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Grade 3 · 2.MD.D.9 · 3.MD.B.4

Measurement is approximate: precision of a tool

Two honest people measure the same pencil and get two different answers. Which one is right, and what does a measurement really tell you?

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Measurement is approximate: precision of a tool
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Why this exists

Anahita measures a pencil from her pencil case and says it is about 6 inches long. Babak measures the same pencil and says it is \displaystyle 5\frac{3}{4} inches. Neither of them made a slip. They used different rulers.

That happens with every kind of measuring: rulers, scales, thermometers, clocks. A tool can only answer with the marks it has, and the thing being measured almost never stops exactly on a mark. So every measurement is a choice of the nearest mark, and the answer you get depends on how close together the marks are.

This lesson is about what that means. It explains why a measurement is always “about”, why two objects can read the same and still be different, and how to pick a tool with marks fine enough for the job in front of you. It is taught over two days: the first is about the nearest mark, the second about what a tool can and cannot tell apart.

Definition

Measuring a length means counting how many copies of a unit fit along it, end to end, which is iterating one unit. A ruler is those copies already laid out, with a numbered mark where each one ends.

The end of an object can stop anywhere, but the tool has marks only at certain places. So:

  • The reading of a measurement is the mark on the tool nearest to where the object ends (or where the pointer or the liquid stops).
  • How finely a tool is marked is the space between two neighboring marks. A ruler marked in quarter inches is finer than one marked in whole inches. A scale marked every gram is finer than one marked every 10 grams.

Taking the nearest mark is the same move as Rounding: replacing a number with a nearby one that is easier to say. A scale marked every 10 grams rounds to the nearest ten grams, exactly as in Rounding to the nearest ten or hundred.

Intuition

Picture each mark on a ruler owning a stretch of the ruler around it: everything from halfway back to the mark before it to halfway on to the mark after it. Any object whose end lands in that stretch gets that mark’s number.

On a ruler marked only in whole inches, each stretch is a whole inch wide. A pencil that ends a little before 6 and a pencil that ends a little after 6 both land in the stretch owned by 6, so both read “about 6 inches”. The ruler cannot tell them apart.

Cut every inch into 4 equal parts and there are four times as many marks, each owning a stretch only a quarter inch wide. Ends that were lumped together on the coarse ruler now land in different stretches and get different numbers.

But the stretches never shrink to nothing. However many marks a ruler has, there is still room between two of them for an end to stop.

Results

Every measurement is approximate. A reading names the nearest mark, not the exact amount. That is why people say “about 6 inches”.

The space between marks is the smallest difference a tool can show. Two different readings on a ruler marked in whole inches are at least 1 inch apart. On a ruler marked in quarter inches they can be just \displaystyle \frac{1}{4} inch apart.

Two objects that read the same can still be different. If both ends are nearest the same mark, both get the same number, and only a finer tool can tell them apart.

Finer marks give a reading closer to the true amount, but never an exact one.

A tool is fine enough for a job when its marks are at least as close together as the difference the job needs to see.

Why it is true

A ruler has marks only at certain places, but the end of an object can stop anywhere along it. Between any two marks there is always room for an end to land, however close the marks are drawn.

The ruler can only answer with one of its marks, so it answers with the nearest one. Every object whose end lands in the same stretch gets the same answer, even though the ends are in different places. That is the whole reason a reading is approximate, and the reason two objects can read the same.

The stretches are as wide as the space between marks. Two ends closer together than that can share a stretch, so the tool cannot always tell them apart. Two ends at least one space apart can never share a stretch (unless an end sits exactly halfway between two marks, the case in Boundaries), so they read differently. That is why the space between marks is the smallest difference a tool can show, and why a job needs marks at least as close as the difference it cares about.

Making the marks closer makes every stretch narrower, so the reading lands nearer the true end. It never makes the stretch zero wide, because there is still room between two marks. So no tool gives an exact length.

Worked examples

Example 1

Part 1: a pencil on a whole-inch ruler

Anahita takes a pencil from her pencil case and lays it on a ruler marked only in whole inches, as in Figure 1. Its flat end is at 0. How long is the pencil?

pencil 0 1 2 3 4 5 6 7 inches
Figure 1The pencil’s end falls between two marks. The nearest mark is 6.
Solution

The ruler has marks only at 0, 1, 2 and so on. The pencil’s point stops between the 5 and 6 marks, so there is no mark right at its end.

The point is much closer to 6 than to 5: the gap back to 5 is most of an inch, and the gap on to 6 is a small piece of one. The nearest mark is 6.

The pencil is about 6 inches long. The word “about” is not vagueness. It says exactly what the ruler can say: of all its marks, 6 is the nearest.

Example 2

Part 1: the same pencil on a finer ruler

Anahita lays the same pencil on a ruler whose inches are each cut into 4 equal parts, as in Figure 2. What does this ruler read?

pencil 0 1 2 3 4 5 6 7 inches
Figure 2The same pencil on a ruler with more marks. Now there is a mark right at its end.
Solution

Each inch now has 4 equal parts, so each small space is a quarter inch. The point of the pencil lies past the 5 inch mark, at the third small mark after it, the last one before 6. That is 5 whole inches and 3 quarter inches more:

5\frac{3}{4} \text{ inches}

Both readings are honest. The first ruler could only say “about 6”; this one can say more, because it has a mark much nearer the end of the pencil. It also shows how far the first reading was from the end: one quarter inch short of 6.

Even \displaystyle 5\frac{3}{4} is a nearest mark. A ruler with still more marks might find the point a hair past it or a hair short of it.

Example 3

Part 1: a bag of screws on two scales

At the tool bench, Babak weighs a bag of screws on two scales, shown as strips in Figure 3. The top scale has a mark every 10 grams; the bottom one has a mark for every gram. What does each read?

marked every 10 grams 200 210 220 230 240 250 260 marked every gram 200 210 220 230 240 250 260
Figure 3One bag of screws, two scales. The top scale can only name tens of grams.
Solution

On the top scale the pointer sits between the 230 and 240 marks. It is closer to 240, so the top scale reads 240 grams.

On the bottom scale every gram has a mark. Counting the small marks after 230, the pointer is on the sixth one, so it reads 236 grams.

The top scale did exactly what rounding does: 236 rounded to the nearest ten is 240. A scale marked every 10 grams can only ever answer with a ten, so its reading is always a whole number of tens of grams.

Example 4

Part 2: two strips that read the same

Zayan cuts two wood strips at the tool bench. On a ruler marked only in whole inches (Figure 4), how long is each? Then measure them on a quarter-inch ruler (Figure 5). How far apart are their ends?

strip A strip B 0 1 2 3 4 5 inches
Figure 4Two different strips, one reading. Both ends are closest to the 4 inch mark.
Solution

On the whole-inch ruler, strip A ends a little past the 4 mark and strip B ends a little before it. Both ends are nearest to 4, so both strips read about 4 inches.

strip A strip B 0 1 2 3 4 5 inches
Figure 5The finer ruler has marks between the two ends, so it can tell the strips apart.

On the quarter-inch ruler, strip A ends at the first small mark after 4, which is \displaystyle 4\frac{1}{4} inches. Strip B ends at the last small mark before 4, which is \displaystyle 3\frac{3}{4} inches.

Count the quarter-inch spaces between the two ends: from \displaystyle 3\frac{3}{4} to 4 is one, from 4 to \displaystyle 4\frac{1}{4} is another. That is 2 quarter inches, or half an inch. The strips really are different, by half an inch, and the whole-inch ruler could not show it, because it has no mark between the two ends. Both ends sat in the stretch owned by 4.

Example 5

Part 2: a thermometer marked every 2 degrees

The thermometer hanging by the tool bench has a mark every 2 degrees. What readings can it give from 60 to 70 degrees? What is the smallest difference it can show between two readings?

Solution

Its marks in that range are at 60, 62, 64, 66, 68 and 70. Those are the only answers it can give there: 6 readings, all even numbers.

Two different readings are at least one space apart, so the smallest difference it can show is 2 degrees.

A small change can go unseen. Say the liquid stops just above the 66 mark in the morning, and by lunch it has risen a little, but its top is still nearer 66 than 68. The thermometer reads 66 both times, though it is warmer at lunch.

Example 6

Part 2: choosing a ruler

Zayan cuts small pegs and must sort them into two piles: pegs \displaystyle 2\frac{1}{2} inches long and pegs \displaystyle 2\frac{3}{4} inches long. He has three rulers: one marked in whole inches, one in half inches, and one in quarter inches. Which of them can do the job?

Solution

The two lengths differ by one quarter inch. Check each ruler for marks at both lengths.

The whole-inch ruler has marks only at 2 and 3. A \displaystyle 2\frac{3}{4}-inch peg reads 3, and a \displaystyle 2\frac{1}{2}-inch peg ends exactly halfway between 2 and 3, so the ruler gives no clear reading for it. Not fine enough.

The half-inch ruler has marks at 2, \displaystyle 2\frac{1}{2} and 3. The shorter peg ends on a mark, but the longer one ends exactly halfway between \displaystyle 2\frac{1}{2} and 3, so its reading depends on the class’s rule for a tie, and a long peg cut a hair short reads \displaystyle 2\frac{1}{2}, the same as a short peg. Not fine enough either.

The quarter-inch ruler has a mark at \displaystyle 2\frac{1}{2} and a mark at \displaystyle 2\frac{3}{4}, so each peg ends on its own mark. It can do the job.

The space between its marks is one quarter inch, the same as the difference Zayan needs to see. Coarser rulers have spaces wider than that difference, and both pegs can fall near the same mark or between marks.

Example 7

Part 2: when a coarse tool is enough

Babak has the two scales from Figure 3. He has two jobs.

(a) Find which of two boxes of nails is heavier. Their true masses are 648 grams and 652 grams.

(b) Find which of two bags of washers is heavier. Their true masses are 320 grams and 480 grams.

Which scale does each job need?

Solution

(a) On the scale marked every 10 grams, 648 is nearest 650 and 652 is nearest 650. Both boxes read 650 grams, so that scale cannot say which is heavier. On the gram scale they read 648 and 652, so the second box is heavier, by 4 grams. The job needs the gram scale: the difference is 4 grams, smaller than the 10-gram space.

(b) On the scale marked every 10 grams, the bags read 320 and 480 grams. That already answers the question: the second bag is heavier. The gram scale would give the same two numbers and no more help.

A finer tool is never wrong, but it is not always needed. What matters is the size of the difference the job has to see.

Boundaries

An end exactly halfway between two marks has no single nearest mark. It is equally close to both. A class has to agree on a rule for that case and use it every time; taking the larger mark, as in Rounding, is one such rule. A finer tool often helps too, since it may have a mark right there.

Counting is exact; measuring is not. Seven pencils in a pencil case is exactly 7. A length, a mass, a temperature or a time can come in any size at all, so a tool with marks can only get near it.

A difference in readings does not give the size of the real difference. Strips of \displaystyle 3\frac{1}{4} and \displaystyle 3\frac{3}{4} inches read 3 and 4 on a whole-inch ruler: the readings are 1 inch apart, but the strips are half an inch apart. A coarse tool can make a difference look bigger, as well as hide one.

Finer is never exact. A ruler marked in quarter inches, or in sixteenths, still names a nearest mark.

Where this goes next

The finer rulers in this lesson are the subject of measuring to the half and quarter inch, which reads them carefully and gathers many measurements into a line plot. Tools whose marks are not one unit apart, such as a scale marked every 5 grams, are read in measuring tools and reading their scales, and grams themselves are in Metric mass units. Reading the liquid in a thermometer is Reading a thermometer.

Later, the idea gets numbers attached. Measurement error asks how far a reading can be from the true amount and how to say so, and Significant figures asks which digits of a recorded measurement the tool really supported.

Exercises

Part 1: the nearest mark

For Exercises 1 to 4, an object is measured on a quarter-inch ruler. What does a ruler marked only in whole inches read for it?

  1. An eraser, \displaystyle 2\frac{1}{4} inches long.
  2. A marker, \displaystyle 4\frac{3}{4} inches long.
  3. A glue stick, \displaystyle 3\frac{1}{4} inches long.
  4. A paintbrush, \displaystyle 7\frac{3}{4} inches long.

For Exercises 5 and 6, an object is weighed on the gram scale. What does the scale marked every 10 grams read for it?

  1. A box of washers, 187 grams.
  2. A bag of tacks, 302 grams.
  3. A thermometer by the tool bench has a mark every 5 degrees. A thermometer with a mark for every degree reads 68 degrees. What does the first thermometer read?
  4. On a whole-inch ruler a nail reads about 3 inches. Which of these quarter-inch readings could it have: \displaystyle 2\frac{1}{4}, \displaystyle 2\frac{3}{4}, \displaystyle 3\frac{1}{4}, \displaystyle 3\frac{3}{4}?

Part 2: what a tool can and cannot show

  1. Two nails measure \displaystyle 1\frac{3}{4} and \displaystyle 2\frac{1}{4} inches on a quarter-inch ruler. What does a whole-inch ruler read for each? How many quarter-inch spaces apart are their ends?
  2. Two boxes of screws have masses of 262 grams and 264 grams. What does the scale marked every 10 grams read for each?
  3. What is the smallest difference a ruler marked in half inches can show between two readings?
  4. A scale has a mark every 5 grams. List every reading it can give from 100 to 120 grams.
  5. A thermometer has a mark every 2 degrees. How many different readings can it give from 50 to 60 degrees, counting both?

For Exercises 14 to 16, which is the coarsest of the three rulers (whole inches, half inches, quarter inches) that has a mark at both lengths?

  1. 6 inches and 9 inches.
  2. \displaystyle 3\frac{1}{2} inches and 4 inches.
  3. \displaystyle 2\frac{1}{4} inches and \displaystyle 2\frac{1}{2} inches.
  4. Babak must tell a box of nails of 721 grams from one of 724 grams. Is the scale marked every 10 grams fine enough? Explain with its readings.
  5. Babak must tell a bag of 300 grams from a bag of 500 grams. Is the scale marked every 10 grams fine enough?

Reasoning

  1. A length reads \displaystyle 4\frac{1}{4} inches on a quarter-inch ruler. Is the length exactly \displaystyle 4\frac{1}{4} inches? Explain.
  2. Two strips both read about 5 inches on a whole-inch ruler. Can they have different lengths? Can they differ by a whole inch? Explain.
  3. Two strips read 5 inches and 6 inches on a whole-inch ruler. Must their lengths differ by exactly 1 inch? Give an example.
  4. A thermometer marked every 2 degrees reads 70 degrees on Monday morning and on Tuesday morning. Was it exactly as warm both mornings? Explain.
  5. Describe one job at a tool bench where a ruler marked in whole inches is fine enough, and one where it is not.
  6. Explain why no ruler, however many marks it has, can give the exact length of every object.
Answers
  1. About 2 inches. \displaystyle 2\frac{1}{4} is one quarter inch past 2 and three quarters short of 3, so 2 is the nearest mark.
  2. About 5 inches. \displaystyle 4\frac{3}{4} is one quarter inch short of 5.
  3. About 3 inches. \displaystyle 3\frac{1}{4} is one quarter inch past 3.
  4. About 8 inches. \displaystyle 7\frac{3}{4} is one quarter inch short of 8.
  5. 190 grams. 187 is 3 grams short of 190 and 7 grams past 180; it rounds to 190.
  6. 300 grams. 302 is 2 grams past 300 and 8 short of 310.
  7. 70 degrees. Its marks near there are 65 and 70; 68 is 2 from 70 and 3 from 65.
  8. \displaystyle 2\frac{3}{4} and \displaystyle 3\frac{1}{4}. Each is one quarter inch from 3. \displaystyle 2\frac{1}{4} is nearest 2, and \displaystyle 3\frac{3}{4} is nearest 4.
  9. Both read about 2 inches. Their ends are 2 quarter-inch spaces apart: from \displaystyle 1\frac{3}{4} to 2, then from 2 to \displaystyle 2\frac{1}{4}. That is half an inch.
  10. Both read 260 grams. Each is nearer 260 than 270; only the gram scale shows they differ, by 2 grams.
  11. \displaystyle \frac{1}{2} inch, the space between two neighboring marks.
  12. 100, 105, 110, 115 and 120 grams: the marks in that range.
  13. 6 readings: 50, 52, 54, 56, 58 and 60.
  14. The whole-inch ruler. Both lengths are whole inches, so it has a mark at each.
  15. The half-inch ruler. The whole-inch ruler has no mark at \displaystyle 3\frac{1}{2}; the half-inch ruler has marks at both.
  16. The quarter-inch ruler. Neither coarser ruler has a mark at \displaystyle 2\frac{1}{4}.
  17. No. It reads 720 grams for both boxes, since 721 and 724 are both nearer 720 than 730. The difference is 3 grams, smaller than its 10-gram space; he needs the gram scale.
  18. Yes. It reads 300 and 500 grams, which tells the two bags apart.
  19. Not necessarily. \displaystyle 4\frac{1}{4} is the nearest quarter-inch mark to the end. The end could be a little before or a little after that mark, and a ruler with more marks might show it.
  20. Yes, they can differ, but by less than a whole inch. Each end is nearer 5 than 4 or 6, so each lies between the halfway points \displaystyle 4\frac{1}{2} and \displaystyle 5\frac{1}{2}, which are 1 inch apart. Strips of \displaystyle 4\frac{3}{4} and \displaystyle 5\frac{1}{4} inches both read 5 and differ by half an inch.
  21. No. Strips of \displaystyle 5\frac{1}{4} and \displaystyle 5\frac{3}{4} inches read 5 and 6 but differ by only half an inch. Strips of \displaystyle 4\frac{3}{4} and \displaystyle 6\frac{1}{4} read 5 and 6 but differ by \displaystyle 1\frac{1}{2} inches.
  22. Not necessarily. Any temperature whose nearest mark is 70 reads 70, so one morning could have been a little warmer than the other.
  23. Answers vary. Fine enough: telling a 3-inch board offcut from a 9-inch one, since the readings 3 and 9 are far apart. Not fine enough: telling apart screws of \displaystyle \frac{3}{4} inch and \displaystyle 1\frac{1}{4} inches that must go in different holes. Both read about 1 inch, since the half-inch difference is smaller than the ruler’s 1-inch space; a quarter-inch ruler is needed.
  24. A ruler has marks only at certain places, but an end can stop anywhere. However close the marks are, there is room between two of them, so some ends fall between marks and the ruler can only give the nearest one.
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