Comparing with a middle object
A toy boat is at home and the cubby shelf is at school, so they can never be set side by side. How can we still find out whether the boat will fit?
Where you are 14 in What measurement is
When you reach for it
Two pencils are easy to compare. Stand them side by side on the table with their bottom ends level, and whichever one sticks out past the other at the top is longer. That is comparing two objects: line up the ends and look at the other end.
Some things cannot be lined up. A cubby shelf is built into the classroom wall. A poster is pinned high on the wall, out of reach of the floor. A toy boat is at home while the shelf it is meant for is at school. Nobody can hold the shelf against the boat. Looking across the room and guessing is not good enough either, because things that are far apart are hard to judge by eye.
The way out is to bring a third object that can travel: a piece of yarn, a strip of paper, a stick. Compare the first thing with the go-between, carry the go-between over, and compare it with the second thing. Two direct comparisons, each with the ends properly lined up, answer the question that one comparison could not.
The same idea answers a second kind of question. If one ribbon is longer than a second, and the second is longer than a third, then the first is longer than the third, even though those two were never held together. That lets a child put three things in order from only two comparisons.
This lesson runs over three days. Day 1 is the go-between itself. Day 2 is three objects and their order. Day 3 is what a go-between needs in order to tell the truth, and how it leads to measuring with units.
The procedure
Day 1: a go-between that travels
Aada wants to know whether her toy boat will fit, lying flat, in her cubby shelf at school. The boat is at home. She lays a piece of yarn along the boat, with one end of the yarn level with the back of the boat, and cuts the yarn exactly where the front of the boat ends. Then she takes the yarn to school and lays it straight along the floor of the shelf, one end against the left wall.
The yarn stops before it reaches the right wall. So the shelf is longer than the yarn. And the yarn is exactly as long as the boat, because it was cut that way. So the shelf is longer than the boat, and the boat will fit.
Why the yarn can speak for the boat. When Aada cut the yarn, she was making a direct comparison: yarn and boat side by side, ends lined up at both ends. That comparison said “the same length”. From then on, the yarn carries the boat’s length with it. Whatever the yarn shows about the shelf, the boat would show too, if only it could be brought there.
There are three things the yarn can show at the shelf, and each one tells us something:
| The yarn in the shelf | So the boat is |
|---|---|
| stops before the far wall | shorter than the shelf: it fits |
| reaches the far wall exactly | as long as the shelf: it just fits |
| runs past the far wall | longer than the shelf: it does not fit |
A mark instead of a cut. The yarn does not have to be cut. A long piece of yarn can be laid along the boat and pinched, or knotted, or marked with a pen at the place where the boat ends. Then only the part from the start of the yarn to the mark stands for the boat, and that part is the go-between. A strip of paper with a pencil line across it works the same way, and it can be used again.
The go-between can turn around. A length does not care which way it points. Yarn cut to how tall a child stands can be laid flat in a shelf, and yarn cut along a shelf can be held up against a wall. A height, a length and a width are all lengths, so any of them can be carried by the same piece of yarn.
Day 2: three things in order
Here are two comparisons, each made the ordinary way with the left ends lined up. Ribbon A is longer than ribbon B. Ribbon B is longer than ribbon C. Ribbons A and C are never held together.
Why A must be longer than C. Picture all three lying together with their left ends lined up. Ribbon C ends somewhere. Ribbon B goes past that place, because B is longer than C. Ribbon A goes past the end of B, because A is longer than B. So A goes past the end of B, which is already past the end of C. A reaches further than C, so A is longer.
Said in plain words, as a child can say it:
If the first is longer than the second, and the second is longer than the third, then the first is longer than the third.
Adults call this the transitive property of “longer than”. It is what made Day 1 work: there the middle object was the yarn. It holds just as well for “shorter than”, and for “as long as”: if the boat is as long as the yarn and the yarn is as long as the shelf, the boat is as long as the shelf. It holds when the facts are mixed, too. If the boat is as long as the yarn and the yarn is shorter than the shelf, the boat is shorter than the shelf, which is exactly what Aada found.
Putting three in order. To order three things by length, find the one that is longer than each of the others: that is the longest. Find the one that is shorter than each of the others: that is the shortest. The one left is the middle one, and it is the only one that is longer than one thing and shorter than another. With A, B and C above, the order from longest to shortest is A, B, C. The middle one, B, was in both comparisons, and that is no accident: the shared object is what ties the two facts together.
When two facts are not enough. Two comparisons only decide the third if the shared object sits between the other two. Suppose instead that A is longer than B and C is also longer than B. Then B is the shortest, but A and C have only been compared with something shorter than both of them. Either one might be longer. To order them, they have to be compared with each other, or through some other go-between.
Day 3: keeping the go-between honest
A go-between only works if the length it carries is the same length at both places. Four things can spoil it, and each one has a reason.
It must not stretch. Yarn pulled very hard can stretch a little. Then it is longer at the shelf than it was at the boat, and it no longer stands for the boat. Pull it just enough to make it straight. A strip of paper or a stick cannot stretch at all, which makes it a safer go-between.
It must not change on the way. If a bit is cut off, or a knot is tied in the part that stands for the boat, the yarn at the shelf is shorter than the yarn that was matched to the boat. The comparison at the shelf is then a comparison with a different length.
It must lie straight. A bent or wavy piece of yarn spends some of its length going from side to side, so it reaches less far than it would if it were straight. It looks shorter than it is.
The picture shows one piece of yarn laid twice under the bottom edge of the polar bear poster, once in waves and once pulled straight. The two tries disagree; the first Day 3 example works out which one to believe.
Both ends must be lined up both times. At the boat, one end of the yarn is level with the back of the boat. At the shelf, the same end must be against the wall. A yarn that starts a little way in from the wall reaches further than it should.
A go-between is the first step toward a unit. The yarn, the stick and the paper strip are all carrying a length from one place to another. A go-between that is shorter than both things is not much use on its own: it fits along the boat and along the shelf, and tells us nothing about which of those is longer. But laid down again and again along each one, end after end, and counted, a short go-between becomes a unit, and the counts can be compared. That is the next lesson.
Worked examples
Day 1: will the train fit?
Aada also wants to keep her toy train in the same cubby shelf. At home she cuts a new piece of yarn exactly as long as the train. At school she lays it straight on the floor of the shelf, one end against the left wall, and the yarn runs past the right wall. Will the train fit in the shelf, lying flat?
SolutionThere are two comparisons, each made with the ends lined up. At home, the yarn and the train were side by side and the yarn was cut where the train ended, so the yarn is as long as the train. At school, the yarn runs past the far wall, so the yarn is longer than the shelf.
The yarn stands for the train. The train is as long as the yarn, and the yarn is longer than the shelf, so the train is longer than the shelf. It does not fit. This is the bottom row of the table above; the boat in the picture was the top row.
A “fits” answer, like the boat’s, settles length only. A toy also has a height, and so does the shelf. To be sure a toy fits, the same check would have to be made for those, with a second piece of yarn.
Day 1: a poster high on the wall
The polar bear poster is pinned high on the classroom wall, so its bottom edge is well above the floor. Is Kaapo taller than the poster? Iida holds a piece of yarn from the floor at Kaapo’s heels, straight up to the top of his head, and cuts it there. Then she climbs on a step and holds the yarn up against the side of the poster, one end level with the poster’s bottom edge. The yarn runs past the top of the poster.
SolutionKaapo cannot stand beside the poster with his feet level with its bottom edge, because the poster starts high above the floor. So the comparison cannot be direct.
The yarn was cut to Kaapo’s height, so it is as long as Kaapo is tall. Held against the poster, it runs past the top, so the yarn is longer than the poster is tall. Kaapo is as tall as the yarn, and the yarn is longer than the poster’s height, so Kaapo is taller than the poster.
The yarn was cut standing up and used standing up, but it did not have to be. Heights are lengths, and the yarn carries a length whichever way it points.
Day 1: a knot instead of a cut
Kaapo wants to know which is longer, the toy boat or the bottom edge of the polar bear poster. He does not want to cut his long piece of yarn. He lays it along the boat, starting level with the back, and ties a small knot where the front of the boat ends. Then he holds the yarn along the poster’s bottom edge, the end of the yarn at the poster’s left corner. The knot lands before the right corner.
SolutionOnly the stretch of yarn from its end up to the knot matters. That stretch is as long as the boat, because the knot was tied exactly where the boat ended. The rest of the yarn, past the knot, is simply not used.
Along the poster, the knot lands before the right corner, so the poster’s bottom edge is longer than the stretch of yarn up to the knot. That stretch is as long as the boat, so the poster’s bottom edge is longer than the boat.
The knot does the same job as the cut on Day 1. Tying it in a long yarn has one cost: the knot must sit right where the boat ends, and the yarn’s end must start right at the poster’s corner, or the stretch being compared is not the boat’s length.
Day 2: three ribbons that never touch
Ribbon P is held beside ribbon Q, left ends lined up, and P is shorter. Ribbon Q is then held beside ribbon R, left ends lined up, and Q is shorter. Which is longer, P or R? Put the three in order from longest to shortest.
SolutionP and R were never side by side, but Q was beside each of them, so Q is the go-between. P is shorter than Q, and Q is shorter than R. “Shorter than” passes along just as “longer than” does, so P is shorter than R, and R is the longer of the two.
For the order, R is longer than both of the others, so it is the longest. P is shorter than both of the others, so it is the shortest. Q is in the middle. From longest to shortest the order is R, Q, P: the facts were given shortest first, so the order comes out the other way around from the way they were said.
A child can check this by laying all three ribbons down together with their left ends lined up. That last step is a check, not the answer: the answer was already settled by the two comparisons.
Day 2: three pieces of yarn
Aada, Kaapo and Iida each cut a piece of yarn.
(a) Iida’s yarn is longer than Aada’s. Aada’s yarn is longer than Kaapo’s. Put the three pieces in order from shortest to longest.
(b) On another day, Aada’s yarn is longer than Kaapo’s, and Aada’s yarn is also longer than Iida’s. Which is longer, Kaapo’s yarn or Iida’s?
Solution(a) Aada’s yarn is in both facts, so it is the go-between. Iida’s is longer than Aada’s, which is longer than Kaapo’s, so Iida’s is longer than Kaapo’s too. Kaapo’s is shorter than both others, so it is the shortest; Iida’s is longer than both others, so it is the longest. From shortest to longest: Kaapo, Aada, Iida.
(b) Aada’s yarn is the longest, since it is longer than each of the other two. But Kaapo’s and Iida’s pieces have both been compared only with something longer than both of them. That does not decide anything between them: Kaapo’s might be longer, or Iida’s might, or they might be the same length. We cannot tell. One more comparison, Kaapo’s yarn held against Iida’s, is needed.
The difference between (a) and (b) is where the shared piece sits. In (a) Aada’s yarn is in the middle, longer than one and shorter than the other, so it ties the two facts together. In (b) it is longer than both, so it ties nothing together.
Day 3: bent or straight?
Kaapo cuts a piece of yarn exactly as long as Iida is tall. He wants to know whether Iida is taller than the polar bear poster is wide. He lays the yarn under the poster’s bottom edge, one end level with the left corner. The first time the yarn lies in waves, and it stops before the right corner. The second time he pulls it straight, without stretching it, and it runs past the right corner. Which is longer, Iida’s height or the poster’s width?
SolutionThe yarn is the same piece both times, so its length did not change. What changed is how it lay. Laid in waves, part of the yarn goes from side to side instead of along the poster, so it reaches less far than its own length. That first try showed something about the waves, not about the yarn’s length.
Pulled straight, all of the yarn runs along the poster’s edge, so how far it reaches is its real length. It runs past the right corner, so the yarn is longer than the poster is wide. The yarn is as long as Iida is tall, so Iida is taller than the poster is wide.
When two tries disagree, the straight one wins, because a straight go-between is the only kind whose reach is its length.
Day 3: a go-between that is too short
Aada has only a short piece of yarn, shorter than the toy boat and shorter than the cubby shelf. Laid along each of them, it stops before the far end both times. What does that tell her about the boat and the shelf?
SolutionTwo facts: the boat is longer than the yarn, and the shelf is longer than the yarn. The yarn is shorter than both, just like Aada’s yarn in part (b) of the last Day 2 example was longer than both. A go-between that is on the same side of both things does not tie them together. The boat might be longer than the shelf, or shorter, or the same; this yarn, used once, cannot say.
There are two ways forward. One is a longer go-between, such as a piece of yarn as long as the boat itself. The other is to keep the short yarn and lay it down again and again along the boat, end after end with no gaps, counting as it goes, and then do the same along the shelf. The bigger count belongs to the longer one. That second way turns the short yarn into a unit, and it is where the next lesson, what a unit is, begins.
Boundaries
The go-between must keep its length from the first comparison to the second. No stretching, no cutting, no knot tied in the part being used, and laid straight both times. If its length changes in between, the two comparisons are about two different lengths, and nothing follows from putting them together.
The ends must be lined up in both comparisons. Each of the two comparisons is an ordinary direct comparison, and it needs one end of each object level, just as when two pencils stand side by side.
The middle object has to sit between the other two. “A is longer than B, and B is longer than C” decides A against C. “A is longer than B, and C is longer than B” does not, and neither does “B is longer than A, and B is longer than C”. A go-between longer than both things, or shorter than both, settles nothing unless it is marked, or laid again and again and counted.
Length is compared with length. A height, a width and a length along the floor are all lengths, and the yarn can carry any of them. It cannot compare how long one thing is with how heavy another is. Weight has a go-between of its own, the balance, in balancing to compare mass.
The facts must be exact, not “about the same”. “Longer than”, “shorter than” and “as long as” each pass along from one comparison to the next. “About as long as” does not. If each ribbon in a long line is only a tiny bit longer than the next, every neighbor looks about the same, yet the first and last ribbons can be very different. Small differences add up.
One length is not the whole story of fitting. A boat that is shorter than a shelf fits only if it is also not too tall or too wide. Each of those is a separate comparison of lengths.
For adults: written with the symbols of comparing numbers, the property is: if A > B and B > C, then A > C. The same is true of < and of =.
Where this goes next
Every go-between in this lesson carried one length across a room, and each one was used only once. The next step is to keep one small go-between and use it over and over: lay it along the boat, move it on, lay it again, and count. That count is a measurement, and the small go-between is a unit. What a unit is starts there, with cubes, paper clips and hand-spans, and with why the copies must touch with no gaps or overlaps. Iterating a single unit moves one copy along end to end, and non standard units measures with things at hand before rulers arrive.
The idea of the middle object never goes away. A ruler is a go-between with the counting already marked on it: hold it against the boat, read a number, hold it against the shelf, read another, and compare the numbers. Comparing two numbers then rests on the very same property: if one number is greater than a second and the second is greater than a third, the first is greater than the third. The balance used to compare weights works the same way, with one object weighed against a middle one, and so does every chain of “bigger than” a child meets later in mathematics.
Exercises
Day 1: a go-between that travels
- Aada cuts a piece of yarn as long as the cubby shelf. She holds it up against the side of the polar bear poster, one end level with the poster’s bottom edge, and the yarn stops below the top of the poster. Which is longer, the shelf or the poster’s height?
- Iida cuts a piece of yarn as long as the bottom edge of the polar bear poster. She lays it straight in the cubby shelf, one end against the left wall, and its other end touches the right wall exactly. What does that tell her?
- Kaapo cuts a piece of yarn as long as Iida is tall. He lays it straight in the cubby shelf, one end against the left wall, and the yarn runs past the right wall. Which is longer, Iida’s height or the shelf?
- Aada lays a long piece of yarn along the polar bear on the poster, from its nose to its tail, and ties a knot where the tail ends. She lays the yarn along the toy boat, the yarn’s end level with the back of the boat, and the knot lands before the front of the boat. Which is longer, the bear on the poster or the boat?
- The toy boat is at home and the cubby shelf is at school. Aada has yarn and scissors. Say what she does at home, what she carries to school, and what she looks for there.
- Iida cuts a piece of yarn as long as the toy boat. She holds it straight up beside Aada, one end on the floor at Aada’s heels, and the top of the yarn is far below the top of Aada’s head. Which is longer, the boat or Aada’s height?
Day 2: three things in order
- The red ribbon is shorter than the blue ribbon. The blue ribbon is shorter than the green ribbon. Which is longer, the red ribbon or the green ribbon?
- Kaapo is taller than Aada. Aada is taller than Iida. Put the three children in order from tallest to shortest.
- The yellow ribbon is longer than the white ribbon. The pink ribbon is longer than the yellow ribbon. Put the three ribbons in order from shortest to longest.
- The polar bear on the poster is shorter than the toy boat. The toy boat is shorter than the cubby shelf. Which is longer, the polar bear or the shelf?
- The red ribbon is longer than the blue ribbon. The red ribbon is also longer than the green ribbon. Can you tell which is longer, the blue ribbon or the green ribbon?
- The gray ribbon is as long as the black ribbon. The black ribbon is longer than the white ribbon. Which is longer, the gray ribbon or the white ribbon?
Use the two pictures for Exercises 13 and 14.
- Put ribbons D, E and F in order from longest to shortest.
- Ribbons E and F were never held side by side. Which of them is longer, and how do you know?
Day 3: keeping the go-between honest
- Aada cuts a piece of yarn as long as the toy boat. On the way to school a short bit is cut off the end. At the shelf the shorter yarn runs past the right wall. Can she still tell whether the boat fits? What if the shorter yarn had stopped before the right wall?
- Kaapo has a very long piece of yarn, longer than the toy boat and longer than the polar bear poster is tall. He lays it beside each of them but makes no mark. What can he tell about the boat and the poster’s height?
- Which makes the safer go-between, a strip of paper or a piece of yarn? Why?
- The silver ribbon is as long as the gold ribbon. The gold ribbon is as long as the purple ribbon. What can you say about the silver ribbon and the purple ribbon?
Explain
- Why must the yarn be laid straight when it is used as a go-between?
- Use a picture of three ribbons with their left ends lined up to explain why, if the first is longer than the second and the second is longer than the third, the first is longer than the third.
- Is it always true that if A is longer than B and C is longer than B, then A is longer than C? Give an example.
- A paper clip is much shorter than the toy boat and the cubby shelf. How could it still help compare them?
Answers
- The poster’s height is longer. The yarn is as long as the shelf, and the poster is taller than the yarn, so the poster’s height is longer than the shelf.
- The poster’s bottom edge is as long as the shelf. The yarn is as long as the edge and as long as the shelf, and two things each as long as the same yarn are as long as each other.
- Iida’s height is longer. The yarn is as long as Iida is tall and longer than the shelf, so Iida is taller than the shelf is long.
- The boat is longer. The yarn up to the knot is as long as the bear, and the boat is longer than that stretch of yarn, so the boat is longer than the bear.
- At home she lays the yarn along the boat, one end level with the back, and cuts it where the front ends. She carries the yarn, which is now as long as the boat. At school she lays it straight in the shelf, one end against the left wall, and looks at the right wall: if the yarn stops before it or just touches it, the boat fits; if it runs past, the boat does not fit.
- Aada’s height is longer. The yarn is as long as the boat, and Aada is taller than the yarn, so Aada is taller than the boat is long. The yarn turned from lying flat to standing up, which is allowed: both are lengths.
- The green ribbon is longer. Red is shorter than blue and blue is shorter than green, so red is shorter than green.
- Kaapo, Aada, Iida. Kaapo is taller than both others and Iida is shorter than both, since Kaapo is taller than Aada, who is taller than Iida.
- White, yellow, pink. Yellow is the shared ribbon in the middle: white is shorter than it and pink is longer.
- The shelf is longer. The bear is shorter than the boat and the boat is shorter than the shelf, so the bear is shorter than the shelf.
- No, we cannot tell. Red is longer than both of them, so red ties nothing together. Blue and green must be compared with each other.
- The gray ribbon is longer. Gray is as long as black, and black is longer than white, so gray is longer than white.
- F, D, E. The first picture shows D longer than E; the second shows F longer than D. D is in both pictures and sits in the middle.
- F is longer. F is longer than D, and D is longer than E, so F is longer than E. Ribbon D is the go-between.
- If the shorter yarn runs past the wall, yes: the boat does not fit. The boat is longer than the shortened yarn, and the shortened yarn is longer than the shelf, so the boat is longer than the shelf. If the shorter yarn had stopped before the wall, she could not tell, because the missing bit might be what the boat needs. She would have to cut a new yarn at the boat.
- Nothing about which of the two is longer. The yarn is longer than both, so it does not sit between them. A knot or mark at the end of the boat, or at the top of the poster, would fix it.
- The strip of paper. It cannot stretch, and it stays straight on its own, so its length is the same at both places. Yarn works too, if it is laid straight and not pulled hard.
- They are the same length. Silver is as long as gold, and gold is as long as purple, so silver is as long as purple.
- A bent yarn spends some of its length going side to side, so its far end lands short of where it would if it were straight. Then the yarn seems shorter than it is, and a comparison with it can come out wrong, as when the wavy yarn under the poster stopped short but the straight one ran past.
- Line up the left ends of all three. The third ribbon ends somewhere. The second goes past that place, because it is longer than the third. The first goes past the end of the second, because it is longer than the second. So the first goes past a place that is already past the end of the third: it is longer than the third.
- No. B can be shorter than both and still leave either one longer. For example, A could be a short ribbon, C a long ribbon, and B a tiny ribbon shorter than both: then A is longer than B and C is longer than B, but C is longer than A.
- Laid once, the paper clip only shows that both things are longer than it. But laid again and again along the boat, end after end with no gaps, and counted, and then the same along the shelf, it gives two counts, and the bigger count belongs to the longer one. That makes the paper clip a unit.