PrepShorts · Study sheet · Class 7 Mathematics · Chapter 3, A Peek Beyond the Point
Chapter 3 · A Peek Beyond the Point
Why whole units are not enough to measure with
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Most first lessons on tenths present them as notation to be learned. This one opens with a problem no whole-number ruler can answer.
The idea
A measurement is not a number, it is a claim about where something lies — and a coarser ruler can only make a weaker claim, never a wrong one. The chapter opens with one screw measured three times on three rulers (Part I, §3.1, p.47); the screw never changes, and none of the three answers is false. What changes is how much each answer rules out. That is why the unit gets cut up: not to correct the earlier answer, but to buy a narrower one.
What you should be able to do
- Explain why a difference too small to see can still decide whether an object works
- Read the same length on rulers marked in whole units, in halves and in tenths, and write down what each ruler entitles you to say
- Say why a coarse reading is not a wrong reading, and what a finer ruler adds
- Write a measured length as a whole number of units plus a number of tenths
- Read such a length aloud in the form the chapter uses
- Given a length that falls between two marks, state the two numbers it lies between
- Describe, in general, the move the chapter makes when an answer must be sharper: cut the unit into equal parts rather than change the unit
Words to know
| Term | Definition in one line | First introduced |
|---|---|---|
| unit | the one whole length that the marks on a ruler count off | printed throughout Part I, §3.1, p.47 |
| scale | the marked ruler a length is read against | printed in Part I, §3.1, p.47 |
| marking | one of the lines cut into a scale, which is what a reading is read against | printed in Part I, §3.3, p.53, where their absence is the problem |
| centimeter | the length unit used on the opening rulers; the book uses this spelling | printed in Part I, §3.1, p.47 |
| one-tenth | one of ten equal parts of a unit | printed in Part I, §3.2, p.48 |
| tenths | the parts you get after one split, counted | printed in Part I, §3.2, p.48 |
| fraction | a number written as so many of so many equal parts | carried in from Class 6 and from Part I, Chapter 8 vocabulary |
| exact | pinned down with nothing left open | printed in Part I, §3.1, p.48 and again in the chapter summary, p.80 |
| accurate | matching the thing measured as closely as the scale allows | printed as accurately in Part I, §3.1, p.47, and as accurate in the chapter summary, p.80 |
| precise measure | the book's phrase for a reading that a finer split makes possible | printed in Part I, §3.4, p.59 |
| interval | the stretch between two neighbouring marks, inside which the true length sits | an added term; the chapter states the idea by naming both endpoints instead |
| graduation | the spacing of the marks on a scale | an added term, not printed in this chapter |
Where people slip up
- "The first two readings were wrong and the third was right." All three are true of the same screw. The screw is between 2 cm and 3 cm; it is also more than 2½ cm and less than 3 cm; it is also 2 and 7 tenths. Each reading rules out more than the last. Getting this backwards turns the whole chapter into a story about correcting errors, which it is not.
- "2 and 7 tenths is the exact length." It is exact to a tenth. The very next section takes the same move again and splits each tenth into ten, for the same reason. A student who thinks the tenth-scale ends the story will find §3.3 unmotivated.
- "Small differences do not matter." The opening scene exists to kill this. The two screws differ by an amount nobody can see, and one of them fails.
- "To measure something small you need a smaller unit." The chapter's move is to cut the unit you already have into equal parts, and to keep the unit's name. A separate small unit — the millimetre — does arrive, but not until §3.5, and it turns out to be one of these parts rather than an independent idea.
- "2½ cm and 2 and 5 tenths cm are different lengths." They are the same length read off differently marked rulers. The half-marked ruler is not wrong, it is just cut a different way — a point the chapter returns to head-on at Part I, §3.4, p.59.
- "The number of marks is what makes a ruler good." What matters is that the parts are equal. Every split in this chapter is into equal parts, and the word does real work.
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Worked answers to this chapter’s exercises
Transcript1,342 words
A toy comes apart, and there is one screw left over to put it back together. The screw goes in, and turns, and never quite bites. It sits there loose, and the toy still wobbles. So somebody fetches another screw out of the same box, and that one holds perfectly. Now put the two of them side by side and look. Same colour. Same thread. Same head. Laid next to each other, you would swear they were the same screw twice.
But one of them works and one of them does not. So there is a difference between these two, it is real, and you cannot see it. The only way to settle it is to measure them, and that turns out to be a much better question than it sounds. Here are the two screws again, lined up from the same end. The upper one is very slightly the longer.
Not by much. About the thickness of a coin. If they were loose in a drawer together, you would never pick the right one on purpose. But the hole in the toy does not care what you can see. It cares about the actual length, and the actual lengths are different. So the real question here is not about screws at all. It is this. How do you get an answer sharp enough to tell these two apart?
Because a measurement is not a fact you look up. It is a claim you make. And a claim can be perfectly true and still be no use to you whatsoever. Start with an ordinary ruler, marked at every whole centimetre and nowhere else. Lay the first screw along it, with its tip at zero. The far end falls somewhere past the two mark, and short of the three. So what is the length?
There is no honest single number to give here. The end does not land on a mark, so this ruler cannot name it. What it can tell you is this. The screw is between two centimetres and three. And that is the whole answer. Not a number. A gap. Notice that it is not a guess, and it is not an approximation either. It is exactly true. The screw really is longer than two, and really is shorter than three.
Now take a second ruler, with one extra mark halfway between every pair. Same centimetres as before. Each one simply cut in two. Lay the same screw on it, tip at zero again. The end is past the two mark, exactly as it was. But now you can also see that it is past the two and a half mark. And it is still short of three. So this ruler says: between two and a half, and three.
That is a better answer, and it is worth being precise about what better means. The first ruler left a whole centimetre of room for the true length to hide in. This one leaves half a centimetre. The screw has not changed at all. The answer has just got narrower. Third ruler. This time each centimetre is cut into ten equal parts. Ten of them, all exactly the same width, filling one centimetre between them.
Lay the screw down one more time. Travel from zero out to the two mark, which is the easy bit. Then count the small parts past it. One, two, three, four, five, six, seven. The end of the screw arrives at the seventh one. So the length is two whole centimetres, and seven of the ten parts of the next one. Two and seven tenths of a centimetre. And for the first time, the answer is a single number instead of a pair.
That happened because the end landed on a mark. Hold on to that, because it comes back at the end. Now go back to the two screws, and put them both on all three rulers. On the whole centimetre ruler, the first screw reads between two and three. And the second screw reads between two and three. The same answer twice. That ruler cannot see any difference between them at all.
On the half ruler, the first one reads between two and a half and three. And the second one reads between two and a half and three. Twice as many marks, and still the identical answer for both screws. Now the tenth ruler. The first screw is two and seven tenths. And the second screw is two and eight tenths. There it is. That is the ruler that could tell them apart, and telling things apart is what a finer scale actually buys you.
It is very tempting to say the first two rulers got it wrong, and the third one got it right. They did not. Put the three answers up together and look at them. Between two and three. Between two and a half and three. Two and seven tenths. Every single one of those is true of the same screw. Draw them as three bands over a line and you can see what is really going on.
The widest band is a whole centimetre across. The next one is half a centimetre, and the third is a tenth of a centimetre. And each band sits entirely inside the one before it. Nothing is being corrected here. Each answer simply rules out more than the one before did. The first ruled out nothing extra. The second ruled out half the room. The third ruled out nine tenths of it.
Now look properly at what that written length is actually saying. Two and seven tenths of a centimetre. The two is a count of whole centimetres. Two of them, laid end to end. The seven is a count as well, but of something much smaller. It counts parts of the next centimetre along. Seven of those parts. And the word tenths is what tells you how big one part is.
One tenth, because that centimetre was cut into ten equal pieces. The word equal is doing real work in that sentence, and it is worth stopping on. If the parts had different sizes, then saying seven parts would not pin down a length at all. You can build a ruler with just as many marks on it, space them unevenly, and get a far vaguer answer out of it. Say it out loud the way it is built. Two, and seven tenths.
Here is a second one to read. Three and two tenths of a centimetre. Three whole centimetres, and then two of the ten parts of the next. On the coarse ruler that same length would only have been between three and four. Now try the tenth ruler on some things that are lying around anyway. An eraser. Four whole centimetres, then one part. Four and one tenth. A pencil. Twelve whole centimetres, and six parts. Twelve and six tenths.
A piece of chalk. Six whole centimetres and three parts, so six and three tenths. Every time, you read off the whole units first, and then you count the parts. That is the entire skill, and it works on anything you care to point it at. So step back and look at what was actually done here. The answer needed to be sharper, and there were two different things that could have changed.
You could have gone off and found a smaller unit to measure in instead. That is not what happened. The centimetre stayed exactly as it was. It simply got cut into equal parts. And a count of those parts got written down after the whole ones. That is the move, and it is available to you again any time you need it. Because two and seven tenths is not really the end of this.
It is exact to a tenth. A screw one hundredth of a centimetre longer would give you that very same reading. So cut each tenth into ten, and go round again. The unit never changes. Only how finely you have cut it up.
Where this fits
Taken from the notes each video was made from, not from the reading order — these are the ideas this one rests on and the ones that later rest on it.
Builds on
- When an approximate answer is the better answerClass 7 · Ch 1, Large Numbers Around Us
Comes up again in
- A tenth partClass 7 · Ch 3, A Peek Beyond the Point
- A hundredth part, and continuing the splitClass 7 · Ch 3, A Peek Beyond the Point
- Odd and even as "can this be arranged in pairs"Class 7 · Ch 6, Number Play
Either side of this one
- The distributive property, and using it to compute fasterClass 7 · Ch 2, Arithmetic Expressions