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Grade 1 · 1.MD.A.2 · 2.MD.A.1

Measuring length

A ruler is covered in marks and numbers. Which number says how long a crayon is, and what if the crayon does not start at the end, or the thing to measure is not straight at all?

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When you reach for it

A length is the number of whole units that fit along a thing, laid end to end with no gaps and no overlaps. What a unit is showed this with a row of many copies, and iterating one unit showed it with one copy moved along and marked. Both work. Both are slow, and every placement is a chance to leave a gap.

A ruler does the laying out ahead of time. It is a row of equal units already stuck together on a stick, with no gaps and no overlaps, and its numbers count the units for us. This lesson is about reading that row: which number is the length, why, and what to do when the thing does not sit at the start of the ruler or does not lie straight.

Here the units have names. A centimeter and an inch are just two units, each the same size on every ruler. Only whole units are used: a length comes out as a whole number of units, or as “about” a whole number.

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

  • Part 1, what the marks mean: a ruler as a row of units; reading a length that starts at 0.
  • Part 2, starting somewhere else: counting the spaces, not the marks.
  • Part 3, about: an end that falls between two marks.
  • Part 4, a curvy path: measuring with a string.

Do every part with real things: a centimeter ruler, an inch ruler, crayons, a few pieces of dry spaghetti, paper squares to cut out, and a long piece of string.

The procedure

Part 1: What the marks mean

Kielo cuts 10 paper squares, all the same size, and glues them in a row on a strip of card, touching, end to end. That is a row of units with no gaps and no overlaps, the same row Iterating a single unit built one placement at a time. Now she wants to number it so nobody has to count the squares each time.

She could write a number inside each square. A ruler does something cleverer: it puts a mark at the end of each unit and writes the number there. The mark at the end of the first square is 1, because one whole unit lies before it. The mark at the end of the second square is 2, because two whole units lie before it. And so on along the row, as in Figure 1.

1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10
Figure 1A ruler is a row of equal units stuck together. Each number sits at the END of a unit, so 0 is where the first unit starts.

Each number counts the units that end there. The number on a mark says how many whole units lie between the start of the row and that mark. That is the whole design.

So the start is 0. At the very beginning of the row, before the first square, no units have been laid yet. The mark there is 0. On a real ruler this is the 0 mark, and it is where the first unit starts, not where it ends. Some rulers print the 0; some leave it off and just draw the mark. Some rulers have a blank bit of plastic or wood before the 0 mark, so the end of the stick is not always the start of the units.

Reading a length. Put one end of the thing exactly at the 0 mark, lying along the ruler’s edge. Look at the mark at the other end. If that mark says 8, then 8 whole units lie between the 0 mark and the end: the thing is 8 units long. The number at the end is the length, because the ruler did the counting.

0 1 2 3 4 5 6 7 8 9 10
Figure 2The crayon starts at 0 and ends at 8. It covers 8 centimeters.

Why line up with 0. If the end of the thing sits at 0, the units the ruler counts are exactly the units along the thing. Line it up anywhere else and the ruler’s count includes units that are not along the thing. Part 2 deals with that.

The unit has to be said. “The crayon is 8” is not a length. On a centimeter ruler it is 8 centimeters. The same crayon on an inch ruler gives a different, smaller number, because an inch is a longer unit than a centimeter (What a unit is, and why we agree on standard ones).

Part 2: Starting somewhere else

Sometimes a thing is not lined up with 0. The end of the ruler is broken off, or the thing was put down in the middle, or it is easier to start at a mark you can see. The length can still be read. It just is not the number at the end anymore.

Oiva lays a spaghetti noodle on a centimeter ruler. Its left end is at the 3 mark and its right end is at the 10 mark (Figure 3).

0 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7
Figure 3The noodle covers the spaces from 3 to 10. There are 7 of them.

The noodle covers spaces, not marks. Each unit on the ruler is the space between two neighboring marks. The noodle lies over the space from 3 to 4, the space from 4 to 5, and so on up to the space from 9 to 10. Counting those spaces, as in Figure 3, gives 7. So the noodle is 7 centimeters long.

Why the end number is not the length here. The number 10 counts every unit from the 0 mark to the 10 mark. The noodle does not cover the first 3 of those, the ones from 0 to 3. Ten units take away the 3 it does not cover leaves 7:

10 - 3 = 7

Why counting marks gives the wrong number. The noodle touches the marks 3, 4, 5, 6, 7, 8, 9, 10. That is 8 marks, one more than the number of spaces, because a row of spaces always has a mark at each end. A mark is a place, with no length of its own. A space is one unit of length. Count the spaces.

Counting on works too. Put a finger on the 3 mark and hop to the next mark, saying “one”, then “two” at the 5, and so on. The hop that lands on 10 is “seven”. Each hop is one space, so the last word said is the number of units.

Wherever it sits, a length is the same. Slide the noodle so its left end is at 0 and its right end lands at 7. The noodle did not change; only where it sits on the ruler did. Starting at 0 is just the case where the end number and the length happen to agree.

Part 3: About

Things are not made to fit a whole number of units. Often an end falls between two marks. With whole units only, there are three honest ways to say the length, the same three Iterating a single unit used for a last placement that did not fit.

Usva lays a short piece of noodle from the 0 mark. Its other end falls a little past the 9 mark, well before the 10 (Figure 4, row A).

0 1 2 3 4 5 6 7 8 9 10 11 12 A 0 1 2 3 4 5 6 7 8 9 10 11 12 B
Figure 4Both ends fall a little past a mark. “About” names the nearest whole number of units.
  • “a bit more than 9 centimeters”: nine whole units fit and some noodle is left.
  • “between 9 and 10 centimeters”: the same, using both whole numbers.
  • “about 9 centimeters”: the end is nearer the 9 mark than the 10 mark, so the nearest whole number of units is 9.

“About” means the nearest whole unit. Look at where the end falls. If it is nearer the mark below, say “about” that mark’s number of units. If it is nearer the mark above, say “about” that one. A noodle ending just short of the 10 mark would be about 10 centimeters.

When the start is not 0. Count the whole spaces from the start to the last mark the thing passes, then look at the bit left over. Example 6 does this.

Part 4: A curvy path

A ruler is straight and stiff. Kielo finds a garden snake curled in a wave in the grass, and wants to know how long it is: not how far its head is from its tail, but how long its body is. A ruler cannot bend along a curve.

A string can. Lay the string along the snake from the tip of the head to the tip of the tail, following every bend, and pinch it where the tail ends. Then pull the string straight along a ruler, with the start at the 0 mark, and read the pinch (Figure 5).

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Figure 5The string bends with the snake. Pulled straight, it reaches the 14 mark. The straight line from head to tail would reach only 9.

Why the string’s length is the snake’s length. Every bit of the snake has a bit of string lying right beside it, and every bit of string from the start to the pinch lies beside a bit of snake. Straightening the string changes its shape, not its length: no string was added and none was cut off. So the straightened string is exactly as long as the path along the snake, and the ruler can measure the string.

The string must not stretch, and must follow every bend. A stretchy string pulled tight on the ruler reads too long. A string that cuts across a bend instead of following it misses some of the snake and reads too short.

Not the straight line. The dashed line in Figure 5, straight from head to tail, is much shorter than the string. It measures how far apart the ends are, not how long the body is. A curvy path is always longer than the straight line between its ends, unless it is already straight.

A string is not the only way: paper squares can be laid along a straightened string and counted, exactly as in What a unit is, and why we agree on standard ones. The string just turns a curved length into a straight one that units can be laid along.

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

A crayon from 0 to 8

Kielo lays a crayon along a centimeter ruler. Its flat end is at the 0 mark and its point lines up exactly with the 8 mark, as in Figure 2. How long is the crayon?

Solution

The crayon starts at 0, where the first unit on the ruler starts. Its point is at the mark numbered 8. On a ruler each number counts the whole units that lie between the 0 mark and that mark, so 8 whole centimeters lie along the crayon, end to end, with no gaps and no overlaps.

The crayon is 8 centimeters long.

To check, count the spaces under the crayon: 0 to 1 is one, 1 to 2 is two, and the last, 7 to 8, is eight.

Example 2

A paper cutout on Kielo’s ruler

Kielo’s paper ruler is Figure 1: 10 squares glued end to end, with a mark at the end of each. Usva lays a paper cutout of a fish along it, nose at the start of the row. Its tail is exactly at the mark where the sixth square ends. What number goes on that mark, how long is the fish in squares, and what number goes on the mark at the start?

Solution

Each mark is numbered by how many whole squares end at or before it. At the end of the sixth square, six squares lie behind it: the mark says 6.

The fish runs from the start of the row to that mark, so it lies along exactly those six squares. The fish is 6 squares long. The number on the end mark is the length because the fish starts where the counting starts.

At the start of the row, before the first square, no square has been laid yet. That mark says 0. This is why a ruler begins at 0 and not at 1: the 1 is at the END of the first unit.

Example 3

The noodle from 3 to 10

Oiva’s spaghetti noodle lies on a centimeter ruler from the 3 mark to the 10 mark (Figure 3). How long is it?

Solution

The noodle does not start at 0, so the number at its end is not its length. Count the spaces it covers. Each space between neighboring marks is one centimeter:

3 \text{ to } 4,\ 4 \text{ to } 5,\ 5 \text{ to } 6,\ 6 \text{ to } 7,\ 7 \text{ to } 8,\ 8 \text{ to } 9,\ 9 \text{ to } 10

That is 7 spaces, so the noodle is 7 centimeters long.

The noodle touches 8 marks, from 3 through 10, but marks have no length; there is one more mark than there are spaces. Another way to see 7: the 10 counts all the units from 0, and the noodle misses the first 3, so 10 - 3 = 7. To check, count on: 3 + 7 = 10.

Example 4

The same crayon in two places

Oiva puts his crayon on a centimeter ruler with its flat end at the 4 mark. Its point is at the 10 mark. Then Usva slides it so its flat end is at 0. Where will its point be, and how long is the crayon?

Solution

From 4 to 10, count the spaces: 4 to 5, 5 to 6, 6 to 7, 7 to 8, 8 to 9, 9 to 10. That is 6 spaces. Counting on gives the same: from 4, six hops reach 10, since 4 + 6 = 10.

So the crayon is 6 centimeters long. Sliding it does not change its length. With the flat end at 0, the point covers 6 spaces from the start and lands on the 6 mark, where the number at the end is the length.

Both readings agree, as they must: 10 - 4 = 6, and 6 - 0 = 6.

Example 5

A noodle a bit more than 9

Usva lays a short noodle on a centimeter ruler with one end at the 0 mark. The other end falls a little past the 9 mark, much nearer 9 than 10, as in row A of Figure 4. How long is the noodle?

Solution

The noodle starts at 0, so the marks count its whole units. It passes the 9 mark, so 9 whole centimeters fit along it. It does not reach the 10 mark, so a tenth whole centimeter does not fit.

The noodle is between 9 and 10 centimeters long, or a bit more than 9 centimeters. Its end is much nearer the 9 mark than the 10 mark, so it is about 9 centimeters long.

Saying “10 centimeters” would count a whole unit that is not all there. Saying just “9 centimeters” would leave out the bit at the end. “About 9” says the nearest whole number and admits it is not exact.

Example 6

A paper cutout starting at 2

Oiva lays a paper cutout on a centimeter ruler with its left edge at the 2 mark. Its right edge falls a little past the 8 mark, much nearer 8 than 9 (Figure 4, row B). About how long is the cutout?

Solution

It does not start at 0, so count spaces. The whole spaces it covers run from 2 to 8:

2 \text{ to } 3,\ 3 \text{ to } 4,\ 4 \text{ to } 5,\ 5 \text{ to } 6,\ 6 \text{ to } 7,\ 7 \text{ to } 8

That is 6 whole centimeters, and 8 - 2 = 6 agrees. Then there is a bit more, past the 8 mark, which does not reach the 9.

So the cutout is between 6 and 7 centimeters long. The bit past the 8 mark is small, much less than half a space, so it is about 6 centimeters long.

Reading “about 8” from the end mark would be wrong for the same reason as in Example 3: the 2 units before the cutout’s left edge are not along the cutout.

Example 7

The garden snake

Kielo lays a string along the curvy garden snake, from the tip of its head to the tip of its tail, following every bend. She pinches the string at the tail. Then she pulls it straight on an inch ruler with its start at 0, and the pinch lands exactly on the 14 mark (Figure 5). How long is the snake?

Solution

The string lay beside every bit of the snake, from head to tail, and nowhere else. So the string from its start to the pinch is exactly as long as the snake’s body.

Pulling the string straight changed its shape, not its length. On the ruler it starts at the 0 mark and ends at the 14 mark, so 14 whole inches lie along it.

The snake is 14 inches long.

The string must not stretch when it is pulled straight; a stretched string would reach past 14 and make the snake look longer than it is.

Example 8

Head to tail, or along the body

Before using the string, Oiva lays the inch ruler straight from the snake’s head to the tip of its tail, across the curves, with the head at 0. The tail is at the 9 mark (the dashed line in Figure 5). Which number is the snake’s length, 9 inches or Kielo’s 14 inches, and how much longer is one than the other?

Solution

Oiva’s 9 inches is how far apart the head and the tail are. The snake does not lie along that straight line; its body bends out to one side and back, again and again. The ruler skipped every bend.

Kielo’s string followed the body, so 14 inches is the snake’s length. If the snake stretched out straight, its tail would be 14 inches from its head.

The difference is the length the bends add:

14 - 9 = 5

The path along the snake is 5 inches longer than the straight line from head to tail. A curvy path is always longer than the straight line between its ends.

Boundaries

Line one end up with a mark. Reading the end mark works only when the thing starts exactly at 0. Starting at another mark works by counting spaces. Starting between two marks, with whole units only, the count is no longer exact.

The 0 mark, not the end of the stick. On many rulers the units start a little way in from the end. Lining up with the end of the stick adds a bit of blank ruler to the count.

Lie along the ruler, not across it at a slant. A crayon laid at an angle across the ruler is read against marks it does not run along, and the reading is not its length.

Count spaces, never marks. A mark is a place; a space between two marks is one unit. For anything not starting at 0, the marks it touches are one more than its length.

Whole units, and “about”. In this lesson a length is a whole number of units or “about” a whole number, the nearest one. An end that falls exactly halfway between two marks is not settled by “nearest”: both marks are equally close. “Between” still says it honestly, and finer marks are the way to say more (Measuring to the half and quarter inch).

A string must follow the path and must not stretch. Cutting across a bend reads too short; stretching reads too long.

Lengths compare only in the same unit. 8 centimeters and 8 inches are not the same length. Two rulers, two units: measure both things with the same one.

Where this goes next

A ruler is a number line made of units. Reading from 3 to 10 is finding the distance between two numbers, which grows into Length on a number line. Lengths that start at 0 and are added end to end grow into adding lengths, and around a shape into Perimeter.

The “about” of Part 3 points at a larger idea: no ruler makes every length exact. Finer marks, halves and fourths of an inch, get closer (Measuring to the half and quarter inch), and how finely a tool is marked sets how closely it can measure (Measurement is approximate: precision of a tool). Centimeters and inches get their families of bigger and smaller units in Metric length units and Customary length units, and a feel for how long each one is in Benchmarks, estimating and choosing a unit.

The string of Part 4 comes back in a big way: the distance around a circle is a curved path, and wrapping a string around a can is the first way to measure it (Circumference).

Exercises

What the marks mean (Part 1)

Exercises 1 and 2 use Figure 6. Every ruler in it is a centimeter ruler.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 B 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 C 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 D 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 E 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 F
Figure 6Rulers for Exercises 1, 2, 6, 7, 12 and 13.
  1. How long is the crayon in row A?
  2. How long is the paper cutout in row B?
  3. Usva lays a noodle along a centimeter ruler with one end at the 0 mark. The other end is exactly at the 13 mark. How long is the noodle?
  4. Oiva glues 12 equal paper squares end to end and puts a mark at the end of each square. What number goes on the mark at the end of the last square? What number goes on the mark at the very start?
  5. A crayon lies on a centimeter ruler from the 0 mark to the 4 mark. How many marks does it touch, counting both ends? How long is it?

Starting somewhere else (Part 2)

  1. How long is the noodle in row C of Figure 6?
  2. How long is the crayon in row D of Figure 6?
  3. A paper cutout lies on an inch ruler from the 1 mark to the 7 mark. How long is it?
  4. A long noodle lies on a centimeter ruler from the 8 mark to the 20 mark. How long is it?
  5. Oiva’s ruler has its end broken off, and the first mark left is 4. He lays a crayon that is 5 centimeters long with its flat end on the 4 mark. At which mark is its point?
  6. One crayon lies from the 0 mark to the 9 mark. Another lies from the 5 mark to the 13 mark. Both rulers are centimeter rulers. Which crayon is longer, and by how much?

About (Part 3)

  1. About how long is the crayon in row E of Figure 6? Say it two ways.
  2. About how long is the paper cutout in row F of Figure 6? Say it two ways.
  3. A noodle starts at the 0 mark of a centimeter ruler and ends just past the 15 mark. About how long is it?
  4. A crayon starts at the 2 mark of a centimeter ruler and ends just before the 9 mark, between 8 and 9 but much nearer 9. About how long is it?
  5. Usva’s noodle starts at the 0 mark and ends between the 10 and 11 marks, much nearer 11. About how long is it?

A curvy path (Part 4)

  1. Kielo lays a string along a different curvy snake, pinches it at the tail, and pulls it straight on an inch ruler from the 0 mark. The pinch is at the 17 mark. How long is the snake?
  2. Oiva lays a ruler straight from the head to the tail of a curled-up snake and reads 10 inches. Is the snake’s body longer than 10 inches, shorter, or exactly 10?
  3. Usva straightens her string on an inch ruler but starts it at the 3 mark. The pinch lands on the 18 mark. How long is the path she measured?
  4. Kielo lays a string along a wiggly line drawn on paper, then straightens the string and lays paper squares along it, end to end. Exactly 9 squares fit. How long is the wiggly line?
  5. One snake’s string reaches 12 inches. Another’s reaches 15 inches. Which snake is longer, and by how much?

Explain

  1. Why is the number at the end of a crayon its length when the crayon starts at the 0 mark?
  2. A crayon lies from the 6 mark to the 11 mark. Why is it 5 centimeters long and not 6, though it touches six marks?
  3. Why does a string give the length of a curvy snake, when the ruler laid straight from head to tail does not?
  4. A crayon lies from the 0 mark to the 7 mark. Kielo slides it so its flat end is at the 3 mark. At which mark is its point now, and why is it still the same length?
Answers
  1. 5 centimeters. It starts at 0 and ends at the 5 mark, and the 5 counts the whole units from 0.
  2. 11 centimeters. It starts at 0 and its right edge is at the 11 mark.
  3. 13 centimeters. It starts at 0, so the end mark is the length.
  4. 12 at the end of the last square, 0 at the start. Each mark counts the squares that end at or before it: twelve at the end, none at the start.
  5. It touches 5 marks (0, 1, 2, 3, 4) and is 4 centimeters long. It covers 4 spaces; there is one more mark than spaces.
  6. 5 centimeters. It covers the spaces from 4 to 9: five of them. Or 9 - 4 = 5.
  7. 8 centimeters. From 6 to 14 there are 8 spaces, and 14 - 6 = 8.
  8. 6 inches. From 1 to 7 there are 6 spaces, and 7 - 1 = 6.
  9. 12 centimeters. Count on from 8: twelve hops reach 20, since 8 + 12 = 20.
  10. The 9 mark. Five spaces on from 4: 4 + 5 = 9.
  11. The first crayon, by 1 centimeter. The first is 9 centimeters. The second covers 13 - 5 = 8 spaces, so it is 8 centimeters, and 9 - 8 = 1.
  12. About 7 centimeters; between 7 and 8, or a bit more than 7. It starts at 0 and ends a little past the 7 mark, nearer 7 than 8.
  13. About 4 centimeters; between 3 and 4, or a bit less than 4. Its edge is most of the way from the 3 mark to the 4 mark, so nearer 4.
  14. About 15 centimeters. Fifteen whole units fit and a little more; the end is nearer 15 than 16.
  15. About 7 centimeters. The whole spaces from 2 to 8 are 6 (8 - 2 = 6), and the end goes most of the way through one more, so it is nearer 7 than 6.
  16. About 11 centimeters. It is between 10 and 11, and its end is nearer the 11 mark.
  17. 17 inches. The straightened string starts at 0 and ends at 17, and it is exactly as long as the snake’s body.
  18. Longer than 10 inches. The straight line from head to tail skips every bend; the body is longer than the straight line between its ends.
  19. 15 inches. From 3 to 18 there are 15 spaces, and 18 - 3 = 15.
  20. 9 squares long. The string is as long as the line, and 9 squares fit along the string with no gaps and no overlaps.
  21. The second snake, by 3 inches. Both are in inches, and 15 - 12 = 3.
  22. Each number on a ruler counts the whole units between the 0 mark and that mark. A crayon starting at 0 lies along exactly those units, so the number at its end counts its units.
  23. Its length is the number of spaces it covers: 6 to 7, 7 to 8, 8 to 9, 9 to 10, 10 to 11, which is five. The six marks are places at the ends of those spaces, one more than the spaces, and a mark has no length.
  24. The string lies beside every bit of the snake’s body, so it is as long as the body, and straightening it keeps its length. The straight ruler measures how far apart the head and tail are, which leaves out every bend.
  25. The 10 mark, since 3 + 7 = 10. The crayon covers 7 spaces wherever it lies; sliding it changes where it sits, not how many units fit along it.
kg1128 · updated 2026-10-21 · CC-BY-SA-4.0