7. Fractions

9 topics1 h

Chapter 7 of NCERT Mathematics for Class 6: Fractions. Four sections — What a fraction is, Fractions as numbers on a line, Comparing fractions and Adding and subtracting, and where the notation came from. Nine videos, 1 h in all.

Worked answers to the 42 exercise questions in this chapter →

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Part 1

What a fraction is

10 minA fraction as an equal share, and why one-ninth is less than one-fifthThe number underneath a fraction counts the people, not the pieces you own — which reverses five years of instinct. With whole numbers, bigger written means bigger. With shares, bigger written underneath means smaller in the hand. Nine beats five and one-ninth loses to one-fifth, and both are true for exactly the same reason.10 min · 12 partsहिंदी10 minA fraction as a part of one wholeA rectangle and a triangle that look nothing alike carry the same name, because the name of a piece is fixed by how many equal pieces the whole was cut into and by nothing else. And once one piece has a name, no other piece needs a new one — the large piece is not a new idea, it is three of the small one. Naming and measuring turn out to be the same act.10 min · 12 partsहिंदी9 minMeasuring a length by choosing a smaller unitWhen a length will not sit whole inside your unit, you do not need a cleverer number — you need a smaller unit. Fold the strip, and the awkward length becomes an ordinary count of steps. Read a fraction that way, as a count rather than a division, and its size becomes visible without any arithmetic at all: the bottom number sets the step, the top number counts them.9 min · 12 partsहिंदी

Part 2

Fractions as numbers on a line

10 minEvery fraction has one place on the number lineA fraction is not a pair of numbers stacked up. It is a length from zero, and it has exactly one place on the line.10 min · 12 partsहिंदी10 minMixed fractions: naming what lies past 1Some fractions pass one whole and some do not, and you can tell which without working anything out.10 min · 12 partsहिंदी

Part 3

Comparing fractions

10 minEquivalent fractions: many names, one number, and lowest termsOne length, three different measuring steps, three different names — and all three are the same number.10 min · 12 partsहिंदी10 minComparing two fractions by giving them a common fractional unitTwo counts made in different step sizes cannot be ranked. Give them a shared step and the comparison becomes counting.10 min · 12 partsहिंदी

Part 4

Adding and subtracting, and where the notation came from

9 minBrahmagupta's method: you can only add units of the same sizeYou can only add units of the same size. Add the tops and the bottoms and see exactly what goes wrong.9 min · 12 partsहिंदी10 minWhere our way of writing fractions comes fromThe stacked form you write fractions in is already sitting in an Indian manuscript. The line between them arrived 850 years later, from Morocco.10 min · 12 partsहिंदी

What you will be able to do

  • State what the number below the line in a fraction counts
  • Identify the whole in a picture before naming any piece of it
  • Declare a chosen object to be one unit and measure other lengths against it
  • Divide the gap between two consecutive whole numbers into a stated number of equal parts and label every new mark
  • Sort a list of fractions into those under one whole and those over it
  • Show that two fractions are equal by measuring one length with two different fractional units
  • Compare two fractions that already share a bottom number, and say why it is easy
  • Add and subtract two fractions that already share a fractional unit, and explain why only the counts change
  • Give the Sanskrit words the chapter reports for a fraction, and what they mean

What this chapter assumes you already know

  • Whole-number order: which of two whole numbers is the larger
  • Sharing a set of objects equally between two or four people, from primary school
  • The words half and quarter used in everyday speech
  • The idea of "the same amount" applied to a solid object, from primary school
  • Recognising a rectangle and a right-angled triangle by sight
  • Measuring a length in whole units with a ruler or a marked strip
  • Measuring a length against a stated unit
  • Division with a remainder, informally

It builds directly on Mathematics is the search for patterns and for why they hold, Counting the parts of a shape sequence produces a number sequence, Every number sequence is a rule, not a list, The same children, different numbers: a number depends on what is being counted, Common factors: which jump sizes can land on a number and Common multiples: why the first "idli-vada" is 15.

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.

  • "Nine is bigger than five, so one-ninth is bigger than one-fifth."
  • "The number underneath counts pieces I have."
  • "A fraction is two numbers, so compare them like two numbers."
  • "A share is always one of the equal pieces."
  • "The parts only count as equal if they are the same shape."
  • "Fractions are a modern school topic."
  • "Different shape means different fraction."
  • "Any four pieces make quarters."