than the other tables’ counts?
Explain
- Why must the cubes touch each other, with no gaps, when we measure?
- Why does “the pencil is 6” not tell us how long the pencil is?
- A stick is longer than 5 cubes and shorter than 6 cubes. Give two ways to say how long it is.
- Why must every copy of a unit be the same size? Use “5 clips” in your answer.
- Two people in different countries each measure a table in centimeters and both get 120. Why can they be sure their tables are the same length, when two hand-span counts of 8 could not tell them that?
Answers
- 7 cubes. Count the cubes from one end to the other: one to seven.
- 4 paper clips. Count the clips: one, two, three, four.
- 3 rods. Count the rods: one, two, three.
- 9 cubes. Each rod could be swapped for 3 cubes, so 3 rods are 3 + 3 + 3 = 9 cubes. The rod count is smaller because a rod is the bigger unit.
- The box, by 3 cubes. Both are in cubes, 12 is more than 9, and 12 - 9 = 3.
- The sock, by 2 cubes. 6 is less than 8, and 8 - 6 = 2.
- They are the same length. Both are 5 of the same clips, so side by side they would match end to end: 5 - 5 = 0 clips of difference.
- 5 rods. Every 2 cubes make one rod, and 2 + 2 + 2 + 2 + 2 = 10, which is five 2s.
- 12 cubes. 3 + 3 + 3 + 3 = 12.
- 2 cubes. Four equal rods cover the same 8 cubes, and 2 + 2 + 2 + 2 = 8.
- More than 6. The child’s foot is shorter, so each heel-to-toe step covers less of the mat, and more steps are needed to get across.
- The child who counted 6. A bigger unit fits fewer times along the same rug, so the hand that fitted only 6 times is the bigger one.
- 9 cubes. The cubes that fit along each ribbon together fit along both: 4 + 5 = 9.
- 3 more hand-spans. 6 + 3 = 9, so after 6 spans there are 3 left to go.
- 9 cubes. 7 + 2 = 9.
- Footsteps. A classroom is long, so paper clips or crayons would have to be laid down a great many times, and each placement is a chance to leave a gap. A footstep is still a good deal smaller than the room, so it fits a fair number of times and gives a useful count.
- Longer. Putting her thumb down past where her little finger had been left a gap, a part of the bench that no hand-span covered. That part was never counted, so 6 is too small: the bench is longer than 6 of her hand-spans.
- A smaller unit, such as a paper clip or a cube. The footstep is bigger than the crayon, so “less than 1 footstep” cannot tell this crayon from any other short thing. A unit smaller than the crayon fits several times and gives a count that says how long it is.
- 5 cubes. Color does not change how far a cube reaches; all five are the same size, so they are five copies of one unit.
- 7 small clips. Swap each of the 2 big clips for 2 small ones and keep the 3 small clips: 2 + 2 + 3 = 7.
- The clips are not all the same size. “5 clips” could be 5 small clips, 5 big clips or a mix, and those are different lengths (Figure 2). A count names a length only when every unit is the same size.
- Shorter. Three full straws and one short straw cover the stick exactly. Four full straws reach further than that, because the fourth full straw is longer than the short one.
- Pia’s, by 2 small clips. 4 big clips are 2 + 2 + 2 + 2 = 8 small clips, and 8 - 6 = 2.
- No. Each of them counted with their own hand, so the two counts are in different units even though both are called hand-spans. If the cousin’s hand is bigger, her 7 spans could be longer than Aiden’s 9.
- Centimeter, inch and meter. Each is fixed once and the same for everyone. A hand-span and a footstep change from person to person, and a paper clip from box to box.
- Bigger than 9. A centimeter is a smaller unit than an inch, so more centimeters fit along the same book. (In fact 9 inches is about 23 centimeters.)
- About 5 centimeters. 2.54 + 2.54 = 5.08, a little more than 5 and much nearer 5 than 6.
- Bigger. A shorter stick is a smaller unit, and a smaller unit fits more times along the same rug. If the copy is only a hair too short, the difference may be tiny: the rug might reach just a little past that table’s fifth stick, “a little more than 5” where the others get exactly 5.
- A gap is a part of the object with no cube on it, so it is never counted, and the count comes out smaller than the true length. With the cubes touching, every bit of the object is covered and counted once.
- The number does not say what was counted. The pencil could be 6 cubes, 6 paper clips or 6 hand-spans long, and those are different lengths. A measurement needs its unit.
- “Between 5 and 6 cubes long”, or “longer than 5 cubes and shorter than 6”. Both say exactly what we saw. We can also say “about” and the nearer whole number: about 5 cubes if the stick’s end is nearer the end of the fifth cube, about 6 if it is nearer the end of the sixth.
- If every clip is the same size, two rows of 5 clips laid side by side match clip for clip, so “5 clips” always means one length. If the clips are different sizes, 5 small clips and 5 big clips are both “5 clips” but different lengths, so the count no longer says how long anything is.
- Both tapes were made to match the same agreed centimeter, so their centimeters are the same size, and 120 of one unit is one length. Two people’s hand-spans are two different-sized units, so two counts of 8 can be two different lengths.