Benchmarks, estimating and choosing a unit
Before you measure something, how can you already know about how long it is, and how do you know which unit and which tool to measure it with?
Where you are 14 in What measurement is
Why this exists
A child who can measure a length can lay a ruler along a pencil and read a number. That child can still have no feel for what the number means. Ask how long the classroom is and the answer might be 10 inches or 1{,}000 feet, because “inch” and “foot” are only words until each one is tied to something the child has seen and held.
A benchmark is that tie. It is a familiar thing that is about one unit long: the edge of a floor tile for a foot, a crayon’s thickness for a centimeter. With a benchmark in mind a child can do three things that a ruler alone does not give.
- Estimate. Picture the benchmark laid along the object again and again, and count. That gives a length before any tool comes out.
- Check a measurement. If the estimate says “about 6 feet” and the ruler work says 60, a slip has happened somewhere, and the child knows to look again.
- Choose a unit and a tool. A unit fits a job when a sensible number of copies of it fit along the object. A benchmark shows that at a glance: nobody measures the playground fence in crayon widths.
This lesson takes three days. Day 1 builds a benchmark for each unit a child meets now (inch, foot, yard, centimeter, meter). Day 2 estimates a length, then measures it, then compares the two. Day 3 chooses the unit and the tool that fit a job. All lengths here are whole units.
Definition
Day 1: a benchmark
A benchmark for a unit is a familiar thing whose length is about one of that unit. It can be part of the body, which is always on hand, or an object in the room that every child can point to. A benchmark is never exact. It is close enough to picture the unit.
Day 2: an estimate
An estimate of a length is a length found by thinking, not by measuring: picture a benchmark laid end to end along the object and count the copies. An estimate always carries a unit. It is said with “about” (“about 6 feet”), or as a range, two lengths the true one should fall between (“between 6 and 7 feet”).
After measuring, how far off the estimate was is the gap between the estimate and the measurement, found by subtracting the smaller from the larger. It is information about the benchmark, not a mark against the child.
Day 3: a sensible unit
A unit is sensible for a length when the count it gives is neither tiny nor huge: at least one copy fits, and not so many that counting them is a chore. Often two units are both sensible, and then either one will do. The tool follows the unit: a ruler for inches or centimeters, a yardstick or meter stick for yards and meters, a measuring tape for anything long or bent.
Intuition
Day 1: a picture for each unit
Each unit gets one picture to carry. Put the real tool next to the benchmark once, so the child sees for themselves that they match.
- Inch. The top part of a grown-up’s thumb, from the tip to the first bend. A child’s thumb is shorter, so let each child find their own inch: hold a ruler and see which part of a finger covers one inch.
- Foot. The edge of a floor tile. Many classroom floors are laid in tiles that are one foot on each side. Check yours with a foot ruler before using it; if they match, the floor is a giant ruler that is always there.
- Yard. A grown-up’s reach, from the nose to the fingertips of an arm held straight out to the side. It is also three floor tiles in a row, because 3 feet make 1 yard.
- Centimeter. The thickness of a crayon. Lay ten crayons side by side, touching, and the bundle is about 10 centimeters across, which turns a hard count of tiny units into counting by tens.
- Meter. Stand a meter stick on the floor next to a child. Where it reaches on their body, at the chest or near the shoulder for many children, is that child’s meter. A doorway is a little more than 2 meters tall, which is a picture for two meters that does not change as a child grows.
Body benchmarks differ from person to person, and a child’s change as they grow. That is fine. A benchmark is a private picture, checked once against a real tool and checked again now and then. The unit itself never changes; only the picture of it does.
Day 2: estimate, measure, compare
An estimate is a measurement done in the head. The child picks the benchmark that fits the object, imagines laying it along the object end to end, and counts, exactly as in Iterating a single unit, except that the copies are pictured rather than laid down.
Then the child measures with the real tool and lays the two numbers side by side. If they are close, the benchmark is working. If the measurement is much bigger, the benchmark is shorter in the child’s mind than in real life, and the next estimate will be better for knowing that. The gap is the point of the exercise. Nobody gets a prize for a lucky guess.
A range is often the more honest estimate. “Between 12 and 15 centimeters” says what the child is sure of: longer than one, shorter than the other. A measurement inside the range shows the child’s picture was right; one outside it shows which way the picture was off.
Day 3: the unit that fits the job
Picture the benchmark against the object. If the benchmark is far longer than the object, as a grown-up’s reach is longer than a crayon, the unit does not fit even once. If it is far shorter, as a crayon’s thickness is against the playground fence, the count runs into the hundreds. The sensible unit is the one where a few, or a few dozen, copies cover the object.
The tool follows. Each move of a short tool along a long object is a chance for the tool to slip or for the count to go wrong, so a long object wants a long tool. A bent or round object wants a tape that bends with it.
Results
Days 1 to 3
| Unit | A benchmark | Check it against |
|---|---|---|
| inch | the top part of a grown-up’s thumb | one space on an inch ruler |
| foot | the edge of a floor tile | a foot ruler |
| yard | a grown-up’s reach, nose to fingertips | a yardstick, or three floor tiles |
| centimeter | the thickness of a crayon | one space on a centimeter ruler |
| meter | where a meter stick reaches on the child | a meter stick |
| 2 meters | the height of a doorway | two meter sticks |
An estimate is a length with a unit, said as “about” or as a range.
Both lengths must be in the same unit before subtracting.
A unit is sensible when at least one copy fits along the object and the count stays small enough to keep track of.
Why it is true
Why a benchmark gives a length. Measuring is counting copies of a unit laid end to end with no gaps and no overlaps (What a unit is, and why we agree on standard ones). If a floor tile is 1 foot long, then a row of tiles is that many feet long, because each tile is one copy of a foot. Estimating with a benchmark is the same count with copies that are nearly a unit rather than exactly one, so the answer is nearly the length rather than exactly it.
Why estimates go wrong in a steady way. If a benchmark is a little short, every copy of it is a little short, and the shortfall piles up: the more copies, the further off the estimate. A child who thinks their big step is 1 foot when it is really 2 feet will say “8 feet” for a chalk line 8 steps long. The line is really 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 16 feet, twice the estimate. That is why comparing an estimate with a measurement is so useful: the gap says not just “wrong” but “your step is longer than you thought”.
Why the gap is found by subtracting. The estimate and the measurement are two lengths in the same unit. The gap between them is how much must be added to the smaller to reach the larger, which is a subtraction, exactly as when comparing any two lengths measured in the same unit.
Why some units fit and others do not. The same object can be measured in any unit, and the length does not change. What changes is the count: a smaller unit needs more copies to cover the same object, and a bigger unit fewer (Smaller units give bigger counts). Too big a unit gives a count under one. Too small a unit gives a count too long to keep. The sensible unit sits between.
Worked examples
The examples follow the days: Examples 1 to 3 are Day 1, Examples 4 and 5 Day 2, and Examples 6 and 7 Day 3.
Lights along the floor tiles
Before the class party, Keahi lays a strand of string lights flat along a row of floor tiles, as in the picture. Each tile is 1 foot on each side. About how long is the strand?
Each tile is one copy of a foot, so the tiles are a ruler with a foot on each space. The lights start at the left edge of tile 1, so nothing is missed at the start. Count the whole tiles the lights cover: 1, 2, 3, 4, 5, 6. Then the lights run partway across tile 7 and stop.
So the strand is more than 6 feet and less than 7 feet long. Keahi can say “about 6 feet”, or, saying more, “between 6 and 7 feet”. Both are good estimates; the range also tells a listener which way the extra bit goes.
The top bar of the swing set
At lunch recess, Rumen’s teacher stands under the top bar of the swing set and holds one arm straight out, nose to fingertips. Moving along under the bar, the teacher finds it is about 4 of these reaches long. About how long is the bar in yards? In feet?
SolutionA grown-up’s reach from nose to fingertips is the benchmark for 1 yard. Four reaches are four copies of a yard, so the bar is about 4 yards long.
Each yard is 3 feet, the same as three floor tiles in a row. Four yards are four threes:
So the bar is about 12 feet long. Both answers are estimates of the same length, and both carry the word “about”.
A bundle of crayons
Zaira wants to know how long her lunch box is. She holds a bundle of 10 crayons side by side, touching. Along the lunch box, the bundle fits 2 times, and then 5 more crayons fill the rest. About how many centimeters long is the lunch box?
SolutionA crayon is about 1 centimeter thick, so a bundle of 10 crayons is about 10 centimeters across. Two bundles are about 10 + 10 = 20 centimeters. The 5 extra crayons are about 5 more centimeters:
The lunch box is about 25 centimeters long. The bundle did the work of ten crayons at once, the same way a ten does the work of ten ones in a written number.
The jump rope at recess
Before recess, Rumen checked his big step against a yardstick and found it is about 2 feet. At recess he walks along a jump rope stretched on the blacktop and takes 4 big steps to reach the end. Then he measures the rope with a yardstick, and it is exactly 3 yardsticks long. Find his estimate, find the measurement in feet, and say how far off the estimate was.
SolutionEach step is about 2 feet, so 4 steps are about
feet. That is the estimate: about 8 feet.
The measurement is 3 yards. Each yard is 3 feet, so the rope is 3 + 3 + 3 = 9 feet long.
Both lengths are now in feet, so they can be compared. The gap is the larger minus the smaller:
The estimate was 1 foot short of the measurement. That is close. It also tells Rumen something: over four steps he came up short, so his big step may be a little more than 2 feet.
A leaf at lunch recess
Keahi finds a leaf at lunch recess. He knows a bundle of 10 crayons is about 10 centimeters. The leaf is longer than the bundle by a few crayon widths, he thinks between 2 and 5. Give his estimate as a range. Then he measures the leaf with a centimeter ruler and it is 13 centimeters. Is the measurement inside his range?
SolutionThe bundle is about 10 centimeters. Two to five more centimeters on top of that gives
so the estimate is “between 12 and 15 centimeters”.
On a number line in centimeters, the estimate is the stretch from 12 to 15, and the measurement is the single mark at 13 (Length on a number line takes this picture further).
13 is more than 12 and less than 15, so the measurement lands inside the range. Keahi’s picture of the leaf was right. A range does not have to be narrow to be useful; it has to hold the true length.
Choosing a tool for the party lights
For the class party, the lights will hang along the wall above the windows, a stretch 6 yards long. Rumen has a foot ruler. Zaira has a yardstick. How many moves does each tool need to measure the stretch? Which tool fits the job?
SolutionThe yardstick is one yard long, so it covers the stretch in 6 moves.
A foot ruler is one foot, and three feet make a yard, so each yard takes 3 moves of the foot ruler. Six yards take six threes:
So the foot ruler needs 18 moves.
Both tools give the same length, because the stretch does not change when the tool does. But each move is a chance for the ruler to slip or for a move to be counted twice, and 18 moves give 12 more chances than 6 moves give. The yardstick fits the job better. A measuring tape that reaches 6 yards would need just one move, which is better still.
A smudged sign on the swing set
A sign on the swing set says the top bar is 3 high, but rain has smudged the unit away. Which units could the sign have said: inches, feet, yards, centimeters or meters?
SolutionPicture each unit’s benchmark 3 times and compare with the swing set, which children stand and swing under.
3 inches is three thumb tops, shorter than a crayon. 3 centimeters is three crayons side by side. Neither is anywhere near a swing set.
3 feet is three floor tiles stacked on end. That is lower than a child’s head, so no child could stand under the bar. It does not fit.
3 yards is three grown-up reaches, which is 3 + 3 + 3 = 9 feet: well above a grown-up’s head. 3 meters is a doorway and a half. Either one makes sense for a swing set.
So the sign said yards or meters. The benchmarks rule out three units but cannot choose between these two, because a yard and a meter are close in size. Only a measurement can settle it.
Boundaries
An estimate is not a measurement. Write it with “about”, or as a range. If a number is going to be used for building, cutting or buying, it has to come from a tool.
A benchmark is only as good as its match. Body benchmarks differ between people and change as children grow, and not every floor tile is a foot. Check each benchmark against a ruler before leaning on it, and again when estimates keep missing the same way.
Benchmarks fit lengths you can see and picture copies along. A length far longer than the benchmark, like a road between two towns, cannot be estimated by imagining thousands of floor tiles. Very long distances get their own picture, a walk of so many minutes, in Customary length units and Metric length units.
Whole units here. “A little more than 6 feet” is said in words. Splitting a unit into halves or tenths is later work.
How far off needs one unit. An estimate of 8 feet and a measurement of 3 yards cannot be subtracted as 8 - 3. Put both in feet first, as in Example 4.
A sensible unit is not always one unit. A swing set bar is fine in yards or meters, and a lunch box in inches or centimeters. The choice changes the count, never the length.
Where this goes next
Measuring with a tool, and reading it well, is the work of Measuring tools and reading their scales. The benchmarks in this lesson keep that work honest: a reading far from the estimate is a signal to look again.
Laying lengths along a number line, as Example 5 did with a range, is Length on a number line, where adding and subtracting lengths becomes jumping along a line. Why a smaller unit always gives a larger count is Smaller units give bigger counts.
The same habit of a benchmark, an estimate, then a check carries over to every other measure. Grams and kilograms get their own benchmarks in Metric mass units, and angles get theirs in Measuring and drawing angles with a protractor, where a right angle is the picture that warns whether a reading of 40 or 140 is the sensible one. It also grows into Estimation and mental math, estimating sums and differences before working them, and into Measurement error, which asks how close any measurement can be.
Exercises
Day 1: benchmarks
- Name the unit each benchmark is about the size of: (a) the edge of a classroom floor tile; (b) the top part of a grown-up’s thumb; (c) the thickness of a crayon; (d) a grown-up’s reach, nose to fingertips. Then (e) say what the height of a doorway is a benchmark for.
- A lunch table sits along a row of floor tiles that are each 1 foot on a side. The table is as long as 7 tiles. About how long is the table?
- The fence beside the swing set is about 7 grown-up reaches long. About how many yards long is it? About how many feet?
- A pencil case is as long as 2 bundles of 10 crayons and 3 more crayons, laid side by side. About how many centimeters long is it?
- The top of the climbing net on the playground is about as high as 2 doorways stacked one on the other. About how many meters high is it?
- The top part of a grown-up’s thumb fits 4 times along a glue stick. About how long is the glue stick?
Day 2: estimate, measure, compare
- At lunch recess Zaira estimated a chalk line as 6 feet. She measured it, and it was 8 feet. How far off was the estimate? Was the estimate too long or too short?
- Keahi estimated a gift bag for the class party as 30 centimeters tall. It measured 26 centimeters. How far off was the estimate, and was it too long or too short?
- Rumen estimated that the playground slide is between 3 and 4 meters long. It measured 5 meters. Is the measurement inside his range? If not, how far past the range is it?
- Rumen estimated a long jump rope as 11 feet, and Zaira estimated it as 14 feet. The rope measured 12 feet. How far off was each estimate?
- Keahi estimated a stick as between 8 and 11 centimeters. It measured 10 centimeters. Is that inside his range? How far is it from each end of the range?
- Zaira thought her big step was about 1 foot. She walked a chalk line in 7 steps and estimated it as 7 feet. It measured 14 feet. If her steps were all the same size, about how long is her big step?
Day 3: choosing a unit and a tool
- Choose inches, feet or yards for each: (a) the length of a crayon; (b) the height of a swing seat above the ground; (c) the length of the playground fence.
- Choose centimeters or meters for each: (a) a leaf; (b) the playground slide; (c) a strand of string lights for the class party; (d) a crayon.
- Choose a foot ruler, a yardstick or a measuring tape for each, and say why: (a) a juice box; (b) the top bar of the swing set; (c) the distance around a swing set post.
- The string lights for the classroom door need a stretch 2 yards long. How many moves does a foot ruler need along it? How many moves does a yardstick need?
- A rope is 2 meters long. How many moves does a meter stick need along it? How many bundles of 10 crayons would it take?
- A ladybug on the slide is 1 long, with the unit missing. Which unit makes sense: centimeter, inch, foot or meter?
- A note says a lunch table is 2 long, with the unit missing. Which of inches, feet, yards or meters make sense? Explain with benchmarks.
- Which is the best estimate for the height of a swing seat above the ground: 2 centimeters, 2 feet or 2 yards?
Reasoning
- Rumen and Zaira each estimate the same chalk line with their own big steps and get different estimates. Measured with two yardsticks, the line comes out the same both times. Explain why.
- Which says more, “about 6 feet” or “between 5 and 7 feet”? Why might a child give a range instead of one number?
- One estimate was off by 1 yard and another by 1 foot. Which was off by more? Explain.
- Explain why it is worth estimating before measuring. Give an example of a slip an estimate would catch.
- An estimate of the swing set’s top bar was off by 2 feet. An estimate of a crayon was also off by 2 feet. Can both be good estimates? Explain.
Answers
- (a) foot; (b) inch; (c) centimeter; (d) yard; (e) 2 meters, since a doorway is a little more than 2 meters tall.
- About 7 feet. Each tile is one copy of a foot, and 7 tiles are 7 copies.
- About 7 yards, which is about 21 feet. Each reach is about a yard, and each yard is 3 feet: 3 + 3 + 3 + 3 + 3 + 3 + 3 = 21.
- About 23 centimeters. Each bundle is about 10 centimeters and each crayon about 1: 10 + 10 + 3 = 23.
- About 4 meters. Each doorway is about 2 meters: 2 + 2 = 4.
- About 4 inches. Each thumb top is about an inch.
- Off by 2 feet, too short. 8 - 6 = 2, and the estimate was the smaller number.
- Off by 4 centimeters, too long. 30 - 26 = 4, and the estimate was the larger number.
- No. 5 is past the top of the range, by 5 - 4 = 1 meter.
- Rumen’s was off by 1 foot, since 12 - 11 = 1. Zaira’s was off by 2 feet, since 14 - 12 = 2.
- Yes, 10 is between 8 and 11. It is 10 - 8 = 2 centimeters above the bottom of the range and 11 - 10 = 1 centimeter below the top.
- About 2 feet. Seven equal steps cover 14 feet, and 2 + 2 + 2 + 2 + 2 + 2 + 2 = 14.
- (a) inches: a crayon is a few thumb tops long. (b) feet: a swing seat is a couple of floor tiles up, too short for yards. (c) yards: a fence is many grown-up reaches long, and in inches the count would be huge. Feet also work for a short fence, with a bigger count.
- (a) centimeters; (b) meters; (c) meters; (d) centimeters. A leaf or a crayon is a few crayon widths long but not a whole meter; the slide and the strand of lights are each several meters.
- (a) a foot ruler: a juice box is shorter than a foot, so one ruler covers it. (b) a yardstick or a measuring tape: the bar is several yards, and a foot ruler would need many moves. (c) a measuring tape: the post is round, and only a tape bends around it.
- 6 moves of the foot ruler, since each yard is 3 feet and 3 + 3 = 6. 2 moves of the yardstick.
- 2 moves of the meter stick. Each meter is 100 centimeters, ten bundles of 10 crayons, so 2 meters is about 10 + 10 = 20 bundles.
- Centimeter. A ladybug is about as long as a crayon is thick. An inch, a thumb top, is bigger than a ladybug, and a foot or a meter is far too big.
- Yards or meters. Two grown-up reaches, or two meter sticks, is the length of a table that seats several children. 2 feet is two floor tiles, a table for one, and 2 inches is two thumb tops, smaller than a lunch box.
- 2 feet. Two floor tiles stacked on end is low enough for a child to sit on, which is where a swing seat hangs. 2 centimeters is two crayons thick, and 2 yards is above a grown-up’s head.
- Their steps are different lengths, so each “step” is a different unit, and the counts differ. A yardstick is the same length for everyone, so two careful measurements with it agree.
- “Between 5 and 7 feet” says more: it gives the lengths the child is sure the line is longer and shorter than. A range is honest about what a benchmark can and cannot tell.
- The one off by 1 yard, since a yard is 3 feet, more than 1 foot.
- An estimate gives a length to compare the measurement with. For example, if a child reads a ruler from the wrong end, or counts 13 yardsticks and writes 13 feet, the measurement will be far from the estimate, and the child knows to measure again.
- No, not both. Two feet is a small miss on a bar many feet long, but a crayon is only a few inches long, so a miss of 2 feet means the estimate was many times too big. How far off matters compared with the length itself.