Grade 1 · 1.MD.A.2 · 3.MD.C.5B

Iterating a single unit

There is only one rain boot, and the rug is much longer than one boot. How can one boot say how long the whole rug is?

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What a unit is, and why we agree on standard onesGrade 1

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Smaller units give bigger countsGrade 2 Area as coveringGrade 3 Measuring lengthGrade 1 Measurement is approximate: precision of a toolGrade 3 +1 more

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Iterating a single unit

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When you reach for it

What a unit is measures by laying many copies of one unit along an object, end to end, with no gaps and no overlaps, and counting them. That needs enough copies to reach from one end to the other. Often there are not enough. A box of cubes runs out before the hallway does. And some units come one at a time: there is one hand, one foot, one paper strip cut for the job, one rain boot by the door.

This lesson measures with one copy of the unit. Put it down at one end. Mark where it ends. Pick it up and put it down again so that it starts exactly at the mark. Count each time it is put down. Moving one copy along like this is called iterating the unit, which just means using it again and again.

Iterating is the same measurement as the row of many copies, done one copy at a time. Every rule from What a unit is, and why we agree on standard ones still holds. What is new is that the row of copies is never all there at once. It lives in the marks, and the child has to build it, one placement at a time, without letting the copy skip ahead or slide back.

It is taught over four days, one part a day:

Part 1, mark and move: the move itself, and why the number of placements is the length.
Part 2, gaps and overlaps: what happens to the count when the copy skips ahead or slides back.
Part 3, a bit more: what to say when the last placement does not fit.
Part 4, a bigger unit: why a longer boot gives a smaller count.

Do every part with real objects: a boot, a strip of paper, a rug, some tape and a pencil.

The procedure

Part 1: Mark and move

Peetu has one rain boot and a long rug. He puts the boot down flat on the rug with its heel exactly at one edge, pointing along the rug. He sticks a small piece of tape on the rug right at the toe. He says “one”. Then he picks the boot up and puts it down again with its heel exactly on the tape, and puts a new piece of tape at the new toe. He says “two”. He keeps going, heel on the last piece of tape, new tape at the toe, one counting word each time, until a toe lands on the far edge. Figure 1 shows the seventh placement and, dashed, the six before it.

Figure 1One boot, put down 7 times. The dashed outlines show where it was before; the solid one is where it is now. The rug is 7 boot lengths long.

Each part of the move has a reason.

Start with the heel exactly at the edge. The measurement is from one edge of the rug to the other. A heel set a little way in leaves a strip of rug before it that no boot ever covers.

Mark where the toe ends. Once the boot is picked up, nothing shows where it was. The mark keeps that spot, so the next placement can start there. A piece of tape, a pencil dot, or a finger held on the spot all do the job; a finger is fine for a short distance, tape is better for a long one.

Put the heel exactly on the mark. The heel on the mark means the new placement starts exactly where the last one stopped. That is the “touching” of What a unit is, and why we agree on standard ones: each copy starts where the one before it ended, with no gaps and no overlaps.

Say one counting word for each placement. Each placement covers one boot length of rug. So the count of placements is the count of boot lengths.

Why the count is the length. Look at Figure 1 as a whole. The seven outlines lie end to end, from edge to edge, touching. It is exactly the row of copies that seven identical boots would make if Peetu had seven of them. The only difference is that the row was built one boot at a time. So the rug is 7 boot lengths long, for the same reason a row of 7 cubes means 7 cubes long: every bit of the rug is covered once, by one placement. The last counting word tells how many placements there were, just as the last word of any count tells how many (How counting tells how many).

Count placements, not marks. In Figure 1 there is a piece of tape at every toe and none at the start, so there are 7 pieces of tape and 7 placements. If Peetu had also stuck a piece of tape on the edge where he began, there would be 8 pieces of tape and still 7 placements. The length is the number of boot lengths between the marks, not the number of marks. Example 2 counts this way.

The unit stays the same. It is one boot the whole time, so every placement is the same length. That is why one copy is such a fair unit: nobody can slip in a bigger one by accident.

Part 2: Gaps and overlaps

The rule for many copies was “no gaps and no overlaps”. With one copy, a gap or an overlap does not stay on the table for everyone to see. It is in the past: it happened when the boot was put down a little off the mark. Figure 2 shows three walks across the same rug, with every placement drawn.

Figure 2Row A is the measurement. In row B the boot skipped ahead and left gaps, so the count is too small. In row C it slid back and overlapped, so the count is too big.

Skipping ahead makes gaps. In row B the heel was put down a little past the tape each time. Between one placement and the next is a strip of bare rug. No boot ever covered it, so no counting word was ever said for it. The bare strips add up to part of the rug that was never counted, and the count, 6, is smaller than the length. A gap makes the count too small.

Sliding back makes overlaps. In row C the heel was put down a little before the tape each time, so each placement covers some of the rug the last one already covered. Those parts are counted twice. The count, 8, is bigger than the length. An overlap makes the count too big.

Only row A is the measurement. In row A each bit of rug is covered exactly once. That is what “no gaps and no overlaps” means for a line of placements: every placement starts where the last one ended.

Gaps and overlaps can also creep in at the two ends. A first heel set a little way in from the edge leaves a gap at the start. A last placement that is counted even though it hangs over the edge counts rug that is not there. Part 3 deals with the far end.

These are the same gaps and overlaps that What a unit is, and why we agree on standard ones shows with a row of cubes (its Figure 3). What changes with one copy is where to look for them. They live in the marks: a mark is only good if the next heel lands exactly on it.

Part 3: A bit more

Hertta is measuring a beanstalk drawn on the classroom wall. She has one paper strip. She holds the bottom of the strip on the floor, right where the beanstalk starts, and makes a small pencil mark on the wall at the strip’s top. Then she moves the strip up so its bottom sits on the mark, and marks again. She counts each placement, as in Figure 3.

Figure 3Nine strips fit, and the beanstalk keeps going a little. It is a bit more than 9 strips tall.

After 9 placements the beanstalk keeps going, but only a little. A tenth strip would go well past its top. So the beanstalk is more than 9 strips tall and less than 10.

The last placement does not count when it does not fit, because a placement counts one whole strip of beanstalk, and the beanstalk does not have a whole tenth strip in it. There are three honest ways to say the length:

“a bit more than 9 strips”, which says exactly what was seen;
“between 9 and 10 strips”, which says the same thing using both whole numbers;
“about 9 strips”, which picks the nearer whole number and uses “about” to say it is not exact.

Here the top is much nearer the ninth mark than where the tenth strip would end, so “about 9” is right, not “about 10”. What a unit is, and why we agree on standard ones says this for a row of cubes (its Example 5); the one-copy version is the same, because the marks build the same row.

Measuring up a wall works just like measuring across a rug. The only new care is the start: the bottom of the strip goes exactly on the floor where the beanstalk begins, and every strip is held straight up, not tilted along the stem.

Part 4: A bigger unit

Leevi’s rain boot is longer than Peetu’s. When Leevi measures the same rug with his boot, each placement covers more rug, so fewer placements reach the far edge. Figure 5, in Example 7, shows it: Peetu counts 7, Leevi a bit more than 5.

Both are measurements of the same rug, and both are right. They are in different units. “The rug is 7” is not a full answer: it has to say “7 of Peetu’s boot lengths”.

The rule is the one What a unit is, and why we agree on standard ones states, now seen with one copy: a bigger unit gives a smaller count, and a smaller unit gives a bigger count. The reason is plain in the picture. The same rug is covered either way, and a bigger piece covers it in fewer pieces. Smaller units give bigger counts takes this further.

So two counts can be compared only when they are in the same unit. 7 boots and 9 strips cannot be compared by their numbers, because a boot and a strip are not the same length. Example 8 shows what to do instead.

Worked examples

Examples 1 and 2 go with Part 1, Examples 3 and 4 with Part 2, Examples 5 and 6 with Part 3, and Examples 7 and 8 with Part 4.

Example 1

The rug, seven boots long

Peetu measures the rug with one rain boot, as in Figure 1. On the last placement the toe lands exactly on the rug’s far edge, and the last word he says is “seven”. How long is the rug?

Solution

Each placement started with the heel on the tape left by the toe before it. So the placements touch: no rug was skipped and none was covered twice. The first heel was at one edge and the last toe at the other, so the placements reach exactly across the rug.

That makes the seven placements the same as a row of seven boots laid heel to toe. Each placement is one boot length. Peetu said one word for each, and the last word was “seven”:

1, 2, 3, 4, 5, 6, 7

So the rug is 7 boot lengths long. To check, Peetu can count the pieces of tape: he put one at each toe and none at the start, so there are 7, one for each placement.

Example 2

Counting the spaces between the tape

Leevi measures the small rug by the door with Peetu’s boot. He sticks a piece of tape on the edge where he starts, then one at each toe. The last toe lands exactly on the far edge, where he sticks the last piece. When he is done there are 6 pieces of tape. How long is the small rug?

Solution

Each placement runs from one piece of tape to the next. Picture the six pieces in a row along the rug: the first is the start edge, and every other piece is the toe of one placement. So every piece except the first stands for one placement:

6 - 1 = 5

The small rug is 5 boot lengths long. Another way to see it: count the spaces between the pieces of tape, not the pieces. Six pieces in a row have five spaces between them, and each space is one boot length.

This is why the count is said with the placements, not afterward from the marks. Saying one word per placement gives 5 straight away.

Example 3

Skipping ahead

The rug in Figure 1 is 7 boot lengths long. Leevi measures it with the same boot, but each time he picks the boot up he puts the heel down a little past the tape, as in row B of Figure 2. His last toe lands at the far edge, and he counts 6. What does 6 say about the rug?

Solution

Each time the heel landed past the tape, a strip of rug was left between the old toe and the new heel. Nothing covered those strips, so they were never counted.

The 6 placements cover only part of the rug. The bare strips cover the rest. So the rug is more than 6 boot lengths, and the count is too small. Peetu’s careful count says how much rug went uncounted:

7 - 6 = 1

The bare strips add up to 1 boot length of rug. Skipping ahead always makes the count too small, never too big, because the gaps are rug that no placement covered.

Example 4

Sliding back

Hertta measures the same rug with the same boot. Each time, the boot slides back a little as she puts it down, so the heel lands a little before the tape, as in row C of Figure 2. Her last toe lands at the far edge, and she counts 8. What does 8 say about the rug?

Solution

Each placement that slid back covered some rug the placement before it had already covered. Those bits of rug were counted twice: once in the first placement and again in the next.

So 8 counts the whole rug once, plus the overlapped bits a second time. The count is too big. The rug is shorter than 8 boot lengths; from Example 1 it is 7, so the overlaps added up to

8 - 7 = 1

boot length of rug counted twice. Sliding back always makes the count too big, because the overlaps are rug counted more than once. To get 7, each heel must land exactly on the tape: not past it (Example 3) and not before it.

Example 5

The beanstalk

Hertta measures the beanstalk on the wall with one paper strip, from the floor up, as in Figure 3. After 9 placements the beanstalk keeps going a little. A tenth strip would go well past the top. How tall is the beanstalk?

Solution

Nine placements fit, each starting on the mark left by the one below it, so the beanstalk is at least 9 strips tall. A tenth placement does not fit: it would cover wall above the beanstalk, which is not part of it. So the tenth does not count, and the beanstalk is less than 10 strips tall.

The beanstalk is a bit more than 9 strips tall, or between 9 and 10 strips tall. Its top is only a little above the ninth mark, much nearer the ninth mark than where a tenth strip would end, so it is about 9 strips tall.

Saying “10 strips” would count a whole strip that is not there. Saying just “9 strips” would leave out the bit at the top. The words “a bit more than” and “about” carry exactly what is known.

Example 6

The rug the short way

Peetu measures the rug the short way, across, as in Figure 4. Four placements fit. The fifth placement starts at the fourth toe and sticks out a little past the far edge, which falls most of the way along it. How wide is the rug?

Figure 4Four boots fit, and the edge falls most of the way along the fifth.

Solution

Four whole placements fit, so the rug is more than 4 boot lengths wide. The fifth does not fit, so it is less than 5. The rug is between 4 and 5 boot lengths wide.

The edge falls most of the way along the fifth placement, so it is nearer where a fifth boot ends than where the fourth ended. So the rug is about 5 boot lengths wide, or a bit less than 5.

Compare Example 5. Both lengths fall between two whole numbers. The beanstalk is nearer the smaller one and the rug is nearer the bigger one, and “about” follows whichever is nearer.

Example 7

A bigger boot

Leevi’s rain boot is longer than Peetu’s. Leevi measures the rug from Example 1, heel to toe with his own boot, as in Figure 5. Will he count more than 7, fewer than 7, or exactly 7? He finds 5 placements fit and a sixth sticks out past the edge. Who is right about the rug?

Figure 5One rug, two boots. The longer boot needs fewer placements.

Solution

Each of Leevi’s placements covers more rug than each of Peetu’s, because his boot is longer. Covering the same rug in bigger pieces takes fewer pieces, so he counts fewer than 7.

That is what happens: 5 of his placements fit and the sixth does not, so the rug is a bit more than 5 of Leevi’s boot lengths long. Peetu’s count is 7 of Peetu’s boot lengths. Both are right. The rug did not change; the unit did.

The two counts also show whose boot is longer, even without the picture: 5 and a bit is fewer than 7, and the unit that fits fewer times along the same rug is the bigger one.

Example 8

The rug or the beanstalk?

The rug from Example 1 is 7 boot lengths long. The beanstalk is a bit more than 9 strips tall. Do the counts 7 and 9 say which is longer? Then Hertta lays her paper strip along the rug by mark and move, and counts exactly 12 strips. Which is longer, the rug or the beanstalk, and by about how much?

Solution

No. The first two counts cannot be compared. They are in different units, boots and strips, and a boot is not the same length as a strip. Nine of one thing can be shorter than seven of another.

Measuring the rug with the strip puts both lengths in the same unit. The rug is 12 strips long and the beanstalk is a bit more than 9 strips tall. Since 12 is more than 9 and a bit, the rug is longer. Take the beanstalk’s 9 whole strips away from the rug’s 12:

12 - 9 = 3

The rug is longer by a bit less than 3 strips, since the beanstalk’s extra bit uses up part of the third strip.

One more thing falls out. The same rug is 12 strips and 7 boots. More strips than boots fit, so the strip is the smaller unit: the strip is shorter than Peetu’s boot.

Boundaries

The one copy must not change. A rain boot keeps its length. A paper strip must not be torn, folded or swapped for a different strip partway through. A hand-span is a one-copy unit too, but the fingers can spread a little more or a little less each time, so it only stays about the same.

The marks must stay put. A piece of tape that gets kicked, or a finger that drifts, moves the start of the next placement, which makes a gap or an overlap without anyone seeing it.

The placements must follow a straight line from one end to the other. A boot that wanders across the rug at a slant measures a slanting path, which is longer than the rug. On the wall, each strip is held straight up from the one below, so the count is the beanstalk’s height and not the length of its curvy stem.

Only whole placements are counted. A last placement that does not fit is not counted as a whole one; the length is “a bit more than” the whole placements, as in Example 5.

The object must be longer than the unit. A boot on a crayon fits less than once, and the most that can be said is “less than 1 boot”. A smaller unit is the answer then, as in What a unit is, and why we agree on standard ones.

Counts in different units do not compare. 7 of Peetu’s boots and 5 and a bit of Leevi’s are the same rug, and 7 boots and 9 strips say nothing about which object is longer until both are measured in one unit (Example 8).

One copy and many copies give the same answer. If every placement starts exactly at the last mark, the count matches what a row of identical copies would give. One copy is slower, and every move is a fresh chance for a gap or an overlap, which is why the marks matter so much.

Where this goes next

Mark and move is exactly what a ruler saves us from doing. A ruler is a row of units already laid end to end, with no gaps and no overlaps, printed on a stick (measuring length). The numbers on it count the placements for us.

Part 4 grows into smaller units give bigger counts. The “bit more” of Part 3 grows into a whole idea: no unit makes every length come out exact, and a smaller unit only gets closer (measurement is approximate). Later, the same move tiles a flat shape with one square, moved along row by row, which is area as covering.

Exercises

Mark and move (Part 1)

Exercises 1 and 2 use Figure 6. Every rug starts at its left edge.

Figure 6Rugs for Exercises 1, 2, 8 and 11.

Rug A was measured with one boot moved heel to toe, each heel on the last toe’s tape. The last toe landed on the far edge. How long is rug A?
Rug B shows only the pieces of tape, with no boot: one piece at the start edge, then one at each toe, the last on the far edge. How long is rug B?
Hertta measures a line drawn on the wall with her paper strip, by mark and move, from the bottom of the line. She says “eleven” as the top of the strip reaches exactly the top of the line. How long is the line?
Peetu’s rug is 7 boot lengths long. He has put the boot down 4 times. How many more placements will reach the far edge?
Leevi measures a rug that is 8 boot lengths long. He sticks a piece of tape on the start edge and one at each toe. How many pieces of tape does he use?

Gaps and overlaps (Part 2)

Leevi measures a rug. Twice, he puts the heel down a little past the tape. Every other time he puts it exactly on the tape. His last toe lands at the far edge, and he counts 9. Is the rug longer than 9 boot lengths, shorter, or exactly 9?
Every time Hertta moves her paper strip up the wall, it slips down a little below her mark before she holds it still. She counts 12 strips to the top of a tall line. Is the line taller than 12 strips, shorter, or exactly 12?
Rug C in Figure 6 was measured with a boot that skipped ahead every time. How many placements are there? Is the rug longer than that many boot lengths, shorter, or the same?
Peetu’s rug is exactly 7 boot lengths long. This time he starts with the heel a little way in from the edge, then puts every heel exactly on the tape. Where does the toe of his seventh placement land: before the far edge, on it, or past it?
Hertta measures a line on the wall, but she starts the first strip a little way above the line’s bottom. She puts every strip exactly on her mark, and the top of her ninth strip is exactly at the top of the line. Is the line taller than 9 strips, shorter, or exactly 9?

A bit more (Part 3)

Rug D in Figure 6 was measured by mark and move. Its right edge falls a little way into the seventh placement. How long is rug D? Say it two ways.
Hertta measures from the floor up to the lowest leaf of the beanstalk. Three strips fit, and the leaf is just a little above the third mark. How high is the leaf?
Peetu says a rug is “about 7 boot lengths long”. Leevi, with the same boot, says it is “between 6 and 7 boot lengths long”. Can both be right? When?
Hertta measures a line on the wall. Five strips fit. The top of the line falls almost at the top of the sixth strip, but not quite. How tall is the line? Say it two ways.

A bigger unit (Part 4)

Hertta’s paper strip is shorter than Peetu’s boot. Peetu’s rug is 7 of his boot lengths long. If Hertta measures the rug with her strip, will she count more than 7, fewer than 7, or exactly 7?
A rug is 10 of Peetu’s boot lengths long and 6 of a grown-up’s boot lengths long. Whose boot is longer?
Hertta cuts her paper strip in half and measures the beanstalk again with one half. The beanstalk was a bit more than 9 whole strips tall. At least how many half strips will fit?
Leevi measures two rugs with his boot. One is 5 of his boot lengths long and the other is 8. Which rug is longer, and by how many of Leevi’s boot lengths?
A red rug is 6 of Peetu’s boot lengths long. A blue rug is 6 of Leevi’s boot lengths long, and Leevi’s boot is longer than Peetu’s. Which rug is longer?

Explain

A rug is measured by mark and move, with a piece of tape on the start edge as well as one at every toe. Why is the number of pieces of tape one more than the length in boot lengths?
Why does skipping ahead make the count too small?
Why does sliding back make the count too big?
Leevi measures Peetu’s 7-boot rug. Once, he puts the heel down a little past the tape. Later, he puts it down a little before the tape, by exactly the same amount. Every other placement is on the tape. What does he count? Is it a good way to measure?
Why is the beanstalk “a bit more than 9 strips tall” and not “10 strips tall”?
Why does a longer boot need fewer placements to cross the same rug?

Answers

5 boot lengths. There are 5 placements, touching, from edge to edge, so one word for each gives 5.
6 boot lengths. There are 7 pieces of tape. The first marks the start edge, and each of the others is the toe of one placement: 7 - 1 = 6. Or count the spaces between the pieces: 6.
11 strips. One word was said for each placement, and the last word was “eleven” with the strip ending exactly at the top.
3 more placements. 4 + 3 = 7, so 7 - 4 = 3.
9 pieces. One at the start edge and one for each of the 8 placements: 1 + 8 = 9.
Longer than 9 boot lengths. Each skip left a gap of rug that no placement covered and no word counted, so 9 is too small.
Shorter than 12 strips. Each slip made the strip overlap the one below, so some of the line was counted twice and 12 is too big.
5 placements, and the rug is longer than 5 boot lengths. The bare rug between the placements was never counted, so the count is too small. (Measured carefully, rug C is 6 boot lengths.)
Past the far edge. The gap at the start pushes every placement along by the same small amount, so the seventh toe lands that much past the edge. That would make the rug look like a bit less than 7 boot lengths, though it is exactly 7.
Taller than 9 strips. The part of the line below the first strip is a gap at the start, never covered and never counted, so the line is 9 strips plus that bit.
Between 6 and 7 boot lengths, or a bit more than 6. The edge is only a little way into the seventh, nearer the end of the sixth, so it is also about 6 boot lengths.
A bit more than 3 strips high, or between 3 and 4. The leaf is just a little above the third mark, so it is about 3 strips high.
Yes. “Between 6 and 7” means the rug ends partway along the seventh placement. “About 7” is also true if that end is nearer the end of the seventh placement than the end of the sixth. Both are right for such a rug.
Between 5 and 6 strips, or a bit less than 6. The top is nearly at the end of the sixth strip, so it is about 6 strips tall.
More than 7. The strip is the smaller unit, so each placement covers less rug and more placements are needed.
The grown-up’s. The same rug took fewer of the grown-up’s boots, 6 against 10, so each grown-up boot covers more.
At least 18. Each whole strip is two halves, so 9 strips are 9 + 9 = 18 half strips. The bit more at the top might hold one more half or might not, so at least 18.
The 8-boot rug, by 3 of Leevi’s boot lengths. Both are in the same unit, and 8 - 5 = 3.
The blue rug. The counts are the same, but each of Leevi’s boot lengths is longer, so 6 of them reach further than 6 of Peetu’s.
The piece on the start edge is not the toe of any placement. Every other piece is the toe of one placement, so there is one more piece of tape than there are placements. The length is the number of placements, which is the number of spaces between the pieces of tape.
When the heel lands past the tape, the rug between the old toe and the new heel is left bare. No placement covers it, so no word counts it, and the count leaves that rug out.
When the heel lands before the tape, the new placement covers rug the last one already covered. That rug gets counted twice, so the count is more than the length.
He counts 7. The skip leaves a small gap, and from then on every placement is pushed along by that amount. The slide back moves them back by the same amount, so the later placements are back where they should be and the last toe lands on the edge. The gap and the overlap happen to be the same size, so the count comes out right. It is not a good way to measure: the right answer came by luck, and with a gap and an overlap of different sizes it would be wrong.
Only 9 whole strips fit. A tenth strip would cover wall above the beanstalk, which is not part of it, so counting it would count a whole strip that is not there. “A bit more than 9” says that 9 fit and some beanstalk is left over.
Each placement of the longer boot covers more of the rug. The whole rug has to be covered either way, and bigger pieces cover it in fewer pieces.