Measuring tools and reading their scales
A meter stick prints only 10, 20, 30 and so on, with a crowd of bare marks in between. How do you read a length that ends on one of the bare marks, or between two of them, and which tool should you pick up in the first place?
Where you are 14 in What measurement is
When you reach for it
A child who has measured with a ruler knows that each number on it counts the units that end at that mark. That lesson used a ruler with a number on every mark. Real tools are not so kind. A meter stick has a hundred marks and prints only ten numbers. A measuring tape may print a number only every five marks. Some rulers print only the even numbers. Before any of that, there is a choice to make: a short ruler, a long stick, or a tape that bends.
This lesson covers both. It is taught over three days, one part a day:
- Part 1, the tools: the inch ruler, the yardstick, the meter stick and the measuring tape, and how to pick the one that suits the job (Examples 1 and 2).
- Part 2, marks without labels: working out what one space stands for, then counting on or back from the nearest label (Examples 3 to 5).
- Part 3, ends between marks: naming the two marks, saying which one is nearer, and giving the length as “about” a whole number of units (Examples 6 and 7).
Reach for it whenever a measuring tool is in a child’s hand, which is every time a length is measured from now on. The units here are inches and centimeters, and every length is a whole number of them or “about” one. How big each unit is, and the other units in each family, belong to Customary length units and Metric length units.
Have the real tools on the table: a ruler with inches on one edge and centimeters on the other, a yardstick, a meter stick, a measuring tape, and ribbon, string and paper strips to measure.
The procedure
Part 1: The tools, and choosing one
Every measuring tool for length is the same idea: a row of equal units already laid end to end with no gaps and no overlaps, with a mark at the end of each unit. The tools differ in how long the row is and whether it bends.
- An inch ruler is 12 inches long. Most school rulers carry inches on one edge and centimeters on the other, about 30 centimeters.
- A yardstick is a stiff stick 36 inches long.
- A meter stick is a stiff stick 100 centimeters long, a little longer than a yardstick.
- A measuring tape is a long strip of cloth or thin metal that bends and rolls up. A sewing tape is often 60 inches long; a builder’s tape is much longer.
Three questions pick the tool.
Is the thing longer than the tool? A tool that is longer than the thing reads its length in one go: line up the start, read the end. A tool that is too short has to be laid down, the place marked, and laid down again, which is Iterating a single unit with a big unit. Each move is a chance to leave a gap or an overlap, and the pieces then have to be added. So a long thing calls for a long tool.
Is the path straight? A stiff ruler or stick cannot follow a curve. The way around a pumpkin or a person’s waist needs the tape. In Measuring length a curvy path was measured with a string, pulled straight and laid on a ruler; a measuring tape is a string with the marks already on it.
Which unit is asked for? A ruler with two edges gives two different numbers for the same thing, and both are right. Read the edge whose unit was asked for, read the same edge at both ends, and write the unit with the number. “4” alone says nothing; “4 inches” says a length.
A longer tool is not always better. A yardstick is clumsy on a button, and a floppy tape is hard to hold flat on a short pencil. The best tool is the shortest one that still reaches past the thing in one go, and bends if the path bends.
Part 2: Marks without labels
Look closely at a meter stick. It has a mark for every centimeter, but it prints a number only at 10, 20, 30 and so on. Often the mark at each five, halfway between two labels, is drawn a little longer, to help the eye. Measuring tapes do the same with their own pattern, and some rulers print only every other number.
The thinking needed was laid out in number lines with different scales: every space on the line stands for the same amount, the step, and the step can be found from two labels and the number of spaces between them. A measuring tool is a number line made of real units, and that adds one fact. On a ruler, a yardstick, a meter stick or a measuring tape, the smallest space is one whole unit, because the tool is a row of units with a mark at the end of each. The labels skip, but the marks do not.
It is still worth checking on any tool not seen before, and it takes a moment. Pick two neighboring labels and count the spaces between them. If the labels go up by 10 and there are 10 spaces, each space is 1 unit. If they go up by 2 and there are 2 spaces, each space is 1 unit. A homemade strip may be marked differently, and the same count catches it (Exercise 13).
Then read the end of the thing. Find the label nearest to it and count marks from there: count on from a label on the left, or count back from a label on the right. Both counts land on the same number, because each mark is one unit more than the mark to its left. The nearest label keeps the count short, and a short count is easy to get right. The longer mark at a five is a free label: on a meter stick it says 65 even though nothing is printed there.
Part 3: Ends between marks
The end of a thing does not have to fall on a mark. Then the reading has three parts: the mark just before the end, the mark just after it, and which of the two is nearer. “The ribbon ends between 52 and 53 centimeters, nearer 53” is an exact description of what the tool shows. Since only whole units are used here, the length is given as the nearer whole number: about 53 centimeters. This is the “about” of Measuring length, now read on a tool whose marks are not all labeled.
When the thing does not start at 0, its length is the number of spaces from the start mark to the end mark, not the number at the end. On a meter stick with labels every 10 it is quick to count those spaces by tens to a label, then by ones.
How close a reading can get depends on the marks. A stick with a mark for every centimeter can say “about 38”. A stick with marks only every 10 centimeters can say no more than “between 30 and 40, nearer 40”. That is the start of Measurement is approximate: precision of a tool.
Worked examples
Choosing tools at the costume shop
Ruslana runs a costume shop. She needs three lengths: the width of a button, the way around a customer’s head for a crown, and the long side of her cutting table, which turns out to be 72 inches. Which tool suits each? For the table, how many times would she lay down a 12-inch ruler, and how many times a 36-inch yardstick?
SolutionThe button is short and flat, so the inch ruler reaches past it in one go. The way around a head is a curve, and only the measuring tape can follow it.
The table is straight and long. With the ruler, each placement covers 12 inches:
That is 6 placements, with 5 chalk marks to keep in line. With the yardstick each placement covers 36 inches, and 36 + 36 = 72, so it takes 2 placements and 1 chalk mark. Both tools give 72 inches if every placement is perfect, but the yardstick gives fewer chances to slip. A long tape would read it in one go.
A ribbon longer than the ruler
Ruslana has only a 12-inch ruler. She lays it along a ribbon from the ribbon’s end, puts a pin at the 12 mark, moves the ruler so its 0 is at the pin, pins again at 12, and moves it once more. This time the ribbon ends at the 5 mark. How long is the ribbon, and what would a yardstick have shown?
SolutionThe first two placements each covered a whole ruler, 12 inches. The third covered 5 inches. So the ribbon is
inches long. A yardstick is 36 inches, longer than the ribbon, so laid from the ribbon’s end it would show the other end at its 29 mark in one reading. Same length, one reading instead of three placements and an addition.
A ribbon on a meter stick
Ruslana lays a ribbon along a meter stick with one end at 0. The stick is labeled only every 10 centimeters. The other end of the ribbon is shown below. How long is the ribbon?
First, what one space stands for. From the 60 label to the 70 label there are 10 spaces, and the labels go up by 10, so each space is 1 centimeter.
The end is past 60 and before 70. The nearest label is 60, four marks back. Count on from 60, one centimeter per mark: 61, 62, 63, 64. The ribbon is 64 centimeters long.
Check from the other side. Counting back from 70: 69, 68, 67, 66, 65, 64, six marks. Both counts agree, and the one from 60 was shorter. The longer mark at 65 is a check too: the end is one mark before it.
A ruler that skips numbers
Mawuli keeps stuffed animals on a shelf. He measures a paper cutout of his stuffed rabbit on a ruler that prints only 2, 4, 6, 8, 10 and 12. How long is the cutout?
From the 2 label to the 4 label there are 2 spaces, and the labels go up by 2, so each space is 1 inch. The ruler has a mark for every inch; it just prints every other number.
The cutout starts at 0 and ends on the bare mark between 6 and 8. One mark past 6 is 7, so the cutout is 7 inches long. Every bare mark on this ruler is an odd number of inches: 1, 3, 5 and so on.
A hay bale and a tape
At the farm, Kauri hooks a measuring tape on the far end of a hay bale, at 0, and pulls it along the side. The tape prints a number every 5 inches. Where the bale ends is shown below. How long is the bale?
From 40 to 45 there are 5 spaces, and the labels go up by 5, so each space is 1 inch.
The bale ends between the 40 and 45 labels, two marks before 45. The nearest label is 45, so count back: 44, 43. The bale is 43 inches long.
Counting on from 40 gives 41, 42, 43, three marks, and the same answer. Two counts with the same result are a good habit when the labels are far apart.
Fabric that starts at 20
Ruslana lays a strip of fabric on a meter stick, but the end of the stick is chipped, so she starts the fabric at the 20 mark. The other end is shown below. About how long is the strip?
Each space is 1 centimeter: there are 10 spaces from 40 to 50.
The nearest label is 50, but the long mark at 45 is nearer still. The fabric ends past the second mark after 45, which is 47, and before the third, which is 48, and it is much nearer 48. So the end reads “about 48”.
The fabric starts at 20, not 0, so 48 is not its length. Count the spaces from 20 to 48: by tens from 20 to 40 is 2 tens, that is 20, and from 40 on to 48 is 8 more.
The strip is about 28 centimeters long. Starting at 20 cost nothing, because a length is the number of units the thing covers, wherever they start.
One rope, two sticks
Kauri has a homemade stick with marks only at 0, 10, 20, 30, 40 and 50 centimeters. He lays a rope on it from 0, then lays the same rope on a stick with a mark for every centimeter. What can each stick tell him?
On the homemade stick the rope ends after the 30 mark and before the 40 mark, and by eye it is nearer 40. The stick has no marks inside that space, so all Kauri can say is “between 30 and 40 centimeters, nearer 40”. About 40 is the best whole label this stick offers.
On the second stick, from 30 to 40 has 10 spaces of 1 centimeter each. The rope ends on the eighth mark after 30: count on 31, 32, \ldots, 38. The rope is 38 centimeters long.
Both readings are true, and they agree: 38 is between 30 and 40, and nearer 40. The second stick says more because it has more marks. A tool can never tell a length more closely than its marks allow.
Boundaries
- Whole units only. An end between two marks is reported as “between” the two and “about” the nearer one. Halves of an inch, and the shorter marks some rulers carry for them, come later.
- Exactly halfway has no nearer mark. If an end sits right in the middle of a space, neither mark is nearer. Say “between 4 and 5 inches”; choosing one would be a guess.
- The smallest space is one unit on standard tools, not on every tool. A homemade strip, or the scale on some thermometers and kitchen tools, may step by 2 or more. Count the spaces between two labels before trusting the bare marks.
- The two edges of a ruler are different units. A reading must use one edge at both ends, and the number is meaningless without its unit.
- A stiff tool measures only a straight path. Around a curve it reads the straight distance instead. The tape, or a string, follows the path.
- A tool too short for the thing adds steps. Each extra placement must start exactly where the last one ended, and the pieces are then added.
Where this goes next
The same reading, a step found from labels and a count from the nearest label, works on every marked scale. A thermometer is a vertical scale (Reading a thermometer), a measuring cup or a container marked in milliliters is another (Liquid volume: liters and milliliters), and a protractor is the same scale bent into a half circle (Measuring and drawing angles with a protractor). Rulers marked in halves and fourths of an inch are read the same way once those parts are numbers.
The coarse stick of Example 7 is the start of Measurement is approximate: precision of a tool: finer marks give a closer reading, and no reading is closer than the tool’s smallest space. The units themselves, how long an inch or a centimeter is and how they compare with feet, yards and meters, are in Customary length units and Metric length units.
Exercises
Part 1: Choosing a tool
- Kauri wants to know how far it is around a round hay bale. Which tool should he use, and why?
- Mawuli wants to measure how tall a small stuffed mouse is. Which tool suits best: an inch ruler, a yardstick or a measuring tape? Why?
- A shelf is 24 inches long. How many times must Mawuli lay down a 12-inch ruler to measure it? How many times a 36-inch yardstick?
- A rug is 60 inches long. How many placements does it take with a 12-inch ruler? With a 36-inch yardstick, how many placements, and what does the yardstick read at the end of the last one?
- Ruslana lays a ribbon on a ruler with inches on one edge and centimeters on the other, with the ribbon’s end at 0 on both. The other end is at 4 on one edge and a little past 10 on the other. What should she write down?
Part 2: Marks without labels
Use the scales for Exercises 6 to 9. In A, D and E the thing starts at 0, off to the left.
- Scale A is part of a meter stick. How long is the ribbon?
- Scale B is an inch ruler that prints only the even numbers. How long is the cutout?
- Scale C is part of a measuring tape in inches, starting at 0 off to the left. How long is the fabric?
- Scale D is part of a meter stick. How long is the ribbon?
- A meter stick is labeled every 10 centimeters. A ribbon starts at 0 and ends on the sixth small mark past 50. How long is the ribbon?
- A tape is labeled 35, 40, 45 with a mark for every inch. A strip of fabric starts at 0 and ends on the mark just before 45. Which label would you count from, and how long is the fabric?
- A tape is labeled 10, 20, 30, with 9 small marks between each pair of labels. What does one space stand for?
- Kauri makes his own measuring strip. It is labeled 0 and 10 centimeters, with 4 evenly spaced bare marks between them. What does one space stand for? A twig starts at 0 and ends on the third bare mark. How long is the twig?
- A stuffed giraffe on Mawuli’s shelf is 42 centimeters tall. Measured on a meter stick labeled every 10, which label is nearest the top of its head, and how many marks away?
Part 3: Ends between marks
- Scale E is part of a meter stick. Name the two marks the cutout ends between, and give its length as “about” a whole number.
- A strip of fabric lies on a meter stick from the 30 mark to the long mark halfway between 40 and 50. How long is it?
- A ribbon lies on a meter stick from the 10 mark to a point between 36 and 37, nearer 36. About how long is it?
- Kauri wraps a tape around a round hay bale. The tape is labeled every 5 inches with a mark for every inch. The end falls between the third and fourth marks after 60, nearer the third. About how far around is the bale?
- On Kauri’s homemade stick, marked only every 10 centimeters, a rope that starts at 0 ends between 20 and 30, nearer 20. What can he say about its length?
- The tail of a stuffed mouse ends exactly halfway between the 4 and 5 inch marks, and the mouse starts at 0. What should be written?
Explain
- Why count from the nearest label rather than from 0?
- Before counting bare marks on a tool you have never used, why check what one space stands for?
- Ruslana measures a ribbon on a stick marked only every 10 centimeters and says “between 40 and 50, nearer 50”. Mawuli measures it on a meter stick and says 46 centimeters. Can both be right?
- Is the longest tool always the best one to use? Give an example.
Answers
- The measuring tape. The way around the bale is a curve, and a tape bends to follow it; a stiff ruler or stick cannot.
- The inch ruler. The mouse is short and the ruler reaches past it in one go. A yardstick would work but is clumsy to hold against a small thing, and a tape is floppy on a short straight length.
- 2 times with the ruler, since 12 + 12 = 24. Once with the yardstick, since 24 is less than 36: the yardstick reads 24 in one go.
- 5 placements with the ruler: 12 + 12 + 12 + 12 + 12 = 60. With the yardstick, 2 placements: the first covers 36 inches, and the second reads 24 at the end, since 36 + 24 = 60.
- “4 inches”, or “about 10 centimeters”. Both describe the same ribbon on different edges. A bare “4” or “10” would not say which unit was counted.
- 33 centimeters. Each space is 1 centimeter (10 spaces from 20 to 30). The end is three marks past 30: 31, 32, 33.
- 9 inches. Labels go up by 2 with 2 spaces between, so each space is 1 inch. The end is one mark past 8.
- 24 inches. From 15 to 20 there are 5 spaces, so each is 1 inch. The end is one mark before 25: count back to 24.
- 88 centimeters. The end is two marks before 90: 89, 88.
- 56 centimeters. Count on from 50: 51, 52, 53, 54, 55, 56.
- Count back from 45, the nearer label: one mark back is 44. The fabric is 44 inches long. (From 40 it would be four marks on, to the same 44.)
- 1. Nine bare marks between two labels make 10 spaces, and the labels go up by 10, so each space is 1 unit.
- Each space is 2 centimeters: 4 bare marks make 5 spaces, and 2 + 2 + 2 + 2 + 2 = 10. The third bare mark is 3 spaces from 0, so the twig is 2 + 2 + 2 = 6 centimeters long.
- The 40 label, two marks away: count on 41, 42. The 50 label is eight marks away, so 40 is nearer.
- Between 52 and 53, nearer 53, so about 53 centimeters.
- The long mark halfway between 40 and 50 is 45. From 30 to 45 is 1 ten and 5 ones, so the strip is 15 centimeters long.
- The end reads about 36. From 10 to 36 is 2 tens and 6 ones, so about 26 centimeters.
- Three marks after 60 is 63, four is 64, and the end is nearer 63. About 63 inches.
- Only that it is between 20 and 30 centimeters, nearer 20, so about 20 centimeters. The stick has no marks inside that space, so it cannot say 22 or 23.
- “Between 4 and 5 inches.” The end is the same distance from both marks, so neither is nearer, and choosing one would be a guess.
- The count from the nearest label is short, and a short count is easy to get right. Every mark is one unit more than the one before, so counting from any label gives the same answer; the nearest one just gets there in the fewest steps.
- The labels alone do not say how much a space is worth. Standard tools have one unit per space, but a homemade strip may not (Exercise 13). Counting the spaces between two labels settles it before any marks are counted.
- Yes. 46 is between 40 and 50, and it is 4 from 50 and 6 from 40, so it is nearer 50. The meter stick has more marks, so it says more; the two readings agree.
- No. A yardstick reads a 30-inch scarf in one go where a ruler takes three placements, but it is clumsy on a button, and no stiff stick can measure around a pumpkin. The best tool reaches past the thing in one go, follows its path, and is no bigger than it needs to be.