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Grade 3 · 3.MD.B.4 · 2.MD.D.9

Measuring to the half and quarter inch

A crayon ends between two inch marks on a ruler. How do you say how long it is, and how do you show the lengths of a whole box of crayons at once?

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A fraction as part of a wholeGrade 3Customary length unitsGrade 2Dot plotsGrades 4 to 6
Where you are 7 in Length and units
Measurement12
Length and units7
▴ 2 aboveCustomary length unitsLength on a number lineMetric length units
Measuring to the half and quarter inch
Converting metric units by powers of ten

When you reach for it

A length is easy to say when its end lands right on an inch mark: the crayon is 3 inches, the earthworm is 2 inches. Most things do not end on an inch mark. An earthworm stretched along a ruler ends somewhere past 2 and before 3, and “about 2” or “almost 3” throws away information the ruler is holding. The marks between the inch marks are there to say how far past the last whole inch the end reaches.

This lesson reads those marks. Part 1 uses a ruler with each inch cut in half. Part 2 uses a ruler with each inch cut into fourths, and names lengths such as \displaystyle 3\frac{1}{4} inches. Part 3 takes a pile of such measurements, a dozen earthworms or a box of crayons, and shows them all at once on a line plot whose marks go up by quarter inches, then reads the plot.

Three ideas are already in place. An inch is a fixed length, and 12 of them make a foot (Customary length units). A fraction such as \displaystyle \frac{1}{4} names one of 4 equal parts of a whole (A fraction as part of a whole). A line plot puts one dot above a number line for each measurement (Dot plots).

The procedure

Part 1: halves of an inch

A ruler is a number line made of inches, with 0 at the end you measure from (Locating a fraction on the number line treats any number line the same way). Lay the object straight along the ruler with one end at 0. The length is the number at the mark where the other end lands.

On a ruler marked in halves, a shorter mark sits halfway between each pair of inch marks. It cuts each inch into 2 equal parts, so each part is \displaystyle \frac{1}{2} inch, because 2 of them fill the inch. The short mark after 2 is 2 whole inches and one half inch more, and we write it \displaystyle 2\frac{1}{2} inches. The whole number counts the full inches; the fraction says what part of the next inch is covered.

When the end falls between two marks, the length is not exactly any mark. Measuring “to the nearest half inch” means choosing the half inch mark the end is closest to. The marks to compare are the two on either side of the end, one just before it and one just after it. Whichever is closer is the reading. The real length is a little more or a little less than the reading, and the reading says so by naming the unit: “\displaystyle 3\frac{1}{2} inches, to the nearest half inch.”

Part 2: fourths of an inch

On a ruler marked in fourths, each inch is cut into 4 equal parts, so each part is \displaystyle \frac{1}{4} inch, a quarter inch. Inside each inch there are 3 marks. The middle one is usually drawn a little longer than the other two, because it is also the half inch mark: 2 of the 4 equal parts make half of the inch, so \displaystyle \frac{2}{4} inch and \displaystyle \frac{1}{2} inch are the same length and land on the same mark. Between 3 and 4 the marks are

3\frac{1}{4}, \quad 3\frac{2}{4} = 3\frac{1}{2}, \quad 3\frac{3}{4},

and the fourth quarter after 3 is 4 itself.

A reading on this ruler has the same two parts as before: the whole inches before the end, then how many quarter inches past the last whole inch. “To the nearest quarter inch” means the closest quarter inch mark, and every half inch mark is one of those.

If the object does not start at 0, the length is still the space it covers. Count the whole inches it covers, then the quarter inches left over. The number printed under its far end says where the end is, not how long the object is.

Part 3: a line plot of the lengths

A single length is one number. Twelve earthworms give twelve lengths, and a list of twelve mixed lengths is hard to take in. A line plot puts them on one number line so the shape of the whole collection shows.

Draw a number line and mark it every quarter inch, starting at or just below the shortest length and ending at or just above the longest. Every quarter inch in that stretch gets a mark, even one that no measurement lands on, because the marks must be equally spaced to keep the picture honest: the gap between \displaystyle 2\frac{1}{4} and \displaystyle 2\frac{1}{2} is the same quarter inch as the gap between 3 and \displaystyle 3\frac{1}{4}, and the plot has to show it that way. Then put one dot above the matching mark for each measurement, stacking dots that share a length.

Reading the plot is counting. The tallest stack is the most common length. The dot farthest right is the longest, farthest left the shortest. “How many are longer than 3 inches?” is answered by counting every dot to the right of the 3 mark. The total number of dots is the number of things measured.

Worked examples

Example 1

Part 1: an earthworm on a half inch ruler

Wojtek finds an earthworm in the school garden and lets it stretch out along a ruler marked in half inches, its tail at 0. Its head lines up with the short mark between 2 and 3 (Figure 1). How long is the earthworm?

earthworm 0 1 2 3 4 inches
Figure 1Two whole inches, then one half inch more. The earthworm is \displaystyle 2\frac{1}{2} inches long.
Solution

From 0 to 2 the earthworm covers 2 whole inches. Its head goes past the 2 mark but stops before 3, so the third inch is only partly covered. The short mark it reaches cuts the inch from 2 to 3 into 2 equal parts, and the head is at the end of the first part. That part is \displaystyle \frac{1}{2} inch.

The earthworm is \displaystyle 2\frac{1}{2} inches long: 2 whole inches and one half inch more.

As a check, count the half inch spaces from the tail to the head. Each inch holds 2 of them, so 0 to 2 holds 2 \times 2 = 4, and one more reaches the head: 5 half inches. Two of those make each whole inch, which leaves one half inch over, the same \displaystyle 2\frac{1}{2} inches.

Example 2

Part 1: a crayon to the nearest half inch

Loic lays a crayon along a ruler marked in half inches, one end at 0. The other end does not land on a mark. It is past 3 and before \displaystyle 3\frac{1}{2}, closer to \displaystyle 3\frac{1}{2} (Figure 2). What is the crayon’s length to the nearest half inch?

crayon 0 1 2 3 4 inches
Figure 2The crayon’s end falls between two marks. It is closer to the \displaystyle 3\frac{1}{2} mark.
Solution

The two marks on either side of the end are 3 and \displaystyle 3\frac{1}{2}. Those are the only two candidates: every other half inch mark is farther away, because it lies beyond one of these two.

The end is closer to the \displaystyle 3\frac{1}{2} mark, so the crayon is \displaystyle 3\frac{1}{2} inches long to the nearest half inch. It is really a little shorter than \displaystyle 3\frac{1}{2} inches, and “to the nearest half inch” is the phrase that says so.

Measured only to the nearest whole inch, the same crayon would be 3 inches, since the end is between 3 and 4 and much closer to 3. The half inch marks let the ruler say more.

Example 3

Part 2: an earthworm on a quarter inch ruler

Maebh measures another earthworm on a ruler marked in fourths of an inch. Its tail is at 0 and its head lines up with the first short mark after 3 (Figure 3). How long is it? Name the other marks between 3 and 4 as well.

earthworm 0 1 2 3 4 inches
Figure 3Each inch is cut into fourths. The earthworm ends one quarter inch past 3.
Solution

The earthworm covers the 3 whole inches from 0 to 3. The inch from 3 to 4 is cut by 3 marks into 4 equal parts, each \displaystyle \frac{1}{4} inch. The head is at the end of the first of those parts.

The earthworm is \displaystyle 3\frac{1}{4} inches long: 3 whole inches and one fourth of an inch more.

The other marks in that inch are \displaystyle 3\frac{2}{4} and \displaystyle 3\frac{3}{4}, two and three quarter inches past 3. The middle mark, \displaystyle 3\frac{2}{4}, is also \displaystyle 3\frac{1}{2}: two of the four equal parts are half of the inch. That is why it is drawn longer, and why a ruler marked in fourths still shows every half inch.

Counting quarter inch spaces checks the length. Each inch holds 4, so 0 to 3 holds 3 \times 4 = 12, and one more reaches the head: 13 quarter inches, which is 3 whole inches and 1 quarter inch.

Example 4

Part 2: the same earthworm, two answers

Maebh’s next earthworm lies along the quarter inch ruler from 0, and its head stops between the \displaystyle 2\frac{1}{2} mark and the \displaystyle 2\frac{3}{4} mark, much closer to \displaystyle 2\frac{3}{4} (Figure 4). How long is it to the nearest quarter inch? To the nearest half inch?

earthworm 0 1 2 3 4 inches
Figure 4One earthworm, two questions. To the nearest quarter inch and to the nearest half inch the answers differ.
Solution

To the nearest quarter inch, the marks on either side are \displaystyle 2\frac{1}{2} and \displaystyle 2\frac{3}{4}. The head is much closer to \displaystyle 2\frac{3}{4}, so the reading is \displaystyle 2\frac{3}{4} inches.

To the nearest half inch, only the half inch marks count. They are the inch marks and the longer middle marks: on either side of the head they are \displaystyle 2\frac{1}{2} and 3. The \displaystyle 2\frac{3}{4} mark sits exactly halfway between those two. The head stops before \displaystyle 2\frac{3}{4}, on the \displaystyle 2\frac{1}{2} side of the halfway point, so it is closer to \displaystyle 2\frac{1}{2}. The reading is \displaystyle 2\frac{1}{2} inches.

Both answers are correct; they answer different questions. The quarter inch reading is closer to the true length because its marks are closer together. Why finer marks give finer readings, and how close a reading must be to the true length, is the subject of Measurement is approximate: precision of a tool.

Example 5

Part 2: a crayon that does not start at zero

Wojtek lays a crayon on a quarter inch ruler with its flat end at the 1 inch mark and its tip at the \displaystyle 3\frac{3}{4} mark (Figure 5). How long is the crayon?

crayon 0 1 2 3 4 inches
Figure 5The crayon starts at 1, not at 0. Its length is the space it covers, counted from where it starts.
Solution

The crayon’s length is the space it covers, wherever on the ruler that space is. The number under the tip only says where the tip is.

Count whole inches from the flat end: 1 to 2 is one inch and 2 to 3 is another, so the crayon covers 2 whole inches. From 3 to the tip there are 3 quarter inch spaces, which is \displaystyle \frac{3}{4} inch. The crayon is \displaystyle 2\frac{3}{4} inches long.

Sliding the crayon left by one inch puts its flat end at 0. Every point on it moves the same amount, so its tip moves from \displaystyle 3\frac{3}{4} to the mark one inch earlier, \displaystyle 2\frac{3}{4}, and now the tip reads the length directly. Counting the quarter inch spaces agrees: 2 inches hold 2 \times 4 = 8, and 3 more make 11 quarter inches.

Example 6

Part 3: making the earthworm line plot

Wojtek’s class digs up twelve earthworms in the school garden and measures each to the nearest quarter inch before letting it go. In the order they were dug up, the lengths in inches are

2\tfrac{3}{4},\ 3,\ 2\tfrac{1}{2},\ 3\tfrac{1}{4},\ 2,\ 2\tfrac{3}{4},
3\tfrac{3}{4},\ 2\tfrac{1}{4},\ 3,\ 2\tfrac{1}{2},\ 3\tfrac{1}{4},\ 2\tfrac{3}{4}.

Make a line plot of the lengths.

Solution

The shortest length is 2 inches and the longest is \displaystyle 3\frac{3}{4} inches, so the number line runs from 2 to \displaystyle 3\frac{3}{4}, marked every quarter inch. That is 8 marks:

2,\ 2\tfrac{1}{4},\ 2\tfrac{1}{2},\ 2\tfrac{3}{4},\ 3,\ 3\tfrac{1}{4},\ 3\tfrac{1}{2},\ 3\tfrac{3}{4}.

The \displaystyle 3\frac{1}{2} mark goes in even though no earthworm is that long. Leaving it out would put \displaystyle 3\frac{1}{4} and \displaystyle 3\frac{3}{4} next to each other, as if they were one quarter inch apart instead of two.

Next, count how many earthworms have each length. Going down the list and making a tally:

Length (inches)Earthworms
21
\displaystyle 2\frac{1}{4}1
\displaystyle 2\frac{1}{2}2
\displaystyle 2\frac{3}{4}3
32
\displaystyle 3\frac{1}{4}2
\displaystyle 3\frac{1}{2}0
\displaystyle 3\frac{3}{4}1

The counts add to 1 + 1 + 2 + 3 + 2 + 2 + 0 + 1 = 12, one for every earthworm, so none was missed or counted twice. Each count becomes a stack of that many dots above its mark (Figure 6). The order the worms were dug up is gone from the plot; what it keeps is how many share each length.

2
\displaystyle 2\frac{1}{4}
\displaystyle 2\frac{1}{2}
\displaystyle 2\frac{3}{4}
3
\displaystyle 3\frac{1}{4}
\displaystyle 3\frac{1}{2}
\displaystyle 3\frac{3}{4}
 earthworm length (inches)
Figure 6Twelve earthworms from the school garden, one dot for each, measured to the nearest quarter inch.
Example 7

Part 3: reading the earthworm line plot

Use the plot in Figure 6. What is the most common length? How long is the longest earthworm, and the shortest? How many earthworms are longer than 3 inches? How many more earthworms are \displaystyle 2\frac{3}{4} inches long than \displaystyle 2\frac{1}{2} inches long?

Solution

The tallest stack, 3 dots, stands above \displaystyle 2\frac{3}{4}. The most common length is \displaystyle 2\frac{3}{4} inches.

The dot farthest to the right is above \displaystyle 3\frac{3}{4}, so the longest earthworm is \displaystyle 3\frac{3}{4} inches. The dot farthest to the left is above 2, so the shortest is 2 inches.

“Longer than 3 inches” means the dots to the right of the 3 mark, not the ones on it. There are 2 above \displaystyle 3\frac{1}{4}, none above \displaystyle 3\frac{1}{2} and 1 above \displaystyle 3\frac{3}{4}, so 2 + 0 + 1 = 3 earthworms are longer than 3 inches.

There are 3 dots above \displaystyle 2\frac{3}{4} and 2 above \displaystyle 2\frac{1}{2}, and 3 - 2 = 1, so one more earthworm is \displaystyle 2\frac{3}{4} inches long.

Boundaries

A reading to the nearest half inch or quarter inch is not the exact length. It names the closest mark, and the true length can be a little more or a little less. Two earthworms with the same reading need not be exactly the same length, which is why a reading carries the words “to the nearest quarter inch.”

When an end sits exactly halfway between two marks, “the closest mark” has no answer: both are equally close. A class has to agree on a rule for that case (for example, always take the larger mark) and use it every time.

The ruler reads a length only when the object lies straight and flat along its edge. An earthworm that curls, or a crayon laid at a slant, covers more than the stretch of ruler under it. If the object starts at a mark other than 0, the number at its far end is a position, and the length comes from counting the space covered.

A line plot marked in quarter inches holds measurements made to the nearest quarter inch or to the nearest half inch, since every half inch mark is also a quarter inch mark. It cannot honestly hold lengths read more finely, say to the nearest eighth, without finer marks. Its marks must be equally spaced and include every quarter inch between the ends, even ones with no dots.

A line plot keeps how many measurements share each length. It loses which object had which length and the order they were measured in.

Where this goes next

Lengths such as \displaystyle 3\frac{1}{4} inches are whole numbers with a fraction attached, and Mixed numbers and improper fractions gives them their name and shows how to rewrite them as a single fraction, the way \displaystyle 3\frac{1}{4} is 13 quarter inches. Benchmark fractions puts every mark inside one inch to use as a landmark for comparing fractions. Line plots with fractions carries the line plot to eighths of an inch and uses the plot to add and share the measured lengths. On the metric side, Metric length units cuts the centimeter into ten millimeters instead of halving and halving again. And Measurement is approximate: precision of a tool asks how close a reading must be to the true length, and why finer marks give finer readings.

Exercises

Part 1: halves of an inch

  1. On a ruler marked in half inches, an earthworm’s tail is at 0 and its head lines up with the short mark between 1 and 2. How long is the earthworm?
  2. On a ruler marked in half inches, an earthworm’s tail is at 0 and its head is between 4 and \displaystyle 4\frac{1}{2}, closer to 4. How long is it to the nearest half inch?
  3. On a ruler marked in half inches, a crayon starts at 0 and its end is between \displaystyle 2\frac{1}{2} and 3, closer to 3. How long is it to the nearest half inch?
  4. How many half inch spaces are there from 0 to 3 on a ruler marked in half inches?
  5. An earthworm is \displaystyle 5\frac{1}{2} inches long. With its tail at 0, between which two inch marks is its head, and on which mark?

Part 2: fourths of an inch

  1. On a ruler marked in fourths, a crayon starts at 0 and ends at the first short mark after 1. How long is the crayon?
  2. On a ruler marked in fourths, an earthworm’s tail is at 0 and its head is on the third mark after 1. How long is the earthworm?
  3. Give another name for the mark \displaystyle 4\frac{2}{4} on a ruler marked in fourths.
  4. On a ruler marked in fourths, a crayon starts at 0 and ends between \displaystyle 1\frac{3}{4} and 2, closer to 2. How long is it to the nearest quarter inch?
  5. A crayon lies on a quarter inch ruler with its flat end at the 1 inch mark and its tip at the \displaystyle 3\frac{1}{2} mark. How long is the crayon?
  6. How many quarter inch spaces are there in 1 inch? In \displaystyle 1\frac{3}{4} inches?

Part 3: line plots

Exercises 12 to 17 use Loic’s crayon line plot.

\displaystyle 1\frac{3}{4}
2
\displaystyle 2\frac{1}{4}
\displaystyle 2\frac{1}{2}
\displaystyle 2\frac{3}{4}
3
\displaystyle 3\frac{1}{4}
\displaystyle 3\frac{1}{2}
 crayon length (inches)
Figure 7Loic’s ten crayons, measured to the nearest quarter inch. Used in Exercises 12 to 17.
  1. How many crayons did Loic measure?
  2. What is the most common crayon length?
  3. How long is the longest crayon, and how many crayons are that long?
  4. How many crayons are shorter than \displaystyle 2\frac{1}{2} inches?
  5. Which marks on the plot have no dots above them?
  6. How many more crayons are \displaystyle 2\frac{1}{2} inches long than 2 inches long?
  7. Maebh measures eight earthworms to the nearest quarter inch. Their lengths in inches are \displaystyle 1\frac{3}{4}, 2, \displaystyle 2\frac{1}{4}, 2, \displaystyle 2\frac{1}{2}, \displaystyle 1\frac{3}{4}, 2, 3. Which marks does her line plot need, and how many dots go above each?
  8. On Maebh’s plot from Exercise 18, how many earthworms are longer than 2 inches?

Reasoning

  1. The earthworm plot in Figure 6 has a mark at \displaystyle 3\frac{1}{2} with no dots above it. Why not leave that mark out?
  2. Explain why \displaystyle 2\frac{2}{4} inches and \displaystyle 2\frac{1}{2} inches name the same mark on a ruler.
  3. An earthworm lies along a quarter inch ruler from 0, and its head stops just past the \displaystyle 1\frac{3}{4} mark. What is its length to the nearest quarter inch? To the nearest half inch? Explain.
  4. Can two earthworms have the same length to the nearest half inch but different lengths to the nearest quarter inch? Give an example or explain why not.
  5. A crayon lies from the 1 inch mark to some other mark. Explain why counting the inches and quarter inches it covers gives its length, but the number under its tip does not.
Answers
  1. \displaystyle 1\frac{1}{2} inches. It covers 1 whole inch, then one of the 2 equal parts of the next inch.
  2. 4 inches. The marks on either side are 4 and \displaystyle 4\frac{1}{2}, and the head is closer to 4.
  3. 3 inches. The marks on either side are \displaystyle 2\frac{1}{2} and 3, and the end is closer to 3.
  4. 6 half inch spaces. Each inch holds 2, and 3 \times 2 = 6.
  5. Between 5 and 6, on the short mark halfway between them. It covers 5 whole inches and half of the sixth.
  6. \displaystyle 1\frac{1}{4} inches. One whole inch, then the first of the 4 equal parts of the next inch.
  7. \displaystyle 1\frac{3}{4} inches. The marks after 1 are \displaystyle 1\frac{1}{4}, \displaystyle 1\frac{1}{2} and \displaystyle 1\frac{3}{4}, so the third is \displaystyle 1\frac{3}{4}.
  8. \displaystyle 4\frac{1}{2}. Two of the four equal parts of an inch make half of it, so \displaystyle \frac{2}{4} inch is \displaystyle \frac{1}{2} inch.
  9. 2 inches. The marks on either side are \displaystyle 1\frac{3}{4} and 2, and the end is closer to 2.
  10. \displaystyle 2\frac{1}{2} inches. From 1 to 3 it covers 2 whole inches; from 3 to \displaystyle 3\frac{1}{2} it covers 2 quarter inches, which is \displaystyle \frac{1}{2} inch.
  11. 4 and 7. One inch holds 4 quarter inches; \displaystyle 1\frac{3}{4} inches holds 4 + 3 = 7.
  12. 10 crayons. Counting every dot: 1 + 2 + 0 + 3 + 1 + 1 + 0 + 2 = 10.
  13. \displaystyle 2\frac{1}{2} inches. Its stack, 3 dots, is the tallest.
  14. \displaystyle 3\frac{1}{2} inches, and 2 crayons are that long. That is the mark farthest right with dots.
  15. 3 crayons. The dots to the left of \displaystyle 2\frac{1}{2}: 1 above \displaystyle 1\frac{3}{4}, 2 above 2 and none above \displaystyle 2\frac{1}{4}, and 1 + 2 + 0 = 3.
  16. \displaystyle 2\frac{1}{4} and \displaystyle 3\frac{1}{4}. No crayon has either length.
  17. 1 more. There are 3 dots above \displaystyle 2\frac{1}{2} and 2 above 2, and 3 - 2 = 1.
  18. Marks every quarter inch from \displaystyle 1\frac{3}{4} to 3: \displaystyle 1\frac{3}{4}, 2, \displaystyle 2\frac{1}{4}, \displaystyle 2\frac{1}{2}, \displaystyle 2\frac{3}{4}, 3. Dots: 2 above \displaystyle 1\frac{3}{4}, 3 above 2, 1 above \displaystyle 2\frac{1}{4}, 1 above \displaystyle 2\frac{1}{2}, none above \displaystyle 2\frac{3}{4}, 1 above 3. Check: 2 + 3 + 1 + 1 + 0 + 1 = 8.
  19. 3 earthworms. The dots to the right of 2: 1 + 1 + 0 + 1 = 3.
  20. The marks on a line plot must be equally spaced, every quarter inch. Without the \displaystyle 3\frac{1}{2} mark, \displaystyle 3\frac{1}{4} and \displaystyle 3\frac{3}{4} would sit one space apart although they differ by two quarter inches, and the empty mark is itself information: no earthworm had that length.
  21. Cut an inch into 4 equal parts. The middle mark has 2 parts on each side, so 2 of the 4 parts, \displaystyle \frac{2}{4} inch, is half of the inch, \displaystyle \frac{1}{2} inch. Past 2 whole inches that same mark is \displaystyle 2\frac{2}{4} and \displaystyle 2\frac{1}{2}.
  22. \displaystyle 1\frac{3}{4} inches to the nearest quarter inch, and 2 inches to the nearest half inch. Its quarter inch neighbors are \displaystyle 1\frac{3}{4} and 2, and it is right by \displaystyle 1\frac{3}{4}. Its half inch neighbors are \displaystyle 1\frac{1}{2} and 2, and \displaystyle 1\frac{3}{4} is the halfway point between those, so stopping just past it puts the head nearer 2.
  23. Yes. One earthworm ends exactly at 3, and another ends just short of the \displaystyle 3\frac{1}{4} mark. To the nearest quarter inch they are 3 and \displaystyle 3\frac{1}{4} inches. To the nearest half inch both are 3 inches: \displaystyle 3\frac{1}{4} is halfway between 3 and \displaystyle 3\frac{1}{2}, and the second head stops before that halfway point, so it is nearer 3.
  24. A length is how much ruler the crayon covers. Starting at 1, the inch from 0 to 1 is not covered, so the number under the tip counts one inch too many. Counting the spaces the crayon covers measures only what it covers, wherever it lies.
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