3. A Peek Beyond the Point

10 topics2 h

Chapter 3 of NCERT Mathematics for Class 7: A Peek Beyond the Point. Four sections — Splitting the unit, The decimal place value system, Comparing and combining decimals and Decimals outside the classroom. Ten videos, 2 h in all.

Worked answers to the 19 exercise questions in this chapter →

0 of 10 done — sign in to track this chapter

Part 1

Splitting the unit

9 minWhy whole units are not enough to measure withMost first lessons on tenths present them as notation to be learned. This one opens with a problem no whole-number ruler can answer.9 min · 10 partsहिंदी9 minA tenth partThe lesson that makes decimals possible, taught entirely without the decimal point — which does not arrive for another two sections.9 min · 11 partsहिंदी10 minA hundredth part, and continuing the splitThe second turn of the handle. Split the tenth again and the same argument runs, which is exactly why it will never stop.10 min · 11 partsहिंदी

Part 2

The decimal place value system

9 minExtending Indian place value to the right of the pointThis is the explanation the previous two were saving the point for. Tenths and hundredths were built without it deliberately.9 min · 11 partsहिंदी9 minWhy measurement units are built in tensMetric conversion is usually taught as a table of facts and a rule about moving the point. That is exactly backwards.9 min · 11 partsहिंदी

Part 3

Comparing and combining decimals

10 minLocating a decimal on the number line, and comparing two decimalsPlacing a decimal on the line and comparing two decimals are taught as separate skills. They are the same skill.10 min · 11 partsहिंदी10 minAdding and subtracting decimals by aligning place valueThis is the lesson that feels like it is teaching nothing, because it very nearly is — and that is the point worth making.10 min · 11 partsहिंदी

Part 4

Decimals outside the classroom

9 minWhat a misplaced decimal point costsA string of digits is not yet a quantity. What a misplaced point costs is measured in crashed spacecraft and wrong doses.9 min · 11 partsहिंदी9 minWhen a decimal-looking number is not a decimal: 4.5 hours, 5.5 overs4.5 hours is four and a half. 5.5 overs is not five and a half. The digit after the point does not always count tenths.9 min · 11 partsहिंदी9 minRival ways of splitting off the fractional part, and why the point wonAlmost everyone believes decimals arrived with the decimal point. The fractions came first, by something like eight hundred years.9 min · 11 partsहिंदी

What you will be able to do

  • Explain why a difference too small to see can still decide whether an object works
  • Write one length in both available forms: as units and tenths, and as a plain count of tenths
  • Explain why splitting each tenth into ten equal parts produces a part of which a hundred make the unit
  • State that a unit may be split into any number of equal parts, and give the chapter's reason for always choosing ten
  • State each of the chapter's unit relations and, from it, say what one small unit is worth as a fraction of the larger one
  • Locate a given decimal on a number line by successive splitting, and draw the chain of magnified lines that does it
  • Add and subtract decimal numbers set out in columns, aligning by place rather than by the ends of the digits
  • State the two things a digit string needs before it names a quantity
  • Say what a digit after the point counts in a genuine decimal number
  • Distinguish decimal fractions themselves from decimal notation for them, and say which came first in the chapter's account

What this chapter assumes you already know

  • Reading a length off a ruler against whole-centimetre marks (Class 6)
  • The idea of a fraction as some number of equal parts of one whole
  • Recognising halves and quarters of a length
  • Comparing two fractions with the same denominator
  • Column addition and subtraction of whole numbers, including carrying and borrowing
  • Adding fractions that share a denominator
  • Reading a mixed number such as 3 and 4 tenths
  • Halving a length, and halving a number written with a fractional part

It builds directly on When an approximate answer is the better answer, The Land of Tens: each place is ten of the one before, Commutative and associative: why order and grouping are free and Crores and millions: one number, two naming systems.

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "The first two readings were wrong and the third was right."
  • "2 and 7 tenths is the exact length."
  • "Small differences do not matter."
  • "To measure something small you need a smaller unit."
  • "2½ cm and 2 and 5 tenths cm are different lengths."
  • "The number of marks is what makes a ruler good."
  • "41 tenths and 41 and 1 tenth are near enough the same."
  • "You can compare counts of tenths across pictures."